123

HandbookSpringerof Experimental Solid

MechanicsSharpe (Ed.)

With DVD-ROM, 874 Figures, 58 in four color and 50 Tables

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187

Electrochemo8. Electrochemomechanicsof Ionic Polymer–Metal Composites

Sia Nemat-Nasser

The ionomeric polymer–metal composites (IPMCs)consist of polyelectrolyte membranes, with metalelectrodes plated on both faces and neutral-ized with an amount of counterions, balancing thecharge of anions covalently fixed to the membrane.IPMCs in the solvated state form soft actuators andsensors; they are sometimes referred to as artificialmuscles. Here, we examine the nanoscale chemo-electromechanical mechanisms that underpin themacroscale actuation and sensing of IPMCs, as wellas some of their electromechanical properties.

8.1 Microstructure and Actuation................. 1888.1.1 Composition ................................ 1888.1.2 Cluster Size .................................. 1898.1.3 Actuation .................................... 191

8.2 Stiffness Versus Solvation ...................... 1918.2.1 The Stress Field

in the Backbone Polymer .............. 191

8.2.2 Pressure in Clusters ...................... 1928.2.3 Membrane Stiffness ...................... 1928.2.4 IPMC Stiffness .............................. 192

8.3 Voltage-Induced Cation Distribution ...... 1938.3.1 Equilibrium Cation Distribution ...... 194

8.4 Nanomechanics of Actuation ................. 1958.4.1 Cluster Pressure Change

Due to Cation Migration ................ 1958.4.2 Cluster Solvent Uptake

Due to Cation Migration ................ 1968.4.3 Voltage-Induced Actuation............ 197

8.5 Experimental Verification ...................... 1978.5.1 Evaluation

of Basic Physical Properties ........... 1978.5.2 Experimental Verification .............. 198

8.6 Potential Applications ........................... 199

References .................................................. 199

Polyelectrolytes are polymers that carry covalently-bound positive or negative charges. They occur nat-urally, such as deoxyribonucleic acid (DNA) andribonucleic acid (RNA), or they have been manu-factured for various applications, such as Nafion R©or Flemion R©, which consist of three-dimensionallystructured backbone perfluorinated copolymers of poly-tetrafluoroethylene, having regularly spaced long per-fluorovinyl ether pendant side-chains that terminatein ionic sulfonate (Nafion) or carboxylate (Flemion)groups. The resulting Nafion or Flemion membranes arepermeable to water or other polar solvents and cations,while they are impermeable to anions.

The ionomeric polymer–metal composites (IPMCs)consist of polyelectrolyte membranes, about 200 μmthick, with metal electrodes (5–10 μm thick) plated onboth faces [8.1] (Fig. 8.1). The polyelectrolyte matrixis neutralized with an amount of counterions, balancing

the charge of anions covalently fixed to the membrane.When an IPMC in the solvated (e.g., hydrated) state isstimulated with a suddenly applied small (1–3 V, de-pending on the solvent) step potential, both the fixedanions and mobile counterions are subjected to an elec-tric field, with the counterions being able to diffusetoward one of the electrodes. As a result, the compositeundergoes an initial fast bending deformation, followedby a slow relaxation, either in the same or in the op-posite direction, depending on the composition of thebackbone ionomer, and the nature of the counterion andthe solvent. The magnitude and speed of the initial fastdeflection also depend on the same factors, as well ason the structure of the electrodes, and other conditions(e.g., the time variation of the imposed voltage). IPMCsthat are made from Nafion and are neutralized with al-kali metals or with alkyl-ammonium cations (except fortetrabutylammonium, TBA+), invariably first bend to-

PartA

8

188 Part A Solid Mechanics Topics

Platinumparticles

Flemion

Au

1μm408nm

Au platingAu plating

a) b)Fig. 8.1a,b Cross section of (a) aPt/Au-plated Nafion-117 membraneat electrode region; the length ofthe bar is 408 nm; and (b) an Au-plated Flemion at electrode region;the length of the bar is 1 μm

wards the anode under a step direct current (DC), andthen relax towards the cathode while the applied volt-age is being maintained, often moving beyond theirstarting position. In this case, the motion towards theanode can be eliminated by slowly increasing the ap-plied potential at a suitable rate. For Flemion-basedIPMCs, on the other hand, the initial fast bending andthe subsequent relaxation are both towards the anode,for all counterions that have been considered. WithTBA+ as the counterion, no noticeable relaxation to-wards the cathode has been recorded for either Nafion-or Flemion-based IPMCs. Under an alternating electric

potential, cantilevered strips of IPMCs perform bendingoscillations at the frequency of the applied voltage, usu-ally no more than a few to a few tens of hertz, dependingon the solvent. When an IPMC membrane is suddenlybent, a small voltage of the order of millivolts is pro-duced across its faces. Hence, IPMCs of this kind canserve as soft actuators and sensors. They are sometimesreferred to as artificial muscles [8.2, 3].

In this chapter, we examine the nanoscale chemo-electromechanical factors that underpin the macroscaleactuation and sensing of IPMCs, as well as some of theirelectromechanical properties.

8.1 Microstructure and Actuation

8.1.1 Composition

The Nafion-based IPMC, in the dry state, is about180 μm thick and the Flemion-based one is about160 μm thick (see [8.4–7] for further information onIPMC manufacturing). Samples consist of

1. backbone perfluorinated copolymer of polytetraflu-oroethylene with perfluorinated vinyl ether sul-fonate pendants for Nafion-based and perfluorinated

Metalcoating

Ionic polymer Naflonor Flamion

Nanosizedinterconnected clusters

0.2 mm

Cracked metal overlayer

a) b)

Cluster group

Fig. 8.2 (a) A schematic representation of an IPMC and (b) a trans-mission electron microscopy (TEM) photo of the cluster structure(see Fig. 8.3 for more detail)

propyl ether carboxylate pendants for Flemion-based IPMCs, forming interconnected nanoscaleclusters ([8.8]; primary physical data for fluorinatedionomers have been summarized in [8.9])

2. electrodes, which in Nafion-based IPMCs consistof 3–10 nm-diameter platinum particles, distributedmainly within a 10–20 μm depth of both faces of themembrane, and usually covered with about 1 μm-thick gold plating to improve surface conductivity,while in Flemion-based IPMCs, the electrodes aregold, with a dendritic structure, as shown in Fig. 8.1

3. neutralizing cations4. a solvent

Figure 8.2 shows a schematic representation of a typicalIPMC, including a photograph of the nanostructure ofthe ionomer (Nafion, in this case), and a sketch of ionicpolymer with metal coating. Figure 8.3 shows some ad-ditional details of a Nafion-based IPMC. The dark spotswithin the inset are the left-over platinum crystals.

The ion exchange capacity (IEC) of an ionomerrepresents the amount of sulfonate (in Nafion) andcarboxylate (in Flemion) group in the material, mea-

PartA

8.1

Electrochemomechanics of Ionic Polymer–Metal Composites 8.1 Microstructure and Actuation 189

Platinum diffusion zone (0–25 μm)

Left-over platinumparticles

1μm

Metal overlayer

20 nm

Fig. 8.3 Near-surface structure of an IPMC; the lower in-set indicates the size of a typical cluster in Nafion

sured in moles per unit dry polymer mass. The drybare ionomer equivalent weight (EW) is defined as theweight in grams of dry ionomer per mole of its anion.The ion exchange capacity and the equivalent weightof Nafion are 0.91 meq/g and 1.100 g/mol, and thoseof Flemion are 1.44 meq/g and 694.4 g/mol, respec-tively.

For neutralizing counterions, we have used Li+,Na+, K+, Rb+, Cs+, Mg2+, and Al3+, as well as alkyl-ammonium cations TMA+, TEA+, TPA+, and TBA+.The properties of the bare ionomer, as well as thoseof the corresponding IPMC, change with the cationtype for the same membrane and solvent. In additionto water, ethylene glycol, glycerol, and crown ethershave been used as solvents. Ethylene glycol or 1,2-ethanediol (C2H6O2) is an organic polar solvent thatcan be used over a wide range of temperatures. Glyc-erol, or 1,2,3-propanetriol (C3H8O3) is another polarsolvent with high viscosity (about 1000 times the vis-cosity of water). Crown ethers are cyclic oligomersof ethylene glycol that serve as macrocyclic ligandsto surround and transport cations (Fig. 8.4). The re-quired crown ether depends on the size of the ion.The 12-Crown-4 (12CR4) matches Li+, 15-Crown-5(15CR5) matches K+, and 18-Crown-6 matched Na+and K+. For example, an 18-Crown-6 (18CR6) mol-ecule has a cavity of 2.7 Å and is suitable for potassiumions of 2.66 Å diameter. A schematic configuration ofthis crown with sodium and potassium ions is shownin Fig. 8.4.

a)

O

O

O

O

O

O

b) c)

Fig. 8.4 (a) Chemical structure of 18-Crown-6 (each node is CH2);(b) Na+ ion; and (c) K+ ion within a macrocyclic 18-Crown-6ligand

8.1.2 Cluster Size

X-ray scanning of the Nafion membranes [8.10] hasshown that, in the process of solvent absorption, hy-drophilic regions consisting of clusters are formedwithin the membrane. Hydrophilicity and hydropho-bicity are generally terms used for affinity or lack ofaffinity toward the polar molecule of water. In thepresent work we use these terms for interaction towardany polar solvent (e.g., ethylene–glycol). Cluster for-mation is promoted by the aggregation of hydrophilicionic sulfonate groups located at the terminuses of vinylether sulfonate pendants of the polytetrafluoroethylenechain. While these regions are hydrophilic, the mem-brane backbone is hydrophobic and it is believed thatthe motion of the solvent takes place among these clus-ters via the connecting channels. The characteristicsof these clusters and channels are important factors inIPMC behavior. The size of the solvated cluster radiusaI depends on the cation form, the type of solvent used,and the amount of solvation. The average cluster sizecan be calculated by minimizing the free energy of thecluster formation with respect to the cluster size. Thetotal energy for cluster formation consists of an elasticUela, an electrostatic Uele, and a surface Usur compo-nent. The elastic energy is given by [8.11]

Uela = 3NkT

〈h2〉

(3

√NEWion

ρ∗NA−aI

)2

,

ρ∗ = ρB +wρs

1+w, (8.1)

where N is the number of dipoles inside a typical clus-ter, k is Boltzmann’s constant, T is the temperature,〈h2〉 is the mean end-to-end chain length, ρ∗ is the ef-fective density of the solvated membrane, and NA isAvogadro’s number (6.023×1023). The electrostatic en-

PartA

8.1

190 Part A Solid Mechanics Topics

Model

x-ray

0 0.5 1 1.5 2 2.5 3

a13 (nm3)

η 105

15

10

5

0

Fig. 8.5 Cluster size in Nafion membranes with differentsolvent uptakes

ergy is given by

Uele = −gN2

4πκe

m2

a3I

, (8.2)

where g is a geometric factor, m is the dipole moment,and κe is the effective permittivity within the cluster.The surface energy can be expressed as

Usur = 4πa2I γ , (8.3)

where γ is the surface energy density of the cluster.Therefore, the total energy due to the presence of clus-ters in the ionomer is given by

Utot = n(Uele +Uela +Usur) , (8.4)

1cm

+

–

a) b) c)(c) t= 4m 45s

(d) t= 6m 27s t= 3ms

t= 5ms

t= 24mst= 36mst= 54ms

t= 10ms

t= 15ms

t= 210ms

t=129ms

(a) t= 0

R+

–

(b) t= 0.6 s

Fig. 8.6 (a) A Nafion-based sample in the thallium (I) ion form is hydrated and a 1 V DC signal is suddenly appliedand maintained during the first 5 min, after which the voltage is removed and the two electrodes are shorted. Initial fastbend toward the bottom ((a) to (b), anode) occurs during the first 0.6 s, followed by a long relaxation upward (towardsthe cathode (c)) over 4.75 min. Upon shorting, the sample displays a fast bend in the same upward direction (not shown),followed by a slow downward relaxation (to (d)) during the next 1.75 min. (b) A Nafion-based sample in the sodiumion form is solvated with glycerol, and a 2 V DC signal is suddenly applied and maintained. It deforms into a perfectcircle, but its qualitative response is the same. (c) A Flemion-based sample in tetrabutylammonium ion form is hydrated,and a 3 V DC signal is suddenly applied and maintained, resulting in continuous bending towards the anode (no backrelaxation)

where n is the number of clusters present in the mem-brane. Minimizing this energy with respect to clustersize,

(∂Utot∂aI

= 0), gives the optimum cluster size at

which the free energy of the ionomer is minimum. Inthis manner Li and Nemat-Nasser [8.11] have obtained

aI =⎡⎣γ 〈h2〉EWion

2RT

(w+ΔV )

ρB

×

(1− 3

√4πρB

3ρ∗(w+ΔV )

)−2⎤⎦1/3

,

ΔV = NA Viρd

EWion, (8.5)

where Vi is the volume of a single ion exchange site.Assuming that 〈h2〉 = EWionβ [8.12], it can be seen that,

a3I = γβ

2RTη ,

η = EW2ion(w+ΔV )

ρB

(1− 3

√4πρB

3ρ∗(w+ΔV )

)−2

.

(8.6)

Figure 8.5 shows the variation of the cluster size(a3

I in nm3) for different solvent uptakes. Data fromNafion-based IPMC samples with different cations anddifferent solvent uptakes are considered. The model iscompared with the experimental results on the clus-ter size based on x-ray scanning, shown as circles

PartA

8.1

Electrochemomechanics of Ionic Polymer–Metal Composites 8.2 Stiffness Versus Solvation 191

in Fig. 8.5 for a Nafion ionomer in various cation formsand with water as the solvent [8.10]. We have setβ = 1.547, γ = 0.15, and Vi = 68 × 10−24 cm3 to calcu-late the cluster size [8.12].

8.1.3 Actuation

A Nafion-based IPMC sample in the solvated stateperforms an oscillatory bending motion when an al-ternating voltage is imposed across its faces, and itproduces a voltage when suddenly bent. When thesame strip is subjected to a suddenly imposed andsustained constant voltage (DC) across its faces, aninitial fast displacement (towards the anode) is gen-

erally followed by a slow relaxation in the reversedirection (towards the cathode). If the two faces ofthe strip are then shorted during this slow relax-ation towards the cathode, a sudden fast motion inthe same direction (towards the cathode) occurs, fol-lowed by a slow relaxation in the opposite direction(towards the anode). Figure 8.6 illustrates these pro-cesses for a hydrated Tl+-form Nafion-based IPMC(left), Na+-form with glycerol as solvent (middle), andhydrated Flemion-based in TBA+-form (right), under1, 2, and 3 V DC, respectively. The magnitudes of thefast motion and the relaxation that follows the fastmotion change with the corresponding cation and thesolvent.

8.2 Stiffness Versus Solvation

To model the actuation of the IPMC samples interms of the chemoelectromechanical characteristics ofthe backbone ionomer, the electrodes, the neutraliz-ing cation, the solvent, and the level of solvation, itis first necessary to model the stiffness of the cor-responding samples. This is discussed in the presentsection.

A dry sample of a bare polymer or an IPMC placedin a solvent bath absorbs solvent until the resulting pres-sure within its clusters is balanced by the elastic stressesthat are consequently developed within its backbonepolymer membrane. From this observation the stiffnessof the membrane can be estimated as a function of thesolvent uptake for various cations. Consider first thebalance of the cluster pressure and the elastic stressesfor the bare polymer (no metal plating) and then usethe results to calculate the stiffness of the correspond-ing IPMC by including the effect of the added metalelectrodes. The procedure also provides a way of es-timating many of the nanostructural parameters thatare needed for the modeling of the actuation of theIPMCs.

8.2.1 The Stress Fieldin the Backbone Polymer

The stresses within the backbone polymer may beestimated by modeling the polymer matrix as an incom-pressible elastic material [8.13, 14]. Here, it will proveadequate to consider a neo-Hookean model for the ma-trix material. In this model, the principal stresses σI arerelated to the principal stretches λI by σI = −p0 + Kλ2

I ,

where p0 is an undetermined parameter (pressure) tobe calculated from the boundary data and K is an ef-fective stiffness, approximately equal to a third of theoverall Young’s modulus. The aim is to calculate K andp0 as functions of the solvent uptake w for various ion-form membranes. To this end, examine the deformationof a unit cell of the solvated polyelectrolyte (bare mem-brane) by considering a spherical cavity of initial (i. e.,dry state) radius a0 (representing a typical cluster), em-bedded at the center of a spherical matrix of initialradius R0, and placed in a homogenized solvated mem-brane, referred to as the matrix. In micromechanics, thisis called the double-inclusion model [8.15]. Assume thatthe stiffness of both the spherical shell and the homoge-nized matrix is the same as that of the (as yet unknown)overall effective stiffness of the hydrated membrane.For an isotropic expansion of a typical cluster, the twohoop stretches (and stresses) are equal, and using theincompressibility condition and spherical coordinates,it follows that

σr(r0) = −p0 + K [(r0/a0)−3(w/w0 −1)+1]−4/3 ,

σθ (r0) = σϕ(r0) = −p0

+ K [(r0/a0)−3(w/w0 −1)+1]2/3 , (8.7)

where r0 measures the initial radial length from thecenter of the cluster, and w0 is the initial (dry) vol-ume fraction of the voids. Theeffective elastic resistanceof the (homogenized solvated) membrane balancesthe cluster’s pressure pc which is produced by thecombined osmotic and electrostatic forces within thecluster.

PartA

8.2

192 Part A Solid Mechanics Topics

8.2.2 Pressure in Clusters

For the solvated bare membrane or an IPMC in theM+-ion form, and in the absence of an applied elec-tric field, the pressure within each cluster pc consistsof osmotic Π(M+) and electrostatic pDD components.The electrostatic component is produced by the ionicinteraction within the cluster. The cation–anion conju-gate pairs can be represented as uniformly distributeddipoles on the surface of a spherical cluster, and theresulting dipole–dipole (DD) interaction forces pDDcalculated. The osmotic pressure is calculated by ex-amining the difference between the chemical potentialof the free (bath) solvent and that of the solvent withina typical cluster of known ion concentration within themembrane. In this manner, one obtains [8.16]

pc = νQ−B K0φ

w+ 1

3κeQ−2

B±α2

w2,

K0 = RT

F, Q−

B = ρB F

EWion, (8.8)

where φ is the practical osmotic coefficient, α is the ef-fective length of the dipole, F is the Faraday constant,ν is the cation–anion valence (ν = 2 for monovalentcations), R = 8.314 J/mol/K is the universal gas con-stant, T is the cluster temperature, ρB is the density ofthe bare ionomer, and κe is the effective permittivity.

8.2.3 Membrane Stiffness

The radial stress σr must equal the pressure pc in thecluster at r0 = a0. In addition, the volume average ofthe stress tensor, taken over the entire membrane, mustvanish in the absence of any externally applied loads.This is a consistency condition that to a degree accountsfor the interaction among clusters. These conditions aresufficient to yield the undetermined pressure p0 andthe stiffness K in terms of solvation volume fractionw and the initial dry void volume fraction w0 for eachion-form bare membrane, leading to the following finalclosed-form results:

K = pc1+w

w0 In − (w0/w)4/3,

In = 1+2An0

n0(1+ An0)1/3− 1+2A

(1+ A)1/3, A = w

w0−1 ,

(8.9)

where n0 and w0 = n0/(1 − n0) are the initial (dry)porosity and initial void ratio (volume of voids divided

by the volume of solid), respectively. The membraneYoung’s modulus may now be set equal to YB = 3K (w),assuming that both the elastomer and solvent are essen-tially incompressible under the involved conditions.

8.2.4 IPMC Stiffness

To include the effect of the metal plating on the stiff-ness, note that for the Nafion-based IPMCs the goldplating is about a 1 μm layer on both faces of an IPMCstrip, while platinum particles are distributed throughthe first 10–20 μm surface regions, with diminishingdensity. Assume a uniaxial stress state and average theaxial strain and stress over the strip’s volume to obtaintheir average values, as

εIPMC = fMHεB + (1− fMH)εM ,

σIPMC = fMHσB + (1− fMH)σM , fMH = fM

1+w,

(8.10)

where the barred quantities are the average values ofthe axial strain and stress in the IPMC, bare (sol-vated) membrane, and metal electrodes, respectively(indicated by the corresponding subscripts ‘B’ and‘M’, respectively); and fM is the volume fraction ofthe metal plating in a dry sample. Setting σB = YBεB,σM = YMεM, and σIPMC = YIPMCεIPMC, it follows that

YIPMC = YMYB

BABYM + (1− BAB)YB,

B = (1+ w)(1− fM)

1+ w(1− fM), w = w(1− fM) ,

(8.11)

where AB is the concentration factor, giving the aver-age stress in the bare polymer in terms of the averagestress of the IPMC σIPMC. Here YB = 3K is evaluatedfrom (8.9) at solvation w when the solvation of theIPMC is w. The latter can be measured directly at vari-ous solvation levels.

Experimentally, the dry and solvated dimensions(thickness, width, and length) and masses of each bareor plated sample are measured. Knowing the compo-sition of the ionomer, the neutralizing cation, and thecomposition of the electrodes, all physical quantitiesin equations (8.8) and (8.11) are then known for eachsample, except for the three parameters α, κe, and AB,namely the effective distance between the neutraliz-ing cation and its conjugate covalently fixed anion, theeffective electric permittivity of the cluster, and the con-centration factor that defines the fraction of axial stress

PartA

8.2

Electrochemomechanics of Ionic Polymer–Metal Composites 8.3 Voltage-Induced Cation Distribution 193

carried by the bare elastomer in the IPMC. The par-ameters α and κe are functions of the solvation level.They play critical roles in controlling the electrostaticand chemical interaction forces within the clusters, aswill be shown later in this chapter in connection withthe IPMC’s actuation. Thus, they must be evaluatedwith care and with due regard for the physics of theprocess.

Estimate of κe and α

The solvents are polar molecules, carrying an electro-static dipole. Water, for example, has a dipole momentof about 1.87 D (Debyes) in the gaseous state and about2.42 D in bulk at room temperature, the difference beingdue to the electrostatic pull of other water molecules.Because of this, water forms a primary and a secondaryhydration shell around a charged ion. Its dielectric con-stant as a hydration shell of an ion (about 6) is thus muchsmaller than that in bulk (about 78). The second term inexpression (8.8) for the cluster pressure pc, can changeby a factor of 13 for water as the solvent, dependingon whether the water molecules are free or constrainedto a hydration shell. Similar comments apply to othersolvents. For example, for glycerol and ethylene gly-col, the room-temperature dielectric constants are about9 and 8 as solvation shells, and 46 and 41 when inbulk.

When the cluster contains both free and ion-boundsolvent molecules, its effective electric permittivitycan be estimated using a micromechanical model. Letκ1 = ε1κ0 and κ2 = ε2κ0 be the electric permittivity ofthe solvent in a solvation shell and in bulk, respectively,where κ0 = 8.85 × 10−12 F/m is the electric permit-tivity of free space. Then, using a double-inclusion

model [8.15] it can be shown that

κe = κ2 +2κ1 + f (κ2 −κ1)

κ2 +2κ1 − f (κ2 −κ1)κ1 ,

f = mw −CN

mw, mw = EWionw

MsolventρBν. (8.12)

Here, CN is the static solvation shell (equal to the coor-dination number), mw is the number of moles of solventper mole of ion (cation and anion), and ν = 2. When mwis less than CN, then all solvent molecules are part of thesolvation shell, for which κe = ε1κ0. On the other hand,when there are more solvent molecules, equation (8.12)yields the corresponding value of κe. Thus, κe is cal-culated in a cluster in terms of the cluster’s volumefraction of solvent, w.

Similarly, the dipole arm α in (8.8), which measuresthe average distance between a conjugate pair of anion–cation, is expected to depend on the effective dielectricconstant of the solvation medium. We now calculate theparameter ±α2, as follows. As a first approximation, welet α2 vary linearly with w for mw ≤ CN, i. e., we set

±α2 = a1w+a2 , for mw ≤ CN , (8.13)

and estimate the coefficients a1 and a2 from the experi-mental data. For mw > CN, furthermore, we assume thatthe distance between the two charges forming a pseudo-dipole is controlled by the effective electric permittivityof the their environment (e.g., water molecules), andhence is given by

α2 = 10−20(

κe

κ1

)2

(a1w+a2) . (8.14)

Note that for mw > CN, we have a1w+a2 > 0. An il-lustrative example is given in Sect. 8.5 where measuredresults are compared with model predictions.

8.3 Voltage-Induced Cation Distribution

When an IPMC strip in a solvated state is subjectedto an electric field, the initially uniform distribution ofits neutralizing cations is disturbed, as cations on theanode side are driven out of the anode boundary clus-ters while the clusters in the cathode side are suppliedwith additional cations. This redistribution of cationsunder an applied potential can be modeled using thecoupled electrochemical equations that characterize thenet flux of the species, caused by the electrochemi-cal potentials (chemical concentration and electric fieldgradients). The total flux consists of cation migrationand solvent transport. The flux Ji of species i is charac-

terized by [8.17–19],

Ji = −Ci Di

RT

∂μi

∂x+Ciυi , (8.15)

where Di is the diffusivity coefficient, μi is the chem-ical potential, Ci is the concentration, and υi is thevelocity of species i. The chemical potential is givenby

μi = μ0 + RT ln(γiCi )+ ziφF , (8.16)

where μ0 is the reference chemical potential, γi isthe affinity of species i, zi is the species charge, and

PartA

8.3

194 Part A Solid Mechanics Topics

φ = φ(x, t) is the electric potential. For an ideal solu-tion where γi = 1, and if there is only one type of cation,the subscript i may be dropped, as will be done in whatfollows.

The variation in the electric potential field in themembrane is governed by the basic Poisson’s electro-static equations [8.20, 21],

∂(κE)

∂x= z(C −C−)F , E = −∂φ

∂x, (8.17)

where E and κ are the electric field and the electric per-mittivity, respectively; C− is the anion concentration inmoles per unit solvated volume; and C = C(x, t) is thetotal ion (cation and anion) concentration. Since the sol-vent velocity is very small, the last term in (8.15) maybe neglected and, in view of (8.17) and continuity, itfollows that

J(x, t) = −D

[∂C(x, t)

∂x

− zC(x, t)

(F

RT

)E(x, t)

],

∂C(x, t)

∂t= −∂J(x, t)

∂x,

∂E(x, t)

∂x= z

(F

κ

)(C(x, t)−C−) . (8.18)

Here κ is the overall electric permittivity of the solvatedIPMC sample that can be estimated from its measuredeffective capacitance. The above system of equationscan be directly solved numerically, or they can be solvedanalytically using approximations. In the following sec-tions, the analytic solution is considered.

First, from (8.18) it follows that

∂

∂x

[∂(κE)

∂t− D

(∂2(κE)

∂x2− C−F2

κRT(κE)

)]= 0 .

(8.19)

This equation provides a natural length scale � anda natural time scale τ that characterize the ionredistribution,

� =(

κRT

C−F2

)1/2

, τ = �2

D, (8.20)

where κ is the overall electric permittivity of the IPMC.If Cap is the measured overall capacitance, then we setκ = 2HCap. Since � is linear in

√κ and κ is propor-

tional to the capacitance, it follows that � is proportionalto the square root of the capacitance.

8.3.1 Equilibrium Cation Distribution

To calculate the ion redistribution caused by the applica-tion of a step voltage across the faces of a hydrated stripof IPMC, we first examine the time-independent equi-librium case with J = 0. In the cation-depleted (anode)boundary layer the charge density is −C−F, whereas inthe remaining part of the membrane the charge densityis (C+ −C−)F. Let the thickness of the cation-depletedzone be denoted by �′ and set

Q(x, t) = C+ −C−

C− , Q0(x) = limt→∞ Q(x, t) .

(8.21)

Then, it follows from (8.17) and boundary and continu-ity conditions that [8.16] the equilibrium distribution isgiven by

Q0(x) ≡

⎧⎪⎪⎨⎪⎪⎩

−1 for x ≤ −h +�′F

RT [B0 exp(x/�)− B1 exp(−x/�)] ,

for −h +�′ < x < h ,

(8.22)

B1 = K0 exp(−a′) ,�′

�=

√2φ0 F

RT−2 ,

B2 = φ0

2− 1

2K0

[(�′

�+1

)2

+1

],

B0 = exp(−a)[φ0/2+ B1 exp(−a)+ B2

],

K0 = F

RT, (8.23)

where φ0 is the applied potential, a ≡ h/�, and a′ ≡ (h −�′)/�. Since � is only 0.5–3 μm, a ≡ h/�, a′ ≡ h′/� �1, and hence exp(−a) ≈ 0 and exp(−a′) ≈ 0. The con-stants B0 and B1 are very small, of the order of10−17 or even smaller, depending on the value of thecapacitance. Therefore, the approximation used to ar-rive at (8.22) does not compromise the accuracy ofthe results. Remarkably, the estimated length of theanode boundary layer with constant negative chargedensity −C−F, i. e., �′ = (√

2φ0 F/RT −2)�, depends

only on the applied potential and the effective capac-itance through the characteristic length. We use � asour length scale. From (8.23) it can be concluded that,over the most central part of the membrane, there ischarge neutrality within all clusters, and the charge im-balance in clusters occurs only over narrow boundarylayers, with the anode boundary layer being thicker

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Electrochemomechanics of Ionic Polymer–Metal Composites 8.4 Nanomechanics of Actuation 195

than the cathode boundary layer. The charge imbal-ance in the clusters is balanced by the correspondingelectrode charges. Therefore, the charge imbalance de-fined by (8.23) applies to the clusters within eachboundary layer and not to the boundary layer itself,since the combined clusters and the charged metalparticles within the boundary layers are electrically bal-anced.

We now examine the time variation of the chargedistribution that results upon the application of a stepvoltage and leads to the equilibrium solution givenabove. To this end, rewrite equation (8.19) as

∂Q

∂t− D

(∂2 Q

∂x2− Q

�2

)= 0 , (8.24)

and set Q = ψ(x, t) exp(−t/τ)+ Q0(x) to obtain thefollowing standard diffusion equation for ψ(x, t):

∂ψ

∂t= D

∂2ψ

∂x2, ψ(x, 0) = −Q0(x) , (8.25)

from which it can easily be concluded that, to a gooddegree of accuracy, one may use the following sim-ple approximation in place of an exact infinite series

solution [8.16]:

Q(x, t) ≈ g(t)Q0(x) , g(t) = 1− exp(−t/τ) . (8.26)

Actually, when the electric potential is suddenly ap-plied and then maintained, say, at a constant level φ0,the cations of the anode clusters are initially depletedat the same rate as cations being added to the clusterswithin the cathode boundary layer. Thus, initially, thecation distribution is antisymmetric through the thick-ness of the membrane, as shown by Nemat-Nasser andLi [8.22],

C(x) = C− + κφ0

2F�2 sinh(h/�)sinh(x/�) . (8.27)

After a certain time period, say, τ1, clusters near the an-ode face are totally depleted of their cations, so only thelength of the anode boundary layer can further increase,while the cathode clusters continue to receive additionalcations. Recognizing this fact, recently Nemat-Nasserand Zamani [8.23] have used one time scale for the firstevent and another time scale for the remaining processof cation redistribution. To simulate the actuation, how-ever, these authors continue to use the one-time-scaleapproximation similar to (8.26).

8.4 Nanomechanics of Actuation

The application of an electric potential produces twothin boundary layers, one near the anode and the othernear the cathode electrodes, while maintaining overallelectric neutrality in the IPMC strip. The cation imbal-ance within the clusters of each boundary layer changesthe osmotic, electrostatic, and elastic forces that tend toexpand or contract the corresponding clusters, forcingthe solvents out of or into the clusters, and producingthe bending motion of the sample. Thus, the volumefraction of the solvent within each boundary layer iscontrolled by the effective pressure in the correspond-ing clusters produced by the osmotic, electrostatic, andelastic forces. These forces can even cause the cathodeboundary layer to contract during the back relaxationthat is observed for Nafion-based IPMCs in various al-kali metal forms, expelling the extra solvents onto theIPMC’s surface while cations continue to accumulatewithin the cathode boundary layer. This, in fact, is whathas been observed in open air during the very slow backrelaxation of IPMCs that are solvated with ethylene gly-col or glycerol [8.24]. To predict this and other actuationresponses of IPMCs, it is thus necessary to model the

changes in the pressure within the clusters in the anodeand cathode boundary layers, and the resulting diffusiveflow of solvent into or out of the corresponding clusters.

8.4.1 Cluster Pressure ChangeDue to Cation Migration

The elastic pressure on a cluster can be calculatedfrom (8.7) by simply setting r0 = aI, where aI is theinitial cluster size just before the potential is applied,and using the spatially and temporally changing volumefraction of solvent and ion concentration w(x, t) andν(x, t), since the osmotic pressure depends on the clus-ter’s ion concentration and it is reduced in the anode andis increased in the cathode clusters. The correspondingosmotic pressure can be computed from

Π(x, t) = φQ−B K0

w(x, t)ν(x, t) , ν(x, t) = C+(x, t)

C− +1 ,

(8.28)

where C+(x, t) is the cation concentration; notefrom (8.21) that ν(x, t) = Q(x, t)+2. The pressure pro-

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8.4

196 Part A Solid Mechanics Topics

duced by the electrostatic forces among interactingfixed anions and mobile cations within each clustervaries as the cations move into or out of the cluster inresponse to the imposed electric field.

As the cations of the clusters within the anodeboundary layer are depleted, the dipole–dipole inter-action forces diminish. We model this in the anodeboundary layer by calculating the resulting pressure,pADD, as a function of the cation concentration, asfollows:

pADD(x, t) = 1

3κ(x, t)Q−2

B

(α(x, t)

w(x, t)

)2(C+(x, t)

C−

).

(8.29)

Parallel with the reduction in the dipole–dipole in-teraction forces is the development of electrostaticinteraction forces among the remaining fixed anions,which introduces an additional pressure, say pAA,

pAA(x, t) = 1

18κ(x, t)Q−2

B

×R2

0

[w(x, t)]4/3

(1− C+(x, t)

C−

), (8.30)

where R0 is the initial (unsolvated, dry) cluster size.This expression is obtained by considering a spheri-cal cluster with uniformly distributed anion charges onits surface. The total pressure within a typical anodeboundary layer cluster hence is

tA(x, t) = −σr(a0, t)+Π(x, t)+ pAA(x, t)

+ pADD(x, t) . (8.31)

Consider now the clusters in the cathode boundarylayer. In these clusters, in addition to the osmotic pres-sure we identify two forms of electrostatic interactionforces. One is repulsion due to the cation–anion pseudo-dipoles already present in the clusters, and the otheris due to the extra cations that migrate into the clus-ters and interact with the existing pseudo-dipoles. Theadditional stresses produced by this latter effect maytend to expand or contract the clusters, depending onthe distribution of cations relative to fixed anions. Weagain model each effect separately, although in fact theyare coupled. The dipole–dipole interaction pressure inthe cathode boundary layer clusters is assumed to bereduced as

pCDD(x, t) = 1

3κ(x, t)Q−2

B

(α(x, t)

w(x, t)

)2

×

(C−

C+(x, t)

), (8.32)

while at the same time new dipole–cation interactionforces are being developed as additional cations enterthe clusters and disturb the pseudo-dipole structure inthe clusters. The pressure due to these latter forces isrepresented by

pDC(x, t) = 2Q−2

B

9κ(x, t)

R0α(x, t)

[w(x, t)]5/3

(C(x, t)

C− −1

).

(8.33)

However, for sulfonates in a Nafion-based IPMC, weexpect extensive restructuring and redistribution of thecations. It appears that this process underpins the ob-served reverse relaxation of the Nafion-based IPMCstrip. Indeed, this redistribution of the cations withinthe clusters in the cathode boundary layer may quicklydiminish the value of pCD to zero or even render it nega-tive. To represent this, we modify (8.33) by a relaxationfactor, and write

pDC(x, t) ≈ 2Q−2

B

9κC(x, t)

R0αC(x, t)

[wC(x, t)]5/3

×

(C+(x, t)

C− −1

)g1(t) ,

g1(t) = [r0 + (1− r0) exp(−t/τ1)], r0 < 1 ,

(8.34)

where τ1 is the relaxation time and r0 is the equilibriumfraction of the dipole–cation interaction forces. The to-tal stress in clusters within the cathode boundary layeris now approximated by

tC = −σr(a0, t)+ΠC(x, t)+ pCDD(x, t)+ pDC(x, t) .

(8.35)

8.4.2 Cluster Solvent UptakeDue to Cation Migration

The pressure change in clusters within the anode andcathode boundary layers drives solvents into or out ofthese clusters. This is a diffusive process that can bemodeled using the continuity equation

w(x, t

)1+w

(x, t

) + ∂υ(x, t

)∂x

= 0 , (8.36)

and a constitutive model to relate the gradient of thesolvent velocity, ∂υ(x, t)/∂x, to the cluster pressure. Wemay use a linear relation and obtain

wi(x, t

)1+wi

(x, t

) = DBLti(x, t

), i = A, C , (8.37)

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Electrochemomechanics of Ionic Polymer–Metal Composites 8.5 Experimental Verification 197

where DBL is the boundary-layer diffusion coefficient,assumed to be constant here.

8.4.3 Voltage-Induced Actuation

As is seen from Fig. 8.6, an IPMC strip can undergolarge deflections under an applied electric potential.Since the membrane is rather thin (0.2 mm) even if itdeforms from a straight configuration into a circle of ra-dius, say 1 cm still the radius-to-thickness ratio wouldbe 50, suggesting that the maximum axial strain in themembrane is very small. It is thus reasonable to usea linear strain distribution through the thickness anduse the following classical expression for the maximumstrain:

εmax ≈ ± H

Rb, (8.38)

where Rb is the radius of the curvature of the membraneand H is half of its thickness. The strain in the mem-brane is due to the volumetric changes that occur withinthe boundary layers, and can be estimated from

εv(x, t) = ln[1+w(x, t)] . (8.39)

We assume that the axial strain is one-third of the volu-metric strain, integrate over the thickness, and obtain

L

Rb= L

2H3(3Y IPMC −2YB)

×

h∫−h

YBL(w(x, t))x ln[1+w(x, t)]dx , (8.40)

where all quantities have been defined before and aremeasurable.

8.5 Experimental Verification

We now examine some of the experimental results thathave been used to characterize this material and to ver-ify the model results.

8.5.1 Evaluationof Basic Physical Properties

Both the bare ionomer and the corresponding IPMCcan undergo large dimensional changes when sol-vated. The phenomenon is also affected by the natureof the neutralizing cation and the solvent. This, inturn, substantially changes the stiffness of the ma-terial. Therefore, techniques have been developed tomeasure the dimensions and the stiffness of the ma-

Table 8.1 Measured stiffness of dry and hydrated Nafion/Flemion ionomers and IPMCs in various cation forms

Dry form Water-saturated form

Thickness Density Stiffness Thickness Density Stiffness Hydration

volume

(μm) (g/cm3) (MPa) (μm) (g/cm3) (MPa) (%)

Na+ 182.1 2.008 1432.1 219.6 1.633 80.5 71.3

Bare Nafion K+ 178.2 2.065 1555.9 207.6 1.722 124.4 50.0

⎧⎨⎩

Cs+ 189.1 2.156 1472.2 210.5 1.836 163.6 41.4

Na+ 149.4 2.021 2396.0 167.6 1.757 168.6 42.0

Flemion K+ 148.4 2.041 2461.2 163.0 1.816 199.5 34.7

⎧⎨⎩

Cs+ 150.7 2.186 1799.2 184.2 1.759 150.6 53.7

IPMC Nafion-based Cs+ 156.0 3.096 1539.5 195.7 2.500 140.4 54.1

Flemion-based Cs+ 148.7 3.148 2637.3 184.1 2.413 319.0 58.1

terial in various cation forms and at various solvationstages. The reader is referred to Nemat-Nasser andThomas [8.25, 26], Nemat-Nasser and Wu [8.27], andNemat-Nasser and Zamani [8.23] for the details of var-ious measurements and extensive experimental results.In what follows, a brief account of some of the essentialfeatures is presented.

One of the most important quantities that charac-terize the ionic polymers is the material’s equivalentweight (EW), defined as dry mass in grams of ionicpolymer in proton form divided by the moles of sul-fonate (or carboxylate) groups in the polymer. It isexpressed in grams per mole, and is measured by neu-tralizing the same sample sequentially with deferent

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8.5

198 Part A Solid Mechanics Topics

cations, and measuring the changes in the weight ofthe sample. This change is directly related to the num-ber of anion sites within the sample and the differencein the atomic weight of the cations. Another importantparameter is the solvent uptake w, defined as the vol-ume of the solvent divided by the volume of the drysample. Finally, it is necessary to measure the weightfraction of the metal in an IPMC sample, which is usu-ally about 40%.

The uniaxial stiffness of both the bare ionomer andthe corresponding IPMC changes with the solvent up-take. It is also a function of the cation form, the solvent,and the backbone material. Table 8.1 gives some dataon bare and metal-plated Nafion and Flemion in the in-dicated cation forms. Note that the stiffness can varyby a factor of ten for the same sample depending onwhether it is dry or fully hydrated.

Surface conductivity is an important electrical prop-erty governing an IPMC’s actuation behavior. Whenapplying a potential across the sample’s thickness atthe grip end, the bending of the cantilever is affectedby its surface resistance, which in turn is dependent onthe electrode morphology, cation form, and the level ofsolvation.

An applied electric field affects the cation distri-bution within an IPMC membrane, forcing the cationsto migrate towards the cathode. This change in thecation distribution produces two thin layers, one nearthe anode and the other near the cathode boundaries.In time, and once an equilibrium state is attained,the anode boundary layer is essentially depleted of itscations, while the cathode boundary layer has becomecation rich. If the applied constant electric potentialis V and the corresponding total charge that is ac-cumulated within the cathode boundary layer oncethe equilibrium state is attained is Q, then the effec-tive electric capacitance of the IPMC is defined asC = Q/V . From this, one obtains the correspondingareal capacitance, measured in mF/cm2, by dividing bythe area of the sample. Usually, the total equilibriumaccumulated charge can be calculated by time integra-tion of the measured net current, and using the actualdimensions of the saturated sample. For alkali-metalcations, one may have capacitance of 1–50 mF/cm2 foran IPMC.

8.5.2 Experimental Verification

As an illustration of the model verification, considerfirst the measured and modeled stiffness of the bare andmetal-plated Nafion-based samples (Fig. 8.7). Here, the

0 10 20 30 40 50 60 70 80

Stiffness (MPa)

Hydration (%)

2500

2000

1500

1000

500

0

Experimental_bare NaflonExperimental_IPMC (Sh2)Experimental_IPMC (Sh5)Model_bare NaflonModel_Naflon IPMC

Fig. 8.7 Uniaxial stiffness (Young’s modulus) of bareNafion-117 (lower data points and the solid curve, model)and an IPMC (upper data points and solid curve) in theNa+-form versus hydration water

lower data points and the solid curve are for the bareionomer in Na+-form, and the upper data points andthe solid curve are for the corresponding IPMC. Themodel results have been obtained for the bare Nafionusing (8.9) and for the IPMC from (8.11). In (8.9), φ

is taken to be 1 and n0 to be 1%. The dry density ofthe bare membrane is measured to be 2.02 g/cm3, theequivalent weight for Nafion-117 in Na+-form (with23 atomic weight for sodium) is calculated to be 1122,the electric permittivity is calculated from (8.12) withCN = 4.5, the temperature is taken to be 300 K (roomtemperature), and κe is calculated using (8.12), with α2

–10 0 10 20 30 40 50 60

Tip displacement/gauge length

Time (s)

0.12

0.08

0.04

0

–0.04

–0.08

–0.12

–0.16

ExperimentModel

Fig. 8.8 Tip displacement of a 15 mm cantilevered strip ofa Nafion-based IPMC in Na+-form, subjected to a 1.5 Vstep potential for about 32 s, then shorted; the solid curveis the model and the geometric symbols are experimentalpoints

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8.5

Electrochemomechanics of Ionic Polymer–Metal Composites References 199

being calculated from (8.13) or (8.14) depending on thelevel of hydration, i. e., the value of w. The only freeparameters are then a1 and a2, which are calculated tobe a1 = 1.5234 × 10−20 and a2 = −0.0703 × 10−20, us-ing two data points. In (8.11), fM is measured to be0.0625, YM is 75 GPa, and AB (the only free parameter)is set equal to 0.5.

To check the model prediction of the observed ac-tuation response of this Nafion-based IPMC, considera hydrated cantilevered strip subjected to a 1.5 V steppotential across its faces that is maintained for 32 s andthen shorted. Figure 8.8 shows (geometric symbols) themeasured tip displacement of a 15 mm-long cantileverstrip that is actuated by applying a 1.5 V step potentialacross its faces, maintaining the voltage for about 32 sand then removing the voltage while the two faces areshorted. The initial water uptake is wIPMC = 0.46, andthe volume fraction of metal plating is 0.0625. Hence,the initial volume fraction of water in the Nafion partof the IPMC is given by wI = wIPMC/(1− fM) = 0.49.The formula weight of sodium is 23, and the drydensity of the bare membrane is 2.02 g/cm3. WithEWNa+ = 1122 g/mol, the initial value of C− for the

bare Nafion becomes 1208. The thickness of the hy-drated strip is measured to be 2H = 224 μm, and basedon inspection of the microstructure of the electrodes,we set 2h ≈ 212 μm. The effective length of the anodeboundary layer for φ0 = 1.5 V is LA = 9.78�, and inorder to simplify the model calculation of the solventflow into or out of the cathode clusters, an equiva-lent uniform boundary layer is used near the cathodewhose thickness is then estimated to be LC = 2.84�,where � = 0.862 μm for φ0 = 1.5 V. The electric per-mittivity and the dipole length are calculated usinga1 = 1.5234 × 10−20 and a2 = −0.0703 × 10−20, whichare the same as for the stiffness modeling. The mea-sured capacitance ranges from 10 to 20 mF/cm2, andwe use 15 mF/cm2. Other actuation model parametersare: DA = 10−2 (when pressure is measured in MPa),τ = 1/4 and τ1 = 4 (both in seconds), and r0 = 0.25.Since wI ≈ (a/R0)3, where a is the cluster size at wa-ter uptake wI, we set R0 ≈ w

−1/30 a, and adjust a to fit

the experimental data; here, a = 1.65 nm, or an averagecluster size of 3.3 nm, prior to the application of the po-tential. The fraction of cations that are left after shortingis adjusted to r = 0.03.

8.6 Potential Applications

The ultimate success of IPMC materials depends ontheir applications. Despite the limited force and fre-quency that IPMC materials can offer, a number of ap-plications have been proposed which take advantage ofIPMCs’ bending response, low voltage/power require-ments, small and compact design, lack of moving parts,and relative insensitivity to damage. Osada and cowork-ers have described a number of potential applicationsfor IPMCs, including catheters [8.28, 29], elliptic fric-tion drive elements [8.30], and ratchet-and-pawl-basedmotile species [8.31]. IPMCs are being considered forapplications to mimic biological muscles; Caldwell hasinvestigated artificial muscle actuators [8.32, 33] andShahinpoor has suggested applications ranging fromperistaltic pumps [8.34] and devices for augmenting hu-man muscles [8.35] to robotic fish [8.36]. Bar-Cohenand others have discussed the use of IPMC actua-

tors in space-based applications [8.2, 37], as their lackof multiple moving parts is ideal for any environ-ment where maintenance is difficult. Besides these,IPMCs find applications in other disciplines, includingfuel-cell membranes, electrochemical sensing [8.38],and electrosynthesis [8.2, 37, 39], as their lack ofmultiple moving parts is ideal for any environmentwhere maintenance is difficult. Besides these, IPMCsfind applications in other disciplines, including fuel-cell membranes, electrochemical sensing [8.38], andelectrosynthesis [8.39]. The commercial availability ofNafion has enabled alternate uses for these mater-ials to be found. Growth and maturity in the field ofIPMC actuators will require exploration beyond thesecommercially available polymer membranes, as the ap-plications for which they have been designed are notnecessarily optimal for IPMC actuators.

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