Modeling of hydro-thermo-mechanical behavior of Nafion NRE212 for ...

Mech Time-Depend Mater (2013) 17:331–347 DOI 10.1007/s11043-012-9187-8

Modeling of hydro-thermo-mechanical behavior of Nafion NRE212 for Polymer Electrolyte Membrane Fuel Cells using the Finite Viscoplasticity Theory Based on Overstress for Polymers (FVBOP) Ozgen U. Colak · Alperen Acar

Received: 23 January 2012 / Accepted: 9 August 2012 / Published online: 4 September 2012 © Springer Science+Business Media, B. V. 2012

Abstract The primary aim of this work is to present the modifications made to the Finite Viscoplasticity Theory Based on Overstress for Polymers (FVBOP). This is a unified state variable theory and the proposed changes are designed to account for humidity and temperature effects relevant to the modeling of the hydrothermal deformation behavior of ionomer membranes used in Polymer Electrolyte Membrane Fuel Cells (PEMFC). Towards that end, the flow function, which is responsible for conferring rate dependency in FVBOP, is modified. A secondary objective of this work was to investigate the feasibility of using the storage modulus obtained by Dynamic Mechanical Analysis (DMA) in place of the elasticity modulus obtained from conventional tensile/compressive tests, and find the correlation between the storage modulus and the elasticity modulus. The numerical simulations were juxtaposed against data from tensile monotonic loading and unloading experiments on perfluorosulfonic acid (PFSA) membrane Nafion NRE212 samples which are used extensively as a membrane material in PEMFC. The deformation behavior was modeled at four different temperatures (298, 323, 338, and 353 K—all values below the glass transition temperature of Nafion) and at three water content levels (3, 7 and 8 % swelling). The effects of strain rate, temperature, and hydration were captured well with the modified FVBOP model. Keywords Nafion · VBO · Temperature · Humidity · Strain rate dependency · PEMFC

1 Introduction Increasing demands for energy and various factors such as economic stability, burgeoning costs for identifying and extracting newer reserves of fossil fuels, and environmental factors are driving the search for alternative fuels. One of these alternative fuels is hydrogen, which, in conjunction with fuel cell technologies, presents a clean and efficient energy option. Even though there are many types of fuel cells, Proton Exchange Membrane Fuel Cells (PEMFC), O.U. Colak () · A. Acar Department of Mechanical Engineering, Yildiz Technical University, Istanbul, Turkey e-mail: [email protected]

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Mech Time-Depend Mater (2013) 17:331–347

Fig. 1 Schematic of PEMFC (Tang et al. 2006)

which convert hydrogen energy to electrical energy, have been at the forefront of extensive industrial and academic research for some time. The main component of a PEMFC is the membrane electrode assembly (MEA), which consists of a polymer electrolyte membrane with a catalyst layer and porous carbon electrode support on either side. The role of MEAs in PEMFCs is to conduct protons from anode electrode to the cathode electrode, and at the same time to act as an electronic insulator and gas barrier to prevent mixing of oxygen and hydrogen. Schematic of PEMFC is given in Fig. 1. To enable proton conduction, the polymer electrolyte membrane needs to be humidified. In response to changes in temperature and moisture, polymer electrolyte membrane experiences expansion and contraction. The different thermal expansion and swelling coefficients of these various materials in MEA introduce hygrothermal stresses during the operation cycles (start/shut down) of the fuel cell. Hygrothermal stresses play an important role in the failure of the MEA by creating pinholes in the membrane or delamination of the polymer membrane and gas diffusion layers (Kundu et al. 2005). Strength and stability problems of polymer electrolyte membranes are the main impediment for their unsuitability for use in the automotive industry. Therefore, the characterization of the mechanical behavior of PEMs experimentally and the ability to perform reliable lifetime predictions through numerical modeling which incorporates the effects of temperature and hydration are of interest for the design of robust PEMFCs (Tang et al. 2006). Membranes commonly used in PEM fuel cells are composed of perfluorosulfonic acid (PFSA) membranes, and the most common commercially available membrane is Nafion produced by DuPont. Even though, a large volume of the experimental data on polymer electrolyte membranes used in PEMFCs is available in the scientific literature, the number of works devoted to modeling is quite limited. Also, most of the modeling efforts assume that the material behavior is rate independent for simplification, whereas the polymeric materials used in polymer electrolyte membranes exhibit rate dependence. The mechanical properties of ionomer membranes Nafion 112, 117, NR111, N1110, 115 and NRE212 in tensile test conditions and using a Dynamic Mechanical Analysis (DMA) are reported in Silberstein and Boyce (2010), Tang et al. (2006), Kundu et al. (2005), Solasi et al. (2007), Satterfield and Benziger (2009), Majsztrik et al. (2007, 2008). The effect of temperature and hydration has been investigated in these works. The modeling of the mechan-


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ical behavior of ionomer membranes in polymer electrolyte membrane fuel cells (PEMFC) has been done mostly using a linear elasticity (Tang et al. 2006), and linear viscoelasticviscoplastic (Solasi et al. 2007), and elasto-plastic (Kusoglu et al. 2006) constitutive models. Recently, some researchers have applied nonlinear viscoelastic-viscoplastic models for modeling uniaxial behavior of polymer membranes (see Silberstein and Boyce 2010; Yoon and Huang 2011). The mechanical response of the membranes subjected to a hygrothermal cycle has been investigated by Kusoglu et al. (2006). Linear-elastic, perfectly plastic laws are used as the material model for Nafion with the added thermal and swelling expansion coefficients. In the work of Solasi et al. (2007) mentioned above, in addition to the experimental work, modeling work similar to that by Kusoglu et al. (2006) was performed. While a linear viscoelasticplastic model was used for the membranes, the authors do comment that a more complete nonlinear viscoplastic model is desirable in order to predict the stress–strain values more precisely. A nonlinear viscoelastic-viscoplastic model by Bergström–Boyce was used to simulate monotonic tensile behavior of Nafion NR 111 by Yoon and Huang (2011). In this work, hydration and temperature dependent empirical equations for elastic modulus are introduced and only the monotonic loading behaviors of Nafion NR 111 are simulated at different temperature and humidity values. Silberstein and Boyce (2010) conducted uniaxial tension tests at the strain rate of 0.01 s−1 at temperatures from 298 to 373 K and swelling percentages from 3 to 9 %. The elasto-viscoplastic models, named Model I and II, are used. Model I was used to simulate only monotonic loading behavior at different temperatures, hydration levels, and strain rates. By including a back stress feature to the model (Model II), loadingunloading and reloading behavior of Nafion NRE212 is modeled at two different strain rates (1 × 10−1 s−1 and 1 × 10−3 s−1 ) and temperatures (353 and 298 K) even though experiments were performed at three different strain rates and five various temperature and humidity values. Using Model II somewhat improved the unloading behavior; however, neither model could collectively capture the strain rate, temperature and humidity dependence of Nafion very well. Some inconsistencies are also observed in the experimental results. Dynamic mechanical analysis (DMA) is a widely used characterization technique for polymeric materials. Here, the samples are typically subjected to a periodic mechanical strain or stress, and temperature ramps can also be implemented simultaneously. The material parameters measured by means of DMA are the storage modulus E , which is related to stiffness of the material (elastic response), and the loss modulus E , which is the measure of energy dissipation (viscous response) of polymers. The E /E ratio labeled tan δ provides a measure of the mechanical damping. These different moduli allow better characterization of the material, because a material’s propensity to return or store energy (E ) and lose energy (E ) can be quantified individually leading to a calculation of the damping (Menard 1999). These three parameters vary significantly with temperature and frequency, especially around polymer relaxation phenomena, such as glass transition or sub-glass transition (Menard 1999). Therefore, DMA is used to determine the glass transition temperature (Tg ) of materials as well. One of the aims in this work was to investigate applicability of the storage modulus obtained by DMA as the elasticity modulus obtained from the conventional tests and to find the correlation between the storage modulus and the elasticity modulus. DMA has been used extensively to characterize parameters mentioned above (Tg , loss and storage modulus, etc.) (see Kundu et al. 2005; Osborn et al. 2007; Sgreccia et al. 2010). However, a few works relate the modulus obtained by DMA to the conventional mechanical properties (Deng et al. 2007; Richeton et al. 2005). Stiffness variations (storage modulus) for var-

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ious polymers (thermoset, thermoplastics, amorphous, semi-crystalline, linear, crosslinked, etc.) over a wide range of temperature ranging from the glassy to rubbery state have been reported by Mahieux and Reifsnider, Richeton and others (see Richeton et al. 2005; Mahieux and Reifsnider 2001; Mahieux et al. 2001, 2002). A modulus (storage modulus) versus temperatures curve is typically obtained by DMA. All transitions (beta, glassy, and flow) can be seen in these curves. Therefore, Mahieux and Reifsnider (2001, 2002), Mahieux et al. (2001) introduced the temperature-dependent storage modulus in terms of three major property transitions observed in polymers and polymer matrix composites using DMA. However, in their work, this stiffness parameter has not been used for simulating stress–strain behavior of polymers or polymer matrix composites. Considering the work by Mahieux and Reifsnider (2001, 2002), Mahieux et al. (2001), Richeton et al. (2005) proposed a model for the prediction of the stiffness modulus for a wide range of temperature and frequencies/strain rates. This new formulation has been validated for two amorphous polymers (polymethyl-methacrylate, PMMA and polycarbonate, PC) using DMA and uniaxial compression testing. The elasticity modulus obtained from uniaxial compression tests at different temperature and strain rates are simulated by a proposed rate and temperature-dependent formulation. However, similarly to Mahieux and Reifsnider (2002), this formulation has not been used in a constitutive model for simulating stress–strain behavior. Temperature-dependent moduli of two cured epoxy systems and two silica-epoxy nanocomposites were experimentally measured by DMA and mechanical tests at various temperatures and correlation between DMA and mechanical testing data is established in Deng at al. (2007). In this work, a constitutive model for modeling of hydro-thermo-mechanical behavior of Nafion NRE212, used as membrane material in PEMFCs, over a wide range of temperature, humidity, and strain rate is proposed. The modified constitutive model is the Finite Viscoplasticity Theory Based on Overstress for Polymers (FVBOP), see Krempl and Gleason (1996), Krempl (1996), Krempl and Tachiban (1998), Colak and Krempl (2005), Colak (2005), Hassan et al. (2011). The modifications have been performed such that temperature and humidity effects can be incorporated. Recognizing that the working conditions of PEMFCs include cyclic variations in temperature and hydration, the monotonic uniaxial loading and unloading behavior of Nafion NRE212 is modeled at different strain rates (0.1, 0.01 and 0.001 s−1 ), temperatures (25, 50, 65, 80 ◦ C) and hydrations (3, 7 and 8 % swelling) using the modified unified state variable theory (VBOP). The storage modulus curve of Nafion NRE212 is modeled using the equation given by Mahieux and Reifsnider (2002) and used as the elastic parameter which is dependent on temperature. This approach of modeling using shifting modulus value is inspired by the work of Deng et al. (2007). Experimental data obtained by Silberstein and Boyce (2010) are compared to the simulation results.

2 Nafion NRE212 The most commonly used membrane material in PEM fuel cells is perfluorosulfonic acid (PFSA) membrane. Among the PFSA polymers used in fuel cells, the most favored one has been the sulfonated tetrafluoroethylene-based fluoropolymer, a copolymer with the trade name Nafion® , developed and manufactured by Walther Grot of DuPont in the 1960s. The material is generated by the copolymerization of a perfluorinated vinyl ether comonomer with tetrafluoroethylene (Mauritz and Moore 2004).


335

Fig. 2 Chemical structure of Nafion

The chemical structure of Nafion is shown Fig. 2. The fluorinated backbone of the polymer is essentially polytetrafluoroethylene (PTFE) (Teflon® ), which gives Nafion good mechanical strength and resistance to harsh chemical environments. Fluorinated ether linkages terminate in sulfonate groups. Despite much research, Nafion membrane morphology is still a subject of debate, see Mauritz and Moore (2004) and Galperin et al. (2009). There are two distinct hydrophobic and hydrophilic regions in Nafion. The hydrophobic region is a semi-crystalline region which is Teflon® -like. The hydrophilic regions consist of sulfonate groups, which swell and change size/shape with water absorption and allow water and proton transport (Majsztrik et al. 2007, 2008). The equivalent weight (EW), which is the ratio of the number of grams of polymer per mole of sulfonic acid groups of the material in the acid form and when completely dry, can be varied and strongly affects mechanical properties. Increasing EW, which means decreasing sulfonic acid group concentration, improves mechanical properties but decreases proton conductivity. Nafion with 1100 EW is typically used in PEM fuel cell applications since it has a reasonable balance of proton conductivity and mechanical integrity (Mauritz and Moore 2004). Nafion’s specifications are designated by a system of numbers. The first two numbers denote EW while the third and possible fourth digits denote dry membrane thickness in thousandths of an inch. (For example, Nafion NRE212 is non-reinforced dispersion cast film based on Nafion with a 2100 EW and 0.002 in thickness.)

3 Finite Viscoplasticity Theory Based on Overstress for Polymers (FVBOP) There are two classes of continuum theories. The first includes classical plasticity theories in which all time effects, such as rate sensitivity, creep, relaxation and strain recovery, are excluded. The second contains viscoplasticity theories which assume that inelastic deformation is rate-dependent even at low homologous temperatures. Viscoplasticity models represented by unified state variable theories do not permit the separation of creep and plasticity. One of the unified state variable theories is Viscoplasticity Theory based on Overstress (VBO) developed by Krempl and his co-workers for metallic materials (see Krempl 1996; Krempl and Tachiban 1998; Colak and Krempl 2003, 2005; Colak 2004). State variables in the model are defined as macroscopic integrators of events associated with microstructure changes and cannot be directly measured or controlled. VBO model is based on standard linear solid (SLS) model with extensive modifications to the properties of each element. Rheological representation of SLS is given in Fig. 3. Governing equation for SLS model is given in Eq. (1): 1 dσ σ E2 dε E2 + ε= + (1) 1+ dt η E1 dt η E1

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Fig. 3 Standard linear solid model (SLS)

When Eq. (1) is written in overstressed form, Eq. (2) is obtained: ε˙ =

σ − Bε σ˙ + E1 C

(2)

E2 1 and C = E1ηE , E1 , and E2 are stiffnesses of springs and η is viscosity where B = EE11+E +E2 2 function. In Eq. (2), (σ − Bε) is called overstress. This overstress concept is used to develop VBO model. It is initially developed for modeling the mechanical behavior of metallic materials. However, due to the similarities observed in the polymeric and metallic materials’ behavior, VBO is modified to capture mechanical behavior of polymers as well, see Colak (2005), Dusunceli and Colak (2008, 2006). With changes in temperature and frequency (or rate), significant variations are observed in the viscoelastic parameters which are strongly dependent on molecular motions and segmental mobility. Therefore, any factor which affects macromolecular mobility, such as ageing and crystallinity, leads to significant changes in viscoelastic properties of polymers. Increasing strain rate increases the strength of the material, since the polymer chains cannot find enough time for recovery. When temperature is increased, the thermal energy of polymer molecules exceeds that of van der Waals interactions between the molecular chains and weak crosslinking interactions between sulfonic acid groups. Chains are able to move easily when the force is applied. As a result, the yield stress, ultimate stress and elasticity modulus decrease. The effect of hydration on mechanical behavior of polymers is due to the ionic regions which absorb water and change the interactions between chains. For modeling material behavior of polymeric materials, two main mechanisms are needed: molecular interaction (stretching and orientation of molecular network) and intermolecular interaction. In VBOP, resistance to deformation due to secondary bonds, which are present between molecular chains (intermolecular interactions), is captured by including the characteristics of a nonlinear dashpot. Intermolecular resistance depends on rate, temperature and hydration. Characteristics of a nonlinear spring are used in the model to capture the resistance to deformation due to primary bonds. Stretching and changing in the orientation of molecular network are defined by nonlinear spring element. The nonlinear spring and dashpot are in parallel. For elastic behavior, linear spring is added to the parallel system as well. A rheological representation of VBOP model is depicted in Fig. 4. Finite VBO is obtained by replacing the ordinary time derivative by an objective one in the small deformation viscoplasticity theory based on overstress (Colak 2004). The additive decomposition of the rate of deformation tensor D is used for modeling finite deformation. The rate of deformation tensor is decomposed into elastic and viscoplastic parts. The flow law for finite deformation theory of VBO is 3 Γ s−g 1+ν s˚ + F (3) d = d e + d vp = CE 2 D Γ


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Fig. 4 Rheological representation of VBOP model

where s and g are respectively the deviators of the Cauchy stress tensor, σ , and the equilibrium stress, G, which is the stress that the material can sustain at rest. E is Young’s modulus and ν is the elastic Poisson’s ratio; de and dvp are deviators of elastic and viscoplastic rate of deformation tensor D, respectively; ‘◦’ denotes the objective rate. The main difference between VBO and VBOP is the parameter C, given by C = 1 − λ(|G − K|/A)α , where λ and α are model parameters, K kinematic stress and A isotropic stress. For metals, C is usually set to one for representing linear unloading behavior. Γ is the overstress invariant with the dimension of stress defined by 3 Γ 2 = (s − g) : (s − g), 2

(4)

F [ ] is the positive, increasing flow function with the dimension of 1/time and F [0] = 0. The symbol [ ] stands for “the function of”. The flow function F [ ] is set as a power law. It is responsible for modeling nonlinear rate sensitivity and is given by m Γ (5) F[] = B D with B as a universal constant having the dimension of 1/time. D is the drag stress, which can be considered as another state variable with a growth law. However, in this study it is a constant. One of the state variables is the equilibrium stress. The equilibrium stress is similar but not quite the same as the back stress in rate-independent plasticity models. In plasticity models the back stress is considered as the repository for kinematic hardening, whereas in VBOP the repository for kinematic hardening is the kinematic stress. The equilibrium stress is the stress that must be overcome to generate inelastic deformation. The growth law for the deviatoric objective equilibrium stress, which is the rate-independent contribution to hardening, is Γ s−g g−k ψ ˚ s˚ k (6) − + 1− g˚ = ψ + ψF E D Γ A E where ψ is shape function bounded by Et < ψ < E. It affects the transition from the quasielastic to the inelastic region. The isotropic stress A is a scalar state variable for modeling

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rate-independent cyclic hardening (or softening) behavior. Its effect is similar to the isotropic hardening in rate-independent plasticity (Krempl 1996). The evolution of the shape function is given in Eq. (7): ψ = ψ1 +

C2 − ψ 1 , exp(C3 |ε vp |)

ψ 1 = C 1 1 + C4

|G| A + |K| + Γ ζ

(7)

where C1 , C2 , C3 , C4 and ζ are material parameters determined using transition regions from elastic to viscoplastic responses of the stress–strain curves. The difference between Cauchy stress and equilibrium stress is called overstress. In the elastic region of stress–strain curve, bond stretching occurs. When the stress is increased beyond a certain level, some chains in amorphous and crystalline phase overcome the secondary interactions (Van der Walls) and irreversible slippage develops and yielding occurs (Yoon and Huang 2011). The equilibrium stress is overcome to generate inelastic deformation. Whenever yielding occurs, overstress becomes nonzero. Then entangled chains start locking up and strain hardening results. During deformation, hardening (dynamic recovery) occurs due to entanglement (disentanglement) of polymer chains. To simulate dynamic recovery and hardening, the hardening terms, which are the first two terms in the evolution equation of the equilibrium stress, and the dynamic recovery term, the third term in Eq. (6), are included into the evolution equation of the equilibrium stress tensor. A tensor-valued kinematic stress k is introduced to model the Bauschinger effect, and tension/compression asymmetry through initial conditions. It also sets the tangent modulus Et at the maximum inelastic strain of interest. The tangent modulus can be positive, zero or negative. The growth law for the objective kinematic stress is given as follows: ˚k = Et F Γ s − g D Γ

(8)

One of the fundamental principles that all constitutive equations have to satisfy is the principle of objectivity or frame indifference. According to this principle, constitutive equations must be invariant under a change of reference frame. Tensor rates used in constitutive equations need to be objective. A co-rotational objective rate of a tensor A is denoted by ˚ =A ˙ + AΩ − ΩA A

(9)

˚ is objective rate of A and Ω is ˙ is the material rate with respect to the basis of A. A where A a skew-symmetric spin tensor. Various forms of Ω can be evaluated, such Jaumann, Green– Naghdi, Logarithmic rates. Since skew-symmetric spin tensor is zero for uniaxial loading, objective rate of any tensor will be equal to material rate.

4 Modification of FVBOP model to account for temperature and humidity effect The present form of VBOP is capable of modeling uniaxial or multiaxial, monotonic or cyclic behavior of wide classes of polymeric materials (such as thermoset, thermoplastics, semi-crystalline or amorphous, etc.) (see Colak 2005; Hassan et al. 2011; Dusunceli and Colak 2006, 2008) and metallic materials at room temperature (see Colak and Krempl 2003, 2005; Colak 2004). The modifications in FVBOP have been performed such that temper-

Mech Time-Depend Mater (2013) 17:331–347 Table 1 Transition temperatures and modulus values on the onset of transitions (see Almeida and Kawano 1999; Osborn et al. 2007)

339

T1 (Tβ )

T2 (Tg )

T3 (Tf )

253 ◦ K

373 ◦ K

503 ◦ K

E1 = 1550 MPa

E2 = 700 MPa

E3 = 14.2 MPa

ature and humidity effects can be modeled. From the phenomenological point of view, it is known that elasticity and tangent modulus, yield stress and ultimate stress depend on temperature. In the modeling of polymeric material behavior at different temperatures, two methods have been used to account for temperature. In the first, some of the material parameters are defined as varying with respect to temperature (as a function such as parabolic). In the second, some of the state variables or material functions are described through functions of temperature that are physically related. The second method is especially useful for viscous effects (Chaboche 2008). A similar approach can be followed for humidity effects as well. In this work, both methods are used to model hydro-thermo-mechanical behavior of polymeric materials. Departing from currently reported research, this work uses the result of DMA based determination of the storage modulus as a function of temperature to establish the temperature dependence of the modulus of elasticity. First, the applicability of the storage modulus by DMA as the elasticity modulus obtained from conventional tests is investigated. Then, the correlation between storage modulus and elasticity modulus is derived. The storage modulus versus temperature curve of Nafion NRE212 is modeled using the equation given by Mahieux and Reifsnider (2002) and used as the temperature-dependent elastic parameter with the idea of shifting modulus inspired by the work of Deng et al. (2007). 4.1 Correlation between storage modulus and elasticity modulus Temperature-dependent elasticity modulus is determined using the storage modulus data obtained from DMA test results. In order to determine storage modulus versus temperature curve mathematically, Eq. (10) proposed by Mahieux and Reifsnider (2002) is used: m1 m2 T T + (E2 − E3 ) exp − E(T ) = (E1 − E2 ) exp − T1 T2 m3 T + E3 exp − T3

(10)

Temperature-dependent storage modulus equation uses the three transition temperature values T1 , T2 , T3 (sub-glass (β) transition, glass (α) transition, flow) and the modulus values at the onset of transitions E1 , E2 , E3 as parameters. Exponential parameters, m1 , m2 , m3 , are Weibull constants which are statistical parameters. Those parameters are almost constant for all amorphous thermoplastic polymers (m1 = 1.5, m2 = 20, m3 = 20) (Mahieux and Reifsnider 2002). Transitions temperatures of Nafion determined by Almeida and Kawano (1999) and Osborn et al. (2007) are used in this work. Experimental work of Silberstein and Boyce (2010) is used to determine the modulus values and compared to the storage modulus curve obtained by Eq. (10). Transition temperatures and modulus values on the onset of transitions of Nafion are given in Table 1. Comparison of storage modulus obtained by Eq. (10) and DMA result of Silberstein and Boyce (2010) is given in Fig. 5.

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Fig. 5 Comparison of storage modulus obtained by Eq. (10) and the DMA experimental result of Silberstein and Boyce (2010)

Fig. 6 Comparison of elasticity modulus varying with temperature and storage modulus curve obtained from Eq. (10)

The storage modulus is related to the elastic contribution of the total viscoelastic behavior (Menard 1999). The elasticity modulus is defined as resistance to elastic deformation and obtained from the slope of the stress–strain curve in the elastic region. Therefore, these two moduli definitions relate to the same physical phenomena—elastic deformation. Figure 6 shows the comparison of elasticity modulus varying with temperature determined from the uniaxial tension experiments of Silberstein and Boyce (2010) and storage modulus curve obtained from Eq. (10). As seen from the figure, the magnitudes of elasticity modulus and storage modulus are not the same. But it could be noticed that the variation of modulus values with varying temperature follows the same trend. Storage and elasticity moduli relate to the same physical phenomena, as explained above, but the technique and instruments used to determine these moduli are different. Indeed, a DMA device can be considered as a miniature tensile testing machine. But the size of the specimen and the loading frame is much smaller than a conventional tensile testing machine. Therefore, the stiffness of a tensile testing machine is much greater than that of a DMA device. This can lead to a DMA overestimating the value of the material modulus. The work of Deng et al. (2007), mentioned in previous sections, proposes an easy and effective


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Fig. 7 Shifted and unshifted storage modulus curves and elasticity modulus varying with temperature. Experimental data is obtained by uniaxial tension tests at different temperatures by Silberstein and Boyce (2010)

way to neglect those effects originated by experimental technique, therefore the value of the modules becomes slightly equal. If a straightforward vertical shifting is applied on the storage modulus curve, the values of the two moduli overlap. Figure 7 shows a comparison of −600 MPa shifted storage modulus curve and elasticity modulus values varying with temperature. One fact should be clarified as a constraint of this shifting phenomenon. The shifting phenomenon is not valid for temperatures above Tg . For temperatures above Tg , the shifted modulus values will be negative which is physically meaningless. Therefore the shifting is valid until the start of glass transition. Following the procedures explained above, temperature-dependent elasticity modulus is obtained as given in Eq. (11): T 1.5 T 20 E(T ) = 850 exp − + 658.8 exp − 253 372 20 T + 14.2 exp − − 600 503

(11)

This technique of determining the temperature-dependent elasticity modulus by a single DMA test derives a more efficient, easy and thrifty experimental work. Moreover, this technique neglects the necessity to assure a large number of homogenous specimens. 4.2 Variable tangent modulus The viscoplastic region of Nafion NRE212 is highly nonlinear. Moreover, temperature dependency of the viscoplastic region of stress–strain curve cannot be ignored as is seen in Fig. 10. Therefore a nonlinear and temperature-dependent tangent modulus is defined as in Eqs. (12a) and (12b): Et0 (T ) = −0.03∗ T + 16 Et =

Et0 (1 + e 2

(12a)

a|ε vp | β

)

(12b)

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Fig. 8 Stress–strain curves at different strain rates for a hypothetical material. The flow function is given in Eq. (5)

where T is temperature in Celsius. Equation (12a) is determined from the uniaxial tension tests performed at different temperatures at the same loading rate by Silberstein and Boyce (2010), considering of the slopes of the viscoplastic part of the stress–strain curves determined at different locations. Equation (12a) provides the initial temperature-dependent tangent modulus, while Eq. (12b) is determined as a function of inelastic strain in order to model the nonlinear viscoplastic behavior of Nafion NRE212 (a = 3 and β = 0.96). 4.3 Temperature and humidity dependent flow function Defining elasticity modulus and tangent modulus as function of temperature is not enough to model hydro-thermo-mechanical behavior of polymers. Therefore, the flow function F [ ], which is responsible for rate dependency in VBOP model, is modified to account for temperature and humidity effects as well. The power law form of flow function in VBOP is given in Eq. (5). When the effects of temperature, humidity and rate on the mechanical behavior of materials are compared, it is observed that increasing temperature or humidity leads to a decrease in the material properties such as elasticity modulus, yield stress, etc. On the other hand, increasing strain rate increases the mechanical properties. Therefore, the effect of strain rate or temperature–hydration is inversely correlated. In the present form of VBOP, strain rate dependency is captured with flow function which is in the form of power law. With modifications, in addition to strain rate dependency, temperature and hydration dependency are also captured by flow function. This analogy of temperature–hydration and strain rate effects leads to the idea of modifying flow function in FVBOP as given in Eq. (13): δD Γ m ξ D Γ n + F [] = B m δD n ξD

(13)

where δ and ξ are the parameters for temperature and humidity, respectively. Equation (13) is introduced considering the viscoplastic potential assumed as power function in Chaboche (Almeida and Kawano 1999). Since flow function is responsible for rate dependency also, it is needed to investigate how the rate dependency change with the made modifications in flow function, F [ ]. When the present form of flow function is used, the stress–strain behavior for an arbitrary material parameters is obtained as in Fig. 8.


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Fig. 9 Stress–strain curves at different strain rates for a hypothetical material. Modified flow function used is given in Eq. (13)

Table 2 δ temperaturedependent parameter used in the simulations, ξ = 0.5

T (◦ K)

298

323

338

353

δ

1

0.32

0.13

0.001

The difference between Cauchy (σ ) and equilibrium stress (G) is the overstress which is a rate-dependent component. If the modified form of F [ ] is used, stress–strain responses are obtained as shown in Fig. 9. As is seen from figures, the rate dependency is still captured. Depending upon material parameters chosen in flow function, the rate dependency can be simulated.

5 Modeling temperature and humidity dependent deformation response of Perfluorosulfonic Acid (PFSA) membrane Nafion NRE212 For the conduction of protons in PEMFCs, the polymer electrolyte membrane needs to be humidified. As a function of temperature and moisture, polymer electrolyte membrane experiences expansion and contraction. Determination of the mechanical behavior of PEMs experimentally and prediction of the mechanical behavior and its dependence on temperature and hydration and lifetime of the membrane precisely is of interest for the design of robust PEMFCs. Therefore, uniaxial tension behavior of perfluorosulfonic acid (PFSA) membrane Nafion NRE212 is modeled at four different temperatures (25, 50, 65, 80 ◦ C) and at three water contents (3, 7 and 8 % swelling). Large-strain monotonic loading and unloading tensile behavior of Nafion NRE212 is highly rate, temperature and hydration dependent. Uniaxial tensile tests are conducted at the strain rates of 0.001 to 0.1 1/s and temperatures from 25 to 100 ◦ C (298 to 353 K) and 3, 7 and 8 % swelling by Silberstein and Boyce (2010). The experimental data are compared to simulation results. A humidity value of 50 % is used in the uniaxial tension experiments at different temperatures. Since humidity is kept constant, parameter related to humidity is kept constant (ξ = 0.5 (for %50 RH)) and simulations are performed for different δ’s and variable elasticity modulus obtained from DMA and tangent modulus given in Eqs. (12a) and (12b). Temperature-dependent parameters (δ) used in the simulations are given in Table 2. Simulation results of uniaxial stress–strain behaviors of Nafion NRE212 at different temperatures are presented in Fig. 10. As seen in the figure, the modified VBO model is

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Fig. 10 Nafion NRE212 experimental and simulation responses under uniaxial strain controlled loading and unloading (strain rate = 0.01 1/s) at different temperatures and 50 % RH. Experimental data are obtained from Silberstein and Boyce (2010)

Table 3 ξ hydration-dependent parameters used in the simulations

Swelling ratio (%)

Dry

3

7

8

ξ

0.5

0.17

0.1

0.07

now capable of modeling the mechanical response of Nafion which is highly temperaturedependent even in the elastic region. Using the modified flow function, the temperaturedependent elasticity modulus and variable tangent modulus are used to account for temperature dependency, temperature-dependent elastic region, yield stress and the nonlinear viscoplastic region. In addition to temperature-dependent behavior, the modeling of unloading behavior of polymeric materials is still an issue. Most of the material models in the literature are not able to simulate nonlinear unloading behavior. On the other hand, VBOP is capable of capturing nonlinear unloading behavior as seen in Fig. 10. The C function is responsible for this nonlinear behavior in addition to shape function which affects the transition from elastic or viscoelastic to viscoplastic region. For investigating hydration effect on the mechanical response of Nafion NRE212, specimens are left in the water for a variable amount of time in accordance with desired water content. Then uniaxial tension experiments are performed by the method of Silberstein and Boyce (2010). The swelling percentage was calculated from the change in the distance between dots marked on the specimen from the dry state to the hydrated state at the start of the test. A hydration-dependent coefficient ξ (given in Table 3) was used in the modified form of the flow function to model the hydration effects on the mechanical behavior of Nafion NRE212. The change in the flow function due to the hydration-dependent coefficient ξ changes the overstress, and this reduction of overstress captures the onset of yield phenomenon. Hydration dependence of on elasticity and the tangent moduli could be neglected compared to temperature dependency. As seen from Fig. 11, the stress–strain behaviors of Nafion NRE212 under various swelling percentages (humidity) are captured well. Swelling decreases the network density. Therefore the mechanical stiffness decreases. The strain rate in simulations given in Fig. 11 is also 0.01 s−1 as used in the experimental work. Strain rate dependency is also simulated and depicted in Fig. 12. Simulation results are not compared to experimental data obtained by Silberstein and Boyce (2010) since there were some inconsistencies in their experiments about rate dependency. In the aforementioned work by Silberstein and Boyce (2010), in their Fig. 3, which is labeled “True stress–


345

Fig. 11 Nafion NRE212 experimental and simulation responses under uniaxial strain controlled loading and unloading (strain rate = 0.01 1/s) at different swelling percentages at room temperature. Experimental data are obtained from Silberstein and Boyce (2010)

Fig. 12 Rate dependency of Nafion NRE212 under uniaxial tension

true strain behavior at 25 ◦ C at multiple strain rates (inset: logarithmic rate dependence of yield stress)”, the square-dotted line (rate 0.01 s−1 ) and in Fig. 5(a), which is labeled “True stress–true strain curve at 0.01 s−1 and (a) as a function of temperature”, the circle-dotted line (25 ◦ C) do not match. The results are different even though the loading conditions are the same. Therefore, only simulation results of FVBOP are depicted in Fig. 12. Trial and error analysis is used for determination of material parameters. The other material parameters used in the simulations are given in Table 4.

6 Conclusions The Finite Viscoplasticity Theory Based on Overstress for Polymers (FVBOP) is modified to simulate hydro-thermo-mechanical behavior of Nafion NRE212 which is used as membrane material in PEMFCs. The storage modulus versus temperature curve of Nafion NRE212 is modeled using the equation given by Mahieux and Reifsnider (2002) and used as the elastic parameter which is dependent on temperature in the modeling with the idea of shifting modulus values inspired by the work of Deng et al. (2007). Uniaxial tension behavior of

346 Table 4 VBO parameters used in the simulations

Mech Time-Depend Mater (2013) 17:331–347 Young’s Modulus (E)

Varies with temperature

Tangent Modulus (Et )

Varies with temperature

Shape Function

Flow Function

c1

135 MPa

c2

55 MPa

c3 and c4

0.31, 0.2

ζ

2

B

1 1/s

D

38 MPa

m

6

n

10

˙Izotropic Stress, A

5 MPa

λ, α

0.25 and 0.15

perfluorosulfonic acid (PFSA) membrane Nafion NRE212 is modeled at four different temperatures (25, 50, 65, 80 ◦ C) and three swelling percentages (3, 7 and 8 %). Experimental data obtained by Silberstein and Boyce (2010) are compared with simulation results. Strain rate, temperature and hydration dependent monotonic loading and unloading large deformation behavior of Nafion NRE212 membrane is captured well with the modified FVBOP. Acknowledgement The support of the Scientific and Technological Research Council of Turkey (TUBITAK) is gratefully acknowledged. The Project No. 108M521.

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