Discrete Element Modeling of Graphene Using PFC 2D

Transcription

1 Discrete Element Modeling of Graphene Using PFC 2D Mohammad Jamal Khattak 1 and Ahmed Khattab 2 1 Infrastructure and Transportation Materials Laboratory, Department of Civil Engineering, University of Louisiana at Lafayette, USA. Khattak@louisiana.edu 2 Laboratory of Composite Materials, Department of Industrial Technology, University of Louisiana at Lafayette, USA. Khattab@louisiana.edu Abstract Discrete element modeling (DEM) technique is currently used for large-scale simulations of soil and rock mechanics. However, in this study, the DEM has been successfully applied to graphene sheet. Carbon atoms were clustered to form a single discrete element, which was then allowed to interact with other similar discrete elements through contact bonds. The parameters for discrete elements and associated contact bonds were derived from previously established atomiclevel fundamental models for stiffness, covalent bond forces and van der Waals bond forces. Even though the developed DEM model is greatly simplified, the elastic behavior of the graphene sheet was well captured through the use of DEM. The predicted tensile and compressive strength values were comparable with other reported studies. It is believe that such developed model showed the high potential of DEM to be utilized for the simulations of nanomaterials such as carbon nanotubes and fibers. Keywords: discrete element modeling; grapheme; carbon nanotubes; elastic modulus; PFC 2D Introduction The discrete element modeling (DEM) is a numerical technique which has been successfully applied for the analysis of rocks and soil mechanics, and other solid materials with bonded contact models [1-4]. In this technique, a finite-difference scheme and Newton s second 1

2 law of motion are employed to study the interaction among discrete particles in contact [3]. Newton s second law is used to determine the translational and rotational motion of each particle arising from the contact forces, applied forces and body forces acting upon it, while the force displacement law is used to update the contact forces arising from the relative motion at each contact. The dynamic behavior is represented numerically by a time-stepping algorithm in which the velocities and accelerations are assumed to be constant within each time step [3, 4]. Amongst various discrete elements codes [5-11] the particle flow code (PFC) 2D/3D has higher computation efficiency and the ability to model fracture behavior and element interaction, as well as the interface conditions (adhesion) between the various phases of composite materials [3]. PFC2D has also been used to developed discrete element models to predict the stiffness, strength and stressstrain response of the asphalt concrete, Portland cement concrete and fiber-reinforced polymer composite materials [12-15]. Even though DEM is commonly used for large-scale simulations of soil and rock mechanics, in this study, it is shown that the DEM can be successfully applied to graphene sheet. In this study, carbon atoms are clustered to form a single discrete element, which is then allowed to 45 interact with other similar discrete elements through contact bonds using PFC 2D code. The parameters for discrete elements and associated contact bonds were derived from previously established atomic-level fundamental models for stiffness, covalent bond forces and van der Waals bond forces. Although the presented DEM model is greatly simplified, not including the acoustic vibrations and other atomic scale effects, the elastic behavior of the graphene sheet is well captured through the use of DEM. It is believed that the developed DEM model can be enhanced and utilized for the simulations of carbon nanotubes (CNT) and carbon nanofibers (CNF) using PFC3D code. 2

3 Discrete Element Model for Graphene Sheet In PFC 2D DEM, each individual element is a rigid body characterized by a mass (m) uniformly distributed in a disk-shaped element of radius (R) and moment of inertia (I). Although the elements are rigid, they can interpenetrate, indicating interaction between the elements. The contact surface of two elements defines the contact plane, which is perpendicular to the axis connecting the two centers of the elements. The mechanical behavior of each rigid element is described by the laws of classical mechanics, including the total force (F) and moments (M) acting on the element that arise due to the interactions with the elements in contact, as well as artificially introduced dissipative forces [3, 16]. 2.1 Contact Stiffness Model Simple contact constitutive models with complex geometrical features are combined in the DEM approach to simulate the complex behavior of a material. Three of the simple contact models, which are Shear and normal stiffness, static and sliding friction, and inter-particle cohesion/adhesion, can be utilized. The elastic relationship between the contact force and relative displacement between particles can be provided by the stiffness model [3, 12, 13]. Two particles A and B in contact, are shown in Figure 1, where k A n and k B n are the normal stiffnesses and k A s and k B s are the shear stiffnesses. The contact stiffness, for a linear contact model, is determined based on the assumption that the stiffness of the two contacting entities acts in series. In a contact-stiffness model, the composite stiffness and force-displacement law of the two particles in contact can be expressed as follows [3]: 74 3

4 75 K n A kn A n B n B kn k = (1) k + 76 K s A ks A s B s B ks k = (2) k + 77 F = nk U (3) n n n 78 Δ F = K ΔU (4) s s s Figure 1. Schematics of Contact Model [3, 12, 13] The relationship between the total normal force (Fn) to the normal displacement (Un) is shown in Equation 3. Equation 4 relates the incremental shear force (DFs) to the incremental shear displacement (Ds). The normal and shear cohesive strength between two contacting balls can be simulated using simple contact bond models, which are applied at the contact point (Figures 1). When the tensile and/or shear stress at a contact exceeds the tensile and shear strengths of the 4

5 bond, the bond breaks and separation and/or frictional sliding can occur. The friction force Fr is given by Fr=mFn, where m is coefficient of friction between the contacting bodies. The elastic constants (k A,B n, k A,B s ) can be related to Young s modulus (E) and Poison s ratio (n) of each constituent of a material. In the normal direction and for column-row array, the following equations can be used [3]. 93 k A B n n 2 = k = Et (5) 94 k s A = k s B = 2Gt (6) 95 E= 2 G(1 + v) (7) 96 Where t is the thickness of a discrete element (disk) and G is the shear modulus of a material Contact Model for Covalent Bonding The individual CNF structure consists of strong covalent carbon-carbon (c-c) bonds. These bonds are formed by sharing of the valence electrons according to the laws of quantum mechanics. However, it is proposed to retain from fundamental atomic scale and low-frequency acoustic vibrations [16]. It has been reported by various studies [17-21] that the nano-mechanics of carbon nanotubes (CNT) can be well represented by linear elastic behavior. Under large stresses these nano-materials fail through plasticity or brittle fracture [18]. Since the elastic and continuum behavior of materials can be well captured by DEM, it is believed that the existing marco-scale scheme of DEM is well suited to study the nano-mechanical behavior of graphene, CNT, and CNF using the current discrete element (DE) contact models [16]

6 Constitutive behavior of interaction between two discrete elements (DE) can be modeled using the parallel-bond. These bonds can be envisioned as a finite-sized disk of elastic massless material with a thickness of xr at the contact and centered on the axis connecting the centers of two DE as shown in Figure 2. In PFC 2D, the parallel bond is also characterized by a radius multiplier (x), a maximum value of 1 indicates that the parallel bond (cementatious effect) extends to the mean diameter of the two contact elements. A set of ideal springs are associated with this DE with normal (k pn ) and shear (k ps ) stiffness uniformly distributed over its cross sectional area. Such bonds establish an elastic interaction between elements that acts in parallel with the slip or contactbond models described earlier [3]. 118 Figure 2. Parallel bond illustration 119 The parallel bonds k pn and k ps can be found using the following equations [3]: 120 k pn E a k n = = and T 2T k ps Ga ks = = (8) T 2T Where, T is the bond width, E a and G a are atomistically determined Young s and shear modulus, and k n and k s are atomistically computed normal and shear force elastic constants, respectively

7 The total force and moment act on the two bonded particles can be related to maximum normal (s m ) and shear (t m ) stresses acting within the bonded material at the bond periphery. The maximum tensile and shear stresses acting on the parallel bond periphery are calculated using the beam theory as follows: 128 σ m = F n A + M s I R m < σ c σ!"# = -!! +!!!! R! < σ! (9) 129 F = τ!"# =!! < τ!! (10) A s τ m < τ c Where, R m is the mean radius of contacting particles, A and I are the area and moment of inertia of the parallel bond cross- section, respectively. M s denote the shear-directed moment, and s c and t c are normal and shear contact forces, respectively. If either of these maximum stresses exceeds its corresponding bond strength (s c and t c ), the parallel bond breaks and it is removed from the model along with the corresponding force, moment and stifnesses. The constitutive behavior relating the bond force and displacement for two particles in contact is shown in Figure 3. At any given time, either the bond model or the slip model is active. When the corresponding component of the force exceeds either of these values, the bond breaks. Additionally, moment may also be acting, as well as the force shown in Figure 3. In PFC 2D code, the normal (f n ) and shear (f s ) bond forces for a unit DE thickness (t) are calculated as : φ n = σ c (2R)t (11) 142 φ s = τ c (2R)t (12) The normal and shear force elastic constants and bond forces for covalent bond interactions can be determined from Morse potential and AMBER force field models [24, 25, 26, 27]. 7

8 145 Tension F n F s Bond breaks Contact bond φ n φ n F r Bond breaks Contact bond Slip model Slip model U n K n 1 a) Normal component of contact force. b) Shear component of contact force. Figure 3. Constitutive behavior for contact bonds 1 K s U s Figure 4 shows DE for a single sheet graphene molecule and its interaction with other similar molecule constructed for conducting DEM simulation for the study. Here six carbon atoms (Figure 4b) are lumped in one DE for a graphene sheet. Higher number of atoms can also be lumped to make a DE, which can save computational time. It should be noted that DE radius (R) is selected as such that the van der Waals forces can also be captured while interacting with other elements. It can be seen from Figure 4 that R of DE is 1.5 times the covalent bond radius (r c ) and the parallel bond width for covalent bonding (T c ) becomes 3r c

9 R r c (b) (a) T c R R: Descrete element radius r c : Covalent bond radius d c : Covalent bond distance T c : Parallel bond width d c Figure 4. a) Single sheet graphene, b) Graphene molecule comprised of six atoms clustered to form DE, c) DE interaction using parallel-bonds formed due to Covalent bonding. 2.3 Contact Model for van der Waals Bonds The interaction of DE due to van der Waals forces can also be simulated using the parallel bonds by utilizing the contact force associated with van der Waals interaction, as shown in Figure T v = d v R (a) d v (b) R Figure 5. a) Graphene atoms clustered in DE showing van der Waals bond distance (d v ), b) Two DE showing parallel bond width representing van der Waals bonding (T v ). 9

10 The figure shows two DE from two adjacent graphene sheets interacting with each other through van der waals force, having a distance of d v. Unlike parallel bond representing covalent bonding, the parallel bond for van der Waals bonding will have the width (T v ) equal to the van der Waals bond distance (d v ). Such interaction can be determined by simple analytical form of Lennard-Jones (LJ) 6-12 potential, which can facilitate determination of normal and shear force elastic constants and bond forces needed for DEM simulations [22]: Since minimum of two graphene sheets are required to model the van der Waals interaction, it becomes impossible to incorporate both the covalent and van der Waals interactions in 2D DEM modeling of graphene sheets. Hence, in this study, the 2D DEM simulation for covalent bond is presented for a signgle graphene sheet. It should be noted that the above approach could be utilized to conduct DEM simulations in 3D code. For PFC 3D DEM code, the DE (discrete element) becomes like a sphere interacting with other DE spheres within a single graphene sheet through covalent bond, as well as with the adjacent graphene sheet through van der Waals bond Model Parameters and Simulation The model parameters for parallel bonds are k pn, k ps (normal and shear stiffnesses), and s c, t c (normal and sheer strengths). In this study, the DEM model parameters are based on the Morse force model for covalent bond [24, 25, 26]. The k n and k s used are 32.6 nn/ o A and 6.5 nn/ o A [24]. Based on DE schematic shown in Figure 4 and covalent bond length of 1.42 o A [28], the DE radius (R) and T c of parallel bond were calculated as 2.13 o A and 4.26 o A, respectively. Graphene sheet consists of carbon atoms bonded with both single and double c-c covalent bonds having bond energies of 473kJ/mole and 618kJ/mole, respectively [28, 29, 30]. The PFC 2D code does not allow associating different parallel bond values for a DE, therefore average bond 10

11 value of 545kJ/mole was used in this study. Using the average bond energy the force required to disassociate a single carbon atom of bond length 1.42 o A was calculated as 45 N. Hence, the parallel bond normal strength for a unit DE thickness (1-m) was determined as 105 GPa using equation (8). The parallel bond shear strength was taken as half of the normal strength. Since six parallel bonds originated from each DE (Figure 4), it was logical to use hexagonal packing of DE to develop synthetic specimen for tensile or compression test simulation in PFC 2D code (Figure 6a). The current DEM virtual test was limited to 2-D analysis techniques and involved the simulation of uniaxial rectangular specimens. The simulated specimens contained up to 6,000 disk-shaped DE. In a 2-D plane, the model is composed of about of 36,000 atoms as shown in Figure 6b. Higher number of atoms clumped in a DE element can also be generated to increase the speed of simulation. Tensile loading was applied to the top and bottom loading plate of the synthetic specimen in PFC 2D. After a certain continuous incremental tensile loading, the response of each DE was monitored (Figure 6b). The results of simulation generated stress and strain data which was plotted as shown in Figure 5c. The stress strain curve shown in Figure 6c exhibits an elastic response and brittle mode of failure for a graphene sheet due to tensile loading. The initial slope of the stress strain curve was used to determine the Young s modulus (E) of the graphene sheet. The stress at failure was obtained as the strength (S) of the graphene sheet. The 215 Poisson s ratio (n) was calculated as the ratio of horizontal to vertical strain and G was determined using the equation 7. It was found that a reasonable values of E, G and S of 1.5 TPa, TPa and 95 GPa, respectively were obtained using the develop DEM model (Figure 6c). These values were in agreement with elastic properties reported in literature for Graphene, see Table 1 [23-24, 30-33]

12 Table 1 Comparison of elastic properties from other studies Model Type E(Tpa) ν G(Tpa) S(Gpa) Brased-truss- (Theoretical) [24] Brased-truss Finite Element (FE) [24] Stretching-hinging [24] Stretching-hinging- (Shear Beam) [24] All Deformation mechanism [24] Potential Energy Minimization FE [24] FE [31] Atomistic simulation [33] Experimental- Nanoindentation-atomic force microscope [32] D-DEM [This study] It was also observed that by creating a random flaw in terms of weak parallel bond or no bond in few DE generated reasonable picture of moments, tensile and shear stresses developed at the vicinity of flaw as shown in Figure 5a. Similarly the test simulation for compressive loading predicted a Young s modulus and compressive strength of 1.45 TPa and 98 GPa, respectively. Figure 7 shows a clear failure in shear due to excessive compressive loading. The aforementioned DEM approach provided a reasonable physical portrayal of the force chains developed in the graphene sheet, which is known to be a critical aspect of mechanical modeling. Such simulations can predict the stress-strain response, modulus/stiffness and potential crack initiation zones. The test simulation for van der Waals bonds could not be performed in PFC 2D as it involves the overlap of at least two graphene sheets, which make the model threedimensional (3-D). Hence, it is recommended to utilize the approach outlined in this study to 12

13 develop a 3-D model of graphene sheets for van der Waals bonds. Such a model will further the understanding to develop 3-D DEM models of CNT and CNF using PFC 3D codes (6a) (6b) 13

14 Tensile Stress, Gpa Strain, mm/mm (6c) Figure 6. a) DEM tensile test simulation of graphene sheet, (b) effect of random flaw in terms of weak bond showing high tensile and shear stresses in the flawed region Figure 7. DEM compressive test simulation of graphene sheet 14

15 Conclusions and Recommendations Discrete element model was presented for armed chair graphene sheet. The DEM parameters were characterized based on atomic-scale fundamental stiffness and bond models such as covalent and van der Waals bonds. A synthetic homogenous model of graphene in PFC 2D was constructed based on elastic constitution models. Uniaxial virtual tensile and compressive test simulations were conducted. Based on the simulation results the following conclusions and recommendation were drawn. 1. The elastic constitutive behavior of graphene sheet was well captured by the developed model using PFC 2D. The contact force chains between particles exhibited a reasonable distribution of moments, shear and normal contact forces. 2. The elastic modulus and tensile strength of graphene obtained from the simulation were comparable to other reported studies. 3. It is recommended to utilize the approach outlined in this study to develop a 3-D model of graphene sheets for van der Waals bonds. It is believed such a model will further the understanding of developing 3-D DEM models for CNT and CNF using PFC 3D code Acknowledgments The authors wish to express their sincere thanks to the University of Louisiana at Lafayette for using their facility and financial support

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