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Research Papers

Cohesive Zone Model for the Interface of Multiwalled Carbon Nanotubes and Copper: Molecular Dynamics Simulation OPEN ACCESS

[+] Author and Article Information
Ibrahim Awad

Department of Mechanical Engineering,
University of Connecticut,
191 Auditorium Road, Unit 3139,
Storrs, CT 06269
e-mail: ibrahim.awad@uconn.edu

Leila Ladani

Department of Mechanical Engineering,
University of Connecticut,
191 Auditorium Road, Unit 3139,
Storrs, CT 06269
e-mail: lladani@engr.uconn.edu

Manuscript received August 22, 2014; final manuscript received December 19, 2014; published online January 20, 2015. Assoc. Editor: Abraham Wang.

J. Nanotechnol. Eng. Med 5(3), 031007 (Aug 01, 2014) (7 pages) Paper No: NANO-14-1056; doi: 10.1115/1.4029462 History: Received August 22, 2014; Revised December 19, 2014; Online January 20, 2015

Due to their superior mechanical and electrical properties, multiwalled carbon nanotubes (MWCNTs) have the potential to be used in many nano-/micro-electronic applications, e.g., through silicon vias (TSVs), interconnects, transistors, etc. In particular, use of MWCNT bundles inside annular cylinders of copper (Cu) as TSV is proposed in this study. However, the significant difference in scale makes it difficult to evaluate the interfacial mechanical integrity. Cohesive zone models (CZM) are typically used at large scale to determine the mechanical adherence at the interface. However, at molecular level, no routine technique is available. Molecular dynamic (MD) simulations is used to determine the stresses that are required to separate MWCNTs from a copper slab and generate normal stress–displacement curves for CZM. Only van der Waals (vdW) interaction is considered for MWCNT/Cu interface. A displacement controlled loading was applied in a direction perpendicular to MWCNT's axis in different cases with different number of walls and at different temperatures and CZM is obtained for each case. Furthermore, their effect on the CZM key parameters (normal cohesive strength (σmax) and the corresponding displacement (δn) has been studied. By increasing the number of the walls of the MWCNT, σmax was found to nonlinearly decrease. Displacement at maximum stress, δn, showed a nonlinear decrease as well with increasing the number of walls. Temperature effect on the stress–displacement curves was studied. When temperature was increased beyond 1 K, no relationship was found between the maximum normal stress and temperature. Likewise, the displacement at maximum load did not show any dependency to temperature.

FIGURES IN THIS ARTICLE
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Nowadays, researchers endeavor to enhance some property or functionality of a wide range of applications from composite materials and biomedical applications to microelectronics and interconnect. Apparently, CNTs have shown many outstanding mechanical properties such as Young's modulus and tensile strength [1-3] since they were reported by Iijima [4]. Therefore, CNTs have captured the attention of many researchers to take advantage of their superiority and to employ them in some of the most demanding applications, e.g., nano-/micro-electronic applications. One of the most promising of those applications are interconnects [5-8]. Although researchers proposed replacing metals with CNTs in interconnects because of their superior properties, CNTs showed improved performance when they are mixed with other metallic materials [9]. CNT/Cu composites have a great potential to fill interconnects with [10,11]; however, issues with electrical and mechanical integrity of CNTs at the interface of existing materials have been prohibitive. Due to the very small scale (nano- and micro-scale), mechanical and electrical testing of a single CNT strand is overly challenging and not very accurate. At this scale, simulation can be used as guiding tool to provide insights on mechanical integrity and long-term reliability of these interconnects. The forces at the interface of CNTs and Cu are governed by molecular interactions. Therefore, regular finite element simulations at the continuum level are not capable of modeling this interface behavior. MD simulation is used, initially, to determine the interface behavior of CNTs and Cu material.

Previous studies conducted on interaction between copper and CNT suggest that the second-order interaction effect, the effect of second layers of CNTs on the interface strength, is negligible when multiple layers of CNTs are studied at the interface of copper [12]. Additional analysis indicated that pull out force is linearly related to the number of CNTs being pulled out. The same study proposes an analytical relationship for the interface shear strength. Realistic modeling on a hybrid structure as suggested needs a complete characterization of the interface. The process proposed here does not use adhesives to attach the CNTs and copper. Therefore, any forces developed at the interface are natural forces formed at molecular level between materials. Experimental characterization of the interface is very difficult and challenging. MD simulation is proposed here to characterize this interface.

These interface properties at molecular level can be related to mesoscale using available CZM.

Vast research has been done on the CZM of CNTs with different materials, mostly polymers. Jiang et al. [13], analytically, established the cohesive law for interfaces between a CNT and polymer that are characterized by the vdW force. For a CNT in an infinite polymer, the tensile cohesive stress depends only on the normal displacement and the shear cohesive stress vanishes. On the other hand, in the same study, for a CNT in a finite polymer matrix, the shear cohesive stress depends on both opening and sliding displacements, i.e., the tension/shear coupling and the tensile cohesive stress remains the same. The simple, analytical expressions of the cohesive law are useful to study the interaction between CNT and polymer, such as in CNT-reinforced composites. The effect of polymer surface roughness on the cohesive law is also studied and shear cohesive stress was found to vanish for a wavy polymer surface and flat graphene. Tan et al. [14] incorporated the nonlinear cohesive law derived from the weak vdW force for CNT/polymer interfaces in a micromechanics model in order to study CNT-reinforced composites. Although CNTs can improve the mechanical behavior of composite at the small strain, such improvement disappears at relatively large strain because the completely debonded CNTs may weaken the composite as they behave like voids in the matrix. They proposed that the increase of interface adhesion between CNTs and polymer matrix may significantly improve the composite behavior at the large strain. Samuel and Kapoor [15] used an inverse iterative finite element (FE) approach to estimate the CZM parameters, viz., interfacial strength and interfacial fracture energy. A microstructure-level three-dimensional (3D) FE model for nano-indentation simulation has been developed where the composite microstructure is modeled using three distinct phases, viz., the CNT, the polymer, and the interface. The unknown CZM parameters of the interface are then determined by minimizing the root mean square error between the simulated and the experimental nano-indentation load–displacement curves for a 2 wt.% CNT–polyvinyl alcohol composite sample at room temperature. Namilae and Chandra [16] developed a hierarchical multiscale methodology linking MD and FE method in order to study the atomic scale interface effects on composite behavior of CNT/fiber matrix. Pull-out simulation has been done by MD. Then, the results of the pull-out simulations are used to extract the parameters of CZM. These parameters are then used in FE in order to study the macroscopic mechanical response of the composite. This developed methodology is employed to study the effect of interface strength on stiffness of CNT-based composites. It is found that interfaces significantly alter the elastic response of CNT-based composites.

More attention was paid to investigate the interaction between polymers and CNTs than metals and CNTs. However, metal composites with CNTs have promising mechanical and electrical properties that make them potentially suitable for many micro-/nano-electronic applications, e.g., TSVs, interconnects, transistors, etc. Therefore, a study of the mechanical interactions between the CNTs and the metallic material typically used for interconnects such as Cu is of paramount importance. In this study, MD simulation is used to study the cohesive zone strength of MWCNT/Cu interface. The interface properties extracted from this study can be beneficial in multiscale modeling of the hybrid or composite material made using MWCNTs and Cu.

Large-scale atomic/molecular massively parallel simulator (lammps) code was employed to simulate the stress–displacement relations in a simple model that consists of MWCNT interfaced with a Cu slab. Figure 1(a) shows a cylindrical TSV made of vertically oriented MWCNTs embedded inside an annular cylinder of Cu. A bundle of embedded MWCNTs contains thousands of them (Fig. 1(b)), making it practically impossible to model the whole bundle and copper cylinder. In order to make the model computationally feasible, only a single strand of the CNT is modeled at the interface of the copper (Fig. 1(c)). Since the bundle consist of many CNTs that are periodically arranged at the interface of copper, it is reasonable to model only one strand and assume periodicity in x-directions, as shown in Fig. 1(b). Thus, just a single unit cell is modeled as shown in Fig. 2(c). It is also assumed that the CNTs are long enough and behave consistent in z-direction. Therefore, the same assumption is made in z-direction. Displacement controlled loading (shown by d arrow in Fig. 1(c)) has been applied to pull the MWCNT out of the Cu slab in a direction normal to the axes of the nanotubes. The resulted stress–displacement curves are obtained and related to available CZMs in literature.

The structure is built by atomic simulation environment [17] using python coding software. MWCNTs used in this study contain up to seven walls. Their chiralities are (5,5), (10,10), (15,15), (20,20), (25,25), (30,30), and (35,35), armchair CNTs, and a length of 27 Å have been modeled. Cu slab's thickness in y-direction is 15 Å and is chosen to be larger than C–Cu cut-off distance which is 7.7 Å (2.5σ which is the finite distance at which the interparticle potential is zero [18]). The length in x-direction ranges from 29 to 50 Å based on the outer diameter of the CNT strand. Cu slab is assumed to be FCC (100) in the direction of CNT axis. Embedded atom method potential [19] is used for interatomic potential between Cu atoms. Adaptive intermolecular reactive empirical bond order (AIREBO) [20,21] interatomic potential is used for carbon atoms as [22-24]. AIREBO model is a function to calculate the potential energy of covalent bonds, interatomic force, and the long-ranged interaction between atoms. For the interatomic potential between Cu and carbon atoms, only vdW potential occurs and that has been implemented in lammps by the use of Lennard–Jones (LJ) interatomic potential [25,26]. The LJ parameters used for C–Cu interactions are (σ = 3.088 Å and ε = 0.025 eV), where σ is the finite distance and ε is the depth of the potential well.1 For the intertube (CNT–CNT) potential, only vdW interaction is considered as in Refs. [27-29]. LJ parameters used for CNT–CNT interaction are (σ = 3.4 Å and ε = 0.00284 eV).

After building the structure of the atoms, the system was then equilibrated at the desired temperature 1 K for 50–250 ps, dependent on the simulation size, using Langevin thermostat accompanied with NVE ensemble to perform time integration and update the positions and velocities of the atoms, where N, the number of atoms, V, the volume of the system, and E, the energy are kept constant. Displacement was applied on carbon atoms using discrete displacements in a displacement controlled manner. As depicted in Fig. 1, the farthest column of atoms in y-direction of the MWCNT was subjected to discrete displacements of 0.1 Å until the MWCNT strand is completely pulled out. The system is equilibrated using NVE ensemble and Langevin thermostat for 10 ps after each loading step in order to avoid energy build-up and destabilization.

lammps code [30] was employed to generate the CZM for MWCNT and copper interface. Before discussing the MD simulation results, stress calculation in MD simulation is discussed first because it is a matter of controversy. Atomic level stress, aka virial stress, is developed from the virial theorem of Clausius [31] and Maxwell [32] by Swenson [33] and Tsai [34] and it is given as follows: Display Formula

(1)σij=1VαV[12β=1N(riβ-riα)fjαβ-mαviαvjα]

where r indicates the position and subscripts iandj are the direction indices and take the values x,y,andz, α and β are atoms in the domain, V is the volume of the domain, N number of atoms neighboring atom α, fjαβ is the component of the force applied on atom α by atom β, mα is the mass of atom α and viαandvjα are the ith and jth velocity components of atom α, respectively. The above expression contains two parts: potential energy and kinetic energy.

The controversy was about whether virial stress tensor is equivalent to Cauchy's mechanical stress tensor or not. Costanzo et al. [35,36] derived Cauchy stress from Lagrangian-based MD method. They have found their derived Cauchy stress does not coincide with virial stress because Cauchy stress has no explicit dependence on velocity terms while virial stress does. Knowing that the involved instantaneous velocities in the above expression of the virial stress are due to thermal fluctuation, virial and Cauchy stresses only coincide at absolute zero temperature because the velocity term vanishes in the virial stress expression. Costanzo et al. also discussed the assumptions under which both stresses can be said to have the same meaning. They are only equivalent if they are time averaged for all times.

Zhou [37] and Shen and Atluri [38] have another opinion about the relationship between virial and Cauchy stress. They derived their own stress definition and proved through examples that virial stress is equivalent to Cauchy stress when the velocity term in virial stress is ignored. Sun et al. [39], Tschopp [40], and Yang et al. [41] and others have used Zhou's interpretation to justify using only the potential part of the virial stress. This point of view contradicts with the opinion of Subramaniyan and Sun [42]. In that study, they have proven through some examples that the kinetic energy term that contains velocities should not be ignored.

In the same time, Costanzo and coworkers [35], Subramaniyan and Sun [42], and Gao and Weiner [43] also mentioned that the kinetic part in the virial stress in unnecessary when Lagrangian frame of reference is considered and velocity terms vanish. Lagrangian frame is implied when atoms do not cross the periodic boundaries of the simulation box.

In the current work, simulation box is large enough that atoms do not cross the periodic boundary conditions. Consequently, considering the later point of view, Lagrangian frame of reference is implied and kinetic part vanishes from Eq. (1). Thus, the virial stress will be given as Display Formula

(2)σij=1VαV[12β=1N(riβ-riα)fjαβ]

which agrees with the point of view of Zhou and the others.

First, the effect of the number of MWCNT's walls is explored. Figure 2(a) depicts the simulation models for 2–7-wall MWCNTs right after equilibration at 1 K. Figure 2(b) shows the different stages during the simulation: before equilibration, after equilibration, during loading, and after separation. Cu atoms are fixed in all directions during simulation. In order to calculate stress, carbon atoms influenced by C/Cu vdW interaction must be determined first. Carbon atoms that lie within C/Cu cut-off distance (distance within which C atoms are influenced by vdW interaction [18]), which is approximately 7.7 Å, are believed to be the ones influenced by the vdW interaction. That is why only the first and second wall of the MWCNT at the interface with Cu are studied. The average stress in the direction of loading, y-direction, is calculated over the atoms enclosed in the red box indicated in Fig. 2(b). Stress per atom is calculated using Eq. (2) for each atom inside the box then the average is computed in order to give the average stress per atom. The average displacement of the same group of atoms in that region is calculated in the same manner as stress.

Figure 3 shows the resulted normal stress–displacement curves. The common behavior of the simulated MWCNTs is that the stress (σ) starts initially at some negative value, compressive stress due to equilibration, and increases until it becomes zero at a certain intermolecular distance (b0.2Å) then it increases to a maximum value at displacement (δn). After this point, stress drops rapidly with increasing displacement until separation takes place. As shown in Fig. 3, dropping down behavior changes and depends on the number of walls. By tracking C/Cu vdW potential energy, separation always takes place at δ5.8Å, marked point in Fig. 3; however, this in not the point at which normal stress stabilizes when the number of walls goes beyond three walls. Even after MWCNT strand is separated and stabilized, some compressive stress remains in it due to intertubular interaction and it is increasing with increasing number of walls.

Figure 4 shows the five-wall MWCNT at different instants: (1) before separation, (2) at the instant of separation, and (3) after separation. After the second zero-stress is attained, the inner layers of MWCNT strand are squeezed due to pulling of the outer layer. Thus, as the strand is squeezed, inner walls push the outer walls farther toward Cu causing a gradual increase in the compressive stress of the studied group of atoms until maximum compressive stress occurs at point 1. Then, compressive stress starts to decrease as C/Cu vdW interaction starts to deteriorate until the interatomic potential reaches to zero eV at point 2. Between points 1 and 2, the MWCNT tries to retrieve its cylindrical configuration and relieve the gained compressive stress until it stabilize at point 3. Even though the MWCNT is stable, there is some compressive stress residue due to the effect of C/C vdW interaction during loading.

As noticed from Fig. 3, dropping down behavior, maximum normal stress, and the accompanied displacement are changing with changing the number of walls. There are many CZM models proposed by scientists and Barenblatt model [44] is found to best fit the results in this study except for the compressive stress region at the end of the pull-out. Maximum normal stress, normal cohesive strength (σmax), and the displacement at this point (δn) are the key parameters in the study of CZM. Thus, the effect of number of walls on those parameters is studied. From Fig. 5, σmax decreases nonlinearly, polynomial fit of degree two with a correlation coefficient of 0.97, with the increase of the number of walls from two- to seven-wall MWCNTs. Our hypothesis is that as the number of walls increases, C/C vdW interaction increases between the walls resulting in pulling the outermost walls that are in the interface with Cu toward the centers of the MWCNT. Thus, the C/Cu interface becomes weaker.

The relationship between δn and number of Walls is shown in Fig. 6. As the number of walls increases from two to seven, δn decreases nonlinearly, logarithmically with a correlation coefficient of 0.9. From Figs. 3, 5, and 6, as the number of walls increases, interfacial stiffness remains constant but σmax and δn decrease, which means that the CZM gets weaker.

Temperature effect was also studied. Three-wall MWCNT was simulated at different temperatures (from 1 K to 398 K). First, the system of MWCNT-Cu was equilibrated at the desired temperature and then load is applied in the same manner. Resulted normal stress–displacement curves (CZM models) for the temperatures are shown in Fig. 7. As depicted from the curves, there is an abrupt change in the CZM when temperature changes from 1 K to 75 K.

Figures 8 and 9 summarize the relationship between temperature and both σmax and δn, respectively. To evaluate the results systematically, a statistical analysis was conducted using ibm spss software [45]. An inverse functional relationship [σmax=c1+(c2/Temperature)andδn=c3+(c4/Temperature)] was found to best fit the relationship between temperature and both σmax and δn after conducting F-tests. The coefficients of determination (r2), which indicates how much the model explains the variability of the response data (σmax or δn) around their mean, were evaluated for both sets of data. The closer this coefficient is to unity, the stronger evidence it is that there is a relationship between the temperature and each parameter. They have been found to be 0.35 and 0.68 for σmax and δn, respectively. These values are small, indicating that there is no true relationship between temperature and both σmax and δn. These small values also indicate that the variation on the results is most likely caused by the randomness in the test. As the temperature increases, the kinetic energy increases and atoms oscillates with higher frequencies introducing some noise to the extracted results. Nonetheless, there seems to be a jump in the values as the temperature is increased from 1 K to 75 K. For the temperatures higher than 1 K, relative standard deviation was calculated for σmax and δn. It shows 4% and 6% for σmax and δn, respectively. Thus, temperature has no significant effect when temperature is higher than 1 K.

MD simulation using lammps software is employed in order to evaluate the CZM parameters and the effect of the number of walls and temperature. Virial stress with eliminating the kinetic part (from MD simulation) is used to represent the Cauchy stress. The CZM obtained from two-, three-, four-, five-, six-, and seven-wall MWCNT was found to be similar to Barenblatt model. The key parameter for the model were generated and reported here. The number of walls has a great effect on the CZM. As the number of wall increases, the interface gets weaker due to the intertubular effect of the inner walls. Inner walls attract the outer walls toward the center of the MWCNT strand and away from the interface with Cu. Thus, less load will be required to separate the MWCNT strand from Cu. Nonetheless, after separation, outer walls still have some compressive stress residue due to inner walls interaction and this residual compressive stress increases with increasing the number of walls.

The normal cohesive strength decreases nonlinearly with increasing the number of walls. Furthermore, the displacement at the maximum stress also decreased nonlinearly when the number of walls increased.

Effect of temperature on CZM parameters shows an abrupt change in the CZM when temperature changes from 1 K to 75 K. However, statistical analysis shows no true relationship between temperature and either σmax or δn. The variation in the results is believed to be greatly affected by the noise induced by atoms' oscillations. Furthermore, when relative standard deviation was calculated for temperatures above 1 K, temperature was found to have no significant effect on CZM parameters.

This paper was based on work supported by the National Science Foundation under CMMI Grant Nos. 1415165, 1242141, 1416682, and 0927319. The authors greatly appreciate the support from NSF. The authors also thank University of Connecticut for the use of their computational facilities.

 

 Nomenclature
  • b =

    intermolecular distance when stress is first zero GPa

  • fjαβ =

    component of the force applied on atom α by atom β

  • i,j,andk =

    direction indices and take the values x,y,andz

  • mα =

    mass of atom α

  • N =

    number of neighbors of atom α

  • r2 =

    coefficient of determination

  • riα =

    position of atom α along in i direction

  • riβ =

    position of atom β along in i

  • V =

    volume of the domain

  • viα =

    ith velocity components of atom α

  • vjα =

    jth velocity components of atom α

  • α =

    one type of the atom in the domain

  • β =

    other atom in the domain

  • δ =

    displacement

  • δn =

    displacement at normal cohesive strength

  • σ =

    normal cohesive stress

  • σij =

    virial stress

  • σmax =

    normal cohesive strength

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Barenblatt, G., 1959, “The Formation of Equilibrium Cracks During Brittle Fracture. General Ideas and Hypotheses. Axially-Symmetric Cracks,” J. Appl. Math. Mech., 23(3), pp. 622–636. [CrossRef]
IBM, spss.
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IBM, spss.

Figures

Grahic Jump Location
Fig. 1

Schematic of the simulation: (a) cylindrical TSV; (b) top view of the interface; and (c) a sample of the MDs structure used in our study

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Fig. 2

(a) Simulation of MWCNT and (b) simulation stages of four-wall MWCNT

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Fig. 3

Number of walls effect

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Fig. 4

Squeezing of five-wall MWCNT

Grahic Jump Location
Fig. 5

Normal cohesive strength—number of walls curve

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Fig. 6

δn-number of walls

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Fig. 7

Temperature effect on CZM

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Fig. 8

Temperature effect on σmax

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Fig. 9

Temperature effect on δn

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