Discrete element simulation of micromechanical removal processes during wire sawing

Wear 304 (2013) 77–82

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Discrete element simulation of micromechanical removal processes during wire sawing T. Liedke, M. Kuna n Freiberg University of Mining and Technology, Institute of Mechanics and Fluid Dynamics, Lampadiusstraße 4, D-09596 Freiberg, Germany

art ic l e i nf o

a b s t r a c t

Article history: Received 18 September 2012 Received in revised form 17 April 2013 Accepted 24 April 2013 Available online 11 May 2013

The material removal rate in the wire sawing technology depends on several features of the used abrasives and the complex process of particle movement and interaction inside the kerf. To investigate these micro-mechanical aspects of the wire sawing process, a numerical model is developed in the present paper. The model is based on the discrete element method adjusted to use sharp edged polygonal particles. The crucial procedure of material removal is implemented through concepts of the fracture mechanics of sharp edged indenters. The model allows to study the influence of essential parameters like wire speed, lapping pressure, particle shape, particle size distribution a.o. on the material removal process. A parametric study towards the influence of wire speed and lapping pressure on the amount of material removal is also presented. The parametric numerical studies carried out confirm the phenomenological law of Preston for the removal rate and enable correlations between the Preston coefficient and micro-mechanical process parameters. & 2013 Elsevier B.V. All rights reserved.

Keywords: Wire sawing Wire lapping Discrete element method (DEM) Contact fracture mechanics Indenter

1. Introduction – the wire sawing process Today, industrial wire sawing has been approved as the most efficient and economic technology to cut brittle materials like semiconductors and others into thin wafers. An important field of application of these wafers is the production of photo-voltaic cells, where mostly silicon wafers are used to gain electricity from solar radiation. Likewise, wafers made from mono-crystalline silicon are required to produce electronic circuits and MEMS of all kinds, which makes it essential to almost all present-day technical equipment. For the cutting process a single wire with a diameter of about 120 μm is wounded around two or four rotating wire guides. These wire guides are grooved with a constant pitch, so that the wire forms a neat web of parallel wire segments with equally spaced distance. A rotation of the wire guides then leads to an axially movement of the wire web. Into this moving wire web the ingot is feed while additionally a suspension of a fluid with abrasive particles, the so called slurry, is supplied. Due to the abrasive particles the moving wire slices into the ingot and leaves thin disks of remaining material between them – the wafers. Fig. 1 gives a schematic view of the described procedure. The material removal process is similar to the common lapping of flat surfaces. Besides that it is a suitable procedure to machine

n

Corresponding author. Tel.: +49 3731392092. E-mail address: [email protected] (M. Kuna).

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hard and brittle materials it is quite productive. The reason therefore is the usage of several hundreds of parallel wire segments per web which allow to cut many wafers at the same time. The actual cutting process in the sawing kerf is hard to observe because of the small particle size of about 15 μm combined with a high wire velocity of about 15 m/s. First attempts thereto can be found e.g. in Liedke et al. [24] (see also Fig. 2) or in Hejazialhosseini et al. [17]. Nevertheless there are models to describe the microscopic removal procedure which are mainly based on examinations of the wafer surface (e.g. [30,16,4]) or on theoretical considerations of the abrasive process (e.g. [20,19,3,2]). It turned out that the removal procedure during the wire sawing process can be described by the “free abrasive machining” (FAM) process and “three-body-abrasion”, which is well known from other surface processing technologies as e.g. polishing [9] or lapping of glass [6–8]. The main idea is that individual particles are trapped between wire and ingot and perform a rolling motion due to the velocity of the wire. During rolling the sharp edges of the particles are indented into the surface and lead to fracture and chipping of the ingot material (see Fig. 3). This entire process is denoted by ‘rolling indentation’. The elementary fracture process of brittle material caused by a single indenter tip pressed into the surface is well known in the literature (e.g. [10,28]). However, the complex behavior of multiple abrasive particles in the kerf is still less investigated. Originally developed for rock-mechanics [13], the discrete element method (DEM) offers the possibility to capture a large number of discrete

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Fig. 1. Schematic view of a wire saw and a close view showing the wire and abrasive particles.

Fig. 2. Example of experimentally observed particles in interaction with wire and ingot. Ingot material: glass, wire speed 1 m/s. Fig. 4. Contact of two particles and equivalent mass-spring-system used to calculate contact forces.

Fig. 3. Schematic view to the process area.

particles and is therefore suitable to model the motion and interaction of abrasive particles during the wire lapping procedure [24,5,38].

2. Preliminary considerations The special in wire sawing is the use of a flexible tool, i.e. the wire. While the ingot is fed in with a certain feed speed the wire does change its shape and forms the typical wire bow. The exact shape of the bow depends on certain machining parameters like wire speed, feed speed and the wire tension as well as the properties of the slurry. A macroscopic mechanical model, which describes the temporal development of this wire deflection and the cutting forces during the sawing process, is presented by Liedke and Kuna [26]. Therein, the system ingot-slurry-wire is decomposed into two parts, the wire and the ingot. The connection between both is a pressure distribution inside the kerf which is acting on both. This twofold consideration allows on the one hand the calculation of macroscopic phenomena like the temporal development of the wire bow without considering the procedures inside the kerf and gives on the other hand the possibility to exclude wire bow, wire tension and feed speed from the microscopic model. Thus, the following DEM-model will represent a small region inside the kerf, which is small enough to reasonably neglect the bow of the wire and assume a constant pressure p0 acting on it. The influence of the slurry fluid as well as the material removal due to indentation processes is considered while the wire bow,

wire tension and feed speed are replaced by the according lapping pressure (see Fig. 3). The abrasive grains are modeled by means of the Discrete Element Method. However, while most of the current DEM software uses spheres to model rigid particles, for lapping processes sharp edged particles are inevitable for a realistic simulation, i.e. polyhedral particles are both essential and superior. Because of higher computational effort only a few DEM-codes exists, which allow for non-spherical particles [14,27]. For the present study, a two-dimensional DEM-code based on polygonal shaped particles as described in e.g. [37,29,33] was extended to meet the requirements of a wire saw simulation. Because the software is restricted to two dimensions, the simulation is limited to observe a cross-sectional cut along the sawing direction. That means the hemi-cylindrical wire is reduced to a flat line and material removal only takes places right below the wire. Nevertheless this gives the possibility to examine the sawing process along the main cutting direction, which is explained below.

3. Discrete element simulation 3.1. Particle representation and interaction The wire, the abrasive particles and the machined material surface are represented by two-dimensional, polygonal shaped, ! rigid bodies having in principle two translatoric ( r i ) and one rotational (ϕi ) degree of freedom. While the particles representing the abrasives are able to move freely, the translational and rotational degrees of freedom of wire and surface particles are restricted to match the boundary conditions of the sawing process. The contact behavior between two particles and Pi and Pj is realized by a visco-elastic rheological model. That means, the ! repulsive force F ijc between two interpenetrating particles is calculated by considering a damped mass-spring-system (see Eq. (1)), using the overlapping area A0 related to the mean distance ai þ aj as a measure for penetration [37,33]. Fig. 4 gives a

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schematic view of two particles in contact, where ci and cj denote ! ! their centers of mass at actual positions r i and r j . The direction of the resulting contact forces is defined through a straight line between the intersection points of the overlapping area while the point of application is the associated center of mass for this area. !n !t The interaction force in normal e and tangential e direction are then calculated from the normal penetration, the tangential sliding strel and the relative velocities vðn;tÞ between both the rel particles, n

F nijc ¼ −k 

Ao n −b  vnrel ai þ aj

t

t

F tijc ¼ þ k  strel −b  vtrel −F nijc μ sgnðvtrel Þ;

ð1Þ

whereby the indices denote: n – normal direction, t – tangential direction. Further variables mean Ao – overlap area, n t n t k and k – contact stiffnesses, b and b – contact damping constants. In tangential direction also a dry Coulomb's friction is assumed with friction coefficient μ. Thus, the resulting overall force due to contact can be written as ! !n !t F ijc ¼ F nijc e þ F tijc e :

ð2Þ

Analogous contact forces are formulated if one interacting particle belongs to the wire or the silicon surface. For simplicity, the carrier fluid is modeled as a stationary laminar shear flow, which induces three phenomena on the particle Pi: a drag force (3), a lift force (4) [35] and a moment of torque (5). As a result the particles translate in axial wire direction, translate towards the wire and rotate around their axis. A feedback effect of the particles to the carrier fluid, which may result in a turbulent flow, is not considered Z 1 2 F dragi ¼ ρf vrel ðyÞcd di dy ð3Þ hi 2 F lifti ¼ 6:46 signð_γ Þ Z M sheari ¼

hi

2 pffiffiffiffiffiffi ηjvrel jdi ρf j_γ j qffiffiffiffiffiffiffiffiffi η=ρf

1 ðy−ysi Þ ρf v2rel ðyÞcd di dy 2

ð4Þ

ð5Þ

Here, the annotations are introduced: hi – projected height, di – depth of particle, cd ¼1.1 – coefficient of drag for a flat plate, ysi – center of mass, ρf – density of fluid, vrel – relative velocity between particle and fluid, η – dynamic viscosity of the carrier fluid. The shear rate γ_ is approximately given by the ratio of the velocity vwire of the wire and the distance between wire and surface hkerf, γ_ ¼ ∂vr =∂y≈vwire =hkerf . The translational and rotational motion of the particles is derived by solving Newton's and Euler's equations ! ! ! € ! mi r ¼ ∑ F ijc þ F dragi þ F lifti

ð6Þ

I i ϕ€ ¼ ∑M ijc þ M sheari

ð7Þ

j

j

Here, mi is the mass of particle Pi and Ii denotes the momentum of inertia and ϕi its rotation angle. To solve Eqs. (6) and (7), a fifth order Predictor–Corrector scheme [1] is used. 3.2. Indentation fracture mechanics The single indentation process of a hard indenter into a brittle material is well studied in the literature (see e.g. [28,32,10]) and occurs as shown in Fig. 5. The indenter is pressed against a surface and causes high local stresses which are proportional to the normal force FN. At a critical load, which depends on the indenter

Fig. 5. Schematic view of fracture process.

geometry, median and radial cracks are formed. During unloading lateral cracks are formed beneath the plastic zone and, if they reach the surface, material is chipped away. To implement the material removal process into the DEM simulation, the fragment size due to fracture has to be calculated. As a first approach, the following simplified correlation between fragment size and normal force is used [21,25]. According to Marshall et al. [28], the length lc and depth hc of the lateral cracks (see Fig. 5) are given by 0 11=2 ! 3=4 B ðE=H Þ C 5=8 V 5=6 ð8Þ lc ∼B A FN @cotðψÞ 1=4 K Ic H V

hc ∼ cotðψÞ1=3

ðE=H V Þ1=2 1=2

HV

!! 1=2

FN ;

ð9Þ

where lc and hc depend on the Vickers hardness HV, the wedge angle ψ of the indenter and the fracture toughness KIc. The proportionality constant for the crack length lc was found experimentally by Moeller et al. [31] for silicon: n

lc ¼ βn F nN

n

βn ¼ 13:75 μm=Nn and nn ¼ 0:85:

with

ð10Þ

To determine the depth hc of the lateral cracks, it is assumed that these cracks are formed beneath the plastic zone [28,15,12]. To calculate the radius of this plastic zone rpl, the relation to the radius b of a spherical indenter can be used. Chiang et al. [11,12] gives for silicon the value β ¼ r pl =b ¼ f ðE=HÞ;

βSi ¼ 2:65:

ð11Þ

By geometrical considerations, this can be transformed into the indentation diagonal length b of a Vickers indenter 0 pffiffiffi 11=3 π 2 A : ð12Þ b ¼ 2b @ cotðψ Þ In combination with the definition of the Vickers hardness HV ¼ α

FN 2

b

and

αVickers ¼ 2;

ð13Þ

a correlation between the indenting normal force FN and the depth of the lateral crack hc is obtained 0 pffiffiffi 1−1=3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi αF N =H V B π 2 A ð14Þ hc ¼ r pl ¼ β @ cotðψ Þ 2 As a first approximation the particles in the DEM-Simulation are considered to behave like Vickers indenters with a wedge angle of ψ ¼ 451.

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Fig. 6. Schematic view of the simulated fracture process. At a collision between abrasive particle and surface particle the acting forces are determined. Than the length and depth of the cracks are calculated and in the last step material is removed by replacing the involved particle by new, differently shaped, ones.

Table 1 Particle size distribution according to ISO 8486-2 (extract).

Table 2 Material properties used in the DEM simulation. Material ρ (kg/m3) Y (Pa)

Designation

ds3 (μm)

ds50 (μm)

ds94 (μm)

μ

s

F400 F500 F600 F800

32 25 19 14

17.3 7 1.0 12.8 71.0 9.3 7 1.0 6.5 7 1.0

8 5 3 2

2.84 2.53 2.22 1.86

0.43 0.49 0.51 0.56

SiC

3210.0

Wire

7880.0

Si

2300.0

Fluid

–

μ

KIc pffiffiffiffiffi (Pa m)

3:30  1011 0.7 0.0 2:10  1011 0.7 0.0

Fcrit (N)

HV (Pa)

η (Ns/m2)

0

0

–

0

0

–

1:27  1011 0.5 9:0  105 0.03 – – – –

10:6  109 – – 0.06

The depth of the median cracks hc;m (see Fig. 5), which do partly remain in the material surface, can be obtained from the definition of the fracture toughness KIc [31]. The median cracks are not included in the numerical model at the moment 0 12=3 B FN C hc;m ¼ B @ε K Ic A

and

εVickers ¼

1 7

ð15Þ

By means of these equations it is possible to calculate the fragment size, i.e. the depth and the length of a fragment, induced by a normal force during the DEM simulation (see Fig. 6). These fragments are simply deleted from the simulation process.

4. Simulation of the wire sawing process 4.1. Particle size distribution and particle properties Real abrasive particle powders which are used for wire sawing processes obey a certain size distribution – commonly used is a particle size distribution according to the FEPA-Standard 422:2006 and ISO 8486 (see Table 1). The simulated particle collectives are created by the use of a Voronoi algorithm, which leads to almost equally sized polygons. To transform these into an arbitrary amount of DEM particles of different sizes, which match a certain size distribution, a functional relation is necessary. From comminution processes it is known that a lognormal distribution meets this requirement [22,18]. Therefore, the values given by the FEPA standards are used to fit a lognormal distribution, which is here used to scale the Voronoi particles to different sizes. The mechanical properties used in the simulation are given in n Table 2. The normal stiffness constant is estimated to k ¼ Ydi =10 t n and the tangential stiffness is assumed to be k ¼ k =2ð1 þ νÞ. Here Y denotes the Young's modulus, ν is the poisson's ratio and di the size of the particle. The simulation uses approximately forty free particles and the length of the simulated area is about 500 μm long, see Fig. 7. The sides of the simulated area act as periodic boundaries. 4.2. Determination of the removal rate and surface roughness As can be seen in Fig. 7, the machined surface is represented by static polygons which, in case of material removal, are replaced by

Fig. 7. Example view of the simulation.

Fig. 8. Example for the development of the mean machined surface during the cutting process for four different particle distributions.

new polygons with a different shape and at a different position. Thus, the overall mean position of all surface polygons represents the macroscopic surface position. From the change of the surface position during the simulated cutting process the material removal rate rr is deduced. Fig. 8 shows the simulated removal process in time leading to a stepped curve which can be approximated by a linear line. The slope of this line is equivalent to the mean material removal rate. In Fig. 8 the mean surface position and the according slope are shown for four different particle size

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distributions. With increasing mean diameter of the size distribution the gradient of the slope increases too, which means more material is removed per time. This result is in good agreement with experimental and empiric observations. The roughness Rz of the surface is calculated from the surface of the polygons by calculating the average distance Rti between the highest peak and lowest valley of five sequent samples i: Rz ¼ 15 ∑5i ¼ 1 Rti according to technical standards. 4.3. Parametric study The numerical model was used to study the influence of the wire speed vc and the lapping pressure p0 on the material removal rate rr. The results are depicted in Figs. 9–12. The wire speed was varied in the range of 1 m/s to 50 m/s and the lapping pressure was varied in the range of 0:1  105 −2:0  105 N=m2 . In the first case the lapping pressure was fixed at constant value p0 ¼ 1:0  105 N=m2 and in the second case the wire speed was fixed at 10 m/ s. Each point in the diagrams results from a complete simulation using one set of parameters. It can be seen that close sets of parameters not necessarily lead to exactly the same result. The reason lies in the stochastic nature of the process. So, similar to real experiments, a great number of simulations has to be performed to reveal the underlying correlation. Fig. 9 shows that the removal rate is just proportional to the speed of the wire, which suggests that hydrodynamic effects may be irrelevant. At least for low velocities this result is clear and is in agreement with the experimental results (e.g. [23]). As can be seen from Fig. 11 the simulated removal rate also depends on the lapping pressure. But unexpectedly there seem to exist at least two separate regions. The empirically known relation between removal rate and lapping pressure can only be found in the second one. A possible reason for this is that fracture is only

Fig. 9. Simulated removal rate of the machined surface with respect to wire velocity.

Fig. 11. Simulated removal rate with respect to lapping pressure.

Fig. 12. Simulated roughness of the machined surface with respect to lapping pressure.

induced if the indentation force exceeds a critical force Fcrit and below this threshold value no material is removed. This behavior sounds realistic, but because of the low particle number used in the simulation this effect is possibly stronger than in real sawing processes where thousands of abrasive particles are involved. Nevertheless, the simulation indicates that with increasing lapping pressure the number of indentation events actually leading to material removal increases. The numerical results can be compared with empirically known correlations as e.g. the Preston law. Preston [34] found a linear correlation between removal rate and relative velocity times pressure r r ∝vc p, which is in good agreement with the numerical result. Using the slopes of Figs. 9 and 11, the proportional constant, the so called Preston coefficient, can be derived and yields kp ≈50 μm2 =N for the presented example. The calculated roughness (Figs. 10 and 12) of the machined surface increases with both wire speed and lapping pressure. This results fits well to empirical observations published recently by Teomete [36]. It should be mentioned that, although the values are close to experimentally observed ones, the results are still of qualitative manner.

5. Conclusions

Fig. 10. Simulated roughness of the machined surface with respect to wire velocity.

In order to study the micromechanical process of material removal during wire sawing, the model of a moving and interacting particle system in a viscous shear flow was developed. The numerical realization of the model is based on the discrete finite element method, which has been extended by sharp edged particles and by concepts of indentation mechanics to account for elementary fracture processes. It could be shown that the numerical simulations are capable of reproducing fundamental

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aspects of the sawing process. Qualitatively, the idea of “three body abrasion” is in principle verified to happen as basic phenomenon in the complex particle system, too. Moreover, this view could be generalized to the multiple interaction of abrasive particles in the sawing kerf with the silicon surface and among each other, driven by the slurry and the moving wire. The results obtained by rather extensive and expensive simulations agree qualitatively quite well with the experimental observations and substantiate the classical Preston law stating that the removal rate is proportional to wire speed and lapping pressure. According to these achievements one can be optimistic about better understanding of sawing processes by numerical simulations of this kind. By means of such a simulation tool, many other micromechanical parameters influencing the sawing process can be investigated as: preferable shape of abrasive particles, optimum particle size distribution or particle content, viscosity (temperature) of the slurry a.s.o. The simplified implementation of fracture processes is considered as a first step towards more sophisticated numerical predictions of removal processes and the generation of surface roughness. Also, the restriction to two-dimensional particles and a two-dimensional modelling domain has to be overcome, since the real sawing kerf is three-dimensional. This questions will be tackled by future research. Acknowledgments This work was performed within the Cluster of Excellence “Structural Design of Novel High-Performance Materials via Atomic Design and Defect Engineering (ADDE)” that is partially supported by the European Regional Development Fund and by the Ministry of Science and Art of Saxony. Especially, the extensive computations were performed by means of the HPC-computer cluster of ADDE. References [1] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford University Press, 2003. [2] M. Bhagavat, I. Kao, Computational Model for Free Abrasive Machining of Brittle Silicon Using a Wiresaw, vol. 104, American Society of Mechanical Engineers, Design Engineering Division (Publication) DE, 1999, pp. 21–30. [3] M. Bhagavat, V. Prasad, I. Kao, Elasto-hydrodynamic interaction in the free abrasive wafer slicing using a wiresaw: modeling and finite element analysis, Journal of Tribology 122 (2000) 394–404. [4] S. Bhagavat, I. Kao, Ultra-low load multiple indentation response of materials: in purview of wiresaw slicing and other free abrasive machining (fam) processes, International Journal of Machine Tools and Manufacture 47 (2007) 666–672. [5] C. Bierwisch, B. Weber, R. Kuebler, M. Moseler, G. Kleer. Contact regimes in the wire sawing process: explicit 3d modeling of peg/sic slurry, in: 23rd European Photovoltaic Solar Energy Conference, 2008. [6] M. Buijs, K. Korpel-van Houten, A model for lapping of glass, Journal of Materials Science 28 (1993) 3014–3020. [7] M. Buijs, K. Korpel-van Houten, A model for three-body abrasion of brittle materials, Wear 162–164 (1993) 954–956. [8] M. Buijs, K. Korpel-van Houten, Three-body abrasion of brittle materials as studied by lapping, Wear 166 (1993) 237–245. [9] R. Chauhan, C. Ahn, T.N. Farris, Role of indentation fracture in free abrasive machining of ceramics, Wear (1993) 246–257.

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