doi: 10.1098/rspa.2003.1124, 2121-2130459 2003 Proc. R. Soc. Lond. A

P.A Voltairas, D.I Fotiadis and C.V Massalas A theoretical study of the hyperelasticity of electro?gels

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10.1098/rspa.2003.1124

A theoretical study of the hyperelasticityof electro-gels

By P. A. Voltairas1, D. I. Fotiadis

1and C. V. Massalas

2

1Department of Computer Science and2Department of Material Science, University of Ioannina,

45110 Ioannina, Greece (pvolter@cs.uoi.gr)

Received 14 March 2002; revised 20 November 2002; accepted 14 January 2003;published online 23 June 2003

The continuum theory of electro-elasticity is used in order to describe the largedeformations observed in gels endowed with electric properties when they are placedin electric fields. The analytical solution of the properly constructed boundary-valueproblem agrees quantitatively with available experimental data.

Keywords: hyperelasticity; electro-gels; biomimetic materials;MEMS; artificial muscles

1. Introduction

Solid metal alloys and plastics have found many applications in medical practice,as biomaterials in prosthetics, artificial hearts, valves, stents, etc. They include,among others, piezoelectric and magnetostrictive materials as bone-growth stimu-lators (Polk 1996) and thermosensitive shape-memory alloys in (endo-)orthodontics,in vascular circulation (stents, valves) or as surgical tools (Lagoudas et al . 2000).There is an increasing need for new materials in many medical applications, withbiomimetic functionalities in eye, ear, muscles and valve operations, drug target-ing, etc., that combine low cost, high efficiency and minor side effects. Hydrogelswith thermo-electromagnetic or chemical properties are good candidates, since theycombine better biocompatibility (hydrophilic) along with more efficient actuationor sensory mechanisms (large deformation). Experiments on pH-sensitive (Qing etal . 2001) and thermosensitive hydrogels (Lindlein et al . 2001) have been performedrecently. In a series of experiments, Zrinyi et al . (2000), Zrinyi (2000), Filipcsei etal . (2000) and Feher et al . (2001) prepared electric and magnetic gels and testedtheir response to electromagnetic fields, confirming their capability to mimic musclecontraction (artificial muscles).

Our previous research on the physical principles of micromagnetic and magneto-elastic phenomena in solids (Voltairas et al . 1999a, b, 2000a–d) directed our motiva-tion to examine possible applications of magnetic carriers in medicine. In a recentpaper (Voltairas et al . 2000e), we proposed a phenomenological model in order toestimate whether a ferrofluid internal tamponade along with a semi-solid magneticsilicon band (magnetic scleral buckle) can contribute to retinal reattachment inretinal-detachment surgery. A condition for the elastic stability of the ferrofluid was

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2122 P. A. Voltairas, D. I. Fotiadis and C. V. Massalas

derived. In an attempt to quantify conditions for magnetic drug targeting and deliv-ery, we developed a hydrodynamic model and compared its predictions with availableexperimental data (Voltairas et al . 2002).

The present work constitutes the first part in a project aimed at examining underwhat conditions electro- or ferrogels can replace non-recoverable injured muscles. Inthis paper, we focus on the control of the experimental apparatus that measures thedeformation of the electro-gels as a function of the external electric stimulus (Filipcseiet al . 2000; Feher et al . 2001), by proposing a suitable theory for the phenomenon.The general continuum theory of nonlinear electro-elasticity is summarized in § 2.In § 3 we introduce the basic model assumptions for: the geometry of the problem,the displacement field (plane deformations), the electric field (collinearity, unifor-mity) and the material properties (isotropy, homogeneity, etc.). The constitutivelaws are derived and comparison with experimental data is performed. Finally, in§ 4, we conclude with the limitations of the model, the possible improvements andfuture generalizations to magnetic-field-sensitive gels (ferrogels) and similar biophys-ical phenomena.

2. General equations and notation

When a polymer hydrogel containing electrically polarized micro- or nanoparticles,hereafter called an electro-gel, is exposed to an electric field, it deforms. The observedlarge strains can be measured experimentally as a function of the applied electricfield. The process is usually reversible for suitable applied fields and gel densities;thus the elasticity of the electro-gel is analogous to the mechanical deformationsof hyperelastic materials. Hence, we prefer the phenomenological description basedon the continuum theory of nonlinear electro-elasticity. In similar problems, withlarge deformations in human soft tissues, the continuum approach proved success-ful for modelling purposes (Holzapfel et al . 2000). Electro-elasticity theory is welldocumented (see, for example, Eringen (1962) and Eringen & Maugin (1989) on thebroader field of thermo-electro-elasticity).

Hereafter, bold and blackboard bold characters will denote vector and tensor fields,respectively. We consider that the electro-gel deforms as a continuous body, whichin the reference (undeformed) configuration occupies a region, Ω ⊂ S, of the wholespace, S, inside the closed surface, ∂Ω. The material points are identified by theirposition vectors, X, in Ω, with Cartesian coordinates, XA (A = 1, 2, 3). After thedeformation, the electro-gel occupies the region, Ωd, and a point originally denotedby X is deformed to the position, x, with coordinates xi (i = 1, 2, 3). In the following,⊗ and · will denote tensor and inner product, respectively. The deformation gradient,F ≡ ∇X ⊗x(X), has Cartesian components given by FiA = ∂xi/∂XA, with gradientvector operators, ∇X , and ∇, in the reference, Ω, and present, Ωd, configurations,respectively. For an electric field, E, a polarization field per unit volume, P , and anelectric permeability of vacuum, ε0, the displacement vector, D, is given by

D = ε0E + P . (2.1)

Let us introduce the Maxwell stress tensor, due to Einstein and Laub (see Toupin1956),

Tm = D ⊗ E − 12ε0E

2I, (2.2)

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A theoretical study of the hyperelasticity of electro-gels 2123

where I is the identity tensor. Then for free body charges, q f, and for J = det F, theelectric body force, fe, is defined by

fe = ∇X · (JF−1

Tm) = J∇ · Tm = J(P · ∇)E + q fE. (2.3)

For a Cauchy stress tensor, T, the nominal stress tensor, S, due to Ogden (1997), isgiven by

S = JF−1

T. (2.4)

Let the outward normal unit vectors, N , and n, on the reference, ∂Ω, and present,∂Ωd, boundary surfaces, respectively, be related by

n = J(F−1)TN , (2.5)

and consider the unit tangential vector, t, on ∂Ωd. Also assume that on the boundarysurface ∂Ωd, free electric charges, σs, are present. Then, if δ(x) = 1 for x ∈ Ωd andδ(x) = 0 for x ∈ S − Ωd, and for [[A]] ≡ Aout − Ain, the equilibrium problem isdescribed by the partial differential equations

∇X · S + fe = 0, in Ω, (2.6)

∇ · D = q f δ(x), in S, (2.7)

∇ × E = 0, in S, (2.8)

and the jump conditions

[[ST + J(F−1Tm)T]]N = 0, on ∂Ω, (2.9)

[[D]] · n = σs, on ∂Ωd, (2.10)

t · [[E]] = 0, on ∂Ωd, (2.11)

in the reference configuration. Gravitational effects are neglected due to strong elec-tric forces fe. The above formulation of the electro-elastic boundary-value problemis due to Toupin (1956). The only difference is that we have included the externalapplied field, E0, along with the depolarization field, EP, in the Maxwell stress tensorabove, as

E = EP + E0. (2.12)

EP is identical to Toupin’s Maxwell self-field, EMS. The notion of the Lorentz localfield EL = −E is frequently used. If we decompose the electric field E in tangentialand normal components on ∂Ω, neglect mechanical surface tractions and make useof the definitions (2.1)–(2.5) and the jump conditions (2.10) and (2.11), the balanceof electro-mechanical surface tractions (2.9) reduces to

STN =

P 2n − σ2

s

2ε0n + σsE0, Pn ≡ P · n, on ∂Ω. (2.13)

The boundary-value problem (2.6)–(2.11) is derived from an energy variational prin-ciple (Toupin 1956). The derivation is not unique. It depends on the form of theelectrostatic energy. For the great controversy regarding the form of the electrostaticenergy, or equivalently the form of the Maxwell stress tensor, the reader should

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2124 P. A. Voltairas, D. I. Fotiadis and C. V. Massalas

consult the footnotes in Eringen (1962). The variational principle also imposes con-straints on the form of the constitutive relations, which in our case read

S = S(F,Π) ≡ ∂W

∂F, (2.14)

E = E(Π, F) ≡ ∂W

∂Π. (2.15)

The free energy depends both on the strain and the polarization:

W = W (E,Π), (2.16)

where E = 12(FT

F− I) is the Green finite strain tensor and Π = P /ρ is the polariza-tion per unit mass. The density in ρ, in Ωd, is related to the density, ρ0, in Ω, throughthe equation ρ0 = Jρ. For more rigour about the constraints that both the materialsymmetry and the second law of thermodynamics impose on the exact form of theconstitutive laws (2.14), (2.15), see Eringen & Maugin (1989). For small concentra-tions of the charged micro- or nanoparticles (diluted electro-gel), the polarizationcontributions to the free-energy density W are negligible; thus

W W (E). (2.17)

Then equations (2.14) and (2.15) are replaced by the more traditional uncoupledones:

S S(F) =∂W

∂F, (2.18)

P P (E). (2.19)

Henceforth, we will restrict our attention to diluted electro-gels, with constitutiveequations of the form (2.18), (2.19) and vanishing mechanical surface traction. In thatcase, Maxwell’s equations of electrostatics, (2.7) and (2.8), are no longer coupled withthe ‘mechanical’ equation (2.6). Introducing the electrostatic potential Φ in (2.7),(2.8) results in

ε0∇2Φ = ∇ · P − q f δ(x), E ≡ −∇Φ. (2.20)

If we express the nominal stress tensor, S, in terms of the Biot stresses t(1)1 , t

(1)2 , t

(1)3 ,

that is, the principal values of the Biot stress tensor, T(1) (Ogden 1997),

T(1) = 1

2(SR + RT

ST), (2.21)

where R is the finite rotation tensor, the derived problem is a complicated butwell-posed one. It involves four second-order partial differential equations (in t

(1)i ,

(i = 1, 2, 3), Φin) in the region Ω inside ∂Ω, one second-order partial differentialequation (in Φout) in the region S −Ω outside ∂Ω, four boundary conditions on ∂Ω,the continuity [[Φ]] = 0 of the potential on ∂Ω and the usual regularity conditionsat infinity. Even for relatively simple prescribed deformation modes, which satisfy(2.6) and the nonlinearities involved in (2.20) due to (2.19), the resultant potentialproblem is a formidable task. For its solution, in special cases, a number of simplifi-cations must be introduced, guided by the observed geometry of deformation as wellas from physical considerations.

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A theoretical study of the hyperelasticity of electro-gels 2125

electrodesX2

X1Le

E0

θc

PEP

nL

ζu

R

R + ζ

− +

−

−

−

−

+

+

+

+

0

+ −

Figure 1. Problem geometry.

3. Theory and experiment

Consider a homogeneous, isotropic, diluted electro-gel, in the form of a rectangularblock of cubic cross-section, of width ζ and height L, which is placed at a distanceR from the coordinates’ origin, between the two electrodes of figure 1. Let Le be thedistance between the electrodes and E0 the uniform static electric field produced bythem. We assume that the electro-gel admits plane deformations of the form

r = f(X1), θ = g(X2), x3 = X3, (3.1)

as shown in figure 1. The displacement field,

u = X − x(X), (3.2)

then corresponds to the bending of the dashed rectangle of figure 1 into a section ofa circular disc, with radius difference ∆R = ζ. The electro-gel is kept fixed at pointsX = (R, 0) and (R + ζ, 0), that is

u(R, 0) = u(R + ζ, 0) = 0. (3.3)

Since the electro-gel is considered to be incompressible, its volume, v, remains invari-ant during the electrically driven deformation, v(Ω) = v(Ωd). This condition deter-mines the maximum deflection angle θc in terms of the geometrical parameters

θc =2L

2R + ζ. (3.4)

The principal stretches λi (i = 1, 2, 3), due to (3.1), are given by the relations

λ1 = f ′(X1), λ2 = f(X1)g′(X2), λ3 = 1, (3.5)

where the prime denotes differentiation with respect to the argument. Then theincompressibility constraint

λ1λ2λ3 = 1, (3.6)

due to (3.5), is solved for r and θ, by the method of separation of variables, with

r =√

2B1X1 + B2, θ =X2

B1 + B3. (3.7)

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2126 P. A. Voltairas, D. I. Fotiadis and C. V. Massalas

The unknown constants Bi (i = 1, 2, 3) are determined from conditions (3.3) suchthat

B1 = R + ζ/2 = L/θc,

B2 = −R(R + ζ) = (ζ/2)2 − (L/θc)2,

B3 = 0.

(3.8)

In order to treat the electrostatic problem in a relatively easy manner, we assumethat the polarization vector, P , and the electric field, E, either inside or outside theelectro-gel, are constant and collinear to the external uniform applied field E0,

Ein = χεE0, (3.9)Eout E0, (3.10)

P = ε0χπEin = ε0χπχεE0, (3.11)

E0 = −EE1, E = const. > 0, (3.12)

where χε and χπ are dimensionless functions of the electric field, E, and the shape ofthe electro-gel L/ζ, and Ei and ei (i = 1, 2) are the unit vectors in the reference andpresent configuration, respectively. The above assumptions are valid for the bulk ofthe electro-gel but not close to the boundary ∂Ω. The solution (3.9)–(3.12) satisfiesthe electrostatic problem (2.7), (2.8), while the jump condition (2.10) reduces to

ε0(1 − εr)E0 · n = σs on ∂Ωd, (3.13)

with the relative dielectric permeability, εr, given by

εr = εr(E, L/ζ) = χε(1 + χπ). (3.14)

εr can be expressed in terms of the dielectric susceptibility, κ, through the relationεr = 1+κ. In general, Ein < Eout, due to the presence of polarization effects, so 0 χε 1 and χπ > 0. For vanishing free-body charges, q f = 0, and due to assumptions(3.9)–(3.12), equation (2.3) gives fe = 0. Then, due to the constitutive laws (2.18)and (2.19), the equilibrium problem (2.6) reduces to the purely mechanical one:

∇X · S = 0. (3.15)

Moreover, since the electro-gel is considered to be isotropic and, due to the deforma-tion mode considered (see (3.1)–(3.8)), the nominal stress tensor S admits, accordingto Ogden (1997), the decomposition

S = T−1

RT = t

(1)1 E1 ⊗ er + t

(1)2 E2 ⊗ eθ, (3.16)

with t(1)i ≡ ∂W/∂λi (i = 1, 2). If we substitute (3.7) and (3.8) into (3.5), we obtain

λ1 =1λ2

= λ =B1

r=

[2θcX1

L+

ζ2θ2c

4L2 − 1]−1/2

, (3.17)

and the Biot stresses t(1)i (i = 1, 2) become

t(1)1 = τ(λ) = W ′(λ), (3.18)

t(1)2 = −λ2τ(λ), (3.19)

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A theoretical study of the hyperelasticity of electro-gels 2127

where the prime denotes differentiation with respect to the argument. Then, with theaid of (3.16), (3.18) and (3.19), the mechanical equilibrium equation (3.15) reducesto

λτ ′(λ) − τ(λ) = 0, (3.20)

since, from (3.17), ∂t(1)i /∂X2 = 0. The general solution of (3.20) is

τ(λ) = W ′(λ) = C1λ, W (λ) = 12C1λ

2 + C2, (3.21)

where Ci, (i = 1, 2) are undetermined constants. For vanishing free-surface charges(σs = 0), in accordance with our assumption for a diluted electro-gel, the balanceof the electro-mechanical surface tractions (2.13) results, due to the solutions (3.9)–(3.12), (3.16), (3.18) and (3.19) to τ(λ) = P 2

n/(2ε0), or equivalently, due to (3.21)to

2λ − e2(1 − χε)2 cos2 θ = 0, e ≡ E

Es, Es ≡

(2C1

ε0

)1/2

, (3.22)

on the boundary X1 = R + ζ. The assumption σs = 0 implies, from (3.13), thatEin = Eout or εr = 1, since E0 · n = 0 on ∂Ωd. In order to preserve the positivedefinite character of the strain-energy function, W (λ), we must have C1 > 0. Noticethat equation (3.22) does not hold for every θ on ∂Ωd, but since our primary concernis to model the available experimental data, we have to satisfy (3.22) only for themaximum deflection angle θc, since what is measured in the experiments is ux =ux(R + ζ, L) for given E and L/ζ. From (3.17) and (3.22) we obtain, for X1 = R + ζand θ = θc,

e2(1 − χε)2[1 +

ζθc

2L

]cos2 θc = 2. (3.23)

If we solve (3.23) for θc and substitute the result in

u ≡ ux/(R + ζ) = 1 − cos θc, (3.24)

we derive the displacement u, as a function of the applied field e and its shape L/ζ,provided that the function χε = χε(e, L/ζ) is specified. Feher et al . (2001) observedthat u(e = 0) = 0. Unfortunately, the solution θc of (3.23) and thus u of (3.24)are singular at e = 0, for a diluted electro-gel, χε(0, L/ζ) = 0. This singularityis a consequence of isotropy, collinearity and especially homogeneity introduced in(3.9)–(3.12). Since the main physical mechanisms observed in the experiments arepresent in our model, we can overthrow the singularity at e = 0 by neglecting (3.4)and replacing the geometrical definition of λ, (3.17), with a suitable function of eand L/ζ. Thus, if we substitute,

λ =λ0(1 + χε)1 + γχε

, λ0 ≡ 12e2(1 − χε)2, (3.25)

with γ 1, in (3.22) and solve for cos θc, the displacement (3.24) reduces to

u ux

Le= 1 −

[1 + χε

1 + γχε

]1/2

. (3.26)

The introduction of relation (3.25) for the stretch, λ, is twofold. The first multipli-cation factor, λ0, eliminates the singularity at the origin, and the square root of the

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2128 P. A. Voltairas, D. I. Fotiadis and C. V. Massalas

2.0

1.5

1.0

0.5

0 1 2 3 4 5E (kV cm−1)

u x (c

m)

0 0.5 1.0 1.5L4V2 × 103 (kV2 cm4)

L = 2.98 cmL = 2.06 cmL = 1.58 cm

(a) (b)

Figure 2. The elastic displacement (or inflection), ux, as a function of (a) the applied electricfield, E, and (b) the product L4V 2. The full lines correspond to equations (3.26) and (3.27) withthe parameters listed in table 1. Circles, triangles and squares are experimental data from Feheret al . (2001), for varying lengths of the electro-gel, as indicated in the enclosed box in (b). Thedashed line in (b) corresponds to equation (3.29).

second, (1+χε)/(1+γχε), behaves like a cosine function, due to the constraint γ 1.The longer the surface of the electro-gel is exposed to the uniform external field, thelarger the depolarization effects induced on it, and as a consequence, the smaller thetotal electric field inside. In order to take into account this shape dependence of theconstitutive law (3.11), we admit for χε the simple power law

χε = α(L/ζ)2βeβ , (3.27)

where all α, β and γ are dimensionless constants. In general, χε should also dependon the concentration of the charged microparticles inside the electro-gel. Since thiscontribution to the deformation was not measured in experiments, we will not discussit further. It can be taken into account by an additional multiplication factor in(3.27). Although (3.25) has the drawback that λ(e = 0) = 0, compared with theexpected λ(e = 0) = 1, it recovers the observed u(e = 0) = 0, due to (3.27).According to experiment, increase in the height of the electro-gel resulted in largerdeformations for the same applied electric field. This is predicted by our model, dueto shape dependence introduced in (3.27). The comparison between our model (3.26),(3.27) and the experiments of Feher et al . (2001) is shown in figure 2, for ζ = 0.1 cm,Le = 3 cm and for three values of the length, L, of the electro-gel.

The evident linearity in figure 2b, as the applied DC voltage, V , vanishes, wasmeasured in experiments and is predicted by our analysis. We assume that β = 2.Then, in the limit of E → 0, the displacement (3.26) reduces, due to (3.27), to

ux A

(L

ζ

)4

E2, A = α(γ − 1)Le

2E2s

. (3.28)

With the substitution E = V/Le in (3.28) this law becomes

ux = GL4V 2, G = 9.1 × 10−3 kV−2 cm−3, (3.29)

and it has been observed by Feher et al . (2001) (the dashed line in figure 2b). Thus,with the substitution E = V/Le in (3.28) and comparing the result with (3.29), weexpress Es, or equivalently C1, due to (3.22), in terms of α, γ and G as

C1 =αε0(γ − 1)4Gζ4Le

. (3.30)

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A theoretical study of the hyperelasticity of electro-gels 2129

Table 1. Calculated parameters from experimental data

L (cm) γ α × 10−7 Es (kV cm−1) C1 (mPa)

2.98 20.6 0.473 51 0.412 28 7.521 382.06 7.0 1.866 57 0.452 90 9.076 421.58 4.0 3.085 80 0.411 76 7.502 42

The solid curves in figure 2 correspond to the values of the parameters in table 1.The constant C1 is identified as Young’s modulus of elasticity for infinitesimaldeformations (λ → 1). Better agreement with the experimental data, for E 2 kV cm−1, could be obtained by allowing β to vary. The above calculations, forfixed β = 2, reveal the observed linear dependence (3.29) that is emphasized in fig-ure 2b. Moreover, our theoretical analysis interprets accurately either the deformationat low electric fields or the saturation effects at high fields.

4. Discussion

In this communication we have developed the theoretical framework for treatinglarge deformations in electric-field-sensitive gels, based on the continuum theory ofelectro-elasticity. The simplified model proposed in § 3 includes all the informationfor quantitative interpretation of electric-field-dependent deformation measurements,since it captures the main physical mechanisms: shape, charged micro- or nano-particle concentration of the electro-gel and nonlinearities on constitutive laws. Thepreservation of the basic features of the geometrical character of the observed defor-mation (maximum deflection angle, aspect ratio) were essential for the success ofthe theoretical approach. Drawbacks, such as the limitation to unit relative dielec-tric permeabilities (εr = 1), are consistent with diluted electro-gel assumptions andnot of major concern, due to biocompatibility issues. Stability topics, either exper-imental or theoretical, related to applied electric field strength and external loads(mechanical, hydrodynamical, gravitational, chemical, etc.) should be examined forfuture biomedical applications. Electrophoretic forces, due to inhomogeneities inelectric-field distribution (fe = 0) can also be taken into account, with the costof complicating the solution procedure. Due to its generality, the presented theoret-ical analysis easily conforms with similar observed mechanisms in biophysics (para-and diamagnetic elastic responses of plants and biological tissues in magnetic fields(see Kuznetsov et al . 1999)), as well as in prototypes in the rapidly developingfield of micro-electro-mechanical systems (MEMS) and their medical counterparts,biomedical microdevices (Bio-MEMS). With minor modifications in the general the-ory, similar deformation phenomena in ferrogels in external magnetic fields can beexplained. Results will be published in the near future.

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