- Email: [email protected]
Elastic behavior of an edge dislocation inside the wall of a nanotube
Available online at www.sciencedirect.com
Scripta Materialia 64 (2011) 709–712 www.elsevier.com/locate/scriptamat
Elastic behavior of an edge dislocation inside the wall of a nanotube S.S. Moeini-Ardakani,a M.Yu. Gutkinb,c and H.M. Shodjad,e,⇑ a
Department of Civil and Environmental Engineering, Massachusets Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA b Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Bolshoj 61, Vasil. Ostrov, St. Petersburg 199178, Russia c Department of Physics of Materials Strength and Plasticity, St. Petersburg State Polytechnical University, Polytekhnicheskaya 29, St. Petersburg 195251, Russia d Department of Civil Engineering, Sharif University of Technology, 11155-9313 Tehran, Iran e Institute for Nanoscience and Nanotechnology, Sharif University of Technology, 11155-9161 Tehran, Iran Received 24 October 2010; revised 14 December 2010; accepted 17 December 2010 Available online 22 December 2010
The problem of edge dislocation inside the wall of a multi-walled nanotube accounting for the surface effects is addressed. Within the framework of surface elasticity the stress field is obtained, using complex potentials. Furthermore, the stress field and image forces acting on the dislocation, with and without an account of the surface stress, are compared together and discussed. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Multi-walled nanotubes, Edge dislocation; Surface elasticity; Elasticity
Since the early 1990s, the deformation behavior of multi-walled nanotubes (MWNTs) has represented a rapidly growing field of research in nanoscience. Different deformation modes, such as localized kinks and bends [1], radial deformation of adjacent MWNTs [2], axial compression, buckling and collapse [3], tension [4], radial compression [5], torsion [6], bending and torsion with rippling [7], and high-velocity impact [8], have been observed and studied in detail. Based on our knowledge of the mechanisms of inelastic deformation in conventional solids, we have to assume that at least several of these deformation modes must include nucleation, motion and multiplication of various defects like vacancies, dislocations and disclinations. Although direct experimental observations of defects in MWNTs are still rather rare, there are some clear examples of their presence, such as edge dislocations, whose lines are parallel to the MWNT axes [9]. Since dislocations could greatly affect not only mechanical but also other physical properties of MWNTs, as is the case with other nanostructures, the dislocation behavior in MWNTs represents an important topic to study.
⇑ Corresponding author at: Department of Civil Engineering, Sharif University of Technology, 11155-9313 Tehran, Iran. Tel.: +98 21 66164209; fax: +98 21 66072555; e-mail: [email protected]
Any theoretical description of defects within a continuum approach is based on solutions of appropriate elastic problems. For straight dislocations in walls of NTs, the case of a screw dislocation has been considered within both the classical theory of elasticity [10,11] and the theory of elasticity with surface stresses [12]. A rigorous account for surface stresses [12] has shown that, unlike the case of classical elasticity [10,11], the screw dislocation can be repelled by free surfaces and can occupy two stable equilibrium positions near them. Encouraged by these unusual results, we address in the present paper the case of a straight edge dislocation inside the wall of an NT. Using the theory of elasticity with surface stresses [13], we obtain and study in detail the dislocation stresses and image forces acting on the dislocation from the inner and outer NT surfaces. We demonstrate that the account for the surface-stress effect leads to qualitative changes in stress distribution and image forces in the subsurface layers of NTs. Consider an infinite elastically isotropic hollow cylinder with inner and outer radii r1 and r2, respectively, which is an elastic model of an NT (Fig. 1). The bulk of the cylinder is characterized by the shear modulus l and Poisson’s ratio m, while its surfaces are characterized by the surface Lame´ constants ls and ks [13]. Let an edge dislocation with the Burgers vector b ¼ bx ex þ by ey be located inside the wall of the cylinder at a distance c from
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.12.022
710
S. S. Moeini-Ardakani et al. / Scripta Materialia 64 (2011) 709–712 y
r r-1
b x
c
s, s
r+1
, s, s
Figure 1. An edge dislocation inside the wall of a nanotube.
its axis. Right-handed Cartesian coordinates (x, y, z), in which the z-axis coincides with the cylinder axis are used. Although the surface-stress elasticity gives us unconventional force balance equations within the free surfaces/interfaces [13], the equilibrium equations inside the bulk remain the same as those defined by the classical theory of elasticity. The surface stress tensor rsij is determined by the usual relation [13,14]: rsij ¼ r0 dij þ @C=@eij , where r0 is the residual surface tension, dij is the Kronecker delta, C ¼ Cðeij Þ is the surface energy and eij is the 2 2 strain tensor for surfaces. Hereinafter we will consider the special case of r0 0. Due to the surface stresses, the constitutive and equilibrium equations for the inner and outer surfaces of the cylinder in the plane strain state are adapted from [14] as 2lgs @uh ðr1 ; hÞ þ ur ðr1 ; hÞ; ½ ð1Þ rshh ðr1 ; hÞ ¼ r1 @h rrr ðr1 ; hÞ ¼
rshh 1 @rshh ; ; rrh ðr1 ; hÞ ¼ r1 @h r1
ð2Þ
respectively. Here r, h and z are the cylindrical coordinates, r1 < r < r+1, gs = (2ls + ks)/(2l), ui is the displacement field, and rij denote the bulk stress tensor. In the limiting case ðls ; ks Þ ! 0, the surface stress (1) vanishes and the equilibrium Eqs. (2) are transformed to the classical boundary conditions, rrj ðr1 ; hÞ ¼ 0: The bulk stress components can be interpreted [15] through two potential functions pffiffiffiffiffiffiffi/ðzÞ and wðzÞ of complex variable z ¼ x þ iy (i ¼ 1) as follows rrr þ rhh ¼ 2 ½/ðzÞ þ /ðzÞ
ð3Þ
rrr irrh ¼ /ðzÞ þ /ðzÞ z/0 ðzÞ ðz=zÞ wðzÞ ð4Þ where the over bar denotes the complex conjugate. Taking the constitutive equations for the bulk into account, one can recast Eq. (1) as ð5Þ rshh ðr1 ; hÞ ¼ gs ½ð1 mÞrhh ðr1 ; hÞ mrrr ðr1 ; hÞ Substituting Eq. (5) in Eq. (2), we obtain the boundary conditions as follows: ð1 mÞ gs rhh ðr1 ; hÞ ¼ ðm gs r1 Þrrr ðr1 ; hÞ ð6Þ @rrr ðr1 ; hÞ ¼ 0: ð7Þ @h In view of Eqs. (3)–(7), the boundary conditions read h i K 1 /ðt1 Þ þ /ðt1 Þ t1 /0 ðt1 Þ rrh ðr1 ; hÞ þ
þ ðK 1 þ r21 Þ t1 /0 ðt1 Þ ¼ t21 wðt1 Þ
ð8Þ
and K 1 =r21 ¼ 2ð1 mÞ gs = where t1 ¼ r1 eih ðgs r1 Þ 1. Due to the presence of the edge dislocation inside the cylinder wall, the complex potentials /ðzÞ and wðzÞ can be expressed as [16] /ðzÞ ¼ c=ðz cÞ þ /0 ðzÞ, and 2 wðzÞ ¼ c=ðz cÞ þ c c=ðz cÞ þ w0 ðzÞ, where /0 ðzÞ and w0 ðzÞ are two holomorphic functions inside the annulus, and c ¼ l ðby ibx Þ=½4pð1 mÞ: The terms P /0 ðzÞ and 1 n w0 ðzÞ are searched as power series, / ðzÞ ¼ 0 n¼1 An z P1 n2 and w0 ðzÞ ¼ n¼1 Bn z , where the unknown coefficients An and Bn are determined from the boundary conditions (8). To this end, the non-analytical parts of /ðzÞ and wðzÞ should be written in form of Fourier series at z ¼ r1 eih . Subsequently, one can either exploit the orthogonality of trigonometric functions use the folPor 1 1 n1 n lowing Maclaurin series: ðz cÞ ¼ c z and n¼0 P1 2 1 n1 n1 ðz cÞ ¼ nc z at jzj < c ; and ðz cÞ ¼ P1 P1 n1 n n¼1 2 n1 n1 c z and ðz cÞ ¼ nc z at jzj > c : n¼1 n¼1 After some manipulations, we find the coefficients An and Bn as follows A1 ¼ 0; A0 ¼ c ðc2 þ K 1 Þ=½c ðK 1 þK þ1 Þ; A1 ¼ C 1 =D1 ; An ¼ ½ðK þ1 K 1 Þð1 þ nÞ C n 2 Dn C n =½ðK þ1 K 1 Þ ð1 n2 Þ Dn Dn ; n ¼ 2; 3; . . ., B0 ¼ 2K þ1 Re½A0 2cRe½c, Bn ¼ K sgnðnÞ ðn 1Þ An þ ½ðn 1ÞK sgnðnÞ þ nr2sgnðnÞ ½An þ sgnðnÞccn1 r2n sgnðnÞ ; n ¼ 1; 2; . . ., where C n ¼ ½cðn 1Þð1 þ K sgnðnÞ c2 Þ 2 2n c cnþ1 þ c ½ðn 1ÞK sgnðnÞ þ nrsgn ðnÞ cn1 rsgnðnÞ ; Dn ¼ 2 2n 2 2n ½ðn 1ÞK 1 þ nr1 r1 ½ðn 1ÞK þ1 þ nrþ1 rþ1 ; n ¼ 1; 2; . . .. Thus, we have found the stress field (3) and (4) of an edge dislocation placed inside the wall of an NT (Fig. 1) with account for the surface-stress effect. The limiting transition gs ! 0 gives the corresponding solution within the classical theory of elasticity. The differences between the surface-stress and classical solutions are illustrated in Figure 2, where the rrr stress maps are plotted in region 0 < h < p for the case of bx ¼ 0:25 nm, by ¼ 0, c ¼ 3 nm, r1 ¼ 1 nm, r+1 = 4 nm and m ¼ 0:3. The surface-stress solution (Fig. 2a) is given for gs ¼ 0:1289 nm (calculated for ls =l ¼ 0:17915 nm and ks =l ¼ 0:1005 nm taken for Al [1 0 0] surface from Ref. [16]), while the classical one (Fig. 2b) is given for gs ! 0. As is seen, near the dislocation line these solutions practically coincide, while far from it they are noticeably different. This mainly concerns the subsurface layers with the thickness of about 1 nm. The classical stress is negative and decreases in magnitude monotonously with h in the layer of thickness 0.5 nm under the outer surface, while the non-classical stress demonstrates 11 periods of oscillations in the range from approximately l=50 to l=50 there. Similarly, the classical stress changes its sign in the layer of thickness 1 nm under the inner surface once or twice, while the non-classical stress demonstrates three periods of oscillations in the range from approximately l=100 to l=100. These non-classical oscillations could be explained by the NT suffering surface rippling due to the presence of edge dislocation. It is worth noting that such a “rippling effect” is also typical for all other non-classical stress components and for any orientation of the dislocation Burgers vector (the corresponding stress maps are not shown here).
S. S. Moeini-Ardakani et al. / Scripta Materialia 64 (2011) 709–712 a
711
b
c
Figure 2. The rrr stress distribution in a nanotube with an inner radius r1 ¼ 1 nm and an outer radius rþ1 ¼ 4 nm, which contains an edge dislocation at the point ðx ¼ 3 nm;y ¼ 0Þ. (a) Non-classical stress map, (b) classical stress map, (c) azimuthal stress distribution in the central and surface layers of the nanotube. The stress values are given in units of l=100 .
The common feature in the classical and surface-stress solution is the double change of the stress sign with increasing h in the central layer of the NT. This effect appears when two free surfaces are close to dislocations, as is the case with edge dislocations in thin plates [17] and screw dislocations in between two cylindrical voids [18]. The aforementioned features are also seen in the plots rrr ðr1 ; hÞ and rrr ððr1 þ rþ1 Þ=2; hÞ, demonstrating the stress distribution over the NT surfaces and in its medial layer, respectively (Fig. 2c). It is evident that (i) the classical solution satisfies the boundary condition rrr ðr1 ; hÞ ¼ 0; (ii) the non-classical surface stress oscillations have a higher amplitude and a lower frequency on the inner surface than on the outer surface; and (iii) the classical and non-classical solutions almost coincide in the central layer of the NT and twice change their signs there. Consider now the image force acting on the dislocation from the NT surfaces. According to the Peach–Koehler formula, where, following Weertman [19], we replace the stress tensor by its deviator part to take into account the inelastic change in the solid volume accompanying the dislocation climb, the image force components can be calculated as fx ify ¼ ~yy ðc; 0Þby þ i ½~ ~xy ðc; 0Þby rxx ðc; 0Þbx þ r ½~ rxy ðc; 0Þbx þ r ~xx ðc; 0Þ=3. Here r ~ij ðc; 0Þ ryy ðc; 0Þ þ r ð1 þ mÞðby þ ibx Þ½~ is the perturbation stress component at the dislocation point. In terms of the aforementioned complex potentials, the climb ðfc Þ and glide ðfg Þ image forces (per unit dislocation length) read fc ifg ¼ b fð2 mÞ½/ 0 ðcÞ ffiþ /0 ðcÞ= qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 þ ðc=cÞ½c/00 ðcÞ þ w0 ðcÞg where b ¼ b2x þ b2y . For the numerical study of these image forces, we have taken two sets of surface elastic moduli used previously by Fang and Liu [16]. The first set, ls =l ¼ 0:17915 nm and ks =l ¼ 0:1005 nm, is characteristic for the Al [1 0 0] surface and gives gs 0:1289 nm; the second set, ls =l ¼ 0:01082 nm and ks =l ¼ 0:19713 nm, is characteristic for the Al [1 1 1] surface and gives gs 0:0877 nm. In Figure 3, both the glide and climb image forces are presented as
functions of the dislocation position c (a and b) and the angle a of the Burgers vector orientation (c and d) in the case of b ¼ 0:25 nm, r1 ¼ 1 nm, r+1 = 4 nm and m ¼ 0:3. It is seen that the classical and non-classical image forces practically coincide when the dislocation glides or climbs within the central layer of the NT, where it can occupy an unstable equilibrium position closer to the inner surface (Fig. 3a and b). Moreover, these forces are also very close to each other in subsurface regions if gs 0:0877 nm, in contrast with the case of gs 0:1289 nm, when the difference between the classical and non-classical solutions is rather significant. Let us consider in more detail the inner and outer and subsurface layers r1 þ b < r < r1 þ 1 nm rþ1 1 nm < r < rþ1 b, respectively, ignoring the atomically thin surface layers of thickness b. When gs 0:1289 nm, the non-classical glide image force changes its sign and reaches its maximum (minimum) value near the point c r1 þ b 1:25 nm (rþ1 b 3:75 nm). The points c 1:55 nm and c 3:45 nm, where fg ¼ 0, are additional non-classical points of stable equilibrium of the edge dislocation near the NT surfaces (Fig. 3a). Thus, in this case, the edge dislocation is repelled by the NT surfaces within the thin subsurface layers. Recently we observed a similar non-classical effect in the case of a screw dislocation placed in the wall of an NT [12]. The non-classical climb image force also demonstrates abnormal behavior near the outer surface at gs 0:1289 nm (Fig. 3b). In the subsurface layer rþ1 1 nm < r < rþ1 b, it twice changes its sign and reaches a minimum value at c 3:6 nm. This means that the climbing dislocation can occupy two additional nonclassical equilibrium positions, one stable and one unstable, near the outer NT surface. Near the inner NT surface, the classical and non-classical climb image forces show a similar behavior. As can be seen from the graphs in Fig. 3c and d, plotted for the dislocation position c ¼ 3:5 nm, the effect of the Burgers vector orientation on the image force depends strongly on the value of gs . When it is positive and relatively small (here gs 0:0877 nm), this effect is quite
712
S. S. Moeini-Ardakani et al. / Scripta Materialia 64 (2011) 709–712
Figure 3. The normalized glide (a, c) and climb (b, d) image forces via (a, b) the dislocation position c at (a) a ¼ 0, (b) a ¼ p=2; (c, d) the angle a of the Burgers vector orientation at c ¼ 3:5 nm for different values of ratio gs , shown in nanometers.
similar for the classical and non-classical image forces. When it is negative and relatively larger in magnitude (here gs 0:1289 nm), the effect is very different for them: while the classical image forces are always positive in the interval 0 < a < p=2, the non-classical image forces can be positive or negative, or even vanish there. In summary, we have obtained for the first time the stress field of an edge dislocation placed inside the wall of an NT within both the classical theory of elasticity and non-classical theory of elasticity with surface stresses. We have shown that use of the latter theory leads to qualitative changes in stress distribution and image forces in the subsurface layers of NTs. Instead of a smooth and monotonous classical solution, we have got stress oscillations along both the inner and outer surfaces, which can be treated as if caused by surface rippling due to the presence of dislocations. We also show that the dislocation is attracted to the inner and outer NT surfaces when it is localized in the central layer of the NT, where the surface-stress effect is negligibly small, and has a classical unstable equilibrium position there. If the dislocation is placed in a subsurface layer, the surface-stress effect depends strongly on the ratio gs of the surface and bulk elastic moduli. When gs is negative and relatively large in magnitude, the dislocation can be repelled from the inner and outer NT surfaces and can have two additional stable equilibrium positions near of them. Based on the present study, there is no special need to use the more complicated surface elasticity theory for describing the dislocation field and behavior in the bulk of a nanoscale solid at distances larger than about 1 nm from the free surface. When studying the situation near the free surface, the choice between the classical and surface theories of elasticity depends on the values of surface elastic moduli. In the case of an edge dislocation, the latter can be unified in a single surface parameter, gs . If gs is negative and not very small in magnitude, one must use the surface elasticity theory, otherwise
one can use the classical theory of elasticity. For further clarification, experimental measurements and computer simulations would be beneficial. To the best of our knowledge, no such studies at currently ongoing. The work of H.M.S. and S.S.M.-A. was supported in part by Sharif University of Technology. M.Yu.G. appreciates the support of the Russian Foundation of Basic Research (Grant 08-02-00304-a). [1] J.F. Despres, E. Daguerre, K. Kafdi, Carbon 33 (1995) 87. [2] R.S. Ruoff, J. Tersoff, D.C. Lorents, S. Subramoney, B. Chan, Nature 364 (1993) 514. [3] O. Lourie, D.M. Cox, H.D. Wagner, Phys. Rev. Lett. 81 (1998) 1638. [4] M.F. Yu, O. Lourie, M.J. Dyer, K. Moloni, T.F. Kelly, R.S. Ruoff, Science 287 (2000) 637. [5] M.F. Yu, T. Kowalewski, R.S. Ruoff, Phys. Rev. Lett. 85 (2000) 1456. [6] A.M. Fennimore, T.D. Yuzvinsky, W.Q. Han, M.S. Fuhrer, J. Cumings, A. Zettl, Nature 424 (2003) 408. [7] M. Arroyo, I. Arias, J. Mech. Phys. Solids 56 (2008) 1224. [8] S.R. Bakshi, V. Singh, D.G. McCartney, S. Seal, A. Agarwal, Scripta Mater. 59 (2008) 499. [9] Z.E. Zhang, Microscopy Res. Tech. 40 (1998) 163. [10] V.A. Lubarda, J. Elasticity 52 (1999) 289. [11] M.Yu. Gutkin, A.G. Sheinerman, Phys. Solid State 49 (2007) 1672. [12] H. Shodja, M.Yu. Gutkin, S.S. Moeini-Ardakani, Phys. Status Solodi B (2011), doi:10.1002/pssb.201046417. [13] M.E. Gurtin, A.I. Murdoch, Arch. Rational Mech. Anal. 57 (1975) 291. [14] P. Sharma, S. Ganti, N. Bhate, Appl. Phys. Lett. 82 (2003) 535. [15] N.L. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, Noordhoff, Leyden, 1975. [16] Q.H. Fang, Y.W. Liu, Acta Mater. 54 (2006) 4213. [17] M.Yu. Gutkin, A.E. Romanov, Phys. Status Solidi A 125 (1991) 107. [18] M.Yu. Gutkin, A.G. Sheinerman, M.A. Smirnov, Mech. Mater. 41 (2009) 905. [19] J. Weertman, Phil. Mag. 11 (1965) 1217.
Scripta Materialia 64 (2011) 709–712 www.elsevier.com/locate/scriptamat
Elastic behavior of an edge dislocation inside the wall of a nanotube S.S. Moeini-Ardakani,a M.Yu. Gutkinb,c and H.M. Shodjad,e,⇑ a
Department of Civil and Environmental Engineering, Massachusets Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA b Institute of Problems of Mechanical Engineering, Russian Academy of Sciences, Bolshoj 61, Vasil. Ostrov, St. Petersburg 199178, Russia c Department of Physics of Materials Strength and Plasticity, St. Petersburg State Polytechnical University, Polytekhnicheskaya 29, St. Petersburg 195251, Russia d Department of Civil Engineering, Sharif University of Technology, 11155-9313 Tehran, Iran e Institute for Nanoscience and Nanotechnology, Sharif University of Technology, 11155-9161 Tehran, Iran Received 24 October 2010; revised 14 December 2010; accepted 17 December 2010 Available online 22 December 2010
The problem of edge dislocation inside the wall of a multi-walled nanotube accounting for the surface effects is addressed. Within the framework of surface elasticity the stress field is obtained, using complex potentials. Furthermore, the stress field and image forces acting on the dislocation, with and without an account of the surface stress, are compared together and discussed. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Multi-walled nanotubes, Edge dislocation; Surface elasticity; Elasticity
Since the early 1990s, the deformation behavior of multi-walled nanotubes (MWNTs) has represented a rapidly growing field of research in nanoscience. Different deformation modes, such as localized kinks and bends [1], radial deformation of adjacent MWNTs [2], axial compression, buckling and collapse [3], tension [4], radial compression [5], torsion [6], bending and torsion with rippling [7], and high-velocity impact [8], have been observed and studied in detail. Based on our knowledge of the mechanisms of inelastic deformation in conventional solids, we have to assume that at least several of these deformation modes must include nucleation, motion and multiplication of various defects like vacancies, dislocations and disclinations. Although direct experimental observations of defects in MWNTs are still rather rare, there are some clear examples of their presence, such as edge dislocations, whose lines are parallel to the MWNT axes [9]. Since dislocations could greatly affect not only mechanical but also other physical properties of MWNTs, as is the case with other nanostructures, the dislocation behavior in MWNTs represents an important topic to study.
⇑ Corresponding author at: Department of Civil Engineering, Sharif University of Technology, 11155-9313 Tehran, Iran. Tel.: +98 21 66164209; fax: +98 21 66072555; e-mail: [email protected]
Any theoretical description of defects within a continuum approach is based on solutions of appropriate elastic problems. For straight dislocations in walls of NTs, the case of a screw dislocation has been considered within both the classical theory of elasticity [10,11] and the theory of elasticity with surface stresses [12]. A rigorous account for surface stresses [12] has shown that, unlike the case of classical elasticity [10,11], the screw dislocation can be repelled by free surfaces and can occupy two stable equilibrium positions near them. Encouraged by these unusual results, we address in the present paper the case of a straight edge dislocation inside the wall of an NT. Using the theory of elasticity with surface stresses [13], we obtain and study in detail the dislocation stresses and image forces acting on the dislocation from the inner and outer NT surfaces. We demonstrate that the account for the surface-stress effect leads to qualitative changes in stress distribution and image forces in the subsurface layers of NTs. Consider an infinite elastically isotropic hollow cylinder with inner and outer radii r1 and r2, respectively, which is an elastic model of an NT (Fig. 1). The bulk of the cylinder is characterized by the shear modulus l and Poisson’s ratio m, while its surfaces are characterized by the surface Lame´ constants ls and ks [13]. Let an edge dislocation with the Burgers vector b ¼ bx ex þ by ey be located inside the wall of the cylinder at a distance c from
1359-6462/$ - see front matter Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2010.12.022
710
S. S. Moeini-Ardakani et al. / Scripta Materialia 64 (2011) 709–712 y
r r-1
b x
c
s, s
r+1
, s, s
Figure 1. An edge dislocation inside the wall of a nanotube.
its axis. Right-handed Cartesian coordinates (x, y, z), in which the z-axis coincides with the cylinder axis are used. Although the surface-stress elasticity gives us unconventional force balance equations within the free surfaces/interfaces [13], the equilibrium equations inside the bulk remain the same as those defined by the classical theory of elasticity. The surface stress tensor rsij is determined by the usual relation [13,14]: rsij ¼ r0 dij þ @C=@eij , where r0 is the residual surface tension, dij is the Kronecker delta, C ¼ Cðeij Þ is the surface energy and eij is the 2 2 strain tensor for surfaces. Hereinafter we will consider the special case of r0 0. Due to the surface stresses, the constitutive and equilibrium equations for the inner and outer surfaces of the cylinder in the plane strain state are adapted from [14] as 2lgs @uh ðr1 ; hÞ þ ur ðr1 ; hÞ; ½ ð1Þ rshh ðr1 ; hÞ ¼ r1 @h rrr ðr1 ; hÞ ¼
rshh 1 @rshh ; ; rrh ðr1 ; hÞ ¼ r1 @h r1
ð2Þ
respectively. Here r, h and z are the cylindrical coordinates, r1 < r < r+1, gs = (2ls + ks)/(2l), ui is the displacement field, and rij denote the bulk stress tensor. In the limiting case ðls ; ks Þ ! 0, the surface stress (1) vanishes and the equilibrium Eqs. (2) are transformed to the classical boundary conditions, rrj ðr1 ; hÞ ¼ 0: The bulk stress components can be interpreted [15] through two potential functions pffiffiffiffiffiffiffi/ðzÞ and wðzÞ of complex variable z ¼ x þ iy (i ¼ 1) as follows rrr þ rhh ¼ 2 ½/ðzÞ þ /ðzÞ
ð3Þ
rrr irrh ¼ /ðzÞ þ /ðzÞ z/0 ðzÞ ðz=zÞ wðzÞ ð4Þ where the over bar denotes the complex conjugate. Taking the constitutive equations for the bulk into account, one can recast Eq. (1) as ð5Þ rshh ðr1 ; hÞ ¼ gs ½ð1 mÞrhh ðr1 ; hÞ mrrr ðr1 ; hÞ Substituting Eq. (5) in Eq. (2), we obtain the boundary conditions as follows: ð1 mÞ gs rhh ðr1 ; hÞ ¼ ðm gs r1 Þrrr ðr1 ; hÞ ð6Þ @rrr ðr1 ; hÞ ¼ 0: ð7Þ @h In view of Eqs. (3)–(7), the boundary conditions read h i K 1 /ðt1 Þ þ /ðt1 Þ t1 /0 ðt1 Þ rrh ðr1 ; hÞ þ
þ ðK 1 þ r21 Þ t1 /0 ðt1 Þ ¼ t21 wðt1 Þ
ð8Þ
and K 1 =r21 ¼ 2ð1 mÞ gs = where t1 ¼ r1 eih ðgs r1 Þ 1. Due to the presence of the edge dislocation inside the cylinder wall, the complex potentials /ðzÞ and wðzÞ can be expressed as [16] /ðzÞ ¼ c=ðz cÞ þ /0 ðzÞ, and 2 wðzÞ ¼ c=ðz cÞ þ c c=ðz cÞ þ w0 ðzÞ, where /0 ðzÞ and w0 ðzÞ are two holomorphic functions inside the annulus, and c ¼ l ðby ibx Þ=½4pð1 mÞ: The terms P /0 ðzÞ and 1 n w0 ðzÞ are searched as power series, / ðzÞ ¼ 0 n¼1 An z P1 n2 and w0 ðzÞ ¼ n¼1 Bn z , where the unknown coefficients An and Bn are determined from the boundary conditions (8). To this end, the non-analytical parts of /ðzÞ and wðzÞ should be written in form of Fourier series at z ¼ r1 eih . Subsequently, one can either exploit the orthogonality of trigonometric functions use the folPor 1 1 n1 n lowing Maclaurin series: ðz cÞ ¼ c z and n¼0 P1 2 1 n1 n1 ðz cÞ ¼ nc z at jzj < c ; and ðz cÞ ¼ P1 P1 n1 n n¼1 2 n1 n1 c z and ðz cÞ ¼ nc z at jzj > c : n¼1 n¼1 After some manipulations, we find the coefficients An and Bn as follows A1 ¼ 0; A0 ¼ c ðc2 þ K 1 Þ=½c ðK 1 þK þ1 Þ; A1 ¼ C 1 =D1 ; An ¼ ½ðK þ1 K 1 Þð1 þ nÞ C n 2 Dn C n =½ðK þ1 K 1 Þ ð1 n2 Þ Dn Dn ; n ¼ 2; 3; . . ., B0 ¼ 2K þ1 Re½A0 2cRe½c, Bn ¼ K sgnðnÞ ðn 1Þ An þ ½ðn 1ÞK sgnðnÞ þ nr2sgnðnÞ ½An þ sgnðnÞccn1 r2n sgnðnÞ ; n ¼ 1; 2; . . ., where C n ¼ ½cðn 1Þð1 þ K sgnðnÞ c2 Þ 2 2n c cnþ1 þ c ½ðn 1ÞK sgnðnÞ þ nrsgn ðnÞ cn1 rsgnðnÞ ; Dn ¼ 2 2n 2 2n ½ðn 1ÞK 1 þ nr1 r1 ½ðn 1ÞK þ1 þ nrþ1 rþ1 ; n ¼ 1; 2; . . .. Thus, we have found the stress field (3) and (4) of an edge dislocation placed inside the wall of an NT (Fig. 1) with account for the surface-stress effect. The limiting transition gs ! 0 gives the corresponding solution within the classical theory of elasticity. The differences between the surface-stress and classical solutions are illustrated in Figure 2, where the rrr stress maps are plotted in region 0 < h < p for the case of bx ¼ 0:25 nm, by ¼ 0, c ¼ 3 nm, r1 ¼ 1 nm, r+1 = 4 nm and m ¼ 0:3. The surface-stress solution (Fig. 2a) is given for gs ¼ 0:1289 nm (calculated for ls =l ¼ 0:17915 nm and ks =l ¼ 0:1005 nm taken for Al [1 0 0] surface from Ref. [16]), while the classical one (Fig. 2b) is given for gs ! 0. As is seen, near the dislocation line these solutions practically coincide, while far from it they are noticeably different. This mainly concerns the subsurface layers with the thickness of about 1 nm. The classical stress is negative and decreases in magnitude monotonously with h in the layer of thickness 0.5 nm under the outer surface, while the non-classical stress demonstrates 11 periods of oscillations in the range from approximately l=50 to l=50 there. Similarly, the classical stress changes its sign in the layer of thickness 1 nm under the inner surface once or twice, while the non-classical stress demonstrates three periods of oscillations in the range from approximately l=100 to l=100. These non-classical oscillations could be explained by the NT suffering surface rippling due to the presence of edge dislocation. It is worth noting that such a “rippling effect” is also typical for all other non-classical stress components and for any orientation of the dislocation Burgers vector (the corresponding stress maps are not shown here).
S. S. Moeini-Ardakani et al. / Scripta Materialia 64 (2011) 709–712 a
711
b
c
Figure 2. The rrr stress distribution in a nanotube with an inner radius r1 ¼ 1 nm and an outer radius rþ1 ¼ 4 nm, which contains an edge dislocation at the point ðx ¼ 3 nm;y ¼ 0Þ. (a) Non-classical stress map, (b) classical stress map, (c) azimuthal stress distribution in the central and surface layers of the nanotube. The stress values are given in units of l=100 .
The common feature in the classical and surface-stress solution is the double change of the stress sign with increasing h in the central layer of the NT. This effect appears when two free surfaces are close to dislocations, as is the case with edge dislocations in thin plates [17] and screw dislocations in between two cylindrical voids [18]. The aforementioned features are also seen in the plots rrr ðr1 ; hÞ and rrr ððr1 þ rþ1 Þ=2; hÞ, demonstrating the stress distribution over the NT surfaces and in its medial layer, respectively (Fig. 2c). It is evident that (i) the classical solution satisfies the boundary condition rrr ðr1 ; hÞ ¼ 0; (ii) the non-classical surface stress oscillations have a higher amplitude and a lower frequency on the inner surface than on the outer surface; and (iii) the classical and non-classical solutions almost coincide in the central layer of the NT and twice change their signs there. Consider now the image force acting on the dislocation from the NT surfaces. According to the Peach–Koehler formula, where, following Weertman [19], we replace the stress tensor by its deviator part to take into account the inelastic change in the solid volume accompanying the dislocation climb, the image force components can be calculated as fx ify ¼ ~yy ðc; 0Þby þ i ½~ ~xy ðc; 0Þby rxx ðc; 0Þbx þ r ½~ rxy ðc; 0Þbx þ r ~xx ðc; 0Þ=3. Here r ~ij ðc; 0Þ ryy ðc; 0Þ þ r ð1 þ mÞðby þ ibx Þ½~ is the perturbation stress component at the dislocation point. In terms of the aforementioned complex potentials, the climb ðfc Þ and glide ðfg Þ image forces (per unit dislocation length) read fc ifg ¼ b fð2 mÞ½/ 0 ðcÞ ffiþ /0 ðcÞ= qffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 þ ðc=cÞ½c/00 ðcÞ þ w0 ðcÞg where b ¼ b2x þ b2y . For the numerical study of these image forces, we have taken two sets of surface elastic moduli used previously by Fang and Liu [16]. The first set, ls =l ¼ 0:17915 nm and ks =l ¼ 0:1005 nm, is characteristic for the Al [1 0 0] surface and gives gs 0:1289 nm; the second set, ls =l ¼ 0:01082 nm and ks =l ¼ 0:19713 nm, is characteristic for the Al [1 1 1] surface and gives gs 0:0877 nm. In Figure 3, both the glide and climb image forces are presented as
functions of the dislocation position c (a and b) and the angle a of the Burgers vector orientation (c and d) in the case of b ¼ 0:25 nm, r1 ¼ 1 nm, r+1 = 4 nm and m ¼ 0:3. It is seen that the classical and non-classical image forces practically coincide when the dislocation glides or climbs within the central layer of the NT, where it can occupy an unstable equilibrium position closer to the inner surface (Fig. 3a and b). Moreover, these forces are also very close to each other in subsurface regions if gs 0:0877 nm, in contrast with the case of gs 0:1289 nm, when the difference between the classical and non-classical solutions is rather significant. Let us consider in more detail the inner and outer and subsurface layers r1 þ b < r < r1 þ 1 nm rþ1 1 nm < r < rþ1 b, respectively, ignoring the atomically thin surface layers of thickness b. When gs 0:1289 nm, the non-classical glide image force changes its sign and reaches its maximum (minimum) value near the point c r1 þ b 1:25 nm (rþ1 b 3:75 nm). The points c 1:55 nm and c 3:45 nm, where fg ¼ 0, are additional non-classical points of stable equilibrium of the edge dislocation near the NT surfaces (Fig. 3a). Thus, in this case, the edge dislocation is repelled by the NT surfaces within the thin subsurface layers. Recently we observed a similar non-classical effect in the case of a screw dislocation placed in the wall of an NT [12]. The non-classical climb image force also demonstrates abnormal behavior near the outer surface at gs 0:1289 nm (Fig. 3b). In the subsurface layer rþ1 1 nm < r < rþ1 b, it twice changes its sign and reaches a minimum value at c 3:6 nm. This means that the climbing dislocation can occupy two additional nonclassical equilibrium positions, one stable and one unstable, near the outer NT surface. Near the inner NT surface, the classical and non-classical climb image forces show a similar behavior. As can be seen from the graphs in Fig. 3c and d, plotted for the dislocation position c ¼ 3:5 nm, the effect of the Burgers vector orientation on the image force depends strongly on the value of gs . When it is positive and relatively small (here gs 0:0877 nm), this effect is quite
712
S. S. Moeini-Ardakani et al. / Scripta Materialia 64 (2011) 709–712
Figure 3. The normalized glide (a, c) and climb (b, d) image forces via (a, b) the dislocation position c at (a) a ¼ 0, (b) a ¼ p=2; (c, d) the angle a of the Burgers vector orientation at c ¼ 3:5 nm for different values of ratio gs , shown in nanometers.
similar for the classical and non-classical image forces. When it is negative and relatively larger in magnitude (here gs 0:1289 nm), the effect is very different for them: while the classical image forces are always positive in the interval 0 < a < p=2, the non-classical image forces can be positive or negative, or even vanish there. In summary, we have obtained for the first time the stress field of an edge dislocation placed inside the wall of an NT within both the classical theory of elasticity and non-classical theory of elasticity with surface stresses. We have shown that use of the latter theory leads to qualitative changes in stress distribution and image forces in the subsurface layers of NTs. Instead of a smooth and monotonous classical solution, we have got stress oscillations along both the inner and outer surfaces, which can be treated as if caused by surface rippling due to the presence of dislocations. We also show that the dislocation is attracted to the inner and outer NT surfaces when it is localized in the central layer of the NT, where the surface-stress effect is negligibly small, and has a classical unstable equilibrium position there. If the dislocation is placed in a subsurface layer, the surface-stress effect depends strongly on the ratio gs of the surface and bulk elastic moduli. When gs is negative and relatively large in magnitude, the dislocation can be repelled from the inner and outer NT surfaces and can have two additional stable equilibrium positions near of them. Based on the present study, there is no special need to use the more complicated surface elasticity theory for describing the dislocation field and behavior in the bulk of a nanoscale solid at distances larger than about 1 nm from the free surface. When studying the situation near the free surface, the choice between the classical and surface theories of elasticity depends on the values of surface elastic moduli. In the case of an edge dislocation, the latter can be unified in a single surface parameter, gs . If gs is negative and not very small in magnitude, one must use the surface elasticity theory, otherwise
one can use the classical theory of elasticity. For further clarification, experimental measurements and computer simulations would be beneficial. To the best of our knowledge, no such studies at currently ongoing. The work of H.M.S. and S.S.M.-A. was supported in part by Sharif University of Technology. M.Yu.G. appreciates the support of the Russian Foundation of Basic Research (Grant 08-02-00304-a). [1] J.F. Despres, E. Daguerre, K. Kafdi, Carbon 33 (1995) 87. [2] R.S. Ruoff, J. Tersoff, D.C. Lorents, S. Subramoney, B. Chan, Nature 364 (1993) 514. [3] O. Lourie, D.M. Cox, H.D. Wagner, Phys. Rev. Lett. 81 (1998) 1638. [4] M.F. Yu, O. Lourie, M.J. Dyer, K. Moloni, T.F. Kelly, R.S. Ruoff, Science 287 (2000) 637. [5] M.F. Yu, T. Kowalewski, R.S. Ruoff, Phys. Rev. Lett. 85 (2000) 1456. [6] A.M. Fennimore, T.D. Yuzvinsky, W.Q. Han, M.S. Fuhrer, J. Cumings, A. Zettl, Nature 424 (2003) 408. [7] M. Arroyo, I. Arias, J. Mech. Phys. Solids 56 (2008) 1224. [8] S.R. Bakshi, V. Singh, D.G. McCartney, S. Seal, A. Agarwal, Scripta Mater. 59 (2008) 499. [9] Z.E. Zhang, Microscopy Res. Tech. 40 (1998) 163. [10] V.A. Lubarda, J. Elasticity 52 (1999) 289. [11] M.Yu. Gutkin, A.G. Sheinerman, Phys. Solid State 49 (2007) 1672. [12] H. Shodja, M.Yu. Gutkin, S.S. Moeini-Ardakani, Phys. Status Solodi B (2011), doi:10.1002/pssb.201046417. [13] M.E. Gurtin, A.I. Murdoch, Arch. Rational Mech. Anal. 57 (1975) 291. [14] P. Sharma, S. Ganti, N. Bhate, Appl. Phys. Lett. 82 (2003) 535. [15] N.L. Muskhelishvili, Some Basic Problems of Mathematical Theory of Elasticity, Noordhoff, Leyden, 1975. [16] Q.H. Fang, Y.W. Liu, Acta Mater. 54 (2006) 4213. [17] M.Yu. Gutkin, A.E. Romanov, Phys. Status Solidi A 125 (1991) 107. [18] M.Yu. Gutkin, A.G. Sheinerman, M.A. Smirnov, Mech. Mater. 41 (2009) 905. [19] J. Weertman, Phil. Mag. 11 (1965) 1217.