Thermodynamically based multiphysic modeling of ionic polymer metal composites

Abstract

The modeling of the complex response of the IPMC-like body to electrical and mechanical stimuli is set within the context of the 3-D theory of linear elasticity. A field of chemically induced distortions is included in the model; these mechanical distortions and the derivation of the final PDE equations of the multiphysics problem are thermodynamically consistent. Some results of the numerical experiments are revisited through an original analysis of the stress distribution along the IPMC-like body.

Keywords

electroactive polymers sensor actuator

Introduction

Ionic polymer metal composites (IPMCs) have potential applications in many engineering fields (Kim et al., 2005; Shahinpoor and Kim 2005). An IPMC is a porous charged polymer, saturated with an electrolytic solvent and plated by two metallic electrodes. A difference in voltages between the electrodes generates mechanical deformations (actuation mechanism) which in turn yield a difference in voltages between the electrodes (sensing mechanism).The electrolytic solution comprises mobile ions and an uncharged solvent. The actuation mechanism is based on ion-mobility. Because of the difference in voltages between the electrodes, mobile ions (due to their hydrophilicity) displace the uncharged solvent; the concentration gradient of the uncharged solvent causes a volumetric dilatation of the hydrophobic polymer which is not uniform along the thickness, hence a dilatation gradient. Thus, the polymer is deformed without any mechanical forces acting on it during the actuation mechanism. On the other hand, due to the polymer hydrophobicity, the uncharged solvent is attracted to the regions where the polymer mass density is lower. The same qualitative displacement behavior is performed by the ions because of their hydrophilicity. The non-uniform distribution of the charged ions is responsible for the difference in voltages between the electrodes during the sensing mechanism. Therefore, sensing and actuation are non-symmetric, i.e., if a given difference in voltages between the electrodes induces a certain deformation during the actuation mechanism, then the same deformation produces a lower difference in voltages during the sensing mechanism.

According to Wallmersperger et al. (2009), the active and sensing behavior of these ionomeric materials was demonstrated by Oguro et al. (1992) and Sadeghipour et al. (1992). Among the increasing number of publications on this subject, the first physical models completed and compared to experimental results were not published until 2000 (Nemat Nasser and Li 2000). Then, Nemat–Nasser (2002) developed a micromechanical model which took into account the coupled ion transport, the electric field and the elastic deformation and predicted qualitatively and quantitatively the response of the IPMC. In order to implement and analytically model a force-sensor including electrical dissipation, we recall two further papers: Farinholt and Leo (2004) and Chen et al. (2007).

An interesting phenomenological continuum model of IPMC actuation is proposed by Costa Branco and Dente (2006). The model assumes that the driving mechanism for IPMC actuation is the electrostatic force exerted on the polymer as a consequence of local charge imbalances throughout the polymer thickness. Del Bufalo et al. (2008) adopted a mixture framework theory to describe the mechanical actuation of IPMCs, i.e.: (a) one solid species to model the polymer backbone matrix, (b) one species to model the uncharged solvent, and (c) one species to model the gaseous charged ions. The model accounts for osmotic pressure, electrostatic forces, polymer swelling, and analytically derived boundary layer formation. Porfiri (2009) presented the same mixture-based model for the sensing and actuation of IPMCs undergoing small deformations and a two-dimensional (2-D) plate-like model (equivalent to traditional models of piezoelectric bimorph plates) is used to show that the IPMC electric capacitance depends only on the IPMC dielectric constant and the depth of the counterion boundary layers.

Another important set of papers for the development of this study is that of Wallmersperger et al. (2007, 2008, 2009). In particular, we refer to the experimental tests discussed in Wallmersperger et al. (2007) to estimate the hydrophylicity parameter which we have used to link the chemical and mechanical physics. Moreover, we share with Wallmersperger et al. (2009) the aim of setting the IPMC's basic equations within the limit of a consistent generalized thermodynamical theory.

Here, the (ionic polymer metal) composite is viewed as a thin, 3-D body resembling the characteristics of a hydrated bare polymer (Naflon) sandwiched between the two metallated layers. The modeling of the complex response of the IPMC-like body to electrical and mechanical stimuli, is set within the context of the 3-D theory of linear elasticity. In the language of solid mechanics, the chemically induced deformations of the IPMC–like body are described in terms of a volumetric distortion field induced by the redistribution of the ions which carry with them the solvent's molecules. The distortion field determines locally a change in the zero-stress state of the body which is not free to be realized and results in elastic deformations and stresses. The electric and chemical physics are described through standard equations of the electrostatics and of mass conservation, respectively. The requirement that all the admissible multiphysics processes satisfy the dissipation inequality allows us to determine the basic constitutive equations of the model. Interestingly, a contribution of mechanical physics to the Nernst–Planck equation, commonly used to describe the motion of chemical species in a fluid medium, turns out; it is fundamental and found to be numerically relevant in the analysis of the sensing behavior of the body.

We perform a set of numerical tests concerning both the actuation and sensing behavior of the body and find the well-known results concerning the step and harmonic response analysis of an IPMC body. Moreover, the standard numerical experiment concerning the analysis of the actuation performances under a fixed voltage difference shows that the steady process is characterized by a special distribution of stress and strain on the cross-section of the body. This evidence inspires an original analysis on the stress distribution along the IPMC-like body, which is presented in the last section.

Chemo-electro-mechanical formulation

We have already pointed out that the mechanisms at the basis of the IPMC's behavior are driven by the change in solvent concentration due to the electrically and mechanically driven migration of the mobile cations which drag solvent molecules along with them. The starting point of our modeling is represented by the assumption that the change in solvent concentration induces a distortion field in the body, which then influences the elastic response of the body. The key points are: taking into account the dependence of this distortion field on the chemical fields and requiring that the key equations of the model are admissible in a thermodynamical sense.

The Mechanics

Let u : $C$ × $R$ → $V$ be the vector field describing the displacement suffered by the IPMC-like body $C$ , a smooth region of the 3-D Euclidean space ℰ with associated translation space $V$ . The deformation field corresponding to u is E = sym ∇u; let us call E the visible deformation and decompose it additively into the distortion E_o and the elastic deformation E_e, respectively:

(1)

We assume that the distortion field E_o is induced by solvent migration occurring in the IPMC when hydrated and stimulated (electrically and mechanically). The form of such a field ensures the model includes the solvent transport within the flux of the cations carriers through the introduction of a hydrophilicity coefficient α. Only one mobile ion species is employed in the model; thus, the distortion E_o is represented as:

(2)

with the fields c :

C

R

→

R

measuring the concentration of the mobile cation and the concentration c_o :

C

→

R

of the fixed charges assumed as constant (in time).

The structure of equation (2) says that the solvent migration induces volumetric deformations. The distortion E_o would be visible if the boundary layers consisting of clusters embedded in the elastic polymer matrix were free from the constraint of the remaining part of the IPMC. In general, such distortions are not compatible, that is, it does not exist a field u_o such that E_o = sym ∇u_o; hence, they are not realizable. The deformation E of the body corresponding to a displacement u is reached through the rising of the elastic deformation E_e which added up to E_o defines, as (1) shows, the compatible field E.

In absence of forces per unit volume and neglecting the inertial forces, the local balance equations of mechanics prescribe that:

(3)

C

, with T :

C

R

→ 𝕃in denoting the stress field, 𝕃in the vector space of the linear transformation of

V

, and skw the operator such that skw A = A − A^T for any A ∈ 𝕃in. Equation (3) has to be equipped with the suitable boundary conditions of kinematical and dynamical nature if the displacement u_o and the traction

\overset{\land}{t}

are prescribed on complementary parts of ∂

C

Given any part $P$ ⊂ $C$ and introduced the unit normal vector field n to ∂ $C$ , the mechanical contribution $W$ _m( $P$ ) to the working expended on $P$ is given by the rate at which work is done by the surface traction Tn acting across ∂ $P$ :

(4)

where the superscript dot denotes derivative with respect to time and equation (3) and Gauss's theorem have been taken into account. As the deformation E consists of the elastic part E_e and the distortion E_o depending on the field c, we have from (1) and (2) that:

(5)

and

(6)

where the opposite p of the spherical part of the stress tensor T, also called pressure, has been introduced as:

(7)

Electro-chemistry

Due to the much larger propagation speed in the electric field compared to the diffusion speed in the chemical field, the equations of the electrostatics are used to describe the electrical physics of the body. Hence, the variation of the electric potential field Φ : $C$ × $R$ → $R$ in the IPMC-like body is governed by the basic electrostatics equation prescribing that in $C$ :

(8)

Here, the vector field d :

C

R

→

V

describes the electric displacement and the scalar field q :

C

R

→

R

the total electric charge density (per unit volume) contributed from electrons, ions, and fixed charges; thus, denoting as usual with F the Faraday's constant, we have:

(9)

where q_o is the electronic charge density. In the subsequent computational analysis, equation (8) will be equipped with Dirichelet and/or Neumann boundary conditions on Φ and on d·n, respectively, depending on the mechanism object of investigation.

According to Hong et al. (2008) and McMeeking and Landis (2005), given the electronic charge density q_o in any part $P$ ⊂ $C$ and the surface charge density on ∂ $P$ due to the flux −d·n, the electrical contribution $W$ _e to the working expended on $P$ is defined as:

(10)

The electrostatics equation (8), the definition of the total charge density (9), and the Gauss theorem allow the rewriting of the working

W

_e as:

(11)

The balance equation for the flux j of the mobile cations per unit time is formulated through the appropriate conservation law. We assume that changes in the concentration c of the cations in any part $P$ of the hydrated body are most generally brought about by diffusion across the boundary ∂ $P$ if any internal sources are absent. Thus, the flux j satisfies the following:

(12)

C

. Equation (12) has to be equipped with suitable initial and boundary conditions depending on the problem under study. A key element in the electrochemistry of the IPMC–like body is the electrochemical potential μ(X, t) of the cations representing the energy the body would gain per unit time by adding one unit of cations at X ∈

C

. According to Hong et al. (2008) and Fried and Gurtin (1999) and in absence of any bulk sources of cations, the chemical contribution

W

_c(

P

) to the working expended on

P

is defined as:

(13)

Granted for the balance equation (12) and the Gauss theorem, we find:

(14)

Equations (6), (11), and (14) provide the rate at which work is done in any part

P

⊂

C

due to the mechanical and electro-chemical processes involved; it is related to the rate of increase of internal free energy by the thermodynamic issues employed in the following section.

Equations (6), (11) and (14) provide Dirichlet and/or Neumann boundary conditions when $P$ = $C$ ; on the part of the boundary ∂ $C$ where displacement, electric displacement, or chemical flux is prescribed, we have the correspondig Dirichlet boundary conditions and Neumann boundary conditions on the rest of the boundary.

Thermodynamically based constitutive equations

Here, we derive from basic thermodynamical issues the equations providing the stress T, the electric field e, and the chemical flux j in the IPMC-like body. The same approach may be employed when considering large deformations of the IPMC-like body which induce more complex couplings among the mechanical and electro-chemical fields.

Our starting point is the assumption of the existence of a free internal energy φ measuring the energy available to any part of the body. The dissipation principle requires that the rate of energy dissipation (defined as the difference between the working expended along a process and the time derivative of the free energy) should be non-negative, for all $P$ ⊂ $C$ and all times:

(15)

Using the representation forms of the workings

W

_m(

P

W

_e(

P

), and

W

_c(

P

) defined by the equations (6), (11), and (14), respectively, equation (15) localizes into:

(16)

with

\tilde{α} = 3 α

. Of course, the free energy φ has to be related to the process by an extra constitutive mapping. In all cases, inequality (16) is to be satisfied identically, i.e., along all processes. With regard to the free energy, we assume that, at each point X ∈

C

, the value of the free energy per unit reference volume depends only on the value of the elastic deformation E_e, the electric displacement d, and the cation concentration c at that point:

(17)

Within this constitutive class, the requirement (16) is fulfilled if, and only if, for each X ∈

C

(18)

(19)

(20)

and

(21)

to be abided by in the same sense as (16). In assertion of equation (18), any dissipative contribution has been neglected.

The constitutive model

We assume the following representation form of free energy:

(22)

The first addend measures the (strain) energy density stored in the deformative process from a state which is not, in general, undistorted as the electrically induced distortion E_o has already altered locally the state of the body. We assume that only the elastic deformation E_e determines the value of the strain energy and the fourth-order positive definite tensor ℂ is the elasticity tensor which provides the mechanical characteristics of the body. We assume an isotropic response of the IPMC-like body and denoted with E_Y and ν the Young's and Poisson's modulus, respectively, define the elasticity tensor ℂ as:

(23)

with

(24)

The elastic moduli μ and λ are usually called Lamé moduli. The second addend of the right-hand side of (22) is the contribution to the free energy of the body due to the electrical physics; there, ε is the permittivity of the IPMC-like body. At the end, the positive function f(c) gives the contribution to the free energy due to the redistribution of the cations.

Using equation (22), equation (18) provides the stress T as:

(25)

Equation (25) looks like equation (5) in Wallmersperger et al. (2004); it is sufficient to change our distortion field E_o with the prescribed strain Ē introduced there. The fundamental difference between E_o and Ē is that in Wallmersperger et al. (2004) it is assumed that Ē is due to the osmotic pressure difference arising as a consequence of the differences in concentration of gel and solution.

Also, equation (19) in Wallmersperger et al. (2009) looks like our (25); it uses thermodynamical considerations to derive a constitutive prescription for the stress T as a linear function of the difference between the visible deformation E and a quantity which we interpret as a distortion linearly related to the charge density q. So, the representation of stress is similar to the one we derived here. Actually, in Wallmersperger et al. (2009), a stress component, quadratically depending on the charge density, is added to the linear contribution because experimental tests verified that the actuation mechanism of IPMCs can be well described by such a quadratic contribution. From our point of view, the quadratic contribution would imply the consideration of terms which are not well comprised in a linear modeling.

Defining the electric field e as:

(26)

equation (2) gives:

(27)

In the end, the electrochemical potential μ turns out as:

(28)

The simplest constitutive equation for mass transport satisfying the inequality (21) is the isotropic Fick's law (Fried and Gurtin 1999):

(29)

with M(c), the mobility, assumed to be strictly positive to ensure that mass flows down a chemical-potential gradient. Since

\nabla μ = f^{} (c) \nabla c + F \nabla Φ + \tilde{α} \nabla p

, the definition D(c) = f″(c)M(c) allows us to write:

(30)

with D(c) the diffusivity. We assume that:

(31)

corresponding to the standard representation of the chemical potential of the cations (Nemat–Nasser 2002) with R the universal constant of gas and T the absolute temperature. Then, equation (30) may be written as:

(32)

where a constant diffusivity D(c) = D has been assumed. Equation (32) looks like a Nernst–Planck equation, commonly used to describe the motion of chemical species in a fluid medium. It describes the flux of ions as the sum of a diffusive component, related to the gradient of the concentration c, and a migrative component, related to the influence of both the electric and mechanical fields. The Nernst–Planck equation also provides the convective contribution to the flux of ions, due to the motion of the solvent; here, the physics related to the solvent is neglected, hence this term is missing in equation (32). Nevertheless, the migrative components have a structure which looks like a convective term if one confuses ∇Φ or ∇ p with the vector field v measuring the velocity of the solvent. Hence, the stabilization techniques commonly used to regularize the solution of the multiphysics problem in presence of convective terms in the Nernst–Plank equation will be adopted and discussed in the next section that is devoted to the computational analysis.

Results

We employ a set of numerical experiments through a specialized computational analysis in order to analyze the behavior of the IPMC-like body in different circumstances. The computational analysis is employed with reference to a cantilever IPMC-like body having a length 2l = 17mm, a width of 2mm, and a thickness 2h = 180μm, corresponding to the sample tested in the laboratory in Wallmersperger et al. (2007) which includes Naflon-117 as the base ionomer and lithium as the cation. Figure 1 shows the sample used in the numerical experiments. We distinguish the clamped surface boundary $A_{1}^{-}$ and its opposite $A_{1}^{+}$ of unit normal ∓i₁, respectively; moreover, we denote with $A_{3}^{+}$ and $A_{3}^{-}$ the boundary surfaces of unit normal ±i₃ which, in an experimental test, are covered with metal electrodes. Within this scheme, the thickness of the thin slab is spanned by i₃ and the longitudinal direction by i₁; the corresponding coordinates are z ∈ (−h, +h) and x ∈ (−l, +l).

The multiphysics problem

The full electro-mechanochemical problem is defined through the following coupled sets of field equations. From (8), and (9) with q_o = 0, (26), and (27) we have the electric physics:

(33)

from (12) and (32), we have the chemical physics

(34)

from (2), (3), and (25), we have the mechanical physics:

(35)

The electrical and mechanical dynamics are chemically driven, as the only evolutive physics is the one defined by equation (34). The partial differential equations (PDEs) (33)–(35) need a set of initial and boundary conditions. We assume that c(X, 0) = c_o(X) for any X ∈

C

. Then, boundary conditions for the electric, chemical, and mechanical equations depending on the specific problem under investigation have to be specified to solve the governing equations. For the chemical field, a zero flux boundary condition over all the boundary is specified to model ion blocking surfaces:

(36)

When the behavior of the IPMC-like body as actuator is analyzed, we prescribe:

(37)

on the boundary surfaces

A_{3}^{\mp}

, respectively, and null electric displacement on the rest of the boundary. Dirichlet boundary conditions for the displacement field u are given at the clamped boundary surface

A_{1}^{-}

(38)

and null tractions Tn over the rest of the boundary. When the sensing behavior of the IPMC-like body is studied, we prescribe:

(39)

A_{3}^{-}

and null electric displacement on the rest of the boundary; hence, we calculate the induced electric potential on

A_{3}^{+}

. Moreover, we prescribe Dirichlet boundary conditions for the displacement field u as:

(40)

on the boundary surfaces

A_{1}^{\mp}

, respectively, with

\bar{w}

the prescribed displacement; null tractions Tn are prescribed over the rest of the boundary.

Numerical experiments

The coupled multiphysics problem has been solved through a commercial finite element code (Comsol Multiphysics 3.5a) using a fixed multigrid with a refined mesh along the thickness. To eliminate unstable oscillations of the solution driven by the conservation equation (34), a specific procedure has been employed where an artificial flux j_a has been added to the diffusive and migrative components of the flux. The artificial flux has the form:

(41)

with δ_i a tuning parameter always less then 0.5 and dependent on the kind of instability shown by the solution, λ the characteristic size of the mesh and:

(42)

the parameter η that defines the zero-machine.

All the parameters but the hydrophilicity α have been fixed as presented in Table 1.

Table 1.

Numerical values of the parameters

Parameter	Symbol and value
Young's modulus	E_y = 220 MPa
Poisson's modulus	ν = 0.3
Anion concentration	c_o = 1073 mol/m³
Permittivity	ε = 0.0177 C²/Nm²
Diffusivity	D = 6 · 10⁻¹¹ m²/s
Temperature	T = 300 K
Universal gas constant	R = 8.314472 Nm/mol K
Faraday constant	F = 96485.3415 C/mol
Tuning parameter	δ_i = 0.25

Actuation tests

The behavior of the IPMC-like body as an actuator has been numerically tested in three different ways. First, the response of the system to different voltages Φ⁺ is obtained. Compatibility requirements between boundary and initial conditions are guaranteed through an application of the upper voltage ${\hat{Φ}}^{+}$ (t) starting from zero, as shown in Figure 2 in (b).

The linear context of the mechanical setting makes it essential to limit the ranges of variations of the applied voltages. First, we fix Φ⁺= 0.05 V and study the pattern of the tip displacement w_tip when the hydrophilicity parameter α has values between 1.0 · 10⁻⁵ and 1.5 · 10⁻⁵m³/mol. As it is shown in Figure 3, the steady mechanical solution, i.e., the value attained by the tip displacement in correspondence of the stationary solution for the mobile concentration, c, strongly depends on α. An appropriate estimation of the parameter is performed with reference to the results in Wallmersperger et al. (2007) corresponding to an applied differential voltage of 0.05 V. We find that the value of the hydrophilicity parameter which provides a steady tip displacement corresponding to the one measured in Wallmersperger et al. (2007) corresponds with the estimation given in Nemat-Nasser (2002), where the notion of hydrophilicity is introduced and an estimation is performed.¹ Figure 4 shows the elastic bending of the IPMC-like body when α = 1.18 · 10⁻⁵ m³/mol at different times; it shows that the electro-chemical feedback makes an initial fast electrically driven actuation followed by a slow actuation when the diffusive component of the chemical flux compensates for the effects of the electrical migrative component.

The boundary layers corresponding to the excess of charge related to the difference c − c_o and to the electric potential Φ are shown in the Figures 5 and 6, respectively. The electro-chemical effects have been extensively investigated and Figures 5 and 6 agree with the standard results concerning the evolution of the chemical concentration and the electric potential as well as the formation of the corresponding boundary layers. The results of our model are particularly interesting from the mechanical point of view. Indeed, equations (1) and (2) introduce the chemical distortions into the mechanical problem as usually is performed with reference to the thermal distortions (Boley and Weiner 1996).²

The consequences of this assumption are graphically illustrated in Figure 7 through the representation of the steady pattern of the axial deformations; their gradient along the thickness of the IPMC-like body induces bending of the beam. The visible axial deformation (thick solid line) has a linear pattern along the thickness and the transverse section stay plane during bending. Interestingly, Figure 7 shows that this linear pattern is the sum of two largely non-linear patterns: one due to the axial component ε_o = E_oi₁ · i₁ (thin solid line) of the distortion and one due to the corresponding elastic componenet ε_e = E_ei₁ · i₁ (thin dotted line). Hence, the axial stress which depends on the elastic part of the deformation has a pattern which is very different from the standard one; we will discuss in more detail the consequences of this observation in the last section of the article through an approximate explicit method.

If the upper voltage is quickly set to zero after the steady solution has been attained, we observe the slow relaxation of the IPMC-like body. It is driven by the diffusive component of the chemical flux which, in the absence of electrical input, goes towards a homogeneity condition for the cation concentration. The model shows a slow return to a straight configuration. Of course, our material being completely elastic and neglecting any microscopic remodeling processes, there is no evidence of residual deformations (Bar–Cohen 2001). In particular, Figure 8 shows the pattern of the tip displacement before and after the applied potential is set to zero. The displacement rate is smaller for the discharging than for the charging, as the mechanics is driven by the ion dynamics which is influenced by the gradient of the electric potential. During charging, this gradient grows within the boundary layers in response to two different and related sources: the bulk chemical source F(c − co) and the boundary electrical source Φ⁺ − Φ⁻ which is fixed. During discharging, the boundary electrical source Φ⁺ − Φ⁻ is null and the decrease of the gradient of the electric potential is slower. In the end, the discharging procedure is slower due to the chemo-electro-mechanical coupling.

Harmonic response analysis

The model has also been tested with reference to a harmonic response analysis. The applied boundary potential Φ⁺ is assumed as:

(43)

with A being the amplitude of the applied harmonic excitation and T the corresponding period.

Figure 9 shows the actuation behavior due to the harmonic potential when A = 0.05V and T = 10s. As expected, we find that after a small transient the response of the IPMC–like body has the characteristic period of the forcing potential.

The sensing behavior

The sensing behavior is analyzed through a numerical experiment with the IPMC-like body constrained at the boundary $A_{1}^{+}$ in such a way that:

(44)

The value of the assigned displacement

\bar{w}

is chosen as the one measured at the tip of the body in the numerical experiment with constant applied electric potential

\bar{w} = 0.0005 m

. Figure 10 shows the electric potential calculated at a steady state in correspondence with the upper surface; it is approximatively three orders of scale less then the applied voltage Φ⁺, which, in the actuation mechanism, induces a tip displacement

\bar{w}

. In this context, sensing and actuation are non-symmetric. Moreover, the induced potential is not only much lower than Φ⁺ but also different in sign because it is induced directly by the ion boundary layers. The whole process is slower and the steady state is reached after a time longer than that needed when an electric potential is applied.

Stress analysis

The bending deformations of the IPMC-like body have different patterns, as shown in Figure 7. Indeed, we see an axial deformation E₁₁ which is linear through the section as the classical theory of beams indicates. A more detailed analysis shows that at any time, the value zero is attained at the middle of the section. On the other hand, the axial distortion ε_o has a pattern which follows the cation concentration c as equation (2) prescribes; consequently, we may observe an axial distortion which is zero everywhere except the boundary layers whose size varies with time (Figure 5). Correspondingly, due to equation (1), even the elastic axial deformation ε_e has a complex pattern. Hence, the pattern of the axial stress T₁₁ differs strongly from the linear one. The outcomes of the numerical experiments may be explicitly discussed within the context of the linear theory of elasticity through a simplified model which is absolutely coherent with our expectations.

We assume that c = c(z, t), i.e., the relevant changes of the cation concentration field occurs along the thickness 2h of the body. Then, the corresponding elastic problem is characterized by chemically driven distortions E_o = E_o(z, t) allowing a representation as functions of z and t. Given the substantial independence of the bending problem on the width of the IPMC-like body, we assume that the plane stress hypotheses in the plane orthogonal to the flexure plane:

(45)

hold, and look for solutions of the elastic problem allowing:

(46)

The balance equations of the IPMC-like body are trivially satisfied while the compatibility equations rotrotE = 0 prescribe that:

(47)

Hence,

(48)

Following a Saint Venant approach, the time-dependent functions C₁(t) and C₂(t) may be determined through the generalized boundary conditions:

(49)

which ensure that the resultant force and bending moment are zero everywhere along the cantilever. It means that the bending of the cantilever is driven only by the chemically-induced distortions. With this, the stress component T₁₁ allows the following representation:

(50)

with

(51)

and I_y the moment of inertia of the section of the IPMC-like body. The force N_c per unit length and the moment M_c per unit length describe the mean value and the first moment of the ion distribution along the thickness. It is worth noting that N_c ≡ 0 due to the conservation of the charge; then, we have:

(52)

that is, the axial stress is far from linear along the thickness of the beam (Figure 7). The axial visible deformation E₁₁ is the sum of the elastic component ε_e = T₁₁/E_Y and the distortional part ε_o = α(c − c_o) and allows the following representation:

(53)

Equation (53) corresponds to the thick solid line in Figure 7 and shows that there is just one axial fiber with unchanged length at any time and consequently neutral with respect to the bending deformation.

If we look for the neutral axis with respect to the stress, equation (52) must be investigated. It shows that the axial stress T₁₁ at the time t is zero in correspondence to the place identified by the abscissa z which satisfies the following equation:

(54)

Equation (54) always allows the solution z = 0, as at longer distances from the boundary layers, the concentration of the mobile ions does not differ from the concentration c_o of the fixed anions. Hence, c(0, t) = c_o and the neutral axis with respect to the deformation is also neutral with respect to the stress. Equation (54) also admits two more symmetrical solutions as the numerical analysis has shown: z = ±βh with 0 < β < 1.

Conclusions and future directions

The modeling of the complex response of the IPMC-like body to electrical and mechanical stimuli is set within the context of the 3-D theory of linear elasticity. A field of chemically–induced distortions is included in the model; these mechanical distortions and the derivation of the final PDE of the multiphysics problem are thermodynamically consistent. Finally, some results of the numerical experiments are revisited through an original analysis on the stress distribution along the IPMC-like body.

Future areas of study include the extension of the modeling within the non-linear mechanics to comprehend giant deformations of the IPMC-like body. In our view, this extension should be driven by the same thermodynamical approach which has been used here.

Footnotes

Nomenclature

¹It is worth noting that in Nemat-Nasser (), the estimation is performed with reference to Na⁺ as the cation involved. As the hydrophilicity is a specific characteristic of the cation, it is reasonable to think about different values that are estimable when the involved cations are different. Nevertheless, the variability of the parameter α is not extremely large and an agreement on the order of magnitude of the parameter is enough for our present objectives.

²Actually, here, a feedback is taken into account through the contribution of the spherical component −p of the stress to the migrative component of the chemical flux as prescribed by thermodynamic considerations, but it has also been written that in the analysis of the actuation pattern, this contribution does not influence the solution of the problem.

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