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Dynamic characteristics of nanoindentation using atomistic simulation
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Acta Materialia 57 (2009) 3341–3348 www.elsevier.com/locate/actamat
Dynamic characteristics of nanoindentation using atomistic simulation Te-Hua Fang a,*, Wen-Yang Chang b, Jian-Jin Huang a a
Institute of Mechanical and Electromechanical Engineering, National Formosa University, Yunlin 632, Taiwan b Microsystems Technology Center, Industrial Technology Research Institute, Tainan 709, Taiwan Received 8 January 2009; received in revised form 30 March 2009; accepted 30 March 2009 Available online 4 May 2009
Abstract Atomistic simulations are used to investigate how the nanoindentation mechanism influences dislocation nucleation under molecular dynamic behavior on the aluminum (0 0 1) surface. The characteristics of molecular dynamics in terms of various nucleation criteria are explored, including various molecular models, a multi-step load/unload cycle, deformation mechanism of atoms, tilt angle of the indenter, and slip vectors. Simulation results show that both the plastic energy and the adhesive force increase with increasing nanoindentation depths. The maximum forces for all indentation depths decrease with increasing multi-step load/unload cycle time. Dislocation nucleation, gliding, and interaction occur along Shockley partials on (1 1 1) slip planes. The indentation force applied along the normal direction, a tilt angle of 0°, is smaller than the force component that acts on the surface atoms. The corresponding slip vector of the atoms in the (1 1 1) plane has low-energy sessile stair-rod dislocations in the pyramid of intrinsic stacking faults. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanoindentation; Aluminum; Dynamic phenomena; Molecular dynamics
1. Introduction Molecular dynamics (MD) simulation for nanotechnology and nanoscience analyses of film materials has been used to interpret the deformation mechanism of defect structures [1,2]. Nanoindentation contact plays a key role in engineering reliability for investigating the mechanical properties of MD behavior at the atomic scale. Many nanoindentation [3,4] experiments and nanoscratch [5,6] have been performed to investigate the molecular deformation mechanism in material science. Experimental tests show the z mechanical and processing responses of nano-materials, but they are unsuitable for investigating the defect mechanism at the atomic scale. However, MD simulation can be effective in simulating the atomistic characteristics of deformation and microscopic fracture processes. Kizler et al. [7] reported the deformation mechanism of an ultra-hard carbide layer under nanoindentation with the aid of MD analysis. Oyeon et al. [8] used MD to analyze
*
Corresponding author. Tel.: +886 5 631 5395; fax: +886 5 631 5397. E-mail address: [email protected] (T.-H. Fang).
the dislocation nucleation in various orientations and dislocation deformations at the surface of nickel single crystals. Jin et al. [9] investigated the multiscale simulation of onset plasticity during the nanoindentation of Al. Although MD simulations have recently been used to study the mechanisms of onset plasticity, dislocation nucleation, and elastic–plastic deformation [9–11], few studies have focused on dynamic nanoindentation [12,13]. Youngmin et al. [14] studied the defect nucleation and the evolution of incipient plasticity in Al material under nanoindentation using atomistic simulation. In an experimental nanoindentation test, it is difficult to vertically indent the sample without changing the tilt angle of the indenter. The effect of the tilt angle of the indenter is difficult to obtain from experiments. The phenomenon of the indenter’s tilt angles is more meaningful and interesting to study using MD simulation. Understanding the defect mechanism at the atomic scale through accurate modeling and simulation is an essential premise. This study uses MD simulation to investigate various molecular models, the multi-step load/unload cycle, the deformation mechanism of atoms, tilt angles of the indenter, and slip vectors. The aluminum (Al) model is chosen
1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.03.048
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for MD simulation because it is widely used in many industries, from aerospace to automotive to microelectronics. For the MD simulation, the effects of the maximum forces and the adhesive forces at the maximum depths of dynamic nanoindentation were investigated. A thermal layer and a thermal layer with a free layer were used to analyze the mechanism of surface plasticity. The effects of the multistep load/unload cycle of nanoindentation with various depths and tilt angles of the indenter were investigated. 2. Simulation methodology The model sizes of the MD simulation in the x, y, and z directions were 10.93, 10.93, and 4.86 nm, respectively, using a face-centered cubic (fcc) bulk crystal. The indenter was a Vickers indenter consisting of 2992 diamond atoms in the form of a square-based pyramid with an angle of 22° between the indenter face and the contact surface. The Vickers indenter is not perfect for MD simulation because of its larger deviation. A correction factor [15] related to the lack of any revolution paraboloid during MD simulation must be taken into account. According to Sakharova et al. [16], the characteristic analyses of molecular deformation after MD simulation are similar compared with Berkovich, Vickers, and conical indenters. For the simulation conditions, the indenter was assumed to be a rigid indenter. The model was initially assumed to have a well-defined atomic structure before thermal equilibration. The indented surface was the (0 0 1) plane of Al for all the numerical experiments. The periodic boundary conditions of the sample were used in the transverse directions. The bottom of the specimen, two layers of atoms, was fixed in space to prevent the substrate from being moved. Periodic boundary conditions were enforced in the x and y directions; there was no periodic boundary condition along the z direction. The velocities of the atoms of the thermal control layers are given by the Maxwell–Boltzmann distribution [17] of the prescribed substrate temperature. The position of the incident atoms was random in the x and y directions, and the z direction of incident atoms was at a 20-fold lattice length above the substrate surface. During the load/unload cycles, the indenter was moved at a constant speed of 100 m s1. The model equilibrated to its minimum energy configuration at 300 K. For most MD simulations of nanoindentation, the indentation velocities are from 1 to 100 m s1 due to the limitations of computation time and power [14]. The dynamics of the model were evaluated by integrating the Newtonian equations of motion using the Verlet algorithm [18] with a time step of 1 fs. The force acting on an individual atom was obtained by summing the forces contributed by the surrounding atoms. The Morse two-body potential and the secondmoment approximation of the tight-binding many-body (TB-SMA) potential were adopted to model the atomic interactions among the Al atoms. The Morse potential only considers the interaction between two atoms without including the simultaneous influence of their neighboring
atoms. The Morse potential energy, UMorse (rij), can be described with three parameters as: ð1Þ U Morse ðrij Þ ¼ D e2aðrij r0 Þ 2eaðrij r0 Þ where D, r0, and rij are the cohesion energy of exchange interaction, the equilibrium distance, and the separation distance between atoms i and j, respectively. a is fitted to the bulk modulus of the material. For C–Al atoms, the values of D, a, and r0 are 0.8092 eV, 0.0186 nm1, and 2.970 nm, respectively. The second-moment approximation of the tight-binding many-body potential, U(rij), contains a repulsive pair potential and a cohesive band energy term, as follows: Uðrij Þ ¼
N X i¼1
"
X j
1=2 # rij rij n exp 2q þ Aexp p 1 r0 r0 2
ð2Þ
where n, r0, and N are the effective hopping integral, the first-neighbor distance, and the number of atoms considered, respectively. The terms n, q, p, and A are fitted to the experimental values of cohesive energy. For Al–Al atoms, the values of n, q, p, r0, and A are 1.316 eV, 2.516, 8.612, 0.2864 nm, and 0.1221 eV, respectively. 3. Results and discussion 3.1. Comparison of molecular models For the comparison of molecular models, a thermal layer (TL) and a thermal layer with a free layer (TFL) were used to analyze the mechanical behavior and the underlying mechanism of surface plasticity. During the MD simulation process, the TL atoms remained at a constant temperature, indicating that all atomic heat is uniform after the heat is induced by indenter indentation. All atoms in the free layer are independent and can vary their temperatures, meaning that temperature depends on the friction force between the indenter and the atoms. The boundary temperature at the thermal layer and the free layer is initially at 300 K. The force versus depth relationships for TL and TFL models are shown in Fig. 1a. The indentation forces for TL model at maximum depths of 0.77, 1.15, 1.44, 1.63, and 1.72 nm are 437, 1168, 2461, 3400, and 4474 nN, respectively. The indentation forces for TFL model at maximum depths of 0.77, 1.15, 1.44, 1.63, and 1.72 nm are 285, 798, 944, 1668, and 2107 nN, respectively. The interatomic bonding forces are relatively weak in TFL because the atoms have more freedom and vibrational energy than those in TL. When a small force was applied to the indenter, it easily penetrated the atoms in the adjustable temperature layer. The Al substrate became softer, indicating that the dislocation was easier to deform. Dislocations mean that the substrate can be rolled into grain boundary layers when the atoms slip past each other. Fig. 1b shows the force versus depth relationship of adhesive forces for TL and TFL. The adhesive forces for
T.-H. Fang et al. / Acta Materialia 57 (2009) 3341–3348
28
500 TL
TFL
Contact pressure [ GPa ]
Force [ nN ]
Probe Thermal layer Fixed layer
300
(TL model) 200
100
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Depth [ nm ]
Probe Free layer Thermal layer Fixed layer
Force [ nN ]
(TFL model)
80
40
0 0.6
TL
0.8
1.0
1.2
20 16 12 8 4 0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Fig. 2. Relationships between the indentation depth and the contact pressure for TL and TFL.
200
160
TFL
Indentation depth [ nm ]
(a)
120
TL
24
400
0 0.6
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1.4
TFL
1.6
1.8
2.0
Depth [ nm ]
(b) Fig. 1. Molecular models of the thermal layer (TL) and the thermal layer with free layer (TFL): (a) the force reaction between the depth and force and (b) the adhesive forces between the depth and force.
TL at maximum depths of 0.77, 1.15, 1.44, 1.63, and 1.72 nm are 360, 638, 1009, 1235, and 1557 nN, respectively. The adhesive forces for TFL at maximum depths of 0.77, 1.15, 1.44, 1.63, and 1.72 nm are 285, 798, 944, 1668, and 2107 nN, respectively. The adhesive force for TL is smaller than that for TFL because the free layer has good atomic mobility. The adhesive effect between the free layer and the indenter increased with increasing indentation depth. The relationship between the indentation depth and the contact pressure for TL and TFL are shown in Fig. 2. The contact pressure applied by the indenter was calculated for an elastic surface using P i2w F i P ð3Þ Pc ¼ Fi 2 i2Z 24:56 ðhmax v S Þ where W represents all atoms of the film belonging to the contact zone, Fi is the out-of-balance force on atom i,
and hmax and S are the maximum indentation depth and unloading stiffness, respectively. v is a geometric constant of the indenter; v = 0.75 for a Vickers indenter. The contact pressure increased with increasing indentation depth because the hardness increased with increasing depth. TL was harder than TFL because it had greater stiffness. In addition, the substrate effect increased when the nanoindentation depth was about 1.6 nm. The contact pressures for TL at maximum depths of 0.77, 1.15, 1.44, 1.63, and 1.72 nm are10.2, 15.2, 19.8, 24.4, and 26 GPa, respectively. The contact pressures for TFL at maximum depths of 0.77, 1.15, 1.44, 1.63, and 1.72 nm are 6.6, 7.2, 7.6, 11.9, and 12.2 GPa, respectively. These contact pressures for TFL are similar to these reported in the literature, as shown in Table 1. Thus, the TFL model was selected for the MD simulation. 3.2. Maximum forces and adhesive effects For the MD simulation, we first investigated the effects of the maximum forces and the adhesive forces at the maximum depths of dynamic nanoindentation. The maximum force is the maximum force required for nanoindentation to indent the maximum depth. Adhesive forces act between the indenter and the sample’s surface atoms, meaning that the external force of the workpiece atoms is exerted onto the indenter. The adhesive force is the force that is required to separate the indentation chip and the indenter atoms. The depth versus force relationships for various maximum forces and adhesive forces of Al films are shown in Fig. 3a. The maximum depths of dynamic nanoindentation include 0.77, 1.15, 1.44, and 1.63 nm at a temperature of 300 K. The results show that the maximum forces of Al at the maximum depths of 0.77, 1.15, 1.44, and 1.63 nm are 437, 1168, 2461, and 3400 nN, respectively. The adhesive forces of Al at the maximum depths of 0.77, 1.15, 1.44, and 1.63 nm are 360, 638, 1009, and 1235 nN, respectively. The maximum forces increased with increasing nanoinden-
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Table 1 Comparison of aluminum characteristics. Item
[19]
[20]
[21]
[22]
[23]
This study
Indenter Indent speed Surface/material Contact pressure (GPa) Max. depth Temperature
Berkovich – Al/0 0 1 7.0 (E)
Sharp edge 500 m s1 Al/1 1 1 4.8 (S)
Sharp tip 300 m s1 Al/1 0 0 6.0 (S)
Wedge-like cylindrical – Al 4.3 (S)
Vickers 100 m s1 Al/0 0 1 6.6 (S)
50 nm 300 K
1.6 nm 293 K
0.5 nm 300 K
Berkovich 10 m s1 Al/1 0 0 4.2 (S) 1.45 (E) 1.5 nm 300 K
7.8 nm –
0.77 nm 300 K
Note: The symbols S, E, and – are the results of simulation, experiment, and unknown, respectively.
400
Maximum loads Adhesive forces
Force [ nN ]
300
200
100
0 0.6
0.8
1.0
1.2
1.4
1.6
1.8
Depth [ nm ] (a) 300
Maximun loads 250
Adhesive forces
Force [ nN ]
200 150
atoms diffusing and stacking around the indenter tip. The stiffnesses of dynamic nanoindentation at the maximum depths of 0.77, 1.15, 1.44, and 1.63 nm are 4059, 4718, 5595, and 5815 N m1, respectively. The stiffness value increased with increasing nanoindentation depth because the atoms became compressed to approach the fixed layer. In addition, the elastic energy of the atomic structure increased with increasing indentation force or depth. The plastic energy of adhesive force depends on the indenter’s geometry, the operation conditions, the material’s hardness, and the maximum force [24]. The velocity versus the force relationships at various maximum forces and adhesive forces are shown in Fig. 3b. The maximum forces for Al film at velocities of 96, 105, 118, 125, and 132 m s1 are 189, 605, 1478, 2342, and 2838 nN, respectively, at a temperature of 300 K. The adhesive forces for Al film at velocities of 96, 105, 118, 125, and 132 m s1 are 186, 445, 727, 976, and 1189 nN, respectively. The maximum forces and the adhesive forces increased with increasing nanoindentation velocity. The force of viscosity in internal atoms increased with increasing indentation load and depth. This is due to a larger structural recovery governed by the dislocation and slip mechanism with increasing nanoindentation velocity.
100
3.3. Multi-step load/unload cycle and dislocation structures 50 0
90
100
110
120
130
140
Velocity [ m/s ]
(b) Fig. 3. Maximum forces and adhesive forces for Al film at a temperature of 300 K: (a) the depth versus force relationship, and (b) the nanoindentation velocity versus force relationship.
tation depth, and the adhesive forces increased with increasing maximum force. This is a result of work hardening, which is the accumulation of pile-up and dislocations of atomic structures. Atomic structure deformation went from elastic to plastic when the maximum forces and the nanoindentation depths were increased. Consequently, adhesive forces and residual depths were enhanced due to
Fig. 4 shows the multi-step load/unload cycle curves with three repetitions for investigating the dynamic behaviors of Al at the maximum depths of 0.76, 1.15, 1.44, and 1.73 nm. There was a small stress drop that produced homogeneous slip planes in defect structures due to the surface relaxation of atoms. The loading forces increased with increasing indentation depth for all indentation depths. However, the maximum forces for all indentation depths decreased with an increasing number of load/unload cycles. After the indenter was removed from the Al film, Al atoms adhered in the vicinity of the indenter tip. The adhesive and elastic forces dominated the unloading behavior. The adhesive forces increased with increasing indentation depth for all indentation depths. However, the adhesive forces decreased with an increasing number of load/unload cycles for all indentation depths. The Al atoms had larger structural recovery when an external force was applied. The
T.-H. Fang et al. / Acta Materialia 57 (2009) 3341–3348
600 500
0.76 nm
1.15 nm
1.44 nm
1.72 nm
Force [ nN ]
400 300
jumpcontact 200
1.72 nm
Slip
1.44 nm 1.15 nm
100
0.76 nm
0 -100
unload
load -200 0
25
50
75
100
Time step [ ps ] Fig. 4. Dynamic behavior of the multi-step load/unload cycle for Al film at maximum depths of 0.76, 1.15, 1.44, and 1.72 nm.
recovery force decreased with an increasing number of multi-step load/unload cycles. For example, the maximum forces for a load/unload cycle with three repetitions at the nanoindentation depth of 1.73 nm are 4474, 3650, and 2575 nN, respectively. The adhesive forces for load/unload cycle with three repetitions at the nanoindentation depth of 1.72 nm are 1577, 1480, and 1237 nN, respectively. The maximum force and the adhesive forces decreased with an increasing number of indentations due to the dislocation and the defect structures formed after the initial indention. The indenter easily indented the molecular surface to the maximum depth at the next indentation test. Fig. 5 shows a partial side view of MD behavior during deformation processes. The Vickers indenter indents and releases the molecular models at different indentation times. The indenter surface has a potential energy that attracts Al atoms when the indenter begins to approach
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the material surface. This is called jump-contact behavior [25], as shown in Fig. 5a. As the indentation depth is increasing, the rupture region initiated from a perfect crystalline is induced by the indenter, as shown in Fig. 5b. Some defects, such as vacancies, dislocations, and plastic indents, can be identified by the evolution of the displacement variation of the indenter. The atoms of amorphous disordered debris piled up around the indented region and a clear side-flow on the lateral side of the indenter was found when the indenter reached the maximum depth, as shown in Fig. 5c. A higher degree of plastic deformation took place after nanoindentation. The indentation shape is formed according to the indenter geometry, with atoms being displaced to the periphery of the contact. The atomic structure of Al is malleable and ductile due to the presence and movement of dislocations within its structure. Malleability and ductility describe the extent to which materials can be deformed upon the applications of compressive and tensile forces, respectively. Fig. 6a shows the top-view atomic configurations of the plastic groove in the XY plane at a temperature of 300 K after Vickers indention. A debris pile-up along the dimple fringe and an amorphous structure around the dimple fringe were found. The gliding of a prismatic dislocation loop mediated permanent deformation far away from the contact surface. The maximum width of the glide bands on the interface of the {1 1 1} and <1 1 0> slip systems was about 1 nm at an angle of 45°. Based on dislocation theory, the slip systems in the Al lattice are the (1 1 1) plane along the ½1 1 2 direction and the ð1 1 1Þ plane along the ½1 1 2 direction. In general, a perfect dislocation, splitting up into two partial dislocations, can be written asb1 = b2 + b3 1 1 1 ½1 1 0 ! ½1 1 2 þ ½1 1 2 3 6 6
Fig. 5. Partial side view of the instantaneous evolution of MD behavior during nanoindentation processes.
ð4Þ
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T.-H. Fang et al. / Acta Materialia 57 (2009) 3341–3348
280
Tilt
Slip direction
Probe
210
0°
5°
10°
15°
30°
Sample
Force [ nN ]
140
Dimple fringe Amorphous groove
70 0 -70
(a) -140 D
[101]
[12 1
]
B
−
δ
1] [21
α
D
−−
C [110]
[110]
(111) −
[110]
−−
b3
(111) b2
(b) Fig. 6. Structure of dislocation nucleation: (a) top-view atomic configurations, including glide band, slip direction, and pile-up region slip; (b) schematic diagram of two Shockley partial dislocations.
According to Frank’s rule, the repulsive force between the two Shockley partials can be approximated by Fr ¼
100
[001]
]
(111)
b3
[10 1
1]
[112]
[01 β
D
1]
75
Fig. 7. The multi-step load/unload cycle curves of Al with indenter tilt angles of 0°, 5°, 10°, 15°, and 30° at the indentation depth of 0.76 nm with three repetitions.
[011]
[110]
[12
50
b2
] γ
(111)
A
25
Time step [ ps ]
[112] 1 [21
0
l ðb2 b3 Þ 2pd
ð5Þ
where d is the spacing between the two Shockley partials, and l is the shear modulus of the material. In Thompson tetrahedron notation, the two Shockley partials are presented as Cd and Dc, respectively, as shown in Fig. 6b. The (1 1 1) planes are stacked on a closely packed sequence ABCABC. Therefore, the Burger’s vectors of the two Shockley partials nucleated beneath the indenter are ½ 1 1 2 in the (1 1 1) plane and ½ 1 1 2 in the ½ 1 1 1 plane. Dislocations of the opposite sign on the same slip plane attract each other, run together, and annihilate each other. There is a tendency for the two Shockley partials to separate, creating a region of a stacking fault between them. The normal ABCABC stacking then changes to an ABCACABC stacking, since a B layer glided into the C-position. 3.4. Tilt angles of the indenter and slip vectors Typically, in an experimental nanoindentation test, it is difficult to vertically indent the sample without change the
tilt angle of indenter. The tilt angle of the indenter influences the indentation force and the contact areas during the experiment. Therefore, it is better to study the effect of the indenter’s tilt angle using MD simulation. Fig. 7 shows the multi-step load/unload cycle curves with three repetitions of Al with tilt angles of 0°, 5°, 10°, 15°, and 30° at the indentation depth of 0.76 nm. A force applied along the normal direction, a tilt angle of 0°, produces a smaller reaction force on the surface atoms. However, the reaction force and adhesive force increased with increasing tilt angle of the indenter. This was due to the contact area between the indenter and the sample increasing with increasing tilt angle. For example, the maximum forces at the tilt angles of 0°, 5°, 10°, 15°, and 30° at the first load/ unload cycle are 43, 53, 65, 69, and 196 nN, respectively. The adhesive forces at the tilt angles of 0°, 5°, 10°, 15°, and 30° are 21, 22, 26, 27, and 76 nN, respectively. The maximum forces and adhesive forces decreased with an increasing number of indentations, indicating that the dislocation and the defect structures of the molecular model are induced by shifted atomic planes after the initial indentation. The indenter then easily indented on the molecular surface at next indentation. A stress that is normal to the slip plane is greater for homogeneous dislocation nucleation in a single crystal. In order to identify and characterize the dislocations nucleated during nanoindentation, we used slip vector analysis, which provides information about Burgers vectors of dislocations. The slip vector approach was first applied to MD studies of nanoindentation by Zimmerman et al. [26]. The slip vector for atom i is defined as n 1 X ðp Rij Þ ð6Þ si ¼ ns j–i ij where ns is the number of slipped neighbors, and pij and Rij are the vector differences in linking atom i and all its n near-
T.-H. Fang et al. / Acta Materialia 57 (2009) 3341–3348
est neighbors j at the current and reference configurations, respectively. The spatial distributions of the slip vector moduli |si| around the nanoindentation trace are shown in Fig. 8. The slip distances include the nanoindentation ˚ for slip vectors of depths of 7.7, 11.5, 14.4, and 16.3 A the Al film. The color indicates the slip distance of Al atoms in angstroms. Green atoms correspond to values of the slip vectors between the stacking fault value and zero, while red and blue atoms have a slip vector ranging between a complete lattice parameter and the stacking fault value. The slip mechanism shows that dislocation loops nucleated on the four (1 1 1) planes and extended into the solid. The sunken shapes in the middle, forming a square-based pyramidal defect structure, are at the maximum depth of indentation. The pyramidal defect structure of fcc is in excellent agreement with experimentally observed permanent deformation structures [27]. The corresponding slip vector of the atoms on one of the slip planes is (1 1 1). The dislocation nucleated on the (1 1 1) plane is therefore the 1=6½ 1 1 2 Shockley partial. Similarly, 1=6½1 1 2 is nucleated on the plane ð 1 1 1Þ. Therefore, there are low-energy sessile stair-rod dislocations in the pyramid of intrinsic stacking faults on (1 1 1) planes. The sessile stair-rods act as barriers to gliding, giving rise to the observed strain hardening during nanoindentation beyond the first yield.
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4. Conclusion The deformation mechanism for defect structures under nanoindentation on an Al (0 0 1) plane was investigated using MD simulation. A series of mechanical behaviors, such as multi-step load/unload cycle, indentation velocity, tilt angles of the indenter, and slip vectors, were explored. The Al material had dislocation slip lines on the (1 1 1) plane along the (1 1 0) direction. A debris pile-up along the dimple fringe and an amorphous structure around the dimple fringe were found. Dislocation caused a minor load drop in the load/unload cycle curve. The nanoindentation force in the normal direction has a smaller force component that acts on the surface atoms. The applied force and adhesive force increased with increasing tilt angle of the indenter. The effects of the tilt angle of the indenter and slip vectors should be considered for nanoindentation experiments. The nucleation split into pairs of Shockley partial dislocations, giving rise to peculiar configurations at the surface of hillocks. Acknowledgements This work was partially supported by the National Science Council of Taiwan, under Grant No. NSC 96-2628-E150-005-MY3.
˚ , (b) 11.5 A ˚ , (c) 14.4 A ˚ , and (d) 16.3 A ˚. Fig. 8. The slip vectors of Al film at a nanoindentation depth of (a) 7.7 A
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Acta Materialia 57 (2009) 3341–3348 www.elsevier.com/locate/actamat
Dynamic characteristics of nanoindentation using atomistic simulation Te-Hua Fang a,*, Wen-Yang Chang b, Jian-Jin Huang a a
Institute of Mechanical and Electromechanical Engineering, National Formosa University, Yunlin 632, Taiwan b Microsystems Technology Center, Industrial Technology Research Institute, Tainan 709, Taiwan Received 8 January 2009; received in revised form 30 March 2009; accepted 30 March 2009 Available online 4 May 2009
Abstract Atomistic simulations are used to investigate how the nanoindentation mechanism influences dislocation nucleation under molecular dynamic behavior on the aluminum (0 0 1) surface. The characteristics of molecular dynamics in terms of various nucleation criteria are explored, including various molecular models, a multi-step load/unload cycle, deformation mechanism of atoms, tilt angle of the indenter, and slip vectors. Simulation results show that both the plastic energy and the adhesive force increase with increasing nanoindentation depths. The maximum forces for all indentation depths decrease with increasing multi-step load/unload cycle time. Dislocation nucleation, gliding, and interaction occur along Shockley partials on (1 1 1) slip planes. The indentation force applied along the normal direction, a tilt angle of 0°, is smaller than the force component that acts on the surface atoms. The corresponding slip vector of the atoms in the (1 1 1) plane has low-energy sessile stair-rod dislocations in the pyramid of intrinsic stacking faults. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Nanoindentation; Aluminum; Dynamic phenomena; Molecular dynamics
1. Introduction Molecular dynamics (MD) simulation for nanotechnology and nanoscience analyses of film materials has been used to interpret the deformation mechanism of defect structures [1,2]. Nanoindentation contact plays a key role in engineering reliability for investigating the mechanical properties of MD behavior at the atomic scale. Many nanoindentation [3,4] experiments and nanoscratch [5,6] have been performed to investigate the molecular deformation mechanism in material science. Experimental tests show the z mechanical and processing responses of nano-materials, but they are unsuitable for investigating the defect mechanism at the atomic scale. However, MD simulation can be effective in simulating the atomistic characteristics of deformation and microscopic fracture processes. Kizler et al. [7] reported the deformation mechanism of an ultra-hard carbide layer under nanoindentation with the aid of MD analysis. Oyeon et al. [8] used MD to analyze
*
Corresponding author. Tel.: +886 5 631 5395; fax: +886 5 631 5397. E-mail address: [email protected] (T.-H. Fang).
the dislocation nucleation in various orientations and dislocation deformations at the surface of nickel single crystals. Jin et al. [9] investigated the multiscale simulation of onset plasticity during the nanoindentation of Al. Although MD simulations have recently been used to study the mechanisms of onset plasticity, dislocation nucleation, and elastic–plastic deformation [9–11], few studies have focused on dynamic nanoindentation [12,13]. Youngmin et al. [14] studied the defect nucleation and the evolution of incipient plasticity in Al material under nanoindentation using atomistic simulation. In an experimental nanoindentation test, it is difficult to vertically indent the sample without changing the tilt angle of the indenter. The effect of the tilt angle of the indenter is difficult to obtain from experiments. The phenomenon of the indenter’s tilt angles is more meaningful and interesting to study using MD simulation. Understanding the defect mechanism at the atomic scale through accurate modeling and simulation is an essential premise. This study uses MD simulation to investigate various molecular models, the multi-step load/unload cycle, the deformation mechanism of atoms, tilt angles of the indenter, and slip vectors. The aluminum (Al) model is chosen
1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.03.048
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for MD simulation because it is widely used in many industries, from aerospace to automotive to microelectronics. For the MD simulation, the effects of the maximum forces and the adhesive forces at the maximum depths of dynamic nanoindentation were investigated. A thermal layer and a thermal layer with a free layer were used to analyze the mechanism of surface plasticity. The effects of the multistep load/unload cycle of nanoindentation with various depths and tilt angles of the indenter were investigated. 2. Simulation methodology The model sizes of the MD simulation in the x, y, and z directions were 10.93, 10.93, and 4.86 nm, respectively, using a face-centered cubic (fcc) bulk crystal. The indenter was a Vickers indenter consisting of 2992 diamond atoms in the form of a square-based pyramid with an angle of 22° between the indenter face and the contact surface. The Vickers indenter is not perfect for MD simulation because of its larger deviation. A correction factor [15] related to the lack of any revolution paraboloid during MD simulation must be taken into account. According to Sakharova et al. [16], the characteristic analyses of molecular deformation after MD simulation are similar compared with Berkovich, Vickers, and conical indenters. For the simulation conditions, the indenter was assumed to be a rigid indenter. The model was initially assumed to have a well-defined atomic structure before thermal equilibration. The indented surface was the (0 0 1) plane of Al for all the numerical experiments. The periodic boundary conditions of the sample were used in the transverse directions. The bottom of the specimen, two layers of atoms, was fixed in space to prevent the substrate from being moved. Periodic boundary conditions were enforced in the x and y directions; there was no periodic boundary condition along the z direction. The velocities of the atoms of the thermal control layers are given by the Maxwell–Boltzmann distribution [17] of the prescribed substrate temperature. The position of the incident atoms was random in the x and y directions, and the z direction of incident atoms was at a 20-fold lattice length above the substrate surface. During the load/unload cycles, the indenter was moved at a constant speed of 100 m s1. The model equilibrated to its minimum energy configuration at 300 K. For most MD simulations of nanoindentation, the indentation velocities are from 1 to 100 m s1 due to the limitations of computation time and power [14]. The dynamics of the model were evaluated by integrating the Newtonian equations of motion using the Verlet algorithm [18] with a time step of 1 fs. The force acting on an individual atom was obtained by summing the forces contributed by the surrounding atoms. The Morse two-body potential and the secondmoment approximation of the tight-binding many-body (TB-SMA) potential were adopted to model the atomic interactions among the Al atoms. The Morse potential only considers the interaction between two atoms without including the simultaneous influence of their neighboring
atoms. The Morse potential energy, UMorse (rij), can be described with three parameters as: ð1Þ U Morse ðrij Þ ¼ D e2aðrij r0 Þ 2eaðrij r0 Þ where D, r0, and rij are the cohesion energy of exchange interaction, the equilibrium distance, and the separation distance between atoms i and j, respectively. a is fitted to the bulk modulus of the material. For C–Al atoms, the values of D, a, and r0 are 0.8092 eV, 0.0186 nm1, and 2.970 nm, respectively. The second-moment approximation of the tight-binding many-body potential, U(rij), contains a repulsive pair potential and a cohesive band energy term, as follows: Uðrij Þ ¼
N X i¼1
"
X j
1=2 # rij rij n exp 2q þ Aexp p 1 r0 r0 2
ð2Þ
where n, r0, and N are the effective hopping integral, the first-neighbor distance, and the number of atoms considered, respectively. The terms n, q, p, and A are fitted to the experimental values of cohesive energy. For Al–Al atoms, the values of n, q, p, r0, and A are 1.316 eV, 2.516, 8.612, 0.2864 nm, and 0.1221 eV, respectively. 3. Results and discussion 3.1. Comparison of molecular models For the comparison of molecular models, a thermal layer (TL) and a thermal layer with a free layer (TFL) were used to analyze the mechanical behavior and the underlying mechanism of surface plasticity. During the MD simulation process, the TL atoms remained at a constant temperature, indicating that all atomic heat is uniform after the heat is induced by indenter indentation. All atoms in the free layer are independent and can vary their temperatures, meaning that temperature depends on the friction force between the indenter and the atoms. The boundary temperature at the thermal layer and the free layer is initially at 300 K. The force versus depth relationships for TL and TFL models are shown in Fig. 1a. The indentation forces for TL model at maximum depths of 0.77, 1.15, 1.44, 1.63, and 1.72 nm are 437, 1168, 2461, 3400, and 4474 nN, respectively. The indentation forces for TFL model at maximum depths of 0.77, 1.15, 1.44, 1.63, and 1.72 nm are 285, 798, 944, 1668, and 2107 nN, respectively. The interatomic bonding forces are relatively weak in TFL because the atoms have more freedom and vibrational energy than those in TL. When a small force was applied to the indenter, it easily penetrated the atoms in the adjustable temperature layer. The Al substrate became softer, indicating that the dislocation was easier to deform. Dislocations mean that the substrate can be rolled into grain boundary layers when the atoms slip past each other. Fig. 1b shows the force versus depth relationship of adhesive forces for TL and TFL. The adhesive forces for
T.-H. Fang et al. / Acta Materialia 57 (2009) 3341–3348
28
500 TL
TFL
Contact pressure [ GPa ]
Force [ nN ]
Probe Thermal layer Fixed layer
300
(TL model) 200
100
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Depth [ nm ]
Probe Free layer Thermal layer Fixed layer
Force [ nN ]
(TFL model)
80
40
0 0.6
TL
0.8
1.0
1.2
20 16 12 8 4 0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Fig. 2. Relationships between the indentation depth and the contact pressure for TL and TFL.
200
160
TFL
Indentation depth [ nm ]
(a)
120
TL
24
400
0 0.6
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1.4
TFL
1.6
1.8
2.0
Depth [ nm ]
(b) Fig. 1. Molecular models of the thermal layer (TL) and the thermal layer with free layer (TFL): (a) the force reaction between the depth and force and (b) the adhesive forces between the depth and force.
TL at maximum depths of 0.77, 1.15, 1.44, 1.63, and 1.72 nm are 360, 638, 1009, 1235, and 1557 nN, respectively. The adhesive forces for TFL at maximum depths of 0.77, 1.15, 1.44, 1.63, and 1.72 nm are 285, 798, 944, 1668, and 2107 nN, respectively. The adhesive force for TL is smaller than that for TFL because the free layer has good atomic mobility. The adhesive effect between the free layer and the indenter increased with increasing indentation depth. The relationship between the indentation depth and the contact pressure for TL and TFL are shown in Fig. 2. The contact pressure applied by the indenter was calculated for an elastic surface using P i2w F i P ð3Þ Pc ¼ Fi 2 i2Z 24:56 ðhmax v S Þ where W represents all atoms of the film belonging to the contact zone, Fi is the out-of-balance force on atom i,
and hmax and S are the maximum indentation depth and unloading stiffness, respectively. v is a geometric constant of the indenter; v = 0.75 for a Vickers indenter. The contact pressure increased with increasing indentation depth because the hardness increased with increasing depth. TL was harder than TFL because it had greater stiffness. In addition, the substrate effect increased when the nanoindentation depth was about 1.6 nm. The contact pressures for TL at maximum depths of 0.77, 1.15, 1.44, 1.63, and 1.72 nm are10.2, 15.2, 19.8, 24.4, and 26 GPa, respectively. The contact pressures for TFL at maximum depths of 0.77, 1.15, 1.44, 1.63, and 1.72 nm are 6.6, 7.2, 7.6, 11.9, and 12.2 GPa, respectively. These contact pressures for TFL are similar to these reported in the literature, as shown in Table 1. Thus, the TFL model was selected for the MD simulation. 3.2. Maximum forces and adhesive effects For the MD simulation, we first investigated the effects of the maximum forces and the adhesive forces at the maximum depths of dynamic nanoindentation. The maximum force is the maximum force required for nanoindentation to indent the maximum depth. Adhesive forces act between the indenter and the sample’s surface atoms, meaning that the external force of the workpiece atoms is exerted onto the indenter. The adhesive force is the force that is required to separate the indentation chip and the indenter atoms. The depth versus force relationships for various maximum forces and adhesive forces of Al films are shown in Fig. 3a. The maximum depths of dynamic nanoindentation include 0.77, 1.15, 1.44, and 1.63 nm at a temperature of 300 K. The results show that the maximum forces of Al at the maximum depths of 0.77, 1.15, 1.44, and 1.63 nm are 437, 1168, 2461, and 3400 nN, respectively. The adhesive forces of Al at the maximum depths of 0.77, 1.15, 1.44, and 1.63 nm are 360, 638, 1009, and 1235 nN, respectively. The maximum forces increased with increasing nanoinden-
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Table 1 Comparison of aluminum characteristics. Item
[19]
[20]
[21]
[22]
[23]
This study
Indenter Indent speed Surface/material Contact pressure (GPa) Max. depth Temperature
Berkovich – Al/0 0 1 7.0 (E)
Sharp edge 500 m s1 Al/1 1 1 4.8 (S)
Sharp tip 300 m s1 Al/1 0 0 6.0 (S)
Wedge-like cylindrical – Al 4.3 (S)
Vickers 100 m s1 Al/0 0 1 6.6 (S)
50 nm 300 K
1.6 nm 293 K
0.5 nm 300 K
Berkovich 10 m s1 Al/1 0 0 4.2 (S) 1.45 (E) 1.5 nm 300 K
7.8 nm –
0.77 nm 300 K
Note: The symbols S, E, and – are the results of simulation, experiment, and unknown, respectively.
400
Maximum loads Adhesive forces
Force [ nN ]
300
200
100
0 0.6
0.8
1.0
1.2
1.4
1.6
1.8
Depth [ nm ] (a) 300
Maximun loads 250
Adhesive forces
Force [ nN ]
200 150
atoms diffusing and stacking around the indenter tip. The stiffnesses of dynamic nanoindentation at the maximum depths of 0.77, 1.15, 1.44, and 1.63 nm are 4059, 4718, 5595, and 5815 N m1, respectively. The stiffness value increased with increasing nanoindentation depth because the atoms became compressed to approach the fixed layer. In addition, the elastic energy of the atomic structure increased with increasing indentation force or depth. The plastic energy of adhesive force depends on the indenter’s geometry, the operation conditions, the material’s hardness, and the maximum force [24]. The velocity versus the force relationships at various maximum forces and adhesive forces are shown in Fig. 3b. The maximum forces for Al film at velocities of 96, 105, 118, 125, and 132 m s1 are 189, 605, 1478, 2342, and 2838 nN, respectively, at a temperature of 300 K. The adhesive forces for Al film at velocities of 96, 105, 118, 125, and 132 m s1 are 186, 445, 727, 976, and 1189 nN, respectively. The maximum forces and the adhesive forces increased with increasing nanoindentation velocity. The force of viscosity in internal atoms increased with increasing indentation load and depth. This is due to a larger structural recovery governed by the dislocation and slip mechanism with increasing nanoindentation velocity.
100
3.3. Multi-step load/unload cycle and dislocation structures 50 0
90
100
110
120
130
140
Velocity [ m/s ]
(b) Fig. 3. Maximum forces and adhesive forces for Al film at a temperature of 300 K: (a) the depth versus force relationship, and (b) the nanoindentation velocity versus force relationship.
tation depth, and the adhesive forces increased with increasing maximum force. This is a result of work hardening, which is the accumulation of pile-up and dislocations of atomic structures. Atomic structure deformation went from elastic to plastic when the maximum forces and the nanoindentation depths were increased. Consequently, adhesive forces and residual depths were enhanced due to
Fig. 4 shows the multi-step load/unload cycle curves with three repetitions for investigating the dynamic behaviors of Al at the maximum depths of 0.76, 1.15, 1.44, and 1.73 nm. There was a small stress drop that produced homogeneous slip planes in defect structures due to the surface relaxation of atoms. The loading forces increased with increasing indentation depth for all indentation depths. However, the maximum forces for all indentation depths decreased with an increasing number of load/unload cycles. After the indenter was removed from the Al film, Al atoms adhered in the vicinity of the indenter tip. The adhesive and elastic forces dominated the unloading behavior. The adhesive forces increased with increasing indentation depth for all indentation depths. However, the adhesive forces decreased with an increasing number of load/unload cycles for all indentation depths. The Al atoms had larger structural recovery when an external force was applied. The
T.-H. Fang et al. / Acta Materialia 57 (2009) 3341–3348
600 500
0.76 nm
1.15 nm
1.44 nm
1.72 nm
Force [ nN ]
400 300
jumpcontact 200
1.72 nm
Slip
1.44 nm 1.15 nm
100
0.76 nm
0 -100
unload
load -200 0
25
50
75
100
Time step [ ps ] Fig. 4. Dynamic behavior of the multi-step load/unload cycle for Al film at maximum depths of 0.76, 1.15, 1.44, and 1.72 nm.
recovery force decreased with an increasing number of multi-step load/unload cycles. For example, the maximum forces for a load/unload cycle with three repetitions at the nanoindentation depth of 1.73 nm are 4474, 3650, and 2575 nN, respectively. The adhesive forces for load/unload cycle with three repetitions at the nanoindentation depth of 1.72 nm are 1577, 1480, and 1237 nN, respectively. The maximum force and the adhesive forces decreased with an increasing number of indentations due to the dislocation and the defect structures formed after the initial indention. The indenter easily indented the molecular surface to the maximum depth at the next indentation test. Fig. 5 shows a partial side view of MD behavior during deformation processes. The Vickers indenter indents and releases the molecular models at different indentation times. The indenter surface has a potential energy that attracts Al atoms when the indenter begins to approach
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the material surface. This is called jump-contact behavior [25], as shown in Fig. 5a. As the indentation depth is increasing, the rupture region initiated from a perfect crystalline is induced by the indenter, as shown in Fig. 5b. Some defects, such as vacancies, dislocations, and plastic indents, can be identified by the evolution of the displacement variation of the indenter. The atoms of amorphous disordered debris piled up around the indented region and a clear side-flow on the lateral side of the indenter was found when the indenter reached the maximum depth, as shown in Fig. 5c. A higher degree of plastic deformation took place after nanoindentation. The indentation shape is formed according to the indenter geometry, with atoms being displaced to the periphery of the contact. The atomic structure of Al is malleable and ductile due to the presence and movement of dislocations within its structure. Malleability and ductility describe the extent to which materials can be deformed upon the applications of compressive and tensile forces, respectively. Fig. 6a shows the top-view atomic configurations of the plastic groove in the XY plane at a temperature of 300 K after Vickers indention. A debris pile-up along the dimple fringe and an amorphous structure around the dimple fringe were found. The gliding of a prismatic dislocation loop mediated permanent deformation far away from the contact surface. The maximum width of the glide bands on the interface of the {1 1 1} and <1 1 0> slip systems was about 1 nm at an angle of 45°. Based on dislocation theory, the slip systems in the Al lattice are the (1 1 1) plane along the ½1 1 2 direction and the ð1 1 1Þ plane along the ½1 1 2 direction. In general, a perfect dislocation, splitting up into two partial dislocations, can be written asb1 = b2 + b3 1 1 1 ½1 1 0 ! ½1 1 2 þ ½1 1 2 3 6 6
Fig. 5. Partial side view of the instantaneous evolution of MD behavior during nanoindentation processes.
ð4Þ
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280
Tilt
Slip direction
Probe
210
0°
5°
10°
15°
30°
Sample
Force [ nN ]
140
Dimple fringe Amorphous groove
70 0 -70
(a) -140 D
[101]
[12 1
]
B
−
δ
1] [21
α
D
−−
C [110]
[110]
(111) −
[110]
−−
b3
(111) b2
(b) Fig. 6. Structure of dislocation nucleation: (a) top-view atomic configurations, including glide band, slip direction, and pile-up region slip; (b) schematic diagram of two Shockley partial dislocations.
According to Frank’s rule, the repulsive force between the two Shockley partials can be approximated by Fr ¼
100
[001]
]
(111)
b3
[10 1
1]
[112]
[01 β
D
1]
75
Fig. 7. The multi-step load/unload cycle curves of Al with indenter tilt angles of 0°, 5°, 10°, 15°, and 30° at the indentation depth of 0.76 nm with three repetitions.
[011]
[110]
[12
50
b2
] γ
(111)
A
25
Time step [ ps ]
[112] 1 [21
0
l ðb2 b3 Þ 2pd
ð5Þ
where d is the spacing between the two Shockley partials, and l is the shear modulus of the material. In Thompson tetrahedron notation, the two Shockley partials are presented as Cd and Dc, respectively, as shown in Fig. 6b. The (1 1 1) planes are stacked on a closely packed sequence ABCABC. Therefore, the Burger’s vectors of the two Shockley partials nucleated beneath the indenter are ½ 1 1 2 in the (1 1 1) plane and ½ 1 1 2 in the ½ 1 1 1 plane. Dislocations of the opposite sign on the same slip plane attract each other, run together, and annihilate each other. There is a tendency for the two Shockley partials to separate, creating a region of a stacking fault between them. The normal ABCABC stacking then changes to an ABCACABC stacking, since a B layer glided into the C-position. 3.4. Tilt angles of the indenter and slip vectors Typically, in an experimental nanoindentation test, it is difficult to vertically indent the sample without change the
tilt angle of indenter. The tilt angle of the indenter influences the indentation force and the contact areas during the experiment. Therefore, it is better to study the effect of the indenter’s tilt angle using MD simulation. Fig. 7 shows the multi-step load/unload cycle curves with three repetitions of Al with tilt angles of 0°, 5°, 10°, 15°, and 30° at the indentation depth of 0.76 nm. A force applied along the normal direction, a tilt angle of 0°, produces a smaller reaction force on the surface atoms. However, the reaction force and adhesive force increased with increasing tilt angle of the indenter. This was due to the contact area between the indenter and the sample increasing with increasing tilt angle. For example, the maximum forces at the tilt angles of 0°, 5°, 10°, 15°, and 30° at the first load/ unload cycle are 43, 53, 65, 69, and 196 nN, respectively. The adhesive forces at the tilt angles of 0°, 5°, 10°, 15°, and 30° are 21, 22, 26, 27, and 76 nN, respectively. The maximum forces and adhesive forces decreased with an increasing number of indentations, indicating that the dislocation and the defect structures of the molecular model are induced by shifted atomic planes after the initial indentation. The indenter then easily indented on the molecular surface at next indentation. A stress that is normal to the slip plane is greater for homogeneous dislocation nucleation in a single crystal. In order to identify and characterize the dislocations nucleated during nanoindentation, we used slip vector analysis, which provides information about Burgers vectors of dislocations. The slip vector approach was first applied to MD studies of nanoindentation by Zimmerman et al. [26]. The slip vector for atom i is defined as n 1 X ðp Rij Þ ð6Þ si ¼ ns j–i ij where ns is the number of slipped neighbors, and pij and Rij are the vector differences in linking atom i and all its n near-
T.-H. Fang et al. / Acta Materialia 57 (2009) 3341–3348
est neighbors j at the current and reference configurations, respectively. The spatial distributions of the slip vector moduli |si| around the nanoindentation trace are shown in Fig. 8. The slip distances include the nanoindentation ˚ for slip vectors of depths of 7.7, 11.5, 14.4, and 16.3 A the Al film. The color indicates the slip distance of Al atoms in angstroms. Green atoms correspond to values of the slip vectors between the stacking fault value and zero, while red and blue atoms have a slip vector ranging between a complete lattice parameter and the stacking fault value. The slip mechanism shows that dislocation loops nucleated on the four (1 1 1) planes and extended into the solid. The sunken shapes in the middle, forming a square-based pyramidal defect structure, are at the maximum depth of indentation. The pyramidal defect structure of fcc is in excellent agreement with experimentally observed permanent deformation structures [27]. The corresponding slip vector of the atoms on one of the slip planes is (1 1 1). The dislocation nucleated on the (1 1 1) plane is therefore the 1=6½ 1 1 2 Shockley partial. Similarly, 1=6½1 1 2 is nucleated on the plane ð 1 1 1Þ. Therefore, there are low-energy sessile stair-rod dislocations in the pyramid of intrinsic stacking faults on (1 1 1) planes. The sessile stair-rods act as barriers to gliding, giving rise to the observed strain hardening during nanoindentation beyond the first yield.
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4. Conclusion The deformation mechanism for defect structures under nanoindentation on an Al (0 0 1) plane was investigated using MD simulation. A series of mechanical behaviors, such as multi-step load/unload cycle, indentation velocity, tilt angles of the indenter, and slip vectors, were explored. The Al material had dislocation slip lines on the (1 1 1) plane along the (1 1 0) direction. A debris pile-up along the dimple fringe and an amorphous structure around the dimple fringe were found. Dislocation caused a minor load drop in the load/unload cycle curve. The nanoindentation force in the normal direction has a smaller force component that acts on the surface atoms. The applied force and adhesive force increased with increasing tilt angle of the indenter. The effects of the tilt angle of the indenter and slip vectors should be considered for nanoindentation experiments. The nucleation split into pairs of Shockley partial dislocations, giving rise to peculiar configurations at the surface of hillocks. Acknowledgements This work was partially supported by the National Science Council of Taiwan, under Grant No. NSC 96-2628-E150-005-MY3.
˚ , (b) 11.5 A ˚ , (c) 14.4 A ˚ , and (d) 16.3 A ˚. Fig. 8. The slip vectors of Al film at a nanoindentation depth of (a) 7.7 A
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