Simulating the buckling deflection of carbon nanotube-made detectors used in medical detections by applying a continuum mechanics model

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1 ie Science Journal 3;() Simulating the buckling delection o carbon nanotube-made detectors used in medical detections by applying a continuum mechanics model Alireza Vahdati *, Mehdi Vahdati,, R. A. Mahdavinejad Mechanical Engineering Department, Naein Branch, Islamic Azad University, Naein, Iran Department o Mechanical engineering, Faculty o Engineering, University o Tehran, Tehran, Iran orresponding author: Vahdati@naeiniau.ac.ir Abstract: arbon nanotubes are increasingly used in abrication o nano-detectors and other nano devices. Herein, the buckling o a carbon nano-tube made detector is simulated. In order to obtain nonlinear constitutive equation o the detectors we assume the distributed electrostatic and Van der Waals attractions along the NT length. By considering the nano orces in a continuum mechanics model we were able to achieve the dierential equation o the NT. In the next step by applying Adomian series solution, we provided an analytical closed-orm solution o NTs. The delection and the buckling parameters are determined and discussed in detail. The analytical obtained results are compared with those o literature as well as numerical solution. The obtained results revealed that the presented continuum based model is in good agreement with experimental result. Moreover it is ound that the analytical solution can be considered as a reliable approach to study the buckling stability o detectors in nanoscales where the presence o van der Waals orce is important. [Vahdati A, Vahdati M, Mahdavinejad R A. Simulating the buckling delection o carbon nanotube-made detectors used in medical detections by applying a continuum mechanics model. ie Sci J 3;():86-9] (ISSN:97-835).. 7 Key words: arbon nanotube (NT), Buckling, ontinuum model, Nano-detector, Electrostatic, Modiied Adomian method (MAD).. Introduction Ater discovering carbons nano-tubes (NTs), these materials are increasingly used in manuacturing small-scale structures. In recent decades these materials are specially or constructing nano-detectors, nanotweezers, nanoelectromechanical switches, etc. (Paradise et al., 7; Baughman et al., 999; Ke et al., 5). It has been demonstrated that the elastic modulus, bending lexibility and tensile strength o carbon nano-tubes are much higher than the conventional metallic materials (Esawi and Farag, 7). These materials have a great potential or medical applications in manuacturing medical detectors, biological sensors etc. Assume a typical cantilever NT detector suspended near electrode surace with a small gap in between. By applying a voltage dierence between the nano-components causes the NT to delect and be attracted toward the electrode surace due to the presence o electrostatic orces. Once this voltage exceeds a critical value, an increase in the electrostatic orce becomes greater than the corresponding increase in the restoring orce, resulting in the unstable collapsing o the NT to the ground position. This behavior is known as the buckling instability and the critical voltage is called the buckling voltage. Predicting this voltage is very important or engineers. While there are several orces such as casimir, capillary and van der Waals orce that are acting in sub-micro distances. As the gap decreases rom micro to nano-scale, the van der Waals interaction occurs. The prediction o the molecular orceinduced instability o NTs nano-detector is a critical subject in design nano-detector: A nanodetector might adhere to its substrate with an applied voltage less than buckling voltage or even without an applied voltage as a result o molecular orce, i the minimum gap between the nanodetector and substrate is not considered (in and Zhao, 5;Abadyan et al., ;Koochi et al., ; Tsai and Tu, ; Tserpes, 7; Desquenes et al., ; Batra and Sears, 7; in and Zhao, 5; Hayt and Buck, ). Thereore predicting the eect o nano-scale orces on perormance o the nano-detector is very important issue or design reliable detectors. There are several approaches or investigate the nano-world. In order to simulate the nanomaterials, several theoretical techniques might be employed by researchers. The most amous molecular dynamics (MD) and molecular mechanics (MM) simulations could be used to study the mechanical behavior o carbon-based nano-materials (Tsai and Tu, ; Tserpes, 7; Desquenes et al., ; Batra and Sears,7). However these methods are very time-consuming and might not be easily used in complex structures. 86 liesciencej@gmail.com

2 ie Science Journal 3;() Although continuum models are more time-saving than MM and MD, their approach oten leads to nonlinear equations that might not be worked out by analytical methods, accurately (Desquenes et al., ; in and Zhao, 5). Thereore analytical approaches are used to solve the constitutive equations o the nano-system. Due to the importance o the instability o NT nano-detectors this work is dedicated to simulating the instability o the nano-detectors. In this paper, the buckling instability o cantilever NT detector has been studied. Modiied Adomian decomposition (MAD) is employed to solve the nonlinear governing equation o the system. The obtained results are veriied by comparing with those rom literature as well as numerical solution. Results will be useul or design the nanodetectors.. Theoretical Model.. Electrostatic interaction et us consider a reestanding multiwalled NT above a ground plane consisted o graphene layers, with interlayer distance d = 3.35 Å, as illustrated in Fig.. When a conductive nanotube is placed over an electrode substrate in the presence o an applied potential dierence between the tube and the electrode, the electrostatic charge would be induced both on the tube and the substrate. To calculate the electrical orces acting on the tube, a capacitance model may be used. For ininitely long metallic cylinders, the capacitance per unit length is given by (Hayt and Buck, ): ( q) () D arccosh( ) Rw Where D is the initial distance between the tube and ground plate, = Nm the permittivity o vacuum so the electrostatic orce per unit length is given by: d ( ( D ) V ) elec d ( D ) () V D D ( D Rw ) arccosh ( ) Rw Where R w the radius o NT and V the applied voltage. By applying external voltage the nanotube delected to ground and the distance between the nano-tube and ground plate reduce to D-U thereore the electrostatic orce per unit length o delected detector can be rewrite as: is V elec (3) D U ( D U )arccosh ( ) RW It s must be noted in this equation we assumed that: D Rw D () Equation (3) can be simpliied by using the ollowing assumption: D U ( D U )arccosh ( ) R D U D U D U ln[( ) ( ) ] RW R W D U D U ln [ ] RW Thereore V elec D U ( D U )ln ( ) R W W (5) (6).. van der Waals interactions ennard-jones potential is a suitable model to describe van der Waals interaction (ennard-jones, 93). It deines the potential between atoms i and j by 6 ij 6 r r (7) ij ij whre r ij is the distance between atoms i and j while 6 and are the attractive and repulsive constants, respectively. For distances higher than 3. Å, such as in this paper, the repulsive term decays extremely ast and can be neglected (Tserpes, 7). For the carbon-carbon interaction, 6 =5. evå 6 (Girialco et al., ). A reliable continuum model has been established to compute the van der Waals energy by doublevolume integral o ennard-jones potential (Ke and Espinosa, 6), that is 6 EvdW ( ) 6 υ n n dυ dυ (8) υ r ( υ, υ ) where and represent the two domains o integration, and n and n are the densities o atoms in these domains, respectively. 87 liesciencej@gmail.com

3 ie Science Journal 3;() The distance between any two points on and is r(, ). Eq. (8) provides acceptable results or explaining the NT-graphene attraction compared to that o direct pair wise summation through molecular dynamics in Eq. (7). For a (5,) single walled carbon nanotube (SWNT) over a graphene surace and or distances larger than 5 Å, the dierence between E vdw speciied by Eq. (8) and molecular dynamics, is less than % (Tserpes, 7). et us consider a reestanding multiwalled NT above a ground plane consisted o graphene layers, with interlayer distance d = 3.35 Å, as illustrated in Fig.. The length o NT is and the initial gap between NT and the ground is D. The boundary condition o the NT is deined as cantilever at one end (with no displacement and rotation) and traction ree at the ree end (with no shear orce and moment). Using Eq. (8), the energy per unit length o nano-tube is simpliied to (Tserpes, 7): Figure. Schematic representation o antilever NT nano-detector Ro D( N) d E vdw 6 σ π RRi rd R( Rr)[3R ( Rr) ] 3.5 [( Rr) R ] (9) where R i and R o are the inner and outer radii o NT, N is the number o graphene sheets and 38nm - is the graphene surace density. Once the van der Waals energy is computed, the corresponding energy terms are employed to derive the component o the intermolecular orce per unit length, vdw, along r-direction as below: vdw EvdW d( ) dr Ro D( N ) d 3 3 R (8r 3r R 7r R 8rR 35 R ) RR ( ) i rd r r R () In most applications it is practically assumed that the diameter o tubes is much smaller than the distance between nanotube and the graphene surace, i.e. (R)<<D. According to this assumption, Eq. () is simpliied to D( N ) d vdw ( r) 6 NW RW 5 rd r () where N W is the number o walls o nanotube and R W is the mean value o their radii. For large number o layers, i.e. D + (N-)d D, substitution o the summation with an integral results: D ( N ) d D ( N ) d dr 5 D 5 r D r d r [ ]. () d D ( D ( N ) d ) dd astly we have: ( D ) N R d D (3) vdw 6 W W.3. Governing equations In order to develop the governing equation o the beams, the constitutive material o the nanotube is assumed to be linear elastic, and only the static delection o the nano-tube is considered. The minimum energy principle was applied, which implies equilibrium when the ree energy reaches a minimum value. By applied the Hamilton principle the governing equilibrium equation can be determined as: W W W W d U d U ( Ee I ) elec U vdw U dx dx dx elas elec vdw 3 d U du d U Ee I E I U dx dx dx e 3 d U e elec vdw ( E I ) UdX dx () Where denotes the variation symbol, X is the position along the nano-tube measured rom the clamped end, U is the beam delection, E e, is the eective Young s modulus o NT which is 88 liesciencej@gmail.com

4 ie Science Journal 3;() typically.9-. TPa (Gupta and Batra, 8) and I is the cross-sectional moment o inertia, equal to (R o - R i )/. By integrating Eq.(), the governing equation o cantilever nano-tube detector is derived as: d U EI elec vdw dx (5.a) With the B.. o: U () du () dx (Geometrical B.. at ixed end), (5.b) and 3 d U d U ( ) ( ) 3 dx dx (Natural B.. at ree end). (5.c) Eqs. (5a-c) can be made dimensionless using the ollowing substitutions, U u, D X x, D, R 6 NW de e ID W V Ee ID These transormations yield, d u dx ( u ( x )) ( u )ln [ ( u )] (6) (7-a) With the B.. : du u () () (7-b) dx (Geometrical B.. at ixed end), and 3 d u d u ( ) ( ) (7-c) 3 dx dx (Natural B.. at ree end). 3. Solution In this section two solving methods has been applied or solving the governing equation. First is MAD and the second is Nemerical solution: 3. Adomian series solution The detail o the MAD can be ind in (Adomian, 983). The analytical MAD solution o equation (7) can be obtained as the ollowing: u( x) x x ( ) x! 3!! ln ( ) 3 ( ) 3 x 6! ln ( ) ln ( ) ( ) x 7! 3 ln ( ) ln ( ) {( )( ) 3 8! ln ( ) ln ( ) ln ( ) [ ( )] } x ln ( ) ln( ) ln ( ) 3 3 [ ( )] x 9! ln ( ) ln( ) ln ( ) (8) Where the constants and can be determined by solving the resulting algebraic equation rom the B. at x= i.e. using equation (7-c). For any given, β and, equation (3) can be used to obtain the buckling parameters o the nano-tube detector. The instability in equation (8) occurs when dβ(x=)/du. The buckling voltage o the system can be determined via plotting the u vs. β. 3. Numerical Solution In order to veriy the analytical results, the cantilever beam-type nano-detector is numerically simulated and the results are compared with those obtained via MAD and PM. The nonlinear governing dierential equation (Eq. (7)) is solved with the boundary value problem solver o MAPE commercial sotware. The step size o the parameter variation is chosen based on the sensitivity o the parameter to the tip delection. The buckling parameters o the system can be determined via the slope o the u-β graphs.. Results and Discussion.. Veriication First we veriy the solution with experiments. In order to veriy the obtained results, the buckling voltage o a typical cantilever NT base nano-detector with the ollowing parameters was compared to experimental data in Table.. The length o the nanotube, = 6:8 m; initial gap between nanotube and electrode, D = 3 m; R w =5nm; E = TPa. As seen the MAD results are in good agreement with experimental results... Simulation o delection 9 89 liesciencej@gmail.com

5 ie Science Journal 3;() u. =.6 u PI.5. Numerical MAD x u...3 (a) x 8(Pull-in) = 7.(Pull-in) (b) Figure. Delection o the cantilever NT or dierent values o when =5 and =. (a) analytical, (b) Numerical u PI Numerical MAD 6 8 Figure 3. Eect o van der Waals orce () on pullin delection (=). 3 5 Figure. Eect o van der Waals orce () on pullin delection (=5) Ater validating the solution by experiments, we simulate the delection o the nano-detector. Figure shows the centerline delection o a typical nano-tube detector under intermolecular orce and external voltage obtained using MAD, and numerical methods. This igure u tip increases rom zero to u tip *, when increases rom zero to *. This igure reveals that the NT nano-detector has initial delection without applying voltage dierence. This is the result o the presence o vdw attraction..3 Eect o van der Waals orce on delection o NT The buckling delection is an important parameter or design the detectors. The relation between buckling delection and van der Waals orce () is presented i igure 3 and or dierent values o radios to initial gap ratio (= and =5) as seen increasing the intermolecular orce the buckling voltage decrease. When no voltage applied (β=) the NT buckling i the van der Waals orce exceeds rom its critical values ( * ) the critical values o can ind rom the horizontal axes o igure 3 and. By comparing Figure 3 and reveal that by increasing the values o, the critical values o increase as well as the buckling voltage. Table. Buckling voltage obtained rom dierent method Method Experimental (Ke et al.,5 ) MAD Numerical Buckling Voltage liesciencej@gmail.com

6 ie Science Journal 3;() 5. onclusions The buckling behavior o a cantilever NT nano-detector has been studied. The obtained results show that: The Adomian series solution is a very power ull method or study the buckling behavior o NT based nano-detectors. In the absence o electrical loading on NT based detector, it can buckling to ground i the van der Waals orce exceeds rom its critical values and this critical values increase i the increase. The van der Waals orce reduces the instability delection o the NT detector. Acknowledgement This work has been ounded as a research project and supported by Islamic Azad University, Naein Branch. orresponding Author: Alireza Vahdati Mechanical Engineering Department, Naein Branch, Islamic Azad University, Naein, Iran Vahdati@naeiniau.ac.ir Reerences. Paradise M, Goswami T. arbon nanotubes Production and industrial applications. Materials & Design 7;8(5): Baughman RH, ui, Zakhidov AA, Iqbal Z, Barisci JN, Spinks GM, Wallace GG, Mazzoldi A, Rossi DD, Rinzler AG, Jaschinski O, Roth S, Kertesz M. arbon nanotube actuators. Science 999; 8: Ke H, Pugno N, Peng B, Espinosa HD. Experiments and modeling o carbon nanotube-based NEMS devices. Journal o Mechanics and Physics o Solids 5;53: Esawi AMK, Farag MM. arbon nanotube reinorced composites: Potential and current challenges. Materials & Design 7;8(9): in WH, Zhao YP. asimir eect on the pullin parameters o nanometer switches. Microsystem Technologies 5;: Abadyan M, Novinzadeh A, Kazemi AS. Approximating the eect o asimir orce on the instability o electrostatic nano-cantilevers. Physica Scripta ;8:589 (pp) 7. Koochi A, Noghrehabadi A, Abadyan M, Roohi E. Investigation o the eect o van der Waals orce on the instability o electrostatic Nano-actuators. International Journal o Modern Physics: B ; 5(9): Tsai J, Tu JF. haracterizing mechanical properties o graphite using molecular dynamics simulation. Materials & Design ;3(): Tserpes KI. Role o intertube spacing in the pullout orces o double-walled carbon nanotubes. Materials & Design 7;8(7):97-.. Desquenes M, Rotkin SV, Alaru NR. alculation o Pull-in Voltages or arbon- Nanotube-Based Nanoelectromechanical Switches. Nanotechnology ;3:-3.. Batra R, Sears A. ontinuum models o multi-walled carbon nanotubes. International Journal o Solids and Structures 7;: in WH, Zhao YP. Nonlinear behavior or nanoscale electrostatic actuators with asimir orce. haos Solitons and Fractals 5;3: Hayt WH, Buck JA. Engineering electromagnetics. 6th ed.. New York: McGrawHill;.. ennard-jones J E. Perturbation Problems in Quantum Mechanics. Proc. R. Soc. A 93; 9: Girialco A, Hodak M, ee R S. arbon nanotubes, buckyballs, ropes, and a universal graphitic potential. Physical Review B ;6(9): Ke. and Espinosa H. D., Handbook o Theoretical and omputational Nanotechnology (American Scientiic Publishers, Valencia A, Gupta S S, Batra R. ontinuum structures equivalent in normal mode vibrations to singlewalled carbon nanotubes. omputational Materials Science 8;3:75-73 (8). 8. Adomian G, Stochastic Systems, Academic Press, ondon, 983. /9/ 9 liesciencej@gmail.com