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Dependence of the electrical conductivity of graphene reinforced epoxy resin on the stress level
International Journal of Engineering Science 120 (2017) 63–70
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Dependence of the electrical conductivity of graphene reinforced epoxy resin on the stress level Daniel Wentzel a,b, Sandi Miller c, Igor Sevostianov a,d,∗ a
Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88001, USA NASA Materials and Components Laboratories Office, JSC White Sands Test Facility, Las Cruces, NM 88004, USA c NASA Ceramic and Polymer Composites Division, Glenn Research Center, Cleveland, OH. 44135, USA d Center for Design, Manufacturing, and Materials, Skolkovo Institute of Science and Technology, Skolkovo, Russia b
a r t i c l e
i n f o
Article history: Received 30 April 2017 Revised 26 May 2017 Accepted 3 June 2017
Keywords: Graphene Epoxy matrix composite Electrical conductivity Elastic stress
a b s t r a c t This paper focuses on electrical conductivity of epoxy based nanocomposites containing graphene particles. We show that adding 0.5 wt% of nanoparticles decreases electrical resistivity of the material more than two orders of magnitude. This effect is modeled with good accuracy using ordinary micromechanical homogenization schemes. Combination of mechanical and electrical tests shows that electrical conductivity of the epoxy-graphene composite depends on the level of tensile stress in the material. To explain this effect we assumed formation of microcracks from the very beginning of the loading process due to very high stress concentration at the particles. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Carbon reinforced composites have been used for more than half a century in applications where high strength to weight ratio is crucial. Due to the high cost and susceptibility to damage, non-destructive evaluation (NDE) techniques that identify damage within a composite structure are highly valued. One issue with many NDE techniques is that they require the material to be conductive. While the carbon fiber reinforcement is conductive, the matrix material holding the reinforcement material together generally is not conductive making many NDE techniques inapplicable. To the best of our knowledge, the idea to estimate strength reduction of a carbon fiber reinforced composites through electrical resistivity measurements has been proposed by Schulte and Baron (1989). The idea was further implemented in works of Wang and Chung (1997a,b; 1998) Wang, et al. (1999, 2006) Abry et al. (1999, 2001), Park, Okabe, Takeda, and Curtin (2002), and applied to carbon fiber yarn by Wentzel and Sevostianov (2013). AC and DC electrical methods have been extensively studied since then and have been used to study a variety of damage mechanisms, e.g. delamination or matrix cracking, under static and dynamic loading conditions. Flandin, Cavaillé, Bréchet, and Dendievel (1999) first used nanoscopic conductive fillers with different aspect ratios in a thermoplastic matrix to monitor the applied macroscopic mechanical strain and the damage evolution during loading. Nanoscaled carbon black particles (Kupke et al., 2001) as well as microscaled carbon black particles (Muto et al., 2001) have also been used to modify the matrix of glass-fiber reinforced thermosets. It was shown that external stress as well as apparent mechanical damage can be detected in these multiphase composites via electrical conductivity methods.
∗
Corresponding author at: Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88001, USA. E-mail address: [email protected] (I. Sevostianov).
http://dx.doi.org/10.1016/j.ijengsci.2017.06.013 0020-7225/© 2017 Elsevier Ltd. All rights reserved.
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D. Wentzel et al. / International Journal of Engineering Science 120 (2017) 63–70
Fig. 1. SEM images of epoxy reinforced with graphene particles.
Böger et al. (2008) developed glass fiber reinforced composites with epoxy matrix containing carbon nanotubes and showed that this material has reasonably high electrical conductivity. Bellucci et al. (2011) studied electrical conductivity of epoxy resin as matrix (Epon 828) containing 0.1%, 0.25% and 0.5% wt carbon black nanofillers or carbon nanotubes. They reported two orders of magnitude decrease of electrical resistivity of such composites as compare to the matrix material. They also showed that epoxy matrix containing carbon nanotubes yields higher values of electrical conductivity than the same matrix containing carbon black particles. Ferreira, Nóvoa, and Marques (2016) provided a review of the state-of-the-art in the area of conducting polymer matrix composites. Macías, D’ Alessandro, Castro-Triguero, Pérez-Mira, and Ubertini (2017) proposed a micromechanical model to explain the conductivity of the polymers containing conducting nanofibers through formation of conducting networks (percolation effect). They assumed that the matrix material is a perfect insulator and did not consider the charge transport across the matric material. However, polymers have low electrical conductivity but it is not zero. Recently, Kang and Snyder (2016) proposed a model for charge transport phenomena in polymers. In the present paper we focus on electrical conductivity of epoxy filled with graphene particles. We show that adding 1.0 wt% of this filler increase the conductivity of the epoxy more than two orders of magnitude. To explain this phenomena, we use ordinary micromechanical models accounting for the strongly oblate shape of the graphene particles. We also observed dependence of the electrical resistivity on the strain level. This phenomenon can be explained from the point of view of microcracks formation and partial debonding between the matrix and particles due to very high stress concentration at the particles tips. The quantitative explanation of the phenomena, however requires more detailed analysis. 2. Materials and methods For this study, a two-part epoxy and electrically conductive graphene particles were used. 0.5 wt% and 1.0 wt% conductive filler were used. Epoxy resin, Epon 862, and ‘W’ amine curing agent were procured from Hexion. Surface functionalized expanded graphite (Miller et al., 2007), TG679, was procured from Adherent Technologies, Albuquerque, NM. All materials were used as received. Resin plaques of baseline Epon 862/W were prepared by mixing 40 g of 862 epoxy resin with 10.56 g of W curing agent. Nanocomposites were prepared through dispersion of either 0.5% or 1.0 wt% of the expanded graphite into graphene layers throughout the base epoxy mixture. Nano-particle dispersion was initiated via a THINKY mixer profile which followed a 4 minute mix/ 4 minutes degas cycle. The mixture was then sonicated for 5 minutes. This process was twice repeated, followed by a cycle in the THINKY mixer, to ensure a final degas. The mixture was poured into a 3 × 8 Teflon mold and cured 125 °C for 2 hrs followed by a 177 °C hold for an additional 2 hr. A scanning electron microscope (SEM) was used to observe how well dispersed the conductive filler was. Dispersion of graphene appeared to be uniform. Fig. 1 illustrates the SEM images of epoxy reinforced with graphene particles. The size of the panels was 75 × 125 mm. From these panels, ASTM D638 Type IV dog bone specimens were machined. The length of the specimens deviated from ASTM D638 guidelines; however, all samples were machined to consistent dimensions. Fig. 2a shows the specimens. Resistance measurements were made using a 4-point method with kelvin clip leads. The leads were spaced 10 mm apart and attached to the center region of the gage section. A calibrated IET Labs Model 1693 RLC Digibridge was used to make the resistance measurements. The basic accuracy of the unit is reported as ± 0.02%. Prior to testing, the unit was tested against a known resistance standard to verify calibration and procedures were followed to zero the unit. This unit was controlled by a custom graphical user interface (GUI) which allowed resistance measurements to be recorded to an Excel spreadsheet. Resistance was measured at a sample rate of 5 Hz.
D. Wentzel et al. / International Journal of Engineering Science 120 (2017) 63–70
65
Fig. 2. (a) Dog-bone specimens of pure epoxy, epoxy reinforced with 0.5 wt% of graphene and epoxy reinforced with 1 wt% of graphene (from left to right); (b) measurement of the electric resistance of a specimen in the process of loading.
Using an Instronۚ Corp. 5569 Series Electromechanical Test Instrument equipped with a 50 kN (11,200 lbf) capacity load cell, self-tightening 25 × 51 mm (1 × 2 in.) wedge action mechanical grips, and Bluehillۚ data acquisition software (version 1.8.289), carbon filled and unfilled epoxy dog bones were subjected to identical stress schedules consisting of a constant strain rate uniaxial tension test. The strain rate was set to 0.33 mm/min. This was selected to match approximate the slow strain rate used in a previous study by Gilat, Goldberg, and Roberts (2005) with the same material. Fig. 2b illustrates the procedure of measurement of the electric resistance of a specimen in the process of loading. 3. Results Resistance, stress, and strain were recorded for each specimen. Resistance measurements preformed on the neat EPON 862 dog bones, as expected, did not conduct well resulting in a resistance measurements on the order of. 5 × 107 . Using the geometry of the specimens we calculated the resistivity as 4.75 × 104 · m. The average stress at failure was 56.58 MPa and the average elongation a failure was 2.44 mm for all the specimens. All samples failed in the gauge section. Based on the ANOVA statistical analysis of the data there was no statistically significant difference in the stress at fracture when compared across the levels of conductive filler. The P-value was equal to 0.33, indicating the effect was not significant. The average stress at failure for 0.5 wt% and 1.0 wt% were 52.99 MPa and 47.94 MPa (54.49 MPa for valid gauge failures) respectively. The average elongation at failure was 2.68 mm (3.07 mm for valid gauge failures). All specimens with graphene inclusions failed in the gauge section. Adding 0.5 wt% of graphene particles reduced the resistivity down to 18 ± 6 · m. Increasing the content of graphene up to 1.0 wt% decreased the resistivity to 13 ± 5 · m. We note that a statistically significant effect has been observed between the levels of conductive filler when comparing the change in resistance as a function of stress/strain (P-value equal to 0.006) and resistivity (P-value equal to 0.003). The data was transformed using a square root function in order to more accurate fit the data; however, a linear fit was also statistically significant. We observed that changes in resistance of samples with graphene inclusions correlate to changes in stress and strain. Samples with 0.5 wt% graphene inclusions appeared to exhibit a linear relationship between elongation and resistance. The samples also appeared to exhibit a polynomial relationship between stress and resistance (Fig. 3). The standard deviation of the baseline resistance measurements for both samples was determined to be 0.004 K. Material exhibited a linear response when comparing sample elongation with resistance. Samples with 1.0 wt% graphene inclusions also appeared to exhibit linear relationship between elongation and resistance and quadratic relationship between stress and resistance. The standard deviation of the baseline resistance measurement of this sample was higher than samples with 0.5 wt% graphene. The standard deviation was determined to be 0.032 K.
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Fig. 3. Dependence of the resistivity of graphene reinforced epoxy on elongation and stress.
3.1. Micromechanical model To model the effective properties of heterogeneous materials, we use the concept of property contribution tensors that express contributions of a given inhomogeneity to the effective properties of interest, under the assumption that the inhomogeneity is placed into a uniform applied field, i.e. this field would have been uniform within the site of the inhomogeneity in its absence. For the elastic properties, these tensors are the compliance or stiffness contribution tensors, H or N; compliance contribution tensors have been first introduced in the context of ellipsoidal pores and cracks in isotropic material by Horii and Nemat-Nasser (1983). For the electrical or thermal conductivity problem, the conductivity or resistivity contribution tensors, K or R have been introduced by Sevostianov and Kachanov (2002) as follows. Assuming a linear relation between electric field E and the electric current density J, the change in E required to maintain the same electric current density if the inhomogeneity is introduced is:
E =
V1 R·J V
(3.1)
where the symmetric second-rank tensor R is called the resistivity contribution tensor of an inhomogeneity. Alternatively, the dual relation can be written as
J =
V1 K ·E V
(3.2)
where K is the conductivity contribution tensor of an inhomogeneity. In the case of the isotropic matrix of conductivity k0 (K = k0 I , R = r0 I , k0 = 1/r0 ),
K = −k20 R
(3.3)
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Table 1 Material constants of the constituents. Constituent
Epoxy EPON 862
Graphene
Density
ρ 1.21 g/cm3 Electrical resistivity r11 = r22 4.75 × 104 ·m r33 4.75 × 104 ·m
2.27 g/cm3 4.424 × 10−8 ·m 1.695 × 10−7 ·m
The R- and K-tensors properly reflect contributions of individual inhomogeneities to the effective resistivity and the effective conductivity and the sums
R(k ) or
K (k )
(3.4)
constitute the general proper microstructural parameters in whose terms the effective resistivity and the effective conductivity should be expressed (Kachanov & Sevostianov, 2005). In the case of a transversely-isotropic spheroidal inhomogeneity, a1 = a2 = a; a3 = γ a, embedded in an isotropic matrix (case of interest in the present work), explicit expression for resistivity and conductivity contribution tensors have the following form (see, for example, Sevostianov & Giraud, 2013):
R=−
1 1 K= [A1 (I − nn ) + A2 nn] k0 k20
(3.5)
where dimensionless factors
A1 =
1 − k111 /k0
,
A2 =
1 − 1 − k111 /k0 f0
1 − k133 /k0
k133 /k0 + 2 1 − k133 /k0 f0
(3.5)
and f0 (γ ) is defined by
⎧ √ ⎨ √1 arctan 1γ−γ 2 , 2 γ 2 (1 − g ) √ ; g ( γ ) = γ 1 −γ f0 = ⎩ √1 ln γ +√γ 2 −1 , 2 γ2 −1 2γ
γ
2 −1
γ−
γ
2 −1
For a thin (strongly oblate) spheroid (a1 = a2 = a,
π g( γ ) → , 2γ
f0 →
πγ
oblateshape (γ < 1 ) prolateshape (γ > 1 )
(3.6)
γ ≡ a/a3 >> 1) (3.7)
4
We now can calculate overall electric conductivity of epoxy/graphene composite and compare them with the experimental results. Material constants of the constituents are given in Table 1. Individual graphene particles are modeled as strongly oblate spheroids with the aspect ratio γ = 10−5 . If interaction between the inhomogeneities is neglected, each inhomogeneity can be assumed to be subjected to the same remotely applied field. Contributions of the inhomogeneities into the effective property of interests can be treated separately and the total gradient of temperature across the representative volume element (RVE) is
1 1 E= J+ k0 V
V (i ) R
·J
(3.8)
i
The summation over inhomogeneities may be replaced by the integration over orientations. In particular, in the case of isotropic orientation distribution of spheroidal inhomogeneities of aspect ratio γ
E=
1 1 [1 + c (2A1 + A2 )/3] J ≡ [1 + c η ] J k0 k0
(3.9)
where c is the volume fraction of graphite particles and parameters A1 and A2 are given by (3.5). Thus the (isotropic) effective conductivity is
re f f =
r0 1 + cη
(3.10)
Parameter η depends on shape of the inhomogeneities and conductivities of the two phases. When explicit expressions for non-interaction approximation are obtained, formulas for overall electric and conductive properties can be also evaluated using standard homogenization techniques, for example Mori-Tanaka-Benveniste (MTB) scheme, (Benveniste, 1986; Mori & Tanaka, 1973) where each inhomogeneity, treated as an isolated one, is placed into the unaltered matrix material and the interactions between different inhomogeneities are accounted for by assuming that the external field acting on them differs from the remotely applied one. This field is taken as the average over the matrix and
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Fig. 4. Comparison of the predictions provided by micromechanical schemes with experimentally measured electrical resistivities of graphene reinforced epoxy before loading.
is the same for all the inhomogeneities. In the case of randomly oriented transversely-isotropic spheroidal inhomogeneities it yields the following formulas for effective electrical conductivity
⎡ re f f = r0 ⎣1 +
⎤−1 cη
( 1 − c )k 0 +
c k0 3
2A1 k 0 k111 −k0
+
A2 k0 k133 −k0
⎦
(3.11)
To compare the results with experimental observations we have to replace the volume fraction by weight fraction. It can be done using the data in Table 1:
1 1 = c cw
ρi ρ i + −1 ρm ρm
(3.12)
Where cw is the wt% of the inhomogeneities. Weight content cw = 0.005 corresponds to volume fraction c = 0.0026, and weight content cw = 0.01 corresponds to volume fraction c = 0.0053 Fig. 4 illustrates the predictions of the Mori-TanakaBenveniste scheme and Non-interaction approximation three schemes for epoxy filled with graphene particles. It is seen that all four lines coincide at the interval of the volume fractions of interest. The experimental data are in good agreement with the micromechanical predictions. To explain observed dependence of the resistivity on the external stress we assume that the microcracks are formed in the process of loading due to the high concentration of stresses at the surface of graphene particles. In this case, we have to consider two types of inhomogeneities – graphene particles (relevant quantities are marked by subscript “g” ) and cracks (relevant quantities are marked by subscript “c” ). For cracks, A1 = 1−1f , A2 = 21f and volume fraction must be replaced by 0
0
the crack density parameter ρ (see Kachanov & Sevostianov, 2005). This parameter depends on the stress level ρ = ρ (σ ). Then, in the framework of non-interaction approximation, the following estimate for the overall resistivity:
re f f =
r0 1 + cg ηg + (8/9 )ρ (σ )
(3.13)
Fig. 5 illustrates the comparison with the experimental data for the case of graphene content 1.0 wt% if the volume fraction of cracks is chosen as a function of stresses shown in the inset. The agreement is reasonably good. The crack density required to match the experimental data is huge. Note, however, that these explanation and comparison serve only as a hypothesis that has to be verified in the follow up research. Different dependence of the crack density on the stress will lead to a different result.
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Fig. 5. Comparison of the dependence of resistivity of graphene reinforced epoxy (0.5 wt%) on stress level calculated by (3.13) with the experimental data. The dependence of crack density on the stress level is shown in the inset.
4. Concluding remarks The addition of 0.5 wt% and 1.0 wt% of graphene particles to EPON 862 W appeared to have noticeable effect on the overall electric resistivity of the material and does not affect its tensile strength. The average stress when compared across the levels and types of conductive filler were not statistically different. The addition of 0.5 wt% and 1.0 wt% graphene demonstrated a linear relationship between elongation and resistance and an apparent higher order polynomial relationship between stress and resistance. The findings indicate that the inclusion of 0.5 wt% graphene can be used to monitor the stress and strain levels of a material by means of electrical resistance measurement. It is shown that the observed changes in electrical properties due to addition of conductive filler can be accounted for using standard homogenization techniques. The micromechanical models characterize the phenomena with sufficient accuracy and do not require any special techniques. We hypothesize that changes in electrical resistance with loading can be described via formation of microcracks. This method has the potential to be scaled in order to develop NDE techniques that could be used to monitor composite structures under load and evaluate stress and strain states. Acknowledgment Financial support from the NASA Cooperative Agreement NNX15AL51H to New Mexico State University are gratefully acknowledged. References Abry, J., Choi, Y., Chateauminois, A., Dalloz, B., Giraud, G., & Salvia, M. (1999). In-situ monitoring of damage in CFRP laminates by means of electrical resistance measurements. Composites Science and Technology, 59, 925–935. Abry, J., Choi, Y., Chateauminois, A., Dalloz, B., Giraud, G., & Salvia, M. (2001). In-situ monitoring of damage in CFRP laminates by means of AC and DC measurements. Composites Science and Technology, 61, 855–864. Bellucci, S., Coderoni, L., Micciulla, F., Rinaldi, G., & Sacco, I. (2011). The electrical properties of epoxy resin composites filled with cnts and carbon black. Journal of Nanoscience and Nanotechnology, 11, 9110–9117. Benveniste, Y. (1986). On the Mori-Tanaka’s method in cracked bodies. Mechanics Research Communications, 13, 193–201. Böger, L., Wichmann, M., Mayer, L., & Schulte, K. (2008). Load and health monitoring in glass fibre reinforced composites with an electrically conductive nanocomposite epoxy matrix. Composites Science and Technology, 68, 1886–1894. Ferreira, A. D. B. L., Nóvoa, P. R. O., & Marques, A. T. (2016). Multifunctional material systems: A state-of-the-art review. Composite Structures, 151, 3–35. Flandin, L., Cavaillé, J. Y., Bréchet, Y., & Dendievel, R. (1999). AC electrical properties as a sensor of the microstructural evolution in nanocomposite materials: Experiment and simulation. Modeling and Simulation in Materials Science, 7, 865–874. Gilat, A., Goldberg, R, K., & Roberts, G. D. (2005). Strain rate sensitivity of epoxy resin in tensile and shear loading. NASA/TM 2005-213595. Horii, H., & Nemat-Nasser, S. (1983). Overall moduli of solids with microcracks: Load-induced anisotropy. Journal of the Mechanics and Physics of Solids, 31, 155–171. Kachanov, M., & Sevostianov, I. (2005). On quantitative characterization of microstructures and effective properties. International Journal of Solids and Structures, 42, 309–336. Kang, S. D., & Snyder, G. J. (2016). Charge-transport model for conducting polymers. Nature Materials, 16, 252–258. Macías, E. G., D’Alessandro, A., Castro-Triguero, R., Pérez-Mira, D., & Ubertini, F. (2017). Micromechanics modeling of the uniaxial strain-sensing property of carbon nanotube cement-matrix composites for SHM applications. Composite Structures, 163, 195–215. Miller, S., Heimann, P., Barlow, J., & Allred, R. (2007). Physical properties of exfoliated graphite nanocomposites by variation of graphite surface functionality. In Proceedings of the 52nd international SAMPE symposium June 3-7. Mori, T., & Tanaka, K. (1973). Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, 21, 571–574. Muto, N., Arai, Y., Shin, S. G., Matsubara, H., Yanagida, H., Sugita, M., et al. (2001). Hybrid composites with self-diagnostic function for preventing fatal failure. Composite Science and Technology, 61, 875–883. Park, J., Okabe, T., Takeda, N., & Curtin, W. (2002). Electromechanical modeling of unidirectional cfrp composites under tensile loading condition. Composites: Part A, 33, 267–275.
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Schulte, K., & Baron, Ch. (1989). Load and failure analyses of CFRP laminates by means of electrical resistivity measurements. Composites Science and Technology, 36, 63–76. Sevostianov, I., & Giraud, A. (2013). Generalization of Maxwell homogenization scheme for elastic material containing inhomogeneities of diverse shape. International Journal of Engineering Science, 64, 23–36. Sevostianov, I., & Kachanov, M. (2002). Explicit cross-property correlations for anisotropic two-phase composite materials. Journal of the Mechanics and Physics of Solids, 50, 253–282. Wang, X., & Chung, D. (1997a). Real-time monitoring of fatigue damage and dynamic strain in carbon fiber polymer-matrix composite by electrical resistance measurement. Smart Materials and Structures, 6, 504–508. Wang, X., & Chung, D. (1997b). Self-monitoring of strain and damage by a carbon-carbon composite. Carbon, 35, 621–630. Wang, X., & Chung, D. (1998). Self-monitoring of fatigue damage and dynamic strain in carbon fiber polymer-matrix composite. Composites: Part B, 29, 63–73. Wang, X., Wang, S., & Chung, D. (1999). Sensing damage in carbon fiber and its polymer-matrix and carbon-matrix composites by electrical resistance measurement. Journal of Materials Science, 34, 2703–2713. Wang, S., Wang, D., & Chung, D. (2006). Method of sensing impact damage in carbon fiber polymer-matrix composite by electrical resistance measurement. Journal of Materials Science, 41, 1135–1147. Wentzel, D., & Sevostianov, I. (2013). Effect of fiber damage on the overall electrical conductivity of bare carbon fiber strand. International Journal of Fracture, 183, 275–282.
Contents lists available at ScienceDirect
International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci
Dependence of the electrical conductivity of graphene reinforced epoxy resin on the stress level Daniel Wentzel a,b, Sandi Miller c, Igor Sevostianov a,d,∗ a
Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88001, USA NASA Materials and Components Laboratories Office, JSC White Sands Test Facility, Las Cruces, NM 88004, USA c NASA Ceramic and Polymer Composites Division, Glenn Research Center, Cleveland, OH. 44135, USA d Center for Design, Manufacturing, and Materials, Skolkovo Institute of Science and Technology, Skolkovo, Russia b
a r t i c l e
i n f o
Article history: Received 30 April 2017 Revised 26 May 2017 Accepted 3 June 2017
Keywords: Graphene Epoxy matrix composite Electrical conductivity Elastic stress
a b s t r a c t This paper focuses on electrical conductivity of epoxy based nanocomposites containing graphene particles. We show that adding 0.5 wt% of nanoparticles decreases electrical resistivity of the material more than two orders of magnitude. This effect is modeled with good accuracy using ordinary micromechanical homogenization schemes. Combination of mechanical and electrical tests shows that electrical conductivity of the epoxy-graphene composite depends on the level of tensile stress in the material. To explain this effect we assumed formation of microcracks from the very beginning of the loading process due to very high stress concentration at the particles. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Carbon reinforced composites have been used for more than half a century in applications where high strength to weight ratio is crucial. Due to the high cost and susceptibility to damage, non-destructive evaluation (NDE) techniques that identify damage within a composite structure are highly valued. One issue with many NDE techniques is that they require the material to be conductive. While the carbon fiber reinforcement is conductive, the matrix material holding the reinforcement material together generally is not conductive making many NDE techniques inapplicable. To the best of our knowledge, the idea to estimate strength reduction of a carbon fiber reinforced composites through electrical resistivity measurements has been proposed by Schulte and Baron (1989). The idea was further implemented in works of Wang and Chung (1997a,b; 1998) Wang, et al. (1999, 2006) Abry et al. (1999, 2001), Park, Okabe, Takeda, and Curtin (2002), and applied to carbon fiber yarn by Wentzel and Sevostianov (2013). AC and DC electrical methods have been extensively studied since then and have been used to study a variety of damage mechanisms, e.g. delamination or matrix cracking, under static and dynamic loading conditions. Flandin, Cavaillé, Bréchet, and Dendievel (1999) first used nanoscopic conductive fillers with different aspect ratios in a thermoplastic matrix to monitor the applied macroscopic mechanical strain and the damage evolution during loading. Nanoscaled carbon black particles (Kupke et al., 2001) as well as microscaled carbon black particles (Muto et al., 2001) have also been used to modify the matrix of glass-fiber reinforced thermosets. It was shown that external stress as well as apparent mechanical damage can be detected in these multiphase composites via electrical conductivity methods.
∗
Corresponding author at: Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88001, USA. E-mail address: [email protected] (I. Sevostianov).
http://dx.doi.org/10.1016/j.ijengsci.2017.06.013 0020-7225/© 2017 Elsevier Ltd. All rights reserved.
64
D. Wentzel et al. / International Journal of Engineering Science 120 (2017) 63–70
Fig. 1. SEM images of epoxy reinforced with graphene particles.
Böger et al. (2008) developed glass fiber reinforced composites with epoxy matrix containing carbon nanotubes and showed that this material has reasonably high electrical conductivity. Bellucci et al. (2011) studied electrical conductivity of epoxy resin as matrix (Epon 828) containing 0.1%, 0.25% and 0.5% wt carbon black nanofillers or carbon nanotubes. They reported two orders of magnitude decrease of electrical resistivity of such composites as compare to the matrix material. They also showed that epoxy matrix containing carbon nanotubes yields higher values of electrical conductivity than the same matrix containing carbon black particles. Ferreira, Nóvoa, and Marques (2016) provided a review of the state-of-the-art in the area of conducting polymer matrix composites. Macías, D’ Alessandro, Castro-Triguero, Pérez-Mira, and Ubertini (2017) proposed a micromechanical model to explain the conductivity of the polymers containing conducting nanofibers through formation of conducting networks (percolation effect). They assumed that the matrix material is a perfect insulator and did not consider the charge transport across the matric material. However, polymers have low electrical conductivity but it is not zero. Recently, Kang and Snyder (2016) proposed a model for charge transport phenomena in polymers. In the present paper we focus on electrical conductivity of epoxy filled with graphene particles. We show that adding 1.0 wt% of this filler increase the conductivity of the epoxy more than two orders of magnitude. To explain this phenomena, we use ordinary micromechanical models accounting for the strongly oblate shape of the graphene particles. We also observed dependence of the electrical resistivity on the strain level. This phenomenon can be explained from the point of view of microcracks formation and partial debonding between the matrix and particles due to very high stress concentration at the particles tips. The quantitative explanation of the phenomena, however requires more detailed analysis. 2. Materials and methods For this study, a two-part epoxy and electrically conductive graphene particles were used. 0.5 wt% and 1.0 wt% conductive filler were used. Epoxy resin, Epon 862, and ‘W’ amine curing agent were procured from Hexion. Surface functionalized expanded graphite (Miller et al., 2007), TG679, was procured from Adherent Technologies, Albuquerque, NM. All materials were used as received. Resin plaques of baseline Epon 862/W were prepared by mixing 40 g of 862 epoxy resin with 10.56 g of W curing agent. Nanocomposites were prepared through dispersion of either 0.5% or 1.0 wt% of the expanded graphite into graphene layers throughout the base epoxy mixture. Nano-particle dispersion was initiated via a THINKY mixer profile which followed a 4 minute mix/ 4 minutes degas cycle. The mixture was then sonicated for 5 minutes. This process was twice repeated, followed by a cycle in the THINKY mixer, to ensure a final degas. The mixture was poured into a 3 × 8 Teflon mold and cured 125 °C for 2 hrs followed by a 177 °C hold for an additional 2 hr. A scanning electron microscope (SEM) was used to observe how well dispersed the conductive filler was. Dispersion of graphene appeared to be uniform. Fig. 1 illustrates the SEM images of epoxy reinforced with graphene particles. The size of the panels was 75 × 125 mm. From these panels, ASTM D638 Type IV dog bone specimens were machined. The length of the specimens deviated from ASTM D638 guidelines; however, all samples were machined to consistent dimensions. Fig. 2a shows the specimens. Resistance measurements were made using a 4-point method with kelvin clip leads. The leads were spaced 10 mm apart and attached to the center region of the gage section. A calibrated IET Labs Model 1693 RLC Digibridge was used to make the resistance measurements. The basic accuracy of the unit is reported as ± 0.02%. Prior to testing, the unit was tested against a known resistance standard to verify calibration and procedures were followed to zero the unit. This unit was controlled by a custom graphical user interface (GUI) which allowed resistance measurements to be recorded to an Excel spreadsheet. Resistance was measured at a sample rate of 5 Hz.
D. Wentzel et al. / International Journal of Engineering Science 120 (2017) 63–70
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Fig. 2. (a) Dog-bone specimens of pure epoxy, epoxy reinforced with 0.5 wt% of graphene and epoxy reinforced with 1 wt% of graphene (from left to right); (b) measurement of the electric resistance of a specimen in the process of loading.
Using an Instronۚ Corp. 5569 Series Electromechanical Test Instrument equipped with a 50 kN (11,200 lbf) capacity load cell, self-tightening 25 × 51 mm (1 × 2 in.) wedge action mechanical grips, and Bluehillۚ data acquisition software (version 1.8.289), carbon filled and unfilled epoxy dog bones were subjected to identical stress schedules consisting of a constant strain rate uniaxial tension test. The strain rate was set to 0.33 mm/min. This was selected to match approximate the slow strain rate used in a previous study by Gilat, Goldberg, and Roberts (2005) with the same material. Fig. 2b illustrates the procedure of measurement of the electric resistance of a specimen in the process of loading. 3. Results Resistance, stress, and strain were recorded for each specimen. Resistance measurements preformed on the neat EPON 862 dog bones, as expected, did not conduct well resulting in a resistance measurements on the order of. 5 × 107 . Using the geometry of the specimens we calculated the resistivity as 4.75 × 104 · m. The average stress at failure was 56.58 MPa and the average elongation a failure was 2.44 mm for all the specimens. All samples failed in the gauge section. Based on the ANOVA statistical analysis of the data there was no statistically significant difference in the stress at fracture when compared across the levels of conductive filler. The P-value was equal to 0.33, indicating the effect was not significant. The average stress at failure for 0.5 wt% and 1.0 wt% were 52.99 MPa and 47.94 MPa (54.49 MPa for valid gauge failures) respectively. The average elongation at failure was 2.68 mm (3.07 mm for valid gauge failures). All specimens with graphene inclusions failed in the gauge section. Adding 0.5 wt% of graphene particles reduced the resistivity down to 18 ± 6 · m. Increasing the content of graphene up to 1.0 wt% decreased the resistivity to 13 ± 5 · m. We note that a statistically significant effect has been observed between the levels of conductive filler when comparing the change in resistance as a function of stress/strain (P-value equal to 0.006) and resistivity (P-value equal to 0.003). The data was transformed using a square root function in order to more accurate fit the data; however, a linear fit was also statistically significant. We observed that changes in resistance of samples with graphene inclusions correlate to changes in stress and strain. Samples with 0.5 wt% graphene inclusions appeared to exhibit a linear relationship between elongation and resistance. The samples also appeared to exhibit a polynomial relationship between stress and resistance (Fig. 3). The standard deviation of the baseline resistance measurements for both samples was determined to be 0.004 K. Material exhibited a linear response when comparing sample elongation with resistance. Samples with 1.0 wt% graphene inclusions also appeared to exhibit linear relationship between elongation and resistance and quadratic relationship between stress and resistance. The standard deviation of the baseline resistance measurement of this sample was higher than samples with 0.5 wt% graphene. The standard deviation was determined to be 0.032 K.
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Fig. 3. Dependence of the resistivity of graphene reinforced epoxy on elongation and stress.
3.1. Micromechanical model To model the effective properties of heterogeneous materials, we use the concept of property contribution tensors that express contributions of a given inhomogeneity to the effective properties of interest, under the assumption that the inhomogeneity is placed into a uniform applied field, i.e. this field would have been uniform within the site of the inhomogeneity in its absence. For the elastic properties, these tensors are the compliance or stiffness contribution tensors, H or N; compliance contribution tensors have been first introduced in the context of ellipsoidal pores and cracks in isotropic material by Horii and Nemat-Nasser (1983). For the electrical or thermal conductivity problem, the conductivity or resistivity contribution tensors, K or R have been introduced by Sevostianov and Kachanov (2002) as follows. Assuming a linear relation between electric field E and the electric current density J, the change in E required to maintain the same electric current density if the inhomogeneity is introduced is:
E =
V1 R·J V
(3.1)
where the symmetric second-rank tensor R is called the resistivity contribution tensor of an inhomogeneity. Alternatively, the dual relation can be written as
J =
V1 K ·E V
(3.2)
where K is the conductivity contribution tensor of an inhomogeneity. In the case of the isotropic matrix of conductivity k0 (K = k0 I , R = r0 I , k0 = 1/r0 ),
K = −k20 R
(3.3)
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Table 1 Material constants of the constituents. Constituent
Epoxy EPON 862
Graphene
Density
ρ 1.21 g/cm3 Electrical resistivity r11 = r22 4.75 × 104 ·m r33 4.75 × 104 ·m
2.27 g/cm3 4.424 × 10−8 ·m 1.695 × 10−7 ·m
The R- and K-tensors properly reflect contributions of individual inhomogeneities to the effective resistivity and the effective conductivity and the sums
R(k ) or
K (k )
(3.4)
constitute the general proper microstructural parameters in whose terms the effective resistivity and the effective conductivity should be expressed (Kachanov & Sevostianov, 2005). In the case of a transversely-isotropic spheroidal inhomogeneity, a1 = a2 = a; a3 = γ a, embedded in an isotropic matrix (case of interest in the present work), explicit expression for resistivity and conductivity contribution tensors have the following form (see, for example, Sevostianov & Giraud, 2013):
R=−
1 1 K= [A1 (I − nn ) + A2 nn] k0 k20
(3.5)
where dimensionless factors
A1 =
1 − k111 /k0
,
A2 =
1 − 1 − k111 /k0 f0
1 − k133 /k0
k133 /k0 + 2 1 − k133 /k0 f0
(3.5)
and f0 (γ ) is defined by
⎧ √ ⎨ √1 arctan 1γ−γ 2 , 2 γ 2 (1 − g ) √ ; g ( γ ) = γ 1 −γ f0 = ⎩ √1 ln γ +√γ 2 −1 , 2 γ2 −1 2γ
γ
2 −1
γ−
γ
2 −1
For a thin (strongly oblate) spheroid (a1 = a2 = a,
π g( γ ) → , 2γ
f0 →
πγ
oblateshape (γ < 1 ) prolateshape (γ > 1 )
(3.6)
γ ≡ a/a3 >> 1) (3.7)
4
We now can calculate overall electric conductivity of epoxy/graphene composite and compare them with the experimental results. Material constants of the constituents are given in Table 1. Individual graphene particles are modeled as strongly oblate spheroids with the aspect ratio γ = 10−5 . If interaction between the inhomogeneities is neglected, each inhomogeneity can be assumed to be subjected to the same remotely applied field. Contributions of the inhomogeneities into the effective property of interests can be treated separately and the total gradient of temperature across the representative volume element (RVE) is
1 1 E= J+ k0 V
V (i ) R
·J
(3.8)
i
The summation over inhomogeneities may be replaced by the integration over orientations. In particular, in the case of isotropic orientation distribution of spheroidal inhomogeneities of aspect ratio γ
E=
1 1 [1 + c (2A1 + A2 )/3] J ≡ [1 + c η ] J k0 k0
(3.9)
where c is the volume fraction of graphite particles and parameters A1 and A2 are given by (3.5). Thus the (isotropic) effective conductivity is
re f f =
r0 1 + cη
(3.10)
Parameter η depends on shape of the inhomogeneities and conductivities of the two phases. When explicit expressions for non-interaction approximation are obtained, formulas for overall electric and conductive properties can be also evaluated using standard homogenization techniques, for example Mori-Tanaka-Benveniste (MTB) scheme, (Benveniste, 1986; Mori & Tanaka, 1973) where each inhomogeneity, treated as an isolated one, is placed into the unaltered matrix material and the interactions between different inhomogeneities are accounted for by assuming that the external field acting on them differs from the remotely applied one. This field is taken as the average over the matrix and
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Fig. 4. Comparison of the predictions provided by micromechanical schemes with experimentally measured electrical resistivities of graphene reinforced epoxy before loading.
is the same for all the inhomogeneities. In the case of randomly oriented transversely-isotropic spheroidal inhomogeneities it yields the following formulas for effective electrical conductivity
⎡ re f f = r0 ⎣1 +
⎤−1 cη
( 1 − c )k 0 +
c k0 3
2A1 k 0 k111 −k0
+
A2 k0 k133 −k0
⎦
(3.11)
To compare the results with experimental observations we have to replace the volume fraction by weight fraction. It can be done using the data in Table 1:
1 1 = c cw
ρi ρ i + −1 ρm ρm
(3.12)
Where cw is the wt% of the inhomogeneities. Weight content cw = 0.005 corresponds to volume fraction c = 0.0026, and weight content cw = 0.01 corresponds to volume fraction c = 0.0053 Fig. 4 illustrates the predictions of the Mori-TanakaBenveniste scheme and Non-interaction approximation three schemes for epoxy filled with graphene particles. It is seen that all four lines coincide at the interval of the volume fractions of interest. The experimental data are in good agreement with the micromechanical predictions. To explain observed dependence of the resistivity on the external stress we assume that the microcracks are formed in the process of loading due to the high concentration of stresses at the surface of graphene particles. In this case, we have to consider two types of inhomogeneities – graphene particles (relevant quantities are marked by subscript “g” ) and cracks (relevant quantities are marked by subscript “c” ). For cracks, A1 = 1−1f , A2 = 21f and volume fraction must be replaced by 0
0
the crack density parameter ρ (see Kachanov & Sevostianov, 2005). This parameter depends on the stress level ρ = ρ (σ ). Then, in the framework of non-interaction approximation, the following estimate for the overall resistivity:
re f f =
r0 1 + cg ηg + (8/9 )ρ (σ )
(3.13)
Fig. 5 illustrates the comparison with the experimental data for the case of graphene content 1.0 wt% if the volume fraction of cracks is chosen as a function of stresses shown in the inset. The agreement is reasonably good. The crack density required to match the experimental data is huge. Note, however, that these explanation and comparison serve only as a hypothesis that has to be verified in the follow up research. Different dependence of the crack density on the stress will lead to a different result.
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Fig. 5. Comparison of the dependence of resistivity of graphene reinforced epoxy (0.5 wt%) on stress level calculated by (3.13) with the experimental data. The dependence of crack density on the stress level is shown in the inset.
4. Concluding remarks The addition of 0.5 wt% and 1.0 wt% of graphene particles to EPON 862 W appeared to have noticeable effect on the overall electric resistivity of the material and does not affect its tensile strength. The average stress when compared across the levels and types of conductive filler were not statistically different. The addition of 0.5 wt% and 1.0 wt% graphene demonstrated a linear relationship between elongation and resistance and an apparent higher order polynomial relationship between stress and resistance. The findings indicate that the inclusion of 0.5 wt% graphene can be used to monitor the stress and strain levels of a material by means of electrical resistance measurement. It is shown that the observed changes in electrical properties due to addition of conductive filler can be accounted for using standard homogenization techniques. The micromechanical models characterize the phenomena with sufficient accuracy and do not require any special techniques. We hypothesize that changes in electrical resistance with loading can be described via formation of microcracks. This method has the potential to be scaled in order to develop NDE techniques that could be used to monitor composite structures under load and evaluate stress and strain states. Acknowledgment Financial support from the NASA Cooperative Agreement NNX15AL51H to New Mexico State University are gratefully acknowledged. References Abry, J., Choi, Y., Chateauminois, A., Dalloz, B., Giraud, G., & Salvia, M. (1999). In-situ monitoring of damage in CFRP laminates by means of electrical resistance measurements. Composites Science and Technology, 59, 925–935. Abry, J., Choi, Y., Chateauminois, A., Dalloz, B., Giraud, G., & Salvia, M. (2001). In-situ monitoring of damage in CFRP laminates by means of AC and DC measurements. Composites Science and Technology, 61, 855–864. Bellucci, S., Coderoni, L., Micciulla, F., Rinaldi, G., & Sacco, I. (2011). The electrical properties of epoxy resin composites filled with cnts and carbon black. Journal of Nanoscience and Nanotechnology, 11, 9110–9117. Benveniste, Y. (1986). On the Mori-Tanaka’s method in cracked bodies. Mechanics Research Communications, 13, 193–201. Böger, L., Wichmann, M., Mayer, L., & Schulte, K. (2008). Load and health monitoring in glass fibre reinforced composites with an electrically conductive nanocomposite epoxy matrix. Composites Science and Technology, 68, 1886–1894. Ferreira, A. D. B. L., Nóvoa, P. R. O., & Marques, A. T. (2016). Multifunctional material systems: A state-of-the-art review. Composite Structures, 151, 3–35. Flandin, L., Cavaillé, J. Y., Bréchet, Y., & Dendievel, R. (1999). AC electrical properties as a sensor of the microstructural evolution in nanocomposite materials: Experiment and simulation. Modeling and Simulation in Materials Science, 7, 865–874. Gilat, A., Goldberg, R, K., & Roberts, G. D. (2005). Strain rate sensitivity of epoxy resin in tensile and shear loading. NASA/TM 2005-213595. Horii, H., & Nemat-Nasser, S. (1983). Overall moduli of solids with microcracks: Load-induced anisotropy. Journal of the Mechanics and Physics of Solids, 31, 155–171. Kachanov, M., & Sevostianov, I. (2005). On quantitative characterization of microstructures and effective properties. International Journal of Solids and Structures, 42, 309–336. Kang, S. D., & Snyder, G. J. (2016). Charge-transport model for conducting polymers. Nature Materials, 16, 252–258. Macías, E. G., D’Alessandro, A., Castro-Triguero, R., Pérez-Mira, D., & Ubertini, F. (2017). Micromechanics modeling of the uniaxial strain-sensing property of carbon nanotube cement-matrix composites for SHM applications. Composite Structures, 163, 195–215. Miller, S., Heimann, P., Barlow, J., & Allred, R. (2007). Physical properties of exfoliated graphite nanocomposites by variation of graphite surface functionality. In Proceedings of the 52nd international SAMPE symposium June 3-7. Mori, T., & Tanaka, K. (1973). Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, 21, 571–574. Muto, N., Arai, Y., Shin, S. G., Matsubara, H., Yanagida, H., Sugita, M., et al. (2001). Hybrid composites with self-diagnostic function for preventing fatal failure. Composite Science and Technology, 61, 875–883. Park, J., Okabe, T., Takeda, N., & Curtin, W. (2002). Electromechanical modeling of unidirectional cfrp composites under tensile loading condition. Composites: Part A, 33, 267–275.
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