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A nonlinear mechanical model for the fatigue life of thin-film carbon nanotube supercapacitors
Accepted Manuscript A nonlinear mechanical model for the fatigue life of thin-film carbon nanotube supercapacitors Giovanni Formica, Walter Lacarbonara PII:
S1359-8368(15)00361-3
DOI:
10.1016/j.compositesb.2015.05.047
Reference:
JCOMB 3635
To appear in:
Composites Part B
Received Date: 25 March 2015 Revised Date:
20 May 2015
Accepted Date: 25 May 2015
Please cite this article as: Formica G, Lacarbonara W, A nonlinear mechanical model for the fatigue life of thin-film carbon nanotube supercapacitors, Composites Part B (2015), doi: 10.1016/ j.compositesb.2015.05.047. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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A nonlinear mechanical model for the fatigue life of thin-film carbon nanotube supercapacitors Giovanni Formicaa , Walter Lacarbonarab,∗ a
Dipartimento di Architettura, University of Roma Tre, 00184 Rome, Italy Dipartimento di Ingegneria Strutturale e Geotecnica, University of Roma La Sapienza, 00198 Rome, Italy
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Abstract
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The effects of cyclic loading on the mechanical performance and fatigue life of a novel carbon nanotube supercapacitor are investigated. The highly flexible supercapacitor is a monolithic, pre-fabricated, fully functional film made of a nanostructured free-standing layer in which ions are stored within two vertically aligned multi-walled carbon nanotube (MWCNs) electrodes that are monolithically interspaced by a solution of microcrystalline cellulose in a room temperature ionic liquid electrolyte. To study the cyclic mechanical response of such nanostructured multilayer composite, an original framework is adopted by combining the equivalent continuum approach of Eshelby-Mory-Tanaka and a Weibull-like approach for the evolution of debonding carbon nanotubes electrodes. One- and three-layer models of the supercapacitor are proposed. Cyclic tests are numerically carried out in strain control. A fatigue life limit is determined by considering a confidence interval for the number of cycles corresponding to the states at which the effective elastic modulus of the partially debonded nanostructured portion of the supercapacitor is reduced by a percentage between 20% and 30%. The simulated cyclic tests yield Wholer-type fatigue curves showing the fatigue life limit as the maximum number of cycles N for each strain amplitude. The sensitivity of the fatigue life with respect to meaningful parameters such as the interfacial strength between the MWCNs and cellulose is investigated. Frequency-response functions of the multilayer nanostructured composite are further computed as function of the strain amplitude during cyclic tests.
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Corresponding author. Email addresses: [email protected] (Giovanni Formica), [email protected] (Walter Lacarbonara)
Preprint submitted to Composites: Part B
17 May 2015
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Keywords: Nano-structures, Thin films, Debonding, Fatigue, Fibre/matrix bond, Damage mechanics. 1. Introduction
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In recent years, substantial improvements in the functionality of a wide spectrum of high-tech engineering applications, such as hybrid electric vehicles, laptops, biomedical devices, unmanned aerial vehicles (UAVs), airships, have been sought resorting to the incorporation of pre-fabricated and free-standing functional composite films that, once connected to current collectors, become lightweight, flexible energy storage devices also known as supercapacitors [1, 2]. In some of these applications (e.g., stratospheric airships, air vehicles), the severe weight constraints are such that this type of energy storage system is the only viable approach. Standard film-based supercapacitors are typically fabricated by assembling two electrodes and a separator impregnated with electrolyte, all sandwiched between two current collectors. The separator is simply devised to avoid electrical contact between the electrodes. However, it causes a significant degradation in the supercapacitor performance during its in-service life because the undesirable initiation/propagation of interfacial debonding between the stacked layers determines a disturbance to ions mobility preventing free travels towards the electrodes. The interfacial problems are solved by making all the key components in the form of a single monolithic layer engineered so as to host inter-spaced multiwalled carbon nanotube electrodes impregnated with electrolyte (see Figure1) [1, 2]. Vertically aligned MWCNs with their percolated pore structure due to their high specific area were optimized in terms of their length and enhanced by an innovative nanoparticle surface coating leading to excellent electrochemical performance, high maximum power density (108.2 kW/kg) and high reaction speed. While the monolithic fabrication of the above mentioned supercapacitor overcomes the separator discontinuity of standard film-based sandwiched supercapacitors, the verticallly aligned carbon nanotube electrodes introduce a new challenge associated with the mechanical performance. When subject to cyclic loading, MWCNs can suffer debonding from the hosting cellullose matrix if the hydrostatic stress states in the plane normal to the CNTs reach the interfacial strength. Therefore, the interfacial discontinuity which is truly local in nature can influence the macroscopic behavior when a nonnegligible volume fraction of MWCNs is subject to such a limit state. Indeed, from a strictly mechanical point of view, the
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cellulose
CNTs
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Figure 1: Schematic geometry of the flexible supercapacitor with the two multi-walled carbon nanotube electrodes immersed in cellulose separated by an inter-space.
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fatigue life of the nanostructure film can be reached when the extent of debonding phenomenon causes a significant loss of average stiffness of the composite. Whether this mechanical limit state can compromise the electrochemical performance of the supercapacitor is still an open question. Certainly, the physical discontinuity between the MWCNs external walls and the electrolyte has the potential to affect the charge mobility and distribution within the percolated pore structure. To investigate the progressive debonding of this class of nanostructured composite films leading to the estimation of their fatigue life, the nonlinear model proposed by [3] is here employed. In the literature, various theories have been proposed to describe damage evolution in composites embedding different types of inclusions, including CNTs [4, 5, 6, 7, 8, 9, 10]. The literature on linear and nonlinear models of carbon nanotube nanocomposite materials employed for micronano plates or beams is wide and covers diverse fields such as homogenization[14], gradient/nonlocal elasticity[16, 17, 18, 19], elasto-plasticity [20]. In the present work, the dynamical formulation presented jn [3] for the macroscopic response of carbon nanotube composites accounting for the cumulative debonding of CNTs due to weakening of the interface is particularly suitable. It relies on a thermodynamically consistent phase flow law for the evolution of the volume fraction of debonded CNTs treated as cylindrical inclusions in a rate form amenable to a full dynamical formulation which enables parametric studies and fatigue life assessment. This flow law is the combination of the Weibull statistics and a law giving the rate of the effective stress measure which drives the debonding progression. As a result of progressive debonding, stiffness degradation occurs in the nanostructured composite up to a state in which the fatigue life is reached. In the literature, various works addressed theoretically and experimen-
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tally fatigue loading in composite laminates by employing stiffness degradation models for graphite/epoxy laminates [11], for cross-ply laminates using a suitable fatigue life measure and shear-lag analysis [12], or for adhesively-bonded double- and stepped-lap joints [13]. The common objective of these approaches was to establish a correlation between the evolution of the residual stiffness (i.e., generalized stiffness degradation) and the number of cycles for a given loading amplitude. When the residual stiffness attains a threshold value called failure stiffness, fatigue life is conventionally assumed to be reached. A typical outcome is the evolution of the modulus reduction ratio with the number of cycles. Such a characterization allows to correlate the loading magnitude with the number of cycles at the fatigue limit (i.e., F-N Wohler-type curves). In the present work, we employ a similar approach for the multilayer nanostructured supercapacitor film to investigate such fatigue limit scenarios. We illustrate the nonlinear mechanical model in Section 2 and the numerical results in Section 3. 2. Nanostructured composite subject to progressive debonding
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The multilayer thin film supercapacitor is made of three domains, the interspace composed of cellulose (isotropic elastic material) and the two adjoining MWNTs electrodes immersed in cellulose (see Figure 1). The latter is a nanostructured layer which is modeled as a three-phase material (see Figure 2): hosting matrix (cellulose), perfectly bonded CNTs, and debonded CNTs. The three phases are described by the associated volume fractions. The nanostructured layer with the perfectly bonded CNTs is described by the equivalent linear elastic tensor obtained via the Eshelby-Mory-Tanaka homogenization method. On the other hand, following [3] a suitable phase law together with a Weibull statistics-like approach is proposed to describe the evolution of the volume fraction of debonded CNTs. Let φ1 denote the volume fraction of perfectly bonded CNTs, φ2 that of debonded CNTs, and φ0 = 1 − φ1 − φ2 that of the elastic hosting matrix. The equivalent constitutive law for the three-phase material is described in terms of the equivalent elastic tensor L according to T=L:E (1)
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with T and E representing the stress and strain tensors, the equivalent elastic tensor is given by L = L0 : [I + B : (I − S : B)−1 ] (2) 4
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perfectly bonded CNTs, φ1 (t)
perfectly bonded CNTs, φ1 hosting matrix, φ0
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debonded CNTs, φ2 (t)
e3 e2 e1
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configuration at time t
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Figure 2: Three-phase nanostructured material: hosting matrix, bonded CNTs and debonded CNTs.
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where S is the Eshelby tensor while the 4th order tensor B is given by: φ p (A p + S)−1 ,
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A p = (L p − L0 )−1 : L0 .
(3)
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The subscripts p = 0, 1, 2 express, respectively, the hosting matrix, the CNTs, and the voids. In particular, the elastic tensor L2 associated with the void phase is assumed to be a zero tensor, whence A2 = −I. The debonding process is modeled as a two-parameter Weibull probability process [3]. Starting from the known volume fraction φo1 of perfectly bonded CNTs at some initial time, the cumulative probability of an increment of CNT debonding is expressed by an exponential function of the effective internal stress for debonding, denoted by T 1y , and the Weibull parameters S 0 ∈ R and M ∈ N+ . In particular, the evolution law of the volume fraction of the perfectly bonded CNTs is recast as * +M 1 T 1y (t, φ1 (t)) ˙ φ1 (t) (4) φ1 (t) = − ∗ t S0 where parameter S 0 is the effective debonding stress between the matrix and the CNTs; the positive integer M is the exponent of the nonlinear law that governs the transition from perfectly bonded to debonded CNTs; the effective stress T 1y is expressed in terms of the trace of T 1 :
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T 1y := 13 [tr(T 1 )]
where T 1 is the averaged internal stresses in the CNT phase expressed as −1 T 1 = L1 : [I − S : (A1 + S)−1 ] : I − S : B : E .
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Moreover, in Eq. (4), the McCauley brackets h · i are used here to ensure that when T 1y ≤ 0, φ˙1 (t) vanishes since the debonding phenomenon evolves only when 5
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the CNTs are subject to hydrostatic tensile stress states. Note that the exponent M in Eq. (4) is introduced to control the progression of debonding in the vicinity of the threshold effective stress. The scalar t⋆ , inversely proportional to the debonding speed, is a characteristic scaling time. Thus, φ1 (t) = φo1 − φ2 (t) exponentially decays to 0 with the scaling time t⋆ , while the volume fraction of debonded CNTs at time t , obtained as φ2 (t) := φo1 − φ1 (t), exponentially tends to φ1o . The equations of motion of the equivalent continuum medium whose constitutive laws are given by (1)–(3) must be complemented with the evolution law (4) to obtain the following PDE problem: x ∈ B, t ∈ [t0 , ∞)
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ρ u¨ = ∇ · T(φ1 ) + b, * + T 1y (φ1 ) M 1 φ1 , φ˙1 = − ∗ t S0
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together with the initial conditions for the volume fraction, displacement and velocity ˙ t0 ) = v0 (x), u(x, t0 ) = u0 (x), u(x, o φ1 (x, t0 ) = φ1 (x),
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The above equations define the initial-boundary-value problem in B for the vectorvalued field u(x, t) and the scalar field φ1 (x, t). For the computational implementations, the carbon nanotubes are taken to be collinear with e3 , whence the material behaviour becomes transversely isotropic with axis e3 . By employing Voigt’s notation, the hstress and strain tensor are i ar⊤ ranged in six-dimensional algebraic vectors: T = T 11 , T 22 , T 33 , T 12 , T 23 , T 13 and h i E⊤ = E11 , E22 , E33 , 2 E12 , 2 E23 , 2 E13 so that the equivalent elastic matrix becomes 0 0 L11 L12 L13 0 L 0 0 12 L22 L23 0 L L L 0 0 0 L = 13 23 33 0 0 L44 0 0 0 0 0 0 0 L55 0 0 0 0 0 0 L66
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2 (L11 + L22 )L33 − 2L13 , (L11 + L22 )
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Note that, in consonance with various works supported by micro-mechanical modeling and experimental observations, the independent elastic coefficients of a CNT-nanocomposite are usually well approximated by five equivalent independent elastic coefficients as done for transversely isotropic materials (i.e., L66 = L55 ). Out of the five elastic constants, the truly meaningful constants are the longitudinal Young modulus E3 and the transverse Young modulus E expressed, respectively, as: E=
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The time evolution of these material constants during the loading history allows to trace the generalized stiffness degradation of the thin-film supercapacitor, hence, the assessment of the fatigue life limit. Indeed, by letting E3o denote the initial value of the effective elastic modulus, the adopted criterion for the fatigue life is such that the limit is reached when the residual modulus is reduced with respect to the initial value by 20 − 30%, i.e, E3 /E3o < r with r = 0.7 − 0.8. 3. Three-dimensional numerical tests
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Figure 3 depicts the two models analyzed, corresponding to a thin body made of one or three layers whose plan size is 104 µm × 104 µm while the thickness of each layer is 200 µm.
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e2 200 × 3 µm e1
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Figure 3: 3D models of the simulated supercapacitor: (a) one-layer, (b) three-layers.
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The mechanical properties are given by the Young modulus of the CNTs and of the hosting matrix, respectively, equal to 970 GPa and 0.135 GPa, while the corresponding Poisson ratios are 0.28 and 0.21, respectively. Strain-driven tests are carried out by subjecting the boundaries of the body with unit normal vectors collinear with e1 and e2 to uniform relative in-plane displacements. The out-of-plane displacement (i.e., collinear with e3 ) is prescribed to be zero at the upper and lower boundary surfaces. Two types of strain tests are conducted, namely, monotonic and cyclic tests given by:
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ε(t) = ǫ˙ t for 0 < t < τ1 ! N c for 0 < t < τ0 , ǫ˙ t sin 2π t τ0 ε(t) = ! Nc ǫ˙ τ0 sin 2π t for τ0 < t < τ1 τ0
(14)
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where ǫ˙ is a reference strain rate, Nc is the number of cycles, τ1 is the total time of the cyclic analysis, τ0 is the time instant at which the slow linear growth of the amplitudes terminates and the steady-state cycles start. The relative in-plane displacements are u1 (t) = 12 ǫ(t) · 104 µm= u2 (t). Unless otherwise stated, we let M = 2, S 0 = 0.5 MPa, t⋆ = τ1 /5, and ǫ˙ = 10−8 1/s, τ1 = 5·105 s and τ0 = 1.1·105 s. As a by-product of the FE computations which yield the field variables (u1 , u2 , u3 , φ1 ), the averages of such quantitites over the body domain are computed. Along the same lines, key variables such as the volume fraction φ1 (t) or the longitudinal Young modulus are computed as space averages according to Z Z 1 1 φ¯ 1 (t) = φ1 (x, t)dV, E¯ 3 (t) = E3 (φ1 )dV (15) V B V B where V is the volume of B.
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3.1. One-layer model In this section, the results of the 3D analysis of the one-layer model are discussed, first, in the context of monotonic tests followed by a detailed illustration of the cyclic tests. The monotonic tests are mostly intended to highlight the sensitivity of the response with respect to the parameters governing the debonding process. On the other hand, the cyclic tests are targeted to the determination of Wholer-type curves so as to quantify the fatigue life limit. Moreover, the cyclic analyses were carried out for different values of the model parameters selected within meaningful ranges as suggested by the results of the monotonic tests.
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One-layer case: monotonic tests. Figure 4 shows the typical decay over time of the perfectly bonded CNTs volume fraction φ¯ 1 . In the same plot, a sequence of space-wise distributions of φ1 is superimposed. Further sensitivity analyses were conducted taking into account variations of the key parameter for the debonding process, namely, the interfacial strength. Additional parametric analyses were carried out by varying the characteristic time t⋆ and the exponent M of the power law.
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Figure 4: Evolution of φ¯ 1 in the monotonic test versus the nondimensional time t/τ1 .The contour plots are snapshots of the volume fraction distribution at discrete times, namely, 15 (1, 2, 3, 4, 5)τ1 .
A more detailed overview of the sensitivity analysis can be found in [3]. Among the various parameters, the interfacial strength S 0 is indeed the most important parameter affecting the evolution of fibers debonding, while such phe9
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nomenon is less sensitive to t⋆ and M. In particular, Figure 5b shows a 20% reduction of the longitudinal equivalent modulus occurring in a halved time when S 0 is reduced from 0.5 MPa to 0.1 MPa. Moreover, as expected, the average volume fraction of perfectly bonded CNTs decreases much more rapidly for S 0 = 0.1 MPa.
Figure 5: Monotonic tests for the one-layer model: evolution of φ¯ 1 (t) (a) and E¯ 3 (t)/E3o (b) for different values of the effective debonding stress S 0 , namely, (0.1, 0.5, 1) MPa.
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It is also worth mentioning that the most sensitive equivalent elastic modulus is the longitudinal modulus E3 rather than the transverse modulus. In [3] it was shown that the transverse elastic modulus E decreases by less than 10%. This result can be justified by the fact that in our model no shear directions of debonding are taken into account, and that along the longitudinal direction the aligned CNTs enhance the effective elastic stiffness of the hosting matrix and thus that of the nanocomposite. Therefore we focused our attention on E3 rather than E for the evaluation of the fatigue life of the composite. One-layer case: cyclic tests. In this section, the results for the cyclic tests are illustrated considering the reference model parameters values adopted in the previous section, with the addition of two parameters regulating the loading history: Nc = 105 (number of cycles) and τ0 = 1.1 · 105 seconds. Considering a greater number of cycles does not affect the fatigue life limit while increasing or decreasing τ0 affects only the rate by which the maximum amplitude is reached and thus the speed by which the onset of the fatigue life limit is attained. In view of the determination of the Wholer-type curves, according to the employed damage criterion the fatigue life is reached when the modulus reduction ratio becomes 80% and 70%, respectively. Such a criterion is considered plausible as a marker of the onset of irreversible mechanical fatigue failure in various previous works dealing with composite materials (see, e.g., [11, 12, 13]). Figure 6
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is the first set of Wholer curves obtained for different interfacial strengths, namely, S 0 = (0.25, 0.5, 0.75, 1, 1.5, 2) MPa and for the two values of modulus reduction ratio. As mentioned, S 0 plays a crucial role in the determination of the fatigue life. To better quantify this aspect, consider for example a strain amplitude ǫ = 0.0021 and the damage criterion E3 /E3o < 0.8. By taking S 0 = 0.4 MPa as baseline value, a 56% increment of S 0 causes a 17% increase of the fatigue life while a 150% strength increment causes a life increment of 72%. The supercapacitor fatigue life is much less sensitive with respect to variations of t⋆ and M than S 0 . By varying t⋆ , increments of the fatigue life by 20-25% at most are found whereas even lower increments (¡ 10 %) are associated with variations of M. One of the parameters of interest is the initial CNT volume fraction φo1 since the constitutive law is directly affected by this parameter. Its effects on the fatigue life are shown in Figure 7. Notwithstanding the fact that this parameter is more meaningful than those mentioned above, its influence is indeed much weaker than that of S 0 .
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Figure 6: Wohler’s confidence curves related to S 0 = [0.25, 0.5, 0.75, 1, 1.5, 2] MPa for the onelayer model: continuous line (E3 /E3o < 0.8) and dashed line (E3 /E3o < 0.7).
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Figure 7: Wohler’s confidence curves related to φo1 = [0.005, 0.05, 0.1, 0.2] for the one-layer model: continuous line ( EE3o < 0.8) and dashed line ( EE3o < 0.7). 3
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3.2. Three-layer model The three-layer model consists of a sequence of three stacked layers: the internal one is made of pure cellulose (see Figure 3) while the two external layers are made of CNT nanostructured composite. Only the external layers are thus affected by the debonding phenomenon. The model is subject to the same loading conditions of the one-layer model, namely monotonic and cyclic.
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Three-layer model: monotonic tests. As expected, the three-layer model yields results similar to those of the one-layer model. Comparing Figures 5 and 8, slight differences can be noted starting only from 60% of the considered nondimensional time. Three-layer model: cyclic tests. The results for the three-layer supercapacitor model subject to cyclic loading are depicted in Figure 9. Alternating signs in the loading conditions make the resulting curves less smooth and definitely lead to a slow down of the debonding process (compare with Figure 8). This point is increasingly more emphasized when increasing values of S 0 are considered. The computations of Wohler’s curves for various values of S 0 yield results (see Figure 6) in terms of sensitivity analysis which emphasize the trends of the one-layer model (see Figure 10). For the same strain amplitude, an increase of S 0 by 50% yields an increment of fatigue life higher than 200%.
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Figure 8: Monotonic tests for the three-layer model: evolution of φ1 (t) (a) and of E3 /E3o (b) for different values of effective debonding stress corresponding to S 0 = [0.5, 0.75, 1, 2, 3, 4, 5] MPa.
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Figure 9: Cyclic tests for the three-layer model: evolution of φ1 (t) (a) and E3 /E3o (b) for different values S 0 equal to [0.25, 0.5, 0.75, 1, 2, 3, 4, 5] MPa.
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Results similar to those of the one-layer model are obtained for the threelayer model when different values of the initial volume fraction φo1 are considered. Figure 11 instead shows the sensitivity analysis with respect to the characteristic time t⋆ . To capture the local/global effects of the debonding process on the dynamic response of the super capacitor, the transfer functions were computed as function of the strain amplitudes during monotonic tests based on 3D simulations. In particular, the changes of the frequencies of the lowest modes were considered during the evolving debonding process. Recent studies and analyses were carried out to unfold the way the frequencies are affected by variations of bonded carbon nanotubes concentrations [14]. Such studies have shown the influence of CNTs volume fraction on the frequencies and mode shapes in the case of optimal alignment of CNTs along the e3 direction. The variation of frequencies during the monotonic tests is important to better
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Figure 10: Cyclic tests for the three-layer model: Wohler’s curves for different values of S 0 equal to [0.25, 0.5, 0.75, 1, 2, 3, 4, 5] MPa.
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understand how the debonding phenomenon influences the modal properties of the material and the fatigue life. Moreover, if the material is designed to work in a selected frequency bandwidth, a progressive variation of debonding can change the modal properties of the material to a level that the system turns out to perform in a frequency range that could be harmful to the material. If we compare the frequencies measured at the initial state with those measured at the final state when debonding is fully achieved (see Fig. 12 for fully clamped boundary conditions), a decrease of about 13 % is found for the lowest five frequencies.
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Figure 11: Cyclic tests for the three-layer model: Wohler’s curves for different values of t⋆ /τ1 equal to [1/10, 2/10, 4/10, 6/10, 8/10, 10/10].
4. Conclusions
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The effects of strain cycles on the fatigue life of a novel carbon nanotube-based highly flexible supercapacitor were investigated. A unique nonlinear modeling approach was devised to describe the progressive debonding of CNTs from the hosting solid electrolyte according to an evolutive law which gives the rate of change of the percentage of debonded CNTs in terms of a Weibull-like probability function. The methodology is based on knowledge of the measured effective stress for the interfacial debonding between CNTs and solid electrolyte, and other constitutive parameters such as the exponent of the nonlinear evolutive law and the rescaling time which can be obtained by fitting a few ad hoc mechanical tests conducted on the investigated multilayer supercapacitor system. The methodology yields Wholer-type fatigue curves in terms of number of cycles after which the fatigue life limit is reached for a given strain amplitude. The fatigue limit is assumed to lie between two bounds corresponding to effective elastic modulus reductions equal to 20 % and 30% with respect to the effective modulus of the undamaged supercapacitor. A rich collection of Wholer’s curves was obtained by explicit calculations using a multi-processor cluster. Under cyclic tests, an increase of the interfacial CNT-matrix strength by 50% was shown to yield an increment of the fatigue life
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Figure 12: Variation of the lowest five frequencies of the 3-Layer 3D model (with all the sides and horizontal surfaces constrained) with the strain amplitude ǫ of the monotonic strain tests.
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higher than 200%. This result depends on the strain amplitude of the cyclic tests by which the supercapacitor is forced. It is clear that the service life of such multilayer nanocomposite structures can be increased by delaying the progressive debonding occurring in the nanostructured electrodes. This can be achieved according to two fundamental strategies: increase the interfacial strength by suitable CNT functionalization and limit the maximum strain amplitude by implementing suitable isolation strategies for the supercapacitor, both topics of future theoretical and experimental investigation.
This work was supported by the Specialized International Collaborative Program (Grant No. UD090080GD) sponsored by the Agency for Defense Development (ADD), Republic of Korea. The technical help of Mr. Angelo Ciminelli for some of the computations is acknowledged.
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DOI:
10.1016/j.compositesb.2015.05.047
Reference:
JCOMB 3635
To appear in:
Composites Part B
Received Date: 25 March 2015 Revised Date:
20 May 2015
Accepted Date: 25 May 2015
Please cite this article as: Formica G, Lacarbonara W, A nonlinear mechanical model for the fatigue life of thin-film carbon nanotube supercapacitors, Composites Part B (2015), doi: 10.1016/ j.compositesb.2015.05.047. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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A nonlinear mechanical model for the fatigue life of thin-film carbon nanotube supercapacitors Giovanni Formicaa , Walter Lacarbonarab,∗ a
Dipartimento di Architettura, University of Roma Tre, 00184 Rome, Italy Dipartimento di Ingegneria Strutturale e Geotecnica, University of Roma La Sapienza, 00198 Rome, Italy
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Abstract
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The effects of cyclic loading on the mechanical performance and fatigue life of a novel carbon nanotube supercapacitor are investigated. The highly flexible supercapacitor is a monolithic, pre-fabricated, fully functional film made of a nanostructured free-standing layer in which ions are stored within two vertically aligned multi-walled carbon nanotube (MWCNs) electrodes that are monolithically interspaced by a solution of microcrystalline cellulose in a room temperature ionic liquid electrolyte. To study the cyclic mechanical response of such nanostructured multilayer composite, an original framework is adopted by combining the equivalent continuum approach of Eshelby-Mory-Tanaka and a Weibull-like approach for the evolution of debonding carbon nanotubes electrodes. One- and three-layer models of the supercapacitor are proposed. Cyclic tests are numerically carried out in strain control. A fatigue life limit is determined by considering a confidence interval for the number of cycles corresponding to the states at which the effective elastic modulus of the partially debonded nanostructured portion of the supercapacitor is reduced by a percentage between 20% and 30%. The simulated cyclic tests yield Wholer-type fatigue curves showing the fatigue life limit as the maximum number of cycles N for each strain amplitude. The sensitivity of the fatigue life with respect to meaningful parameters such as the interfacial strength between the MWCNs and cellulose is investigated. Frequency-response functions of the multilayer nanostructured composite are further computed as function of the strain amplitude during cyclic tests.
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∗
Corresponding author. Email addresses: [email protected] (Giovanni Formica), [email protected] (Walter Lacarbonara)
Preprint submitted to Composites: Part B
17 May 2015
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Keywords: Nano-structures, Thin films, Debonding, Fatigue, Fibre/matrix bond, Damage mechanics. 1. Introduction
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In recent years, substantial improvements in the functionality of a wide spectrum of high-tech engineering applications, such as hybrid electric vehicles, laptops, biomedical devices, unmanned aerial vehicles (UAVs), airships, have been sought resorting to the incorporation of pre-fabricated and free-standing functional composite films that, once connected to current collectors, become lightweight, flexible energy storage devices also known as supercapacitors [1, 2]. In some of these applications (e.g., stratospheric airships, air vehicles), the severe weight constraints are such that this type of energy storage system is the only viable approach. Standard film-based supercapacitors are typically fabricated by assembling two electrodes and a separator impregnated with electrolyte, all sandwiched between two current collectors. The separator is simply devised to avoid electrical contact between the electrodes. However, it causes a significant degradation in the supercapacitor performance during its in-service life because the undesirable initiation/propagation of interfacial debonding between the stacked layers determines a disturbance to ions mobility preventing free travels towards the electrodes. The interfacial problems are solved by making all the key components in the form of a single monolithic layer engineered so as to host inter-spaced multiwalled carbon nanotube electrodes impregnated with electrolyte (see Figure1) [1, 2]. Vertically aligned MWCNs with their percolated pore structure due to their high specific area were optimized in terms of their length and enhanced by an innovative nanoparticle surface coating leading to excellent electrochemical performance, high maximum power density (108.2 kW/kg) and high reaction speed. While the monolithic fabrication of the above mentioned supercapacitor overcomes the separator discontinuity of standard film-based sandwiched supercapacitors, the verticallly aligned carbon nanotube electrodes introduce a new challenge associated with the mechanical performance. When subject to cyclic loading, MWCNs can suffer debonding from the hosting cellullose matrix if the hydrostatic stress states in the plane normal to the CNTs reach the interfacial strength. Therefore, the interfacial discontinuity which is truly local in nature can influence the macroscopic behavior when a nonnegligible volume fraction of MWCNs is subject to such a limit state. Indeed, from a strictly mechanical point of view, the
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Figure 1: Schematic geometry of the flexible supercapacitor with the two multi-walled carbon nanotube electrodes immersed in cellulose separated by an inter-space.
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fatigue life of the nanostructure film can be reached when the extent of debonding phenomenon causes a significant loss of average stiffness of the composite. Whether this mechanical limit state can compromise the electrochemical performance of the supercapacitor is still an open question. Certainly, the physical discontinuity between the MWCNs external walls and the electrolyte has the potential to affect the charge mobility and distribution within the percolated pore structure. To investigate the progressive debonding of this class of nanostructured composite films leading to the estimation of their fatigue life, the nonlinear model proposed by [3] is here employed. In the literature, various theories have been proposed to describe damage evolution in composites embedding different types of inclusions, including CNTs [4, 5, 6, 7, 8, 9, 10]. The literature on linear and nonlinear models of carbon nanotube nanocomposite materials employed for micronano plates or beams is wide and covers diverse fields such as homogenization[14], gradient/nonlocal elasticity[16, 17, 18, 19], elasto-plasticity [20]. In the present work, the dynamical formulation presented jn [3] for the macroscopic response of carbon nanotube composites accounting for the cumulative debonding of CNTs due to weakening of the interface is particularly suitable. It relies on a thermodynamically consistent phase flow law for the evolution of the volume fraction of debonded CNTs treated as cylindrical inclusions in a rate form amenable to a full dynamical formulation which enables parametric studies and fatigue life assessment. This flow law is the combination of the Weibull statistics and a law giving the rate of the effective stress measure which drives the debonding progression. As a result of progressive debonding, stiffness degradation occurs in the nanostructured composite up to a state in which the fatigue life is reached. In the literature, various works addressed theoretically and experimen-
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tally fatigue loading in composite laminates by employing stiffness degradation models for graphite/epoxy laminates [11], for cross-ply laminates using a suitable fatigue life measure and shear-lag analysis [12], or for adhesively-bonded double- and stepped-lap joints [13]. The common objective of these approaches was to establish a correlation between the evolution of the residual stiffness (i.e., generalized stiffness degradation) and the number of cycles for a given loading amplitude. When the residual stiffness attains a threshold value called failure stiffness, fatigue life is conventionally assumed to be reached. A typical outcome is the evolution of the modulus reduction ratio with the number of cycles. Such a characterization allows to correlate the loading magnitude with the number of cycles at the fatigue limit (i.e., F-N Wohler-type curves). In the present work, we employ a similar approach for the multilayer nanostructured supercapacitor film to investigate such fatigue limit scenarios. We illustrate the nonlinear mechanical model in Section 2 and the numerical results in Section 3. 2. Nanostructured composite subject to progressive debonding
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The multilayer thin film supercapacitor is made of three domains, the interspace composed of cellulose (isotropic elastic material) and the two adjoining MWNTs electrodes immersed in cellulose (see Figure 1). The latter is a nanostructured layer which is modeled as a three-phase material (see Figure 2): hosting matrix (cellulose), perfectly bonded CNTs, and debonded CNTs. The three phases are described by the associated volume fractions. The nanostructured layer with the perfectly bonded CNTs is described by the equivalent linear elastic tensor obtained via the Eshelby-Mory-Tanaka homogenization method. On the other hand, following [3] a suitable phase law together with a Weibull statistics-like approach is proposed to describe the evolution of the volume fraction of debonded CNTs. Let φ1 denote the volume fraction of perfectly bonded CNTs, φ2 that of debonded CNTs, and φ0 = 1 − φ1 − φ2 that of the elastic hosting matrix. The equivalent constitutive law for the three-phase material is described in terms of the equivalent elastic tensor L according to T=L:E (1)
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with T and E representing the stress and strain tensors, the equivalent elastic tensor is given by L = L0 : [I + B : (I − S : B)−1 ] (2) 4
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perfectly bonded CNTs, φ1 (t)
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debonded CNTs, φ2 (t)
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configuration at time t
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Figure 2: Three-phase nanostructured material: hosting matrix, bonded CNTs and debonded CNTs.
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where S is the Eshelby tensor while the 4th order tensor B is given by: φ p (A p + S)−1 ,
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(3)
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The subscripts p = 0, 1, 2 express, respectively, the hosting matrix, the CNTs, and the voids. In particular, the elastic tensor L2 associated with the void phase is assumed to be a zero tensor, whence A2 = −I. The debonding process is modeled as a two-parameter Weibull probability process [3]. Starting from the known volume fraction φo1 of perfectly bonded CNTs at some initial time, the cumulative probability of an increment of CNT debonding is expressed by an exponential function of the effective internal stress for debonding, denoted by T 1y , and the Weibull parameters S 0 ∈ R and M ∈ N+ . In particular, the evolution law of the volume fraction of the perfectly bonded CNTs is recast as * +M 1 T 1y (t, φ1 (t)) ˙ φ1 (t) (4) φ1 (t) = − ∗ t S0 where parameter S 0 is the effective debonding stress between the matrix and the CNTs; the positive integer M is the exponent of the nonlinear law that governs the transition from perfectly bonded to debonded CNTs; the effective stress T 1y is expressed in terms of the trace of T 1 :
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T 1y := 13 [tr(T 1 )]
where T 1 is the averaged internal stresses in the CNT phase expressed as −1 T 1 = L1 : [I − S : (A1 + S)−1 ] : I − S : B : E .
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Moreover, in Eq. (4), the McCauley brackets h · i are used here to ensure that when T 1y ≤ 0, φ˙1 (t) vanishes since the debonding phenomenon evolves only when 5
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the CNTs are subject to hydrostatic tensile stress states. Note that the exponent M in Eq. (4) is introduced to control the progression of debonding in the vicinity of the threshold effective stress. The scalar t⋆ , inversely proportional to the debonding speed, is a characteristic scaling time. Thus, φ1 (t) = φo1 − φ2 (t) exponentially decays to 0 with the scaling time t⋆ , while the volume fraction of debonded CNTs at time t , obtained as φ2 (t) := φo1 − φ1 (t), exponentially tends to φ1o . The equations of motion of the equivalent continuum medium whose constitutive laws are given by (1)–(3) must be complemented with the evolution law (4) to obtain the following PDE problem: x ∈ B, t ∈ [t0 , ∞)
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ρ u¨ = ∇ · T(φ1 ) + b, * + T 1y (φ1 ) M 1 φ1 , φ˙1 = − ∗ t S0
subject to the boundary conditions on ∂B f (where force data are prescribed) and ∂Bu (where kinematic data are prescribed) that read: T · n = f, ¯ u = u,
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together with the initial conditions for the volume fraction, displacement and velocity ˙ t0 ) = v0 (x), u(x, t0 ) = u0 (x), u(x, o φ1 (x, t0 ) = φ1 (x),
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The above equations define the initial-boundary-value problem in B for the vectorvalued field u(x, t) and the scalar field φ1 (x, t). For the computational implementations, the carbon nanotubes are taken to be collinear with e3 , whence the material behaviour becomes transversely isotropic with axis e3 . By employing Voigt’s notation, the hstress and strain tensor are i ar⊤ ranged in six-dimensional algebraic vectors: T = T 11 , T 22 , T 33 , T 12 , T 23 , T 13 and h i E⊤ = E11 , E22 , E33 , 2 E12 , 2 E23 , 2 E13 so that the equivalent elastic matrix becomes 0 0 L11 L12 L13 0 L 0 0 12 L22 L23 0 L L L 0 0 0 L = 13 23 33 0 0 L44 0 0 0 0 0 0 0 L55 0 0 0 0 0 0 L66
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Note that, in consonance with various works supported by micro-mechanical modeling and experimental observations, the independent elastic coefficients of a CNT-nanocomposite are usually well approximated by five equivalent independent elastic coefficients as done for transversely isotropic materials (i.e., L66 = L55 ). Out of the five elastic constants, the truly meaningful constants are the longitudinal Young modulus E3 and the transverse Young modulus E expressed, respectively, as: E=
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The time evolution of these material constants during the loading history allows to trace the generalized stiffness degradation of the thin-film supercapacitor, hence, the assessment of the fatigue life limit. Indeed, by letting E3o denote the initial value of the effective elastic modulus, the adopted criterion for the fatigue life is such that the limit is reached when the residual modulus is reduced with respect to the initial value by 20 − 30%, i.e, E3 /E3o < r with r = 0.7 − 0.8. 3. Three-dimensional numerical tests
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Figure 3 depicts the two models analyzed, corresponding to a thin body made of one or three layers whose plan size is 104 µm × 104 µm while the thickness of each layer is 200 µm.
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e2 200 × 3 µm e1
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Figure 3: 3D models of the simulated supercapacitor: (a) one-layer, (b) three-layers.
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The mechanical properties are given by the Young modulus of the CNTs and of the hosting matrix, respectively, equal to 970 GPa and 0.135 GPa, while the corresponding Poisson ratios are 0.28 and 0.21, respectively. Strain-driven tests are carried out by subjecting the boundaries of the body with unit normal vectors collinear with e1 and e2 to uniform relative in-plane displacements. The out-of-plane displacement (i.e., collinear with e3 ) is prescribed to be zero at the upper and lower boundary surfaces. Two types of strain tests are conducted, namely, monotonic and cyclic tests given by:
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ε(t) = ǫ˙ t for 0 < t < τ1 ! N c for 0 < t < τ0 , ǫ˙ t sin 2π t τ0 ε(t) = ! Nc ǫ˙ τ0 sin 2π t for τ0 < t < τ1 τ0
(14)
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where ǫ˙ is a reference strain rate, Nc is the number of cycles, τ1 is the total time of the cyclic analysis, τ0 is the time instant at which the slow linear growth of the amplitudes terminates and the steady-state cycles start. The relative in-plane displacements are u1 (t) = 12 ǫ(t) · 104 µm= u2 (t). Unless otherwise stated, we let M = 2, S 0 = 0.5 MPa, t⋆ = τ1 /5, and ǫ˙ = 10−8 1/s, τ1 = 5·105 s and τ0 = 1.1·105 s. As a by-product of the FE computations which yield the field variables (u1 , u2 , u3 , φ1 ), the averages of such quantitites over the body domain are computed. Along the same lines, key variables such as the volume fraction φ1 (t) or the longitudinal Young modulus are computed as space averages according to Z Z 1 1 φ¯ 1 (t) = φ1 (x, t)dV, E¯ 3 (t) = E3 (φ1 )dV (15) V B V B where V is the volume of B.
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3.1. One-layer model In this section, the results of the 3D analysis of the one-layer model are discussed, first, in the context of monotonic tests followed by a detailed illustration of the cyclic tests. The monotonic tests are mostly intended to highlight the sensitivity of the response with respect to the parameters governing the debonding process. On the other hand, the cyclic tests are targeted to the determination of Wholer-type curves so as to quantify the fatigue life limit. Moreover, the cyclic analyses were carried out for different values of the model parameters selected within meaningful ranges as suggested by the results of the monotonic tests.
1/5 τ1
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One-layer case: monotonic tests. Figure 4 shows the typical decay over time of the perfectly bonded CNTs volume fraction φ¯ 1 . In the same plot, a sequence of space-wise distributions of φ1 is superimposed. Further sensitivity analyses were conducted taking into account variations of the key parameter for the debonding process, namely, the interfacial strength. Additional parametric analyses were carried out by varying the characteristic time t⋆ and the exponent M of the power law.
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Figure 4: Evolution of φ¯ 1 in the monotonic test versus the nondimensional time t/τ1 .The contour plots are snapshots of the volume fraction distribution at discrete times, namely, 15 (1, 2, 3, 4, 5)τ1 .
A more detailed overview of the sensitivity analysis can be found in [3]. Among the various parameters, the interfacial strength S 0 is indeed the most important parameter affecting the evolution of fibers debonding, while such phe9
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nomenon is less sensitive to t⋆ and M. In particular, Figure 5b shows a 20% reduction of the longitudinal equivalent modulus occurring in a halved time when S 0 is reduced from 0.5 MPa to 0.1 MPa. Moreover, as expected, the average volume fraction of perfectly bonded CNTs decreases much more rapidly for S 0 = 0.1 MPa.
Figure 5: Monotonic tests for the one-layer model: evolution of φ¯ 1 (t) (a) and E¯ 3 (t)/E3o (b) for different values of the effective debonding stress S 0 , namely, (0.1, 0.5, 1) MPa.
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It is also worth mentioning that the most sensitive equivalent elastic modulus is the longitudinal modulus E3 rather than the transverse modulus. In [3] it was shown that the transverse elastic modulus E decreases by less than 10%. This result can be justified by the fact that in our model no shear directions of debonding are taken into account, and that along the longitudinal direction the aligned CNTs enhance the effective elastic stiffness of the hosting matrix and thus that of the nanocomposite. Therefore we focused our attention on E3 rather than E for the evaluation of the fatigue life of the composite. One-layer case: cyclic tests. In this section, the results for the cyclic tests are illustrated considering the reference model parameters values adopted in the previous section, with the addition of two parameters regulating the loading history: Nc = 105 (number of cycles) and τ0 = 1.1 · 105 seconds. Considering a greater number of cycles does not affect the fatigue life limit while increasing or decreasing τ0 affects only the rate by which the maximum amplitude is reached and thus the speed by which the onset of the fatigue life limit is attained. In view of the determination of the Wholer-type curves, according to the employed damage criterion the fatigue life is reached when the modulus reduction ratio becomes 80% and 70%, respectively. Such a criterion is considered plausible as a marker of the onset of irreversible mechanical fatigue failure in various previous works dealing with composite materials (see, e.g., [11, 12, 13]). Figure 6
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is the first set of Wholer curves obtained for different interfacial strengths, namely, S 0 = (0.25, 0.5, 0.75, 1, 1.5, 2) MPa and for the two values of modulus reduction ratio. As mentioned, S 0 plays a crucial role in the determination of the fatigue life. To better quantify this aspect, consider for example a strain amplitude ǫ = 0.0021 and the damage criterion E3 /E3o < 0.8. By taking S 0 = 0.4 MPa as baseline value, a 56% increment of S 0 causes a 17% increase of the fatigue life while a 150% strength increment causes a life increment of 72%. The supercapacitor fatigue life is much less sensitive with respect to variations of t⋆ and M than S 0 . By varying t⋆ , increments of the fatigue life by 20-25% at most are found whereas even lower increments (¡ 10 %) are associated with variations of M. One of the parameters of interest is the initial CNT volume fraction φo1 since the constitutive law is directly affected by this parameter. Its effects on the fatigue life are shown in Figure 7. Notwithstanding the fact that this parameter is more meaningful than those mentioned above, its influence is indeed much weaker than that of S 0 .
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Figure 6: Wohler’s confidence curves related to S 0 = [0.25, 0.5, 0.75, 1, 1.5, 2] MPa for the onelayer model: continuous line (E3 /E3o < 0.8) and dashed line (E3 /E3o < 0.7).
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Figure 7: Wohler’s confidence curves related to φo1 = [0.005, 0.05, 0.1, 0.2] for the one-layer model: continuous line ( EE3o < 0.8) and dashed line ( EE3o < 0.7). 3
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3.2. Three-layer model The three-layer model consists of a sequence of three stacked layers: the internal one is made of pure cellulose (see Figure 3) while the two external layers are made of CNT nanostructured composite. Only the external layers are thus affected by the debonding phenomenon. The model is subject to the same loading conditions of the one-layer model, namely monotonic and cyclic.
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Three-layer model: monotonic tests. As expected, the three-layer model yields results similar to those of the one-layer model. Comparing Figures 5 and 8, slight differences can be noted starting only from 60% of the considered nondimensional time. Three-layer model: cyclic tests. The results for the three-layer supercapacitor model subject to cyclic loading are depicted in Figure 9. Alternating signs in the loading conditions make the resulting curves less smooth and definitely lead to a slow down of the debonding process (compare with Figure 8). This point is increasingly more emphasized when increasing values of S 0 are considered. The computations of Wohler’s curves for various values of S 0 yield results (see Figure 6) in terms of sensitivity analysis which emphasize the trends of the one-layer model (see Figure 10). For the same strain amplitude, an increase of S 0 by 50% yields an increment of fatigue life higher than 200%.
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Figure 8: Monotonic tests for the three-layer model: evolution of φ1 (t) (a) and of E3 /E3o (b) for different values of effective debonding stress corresponding to S 0 = [0.5, 0.75, 1, 2, 3, 4, 5] MPa.
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Figure 9: Cyclic tests for the three-layer model: evolution of φ1 (t) (a) and E3 /E3o (b) for different values S 0 equal to [0.25, 0.5, 0.75, 1, 2, 3, 4, 5] MPa.
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Results similar to those of the one-layer model are obtained for the threelayer model when different values of the initial volume fraction φo1 are considered. Figure 11 instead shows the sensitivity analysis with respect to the characteristic time t⋆ . To capture the local/global effects of the debonding process on the dynamic response of the super capacitor, the transfer functions were computed as function of the strain amplitudes during monotonic tests based on 3D simulations. In particular, the changes of the frequencies of the lowest modes were considered during the evolving debonding process. Recent studies and analyses were carried out to unfold the way the frequencies are affected by variations of bonded carbon nanotubes concentrations [14]. Such studies have shown the influence of CNTs volume fraction on the frequencies and mode shapes in the case of optimal alignment of CNTs along the e3 direction. The variation of frequencies during the monotonic tests is important to better
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Figure 10: Cyclic tests for the three-layer model: Wohler’s curves for different values of S 0 equal to [0.25, 0.5, 0.75, 1, 2, 3, 4, 5] MPa.
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understand how the debonding phenomenon influences the modal properties of the material and the fatigue life. Moreover, if the material is designed to work in a selected frequency bandwidth, a progressive variation of debonding can change the modal properties of the material to a level that the system turns out to perform in a frequency range that could be harmful to the material. If we compare the frequencies measured at the initial state with those measured at the final state when debonding is fully achieved (see Fig. 12 for fully clamped boundary conditions), a decrease of about 13 % is found for the lowest five frequencies.
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Figure 11: Cyclic tests for the three-layer model: Wohler’s curves for different values of t⋆ /τ1 equal to [1/10, 2/10, 4/10, 6/10, 8/10, 10/10].
4. Conclusions
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The effects of strain cycles on the fatigue life of a novel carbon nanotube-based highly flexible supercapacitor were investigated. A unique nonlinear modeling approach was devised to describe the progressive debonding of CNTs from the hosting solid electrolyte according to an evolutive law which gives the rate of change of the percentage of debonded CNTs in terms of a Weibull-like probability function. The methodology is based on knowledge of the measured effective stress for the interfacial debonding between CNTs and solid electrolyte, and other constitutive parameters such as the exponent of the nonlinear evolutive law and the rescaling time which can be obtained by fitting a few ad hoc mechanical tests conducted on the investigated multilayer supercapacitor system. The methodology yields Wholer-type fatigue curves in terms of number of cycles after which the fatigue life limit is reached for a given strain amplitude. The fatigue limit is assumed to lie between two bounds corresponding to effective elastic modulus reductions equal to 20 % and 30% with respect to the effective modulus of the undamaged supercapacitor. A rich collection of Wholer’s curves was obtained by explicit calculations using a multi-processor cluster. Under cyclic tests, an increase of the interfacial CNT-matrix strength by 50% was shown to yield an increment of the fatigue life
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Figure 12: Variation of the lowest five frequencies of the 3-Layer 3D model (with all the sides and horizontal surfaces constrained) with the strain amplitude ǫ of the monotonic strain tests.
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higher than 200%. This result depends on the strain amplitude of the cyclic tests by which the supercapacitor is forced. It is clear that the service life of such multilayer nanocomposite structures can be increased by delaying the progressive debonding occurring in the nanostructured electrodes. This can be achieved according to two fundamental strategies: increase the interfacial strength by suitable CNT functionalization and limit the maximum strain amplitude by implementing suitable isolation strategies for the supercapacitor, both topics of future theoretical and experimental investigation.
This work was supported by the Specialized International Collaborative Program (Grant No. UD090080GD) sponsored by the Agency for Defense Development (ADD), Republic of Korea. The technical help of Mr. Angelo Ciminelli for some of the computations is acknowledged.
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