Requirements for Ductile-mode Machining Based on Deformation Analysis of Mono-crystalline Silicon by Molecular Dynamics Simulation

Requirements for Ductile-mode Machining Based on Deformation Analysis of Mono-crystalline Silicon by Molecular Dynamics Simulation 1

H. Tanaka1, S. Shimada1 (1), L. Anthony2 Osaka Electro-Communication University, Neyagawa, Osaka, Japan 2 Waseda University, Shinjuku-ku, Tokyo, Japan

Abstract To obtain scientific guidelines for ductile-mode machining, nano-indentation, nano-bending, and nanomachining of defect-free mono-crystalline silicon are investigated by molecular dynamics simulation. Results show that amorphous phase transformation of silicon is a key mechanism for inelastic deformation, and stable shearing of the amorphous is necessary for ductile-mode machining. Stress analysis suggests that stable shearing takes place under a compressive stress field. In practice, a sharp cutting edge tool with a large negative rake angle should be used for effective ductile-mode machining, and vibration machining should be applied for larger depths of cut as it enlarges the amorphous region in front of the cutting edge. Keywords: Simulation, Silicon, Ductile-mode machining

because the deformation of stress based on conventional continuum mechanics cannot be applied to a discontinuous atomic model. The stress of the atomic model is defined and calculated from the resultant interatomic force between the atoms on both sides of a small interface plane, the area of which is 2Lcu2Lc [4]. Here, Lc=0.543 nm is the lattice constant of silicon. The number of nearest neighbors, Nn, is used as an index of crystallinity of silicon atoms. Here, Nn is defined as the number of atoms located within a sphere whose radius is the average distance from a specific atom to the nearest and the second-nearest atoms. Nn in a crystalline phase of diamond structure is 4. For an amorphous phase, Nn is larger than 4 and on the surface or around a crack it is less than 3. Figure 1 shows the initial model for nano-indentation used to analyze the deformation of silicon under a simple stress distribution. The specimen has the dimensions 40.7 nm u 1.1 nm u 40.7 nm, and contains 90602 silicon atoms. The rigid cylindrical diamond indenter has a tip radius of 7.1 nm. Indentation is performed by a downward movement of the indenter at a speed of V=50 m/s, which corresponds to high speed machining or grinding. The specimen consists of Newtonian atoms, elastic

1 INTRODUCTION The demand for accurate, low-cost ductile-mode machining of brittle materials with improved surface quality is increasing. This is especially the case for key industrial materials such as silicon. On the other hand, it is empirically well known that chips are formed in the ductile-mode when the depth of cut is smaller than the critical value intrinsic to the work material [1]. To achieve ductile-mode machining with all brittle materials, it is essential that we understand the mechanism of material removal at an extremely small depth of cut. This in turn requires that the mechanical behavior under various deformation conditions is fully understood. To obtain scientific guidelines for ductile-mode machining, nanoindentation, three-point nano-bending, and nanomachining of defect-free mono-crystalline silicon are analyzed by molecular dynamics (MD) simulation which is a useful method to analyze the material behavior at the atomic scale [2]. 2 MOLECULAR DYNAMICS MODELING To analyze dynamic deformation and fracture behaviors of defect-free mono-crystalline silicon, MD simulations were carried out using three-dimensional models, as shown in Figures 1 to 3. All the specimens are defect-free mono-crystalline silicon with surfaces composed of {100} planes. Periodic boundaries are applied in the y direction, which corresponds to the two-lattice constant of silicon. Although a Tersoff-type three-body potential [3] is applied for the silicon structure, a Morse type two-body potential [4] is employed to express the interaction between silicon and diamond. For the conversion of the kinetic energy of silicon atoms into an equivalent temperature, the thermal energy derived from the equation of specific heat proposed by Debye is used [5]. In the thermostat layers, which absorb the heat outwards in the model, the kinetic energy of atoms is adjusted for every computation time step so as to maintain the equivalent average temperature at 293 K. In order to investigate deformation behavior, the concept of stress in the atomic model must be examined. This is

Annals of the CIRP Vol. 56/1/2007

Diamond indenter (Rigid, Radius:R) Indentation speed: V Periodic boundary Silicon {100} l z y

Newtonian atoms Elastic supported atoms Thermostat atoms (293 K)

x Figure 1: Initial model for indentation.

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doi:10.1016/j.cirp.2007.05.015

is applied. The dimensions of the models are 65.3 nm x 1.1 nm x 24.6 nm for a depth of cut of 5.2 nm, and 130.5 nm x 1.1 nm x 49.0 nm for a depth of cut of 20.2 nm. For the tool configuration, the cutting edge radii are 2.2 nm and 8.6 nm, and the rake angles are 0, 30 and -30 degrees. The cutting speed is 200 m/s. For vibration cutting, the cutting tool moves on a circle with a radius of 2.0 nm with the period of 21 ps, after a 50 nm conventional cutting distance.

support atoms, and thermostat atoms. The elastic support atoms and thermostat atoms are arranged in two and four layers around the Newtonian atoms, respectively, except at the top surface which is subjected to indentation. In the elastic support layers, atoms are elastically supported from their original positions, whose spring constant k is derived from the elastic stiffness constant of silicon, whose value is c11=1.66 u1011 Pa [5] as expressed in Equation 1. Here, l is the distance between the elastic support atom layers.

k

3

c11 L2c 2l

CRITERIA FOR PHASE TRANSFORMATION AND FRACTURE Figure 4(a) shows a snapshot of the indentation and Figure 4(b) shows a close up view of the rectangular area shown in Figure 4(a). The two types of inelastic deformation shown in the figure take place together with two different phase transformations from diamond to the other crystaline structures, and to amorphous structures in indentation. These phase transformations are termed ‘crystal phase transformation’ and ‘amorphous phase transformation’, respectively in this paper. In the loading process, the amorphous phase transformation occurs on the specimen surface from the interface between the specimen and indenter. At a certain depth of indentation (5.2 nm for the example shown in Figure4), thin layers are formed by crystal phase transformation in the bulk of the specimen and extend beneath the indenter along the (111) planes. In these layers, a shearing deformation is observed on the (111) planes. After a crystal phase is formed, a continuous stable shearing deformation is not observed in the region because it needs a higher stress. For a larger inelastic deformation caused by the crystal phase transformation, the region should be enlarged.

(1)

Figure 2 shows the initial model for three-point nanobending to analyze the brittle fracture of silicon under a simple stress distribution. The specimen has the dimensions 97.7 nm u 1.1 nm u 10.9 nm and contains 58402 silicon atoms. The rigid three cylindrical diamond indenters have tip radii of 2.22 nm. Three-point bending is performed by an upward movement of the supporting pins at both ends of the specimen and a downward movement of the central pin with a speed of 50 m/s. Here, the term ‘indentation’ is used to describe the bending process considering the motion of the indenters. The specimen consists of Newtonian atoms and thermostat atoms that are arranged in 8 layers at both ends of the specimen. Diamond indenter (Rigid) Newtonian atoms Thermostat atoms (293 K)

Loading (V=50m/s) Silicon {100}

A

Diamond indenter Periodic boundary

z y

x

Figure 2: Initial model for three-point bending. Si Figure 3 shows the initial model for nano-machining with a rectangular solid mono-crystalline silicon workpiece, which is cut in the <100> direction by a rigid diamond tool. The workpiece consists of fixed boundary atoms, thermostat atoms, and Newtonian atoms. For steadystate chip removal, the moving control volume method [6]

(a) t =110 ps (b) Atomic structure Figure 4: Two kinds of phase transformation. A : Crystal phase transformation B : Amorphous phase transformation Then the phase transformation from crystal to amorphous follows at the front end of crystal phase, as shown in Figure 4(b). The area of the amorphous region extends as the indentation depth increases. These phase transformations take place due to a severe atomic lattice distortion, which breaks the original atomic bonding among silicon atoms. Thus, the crystalline orientation of the initial model no longer has any meaning. The initiation of such inelastic deformation must have a close relationship with the distortion energy and the atomic structure. Since the distortion energy in a deformable body is proportional to the octahedral shearing stress (IJoct), the IJoct in the specimen was examined over the whole control volume [6].

Calculation area Diamond tool㩷 (Rigid)

V

Vc <100> Silicon {100} Flow in

Flow out z y

B

x

Newtonian atoms Boundary atoms Thermostat atoms (293 K) Periodic boundary

3.1 Criteria for phase transformation The possible deformation directions for crystal phase transformation are toward the three centers of hexagons constructed from six carbon atoms on the adjoining {111} plane. Figure 5(a) shows the crystal phase transformation

Figure 3: Initial model for nano-machining.

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an equivalent size of 0.30 nm due to the amorphous phase transformation. These defects form on the specimen surface and may act as crack nuclei. Crack initiation and extension take place when the crack nuclei are generated in the region where the maximum tensile stress is larger than 30 GPa.

in terms of the atomic structure and the direction of IJoct at (x, y, z)=(41Lc, Lc, 64Lc). Atoms on every two {111} planes are indicated in different colors for easy recognition of the structure. When the value of IJoct reaches 9.0-9.3 GPa and the stress direction fits to one of the three deformable directions, crystal phase transformation takes place. Figure 5(b) shows the amorphous phase transformation in terms of the atomic structure and the direction of IJoct at (x, y, z)=(38Lc, Lc, 58Lc) beneath the center of the indenter. When IJoct reaches 9.0-9.3 GPa and the stress direction is almost parallel to the {111} plane, amorphous phase transformation takes place. Successive transformation from the crystal phase to amorphous phase is caused by the change in direction of IJoct due to lattice distortion by indentation. Whether a crystal or an amorphous phase transformation takes place depends on the direction of the stress in terms of the atomic structure just before the deformation.

Critical ımax (a)

(b)

Figure 6: Crack extension and principal stresses at crack tip. 4 Critical IJoct

4.1 Effect of depth of cut Figure 7 shows the difference in stable chip removal process in terms of the distribution of the octahedral shearing stress around the cutting edge depending on the depth of cut. Red and blue dots show the position of atoms whose Nn is larger than 4 and smaller than 4, respectively. Amorphous phase transformation takes place in front of the cutting edge. At a depth of cut of 5.2 nm , a flow type chip forms as a result of the continuous and stable plastic flow by a shearing deformation in the amorphous region. However, at a depth of cut of 20.2 nm, an intermittent shear type chip is observed. As the distance between the cutting edge and free surface to be cut becomes longer, the shearing plane of the amorphous phase cannot be easily formed. When the amorphous region in front of the cutting edge grows to a certain size, a thin crystal phase transformation or an amorphous phase transformation extends toward the free surface, as shown in Figure 8. A higher tensile stress, which may cause brittle-mode machining, appears in front of the cutting edge between the intervals of the forming shear plane as the depth of cut becomes larger, as shown in Figures 9 and 10. A thin inelastic deformation forms and extends by the crystal phase transformation. However, a continuous shearing deformation in the region of the crystal phase transformation does not take place. Therefore, continuous and stable shearing deformation in the region of the amorphous phase is necessary for ductile-mode machining.

Critical IJoct

(a) Crystal (b) Amorphous Figure 5: Critical octahedral shearing stress and its direction for ‘crystal phase transformation’ and ‘amorphous phase transformation’. 3.2 Criterion for crack extension Figure 6(a) shows a snapshot of the initiation of brittle fracture in three-point bending. The crack extension takes place as follows. First, the atoms which have Nn larger than 5 stochastically appear and disappear on the surface layer on the tensile-stressed side. Then, an amorphous phase transformation takes place in the localized area on the highly tensile-stressed surface. Finally, crack initiation is observed from the small amorphous region and the crack begins to extend toward to the inside of the specimen. As the indenters seem to be separated from the specimen after the beginning of crack extension, the crack appears to extend spontaneously without any extra external force. The maximum tensile stress (ımax) among the three principal stresses in the silicon specimen can be examined over the whole control volume. Figure 6(b) shows the distribution of principal stresses at the crack tip and the atomic structure. The value of ımax reaches approximately 30 GPa just before crack initiation. This is in good agreement with that of the theoretical tensile strength [7] as expressed in Equation 2.

E 2S

29 . 9 GPa

CUTTING MECHANISM OF MONOCRYSTALLINE SILICON

(2)

Chip

Diamond tool

Based on the Griffith theory [8], the critical crack length which extends under a tensile stress of 30 GPa is estimated to be 0.30 nm after Equation 3.

cc

4JE 2 SV 2

5 nm

(3)

Amorphous phase

10 nm

(a) d =5.2 nm (b) d =20.2 nm Figure 7: Difference in chip removal process in terms of distribution of octahedral shearing stress depending on depth of cut.

Here, Ȗ is the specific surface energy of silicon, which is evaluated as 2.18 J/m2 from 3 bonds on the unit cell of the {111} plane. Based on the MD analysis, the stochastic thermal activation process in lattice vibration causes defects with

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cutting as a result of the higher shearing stress that is widely distributed in front of the cutting edge. Because vibration machining enlarges the amorphous region, it is more effective for ductile-mode machining even with a larger depth of cut.

A Cutting edge

10 nm B

Shearing deformaiton

(a) t =135 ps (b) Figure 8: Phase transformations for shearing deformation in front of cutting edge. A : Crystal phase transformation B : Amorphous phase transformation

(a) Conventional cutting (b) Vibration cutting Figure 11: Effect of vibration cutting on distribution of ochahedral shearing stress.

4.2 Effect of cutting edge radius and rake angle Figures 9 and 10 show the effects of cutting edge sharpness and rake angle on the chip removal process in terms of the distribution of maximum principal stress. As the cutting edge sharpness increases, a higher tensile stress appears at the tip of the wedge shaped amorphous region between the intervals of the shear plane forming, and may lead to crack initiation. In Figure 10(a), a crack is observed at a rake angle of 30 degrees due to a higher tensile stress than 30 GPa. In contrast, the shearing deformations take place without cracks at a rake angle of -30 degrees, as shown in Figure 10(b). This is because the stable shearing deformation in the amorphous region takes place toward the free surface under a compressive stress field.

5

SUMMARY

xMD simulation shows that the amorphous phase transformation of silicon is a key mechanism for inelastic deformation, and stable shearing of the amorphous region is necessary for ductile-mode machining. xResults also show that crack extension during ductilemode machining can be avoided by using stable shearing under a compressive stress field. xIn practice, a sharp cutting edge tool with a large negative rake angle should be used for effective ductilemode machining, and vibration machining should be applied for larger depths of cut as it enlarges the amorphous region in front of the cutting edge.

10 nm

6 ACKNOWLEDGMENTS This work was supported by the ‘Academic Frontier Promotion Center’ Project for Osaka ElectroCommunication University (2003-2007). 7 REFERENCES [1] Shimada, S., Ikawa, N., Inamura, T., Takezawa, N., Ohmori,H., Sata, T., 1995, Brittle-Ductile Transition Phenomena in Microindentation and Micromachining, Annals of the CIRP, 44/1:523-526. [2] Shimada, S., Ikawa, N., Tanaka, H., Ohmori, G., Uchikoshi, J., Yoshinaga, H., 1993, Feasibility Study on Ultimate Accuracy in Microcutting Using Molecular Dynamics Simulation, Annals of the CIRP, 42/1:91-94. [3] Tersoff, J., 1989, Modeling solid-state chemistry: Interatomic potentials for multicomponent systems, Physical Review B, 39/8:5566-5568. [4] Zhang, L.-C., Tanaka, H., 1999, On the Mechanics and Physics in the Nano-Indentation of Silicon Monocrystals, JSME Series A. 42/4:546-559. [5] Kittel, C., 1996, Introduction to Solid State Physics 7th ed., John Wiley & Sons, Inc., New York. [6] Cheong, W.-D., Zhang, L.-C., Tanaka, H., 2001, Some Essentials of Simulating Nano-Surfacing Processes Using the Molecular Dynamics Method, Key Engineering Materials, 196:31-41. [7] Taniguchi, N., 1996, Nanotechnology, Oxford University Press, New York. [8] Wulff, J., 1965, Mechanical Behavior, John Wiley & Sons, Inc., New York.

(a) R =2.2 nm (b) R =8.6 nm Figure 9: Difference in chip removal process in terms of distribution of maximum principal stress depending on cutting edge sharpness. Tension

10 nm Crack

Comp.

(a) Į =30 degree (b) Į =-30degree Figure 10: Difference in chip removal process in terms of distribution of maximum principal stress depending on rake angle. 4.3 Effect of vibration machining Figure 11 shows the effects of vibration cutting on the chip removal process in terms of the distribution of octahedral shearing stress. The amorphous region becomes larger in vibration cutting than in conventional

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