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JClub July 2010: Mechanics of Ionic Polymer Metal Composites

Wei Hong's picture

Ionic polymer-metal composite (IPMC) is a polyelectrolyte (usually Nafion or Femion swollen by simple salt solution) strip or membrane with both sides plated with metal electrodes. It is a particular design of electroactive-polymer device rather than a new class of material. When a voltage is applied between its electrodes, it will bend toward either electrode depending on the polarity (anode for a negatively charged gel), and the magnitude of deformation could be controlled by the electric signal.  Reversely, the deformation of an IPMC can generate electric signal or even energy output [1-4].  Therefore IPMC has recently becomes a hot topic in actuation, sensor and energy harvesting applications, especially when integrated with the characteristics of certain gels that are responsive to other environmental stimuli such as pH value or temperature.

Compared to traditional actuation devices, the IPMC is small, simple, low-cost, resilient, noise-free, biocompatible, works at low voltage and capable of large deformation [a].  However, it also suffers from several problems: its actuation response is non-linear and also followed by non-controllable relaxation, its response is relatively slow, its working voltage is limited by the electrolysis of swelling media, its performance deteriorates in the long term as swelling media evaporates, its operation is time and history dependent, and so on [5, 6].  There have been numerous experimental efforts to overcome these problems by changing the swelling media or modifying the structures of polymer matrix and electrodes [7-10].

 

On the other hand, the underlying mechanism of IPMCs has yet to be fully understood.  Deformation theories of polyelectrolyte gel [b] were initiated almost at the same time as when Oguro published his first design of IPMC [11].  By using the same frame work, equilibrium and kinetics of bending of IPMC was then calculated [12, 13].  It is suggested that free ions drift under electric field, the concentration gradient creates osmotic pressure difference that drives the gel to bend, which is balanced by the elastic resistance of the matrix, while the kinetics is formulated by phenomenological law of porelasticity and diffusion rather than viscoelasticity [c].  Later another stress part is added to include the collective behavior of fixed charges [14-17].  A detail model of this stress is presented for Nafion by Nemat-Nasser [18], who assumes that Nafion contains solution clusters and a hydrophilic polymer matrix, with the detailed morphology determined by the level of hydration, as suggested by materials models [19, 20].  A double layer of ions forms at phase boundary where the electro-static interaction of the ordered polarized results in a net excess pressure.  Although the essential microstructural features are captured, more assumptions have to be made in order to incorporate this effect in the general framework of continuum theory.

 

A nonlinear field theory for polyelectrolyte gels recently proposed [21] provides a way of describing the deformation and electrochemistry of polyelectrolyte gels.  The theory suggests that the equilibrium behavior of a polyelectrolyte gel is fully determined by its free-energy density, as a function of strain, electric displacement, and concentration of mobile species.  The concept of osmotic pressure, which has often been used without a physical definition [22, d], is introduced as a Lagrange multiplier for the incompressibility constraint.  The equations that govern the evolution of polyelectrolyte gels in a nonequilibrium state are formulated, based on the conservation law of all mobile species and the kinetic equations that relates the diffusion flux of mobile species to its driving force, the gradient of the chemical potential.  A self-consistent model shall have the chemical potential derived from the free-energy function as well, containing contributions from the elasticity of the polymer, the concentration of mobile species, and the electric field.  If specific combinations of the free-energy function and the kinetic laws are chosen, one could recover the Nernst-Planck equation, an evolution equation often used in various models [15, 16].  Clearly, a gap exists between the microstructure of the material and its free-energy function / kinetic law.  The following questions may need to be answered before we can fill in this gap in theoretical understanding.

 1. The effect of the microstructure and thickness of the electrodes/interfaces

One of the important conclusions by existing models is that there is an ion-depletion region near the electrode and the deformation is determined by this boundary layer [18].  However, most models simply assume a sharp interface between electrode and polymer matrix.  Under such an assumption, in an equilibrium or steady state, the physical laws and a simple dimensional analysis will lead to a result that the bulk of a gel is electroneutral except for the boundary layer, characterized by the Debye length.  While an 1D analysis can estimate a bending moment induced by the ultra high stress in the thin boundary layer, 3D continuum mechanics indicates that a surface compressive stress may rather cause surface instabilities such as wrinkle and crease.  On the other hand, an effective IPMC design actually needs the electrode metal particles to infiltrate into polymer matrix, and experiments also suggest a strong correlation between the actuation strain and the morphology and thickness of electrodes [23, 24]. It is possible that the energy is majorly stored in the “vague” electrode-polymer interface, or the microstructured electrode with finite thickness, rather than in the thin boundary layer between the polyelectrolyte and a mathematically sharp interface?

 2. Are the basic laws of electrostatics still valid in an electrolyte-metal composite?

In most existing models of IMPC, the governing equation for the electric field, namely the Poisson-Boltzmann equation, is derived from the Gauss’s law of electrostatics.  A homogeneous polyelectrolyte mixture is sometimes treated as a dielectric medium when all charged particles are excluded [21].  However, for a material with microstructures of various length scales, the validity of such an assumption has never been discussed.  For example, the mixture of metal, polyelectrolyte, and ionic solution at the interfaces, which seems to play an important role, turns out to be a medium that is both an electric conductor and an ionic conductor.  Even for the case when the structure is random, a proper way of homogenization is a challenge.  Other examples include the microporous structure of Nafion, in which the mobile charges are distributed in order, and the distribution interacts with the macroscopic electric field.

 3. The origin and mechanism of the electrostatic and ionic contributions to force/stress 

The definition and notation of electrostatic forces in solids have always been controversial, as commented by Zhigang in his paper on deformable dielectrics.  In a system containing dielectric polymer and solvent, mobile and immobile ions, and even electronic conductors, the “force” or “stress” in a continuum mechanics manner is even hard to imagine.  Maybe one should rather avoid ambiguous terms like force and stress.  A question that arises naturally is how the charges carried by the polymer network and the mobile ions interact with each other and with external field, and further affect the system as a whole.  The answers to such a question have impacts much broader than just calculating the bending of a polymer strip.  For example, biological tissues are materials of similar or more complex structures.  Qualitatively, it has been argued that the main contributions include the electrostatic repulsions between fixed charges and the osmosis by the concentration difference of mobile ions in the solvent [25].

An approach often used (e.g. in multiphasic theories) is the introduction of the chemical expansion stress [26] or similarly the eigenstrain [e], the thermodynamic validity of which is put into question recently [27].  Another approach is the use of the Maxell stress in dielectrics through homogenizing the charge distribution.

Alternatively, one could start from the microstructure of the material and sum over all the ion-ion interactions, an approach similar to statistical physics or atomistic simulations [28].  However, similar as one calculating Maxwell stress, careful sum over all interaction pairs need to be performed, which also introduces the question of how microstructures (e.g. electric double layers) will evolve in response to the change in macroscopic fields.

 

IPMC is not only interesting in application, but also one of the model systems of natural soft materials with tunable parameters.  Modeling it may sever as the first step towards understanding the mechanics of soft matters.

 

Key References:

[a] M. Shahinpoor and K. J. Kim, “Ionic polymer-metal composite: I Fundamentals”, Smart Mater. Struct. 10, 819 (2001)

[b] M. Doi, M. Matsumoto and Y. Hirose, “Deformation of ionic polymeric gels by electric fields” J. Macrocol. 20, 5504 (1992):

[c] P. G. deGennes, K. Okumura, M. Shahinpoor and K. J. Kim, “Mechanoelectric effects in ionic gels”, Europhys. Lett. 50(4) 513 (2000)

[d] M. Shahinpoor, “Continuum eletromechanics of ionic polymeric gels as artificial muscles for robtotic applications”, Smart. Mater. Struct. 3, 367 (1994)

[e] S. Nemat-Nasser, “Micromechanics of actuation of ionic polymer-metal composites”, J Appl Phys 95 (5): 2899 (2002)

[f] K. J. Kim and M. Shahinpoor, “Ionic polymer-metal composite: II Manufacturing technique”, Smart Mater. Struct. 12, 65 (2003)

[g] M. Shahinpoor and K. J. Kim, “Ionic polymer-metal composite: III Modeling and simulations as biomemetic sensors, actuators, transducers and artificial muscles”, Smart Mater. Struct. 10, 819 (2001)

[h] M. Shahinpoor and K. J. Kim, “Ionic polymer-metal composite: IV industrial and medical applications”, Smart Mater. Struct. 13, 1362 (2004)

 

Other References:

[1] K. Asaka, K. Oguro, Y. Nishimura, M. Mizuhat and H. Takenaka, “Bending of polyelectrolyte membrane-platinum composites by electric stimuli I”, Polymer J. 27(4) , 436 (1995)

[2] K. Sadeghipour, R. Salomon and S. Neogi, “Development of a novel electrochemically active membrane and smart material based vibration sensor/damper” Smart Mater. Struct. 1, 172 (1992)

[3] M. Shahinpoor, Y. Bar-Cohen, J. O. Simpson and J. Smith, “Ionic polymer-metal composite as biomimetic sensors, actuators and artificial muscles – a review”, Smart Mater. Struct. 7, 15 (1998)

[4] M. Aureli, C. Prince, M. Porfiri and S. D. Peterson, “Energy harvesting from base excitation of ionic polymer metal composites in fluid environment”, Smart Mater Struct 19, 015003 (2010)

[5] Y. Bar-Cohen, S. Leary, A. Yavrouian, K. Oguro, S. Tadoro, J. Harrizion, J. Simith and J. Su, “Chanllenges to the application of IPMC as actuators of planerary mechanis”, Proc SPIE Smart Struc. Mater. Sympo. 3987-21 (2000)

[6] X. Bao, Y. Bar-Cohen and S. Lih, “Measurements and macro models of ionomeric polymer-metal composties”, Proc. SPIE Smart Struc. Mater. Sympo. 4695-27 (2002)

[7] S. Nemat-Nasser and Y. Wu, “Comparative experimental study of ionic polymer-metal composites with different backbone ionomer and in various cation forms”, J Appl. Phys. 93(9), 5255 (2003)

[8] B. J. Akle, M. D. Bennett and D. J. Leo, “High-strain ionomeric ionic liquid electroactive actuators”, Sens. Actuators, A 126(1), 173 (2006)

[9] N. Kamamichi, M. Yamakita, T. Kozuki, K. Asaka and Z. W. Luo, “Doping effects on robotic system with ionic polymer-metal composite actuators”, Adv. Rob. 21(1-2), 65 (2007)

[10] S. Liu, R. Montazami, Y. Liu, V. Jain, M. Lin, X. Zhou, J. R. Heflin, Q.M. Zhang, “Influence of the conductor network composites on the electromechanical performance of ionic polymer conductor network composite actuators”, Sensors and Actuators A 157, 267 (2010)

[11] K.Oguro, Y. Kawami and H. Takenaka, “Bending of an ion-conducting polymer film electrode composite by an electric stimulus at low voltage”, J. Micromachine SOC 5, 27 (1992)

[12] D. Segalman, W. Witkowski, D. Adolf, M. Shahinpoor, “Electrically-controlled polymeric gels as active materials in adaptive structures”, Smart Mater Struct 1, 95 (1992)

[13] M. Shahinpoor, “Continuum eletromechanics of ionic polymeric gels as artificial muscles for robtotic applications”, Smart. Mater. Struct. 3, 367 (1994)

[14] J. Firmrite, H. Struchtrup and N. Djilali, “Transport phenomena in polymer electrolyte membranes I medeling framework”, J Electrochem. SOC 152(9), A1804 (2005)

[15] R. Luo, H. Li and K. Y. Lam, “Modeling and simulation of chemo-electro-mechanical behavior of pH-electric-senstive hydrogel”, Anal. Bioanal. Chem. 398, 863 (2007)

[16] M. Porfiri, “Charge dynamics in ionic polymer metal composite”, J Appl Phys 104, 104915 (2008)

[17] T. Wallmersperger, A. Horstmann, B. Kroplin and D. J. Leo, “Thermodynamical modeling of the eletromechanical behavior of ionic polymer metal composites”, J Intell. Mater. Syst. Struct. 20(6), 741 (2009)

[18] S. Nemat-Nasser, “Micromechanics of actuation of ionic polymer-metal composites”, J Appl Phys 95 (5): 2899 (2002)

[19] W. Y. Hsu and T. D. Gierke, “Elastic theory for ionic clustering in perfluorinated ionomers”, Macromol. 15, 101 (1982)

[20] K. A. Mauritz and R. B. Moor, “State of understanding of Nafion”, Chem. Rev. 104, 4535 (2004)

[21] W. Hong, X. Zhao, Z. Suo, “Large deformation and electrochemistry of polyelectrolyte gels” J Mech. Phys. Solids. 58, 558-577 (2010).

[22] P.J. Flory, in “Principles of polymer chemistry”, Ithaca, New York, Come11 University Press (1953)

[23] N. Fujiwara, K. Asaka, Y. Nishimura, K. Oguro and E. Torikai, “Preparation of gold-solid polymer electrolyte composite as electric stimuli-responsive materials”, Chem. Mater. 12, 1750 (2000)

[24] K. Onishi, S. Sewa, K. Asaka, N. Fujiwara and K. Oguro, “Morphology of electrodes and bending response of the polymer electrolyte actuator”, Electrochimica Acta 46, 737 (2000)

[25] V. C. Mow and X. E. Guo, “Mechano-electrochemical properties of aritcular cartilage: their inhomogeities and anisotrpies”, Annu. Rev. Biomed. Eng. 4, 175 (2002)

[26] W. M. Lai, J. S. Hou and V. C. Mow, “A triphasic theory for the swelling and deformation behavior of articular cartilage”, J. Biomech. Engr. 113, 245 (1991)

[27] J. H. Huyghe and W. Wilson, “On the thermodynamical admissibility of the triphasical theory of charged hydrated tissues”, J Biomech. Engr. 131, 044504 (2009)

[28] S. A. Rice and N Nagasawa, in “Polyelectrolyte solution, a theoretical introduction”, Academic Press, New York (1961)

This review is completed with the help of Xiao Wang.

Comments

Lianhua Ma's picture

Dear Weihong,Thank you for your organization of new monthly Jclub.Nice topic As a smart polymer actuator, the IPMC exhibit mechanical deformation behavior when subjected to electrical field.  I struggled with the following question. As mentioned above, the dielectric body between two electrodes is a polyelectrolyte strip or membrane. Different from traditional dielectric polymer, the polyelectrolyte gel is a mixture of polymer chain, water and ions. If the saturated polyelectrolyte gel is subjected to external mechanical loads, I believe the gel drains under this mechanical force, that is to say, the water can migrate out of the gel.  I am wondering whether or not the solvent migrates out when the IPMC is only subjected to electric field (no mechanical loads). From some references, it seems that the solvent in IPMC would not migrate out, although the polyelectrolyte strip deformed under applied voltage. I don't know why. Can somebody help me out with this question?Thanks in advance.Lianhua

xiao_wang's picture

 

Hi, Lianhua,

Solvent migration across the IPMC surface did being observed. For example ( Ref 18, Part II B) a IPMC swelled by glycolor glycerol dries at anode side and wets at cathode side during actuation.

This is because the chemical potential of swelling medium actually relates to swelling ratio. The polymer network has to expand if one adds solvent molecules into it, obviously the work done in this process relates to the how much the polymer network has been deformed. A form of mathematical equation could be found on ref 21, eqn(6.11) .

miu_{solvent}=\frac{\partial W}{\partial C_s} + \Pi*v_s

where W is free energy of the system, C_s is concentration of solvent (or water in this case), \Pi is osmotic pressure, v_s is size of water molecule.

When the chemical potential of water in the gel exceeds that in the gas phase or free water, the gel loses water. The reason why this phenomenon is rare in IPMC is due to the small strain of actuation (~1%) compared to that when one try to squeeze a gel (~100%). Whether this deformation is done by electrical work or mechanical work does not matter.

Xiao

Lianhua Ma's picture

Hi Xiao,

Thank you for your explaination.

I can understand the migration mechanism of solvent when the gel is squeezed by mechanical force, because the chemical potential of water increases due to the effect of compression force ,and the higer potential dirves the gel to lose solvent.

However, I am still confused of the solvent migration induced by electric field. We know that the polyelectrolyte gel has no stress when it is subjected to electric field, and the chemical potential of solvent in gels equilibrates with that of external solvent.   The electrical work has been fully converted to strain energy of gels.

If the solvent can migrate out of gels, what is the driven source?

Thanks

Lianhua

Lianhua Ma's picture

Maybe this migration could be explained as follows,

The chemical potential of solvent comes from three part:  1. mixing energy  2. strain energy of polymer chains 3. electric field energy

At the first equilibrium state(no electric field), the chemical potential of solvent is from two contributions(1 mixing energy+2 strain energy),and equilibrates with that of external solvent.

If we apply voltage across the gel, the electrical filed energy has been fully converted to strain energy of polymer chains and increases the contribution of strain energy. To achive a new equilibrium state, the contribution to chemical potential from mixing energy has to decrease, and then the gel loses solvent to decrease the volume of solvent.

Lianhua

xiao_wang's picture

 

Hi, Lianhua,

It is an interesting thought that contribution from mixing term to total free energy could decrease due to applied electric field, my thought is, for example, some ordered structure formed at the Nafion/solvent phase boundary as suggested by Nermat-Nasser's paper.

Unfortunately, I cannot agree with your idea. It is true to talk about conservation of energy and conversion from one to another, but not necessarily so when dealing with the derivative of it. If you are really interested in the true "source" of the solvent transportation, you will have to take derivative to the chemical potential and write out the kinetic law, and check the leading term in the thermodynamic driving force. By the way, stress could be written as just another derivative of free energy, which is irrelevant to the problem of chemical potential (though one might connect them by Maxwell relation).

Xiao

Lianhua Ma's picture

Hi Xiao,

Thanks for your explain to clarify this problem.

I agree that we should take derivative to the chemical potential and write out the leading term in the thermodynamic driving force. We can get rigorous formulations to describe the migration phenomenon.

 

I think the idea of sources of solvent chemical potential could be used to intuitively explain the migration of solvent

For the initial equilibrium state without electric field, if we take derivative to C (number of solvent molecules), we find the chemical potential results from the tension of polymer chain and mixing of solvent and polymer. i.e. miu=miu1+miu2.  The chemical potential of solvent in gel is identical to that of external solvent.

 

On the base of the initial state, we further apply electric field and get a new equilibrium state. In this case, the chemical potential of solvent in gel is also identical to that of external solvent. Due to the introduction of electric field energy, the chemical potential of solvent in the gel is from three parts: strain energy +mixing +electrical field energy.  i.e. miu=miu1+miu2+miu3

Compared to initial equilibrium state, the contribution of miu2 is relatively small, that is, miu2 decreases

I am not quite sure whether this is correct.

I will check the detailed derivation. 

Thanks again for your comments.

Lianhua

jiangyuli's picture

Thanks Wei for a nice review on IPMC. I had fortune to work
with Sia on this fascinating material system more than ten years ago, and
published two early papers with Sia, JOURNAL OF APPLIED
PHYSICS
Volume: 87 Issue: 7 Pages: 3321-3331, and MECHANICS OF MATERIALS Volume: 32 Issue: 5 Pages: 303-314.
Though I have not worked on this topic after I left UCSD, it always had a
special place in my heart.

Wei touched some issues that we struggled quite bit when
trying to understand the mechanism of IPMC, the microstructures and the
electrostatic interactions. Much progress has been made since then, but some
questions remain unanswered. I will share some of my thought in this process.

IMPC is often mentioned together with
polyelectrolyte gels, but in my view, there are some rather significant
differences. In fact, the bending direction respect to the applied electric of
IPMC is opposite to that of polyelectrolyte gels (if I remember right; I could
not get access to Shiga’s papers on gel now). The differences lie in the
following; (1) IPMC is more or less a closed system, with mass redistribution
within Nafion, but not in and out, while gel in a solution is an open system,
with mass transfer in between. Thus water plays a much more important role in
gel’s deformation; (2) Nafion does have phase separation between hydrophilic
ionic cluster and hydrophobic matrix, and such microstructure does have
substantial influence on IPMC’s response. Because of this, IPMC is also
different from dielectric elastomer, for the existence of mobile ions as well
as ionic channel from charge conduction, though its matrix can indeed be viewed
as dielectric, I think. For these reasons, we thought (and I still think so)
that it is electrostatic interactions, not osmotic pressure, that plays a
dominant role in IMPC’s deformation, as we argued in JOURNAL OF APPLIED PHYSICS Volume:
87 Issue:
7 Pages:
3321-3331.

The actual electrostatic interaction, of course, is much more complicated than the over-simplified continuum picture we presented, and to fully understand the mechanism, I think it is necessary to go one scale down, and model the interactions in terms of
discrete ions, instead of a continuum charge/ion density. With that, I think a
much clearer picture will emerge.

Wei Hong's picture

Thank you Jiangyu for the valuable comment and clarification.  As you can tell that this JClub post is just a note we wrote while reading related literature.  The contents reflect the puzzles we encountered through reading.  There are definitely errors and misunderstanding in some descriptions or statements.  I believe that one of the purposes of this JClub is to bring discussion and hopefully to arrive at a better understanding on the topic.

I agree with you that the material and mechanism of IPMC is different from those of polyelectrolyte gels when used directly as actuators in liquid solvent.  However, in a continuum point of view, I don't see much difference between the matrix material of IPMCs and polyelectrolytes: both are cross-linked polyelectrolytes, both contain solvent (water), both contain mobile ions.  They are different in microstructures.  However, in continuum mechanics, we usually have one set of "universal" field equations for the whole catigory of materials, but a unique "constitutive law" for one individual material (with specific microstructure).  In analogy, I believe the behavior of IPMCs should fit in the general framework of polyelectrolytes.  Through developing a suitable free-energy function (which may be very different from Flory-Renners, since swelling is not an key issue here) and kinetic equations that governs the migration of mobile species, we shall be able to capture the mechanism of IPMCs.

We also believe that the microstructure of the material would influence its properties.  For example, the microstructure of the material near an electrode may be very difference from that in the middle.  If that is the case, modeling the IPMC as a homogeneous material with the same governing equations through out its thickness, and the electrode as mathematical lines may not be a good idea.

We have also realized (just recently) that the limit in the amount of mobile species in an IPMC plays an important role, and maybe it is the major difference between a gel in liquid and an IPMC.

jiangyuli's picture

Wei, I pretty much agree with everything you said in the reply. When I mentioned the difference between IMPC and gel, I focused on the mechanism. In terms of methodology, then of course they bear much similarities. I also agree that the microstructure at metal-Nafion interface is much different from Nafion interior, and that needs to be taken into account in the analysis. For Nafion with the kind of ionic clustering, I also believe that it will be very beneficial if the continuum theory are informed by microscopic studies. I will be very interested in learning your finding on this fascinating materials.

 

Jinxiong Zhou's picture

Hi, Wei,

Sorry for delay to notice this very nice Journal Club you organized. That is a fasinating topic. I have a question regarding your last sentence of your previous reply to Jiangyu's comment. Would you please explain a little bit on the effects of the limit in the amout of mobile species? You mentioned that it plays an important role. What is the limit in the amout of mobile specicies? Why is it an important factor that should be considered?

Regards,

Jinxiong

xiao_wang's picture

 

Hi, Jiangyu

I am Wei's student. Thank you for clarifying the difference between Nafion and a homogeneous polyelectrolyte gel. I have learnt a lot of IPMC by studying the publication of your group, but I still have some questions.

As explained in the paper, the exact shape of cluster is not important. But what if the hydration increases so much (>50%) that the matrix actually forms network structure, and then the geometry of your analysis is completely different, which is somehow closer to a homogeneous polyelectrolyte gel. It seems that simply extend your result based on cluster still agrees with experiment (for example the part on stiffness). Does it imply any relation of your model to a more general explanation to actuation mechanism of polyelectrolyte gel?

I don't understand the microstructure bases of your calculation for the dipole interaction pressure. Because to my knowledge, the sulfonate group should be completely dissociated during hydration, then how is the well oriented dipole pair forms between free ions and the fixed charge on the polymer backbone?

Thank you.

Xiao

jiangyuli's picture

Xiao, I think hydration is not arbitrary, but has an equilibrium governed by osmotic pressure. So for Nafion at least, network structure may not be possible under normal circumstance. I do not believe our model on IMPC can be directly applied to gel. The key difference is that we treat IMPC as a closed system, and thus within it charge is not neutral. On the other hand, gel exchanges ions with solution, and it is safe to assume charge netrality (at a suitable length scales).

 

I am not sure about what you mean by "the sulfonate group should be completely dissociated during hydratio", can you claeify?

Meredith N. Silberstein's picture


Hi,

I work on coupled hygro-mechanical modeling for low temperature fuel cell membranes. I became familiar with the IPMC literature since Nafion is the leading material for both applications.

I agree that from a continuum perspective it is worthwhile to group IPMC and polyelectrolyte gels together since both are electro-chemo-mechanically coupled problems (I would also throw fuel cell membranes and a lot of the biomechanical work in here as well). Establishing a set framework is not going to magically solve the challenge of developing constitutive laws, but it does establish thermodynamic boundaries for those laws and a common language by which various models can be compared. Is anyone working on integrating the more microstructurally based constitutive models of either IPMC or polyelectrolyte gels into a more general electro-chemo-mechanical framework?

A related issue that I noticed while reading through these papers as well as in my own work is that these models are difficult to validate in part because there is no ready made finite element option for their implementation. At least in Abaqus, the coupled hygro-mechanical options (putting aside the extra degree of complication from electric fields and ion flow) seems to be only small deformation.

A side note on an earlier question from Lianhua - when you work with ions and electric fields, it is actually the gradient of the electrochemical potential which drives ion flow rather than the chemical potential that is relevant for neutral species. This may help clarify your understanding.

~Meredith Silberstein

Zhigang Suo's picture

Dear Meredith:  Thank you for your extremely clear statements.  You might have known that recently we found a way to implement pH sensitive polyelectrolyte in ABAQUS.  Our approach deals with finite deformation, but is limited to states of equilibrium.

Romain Marcombe, Shengqiang Cai, Wei Hong, Xuanhe Zhao, Yuri Lapusta, Zhigang Suo, A theory of constrained swelling of a pH-sensitive hydrogel. Soft Matter 6, 784-793 (2010).

Lianhua Ma's picture

Hi Meredith,

 

Thank you for the clarification.

 I think I now understand that the driving mechanism of ion (or solvent) flow derives from the gradient of the electrochemical (chemical) potential, i.e. the difference of the potential, which is similar to pressure difference.

 

My work is also focused on multi-field coupling behaviour. I think the key to investigate this kind of problem is to establish corresponding theoretical framework. Generally, the constitutive model can be derived from either thermodynamics or microstructure. For nonlinear finite deformation issue of multifield material, I think the main difference compared to linear problem lies in:

 

1)      establishing the free energy density function of coupling system.

 

Wei et.al proposed a nonlinear field theory for polyelectrolyte gels by introducing the free-energy density function. Especially, the free energy of polymer chain in gels can be expressed by Flory-Huggins statistical model. However, other materials such as Nafion and biological soft tissue are different from gels in microstructures.  New forms of free energy density function should be further considered.

 

2)      The transformation of Lagrangian and Euler description of governing equations.

 

Most filed governing equations heat transfer ion diffusion, electro-magnetic problems...)are generally described by Euler forms. When coupled with mechanical large deformation, the transformation of the two descriptions of kinetics should be considered.

  

Once we obtain the theoretical models, we can solve the coupled problem by numerical methods such as FEM and FDM.  The implementation of corresponding FEM is another issue. In ABAQUS, most multi-filed coupling modules are only used for small deformation. For finite deformation coupling behaviors, we have to code the derived theoretical formulations by user subroutine (UEL).

Regards

Lianhua

aaam's picture

Dear Colleagues,

 

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

Could we analytically solve the problem of large amplitude vibration for an IPMC beam?

 

Thanks,

Amir

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