Theoretical derivation and a single abrasive particle on material removed of slicing ceramic

Multi-wire saw machining (MWSM) used for slicing hard-brittle materials in semiconductor, is an important material removal process that uses free abrasives. Cutting model of single-wire saw machining (SWSM) is the basis of MWSM, and the material removal mechanism of SWSM can better understand than MWSM. Mathematical model (includes brittle fracture and plastic deformation) is presented in this paper for SWSM ceramic with abrasives. This paper determines the effect of various machining parameters on the removal of hard-brittle materials. For brittle fracture of SWSM ceramics, the minimum of the strain energy density is used as a fracture criterion. The materia l removed of SWSM ceramics due to plastic deformation is calculated using the equations of motion. Actual wire-sawing experiments are conducted to verify the results from the developed mathematica l model. Theoretical results agree with experimental data and practical experience. The developed mathematical model shows that brittle fracture plays a major concern role in material removed of SWSM ceramics. Wire speed and working load have positively correlated with material removed of SWSM ceramics. The coefficient of friction is low, a lateral crack, which propagates almost parallel to the working surface, leads to more brittle fracture and material removed is increased on SWSM ceramics.


Introduction
Ceramics are an important hard-brittle material for the semiconductor and photovoltaic industr ie s .
Wire-sawing technology (contains SWSM and MWSM) is the most efficient and economic tool for machining hard-brittle materials, such as silicon, SiC, ceramic and sapphire for the photovoltaic or semiconductor industries [1][2][3]. A single-wire saw consists of a thin steel wire that is wound around wire-guides, forming a web of parallel wires. The abrasion for single-wire sawing technology can be free or fixed. Wire-saw machining (WSM) is very important work for slicing ceramic ingots. The efficiency of SWSM depends primarily on the cutting action of abrasive particle in the slurry. The ceramic that is removed by the abrasive particles is the major factor in machining efficiency, cost reduction and the quality of the machined surface [4]. Studies [5][6] show that there are both cracks and plastic flow during the cutting of brittle materials. Finnie and McFadden [7] used hard abrasive particles to strike a ductile metal at a low angle and predicted that the volume of material removed is positively correlated to the n th power of speed, for 2<n<3. Evans [8] showed the impact of each abrasive particle on hard-brittle materials is proportional to the penetration depth of the abrasive particle and the square of the crack length using the elastic-plastic theory. Wang and Rajurkar [9] studied the cutting factors on the material removal rate (MRR) of an ultrasonic machining process.
However, the MRR is the amount of weight removed per unit volume of material and is widely used to optimize the wire-sawing process. Slicing ceramics requires a high value for the MRR but the processing conditions must be controlled to avoid instability which causes defects such as cracks, porosity and straying of cutting path.
On the other hand, some studies develop a few material removal models for the wire-sawing process that uses abrasive diamond particles [10][11][12][13]. The removal of hard-brittle material during wire-sawing involves plastic deformation and brittle fracture. During SWSM, the stainless steel wire driven abrasives move against rather than strike the work material. Therefore, this present study determines the theoretical model of MRR for hard-brittle material during SWSM. The governing equations are derived based on the fracture principle of strain energy density (for brittle fracture) and the equation of motion (for plastic deformation), and the effect of various cutting factors (S and P) on material removal is determined. In addition, experiments are conducted for verification the developed theoretical model in this study.

Mechanism of brittle material removal
The mechanism for the removal hard-brittle material during WSM involves the shear force of the thin film in the slurry and the fine cutting from the rolling and impact of abrasives [14]. The shear force of the thin film and the rolling impact have a limited effect due to large abrasives of about 8~27 m are used for WSM. When the depth of cut is less than 1 m, the hard-brittle materials experience ductile-regime grinding [15]. The material experiences plastic deformation due to the plowing of abrasives and piles up either side of the abrasive. Material at the front end of the abrasive cutting edge also undergoes chipping due to plastic deformation. For a depth of cut of 1~4 m, a lateral crack emerges along with debris [16]. This result is similar to that for scribing indentatio n [17], in which the pile-up that is generated by plowing functio n may be removed at the same time.
Figs. 1(a) and 1(b) illustrate the mechanism for the removal hard-brittle materials during SWSM. Fig. 1(a) shows that the material is pushed to the sides and piles up due to plowing by abrasives when the depth of cut is shallow and there is fine chipping at the front end of the abrasives. Fig. 1(b) shows the lateral crack that is formed for a deeper cut, which produces debris and removes a larger quantity of material. The experimental results for driving stainless steel wire to allow SiC slurry to work on a workpiece made of Al2O3 are shown in Figs. 2(a) and 2(b). Position A in Fig. 2(a) shows the plastic 4 deformation and position B shows brittle fracture. In Fig. 2(b) position D denotes the chipping that is generated by brittle fracture. The chip was analyzed using an Energy-dispersive Spectrometer. Fig.   2(c) shows that the chip consists of Al2O3, in which Au is tinted for observation by SEM. This study assumes that a minute volume is removed by abrasive slurry, as shown in Fig. 3, wherein zones Ⅰ and II are the plastic deformation region and the lateral fracture region, respectively. The following descriptions are divided into two parts. Zone Ⅰ: plastic deformation region b : cutting width of the abrasive particle Zone Ⅱ: brittle fracture region P * : minimum threshold load for lateral crack Zone Ⅲ : predicted lateral fracture path tan= Fig. 3 Schematic diagram of the volume removed by a single abrasive particle

Brittle fracture
The stress intensity factor (K), the energy release rate (G) and the J-integral are the parameters that are commonly used to predict fracture within a brittle material. However, they all assume that there is a crack before the analysis. Sih [18] uses the strain energy density criterion to predict fracture initiation and to determine the fracture trajectory. The strain energy density criterion is based on the global energy field, instead of the local stress or energy field for the failure analysis. This method does not require the assumption that a crack exists. The strain energy density ) / ( dV dW can be written as follows: where W denotes the total energy that is stored in an element; V denotes the volume of materia l removed that is removed by an abrasive particle and As r increases gradually, a series of points corresponding to [ dV dW / ] min max are obtained. The fracture path is obtained by connecting these points [19]. There are two different types of crack: a median crack that propagates underneath scribing and a lateral crack, which results in surface fracture. The lateral crack has a significant effect on the removal of brittle material. The minimum threshold load for a lateral crack (P * ) is derived in [16]. In Fig. 3(d), the concentrated forces of the abrasive particle acting in the Z and -Y directions on the sides of workpiece (Z plane) are  cos * P ( ZZ P ) and  sin * P ( ZY P  ), respectively, so the stress in the Z plane, expressed in polar coordinates, is [20]: , and  denotes the coefficient of friction. The strain energy density is [20] where  is Poisson's ratio and E is the modulus of elasticity. Substituting Eq. (2) into Eq. (3) yields: According to the uni-axial tensile test, the critical strain energy density ( dV dW / )C can be represented as:   [7] solved the equations of motion for this particle to predict the volume that is removed by erosion. The equations of motion for abrasive particles are used to derive the volume of plastic deformation, as shown in Eqs. (7)~(9). In Fig. 5(b), the center of mass of the abrasive particle moves parallel to the Z plane and rotates at an angle of  due to the working load ( P ) that is exerted by the stainless wire and the wire speed ( S ). The locus left by the abrasive particle tip cutting into the material surface is (XT, YT). The cutting also leads to a plastic deformatio n chip at the front end.
Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 August 2020 doi:10.20944/preprints202008.0510.v1 (a) Inertial force only (b) Working load (P) and wire speed (S) Fig. 5 Schematic diagram of a two-dimensional cutting process using an abrasive particle The depth of cut is very shallow and the rotational angle is very small, so it is assumed that the movement relative to the abrasive particle tip and the center of mass is as shown in Fig. 5(b). The horizontal and vertical cutting forces at the tip of the abrasive particle are X F and Y F , respectively, as shown in Fig. 5(b), so the plastic flow stress A F X /   is assumed to be constant, where A is the projection area of the contact between the abrasive particle and the chip.

However,
Lb A  , where L is the chip length and b is the width of cut for a single abrasive particle, as shown in Fig. 5(b). From reference [7], Using these assumptions, The equations of motion for the abrasive particle are rewritten as: where m is the mass for a single abrasive particle, as shown in Fig. 3(a), P is the working load, I is the moment of inertia of a particle about its center of gravity and r is the average particle radius, as shown in Fig. 5(b). For the initial conditions, , the solution of Eq. (11) is: into Eq. (10) and using initial conditions can be expressed as : By substituting ) (t Y into Eq. (12) and using initial values 0 ) 0 (   and 0 ) 0 ( '   , the rotation of the particle is:

Material removed by an abrasive particle
The material that is removed by an abrasive particle (V ) is calculated by combining the brittle fracture and the plastic deformation as: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 24 August 2020 doi:10.20944/preprints202008.0510.v1 Therefore, the material that is removed is: If the tip of the abrasive particle is moving horizontally as it leaves the surface, Eq. (10) yields: Substituting Eqs. (13), (14), (15) and (19) into Eq. (18), the volume that is removed is written as: This equation consists of two parts: for a plastic deformation and for a brittle fracture. The material removal increases as the wire speed (S) and working load (P).

Experimental results and discussion
Experiments were conducted to test the proposed theory. The workpiece swings back and forth, as shown in Fig. 6. This vibrating machining model evenly distributes the cutting force of each active grain and makes the disposal of chips more efficient. The main experimental parameters and their values are listed in Table 1. To derive a more precise correlation between the machining parameters and the material that is removed, Eq. (20) is expressed in non-dimensional form as: For a single abrasive particle (GC# 600), the width of cut b ( 3 / D ) is about 9 m, where D is the diameter, which is 27 m, and the mass m ) is 210 -11 kg, where ρ is the density, with a value of 3.22 g/cm 3 . If =0.3 [21], [7], the plastic flow stress  for Al2O3 is about 400 MPa, chip length is L (1.1b) and the horizontal cutting force For a single abrasive particle, the working load is P (total working load/ 0 n ), so if 0 a is the total area of the machining zone (See Fig. 6), the maximum number of particles that participate in the operation is calculated by dividing the area of the grit by the largest cross-section a0 ( 0 0 b L ), where 0 L is the line contact length and 0 b is the line width. Accounting for tiny gaps between the grit, the total number of particles 0 n is: , where 0 C is the gap coefficient, which has a value of 10~20%.  The material V that is removed by a single abrasive particle is ) denotes the total number of machining abrasives per unit time, where 1 C is the wire speed coefficient, which has a value of 70~90%, and S is the wire speed.
The experimental conditions and results are listed in Table 2, which is used to produce the chart in   Eq. (20) allows the contribution of the material that is removed by plastic deformation to be determined. This is plotted as a dashed line in Fig. 8. A small amount of material is removed by plastic deformation and brittle fracture plays the dominant role. This is reasonable since there is only ductile flow when the depth of cut is less than 1 m [16]. For this study, the depth of cut (from Eq. (10)) is 0.5 m to 4 m, so there is lateral cracking and plastic flow. Fig. 9 shows the relations hip between the working load and the ratio of brittle/ductile removal rates. Increasing the working load increases the depth of cut so there is more evidence of lateral brittle fracture and the kerf width increases. For wire-saw machining, the MRR is increased when the working load (P) is increased. If the working load exceeds a threshold, the steel wire can break easily. The cutting path can also become crooked. Therefore, the working load must not exceed a specified threshold value. If the wire speed (S) is increased, the amount of grains that enters the machining zone from the slurry within a specific time increases, so significantly more material is removed.
From Eq. (20), the larger the grit size A, the greater is the machining efficiency. However, it the surface roughness of the wafer also increases. Suwabe et al. [22] showed that the addition of smaller grains in the slurry results in a smoother wafer. However, when all of the particles in the slurry are smaller than #1000, a further increase in the working load or the wire speed produces no increase is the volume that is removed. Fig. 10 shows the theoretical values for material removed for abrasives of mesh size #600 (solid line) and size #1000 (dash line). there is no difference in the amount of material that is removed when the grit is too small. The higher the material hardness, the greater is the normal cutting force ( Y F ), so the MRR is decreased. Higher material hardness results in brittle fracture of the material. The coefficient of friction has an effect on crack propagation [23].

Conclusion
Slicing ceramic technology using WSM still has disadvantages in terms of the MRR deviation.
However, WSM increases productivity and the MRR for slicing ceramics, which decreases the