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Simulation and Experimental Research on Liquid Spreading in a Wire-Sawn Kerf

Abstract

The significance of liquids in abrasive wire sawing has been demonstrated in several studies. However, the performance of its spreading behavior is limited by the current development trend, where the wafer has a larger area and the kerf is narrower. Moreover, there are very few studies on the liquid spreading behavior in wire-sawn kerfs. Therefore, a 3D CFD (computational fluid dynamics) model is presented in this paper and used to simulate the liquid spreading behavior in a kerf based on a VOF (volume of fluid) method with a CSF (continuum surface force) model, which is used to simulate multiphase flow, and an empirical correlation for characterizing the liquid dynamic contact angle using UDF (user defined functions). Subsequently, parametric simulations are performed on the kerf area, kerf width, liquid viscosity, liquid surface tension, and liquid velocity at the inlet area of the kerf, and verification experiments are conducted to determine the validity of the simulation model. From the simulation and experimental results, three typical liquid spreading regimes that exhibit different effects on wire sawing in the kerfs are found, and their limiting conditions are identified using non-dimensional analysis. Subsequently, a prediction model is proposed for the liquid spreading regime based on a set of Weber and Capillary numbers. For wire sawing, an increase in the wafer area does not change the liquid spreading regime in the kerf; however, a reduction in the kerf width significantly hinders the liquid spreading behavior. Thereby, the spreading regime can be effectively converted to facilitate wire sawing by adjusting the physical properties and supply conditions of the liquid.

1 Introduction

Abrasive wire sawing technology has been widely applied in wafer production in the semiconductor and photovoltaic industry. Since the 1990s, multi-wire slurry sawing (MWSS) has been applied in slicing silicon ingots owing to its high throughput, smaller kerf loss, and ability to slice large-sized ingots [1,2,3]. Recently, diamond wire sawing (DWS) was promoted and achieved industrial status owing to its efficient material removal [4, 5]. In abrasive wire sawing, the cutting liquid is directly sprayed onto the kerf or onto a high-speed moving (5–30 m/s) wire web, which then carries it into the sawing channel, as shown in Figure 1(a), (b). Here, the significance of the liquid should not be underestimated owing to the nature of its main functions: lubrication, cooling, and chip removal [6, 7]. Particularly, for MWSS, the cutting fluid is used to carry abrasive particles into the sawing channel to achieve material removal.

Figure 1
figure 1

Two main liquid supply modes in abrasive wire sawing: (a) Spraying into the kerf, (b) Spraying onto a moving wire web

Considerable research efforts have been devoted to the liquid behavior during abrasive wire sawing owing to its indispensable role. Thus far, majority of the studies have focused on MWSS because of its earlier development, which dates back to when Bhagavat et al. [8] proposed a 2D model for simulating the EHD (elasto-hydrodynamic) interaction between wire and slurry. They assumed that the flow is incompressible, steady, and laminar, and used the finite element method to analyze the slurry film thickness and pressure distribution. Moreover, Möller et al. [1] also presented a 2D model for analyzing the EHD behavior of slurry, and several other studies by Bhagavat et al. [8] and Yang et al. [9] have addressed the abrasive behavior of slurry and clarified the prominent impact of the liquid film thickness and pressure distribution on material removal. Furthermore, the significance of the coolant liquid on the thermal environment of wire sawing has also been demonstrated using finite element simulations [10,11,12].

Although it is not clearly stated, in all the above studies, it is acquiesced that the liquid can enter the sawing channel and remain in a continuous and saturated state. However, a crucial question is overlooked: Whether and how the liquid can enter the sawing area. Therefore, Ishikawa et al. [13] observed the formation of the air layer in the sawing channel, and Nassauer et al. [14] observed the phenomenon of emerging and collapsing micro bubbles in the sawing experiments. Based on these, it was confirmed that there is air in the sawing channel and the liquid is not saturated, which may prevent the liquid from performing its function and result in dry cutting.

Recently, advances in fabrication have resulted in the development of abrasive wire sawing technology to produce larger wafer diameters (from 2 inches to 12 inches) and smaller wire diameters with a lower kerf loss (the current minimum wire diameter is up to 60 μm). This trend promotes the use of a larger sawing kerf area and narrower kerf width in the sawing process. The complex fluid environment at small scales creates entirely new challenges to the entering-spreading behavior of the liquid in the kerf. Nevertheless, for abrasive wire sawing, neither the entering-spreading behavior of the liquid nor the introduction of air has been explicitly represented, and the liquid supply mode involving direct spraying on the kerf has not been investigated. Moreover, most of the abovementioned studies employ 2D models for simulation, which have inevitable limitations when simulating actual conditions.

In this paper, a 3D CFD model is constructed to simulate the liquid spreading behavior in a kerf, and parametric simulations are performed on the kerf area, kerf width, liquid viscosity, liquid surface tension, and liquid velocity at the kerf inlet area. Furthermore, the validity of the simulation model is verified using experiments.

2 Simulation and Experimental Methodology

This paper aims to investigate the liquid spreading behavior in a kerf, focusing on a liquid supply mode involving direct spraying into the kerf. To simplify the model, the single wire sawing of a sapphire ingot is taken as an example.

2.1 Simulation Model and Boundary Condition

For the physical phenomenon of interest, a series of 3D geometry models are built in different dimensions for parametric simulation (the geometric dimensions of the models are shown in Table 4, see Appendix), and their structural features are shown in Figure 2. As shown in Figure 2(a), the calculation domain is a cuboid, and the inlet, a square orifice of 4×4 mm2, is set at the center of its top. In the calculation domain, an ingot model, is built 15 mm below the orifice to represent the nozzle position during general sawing. To reduce computational load, the ingot model was simplified to a thickness of 2 mm, and a kerf was processed at the central section of the ingot (Figure 2(a)). Moreover, as presented in Figure 2(d), the sawing area on the kerf surface is simplified to a flat area without the sawing wire.

Figure 2
figure 2

Geometric model and grids: (a) Calculation domain and geometric model, (b) Grid of the ingot with a kerf, (c) Local mesh densification at the ingot and kerf walls, (d) Simplification of the sawing area

The geometric models are then meshed into structured hexahedral grids, as shown in Figure 2(b), and local mesh densification is conducted on the ingot and kerf walls, as shown in Figure 2(c). Furthermore, grid size independence is performed to enhance the accuracy and efficiency of the calculation. When the total number of the grid nodes in the model is greater than 2.2 × 104, and the smallest grid volume is less than 2.3 × 10−12 mm3, the simulation results do not have any significant differences.

The boundary conditions for simulation are shown in Table 1. All walls are assumed to be uniform sand-grain surfaces with certain roughness heights \(K_{s}\) (the determination of the \(K_{s}\) depends on the measured surface roughness). In addition, the operating pressure is 101325 Pa, the gravitational acceleration \(g\) is 9.8 m/s2, and \(v_{0}\) is the inlet velocity.

Table 1 Boundary conditions

2.2 Experimental Setup

The experimental setup (see Figure 3) consists of the following parts: (a) pump, (b) precision valve, (c) float flowmeter, (d) 3D printing nozzle (with a 4 mm×4 mm square outlet), (e) liquid recovery tank, (f) three-axis linkage platform, and (g) high-speed camera (Phantom/V2511). Parts (a)–(d) are connected using water pipes and are located in the liquid recovery tank for flow control and recovery. Additionally, a three-axis linkage platform is fixed at the bottom of the tank, and a 2-inch ingot with a sawn kerf is fixed on the platform (the sawing parameters are shown in Table 5, see Appendix).

Figure 3
figure 3

Experimental setup

The experimental device was simplified based on a diamond wire sawing machine (JXQ-1201) liquid supply system, but the actual sawing process was not considered. Furthermore, various physical parameters were measured and controlled in the experiment to verify the consistency of the experimental and simulated boundary conditions. These include the dimensions of the ingot and its kerf, and the nozzle size (see Table 2) for use in the simulation. In addition, the position of the ingot was adjusted using a three-axis linkage platform (ingot was placed 15 mm below the nozzle). During the experiment, the tank was first filled with the experimental liquid before the pump was turned on, and the liquid flow rate was set using the precision valve and float flowmeter, which allowed liquid flow at a certain flow rate from the nozzle. Simultaneously, the entire process of liquid entering and spreading in the kerf was captured using the high-speed camera.

Table 2 Physical properties of different solutions

To clarify the effect of the physical properties of the liquid, three solutions were prepared for the simulations and experimental studies, as shown in Table 2, where \(\mu_{l}\), \(\rho_{l}\) and \(\gamma_{lv}\) are the liquid dynamic viscosity, liquid density, and liquid surface tension, respectively. Solution A is a 2% aqueous solution of a commercially available cutting fluid, solution B is PEG 200 solution, and solution C is deionized water. The prepared solutions represent the cutting fluids commonly used in DWS (mainly water-based coolant) and MWSS (mainly composed of polyethylene glycol (PEG)) [10], respectively. All the solutions were prepared and measured at a room temperature of 298.15 K, and the liquid viscosity, surface tension, and density were measured using a digital rotor viscometer (NDJ-8S), liquid surface tension measuring instrument (BZY-B), graduated cylinder, and precision electronic scale, respectively.

2.3 Simulation Parameters

The basic parameters influencing the liquid spreading process typically include the liquid properties, flow velocity, and scale parameters of the flow environment [15]. Therefore, the following simulation parameters were chosen assuming that the liquid density \(\rho_{l}\)= 999.12 is a fixed value in the simulation study: \(A_{k}\) is the area of the kerf center section, \(D_{w}\) is the kerf width, \(\mu_{l}\) is the liquid dynamic viscosity, \(\gamma_{lv}\) is the liquid surface tension, and \(v_{i}\) is the liquid velocity at the inlet area of the kerf.

The variations in the simulation parameters are shown in Table 3, where \(A_{k}\) and \(D_{w}\) represent the development trend in the industry (ingot diameter \(D_{i}\)= 2, 6, 12 inch) and the actual processing conditions (sawn depth \(D_{s}\)=\(D_{i} /5\), \(D_{i} /2\), \(4D_{i} /5\)), respectively. Additionally, the liquid properties (physical properties of the solutions) were set within a reasonable range (see Table 2). According to Bernoulli’s theorem [15, 16], the variations in the \(v_{i}\) are determined using the inlet velocity \(v_{0}\) (see Table 1) and distance \(h_{i}\) between the inlet and top of the kerf as follows:

$$v_{i} = \sqrt {v_{0}^{2} + 2gh_{i} } = \sqrt {v_{0}^{2} + 2 \times 9.8 \times 0.015} = \sqrt {v_{0}^{2} + 0.294}.$$
(1)
Table 3 Variations of simulation parameters

Moreover, three non-dimensional numbers, Reynolds number \(Re = \rho_{l} vl_{0} /\mu_{l}\), Weber number \(We = \rho_{l} v^{2} l_{0} /\gamma_{lv}\) and Capillary number \(Ca = v\mu_{l} /\gamma_{lv}\), were introduced to characterize the liquid flow state and analyze simulation results [17]; where \(v\) is the fluid flow velocity and \(l_{0}\) is the characteristic length. For the liquid flow in the kerf, \(l_{0}\) is equal to the kerf width \(D_{w}\), similar to the flow between parallel plates [18]. In this study, the maximum Reynolds number \(Re_{\max }\) in the kerf can be approximately obtained as:

$$Re_{\max } \approx \frac{{\rho_{l\max } v_{i\max } l_{0\max } }}{{\mu_{l\min } }} = \frac{1122.37 \times 1.68 \times 0.0005}{{0.00092}} \approx 1024.7.$$
(2)

Since \(Re_{\max }\) is less than the lower bound of the critical Reynolds number \(Re_{{\text{c}}}\) (2000) [18], the liquid in the kerf is laminar. For a \(Re\)1, the quantity of interest in the effect of the surface tension is determined using the Weber number [19], and the minimum weber number is approximated as follows:

$$We_{{\min }} \approx \frac{{\rho _{{l\min }} v_{{i\min }} {}^{2}l_{{0\min }} }}{{\gamma _{{lv\min }} }} = \frac{{999.12 \times 0.55^{2} \times 0.0001}}{{0.08}} = 0.37.$$
(3)

Hence, the effect of the surface tension should not be neglected since the Weber number \(We\)<1 [18].

2.4 Governing Equations and Solving Method

Since two-phase flow is involved in this study, a volume of fluid (VOF) method was introduced to capture the gas-liquid interface. In the VOF model, the volume fraction of each fluid is tracked down through each cell in the domain [20]. For the fluid \(i\), the volume fraction \(\alpha_{i}\) is defined as follows:

\(\alpha_{i} = 0\): The cell has no fluid \(i\).

\(\alpha_{i} = 1\): The cell is full of the fluid \(i\).

\(\alpha_{i} < 0\): The cell has an interface between the fluid \(i\) and one or more other fluids.

Subsequently, the continuity equation for the volume fraction was solved to track the interface. Assuming that the subscript \(i\) represents the fluid \(i\), the continuity equation has the following form [21]:

$$\frac{{\partial \alpha_{i} }}{\partial t} + \nabla \cdot \left( {\alpha_{i} \vec{v}_{i} } \right) = 0.$$
(4)

Moreover, the fluid properties in the continuity equation are determined based on their phase components in each control volume. For the two-phase flow in this study, the density and viscosity are given by:

$$\rho = \alpha_{j} \rho_{j} + \alpha_{i} \rho_{i},$$
(5)
$$\mu = \alpha_{j} \mu_{j} + \alpha_{i} \mu_{i},$$
(6)

where the subscript \(j\) represents fluid \(j\). Depending on the fluid properties, a momentum equation is then solved for the velocity field that is shared among the phases [21]:

$$\frac{\partial }{\partial t}\left( {\rho \vec{v}} \right) + \nabla \cdot \left( {\rho \vec{v}\vec{v}} \right) = - \nabla p + \nabla \cdot \left[ {\mu \left( {\nabla \vec{v} + \nabla \vec{v}^{{\text{T}}} } \right)} \right] + \rho \vec{g} + \vec{F},$$
(7)

where \(p\) is the static pressure, \(\rho \vec{g}\) is the gravitational body force and \(\vec{F}\) is the external body force defined using the source terms.

Considering the significance of the surface tension, a continuum surface force (CSF) model [22] was introduced. Here, the surface tension can be expressed according to the volume force \(F_{v}\) across the spreading interface, which was added to the VOF model through the source terms of the momentum equation, and can be given as follows [23]:

$$F_{v} = \gamma_{ij} \frac{{2\rho k\nabla \alpha_{i} }}{{\left( {\rho_{i} + \rho_{j} } \right)}},$$
(8)

where \(\gamma_{ij}\) is the surface tension between the two phases, \(n = \nabla \alpha_{i}\) is the surface normal, \(k = \nabla \cdot \hat{n}\) is the surface curvature, and \(\hat{n} = n/\left| n \right|\) is the unit normal vector to the interface. Moreover, a wall adhesion term is introduced to specify \(\hat{n}\) at the liquid-solid-gas interface [24]:

$$\hat{n} = \hat{n}_{w} \sin \theta_{d} + \hat{t}_{w} \cos \theta_{d},$$
(9)

where \(\hat{n}_{w}\) and \(\hat{t}_{w}\) are the unit vectors normal and tangential to the solid wall, and \(\theta_{d}\) is the dynamic contact angle between the liquid and wall. Based on Hoffman’s systematic study [25] of the \(\theta_{d}\) in glass capillary tubes, Jiang et al. [26] deduced an empirical correlation which fits the data equally well:

$$\frac{{\cos \theta _{d} - \cos \theta _{s} }}{{\cos \theta + 1}} = - \tanh \left( {4.96Ca^{{0.702}} } \right),$$
(10)

where \(\theta_{s}\) is the static contact angle between liquid and wall. In this paper, the free liquid jet flows from the top to the kerf; hence, \(\theta_{s}\) is assumed to be similar to the initial contact angle \(\theta_{0}\) when the liquid just reaches the kerf. As shown in Figure 4, \(\theta_{0}\) can be defined as: \(\theta_{0} \approx 90^{ \circ } + \arcsin (D_{{\text{w}}} /D_{l} )\). Additionally, because \(D_{w}\) is much smaller than the diameter of the liquid jet \(D_{l} = 0.004\), \(\theta_{s} \approx \theta_{0} \approx 90^{ \circ }\). Eq. (10) was then reduced to Eq. (11) to characterize the dynamic contact angle:

$$\cos \theta _{d} = - \tanh \left( {4.96Ca^{{0.702}} } \right).$$
(11)
Figure 4
figure 4

Schematic of the initial contact angle when the liquid just reaches the kerf

The computational model was then implemented into the Ansys Fluent software for calculation. Here, the momentum equations were solved using the pressure-based solver, and the continuity equation was solved via implicit time discretization. For pressure velocity coupling, the pressure implicit splitting of operators (PISO) scheme [27] was used, and to calculate the calculation accuracy, the global Courant number was set to 0.2 using an adaptive time step. In addition, a dynamic contact angle was introduced using user defined functions (UDF).

The following assumptions were made before the simulation setup: (1) both the air and liquid are incompressible fluids, (2) the motion of the wire is not considered and the surface of the kerf is a stationary plane, (3) heat transfer is not taken into account because the focus of this study is the liquid flow regimes in the sawn kerf, and the entire process is thermostatic at a temperature of 298.15 K.

3 Results

3.1 Liquid Entering-Spreading Progress

A simulation research was first conducted under the following parameters, which are close to the general situation: \(D_{i}\)= 50.8 mm, \(A_{k}\)=1013.4 mm2, \(D_{w}\)= 0.3 mm, \(v_{i}\)= 0.4 m/s, and the liquid properties of the solution A.

The 3D liquid flow state outside the kerf is shown in Figure 5(a). From Figure 5(a)i, the liquid first flows from the top to bottom before coming into contact with the ingot, and a liquid layer is then gradually formed at the outer edge of the kerf (Figure 5(a)iii−vi). Moreover, the evolution of the liquid contour in the center section of the kerf is shown in Figure 5(b). As the liquid reaches the upper surface of the ingot (Figure 5(b)i), it enters the kerf (Figure 5(b)ii) and spreads downward with an arc-shaped gas-liquid interface (Figure 5(b)iii). Subsequently, the spreading liquid reaches the kerf surface and gradually covers the sawing area (Figure 5(b)iv). Finally, it saturates the kerf and maintains an equilibrium state (Figure 5(b)v and vi) when \(T \ge 0.45\) s.

Figure 5
figure 5

Liquid entering-spreading process in the kerf: (a) 3D plot of the liquid dynamic flow state, (b) Transient evolution of the liquid contours in the center section of the kerf

To evaluate the spreading behavior, the liquid flow time to reach the equilibrium state \(t_{e}\) is taken as an evaluation index in the following section.

3.2 Parametric Simulation

3.2.1 Effect of Kerf Area

Figure 6 shows the simulation results of various \(A_{k}\) (simulation parameters: \(A_{k1}\)= 223.14 mm2, \(T\)= [0.07, 0.11, 0.21] s; \(A_{k2}\)= 1013.4 mm2, \(T\)= [0.27, 0.35, 0.45] s; \(A_{k3}\)= 1782.94 mm2, \(T\)= [0.49, 0.58, 0.74] s; \(A_{k4}\)= 9120.73 mm2, \(T\)= [1.92, 5.95] s; \(A_{k5}\)= 35342.91 mm2; \(T\)= [2.71, 17.80] s; other simulation parameters: \(D_{w}\)= 0.3 mm, \(v_{i}\)= 0.67 m/s, liquid properties of the solution A). The evolution of the liquid contours is presented using a scatter plot of \(t_{e}\). It should be noted that the liquid spreading interface gradually transforms from arc-shape to a U-shape with an increase in the \(A_{k}\). However, there is no fundamental discrepancy in the liquid spreading regimes: the liquid that flows into the kerf from the top to bottom spreads to both sides after reaching the center of the sawing area before finally reaching an equilibrium state. Moreover, the results of \(A_{k4}\) and \(A_{k5}\) show that the kerf cannot be filled after the liquid contour is stable (Figure 6), which is attributed to the insufficient liquid flow. Further, \(t_{e}\) significantly increases with increasing \(A_{k}\) (Figure 6), which also indicates that a higher liquid flow is required for a larger \(A_{k}\).

Figure 6
figure 6

Plot of the liquid flow time to reach the equilibrium in different kerf areas, and the evolution of the liquid spreading contour

3.2.2 Effect of Kerf Width

The simulation results of various \(D_{w}\) are shown in Figure 7 (simulation parameters: \(D_{w1}\)= 0.5 mm, \(T\)= [0.14, 0.35] s; \(D_{w2}\)= 0.4 mm, \(T\)= [0.20, 0.39] s; \(D_{w3}\)= 0.3 mm, \(T\)= [0.30, 0.45] s; \(D_{w4}\)= 0.2 mm, \(T\)= [0.45, 1.04] s; \(D_{w5}\)= 0.1 mm, \(T\)= [0.35, 4.08] s; \(A_{k2}\)= 1013.4 mm2, \(v_{i}\)= 0.67 m/s, liquid properties of solution A). \(t_{e}\) exhibits an increasing trend as \(D_{w}\) declines, and significantly increases when \(D_{w}\)> \(D_{w4}\). Typically, a decrease in \(D_{w}\) makes it more difficult for the liquid to spread in the kerf. Meanwhile, the spreading contours show a clear discrepancy in the cases of various \(D_{w}\). In cases \(D_{w1}\), \(D_{w2}\) and \(D_{w3}\), the liquid spreads with arc-shaped gas-liquid interfaces with different curvatures. However, in cases \(D_{w4}\) and \(D_{w5}\), distinct air chambers are noted in the kerf.

Figure 7
figure 7

Plot of the liquid flow time to reach an equilibrium for different kerf widths, and the evolution of the liquid spreading contour

The formation process of the air chamber phenomenon is shown in Figure 8 (simulation parameters: \(D_{i}\)= 50.8 mm, \(A_{k}\)=1013.4 mm2, \(D_{w}\)= 0.1 mm, \(v_{i}\)= 0.4 m/s, liquid properties of solution A). It can be observed that for case \(D_{w5}\), the liquid spreads to both sides when it enters the kerf (Figure 8i). Subsequently, the incoming liquid slides along the outer edge of the kerf (Figure 8ii). A small amount of the liquid then gradually seeps into the kerf (Figure 8(b)iii) and wraps it in a liquid layer, forming an air chamber in the kerf. Furthermore, as the liquid slowly spreads and squeezes downward, an opening is created in the wrapped liquid layer (Figure 8iv), from which air is squeezed out. Therefore, the liquid spreads to the kerf surface and gradually expels the air (Figure 8(b)v) before finally reaching an equilibrium state (Figure 8(b)vi).

Figure 8
figure 8

Results of the simulations under a kerf width of 0.1 mm: (a) 3D plot of the liquid dynamic flow regime, (b) Transient evolution of the liquid contours in the center section of the kerf

3.2.3 Effect of Liquid Viscosity

The simulation results for the various liquid viscosities \(\mu_{l}\), and the liquid spreading contours at a liquid flow time of 0.3 s are presented in Figure 9, (simulation parameters: \(D_{i}\)= 50.8 mm, \(A_{k}\)= 1013.4 mm2, \(D_{w}\)= 0.3 mm, \(v_{i}\)= 0.67 m/s, \(\gamma_{lv}\)= 50 mN/m, \(\rho_{l}\)= 999.12), where \(t_{e}\) shows an accelerated increasing trend with \(\mu_{l}\), based on the plotted liquid spreading contours. Moreover, a sharp increase in \(t_{e}\) is seen between \(\mu_{l2}\) and \(\mu_{l3}\) (as indicated by the red arrow). This could be attributed to the formation of the air chamber phenomenon when \(\mu_{l} \ge \mu_{l3}\) (see Figure 9). By comparing the liquid spreading area using the same flow time (0.3 s), it was confirmed that the grow of \(\mu_{l}\) significantly reduces the liquid spreading rate in the kerf and contributes to the air chamber phenomenon.

Figure 9
figure 9

Plot of the liquid flow time to the equilibrium for various liquid dynamic viscosities: \(\mu_{l1}\)= 1 mPa·s, \(\mu_{l2}\)= 1.5 mPa·s, \(\mu_{l3}\)= 3 mPa·s, \(\mu_{l4}\)= 5 mPa·s, \(\mu_{l5}\)= 15 mPa·s, \(\mu_{l6}\)= 30 mPa·s

3.2.4 Effect of Liquid Surface Tension

The simulation results for the various liquid surface tensions \(\gamma_{lv}\), and the liquid spreading contours at a liquid flow time of 0.3 s are presented in Figure 10 (simulation parameters: \(D_{i}\)= 50.8 mm, \(A_{k}\)= 1013.4 mm2, \(D_{w}\)= 0.3 mm, \(v_{i}\)= 0.67 m/s, \(\mu_{l}\)= 3 mPa·s, \(\rho_{l}\)= 999.12 kg/m3). The profile of \(t_{e}\) shows an upward trend as \(\gamma_{lv}\) increases, and a growth spurt is presented between \(\gamma_{lv3}\) and \(\gamma_{lv4}\). Moreover, as \(\gamma_{lv}\) increases, the spreading interface gradually evolves from an arc-shape (\(\gamma_{lv1}\)) to a flatter shape (\(\gamma_{lv2}\) and \(\gamma_{lv3}\)), further forming a liquid layer that wraps the kerf and inducing the occurrence of the air chamber phenomenon in the kerf (\(\gamma_{lv} \ge \gamma_{lv4}\)). Therefore, it is speculated that the different spreading regimes can be mutually-transform when the parameters are altered within a certain limit. Therefore, it can be concluded that the reduction in \(\gamma_{lv}\) facilitates liquid spreading in the kerf and assists in avoiding the formation of an air chamber.

Figure 10
figure 10

Plot of the time required to the reach the equilibrium state under various liquid surface tensions: \(\gamma_{lv1}\)= 20 mN/m, \(\gamma_{lv2}\)= 30 mN/m, \(\gamma_{lv3}\)= 40 mN/m, \(\gamma_{lv4}\)= 50 mN/m, = 60 mN/m, \(\gamma_{lv5}\)= 70 mN/m

3.2.5 Effect of the Liquid Inlet Velocity

Figure 11 presents the simulation results for various liquid velocities \(v_{i}\), and the liquid spreading contours at a liquid flow time of 0.4 s (simulation parameters: \(D_{i}\)= 50.8 mm, \(A_{k}\)= 1013.4 mm2, \(D_{w}\)= 0.2 mm, \(\mu_{l}\)= 3 mPa·s, \(\gamma_{lv}\)= 80 mN/m, \(\rho_{l}\)= 999.12 kg/m3). For the plotted liquid spreading contours, \(t_{e}\) shows an accelerated increasing trend as \(v_{i}\) decreases. Moreover, a special phenomenon is observed when \(v_{i} = v_{i3}\), where almost no liquid could flow into the kerf, and \(t_{{\text{e}}}\) is considered to be infinite. Furthermore, as \(v_{i}\) decreases, the evolution of the spreading interface is observed again, where the spreading interface gradually evolves from an arc-shape (\(v_{i1}\)) to a liquid layer wrapping the kerf (\(v_{i2}\) and \(v_{i3}\)). Eventually, the liquid could barely enter the kerf (\(v_{i4}\)).

Figure 11
figure 11

Plot of the time required to reach the equilibrium state for various liquid velocities at the inlet area of the kerf: \(v_{i1}\)=1.68 m/s, \(v_{i2}\)= 0.97 m/s, \(v_{i3}\)= 0.67 m/s, \(v_{i4}\)= 0.55 m/s

4 Discussion

In the preceding section, numerical simulations were conducted to obtain the liquid spreading regime in the kerfs. By comparing the transient evolution of the liquid spreading contour, three typical liquid spreading regimes were found: (I) The liquid spreads in an up-down direction with an arc-shaped interface; (II) the spreading liquid wraps the outer contour of the kerf, forming an air chamber in the kerf; and (III) the liquid can barely enter the kerf.

4.1 Experimental Verification

Experimental verifications were conducted focusing on the three liquid spreading regimes. Here, the experimental parameters are basically the same as those of the simulations. As shown in Figure 12(a), (b), the simulation and experimental results of solution A have very similar evolutions of the liquid spreading contour with corresponding flow times, which comprise arc-shaped spreading interfaces. This confirms the existence of liquid spreading regime I.

Figure 12
figure 12

Transient evolution of the spreading contours of solution A in the kerf: (a) Simulation results (\(A_{k}\)= 1782.94 mm2, \(D_{i}\)=50.8 mm, \(D_{w}\)= 0.25 mm, \(v_{i}\)= 0.67 m/s), (b) Experimental results (\(A_{k} \approx\) 1780.14 mm2, \(D_{i}\)=50.8 mm, \(D_{w}\)= 0.253 mm, \(v_{i}\)= 0.41 m/s)

As shown in Figure 13(a), (b), similar liquid spreading evolution process can be observed, and the air chamber phenomenon is seen in both the simulation and experimental results of solution B. This confirms the existence of regime II.

Figure 13
figure 13

Transient evolution of the spreading contours of solution B in the kerf: (a) Simulation results (\(A_{k}\)= 1782.94 mm2, \(D_{i}\)=50.8 mm, \(D_{w}\)= 0.25 m, \(v_{i}\)= 0.67 m/s), (b) Experimental results (\(A_{k} \approx\) 1780.14 mm2, \(D_{i}\)=50.8 mm, \(D_{w}\)= 0.253 mm, \(v_{i}\)= 0.403 m/s)

As shown in Figure 14(a)–(c), a very similar evolution of the liquid spreading contour is observed in both the simulation and experimental results, where the liquid barely enters the kerf and just spreads around the outer edge. This proves the existence of liquid spreading regime III.

Figure 14
figure 14

Transient evolution of the spreading contours of solution C in the kerf: (a) Simulation results (\(A_{k}\)= 1782.94 mm2, \(D_{i}\)=50.8 mm, \(D_{w}\)= 0.25 m, \(v_{i}\)= 0.55 m/s), (b) Experimental results (\(A_{k} \approx\) 1780.14 mm2, \(D_{i}\)=50.8 mm, \(D_{w}\)= 0.253 mm, \(v_{i}\)= 0.55 m/s)

4.2 Causes and Influences of Spreading Regimes

To determine the causes of different spreading regimes, typical simulation cases are selected, and the liquid contours, velocity vector, and pressure in the kerf for a certain flow time are shown in Figure 15 (simulation parameters: (I) \(D_{i} \,\) = 50.8 mm, \(A_{k} \,\) =1013.4 mm2, \(D_{w}\)= 0.3 mm, \(v_{i}\)= 0.67 m/s, liquid properties of solution A, \(T\)= 0.3 s; (II) \(D_{i} \,\) = 50.8 mm, \(A_{k} \,\) =1013.4 mm2, \(D_{w}\)= 0.1 mm, \(v_{i}\)= 0.67 m/s, liquid properties of solution A, \(T\)= 0.45 s; (III) \(D_{i} \,\) = 50.8 mm, \(A_{k} \,\) =1013.4 mm2, \(D_{w}\)= 0.2 mm, \(v_{{\text{i}}}\)= 0.58 m/s, \(\mu_{l}\)= 3 mPa·s, \(\gamma_{lv}\)= 80 mN/m, \(\rho_{l}\)= 999.12 kg/m3, \(T\)= 0.3 s).

Figure 15
figure 15

Three typical spreading regimes and their corresponding (a) vector diagrams and (b) total pressure nephograms

For spreading regime I, a significant pressure gradient (approximately 230−0 Pa, see Figure 15(b)-I) is observed between the kerf top and spreading interface, which can cause a large spreading velocity in the kerf [28, 29]. This is attributed to the significant velocity vector in the kerf, both in the liquid and air phases (Figure 15(a)-I). Thus, the liquid can spread with an arc-shaped interface in the kerf.

For spreading regime II, there is a high pressure gradient at the outer edge of the kerf (about 230−30 Pa, Figure 15(b)-II) where the velocity vector is significant. However, the pressure gradient is between the kerf top and the spreading interface is very small (approximately 230−170 Pa, see Figure 15(b)-II), making the velocity vector almost invisible. Hence, the liquid is more likely to seep in from the outer edge and form a liquid layer that wraps the kerf, which results in the generation of an air chamber.

For spreading regime III, there are no liquid pressure gradients and velocity vectors in the kerf; consequently, the liquid barely enters the kerf.

Overall, in spreading regime I, the liquid spreads easier and quicker in the kerf to expel air from the sawing area, infiltrate the processing area, and achieve its important functions. However, in spreading regime II, the air chamber significantly hinders the spreading behavior of the liquid and increases the probability of dry cutting during sawing. In spreading regime III, dry cutting has a high likeliness of occurring [30], especially for MWSS, causing the abrasive to not be carried by the liquid to the sawing area. Therefore, it is very detrimental to wire sawing.

4.3 Non-Dimensional Analysis

These results show that a decrease in the \(A_{k}\), \(\mu_{l}\), \(\gamma_{lv}\), and the growth of the \(D_{w}\) and \(v_{i}\) result in a reduction in the \(t_{e}\), which means that the liquid spreads better in the kerf. However, in-depth analysis is yet to be conducted on the interplay of these parameters and their synergy effects on the spreading behavior. Moreover, it can be seen from Figures 10, 11 that the different spreading regimes can be mutually transformed when the parameters are altered to a certain extent. To determine the mechanism, extensive numerical simulations were conducted (all relevant simulation parameters and results are shown in Table 6, see Appendix) owing to their significant influence on small-scale liquid flow.

Meanwhile, non-dimensional analysis was performed to explore the synergy effect of the parameters using two typical dimensionless numbers, the Weber number and Capillary number, which were used owing to their significant influence on small-scale liquid flow [19, 30]. The Weber and Capillary numbers were each used to represent different simulation cases: \(We_{r} = \rho_{l} v_{i}^{2} l_{0} /\gamma_{lv}\) and \(Ca_{r}\), and the simulations were carried out within certain limits of the \(We_{r}\) (5–300) and \(Ca_{r}\) (0.3–8).

The comprehensive simulation results for the non-dimensional parameters \(We_{r}\) and \(Ca_{r}\) are plotted in Figure 16. The log-log plot shows that there are clear boundaries between the three spreading regimes. This explains the mechanism of the mutual transformation between spreading regimes. Moreover, the two boundary lines were written by fitting the data in the boundary area using (both with \(R^{2} \approx 0.99\)):

$$l_{{1:}} \,We_{r} = 24.8Ca_{r} ^{{0.65}},$$
(12)
$$l_{2:} \,We_{r} = 1.47Ca_{r}^{0.1}.$$
(13)
Figure 16
figure 16

Non-dimensional map of the simulation results

Therefore, a predictive model is proposed for identifying the liquid spreading regime, which is given using a set of Webb and Capillary numbers:

$$\left\{ \begin{gathered} {\text{Regime I}}: \, We_{r} > 24.8Ca_{r}^{0.65} , \hfill \\ {\text{Regime II}}: \, 1.47Ca_{r}^{0.1} < We_{r} < 24.8Ca_{r}^{0.65} , \hfill \\ {\text{Regime III}}: \, We_{r} < 1.47Ca_{r}^{0.1} . \hfill \\ \end{gathered} \right.$$
(14)

Before the actual processing, the Weber (\(Ca_{r} = \mu_{l} v_{i} /\gamma_{lv}\)) and Capillary numbers (\(We_{r} = \rho_{l} v_{i}^{2} l_{0} /\gamma_{lv}\)) were calculated by measuring parameters such as the kerf width \(D_{w}\) (\(l_{0} \approx D_{w}\)), the liquid flow rate \(v_{i}\), the liquid density \(\rho_{l}\), the viscosity \(\mu_{l}\), and the surface tension \(\gamma_{lv}\). For a given set of Weber and Capillary numbers, the liquid spreading regime can be predicted as: When \(We_{r} > 24.8Ca_{r}^{0.65}\), the liquid spreads according to regime I; when \(1.47Ca_{r}^{0.1} < We_{r} < 24.8Ca_{r}^{0.65}\), the liquid spreads according to regime II; and when \(We_{r} < 1.47Ca_{r}^{0.1}\), the liquid spreads according to regime III.

This paper provides insight into the influence of the liquid spreading behavior in kerfs during wire sawing. It also provides a new approach for solving problems that may be encountered during wire sawing. However, additional studies should be conducted to corroborate the application of the dynamic contact angle model in 3D geometry. Furthermore, the high validity simulation method used in this paper may contribute to the research on the filling flow in small scales, such as fluid problems in precision machining, and micro injection molding.

5 Conclusions

  1. (1)

    This study was designed to investigate the liquid spreading behavior in wire-sawn kerfs. A CFD simulation model was established, and parametric simulations were successfully conducted to identify the effect of the kerf area, kerf width, liquid viscosity, liquid surface tension, and liquid velocity at the inlet of the kerf.

  2. (2)

    Based on the development trend in wire sawing, an increase in the wafer area does not change the liquid spreading regime in the kerf, but a reduction in the kerf width significantly hinders the liquid spreading behavior. By adjusting the physical properties and supply conditions of the liquid, the spreading regime can be effectively converted to facilitate wire sawing.

  3. (3)

    Three typical liquid spreading regimes were found in both the simulation and experimental results: (I) The liquid spreads in the up-down direction with an arc-shaped interface; (II) the spreading liquid wraps the outer contour of the kerf and forms an air chamber in the kerf; and (III) the liquid barely enters the kerf. Moreover, the limiting conditions of the three spreading regimes are identified using a non-dimensional analysis, and a prediction model for the liquid spreading regime is proposed using a given set of Weber and Capillary numbers. When \(We_{r} > 24.8Ca_{r}^{0.65}\), the liquid spreads according to regime I; when \(1.47Ca_{r}^{0.1} < We_{r} < 24.8Ca_{r}^{0.65}\), the liquid spreads according to regime II; and when \(We_{r} < 1.47Ca_{r}^{0.1}\), the liquid spreads according to regime III.

Availability of Data and Materials

The datasets supporting the conclusions of this article are included within the article.

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Funding

Supported by National Natural Science Foundation of China (Grant Nos. 51375179, U22A20198).

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Contributions

HH was in charge of the whole trial; LL wrote the manuscript and finished sampling and laboratory analyses. All authors read and approved the final manuscript.

Authors’ information

Lin Lin, born in 1993, is currently a doctor candidate at Xiamen University, China. He received his master degree from Huaqiao University, China, in 2020. His research focus on micro/nano robots' mechanical effects and their associated biomedical applications.

Hui Huang, born in 1974, is currently a professor at Huaqiao University, China. He received his Ph.D degree in 2002 from Nanjing University of Aeronautics and Astronautics, China. His research interests include the machining of brittle material and manufacturing of superabrasive tools.

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Correspondence to Hui Huang.

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Appendix

Appendix

See Tables 4, 5, 6

Table 4 Parameters of the geometric model
Table 5 Wire sawing conditions and parameters
Table 6 Simulation parameters and results in the non-dimensional analysis (R refers to the spreading regimes in the kerf)

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Lin, L., Huang, H. Simulation and Experimental Research on Liquid Spreading in a Wire-Sawn Kerf. Chin. J. Mech. Eng. 36, 149 (2023). https://doi.org/10.1186/s10033-023-00969-4

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