Simulation of Nanoscale Peeling and Adhesion of Single-Walled Carbon Nanotube on Graphite Surface

We have performed molecular mechanics study of nanoscale peeling and adhesion processes of carbon nanotube (CNT) on the rigid graphite surface. First, as a model of CNT, single-walled carbon nanotube (SW-CNT) of the (3,3) armchair type with a length of l = 99.3 Å comprised of 480 carbon atoms is used. In the simulation CNT physically adsorbed on the graphite substrate is peeled (retracted) from the surface and then adsorbed (approached) onto the surface. We have first obtained the vertical force-distance curve with the characteristic hysteresis loop derived from the bistable states between the lineand point-contacts during the peeling and adhesion processes. The analysis of the vertical and lateral force curves reveals that the CNT shows multiscale mechanics – both nanoscale mechanics on the order of CNT’s length (≃ 100 Å) and atomic-scale mechanics on the order of CNT’s diameter (≃ several Å). The deflection of CNT along z direction for some regions can be well explained by theory of elasticity. Next the effect of the CNT length l on the peeling process is studied. As the CNT becomes shorter, discrete jump of the force curve vanishes and the peeling force curve exhibits continuous behavior because the shorter CNT has larger spring constant kz along the vertical direction. The length l dependence of kz in the present simulation exhibits kz ∝ l−2.98, which is in good agreement with theory of elasticity, kz ∝ l−3. Lastly the effect of the chirarity of the CNT on the peeling and adhesion processes is studied for the armchair, zigzag and chiral type CNTs for the length of about 50 Å. The hysteresis of the peeling curve shows the slight difference of the adhesive behavior among different chirality of CNTs. [DOI: 10.1380/ejssnt.2008.72]


I. INTRODUCTION
Adhesion and peeling phenomena appear in all of the surfaces and interfaces of both inorganic and biological materials.Recently nanoscale peeling process has been studied by extending biological polymer chain such as proteins using atomic-force microscopy to clarify mechanical mechanism of unfolding of the polymer chain.Such kind of experiments have attracted many attentions worldwide as a new method of spectroscopy of structural analysis of biological macromolecules [1].
On the other hand, the mechanical, electronic and optical properties of the carbon nanotubes (CNT's) [2] have also recently attracted many attentions as research subjects of not only basic science but also industrial application.Especially, importance of the CNT as a machinery part to build up machines and objects in nano-and micrometer scale, have been rapidly increased.Many challenging studies aiming for application, such as atomic force microscopy (AFM) tip [3], nano-pincette [4], and rotational actuators in microelectromechanical systems [5], have been successfully performed.In such mechanical systems the mechanical properties of CNT plays quite important roles.
Therefore we have focused attention on CNT as a model of inorganic polymer chain, and have performed simulation of peeling of single-walled carbon nanotube (SW-CNT) with a length of 40.3 Å [6].Although preliminary results of peeling process have been obtained, general feature of nanoscale peeling has not been clarified yet.In this work, more general case, both peeling and adhesion processes of SW-CNT with a longer length, l = 99.3Å, physically adsorped on the graphite surface is studied by molecular mechanics simulation in Section III-A.First the characteristic hysteresis appears in the force curve during the nanoscale peeling and adhesion processes.Next it is clarified that the nanoscale line-and point-contacts formed between CNT and graphite surface can be clearly defined by discrete jumps of the force curve.The elastic behavior of CNT as a nanospring with a length of about 100 Å clearly appears in the vertical force curve.In Section III-B, the deflection of CNT along z direction for some regions is compared with that evaluated by theory of elasticity.In Section III-C, it is revealed that the atomic-scale mechanics of the free edge of CNT with a diameter of 4.2 Å, appears in the lateral force curve.Thus the CNT shows both nanoscale mechanics on the order of 100 Å of its length and atomic-scale mechanics on the order of several several Å of its diameter.Thus the CNT exhibits unique multiscale dynamics due to its large aspect ratio, length/diameter.In Section III-D, the effect of the CNT length on the peeling process is discussed.Since the shorter CNT has larger spring constant k z along the vertical direction, the discrete jump of the force curve vanishes and the peeling force curve exhibits continuous behavior.The length l dependence of k z is compared with that obtained by theory of elasticity.In Section III-E, the chirarity dependence of the CNT on the peeling and adhesion processes is discussed.The hysteresis of the peeling curve for each chirality shows the slight difference of the adhesive feature among armchair, zigzag and chiral CNTs.

II. MODEL AND METHOD OF SIMULATION
The model used in the simulation is as follows.As an example of model of the CNT used in Sections III-A -III-D, a single-walled carbon nanotube (SW-CNT) of the (3,3) armchair type with a length of l = 99.3Å and a diameter of 4.2 Å, comprised of 480 carbon atoms, is adopted [Fig.1].This SW-CNT with an open edge is constructed by repeated structures of α and β rings comprised of six carbon atoms as illustrated in dotted inset of Fig. 1.In Section III-D, various armchair type CNTs for 6.3 Å ≤ l ≤ 99.3 Å are used.In Section III-E, the armchair, zigzag and chiral types CNTs, comprised of 240, 240 and 242 carbon atoms, corresponding to the length of l = 49.2,50.0 and 47.8 Å, respectively, are adopted.As a model of the substrate surface, the rigid rectangular graphene with each side of 54 Å × 164 Å, comprised of 3536 carbon atoms, is used [Fig.1].First covalent bonding structures of both the CNT and graphite sheet are separately optimized by minimizing the total energy described by the Tersoff potential [7], using the Polak-Rebiere-type conjugate gradient (CG) method [8].Here the convergence criterion is set so that the maximum of absolute value of all the forces acting on the movable atoms, is lower than 10 −4 eV/ Å, i.e., max 1≤i≤N |F i | ≤ 10 −4 eV/ Å, where N is the total number of movable atoms, and F i is a force acting on the i-th movable atom.
Next, for the case of armchair CNT, the CNT is located on the rigid graphene [Fig.1], so that the AB stacking registry between the bottom part of the CNT and the graphite lattice is conserved.For the zigzag and chiral CNTs, one of the several metastable adsorption position is chosen as an initial position, which will be mentioned later in Section III-E.Then the optimized structure of the CNT physically adsorbed on the graphene is obtained by minimizing the total energy V total = V cov + V vdW , using the CG method.Here, as the covalent bonding interaction potential of the CNT, V cov , and as the nonbonding vdW interaction potential between the CNT and the graphene, V vdW , the Tersoff [7] and Lennard-Jones (LJ) [9] potentials, are used, respectively.
The simulation under the condition of the quasi-static process of moving velocity v → 0 for T = 0 K, is considered.First the red-colored ring on the left edge of CNT [Fig.1] which is assumed to be attached to the AFM tip apex is gradually moved upward along the z direction, parallel to [0001] axis, by 0.1 Å until the displacement ∆z = 60 Å.After that, the left edge is gradually moved downward toward the graphite surface again until the displacement ∆z = 0 Å [Fig.1].Here, for each left-edge position, ∆z, the total energy V total is minimized using the CG method.Thus the optimized positions of the movable carbon atoms of the CNT, (x, y, z), the vertical force F z , and the lateral forces F x and F y , acting on the fixed left edge, are obtained.

A. Peeling and adhesion process of armchair CNT
The series of the typical change of the CNT shape during peeling and adhesion processes within x − z plane are illustrated in Figs.2A-2I, corresponding to Figs. 3A-3I, the vertical force acting on the moving left edge, F z , as a function of ∆z, the displacement from the initial position along z-direction.
At first the CNT takes an initial structure parallel to the substrate surface [Fig.2A: ∆z = 0 Å], which means formation of a line contact between the CNT and the graphite surface.Here vertical force F z is zero [Fig.3(a)A].Just after the beginning of the peeling Then, the CNT is approached to the graphite surface again.Unlike during the peeling (retract) process, the point contact does not appear during the adhesion (approach) process, although the CNT slightly bends toward the graphite surface due to the attractive interaction [Fig.2H: ∆z = 13.0Å] When the CNT is approached to the surface within a certain height, the CNT is suddenly adsorbed onto the surface forming line contact [Fig.2I: ∆z = 12.9 Å], which makes the 3rd discontinuous jump in the F z curve [Figs.3H→3I].Thus the hysteresis loop (C→D→E→F→H→I) with adhesive energy of 6.84 eV appears in Fig. 3 due to the bistable states between the line-and point-contacts.Thus the elastic bending feature of the CNT as a nanospring on the nanoscale order of the CNT's length, appears in the vertical force curve.

B. Comparison with theory of elasticity
Here theory of elasticity is used to explain the deflection of CNT during approaching process [Fig.3(a)G→F→H].If the CNT is regarded as a cantilever of hollow circular cylindar receiving uniformly distributed load f z = F rigid z /l as illustrated in inset of Fig. 3(d), the deflection v(x) of the center axis of the CNT cantilever is described as, where F rigid z is the interaction force between the CNT and graphite when the CNT is assumed as rigid.E and I are the Young modulus and the geometric moment of inertia, respectively.Now, in order to calculate CNT deflection for ∆z = 13.0Å [Fig. 3 1).The thickness of the graphite sheet t is assumed as 3.4 Å.Then the deflection curve v(x) is calculated using E as a fitting parameter as shown in Fig. 3(d).
Fig. 3(d) clearly shows our present molecular mechanical simulation is in good agreement with theory of elasticity.Here E = 0.51 TPa is obtained, which is on the order of magunitude nearly the same as those obtained in the previous various first-principles and molecular dynamics simulations.Although the definition of E, I, and l, and the assumption of uniformly distributed load f z become ambiguous at nanoscale, theory of elasticity can explain the deflection of nanoscale CNT spring to a certain extent.However, since there are several metastable buckling structures between reverse-S character and the shape opening downwards, elastic stability problem must be solved to explain all the peeling and adsorption processes, considering boundary condition of the free edge carefully, which is our future work.

C. Atomic-scale mechanics along lateral direction
On the other hand, atomic-scale mechanics of the CNT on the several Å order of the CNT's diameter, also appears in the lateral force (F y ) curve during the peeling (retract) process [Fig.4(a)].At first the F y curve shows corrugation with a peak to peak amplitude of A PP ≃ 0.005 eV/ Å during the line contact [Fig.4(a)0].However, A PP rapidly decreases to 0.002 eV/ Å just after the point contact formation [Fig.4(a)1].Then, as the peeling is proceeded, both A PP and the period of corrugation decrease [Fig.4(a) [1][2][3][4][5][6][7][8].This is because, as the peeling is proceeded, the interaction force between the CNT edge and the graphite surface becomes smaller with an increase of an angle between CNT and graphite as shown in Figs.3(b)D→3(b)E.Fig. 4(b) shows the corresponding zigzag trajectories of atoms on the free edge of the CNT, conserving AB stacking registry with the graphite lattice.As a comparison, the periodic zigzag trajectories for the lateral scanning process are shown in Fig. 4(c), which clearly exhibit the line contact corresponding to the 0 region in Fig. 4(b).Since F y is on the order of magnitude of several pN, the thermal noise prevents F y from being observed under room temperature.However, friction force microscopy measurement using a cantilever with a high sensitivity on the order of pN and/or the ultra low-temperature measurement will enable us to observe the above atomic-scale information of CNT mechanics.

D. Effect of length l on peeling process
It can be expected that the CNT's length plays an important role in the nanoscale elasticity of the CNT spring.Here we found significant length dependence of armchairtype CNT on the peeling force curve for the length l ≤ 99.3 Å.As shown in Fig. 5(a), peeling force curves for l = 99.3Å and 79.5 Å, have two discrete jumps corresponding to the buckling (transition from the reverse-S character to the shape opening downwards) and the complete peeling.However, for the shorter CNT, the number of the discrete jumps decreases.As shown in Fig. 5(b), the force curves for 24.0 Å ≤ l ≤ 59.3 Å exhibit only single discrete jump corresponding to the complete peeling.As the CNT becomes further shorter than l = 18.9 Å, the force curves have no discrete jumps any longer and show continuous behavior as shown in Fig. 5(c).This significant change of the peeling force curve depending on the CNT length occurs because the z component of the effective spring constant of CNT, k z , increases as the CNT length, l, decreases.Thus, for l ≤ 18.9 Å[Fig.5(c)], little elastic information of the CNT is included in the peeling force curve.On the other hand, for l ≥ 24.0 Å[Figs.5(a In order to clarify the length l dependence of k z , the effective k z is approximately calculated by using where F peeling z (∆z peeling ) and ∆z peeling denote the force acting on the moving edge and the position of the moving edge just before the complete peeling, respectively.Fig. 6 shows k z as a function of l.Here the only cases of  l ≥ 24.0 Å, where the peeling can be well defined, are considered.Fig. 6 exhibits k z ∝ l −2.98 , which is nearly the same as k z ∝ l −3 obtained by theory of elasticity.Thus it is quantitatively clarified that the shorter CNT has a larger spring constant k z .

E. Effect of chirality on peeling and adhesion process
The chirality dependence of the peeling force curve is investigated.Figs.7(a-1)-7(c-1) show force curves of the (3,3) armchair, (5,0) zigzag and (4,2) chiral types CNT during the peeling and adhesion processes.Here the armchair, zigzag and chiral types CNTs are comprised of 240, 240 and 242 carbon atoms, corresponding to approximately the length of l = 49.2,50.0 and 47.8 Å, respectively.The shape of the hysteresis of the force curve clearly becomes different from that of Fig. 3 Although the basic shape of the force curves seems to have little chirality dependence, the adhesive energies corresponding to the area of hysteresis loops are slightly different from each other, which exhibits the difference of the adhesive feature among different chiralities.The armchair-and chiral types CNT have the largest and the smallest adhesive energies ∆E = 1.77 and 1.55 eV, respectively.One of the origins of this difference is the lattice orientation between the bottom part of the CNT and the graphite lattice.Figs.7(a-2)-7(c-2) show the bottom part of the CNT (red-colored lines) and the graphite substrate surface (blue-colored lines).Here the carbon atoms of the CNT located at the distance of less than 4.0 Å above the graphite substrate surface along z-direction are indicated as the bottom part of the CNT.The AB-stacking registry for the case of armchair type is the most commensurate and energetically stable among the registries for three chirality types.Therefore AB stacking registry induces stronger adhesive energy.The other origin of the difference of the peeling feature is the difference of the free edge structure.As shown in Figs.3(b) and 4(b), it can be expected that the atomic structure of the free edge of the CNT during the point contact gives significant influences on the peeling force and peeling position ∆z.
Systematic studies of the effects of the lattice orientation and the free edge structure on the adhesion can lead to the possibility of the classification of the chirarity of CNT using information of the peeling process.Detailed analysis of further studies will be discussed elsewhere.

IV. CONCLUSION
In this work molecular mechanics study of nanoscale peeling and adhesion processes of the CNT has been performed.We have first obtained the vertical force-distance curve with the characteristic hysteresis loop with adhesive energy of 6.84 eV, due to the bistable states between line-and point-contacts.It is clarified that the vertical and lateral force curves show the nanoscale and atomic-scale mechanics of the CNT, respectively.Thus http://www.sssj.org/ejssnt(J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) e-Journal of Surface Science and Nanotechnology the CNT exhibits unique multiscale dynamics on the order of nanoscale CNT's length and atomic-scale CNT's diameter, derived from the large aspect ratio, length/diameter.The deflection of CNT along z direction for some regions can be well explained by theory of elasticity.The CNT length dependence of peeling process is discussed.It is clarified that the effective change of the CNT stiffness induces multiple discrete jumps in the peeling curves.The length l dependence of k z in the present simulation exhibits k z ∝ l −2.98 (24.0 Å ≤ l ≤ 99.3 Å), which is in good agreement with theory of elasticity, k z ∝ l −3 .The difference of adhesive features among different chiralities for the length of about 50 Å is also discussed and the adhesive energy for each chirality is evaluated.Our present simulated results will be compared with experimental results using atomic force miroscopy with CNT tip in the near future.

FIG. 1 :
FIG. 1: The exmaple of model of a single-walled carbon nanotube (SW-CNT) physically adsorbed onto the rigid rectangular graphite sheet used in the simulation discussed in Sections III-A -III-D.The SW-CNT consists of α and β rings comprised of six carbon atoms as shown in the dotted inset.The left edge is moved upward and downward along the z direction, parallel to [0001] axis, by 0.1 Å.

FIG. 2 :
FIG. 2: Typical change of the shape of the armchair CNT for l = 99.3Å during the peeling process (A→B→C→D→E→F→G) and adhesion process (G→H→I) within x − z plane.The red-colored CNT and blue-colored graphite sheet are shown.The displacement of the moving edge from the initial position, ∆z[ Å], is indicated on the upperright positions of each picture.

FIG. 3 :
FIG. 3: (a) The vertical force, Fz, acting on the moving edge, plotted as a function of the displacement ∆z.The positions A-I correspond to those of Fig. 2. (b) Atomic structure of the free edge of the CNT within x − z plane is enlarged.(c) The vertical van der Waals interaction force Fz acting on the atoms of the two arrays on the free edge indicated in the insets in (b), during the peeling process.(d) Comparison of the deflection v(x) between the present molecular mechanical simulation and theory of elasticity.The case of the deflection for ∆z = 13.0Å [Figs.2H and 3(a)H] is considered.

FIG. 4 :( 4 ,
FIG. 4: (a) The lateral force, Fy, acting on the moving edge, plotted as a function of the displacement ∆z during the peeling (retract) process (A→C→D→E→F→G: ∆z = 0.0 Å -60.0 Å).(b) The trajectories of the free edge of the CNT within x − y plane during the peeling process before the CNT is completely peeled from the graphite surface (A→C→D→E: ∆z = 0.0 Å -56.6 Å).The peaks 0-8 correspond to those of (a).(c) The trajectories of the free edge of the CNT within x − y plane during the lateral scanning process compared with (b).
) and 5(b)], as the CNT becomes longer, more elastic information of the CNT is included in the force curve, which induces the complicated behavior with several discrete jumps in the peeling force curve as shown in Figs.5(a) and 5(b).

FIG. 6 :
FIG. 6: Spring constant of CNT along z direction, kz = F peeling z /∆z peeling , as a function of the CNT length l for l ≥ 24.0 Å. Fitting line shows kz has an inverse power law dependence of l −2.98 , which is nearly equivalent to that of the cantilever, l −3 based on theory of elasticity.

FIG. 7 :
FIG. 7: The force curves for the case of (a-1) armchair, (b-1) zigzag and (c-1) chiral CNTs for the length l = 49.2,50.0 and 47.8 Å, respectively, during the peeling and adsorption processes.The orientation of the bottom part of the CNT (red-colored lines) and the graphite substrate surface (blue-colored lines) for (a-2) armchair, (b-2) zigzag and (c-2) chiral CNTs.Here the carbon atoms located at the distance of less than 4.0 Å above the graphite substrate along [0001] direction are indicated as the bottom part of the CNT.The green-colored carbon atoms indicate the left moving edge.