# Study on Influence of Adhesion Properties in Screen Printing Electronic Ink Transfer

Authors: Ye Tian; YingCai Yuan; Lin Zhao; GaoSheng He; TuoKai Peng; Jianlin Xu; LiJie Cao; TianQi Gu; Yan Li
Journal: Shock and Vibration
 (2022)

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## Abstract

Printing electronic components by screen printing with the excellent printing quality, high efficiency, and environmental protection has broad application prospects in additive manufacturing. However, because conductive-ink blocks the screen, printing efficiency and quality decreased in the printing process. Thus, it is pivotal to explore adhesion factors from the interaction of conductive ink and screen. Herein, according to the transfer process of screen printing ink, solid-liquid wetting theory, and adhesion mechanism, we establish a liquid bridge adhesion model between two plates and obtain the functional relationship between the adhesion force and its influencing factors. Fluent software verifies the adhesion force model's conclusion to get the influence trend of the adhesion force on the ink residue. At last, by establishing a liquid bridge tensile fracture test, we gain experimental results consistent with the theoretical model and numerical simulation results. It is demonstrated that solid-liquid contact angle, tensile distance, and liquid volume are the main factors affecting the change of solid-liquid adhesion. The increase of tensile distance and contact angle will affect the adhesion force. The smaller the maximum solid-liquid adhesion during the initial stretching of the liquid bridge, the less the residual amount remained on the solid surface. By means of screen modification, the adhesion between conductive ink and screen can be reduced; meanwhile, the efficiency and quality of printed products can be improved. What's more, it also provides a valuable reference for the modification of screen printing electronics.


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## Body

## 1. Introduction

Printed electronics is an additive manufacturing technology that uses traditional printing methods to manufacture electronic components and circuits and realizes the integration of printing and electronics [1]. Compared with the manufacture of traditional electronic components, the advantage of manufacturing electronic components by printing is flexibility, large format, and low cost [2–5]. In printed electronic manufacturing processes, such as inkjet printing and gravure printing, screen printing to produce electronic products (hereinafter referred to as screen printing electronics) has the advantages of high ink layer thickness, excellent electrical conductivity, low cost and environmental protection. It is the most widely used and mature manufacturing process today. References [6–11].In a printing process of screen printing electronics, the conductive ink is squeezed through the screen image and text area under the horizontal movement of the scraper and spreads on the printing substrate to form a print. The schematic diagram of ink transfer is shown in Figure1. Printing quality requires good use performance and high precision [12–16]. The key to high-quality printing is to establish a printing technology with high resolution and Ink Transferring Rate. Lorenz [17] pointed out that the electrode width of screen-printed silicon solar cells was reduced from 120 μm to 28 μm. The general method to improve printing resolution is to select the screen mesh with a large mesh and small aperture. However, in printing production, the smaller aperture will cause the conductive ink to block the screen frequently, as shown in Figure 2, thus the amount of ink transferred to the substrate (hereinafter referred to as Ink Transferring Rate), the thickness of ink layer was reduced, the conductivity of printed lines became worse either [18–20]. To solve the problem of Ink Transferring Rate reduction, scholars at home and abroad have increased the Ink Transferring Rate by changing the screen structure [20, 21], the rheological properties of conductive ink [20, 22–24], printing process parameters [25, 26], etc., but there are some drawbacks. Taking the knotless net screen printing stencil as an example, this structure reduces the durability of the screen while increasing the ink transfer rate, and the high plate-making cost increases the printing cost.Figure 1
Schematic diagram of ink transfer.Figure 2
Screen printing blocked by conductive ink.The mechanism of ink transfer in printing electronics was studied. Liu et al. explored the mechanism of ink transfer in screen printing and thought that increasing the dynamic pressure in the wedge-shaped area could ensure enough ink drop on the substrate [27]. Lee et al. improved the ink transferring rate and printing stability by exploring the process parameters of gravure printing [28]. Chu analyzed the influence of ink flow and ink transferring rate on the rotation speed of two ink rollers when they roll against each other, ink rollers diameter and ink layer thickness before two ink rollers’ contacting [29]. Kapur established a mathematical model of ink transfer in screen printing and proposed that ink surface energy is the key factor affecting ink transfer [30]. It can be seen that the research on ink transfer mechanism is mostly aimed at the influence of printing process parameters, and few scholars have explored the influence of adhesion between ink and screen in screen printing electronics. Achieving controllable adhesion is an effective way to solve ink-blocking mesh in the electronic field of screen printing and improve the ink transferring rate. In order to independently regulate and control the surface adhesion according to the performance requirements, domestic and foreign researchers define the pulling force required to separate two contact objects as adhesion force, and take adhesion force as an index to measure the adhesion force [31]. From the perspectives of changing the contact area [32–36] and changing the surface energy [37–39], various methods [40] for regulating and controlling the surface adhesion are put forward [41], which are mainly divided into “reshaping” and “modification”. Wu et al. [42] prepared nano-convex-patterned polyimide surface and nano-concave patterned polyimide surface, which effectively reduced the adhesion on the surface of the samples. Shivaprakash et al. [43] prepared nano-silica particles on silicon wafers. It is found that nanoparticles can reduce the adhesion of silicon wafer surface. Qing et al. [44] prepared a titanium diboride coating on a silicon wafer, the surface adhesion with this coating decreased by about 75% compared with that on the bare silicon surface. Liang [45] and others reported a Fluorine-containing onion carbon organic film, which reduced the surface adhesion by about 90% compared with Trifluoroacetic acid.Based on the fact that modification has become the most economical and effective means to regulate the adhesion between conductive ink and screen printing plate, the summation that reduction of surface energy and contact area can reduce adhesion was verified by a considerable number of experiments, while the clear conclusion about the functional relationship between surface energy and adhesion has not been obtained yet. Hence, we constructed an adhesion model of conductive ink to obtain the functional relationship between adhesion and surface energy (characterized by contact angle) and contact area, as and established a simulation analysis model to verify the conclusion of the adhesion model and acquired the influence trend of adhesion on ink residue. It is indicated that the adhesion model and simulation analysis model are consistent with the practical result through the liquid bridge tensile fracture experiment.


## 2. Model of Liquid Bridge Adhesion between Two Plates under Gravity-Free Condition

### 2.1. Adhesion Mechanism in Surface Modification

When the liquid drops are placed on a smooth solid surface, the liquid drops spread on the solid surface, and the solid-vapor interface is replaced by the solid-liquid interface, and the liquid-vapor interface is enlarged. Under isothermal and isobaric conditions, the change of surface energy per unit area before and after the solid surface is wetted, that is, the change of Gibbs free energy function, is shown in equation (1). Shown as equation (2) for spreading parameter S, which means the change of surface energy per unit area before and after surface wetting. According to the second law of thermodynamics, When S≥0, the droplets completely wet the solid surface and spontaneously spread on the solid surface. When S<0, droplets partially wet the solid surface and keep an equilibrium shape in spherical form [46].(1)ΔG=γsl+γlv−γsv,(2)S=−ΔGA=γsv−γsl+γlv,where γsl, γsv, and γlv are the “solid-liquid,” “solid-vapor,” and “liquid-vapor” interfacial tensions, N/m. A is expended area, m2.WhenS<0, the droplets spread on the solid surface and reach equilibrium, the angle formed by the vapor-liquid interface and the solid-liquid interface at the intersection of vapor-liquid-solid three phases is defined as the contact angle θ, the spreading of droplets on the solid surface is shown in Figure 3. The surface energy of solid γsv is generally difficult to measure. Therefore, contact angles θ is used to characterize the wetting difficulty of the solid surface. In 1805, aimed at partial wetting, Thomas Young quantitatively analyzed the relationship between solid surface free energy and contact angle [47], and Young's equation is (3)cosθ=γsv−γslγlv.Figure 3
Spreading of liquid droplets on a solid surface.Under isothermal and isobaric conditions, when the liquid adheres to the solid surface per unit area, as shown in Figure4, the surface tension force is γsl, it is separated into two independent phases by an external force, the surface tension force is γlv and γsv. In this separation process, the work done by the external force is called adhesion work Wa. It is shown as the (4) for the expression.(4)Wa=γlv+γsv−γsl.Figure 4
Schematic diagram of the two-phase separation.Combined with Young's equation Dupre to simplified (3), and the relationship between adhesion work and contact angle is shown in (5) [48].(5)Wa=γlv1+cosθ.From the point of view of energy, the research on the modification of screen printing electronics is considered. If the properties of fixed conductive ink do not change, the wettability of conductive ink on the screen can be affected by changing the surface energy of the screen so as to achieve the purpose of changing the adhesion ability between solid and liquid, and finally solve the problem of ink adhering to the screen.


### 2.2. Force Analysis of Droplets

Generally, when the capillary length is larger than the characteristic scale of liquid, the influence of gravity can be ignored [49], and the capillary length formula is defined as:(6)lc=γρg,where γ is the liquid surface tension, N/m, ρ is liquid density, kg/m3, g is the acceleration of gravity, m/s2.Definition is made as following: the surface tension of conductive silver ink 0.03 N/m, the density is 1950 kg/m3, capillary length is 1.253 mm, this length is taken as the ink droplet diameter, the volume is 1.03 μL. In this volume, the influence of surface tension on ink droplets is greater than that of gravity, and the influence of gravity is negligible.


### 2.3. Model of Liquid Bridge Adhesion between Two Plates

According to the schematic diagram of ink transfer, it is necessary to overcome the adhesion between ink and screen to reduce the residual amount of ink on-screen printing. It is very important to build a solid-liquid adhesion and separation model to research modification of screen printing electronics. Modeling the tensile fracture process of a single screen thread and conductive ink in the screen, when the geometric radius of a single screen thread is unlimited extension, the cylindrical screen thread is simplified as a flat plate, and the model is simplified as the tensile fracture process between conductive ink and two flat plates. The tensile fracture model based on the liquid bridge between two plates is simple and versatile, which provides reference experience for establishing a more reasonable model.It is assumed that the contact angle of ink and upper and lower plates are the sameθ, in addition, in the process of stretching, the influence of gravity of the liquid phase is neglected, in the process of stretching and breaking of the liquid phase, the liquid phase will not evaporate or condense, and the volume is conserved, the two plates are rigid. The schematic diagram of the liquid bridge between two plates is shown in Figure 5. The adhesive force F acting on the plane is mainly affected by capillary force, which consists of the following two parts:(1)
Adhesion caused by the pressure difference between inside and outside the meniscus Δp.(2)
Vertical component of surface tension in the tangential direction of the liquid meniscus.Figure 5
Schematic diagram of liquid bridge between two plates.Therefore,(7)F1=πl22Δp1+πl⋅γ⋅sinθ.In equation (7), l is the diameter of the contact part between the liquid phase and two plates, m, Δp1 is the pressure difference across the vapour-liquid, N. When the liquid bridge is stable, the Young–Laplace equation is satisfied at the liquid phase contour point M, as shown in equation (8).(8)Δp1=γ1r1+1r2.According to the Young-Laplace equation, in general, in order to obtain the expression of liquid bridge capillary force and the appearance model of the liquid bridge, the meniscus is assumed to be a circular arc, sor1 and r2 in the differential equation can be expressed as the two main radii of curvature of the liquid-vapor two-phase interface in liquid bridge, m.(9)r1=R,r2=−D2.The following relations can be obtained from the basic assumptions, as shown in (10) and (11)(10)R=H2cosθ,(11)l=D+2R1−sinθ.In whichH is the height of the liquid bridge, m, D is the diameter of the necking part of the liquid bridge, m.The volume of the liquid bridge is approximated as a cylindrical volume, and the diameter of the cylinder takes the intermediate value between the wetting diameterl of the liquid bridge and the diameter D at the neck of the liquid bridge so that the approximate volume expression of the liquid bridge can be obtained as equation (13).(12)V=πD+l42H,where l is the wetting diameter of the liquid bridge, m, D is the diameter at the neck of the liquid bridge, m, l and D can be obtained from the joint equation (11)-(13). The expression, as shown type (13) and (14)(13)l=4VπH+H1−sinθ2cosθ,(14)D=4VπH−H1−sinθ2cosθ.Therefore, according to the Young-Laplace equation, the pressure difference between inside and outside the liquid-vapor phase can be obtained, as shown in equation (16).(15)Δp=γ2cosθH−24V/πH−H1−sinθ/2cosθ.To sum up, the expression of liquid bridge adhesion between two plates is shown in equation (17).(16)F1=π4V/πH+H1−sinθ/2cosθ22γ2cosθH−24V/πH−H1−sinθ+π4VπH+H1−sinθγsinθ.From the above expression, it can be seen that the liquid bridge adhesion between two plates is related to the liquid volume, solid-liquid contact angle and stretching distance.


### 2.4. Analysis of Calculation Results of Liquid Bridge Adhesion Model between Two Plates

In the theoretical derivation of the liquid bridge model between two plates, when the upper plate moves upwards, it is assumed that the solid contact angle between the liquid phase and the lower plate does not change. The volume of the conductive ink is expressed by equation (13). Under the condition of the same liquid bridge volume calculated by equation (17), the change value of adhesion force with stretching distance under different contact angles is shown in Table 1. According to the model data in Table 1, draw a graph of changes in adhesion and stretching distance, as shown in Figure 6. It can be seen from the figure that (1) the adhesion decreases with the increase of stretching distance, and when the stretching distance reaches 1 mm, the adhesion tends to 0 N, indicating that the liquid bridge has broken at this time. (2) At the stretching distance of 0.1 mm, the adhesion force of the liquid bridge is the largest, and the corresponding adhesion force decreases with the increase of the contact angle. That is, with the increase of contact angle, the more easily the liquid bridge breaks.Table 1
Theoretical value of liquid bridge adhesion under different contact angles.
Stretching distance,H (mm)
Adhesion force,F (x10−4N)
10°
20°
30°
40°
50°
60°
70°
80°
0.1
59.34
56.92
52.77
47.02
39.84
31.44
22.10
12.07
0.2
14.88
14.43
13.53
12.21
10.52
8.51
6.24
3.77
0.3
6.59
6.48
6.18
5.67
4.99
4.14
3.18
2.11
0.4
3.63
3.65
3.55
3.34
3.00
2.57
2.05
1.47
0.5
2.21
2.30
2.31
2.22
2.05
1.81
1.50
1.15
0.6
1.40
1.53
1.59
1.59
1.51
1.37
1.18
0.95
0.7
0.85
1.03
1.14
1.19
1.16
1.10
0.97
0.82
0.8
0.43
0.67
0.82
0.90
0.93
0.90
0.84
0.73
0.9
0.09
0.37
0.57
0.69
0.75
0.76
0.73
0.66
1.0
0
0.12
0.36
0.52
0.61
0.65
0.65
0.60Figure 6
Variation curve of adhesion force with stretching distance under different contact angles.Meanwhile, according to (16), under the premise that the contact angle is constant at 40°, the change value of the adhesion force of the liquid bridge with different liquid bridge volumes versus the stretching distance is shown in Table 2 (the maximum volume is not more than 1 μL). According to Table 2, draw a graph of the change of adhesion force with stretching distance, as shown in Figure 7. It can be seen from Figure 7 that under the same stretching distance, the smaller the volume of the liquid bridge, the smaller the adhesive force, which is caused by the decrease of liquid volume and the area of auxiliary development on the solid surface.Table 2
Theoretical value of liquid bridge adhesion force under different liquid bridge volume.
Stretching distance,H (mm)
Adhesion force,F (x 10−4 N)
Liquid bridge volume (1 v = 1μL)
0.2 v
0.4 v
0.6 v
0.8 v
1 v
0.1
9.66
19.12
28.44
37.73
47.02
0.2
2.60
5.04
7.45
9.84
12.21
0.3
1.24
2.38
3.49
4.59
5.67
0.4
0.73
1.41
2.06
2.70
3.34
0.5
0.46
0.93
1.37
1.80
2.22
0.6
0.28
0.64
0.97
1.28
1.59
0.7
0.14
0.44
0.70
0.95
1.19
0.8
0
0.29
0.51
0.71
0.90
0.9
0
0.15
0.35
0.53
0.69
1.0
0
0
0.21
0.37
0.52Figure 7
Variation curve of adhesion force of different liquid bridge volumes with stretching distance (40°).


## 2.1. Adhesion Mechanism in Surface Modification

When the liquid drops are placed on a smooth solid surface, the liquid drops spread on the solid surface, and the solid-vapor interface is replaced by the solid-liquid interface, and the liquid-vapor interface is enlarged. Under isothermal and isobaric conditions, the change of surface energy per unit area before and after the solid surface is wetted, that is, the change of Gibbs free energy function, is shown in equation (1). Shown as equation (2) for spreading parameter S, which means the change of surface energy per unit area before and after surface wetting. According to the second law of thermodynamics, When S≥0, the droplets completely wet the solid surface and spontaneously spread on the solid surface. When S<0, droplets partially wet the solid surface and keep an equilibrium shape in spherical form [46].(1)ΔG=γsl+γlv−γsv,(2)S=−ΔGA=γsv−γsl+γlv,where γsl, γsv, and γlv are the “solid-liquid,” “solid-vapor,” and “liquid-vapor” interfacial tensions, N/m. A is expended area, m2.WhenS<0, the droplets spread on the solid surface and reach equilibrium, the angle formed by the vapor-liquid interface and the solid-liquid interface at the intersection of vapor-liquid-solid three phases is defined as the contact angle θ, the spreading of droplets on the solid surface is shown in Figure 3. The surface energy of solid γsv is generally difficult to measure. Therefore, contact angles θ is used to characterize the wetting difficulty of the solid surface. In 1805, aimed at partial wetting, Thomas Young quantitatively analyzed the relationship between solid surface free energy and contact angle [47], and Young's equation is (3)cosθ=γsv−γslγlv.Figure 3
Spreading of liquid droplets on a solid surface.Under isothermal and isobaric conditions, when the liquid adheres to the solid surface per unit area, as shown in Figure4, the surface tension force is γsl, it is separated into two independent phases by an external force, the surface tension force is γlv and γsv. In this separation process, the work done by the external force is called adhesion work Wa. It is shown as the (4) for the expression.(4)Wa=γlv+γsv−γsl.Figure 4
Schematic diagram of the two-phase separation.Combined with Young's equation Dupre to simplified (3), and the relationship between adhesion work and contact angle is shown in (5) [48].(5)Wa=γlv1+cosθ.From the point of view of energy, the research on the modification of screen printing electronics is considered. If the properties of fixed conductive ink do not change, the wettability of conductive ink on the screen can be affected by changing the surface energy of the screen so as to achieve the purpose of changing the adhesion ability between solid and liquid, and finally solve the problem of ink adhering to the screen.


## 2.2. Force Analysis of Droplets

Generally, when the capillary length is larger than the characteristic scale of liquid, the influence of gravity can be ignored [49], and the capillary length formula is defined as:(6)lc=γρg,where γ is the liquid surface tension, N/m, ρ is liquid density, kg/m3, g is the acceleration of gravity, m/s2.Definition is made as following: the surface tension of conductive silver ink 0.03 N/m, the density is 1950 kg/m3, capillary length is 1.253 mm, this length is taken as the ink droplet diameter, the volume is 1.03 μL. In this volume, the influence of surface tension on ink droplets is greater than that of gravity, and the influence of gravity is negligible.


## 2.3. Model of Liquid Bridge Adhesion between Two Plates

According to the schematic diagram of ink transfer, it is necessary to overcome the adhesion between ink and screen to reduce the residual amount of ink on-screen printing. It is very important to build a solid-liquid adhesion and separation model to research modification of screen printing electronics. Modeling the tensile fracture process of a single screen thread and conductive ink in the screen, when the geometric radius of a single screen thread is unlimited extension, the cylindrical screen thread is simplified as a flat plate, and the model is simplified as the tensile fracture process between conductive ink and two flat plates. The tensile fracture model based on the liquid bridge between two plates is simple and versatile, which provides reference experience for establishing a more reasonable model.It is assumed that the contact angle of ink and upper and lower plates are the sameθ, in addition, in the process of stretching, the influence of gravity of the liquid phase is neglected, in the process of stretching and breaking of the liquid phase, the liquid phase will not evaporate or condense, and the volume is conserved, the two plates are rigid. The schematic diagram of the liquid bridge between two plates is shown in Figure 5. The adhesive force F acting on the plane is mainly affected by capillary force, which consists of the following two parts:(1)
Adhesion caused by the pressure difference between inside and outside the meniscus Δp.(2)
Vertical component of surface tension in the tangential direction of the liquid meniscus.Figure 5
Schematic diagram of liquid bridge between two plates.Therefore,(7)F1=πl22Δp1+πl⋅γ⋅sinθ.In equation (7), l is the diameter of the contact part between the liquid phase and two plates, m, Δp1 is the pressure difference across the vapour-liquid, N. When the liquid bridge is stable, the Young–Laplace equation is satisfied at the liquid phase contour point M, as shown in equation (8).(8)Δp1=γ1r1+1r2.According to the Young-Laplace equation, in general, in order to obtain the expression of liquid bridge capillary force and the appearance model of the liquid bridge, the meniscus is assumed to be a circular arc, sor1 and r2 in the differential equation can be expressed as the two main radii of curvature of the liquid-vapor two-phase interface in liquid bridge, m.(9)r1=R,r2=−D2.The following relations can be obtained from the basic assumptions, as shown in (10) and (11)(10)R=H2cosθ,(11)l=D+2R1−sinθ.In whichH is the height of the liquid bridge, m, D is the diameter of the necking part of the liquid bridge, m.The volume of the liquid bridge is approximated as a cylindrical volume, and the diameter of the cylinder takes the intermediate value between the wetting diameterl of the liquid bridge and the diameter D at the neck of the liquid bridge so that the approximate volume expression of the liquid bridge can be obtained as equation (13).(12)V=πD+l42H,where l is the wetting diameter of the liquid bridge, m, D is the diameter at the neck of the liquid bridge, m, l and D can be obtained from the joint equation (11)-(13). The expression, as shown type (13) and (14)(13)l=4VπH+H1−sinθ2cosθ,(14)D=4VπH−H1−sinθ2cosθ.Therefore, according to the Young-Laplace equation, the pressure difference between inside and outside the liquid-vapor phase can be obtained, as shown in equation (16).(15)Δp=γ2cosθH−24V/πH−H1−sinθ/2cosθ.To sum up, the expression of liquid bridge adhesion between two plates is shown in equation (17).(16)F1=π4V/πH+H1−sinθ/2cosθ22γ2cosθH−24V/πH−H1−sinθ+π4VπH+H1−sinθγsinθ.From the above expression, it can be seen that the liquid bridge adhesion between two plates is related to the liquid volume, solid-liquid contact angle and stretching distance.


## 2.4. Analysis of Calculation Results of Liquid Bridge Adhesion Model between Two Plates

In the theoretical derivation of the liquid bridge model between two plates, when the upper plate moves upwards, it is assumed that the solid contact angle between the liquid phase and the lower plate does not change. The volume of the conductive ink is expressed by equation (13). Under the condition of the same liquid bridge volume calculated by equation (17), the change value of adhesion force with stretching distance under different contact angles is shown in Table 1. According to the model data in Table 1, draw a graph of changes in adhesion and stretching distance, as shown in Figure 6. It can be seen from the figure that (1) the adhesion decreases with the increase of stretching distance, and when the stretching distance reaches 1 mm, the adhesion tends to 0 N, indicating that the liquid bridge has broken at this time. (2) At the stretching distance of 0.1 mm, the adhesion force of the liquid bridge is the largest, and the corresponding adhesion force decreases with the increase of the contact angle. That is, with the increase of contact angle, the more easily the liquid bridge breaks.Table 1
Theoretical value of liquid bridge adhesion under different contact angles.
Stretching distance,H (mm)
Adhesion force,F (x10−4N)
10°
20°
30°
40°
50°
60°
70°
80°
0.1
59.34
56.92
52.77
47.02
39.84
31.44
22.10
12.07
0.2
14.88
14.43
13.53
12.21
10.52
8.51
6.24
3.77
0.3
6.59
6.48
6.18
5.67
4.99
4.14
3.18
2.11
0.4
3.63
3.65
3.55
3.34
3.00
2.57
2.05
1.47
0.5
2.21
2.30
2.31
2.22
2.05
1.81
1.50
1.15
0.6
1.40
1.53
1.59
1.59
1.51
1.37
1.18
0.95
0.7
0.85
1.03
1.14
1.19
1.16
1.10
0.97
0.82
0.8
0.43
0.67
0.82
0.90
0.93
0.90
0.84
0.73
0.9
0.09
0.37
0.57
0.69
0.75
0.76
0.73
0.66
1.0
0
0.12
0.36
0.52
0.61
0.65
0.65
0.60Figure 6
Variation curve of adhesion force with stretching distance under different contact angles.Meanwhile, according to (16), under the premise that the contact angle is constant at 40°, the change value of the adhesion force of the liquid bridge with different liquid bridge volumes versus the stretching distance is shown in Table 2 (the maximum volume is not more than 1 μL). According to Table 2, draw a graph of the change of adhesion force with stretching distance, as shown in Figure 7. It can be seen from Figure 7 that under the same stretching distance, the smaller the volume of the liquid bridge, the smaller the adhesive force, which is caused by the decrease of liquid volume and the area of auxiliary development on the solid surface.Table 2
Theoretical value of liquid bridge adhesion force under different liquid bridge volume.
Stretching distance,H (mm)
Adhesion force,F (x 10−4 N)
Liquid bridge volume (1 v = 1μL)
0.2 v
0.4 v
0.6 v
0.8 v
1 v
0.1
9.66
19.12
28.44
37.73
47.02
0.2
2.60
5.04
7.45
9.84
12.21
0.3
1.24
2.38
3.49
4.59
5.67
0.4
0.73
1.41
2.06
2.70
3.34
0.5
0.46
0.93
1.37
1.80
2.22
0.6
0.28
0.64
0.97
1.28
1.59
0.7
0.14
0.44
0.70
0.95
1.19
0.8
0
0.29
0.51
0.71
0.90
0.9
0
0.15
0.35
0.53
0.69
1.0
0
0
0.21
0.37
0.52Figure 7
Variation curve of adhesion force of different liquid bridge volumes with stretching distance (40°).


## 3. Simulation Study on Tensile Fracture Process of Liquid Bridge

### 3.1. Model Building and Grid Division

#### 3.1.1. Model Establishment

The Volume of Fluid (VOF) method is used to simulate the tensile fracture of the liquid bridge between two plates by the Computational Fluid Dynamics (CFD), and the Integrated Computer Engineering and Manufacturing code for Computational Fluid Dynamics (ICEMCFD) software in Ansys Workbench is used to model the tensile fracture process of the liquid bridge between two plates in two dimensions. The calculation domain of 5 mm × 10 mm in the x-y plane is established, and the software inZ-axis direction is defined as unit length, as shown in Figure 8. The length of the upper plate is L1 = 2 mm, and the width is L2 = 0.5 mm.Figure 8
Two-plane two-dimensional model.


#### 3.1.2. Grid Division of Two-Dimensional Model between Two Flat Plates

Based on modeling in the previous step, the fluid domain is divided into the fluid region and fluid-move region. The fluid-move area moves upward with the upper plate at a constant speed, and the position of the upper plate changes constantly, so the grid is dynamically processed. The whole model is divided into grids in a mixed grid mode, which has the advantage of improving grid quality, reducing the computation and saving time. The area division is shown as Figure9(a). The hybrid grid is composed of an unstructured grid and a structured grid, in which the structured grid is a quadrilateral grid, which is defined as the upward moving fluid-move region. As shown in Figure 9(b), the minimum side length of the quadrilateral grid is 5.78e−5 m, and the maximum side length of the quadrilateral grid is 5.83e−5 m. Unstructured mesh is composed of triangular meshes, which are defined as a constant fluid region, as shown in Figure 9(c). The minimum side length is 3.08417e−5 m, the maximum side length is 8.6211e−5 m, and the number of corresponding grids in the simulation domain is 41,600.Figure 9
Grid division of ICEMCFD two-dimensional model. (a) Model region division. (b) Structured grid region. (c) Unstructured grid region.
(a)
(b)
(c)


### 3.2. Numerical Simulation Parameter Setting

The ANSYS FLUENT 17.0 computational fluid dynamics software is used for numerical simulation, and the basic parameters used in the simulation experiment are shown in Table3.Table 3
Basic simulation parameters.
Model parameter
Numerical value
Contact angle of upper plate
10°–80°
Contact angle of lower plate
90°
Initial distance between two plates
1.5 mm
Initial movement speed of upper plate
1 mm/s
Ink density
1950 kg/m3
Ink viscosity
9 pa∙s
Surface tension of conductive ink
0.03 N/m
Moving distance of upper plate
4.5 mmIn the process of simulating the tensile fracture of the liquid bridge between two plates, the air phase is defined as the first basic item, the conductive silver ink is defined as the second basic item, and it is defined as an incompressible fluid with a volume of 1μL, which is in contact with the upper and lower plates. Liu [50] verified in experiments that the property of conductive silver ink is pseudoplastic fluid and belongs to laminar flow. Therefore the liquid property in numerical simulation is defined as laminar flow. The upper wall-up of the simulation area is defined as the pressure outlet, and the left, right, and lower walls are defined as the wall surface [51]. VOF method is used to simulate multiphase flow, Geo-Reconstruct scheme is used to calculate phase volume fraction, and the pressure dispersion solution uses PRESTO!, using the Standard model of Realizable k−ε turbulence model. The dynamic mesh model is adopted, and spring smoothing model and local reconstruction model are selected to update the mesh.


### 3.3. Comparison of Liquid Bridge Adhesion with Theoretical Values under Different Contact Angles

The simulation results of liquid bridge adhesion with stretching distance under different contact angles are shown in Table4. In order to visually express the difference between the simulation results and the theoretical results, a comparison diagram of the changing trend of the two results is drawn. The trend of 10°–40° adhesion with stretching distance is shown in Figure 10(a), and the trend of 50°–80° adhesion with stretching distance is shown in Figure 10(b). The maximum deviation rate between simulated values and the theoretical values is 14.9%. There are two reasons for the analysis error: (1) because the mixed grid method is adopted for meshing in numerical simulation, during the upward movement of the structured grid area (fluid-move), the unstructured grid area (fluid) is affected by stretching, and the accuracy at the bottom is not high, and the liquid just adheres with the bottom wall (wall) at this position. (2) In the theoretical derivation part, the liquid volume is approximately replaced by equation (8), and there is an error between the approximate volume and the real volume, which eventually leads to the deviation of the simulation results. The comparison results show that the changing trend of liquid bridge adhesion between two plates affected by stretching distance and contact angle obtained by Ansys Fluent numerical simulation software is consistent with the theoretical results, proving that the numerical simulation has high accuracy.Table 4
Simulation values of adhesion force of liquid bridge under different contact angles.
Stretching distanceH (mm)
Adhesion forceF (x 10−4 N)
10°
20°
30°
40°
50°
60°
70°
80°
0.1
65.21
59.82
55.13
52.14
45.71
33.12
24.41
12.93
0.2
15.13
16.23
14.11
13.03
12.02
9.19
7.00
4.03
0.3
6.87
6.84
6.34
5.92
5.73
4.37
3.40
2.22
0.4
4.17
3.91
3.81
3.63
3.35
2.65
2.26
1.65
0.5
2.54
2.36
2.20
2.43
2.35
1.95
1.63
1.19
0.6
1.57
1.54
1.76
1.62
1.73
1.55
1.25
0.99
0.7
0.97
1.11
1.01
1.22
1.32
1.16
1.00
0.90
0.8
0.49
0.71
0.75
1.00
1.06
1.00
0.86
0.81
0.9
0
0
0.57
0.77
0.86
0.86
0.79
0.75
1.0
0
0
0
0
0
0
0
0Figure 10
(a) Comparison of simulation value and theoretical value of adhesion force change trend at 10°–40°. (b) Comparison of simulation value and theoretical value of adhesion force change trend at 50°–80°.
(a)
(b)


### 3.4. The Relationship between Liquid Bridge Adhesion and Ink Residue under Different Contact Angles

Conductive ink adheres to the upper plate. When the upper plate moves upwards, the liquid bridge conductive ink is stretched to break. At this time, a part of ink will remain on the surface of the upper plate, which is called residual inkmre. In the two-dimensional numerical simulation, it is assumed that the Z-axis direction is constant per unit length, so the change of contrast conductive ink area can reflect the change of volume. Import the simulation data into CFD-POST software for post-processing, carry out dimensionless treatment on the residual ink, and calculate the percentage of the two-dimensional area Sag of the residual ink on the upper plate to the area Sfluid-move of the fluid-move area to obtain the specific amount kre of the proportion of ink residue, as shown in equation (18). Furthermore, the changing trend of ink residue under different contact angles was compared.(17)kre=SAgSfluid−move%.In order to quantitatively analyse the influence of different contact angles on the residual ink on the upper plate, Figure11 shows the nephogram of the residual ink when the upper plate stops moving, and the contact angles are 30°, 50°, and 70°, respectively. According to the simulation results, the maximum adhesion of the liquid bridge and the proportion of the residual ink under different contact angles are measured, as shown in Table 5. According to the data in Table 5, the variation curves of contact angle, ink residue proportion, and maximum adhesion force are drawn, as shown in Figure 12. It can be seen from the figure that (1) under the premise that the liquid volume between two plates and the contact angle of the lower plate are unchanged, the smaller the contact angle of the upper plate, the greater the amount of ink remaining on the upper plate. (2) At the tensile distance of 0.1 mm, the liquid bridge adhesion is the largest, and the corresponding adhesion decreases with the increase of the contact angle. (3) Under the different of contact angle, the maximum adhesive force has the same change trend as the residual amount of ink on the upper plate. The reason for this phenomenon is that the contact angle becomes smaller, the ink is easier to spread on the solid surface, the spreading area increases, and the adhesion force correspondingly increases. Compared with the lower plate with an unchanged contact angle, the upper plate will adhere more ink, that is, the residual amount of ink on the upper plate will increase after the liquid bridge between the two plates is stretched and broken.Figure 11
Cloud picture of numerical simulation of ink residue.Table 5
Simulation values under different contact angles.
Contact angle of upper plate
Maximum adhesion force F of liquid bridge
Proportion of ink residue, kre/(%)
θ/(°)
F/(x10-3 N)
10°
6.52
32.18
20°
5.98
26.51
30°
5.51
21.80
40°
5.21
21.48
50°
4.57
16.59
60°
3.31
7.68
70°
2.44
6.69
80°
1.29
5.05Figure 12
Change trend of ink residue proportion and adhesion under different contact angles.


### 3.5. The Relationship between Liquid Bridge Adhesion and Stretching Distance under the Same Contact Angle and Different Liquid Volume

Figure13 shows the nephogram of marked liquid volume before the upper plate moves, which is 0.2 V, 0.4 V, 0.6 V, 0.8 V, and 1 V respectively, and the initial liquid volume is 1V is 1 μL. Table 6 shows the variation of liquid bridge adhesion and stretching distance under different liquid volumes when the contact angle of the two plates is 40. According to the data in Table 6, the curves of adhesion force with stretching distance under different liquid bridge volumes is shown in Figure 14. It can be seen from the figure that: (1) with the increase of stretching distance, the adhesion of liquid bridge decreases gradually under different liquid volumes. (2) Under the same stretching distance, the larger the volume of liquid, the greater the corresponding adhesive force. The reason for this phenomenon is that under the same stretching distance and solid-liquid contact angle, the larger the liquid volume, the larger the solid-liquid contact area, and the greater the corresponding liquid bridge adhesion. The trend of the simulation results is consistent with that of the theoretical derivation shown in Figure 7. Meanwhile, the change of the solid-liquid adhesion force is observed by changing the liquid volume to affect the liquid-solid spreading area, it is further proved that changing the solid-liquid contact angle in a modified manner affects the solid-liquid spreading area and ultimately changes the liquid bridge adhesion force.Figure 13
Numerical simulation nephogram of different initial ink volumes.Table 6
Simulation values of liquid bridge adhesion under different liquid bridge volumes.
Stretching distance,H (mm)
Adhesion forceF (x10−4 N)
Liquid bridge volume (1 v = 1μL)
0.2 v
0.4 v
0.6 v
0.8 v
1 v
0.1
12.43
21.23
32.24
38.41
52.14
0.2
2.30
5.56
8.52
10.43
13.03
0.3
0.74
2.94
3.70
4.88
5.92
0.4
0.50
1.63
2.47
3.08
3.63
0.5
0
1.04
1.44
1.96
2.43
0.6
0
0.59
1.00
1.34
1.62
0.7
0
0
0.81
1.01
1.22
0.8
0
0
0.57
0.85
1.00
0.9
0
0
0
0.59
0.77
1.0
0
0
0
0
0Figure 14
Variation curve of adhesion force with stretching distance under 40° contact angle and different liquid volume.


### 3.6. Guidance of Simulations and Theoretical Calculations for Practical Screen Printing

From the simulation and theoretical calculation, it can be found that the adhesive force at the first tensile distance (0.1 mm) is the largest and decreases gradually with the increase of tensile distance. This indicates that the liquid adhesion to the solid surface decreases with the increase of the tensile distance, and that when the tensile distance increases to 1 mm at different contact angles, the adhesion force values are equal to or close to 0, which indicates that the liquid bridge is broken or almost broken. Therefore, it is recommended that in actual screen printing production, the off-screen spacing should be at least more than 1 mm. In this way, it can avoid that the screen plate cannot bounce back immediately due to the too close distance from the screen, or the edge of the printed graphics is blurred and the surface roughness is uneven, etc., which will affect the product quality and printing accuracy.Through the simulation calculation, it can be seen that the change trend of the maximum adhesion and ink residue is the same under different contact angles. That is, with the increase of contact angle, the maximum adhesion decreases and the amount of ink remaining on the solid surface also decreases. Therefore, it is suggested that in the actual screen printing production, the surface of the substrate can be modified in a targeted manner, and the solid-liquid contact angle of the substrate surface can be appropriately reduced in combination with the quality analysis of the printed matter. It can increase the adhesion of the substrate to the ink in the mesh. Or modify the screen to increase the contact angle of the screen surface, in order to reduce the adhesion of the screen to the ink.It can also be known from the simulation and theoretical calculation that under the same stretching distance, the larger the liquid volume is, the greater the corresponding solid-liquid adhesion force is. Therefore, in the actual screen printing production, the single mesh can be appropriately increased and the mesh wire diameter can be reduced in the production process of the screen mesh. In this way, the contact area between the conductive ink and the substrate in a single mesh can be increased, the adhesion of the substrate to the ink can be increased, and the ink transfer rate can be increased.


## 3.1. Model Building and Grid Division

### 3.1.1. Model Establishment

The Volume of Fluid (VOF) method is used to simulate the tensile fracture of the liquid bridge between two plates by the Computational Fluid Dynamics (CFD), and the Integrated Computer Engineering and Manufacturing code for Computational Fluid Dynamics (ICEMCFD) software in Ansys Workbench is used to model the tensile fracture process of the liquid bridge between two plates in two dimensions. The calculation domain of 5 mm × 10 mm in the x-y plane is established, and the software inZ-axis direction is defined as unit length, as shown in Figure 8. The length of the upper plate is L1 = 2 mm, and the width is L2 = 0.5 mm.Figure 8
Two-plane two-dimensional model.


### 3.1.2. Grid Division of Two-Dimensional Model between Two Flat Plates

Based on modeling in the previous step, the fluid domain is divided into the fluid region and fluid-move region. The fluid-move area moves upward with the upper plate at a constant speed, and the position of the upper plate changes constantly, so the grid is dynamically processed. The whole model is divided into grids in a mixed grid mode, which has the advantage of improving grid quality, reducing the computation and saving time. The area division is shown as Figure9(a). The hybrid grid is composed of an unstructured grid and a structured grid, in which the structured grid is a quadrilateral grid, which is defined as the upward moving fluid-move region. As shown in Figure 9(b), the minimum side length of the quadrilateral grid is 5.78e−5 m, and the maximum side length of the quadrilateral grid is 5.83e−5 m. Unstructured mesh is composed of triangular meshes, which are defined as a constant fluid region, as shown in Figure 9(c). The minimum side length is 3.08417e−5 m, the maximum side length is 8.6211e−5 m, and the number of corresponding grids in the simulation domain is 41,600.Figure 9
Grid division of ICEMCFD two-dimensional model. (a) Model region division. (b) Structured grid region. (c) Unstructured grid region.
(a)
(b)
(c)


## 3.1.1. Model Establishment

The Volume of Fluid (VOF) method is used to simulate the tensile fracture of the liquid bridge between two plates by the Computational Fluid Dynamics (CFD), and the Integrated Computer Engineering and Manufacturing code for Computational Fluid Dynamics (ICEMCFD) software in Ansys Workbench is used to model the tensile fracture process of the liquid bridge between two plates in two dimensions. The calculation domain of 5 mm × 10 mm in the x-y plane is established, and the software inZ-axis direction is defined as unit length, as shown in Figure 8. The length of the upper plate is L1 = 2 mm, and the width is L2 = 0.5 mm.Figure 8
Two-plane two-dimensional model.


## 3.1.2. Grid Division of Two-Dimensional Model between Two Flat Plates

Based on modeling in the previous step, the fluid domain is divided into the fluid region and fluid-move region. The fluid-move area moves upward with the upper plate at a constant speed, and the position of the upper plate changes constantly, so the grid is dynamically processed. The whole model is divided into grids in a mixed grid mode, which has the advantage of improving grid quality, reducing the computation and saving time. The area division is shown as Figure9(a). The hybrid grid is composed of an unstructured grid and a structured grid, in which the structured grid is a quadrilateral grid, which is defined as the upward moving fluid-move region. As shown in Figure 9(b), the minimum side length of the quadrilateral grid is 5.78e−5 m, and the maximum side length of the quadrilateral grid is 5.83e−5 m. Unstructured mesh is composed of triangular meshes, which are defined as a constant fluid region, as shown in Figure 9(c). The minimum side length is 3.08417e−5 m, the maximum side length is 8.6211e−5 m, and the number of corresponding grids in the simulation domain is 41,600.Figure 9
Grid division of ICEMCFD two-dimensional model. (a) Model region division. (b) Structured grid region. (c) Unstructured grid region.
(a)
(b)
(c)


## 3.2. Numerical Simulation Parameter Setting

The ANSYS FLUENT 17.0 computational fluid dynamics software is used for numerical simulation, and the basic parameters used in the simulation experiment are shown in Table3.Table 3
Basic simulation parameters.
Model parameter
Numerical value
Contact angle of upper plate
10°–80°
Contact angle of lower plate
90°
Initial distance between two plates
1.5 mm
Initial movement speed of upper plate
1 mm/s
Ink density
1950 kg/m3
Ink viscosity
9 pa∙s
Surface tension of conductive ink
0.03 N/m
Moving distance of upper plate
4.5 mmIn the process of simulating the tensile fracture of the liquid bridge between two plates, the air phase is defined as the first basic item, the conductive silver ink is defined as the second basic item, and it is defined as an incompressible fluid with a volume of 1μL, which is in contact with the upper and lower plates. Liu [50] verified in experiments that the property of conductive silver ink is pseudoplastic fluid and belongs to laminar flow. Therefore the liquid property in numerical simulation is defined as laminar flow. The upper wall-up of the simulation area is defined as the pressure outlet, and the left, right, and lower walls are defined as the wall surface [51]. VOF method is used to simulate multiphase flow, Geo-Reconstruct scheme is used to calculate phase volume fraction, and the pressure dispersion solution uses PRESTO!, using the Standard model of Realizable k−ε turbulence model. The dynamic mesh model is adopted, and spring smoothing model and local reconstruction model are selected to update the mesh.


## 3.3. Comparison of Liquid Bridge Adhesion with Theoretical Values under Different Contact Angles

The simulation results of liquid bridge adhesion with stretching distance under different contact angles are shown in Table4. In order to visually express the difference between the simulation results and the theoretical results, a comparison diagram of the changing trend of the two results is drawn. The trend of 10°–40° adhesion with stretching distance is shown in Figure 10(a), and the trend of 50°–80° adhesion with stretching distance is shown in Figure 10(b). The maximum deviation rate between simulated values and the theoretical values is 14.9%. There are two reasons for the analysis error: (1) because the mixed grid method is adopted for meshing in numerical simulation, during the upward movement of the structured grid area (fluid-move), the unstructured grid area (fluid) is affected by stretching, and the accuracy at the bottom is not high, and the liquid just adheres with the bottom wall (wall) at this position. (2) In the theoretical derivation part, the liquid volume is approximately replaced by equation (8), and there is an error between the approximate volume and the real volume, which eventually leads to the deviation of the simulation results. The comparison results show that the changing trend of liquid bridge adhesion between two plates affected by stretching distance and contact angle obtained by Ansys Fluent numerical simulation software is consistent with the theoretical results, proving that the numerical simulation has high accuracy.Table 4
Simulation values of adhesion force of liquid bridge under different contact angles.
Stretching distanceH (mm)
Adhesion forceF (x 10−4 N)
10°
20°
30°
40°
50°
60°
70°
80°
0.1
65.21
59.82
55.13
52.14
45.71
33.12
24.41
12.93
0.2
15.13
16.23
14.11
13.03
12.02
9.19
7.00
4.03
0.3
6.87
6.84
6.34
5.92
5.73
4.37
3.40
2.22
0.4
4.17
3.91
3.81
3.63
3.35
2.65
2.26
1.65
0.5
2.54
2.36
2.20
2.43
2.35
1.95
1.63
1.19
0.6
1.57
1.54
1.76
1.62
1.73
1.55
1.25
0.99
0.7
0.97
1.11
1.01
1.22
1.32
1.16
1.00
0.90
0.8
0.49
0.71
0.75
1.00
1.06
1.00
0.86
0.81
0.9
0
0
0.57
0.77
0.86
0.86
0.79
0.75
1.0
0
0
0
0
0
0
0
0Figure 10
(a) Comparison of simulation value and theoretical value of adhesion force change trend at 10°–40°. (b) Comparison of simulation value and theoretical value of adhesion force change trend at 50°–80°.
(a)
(b)


## 3.4. The Relationship between Liquid Bridge Adhesion and Ink Residue under Different Contact Angles

Conductive ink adheres to the upper plate. When the upper plate moves upwards, the liquid bridge conductive ink is stretched to break. At this time, a part of ink will remain on the surface of the upper plate, which is called residual inkmre. In the two-dimensional numerical simulation, it is assumed that the Z-axis direction is constant per unit length, so the change of contrast conductive ink area can reflect the change of volume. Import the simulation data into CFD-POST software for post-processing, carry out dimensionless treatment on the residual ink, and calculate the percentage of the two-dimensional area Sag of the residual ink on the upper plate to the area Sfluid-move of the fluid-move area to obtain the specific amount kre of the proportion of ink residue, as shown in equation (18). Furthermore, the changing trend of ink residue under different contact angles was compared.(17)kre=SAgSfluid−move%.In order to quantitatively analyse the influence of different contact angles on the residual ink on the upper plate, Figure11 shows the nephogram of the residual ink when the upper plate stops moving, and the contact angles are 30°, 50°, and 70°, respectively. According to the simulation results, the maximum adhesion of the liquid bridge and the proportion of the residual ink under different contact angles are measured, as shown in Table 5. According to the data in Table 5, the variation curves of contact angle, ink residue proportion, and maximum adhesion force are drawn, as shown in Figure 12. It can be seen from the figure that (1) under the premise that the liquid volume between two plates and the contact angle of the lower plate are unchanged, the smaller the contact angle of the upper plate, the greater the amount of ink remaining on the upper plate. (2) At the tensile distance of 0.1 mm, the liquid bridge adhesion is the largest, and the corresponding adhesion decreases with the increase of the contact angle. (3) Under the different of contact angle, the maximum adhesive force has the same change trend as the residual amount of ink on the upper plate. The reason for this phenomenon is that the contact angle becomes smaller, the ink is easier to spread on the solid surface, the spreading area increases, and the adhesion force correspondingly increases. Compared with the lower plate with an unchanged contact angle, the upper plate will adhere more ink, that is, the residual amount of ink on the upper plate will increase after the liquid bridge between the two plates is stretched and broken.Figure 11
Cloud picture of numerical simulation of ink residue.Table 5
Simulation values under different contact angles.
Contact angle of upper plate
Maximum adhesion force F of liquid bridge
Proportion of ink residue, kre/(%)
θ/(°)
F/(x10-3 N)
10°
6.52
32.18
20°
5.98
26.51
30°
5.51
21.80
40°
5.21
21.48
50°
4.57
16.59
60°
3.31
7.68
70°
2.44
6.69
80°
1.29
5.05Figure 12
Change trend of ink residue proportion and adhesion under different contact angles.


## 3.5. The Relationship between Liquid Bridge Adhesion and Stretching Distance under the Same Contact Angle and Different Liquid Volume

Figure13 shows the nephogram of marked liquid volume before the upper plate moves, which is 0.2 V, 0.4 V, 0.6 V, 0.8 V, and 1 V respectively, and the initial liquid volume is 1V is 1 μL. Table 6 shows the variation of liquid bridge adhesion and stretching distance under different liquid volumes when the contact angle of the two plates is 40. According to the data in Table 6, the curves of adhesion force with stretching distance under different liquid bridge volumes is shown in Figure 14. It can be seen from the figure that: (1) with the increase of stretching distance, the adhesion of liquid bridge decreases gradually under different liquid volumes. (2) Under the same stretching distance, the larger the volume of liquid, the greater the corresponding adhesive force. The reason for this phenomenon is that under the same stretching distance and solid-liquid contact angle, the larger the liquid volume, the larger the solid-liquid contact area, and the greater the corresponding liquid bridge adhesion. The trend of the simulation results is consistent with that of the theoretical derivation shown in Figure 7. Meanwhile, the change of the solid-liquid adhesion force is observed by changing the liquid volume to affect the liquid-solid spreading area, it is further proved that changing the solid-liquid contact angle in a modified manner affects the solid-liquid spreading area and ultimately changes the liquid bridge adhesion force.Figure 13
Numerical simulation nephogram of different initial ink volumes.Table 6
Simulation values of liquid bridge adhesion under different liquid bridge volumes.
Stretching distance,H (mm)
Adhesion forceF (x10−4 N)
Liquid bridge volume (1 v = 1μL)
0.2 v
0.4 v
0.6 v
0.8 v
1 v
0.1
12.43
21.23
32.24
38.41
52.14
0.2
2.30
5.56
8.52
10.43
13.03
0.3
0.74
2.94
3.70
4.88
5.92
0.4
0.50
1.63
2.47
3.08
3.63
0.5
0
1.04
1.44
1.96
2.43
0.6
0
0.59
1.00
1.34
1.62
0.7
0
0
0.81
1.01
1.22
0.8
0
0
0.57
0.85
1.00
0.9
0
0
0
0.59
0.77
1.0
0
0
0
0
0Figure 14
Variation curve of adhesion force with stretching distance under 40° contact angle and different liquid volume.


## 3.6. Guidance of Simulations and Theoretical Calculations for Practical Screen Printing

From the simulation and theoretical calculation, it can be found that the adhesive force at the first tensile distance (0.1 mm) is the largest and decreases gradually with the increase of tensile distance. This indicates that the liquid adhesion to the solid surface decreases with the increase of the tensile distance, and that when the tensile distance increases to 1 mm at different contact angles, the adhesion force values are equal to or close to 0, which indicates that the liquid bridge is broken or almost broken. Therefore, it is recommended that in actual screen printing production, the off-screen spacing should be at least more than 1 mm. In this way, it can avoid that the screen plate cannot bounce back immediately due to the too close distance from the screen, or the edge of the printed graphics is blurred and the surface roughness is uneven, etc., which will affect the product quality and printing accuracy.Through the simulation calculation, it can be seen that the change trend of the maximum adhesion and ink residue is the same under different contact angles. That is, with the increase of contact angle, the maximum adhesion decreases and the amount of ink remaining on the solid surface also decreases. Therefore, it is suggested that in the actual screen printing production, the surface of the substrate can be modified in a targeted manner, and the solid-liquid contact angle of the substrate surface can be appropriately reduced in combination with the quality analysis of the printed matter. It can increase the adhesion of the substrate to the ink in the mesh. Or modify the screen to increase the contact angle of the screen surface, in order to reduce the adhesion of the screen to the ink.It can also be known from the simulation and theoretical calculation that under the same stretching distance, the larger the liquid volume is, the greater the corresponding solid-liquid adhesion force is. Therefore, in the actual screen printing production, the single mesh can be appropriately increased and the mesh wire diameter can be reduced in the production process of the screen mesh. In this way, the contact area between the conductive ink and the substrate in a single mesh can be increased, the adhesion of the substrate to the ink can be increased, and the ink transfer rate can be increased.


## 4. Experimental Research

### 4.1. Experimental Instruments

The liquid bridge tensile test bench developed by the laboratory used in the experiment is shown in Figure15. The contact angle-measuring instrument is DSA100 optical contact angle-measuring instrument produced by KRUSS company in Germany.Figure 15
Liquid bridge tensile test bench.


### 4.2. Experimental Materials

The flat materials used in this tensile fracture experiment of the liquid bridge are all 24 mm × 50 mm glass sheets. The surface treatment agents are NC310 super-hydrophobic solution and NC3082 hydrophilic solution produced by Changzhou Narok Coatings Technology Co., Ltd., and Glaco hydrophobic surface treatment agent produced in Japan, as shown in Figure16.Figure 16
Surface treatment agent.


### 4.3. Experimental Scheme Design

Four kinds of glass surfaces with different contact angles were obtained by modifying the glass sheet with three kinds of surface modifiers and leaving the original glass sheet surface unmodified. These four kinds of glass surfaces were used as the upper plate in the tensile fracture experiment of liquid bridge between two plates. By drawing and breaking after adhesion with the ink, the adhesion force of the liquid bridge during static stretching and the amount of ink remaining after the liquid bridge broken were measured.The measurement of adhesion force during quasi-static stretching of liquid bridge adopts the combination of the liquid bridge stretching test bench and electronic balance to measure the change of adhesion force during stretching of liquid bridge. The measurement method is as follows: fix the unmodified lower plate on the electronic balance, drop a constant volume of conductive silver ink on the lower plate, and then measure the mass on the electronic balance asmdown. When the modified upper plate contacts the ink on the lower plate, measure the mass on the electronic balance as mup. Subtract the two values to obtain the corresponding adhesion force values at different stretching distances in the stretching process, as shown in (18), where g is the acceleration of gravity. When the liquid bridge is stretched and broken, record the mass mla of the lower plate at this time and find the difference before and after to obtain the mass mre of the ink remaining on the upper plate, as shown in equation (19). After dimensionless treatment of ink residue, the ratio of mre to the initial ink quality mdown of the lower plate was calculated, and the ink residue ratio k’re was obtained as shown in equation (20).(18)F=mdown−mupg,(19)mre=mdown−mla,(20)kre′=mremdown%.Each group of experiments was repeated 15 times, and the bad values in the measured results of this experiment were removed by the Dixon criterion. According to the Gaussian error law and the principle of the least squares method [52], it can be proved that in a group of observation values of equal precision, the arithmetic mean value is the best reflection of the true value. Therefore, the average value of the experimental data is taken here, and the results are added to the experimental data analysis table, and the maximum relative error (MAX-RE) and standard error (SE) of each group of data are obtained [53]. Finally, the modification scheme corresponding to the minimum ink residue was obtained. Equations for the arithmetic mean and maximum relative error are given in equation (21)-(23).(21)x¯=x1+x2+x3+⋯+xnn=∑i=1nxin,(22)MAXRE=xi−x¯xi×100%,(23)SE=∑i=1nxi−x¯2n2.


### 4.4. Experimental Results and Discussion

Environmental conditions of this experiment: the ambient temperature of the liquid bridge tensile test bench is (22 ± 1) °C, and the relative humidity of air in the ambient environment is (27 ± 5) %.


#### 4.4.1. Analysis of Theoretical and Experimental Results of Adhesion

Figure17 shows the contact angle measurement diagram of conductive ink on four different modified surfaces. Respectively use NC3082 modified solution to make the hydrophilic treatment surface, the contact angle is 31.4°. Use NC310 modification solution to make the hydrophobic treatment surface, the contact angle is 65.3°. Use glaco modification solution to make the hydrophobic treatment surface, the contact angle is 56.4°. The glass surface without any modification treatment has a contact angle of 42.2°.Figure 17
contact angle measurement diagram. (a) Without modification. (b) 3082 modified solution. (c) 310 modified solution. (d) Glaco modified solution.
(a)
(b)
(c)
(d)Quasi-static liquid bridge tensile fracture experiments were performed on four different surfaces. The change value of the adhesion force with the stretching distance and the corresponding maximum relative error value are obtained through the experimental measurement, as shown in Table7. According to the data in Table 7, the variation curve of adhesion force with tensile distance on four surfaces and the corresponding standard error are drawn, as shown in Figure 18. It can be seen from the figure that, (1) with the increase of stretching distance, the adhesion force of four different modified surfaces decreases gradually with stretching distance. (2) Under the same stretching distance, the larger the contact angle is, the smaller the corresponding adhesion is. It can also be seen from the figure that the adhesion force and the theoretical and simulation results are quite different. The reasons for this change are as follows: (1) The minimum amount of ink extruded by the ink titration device is 10 μL, which exceeds the maximum volume limit without the influence of gravity, so the conductive ink is affected by gravity during stretching, thus affecting the numerical change of adhesion force. (2) In the theoretical part, only capillary force is considered as the main force of adhesion force, and the influence of van der Waals force and electrostatic force are not considered. (3) In the theoretical derivation and numerical simulation of the tensile fracture of the liquid bridge between two plates, it is considered that the moving speed in the vertical direction is far greater than that in the transverse direction, and the transverse speed is neglected. In the actual experiment, the transverse speed has an influence on the ink transfer, which eventually leads to the deviation of the experimental results. However, the overall trend of adhesive force change accords with the conclusion of the theoretical model and numerical simulation, that is, the adhesive force of liquid bridge decreases with the increase of stretching distance, and the adhesive force of liquid bridge decreases with the increase of contact angle at the same stretching distance.Table 7
Experimental values of adhesion under different contact surfaces.
Stretching distanceH/(mm)
Adhesion F (x10-2 N)
3082 modified solution
MAX-RE
Without modification
MAX-RE
Glaco modified solution
MAX-RE
310 modified solution
MAX-RE
θ/(31.4°)
(%)
θ/(42.2°)
(%)
θ/(56.4°)
(%)
θ/(65.3°)
(%)
0.1
7.32
0.54
7.21
0.42
6.98
0.43
6.76
0.59
0.2
5.85
0.34
5.28
0.76
4.32
0.92
4.10
0.97
0.3
5.68
0.53
4.84
0.83
4.30
0.91
3.66
1.08
0.4
5.12
0.79
4.42
0.91
3.96
1.25
3.63
0.82
0.5
3.57
1.13
2.96
1.02
2.71
1.12
2.61
1.51
0.6
3.21
0.63
2.93
1.35
2.63
1.13
0
0
0.7
1.42
2.15
1.34
2.89
0
0
0
0
0.8
0
0
0
0
0
0
0
0
0.9
0
0
0
0
0
0
0
0
1.0
0
0
0
0
0
0
0
0Figure 18
Variation curve of adhesion force with stretching distance of liquid bridge under different contact angles.


#### 4.4.2. Analysis of Simulation Results and Experimental Results of the Residual Amount

Quasi-static tensile fracture experiments were carried out on four different surfaces, and the value of the residual ink ratio and the corresponding maximum relative error value under different contact angles were measured in Table8. According to Table 8, the variation trend of ink residual amount in the drawing and simulation results was obtained. The comparison chart and the corresponding standard error are shown in Figure 19. It can be found that with the increase of solid-liquid contact angle, the quality of ink remaining on the upper plate decreases. The changing trend is the same as that obtained by simulation results, which proves the accuracy and reliability of numerical simulation.Table 8
Experimental values of ink residue under different surfaces.
3082 modified solution
MAX-RE
Without modification
MAX-RE
Glaco modified solution
MAX-RE
310 modified solution
MAX-RE
θ/(31.4°)
(%)
θ/(42.2°)
(%)
θ/(56.4°)
(%)
θ/(65.3°)
(%)
Proportion of ink residue,kre'/(%)
41.67
4.09
33.29
3.21
27.78
7.57
16.7
7.58Figure 19
Influence of different contact angles on the proportion of ink residue.


## 4.1. Experimental Instruments

The liquid bridge tensile test bench developed by the laboratory used in the experiment is shown in Figure15. The contact angle-measuring instrument is DSA100 optical contact angle-measuring instrument produced by KRUSS company in Germany.Figure 15
Liquid bridge tensile test bench.


## 4.2. Experimental Materials

The flat materials used in this tensile fracture experiment of the liquid bridge are all 24 mm × 50 mm glass sheets. The surface treatment agents are NC310 super-hydrophobic solution and NC3082 hydrophilic solution produced by Changzhou Narok Coatings Technology Co., Ltd., and Glaco hydrophobic surface treatment agent produced in Japan, as shown in Figure16.Figure 16
Surface treatment agent.


## 4.3. Experimental Scheme Design

Four kinds of glass surfaces with different contact angles were obtained by modifying the glass sheet with three kinds of surface modifiers and leaving the original glass sheet surface unmodified. These four kinds of glass surfaces were used as the upper plate in the tensile fracture experiment of liquid bridge between two plates. By drawing and breaking after adhesion with the ink, the adhesion force of the liquid bridge during static stretching and the amount of ink remaining after the liquid bridge broken were measured.The measurement of adhesion force during quasi-static stretching of liquid bridge adopts the combination of the liquid bridge stretching test bench and electronic balance to measure the change of adhesion force during stretching of liquid bridge. The measurement method is as follows: fix the unmodified lower plate on the electronic balance, drop a constant volume of conductive silver ink on the lower plate, and then measure the mass on the electronic balance asmdown. When the modified upper plate contacts the ink on the lower plate, measure the mass on the electronic balance as mup. Subtract the two values to obtain the corresponding adhesion force values at different stretching distances in the stretching process, as shown in (18), where g is the acceleration of gravity. When the liquid bridge is stretched and broken, record the mass mla of the lower plate at this time and find the difference before and after to obtain the mass mre of the ink remaining on the upper plate, as shown in equation (19). After dimensionless treatment of ink residue, the ratio of mre to the initial ink quality mdown of the lower plate was calculated, and the ink residue ratio k’re was obtained as shown in equation (20).(18)F=mdown−mupg,(19)mre=mdown−mla,(20)kre′=mremdown%.Each group of experiments was repeated 15 times, and the bad values in the measured results of this experiment were removed by the Dixon criterion. According to the Gaussian error law and the principle of the least squares method [52], it can be proved that in a group of observation values of equal precision, the arithmetic mean value is the best reflection of the true value. Therefore, the average value of the experimental data is taken here, and the results are added to the experimental data analysis table, and the maximum relative error (MAX-RE) and standard error (SE) of each group of data are obtained [53]. Finally, the modification scheme corresponding to the minimum ink residue was obtained. Equations for the arithmetic mean and maximum relative error are given in equation (21)-(23).(21)x¯=x1+x2+x3+⋯+xnn=∑i=1nxin,(22)MAXRE=xi−x¯xi×100%,(23)SE=∑i=1nxi−x¯2n2.


## 4.4. Experimental Results and Discussion

Environmental conditions of this experiment: the ambient temperature of the liquid bridge tensile test bench is (22 ± 1) °C, and the relative humidity of air in the ambient environment is (27 ± 5) %.


### 4.4.1. Analysis of Theoretical and Experimental Results of Adhesion

Figure17 shows the contact angle measurement diagram of conductive ink on four different modified surfaces. Respectively use NC3082 modified solution to make the hydrophilic treatment surface, the contact angle is 31.4°. Use NC310 modification solution to make the hydrophobic treatment surface, the contact angle is 65.3°. Use glaco modification solution to make the hydrophobic treatment surface, the contact angle is 56.4°. The glass surface without any modification treatment has a contact angle of 42.2°.Figure 17
contact angle measurement diagram. (a) Without modification. (b) 3082 modified solution. (c) 310 modified solution. (d) Glaco modified solution.
(a)
(b)
(c)
(d)Quasi-static liquid bridge tensile fracture experiments were performed on four different surfaces. The change value of the adhesion force with the stretching distance and the corresponding maximum relative error value are obtained through the experimental measurement, as shown in Table7. According to the data in Table 7, the variation curve of adhesion force with tensile distance on four surfaces and the corresponding standard error are drawn, as shown in Figure 18. It can be seen from the figure that, (1) with the increase of stretching distance, the adhesion force of four different modified surfaces decreases gradually with stretching distance. (2) Under the same stretching distance, the larger the contact angle is, the smaller the corresponding adhesion is. It can also be seen from the figure that the adhesion force and the theoretical and simulation results are quite different. The reasons for this change are as follows: (1) The minimum amount of ink extruded by the ink titration device is 10 μL, which exceeds the maximum volume limit without the influence of gravity, so the conductive ink is affected by gravity during stretching, thus affecting the numerical change of adhesion force. (2) In the theoretical part, only capillary force is considered as the main force of adhesion force, and the influence of van der Waals force and electrostatic force are not considered. (3) In the theoretical derivation and numerical simulation of the tensile fracture of the liquid bridge between two plates, it is considered that the moving speed in the vertical direction is far greater than that in the transverse direction, and the transverse speed is neglected. In the actual experiment, the transverse speed has an influence on the ink transfer, which eventually leads to the deviation of the experimental results. However, the overall trend of adhesive force change accords with the conclusion of the theoretical model and numerical simulation, that is, the adhesive force of liquid bridge decreases with the increase of stretching distance, and the adhesive force of liquid bridge decreases with the increase of contact angle at the same stretching distance.Table 7
Experimental values of adhesion under different contact surfaces.
Stretching distanceH/(mm)
Adhesion F (x10-2 N)
3082 modified solution
MAX-RE
Without modification
MAX-RE
Glaco modified solution
MAX-RE
310 modified solution
MAX-RE
θ/(31.4°)
(%)
θ/(42.2°)
(%)
θ/(56.4°)
(%)
θ/(65.3°)
(%)
0.1
7.32
0.54
7.21
0.42
6.98
0.43
6.76
0.59
0.2
5.85
0.34
5.28
0.76
4.32
0.92
4.10
0.97
0.3
5.68
0.53
4.84
0.83
4.30
0.91
3.66
1.08
0.4
5.12
0.79
4.42
0.91
3.96
1.25
3.63
0.82
0.5
3.57
1.13
2.96
1.02
2.71
1.12
2.61
1.51
0.6
3.21
0.63
2.93
1.35
2.63
1.13
0
0
0.7
1.42
2.15
1.34
2.89
0
0
0
0
0.8
0
0
0
0
0
0
0
0
0.9
0
0
0
0
0
0
0
0
1.0
0
0
0
0
0
0
0
0Figure 18
Variation curve of adhesion force with stretching distance of liquid bridge under different contact angles.


### 4.4.2. Analysis of Simulation Results and Experimental Results of the Residual Amount

Quasi-static tensile fracture experiments were carried out on four different surfaces, and the value of the residual ink ratio and the corresponding maximum relative error value under different contact angles were measured in Table8. According to Table 8, the variation trend of ink residual amount in the drawing and simulation results was obtained. The comparison chart and the corresponding standard error are shown in Figure 19. It can be found that with the increase of solid-liquid contact angle, the quality of ink remaining on the upper plate decreases. The changing trend is the same as that obtained by simulation results, which proves the accuracy and reliability of numerical simulation.Table 8
Experimental values of ink residue under different surfaces.
3082 modified solution
MAX-RE
Without modification
MAX-RE
Glaco modified solution
MAX-RE
310 modified solution
MAX-RE
θ/(31.4°)
(%)
θ/(42.2°)
(%)
θ/(56.4°)
(%)
θ/(65.3°)
(%)
Proportion of ink residue,kre'/(%)
41.67
4.09
33.29
3.21
27.78
7.57
16.7
7.58Figure 19
Influence of different contact angles on the proportion of ink residue.


## 4.4.1. Analysis of Theoretical and Experimental Results of Adhesion

Figure17 shows the contact angle measurement diagram of conductive ink on four different modified surfaces. Respectively use NC3082 modified solution to make the hydrophilic treatment surface, the contact angle is 31.4°. Use NC310 modification solution to make the hydrophobic treatment surface, the contact angle is 65.3°. Use glaco modification solution to make the hydrophobic treatment surface, the contact angle is 56.4°. The glass surface without any modification treatment has a contact angle of 42.2°.Figure 17
contact angle measurement diagram. (a) Without modification. (b) 3082 modified solution. (c) 310 modified solution. (d) Glaco modified solution.
(a)
(b)
(c)
(d)Quasi-static liquid bridge tensile fracture experiments were performed on four different surfaces. The change value of the adhesion force with the stretching distance and the corresponding maximum relative error value are obtained through the experimental measurement, as shown in Table7. According to the data in Table 7, the variation curve of adhesion force with tensile distance on four surfaces and the corresponding standard error are drawn, as shown in Figure 18. It can be seen from the figure that, (1) with the increase of stretching distance, the adhesion force of four different modified surfaces decreases gradually with stretching distance. (2) Under the same stretching distance, the larger the contact angle is, the smaller the corresponding adhesion is. It can also be seen from the figure that the adhesion force and the theoretical and simulation results are quite different. The reasons for this change are as follows: (1) The minimum amount of ink extruded by the ink titration device is 10 μL, which exceeds the maximum volume limit without the influence of gravity, so the conductive ink is affected by gravity during stretching, thus affecting the numerical change of adhesion force. (2) In the theoretical part, only capillary force is considered as the main force of adhesion force, and the influence of van der Waals force and electrostatic force are not considered. (3) In the theoretical derivation and numerical simulation of the tensile fracture of the liquid bridge between two plates, it is considered that the moving speed in the vertical direction is far greater than that in the transverse direction, and the transverse speed is neglected. In the actual experiment, the transverse speed has an influence on the ink transfer, which eventually leads to the deviation of the experimental results. However, the overall trend of adhesive force change accords with the conclusion of the theoretical model and numerical simulation, that is, the adhesive force of liquid bridge decreases with the increase of stretching distance, and the adhesive force of liquid bridge decreases with the increase of contact angle at the same stretching distance.Table 7
Experimental values of adhesion under different contact surfaces.
Stretching distanceH/(mm)
Adhesion F (x10-2 N)
3082 modified solution
MAX-RE
Without modification
MAX-RE
Glaco modified solution
MAX-RE
310 modified solution
MAX-RE
θ/(31.4°)
(%)
θ/(42.2°)
(%)
θ/(56.4°)
(%)
θ/(65.3°)
(%)
0.1
7.32
0.54
7.21
0.42
6.98
0.43
6.76
0.59
0.2
5.85
0.34
5.28
0.76
4.32
0.92
4.10
0.97
0.3
5.68
0.53
4.84
0.83
4.30
0.91
3.66
1.08
0.4
5.12
0.79
4.42
0.91
3.96
1.25
3.63
0.82
0.5
3.57
1.13
2.96
1.02
2.71
1.12
2.61
1.51
0.6
3.21
0.63
2.93
1.35
2.63
1.13
0
0
0.7
1.42
2.15
1.34
2.89
0
0
0
0
0.8
0
0
0
0
0
0
0
0
0.9
0
0
0
0
0
0
0
0
1.0
0
0
0
0
0
0
0
0Figure 18
Variation curve of adhesion force with stretching distance of liquid bridge under different contact angles.


## 4.4.2. Analysis of Simulation Results and Experimental Results of the Residual Amount

Quasi-static tensile fracture experiments were carried out on four different surfaces, and the value of the residual ink ratio and the corresponding maximum relative error value under different contact angles were measured in Table8. According to Table 8, the variation trend of ink residual amount in the drawing and simulation results was obtained. The comparison chart and the corresponding standard error are shown in Figure 19. It can be found that with the increase of solid-liquid contact angle, the quality of ink remaining on the upper plate decreases. The changing trend is the same as that obtained by simulation results, which proves the accuracy and reliability of numerical simulation.Table 8
Experimental values of ink residue under different surfaces.
3082 modified solution
MAX-RE
Without modification
MAX-RE
Glaco modified solution
MAX-RE
310 modified solution
MAX-RE
θ/(31.4°)
(%)
θ/(42.2°)
(%)
θ/(56.4°)
(%)
θ/(65.3°)
(%)
Proportion of ink residue,kre'/(%)
41.67
4.09
33.29
3.21
27.78
7.57
16.7
7.58Figure 19
Influence of different contact angles on the proportion of ink residue.


## 5. Conclusions

In summary, we investigated the effect of solid surface energy on the adhesion between solid and liquid by the classical wetting theory. Combined with the screen printing process, we established a liquid bridge adhesion model between the two plates, and obtained that the contact angle, stretching distance and liquid volume are the main factors affecting the solid-liquid adhesion characteristics during the ink transfer process of screen printed electronic. The simulation calculations and the liquid bridge tensile fracture experiment showed that the solid-liquid contact angle can affect the ink residual amount after liquid bridge tensile fracture. The results confirm the rationality of the modification method proposed in this paper, and at the same time provide a reference for the realization of controllable solid-liquid adhesion and the improvement of ink transfer rate through modification in screen printed electronics.



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*Source: 2229688-2022-04-21.xml*