Open AccessArticle

Thickness Model of the Adhesive on Spacecraft Structural Plate

Yanhui Guo

Peibo Li

^2,*,

Yanpeng Chen

¹,

Xinfu Chi

¹ and

Yize Sun

College of Mechanical Engineering, Donghua University, Shanghai 201620, China

Shanghai Institute of Spacecraft Equipment, Shanghai 200240, China

Author to whom correspondence should be addressed.

Aerospace 2025, 12(2), 159; https://doi.org/10.3390/aerospace12020159

Submission received: 22 November 2024 / Revised: 10 February 2025 / Accepted: 17 February 2025 / Published: 19 February 2025

(This article belongs to the Section Astronautics & Space Science)

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Abstract

This paper establishes a physical model for the non-contact rotary screen coating process based on a spacecraft structural plate and proposes a theoretical expression for the adhesive thickness of the non-contact rotary screen coating. The thickness of the adhesive is a critical factor influencing the quality of the optical solar reflector (OSR) adhesion. The thickness of the adhesive layer depends on the equivalent fluid height and the ratio of the fluid flow rate to the squeegee speed below the squeegee. When the screen and fluid remain constant, the fluid flow rate below the squeegee depends on the pressure at the tip of the squeegee. The pressure is also a function related to the deformation characteristics and speed of the squeegee. Based on the actual geometric shape of the wedge-shaped squeegee, the analytical expression for the vertical displacement of the squeegee is obtained as the actual boundary of the flow field. The analytical expression for the deformation angle of the squeegee is used to solve the contact length between the squeegee and the rotary screen. It reduces the calculation difficulty compared with the previous method. Based on the theory of rheology and fluid mechanics, the velocity distribution of the fluid under the squeegee and the expression of the dynamic pressure at the tip of the squeegee were obtained. The dynamic pressure at the tip of the squeegee is a key factor for the adhesive to pass through the rotary screen. According to the continuity equation of the fluid, the theoretical thickness expression of the non-contact rotary screen coating is obtained. The simulation and experimental results show that the variation trend of coating thickness with the influence of variables is consistent. Experimental and simulation errors compared to theoretical values are less than 5%, which proves the rationality of the theoretical expression of the non-contact rotary screen coating thickness under the condition of considering the actual squeegee deformation. The existence of differences proves that a small part of the colloid remains on the rotary screen during the colloid transfer process. The expression parameterizes the rotary screen coating model and provides a theoretical basis for the design of automatic coating equipment.

Keywords:

thickness model; rotary screen printing; variable cross-section cantilever beam; spacecraft structural plates; optical solar reflector

1. Introduction

Spacecraft structural plates are essential components of the spacecraft power system and provide support for other components [1]. To mitigate radiation exposure from outer space and maintain thermal balance within the spacecraft, optical solar reflectors (OSRs) are affixed to the structural plate. Traditional fastening methods can contribute to increased weight and may compromise the integrity of the OSR, thereby making adhesive bonding technology the preferred solution [2,3].

The thickness of the adhesive is a critical factor influencing the quality of OSR adhesion [4,5]. This study proposes a coating adhesive process that employs non-contact rotary screen printing to achieve a coating adhesive of a large area on a spacecraft structural plate [6]. The contact between the tip of the squeegee and the rotary screen presses the adhesive onto the structural plate, and the rotary screen separates from the structural plate while it rotates. Finally, the translation of the rotary screen forms the adhesive layer [7], as shown in Figure 1.

Kobs and Voigt (1971) [8] listed over 50 variables, selected several of the most important variables and evaluated the effectiveness of over 280 different parameter combinations in comparative adhesive testing. Pan Jianbiao et al. (1998) [9] provided the following quality indicators for screen printing: average thickness of the printing layer, uniformity of the printing thickness, resolution of the printing patterns, the missing rate of printing patterns, etc., and listed the parameters that affect printing quality. They provided a large amount of data for supporting subsequent research, but did not establish a theoretical model of parameters. At present, velocity and pressure models have been studied most frequently in the flow of fluid in front of the squeegee for screen printing. Riemer (1989) [10] proposed one of the earliest theories on the process of screen printing. He mainly analyzed the screen printing model based on the Taylor squeegee model of Newtonian fluid [11]. The ink rolls in front of the squeegee and generates pressure at the squeegee tip, pressing ink into the screen. Kawabata (1993) [12] applied the complex function method to a screen printing flow with a low Reynolds coefficient and obtained the pressure distribution and vorticity streamline of the Taylor flow. Hunter (1994) [13] extended Riemer’s work to fluid flow under squeegees. Riedler (1983) [14] extended the screen model to non-Newtonian fluid. They establish a theoretical foundation for following research on the screen printing process. With the help of simulation software, computational fluid dynamics (CFD) [15,16,17,18,19] is widely applied to the numerical simulation of flow field mathematical models. Durairaj (2002) [20] developed a numerical model based on the finite volume method (FVM) to study the pressure distribution in the flow field of non-Newtonian fluid. Clements (2007) [21] analyzed the effect of fluid volume force on the screen coating process and obtained a mathematical model of fluid pressure distribution. Krammer (2015) [22] designed a finite volume model (FVM) to calculate the pressure distribution of non-Newtonian fluid along the screen surface and analyzed the impact of the contact angle of the squeegee for the pressure in the printing process. Glinski (2001) [23] compared the pressure and velocity distribution in the flow field of Newtonian fluid and non-Newtonian fluid through computational fluid dynamics (CFD) simulation of the movement of solder paste in front of the squeegee during screen gluing of electronic components. Ishak (2020) [24] used CFD simulated in FLUENT by using different squeegee parameters, which are the angle and printing speed. The above-mentioned scholars have all used CFD simulation to obtain the pressure distribution of fluid flow under the squeegee. However, most models do not consider that the real boundary of the flow field is caused by the deformation of the squeegee. It cannot accurately describe the flow of the flow field, which affects the analysis of coating thickness on the substrate.

The majority of squeegee materials are polyurethane. Deformation of the squeegee under pressure changes the shape of the flow field, affecting the distribution of velocity and pressure in the flow field. It can be seen that the deformation of the squeegee plays a vital role in affecting the flow field pressure. The deformation of the squeegee greatly increases the difficulty of model analysis, so the shape of the squeegee has been simplified in the previous models. There are many analysis methods for uniform cantilever beams, such as the finite integral method (FIM) [25], homotopy analysis method (HAM) [26,27], variational iteration method (VIM) [28], and differential transformation method (DTM) [29]. The analytical expression of the tapered cantilever beam is more complex [30], and the finite element method (FEM) is used as an auxiliary analysis method. Li (2022) [28] established a nonlocal numerical model for the large deformation analysis of functionally graded beams (FGB) with variable cross-sections by using the periodic dynamic differential operator (PDDO). Bilancia (2021) [31] described variable cross-section cantilever beams using beam-constraint models under different load conditions. Aghaei (2021) [32] proposed a non-linear dynamic model for variable cross-section beams. Dunkerley (2008) [33] made a critical comparison of the squeegee systems currently used for rotary screen carpet printing. Mannan (1994) [34] compared different types of squeegees used in the screen coating process of reflow solder paste in SMT, analyzed the influence of squeegee deformation on coatings, and considered the influence of the solder paste flow field for non-Newtonian fluid. However, the above methods did not obtain accurate analytical solutions. Liu (2022) [35] deduced the analytical expression of the deformation equation of the more complex variable cross-section squeegee and obtained the integral expressions of the dimensionless speed and dynamic pressure based on the semi-inverse method of linear elasticity. These works provide a preliminary foundation for fluid deposition. The analytical formula is too complex, which increases the difficulty of deriving the thickness formula.

This article focuses on the thickness of the adhesive layer for rotary screen coating. By analyzing the coating process, we can determine the variables that the adhesive thickness depends on. As the basic work continued, Jeong and Kim (1985) [36] hypothesized that there was a gap between the squeegee and the substrate. Owczarek and Howland (1990) [37,38] modeled the power-law fluid flow before and after the linear squeegee based on the lubrication theory and analyzed the influencing factors of the thickness forming. Jewell (2003) [39] independently designed a similar model and considered power-law fluid. White (2005) [40] regards the wire mesh as a permeable film and decomposes the flow field into an external flow field and an internal flow field. The mathematical model of a Newtonian fluid screen coating was established by using the numerical approximation method, and the influence of various physical parameters on the screen coating process was further analyzed. M. Taroni (2011) [41] established a screen printing process model based on power-law fluid, which was inspired by White. Fox (2003) [42] proposed a numerical model of screen coating supported by image transmission data, which combined the film’s hydrodynamic behavior with the deformation of the drum squeegee and the flow through the porous screen. Kanwal (2020) [43] utilized lubrication approximation theory to simplify the blade coating process. The analytical expressions for flow rate, pressure gradient, and velocity were obtained, while the load and coating thickness were computed numerically with the help of the shooting method. Sunar (2022) [44] quantitatively analyzed the effects of particle diameters and screen apertures on the deposition amount, deposition height, and coverage area of PCB screen printing. Hawkyard (2008) [45] monitored the rheological changes of printing paste and studied the dosage of electroformed nickel rotary screen printing slurry and its permeability to cotton cloth using the sample machine with a magnetic rod (Zimmer). Özdemir (2022) [46] optimized the key parameters that affect the printing ink volume in screen printing (the mesh size, paper type, and ink type) to minimize the environmental impact and cost of ink. Ghadiri (2011) [47] studied the transfer of non-Newtonian ink between a flat plate and a groove using CFD. The development of CFD provides convenience for analyzing the flow of adhesive during the coating process, and its widespread application provides reference significance.

Based on previous research, this paper combines the actual deformation of the squeegee and the pressure at the tip of the squeegee to derive a non-contact rotary screen coating thickness equation. Finally, we verify the rationality of the equation by simulation and testing of the coating process. In Section 2, the analytic expression of the deformation curve of the lower surface of the squeegee wedge is given and verified by finite element analysis. In Section 3, the coating process is divided into three regions, and the flow field in each region is analyzed, respectively, thus, the theoretical thickness equation is derived. In Section 4, we simulated the coating process and established coating thickness experiments to verify the reliability of the theoretical thickness expression. In Section 5, we summarized the research results and the innovative points of the paper.

2. Analysis of Squeegee Deformation

This section applies the bending deformation theory to obtain analytical expressions for squeegee deformation and rotation angles and compares them with the numerical solutions of FEM. The squeegee is embedded in the rotary screen under clamping, and the other end is suspended under the vertical upward coating force (F). This study regards the squeegee (the part extending out of the fixture) as a cantilever beam with unit width. The unit width coating force

f = \frac{F c o s θ}{W}

and W is the width of the squeegee. The squeegee model is established in the

x O y

and

x_{1} O y_{1}

coordinate systems, as shown in Figure 2. Considering the irregularity of the squeegee, it is divided into rectangular (a uniform cross-section cantilever beam) and wedge-shaped parts (a variable cross-section cantilever beam) for separate analysis. Table 1 shows the model parameters, where h is the thickness of the squeegee, L is the extended length of the squeegee,

L_{1}

is the length of the wedge-shaped part,

α

is the squeegee angle, and

θ

is the initial contact angle.

In the

x_{1} O y_{1}

coordinates, the approximate differential equation for the deflection curve of a rectangular cross-section is

E I w_{1}^{″} = M_{1} (x_{1})

(1)

where

M_{1} (x_{1}) = f (x_{1} - L_{1})

, is the torque;

I = \frac{{b h}^{3}}{12}

, is the inertial moment of a rectangular cross-section beam [48], b = 1; E is the elastic modulus of the squeegee. With the boundary conditions

w_{1}^{'} |_{x_{1} = L} = 0

w_{1} |_{x_{1} = L} = 0

, the rectangle deformation angle

w_{1}^{'}

and deflection

w_{1}

are integrated

w_{1}^{'} = \frac{6 f}{E h^{3}} (x_{1}^{2} - {2 L}_{1} x_{1} - L^{2} + 2 L_{1} L)

(2)

w_{1} = \frac{2 f}{E h^{3}} [x_{1}^{3} - 3 L_{1} x_{1}^{2} + (6 L L_{1} - 3 L^{2}) x_{1} - 3 L^{2} L_{1} + 2 L^{3}]

(3)

The wedge-shaped part can be regarded as a variable cross-section beam, and the inertia moment of the variable cross-section

I (x_{1})

[48] is

I (x_{1}) = \frac{b}{12} {[h_{0} + \frac{x_{1}}{L_{1}} (h - h_{0})]}^{3}

(4)

where

h_{0} = 0

is the thickness of the squeegee tip.

M (x_{1}) = f x_{1}

and Equation (4) are brought into Equation (1). It is integrated with the boundary conditions

w_{1}^{'} |_{x_{1} = L_{1}} = w^{'} |_{x_{1} = L_{1}}

w_{1} |_{x_{1} = L_{1}} = w |_{x_{1} = L_{1}}

, The deformation angle

w^{'}

and deflection

w

of the wedge-shaped part become

w^{'} = - \frac{12 f L_{1}^{3}}{E h^{3} x_{1}} + \frac{6 f}{E h^{3}} [L_{1}^{2} - L^{2} + 2 L L_{1}] = \frac{f}{E h^{3}} \{6 [L_{1}^{2} - L^{2} + 2 L L_{1}] - \frac{12 L_{1}^{3}}{x_{1}}\}

(5)

w = - \frac{12 f L_{1}^{3}}{E h^{3}} \ln (x_{1}) + \frac{6 f x_{1}}{E h^{3}} [2 L_{1}^{2} - {(L - L_{1})}^{2}] + \frac{2 f}{E h^{3}} (5 L^{3} - 5 L_{1}^{3} {+ 12 L L}_{1}^{2} - 6 L^{2} L_{1} - 6 L_{1}^{3} l n L_{1})

(6)

The flow field is mainly affected by vertical deformation. Therefore, according to geometric relationships, we obtain the following equation:

\{\begin{matrix} y = w \cdot c o s θ + \frac{x_{1}}{c o s \frac{α}{2}} \cdot s i n (θ - \frac{α}{2}) \\ x = \frac{x_{1}}{c o s \frac{α}{2}} \cdot c o s (θ - \frac{α}{2}) \end{matrix}

(7)

The deformation equation of the lower surface of the wedge-shaped part of the squeegee in the vertical direction of the

x O y

coordinate can be expressed as:

y = - \frac{12 f L_{1}^{3} c o s θ}{E h^{3}} \ln (\frac{x \cdot c o s \frac{α}{2}}{c o s (θ - \frac{α}{2})}) + \frac{6 f}{E h^{3}} [2 L_{1}^{2} - {(L - L_{1})}^{2}] \frac{x \cdot c o s \frac{α}{2} \cdot c o s θ}{c o s (θ - \frac{α}{2})} + \frac{2 f}{E h^{3}} (5 L^{3} - 5 L_{1}^{3} {+ 12 L L}_{1}^{2} - 6 L^{2} L_{1} - 6 L_{1}^{3} l n L_{1}) c o s θ + x \cdot t a n (θ - \frac{α}{2})

(8)

In order to verify the correctness of the analytical expression, the deformation of the squeegee was simulated by the finite element method (FEM). The simulation results of the squeegee deformation are shown in Figure 3, where the initial contact angles are 85° and 60° and the adhesive forces are 40 N and 60 N. It can be clearly seen from the figure that the deformation of the squeegee is non-linear, with a maximum deformation at the tip of the wedge-shaped part of the squeegee. The deformation increases with the increase in squeegee pressure and decreases with the increase in squeegee contact angle. Figure 4 shows the comparison between the analytical expression and the finite element simulation (FEM) results of the deformation of the lower edge of the wedge in the vertical direction obtained by expression (8). The analytical solution matches well with the finite element curve; therefore, this expression can be used for subsequent research.

3. Modeling of Adhesive Layer Thickness

The research on the rotary screen coating process ultimately boils down to the thickness of the adhesive layer, and the study of the distribution of the adhesive flow field is a prerequisite. The adhesive flow field can be divided into three regions: pressurization region I, filling region II, and deposition region III, as shown in Figure 5.

In Figure 5,

X_{a q}

is the accumulated length of the adhesive in front of the squeegee.

V_{p}

is the adhesive flow speed under the squeegee.

V_{s}

is the printing speed of the squeegee and rotary screen.

H_{d}

is the adhesive layer thickness.

H_{s}

is the rotary screen thickness.

When coating, the squeegee tip is compressed and deformed. The contact angle decreases relative to the initial state, forming a flat surface that acts on the rotary screen. Owczarek and Howland [38] discussed the importance of contact surface length (

X_{s q}

), but they did not provide a calculation method for the length. Yuegang Liu [35] approximates the contact length as the distance from the origin to the lowest point of the deformation curve in the x-direction, but its expression is too complex. When the contact length of the lower surface of the squeegee wedge is

X_{s q}

, the deformation angle at the rightmost end of the contact area is

(θ - \frac{α}{2})

. Therefore, it can be concluded that

X_{s q} = \frac{12 f L_{1}^{3}}{6 f [L_{1}^{2} - L^{2} + 2 L L_{1}] + E h^{3} (θ - \frac{α}{2})}

(9)

For the study of the flow field in region I, the coordinate system with the origin at

{x = X}_{s q}

on the lower surface of the wire mesh is reconstructed. The deformation function of the lower surface of the squeegee wedge in the new coordinate system is

G (x) = y |_{x + X_{s q}} - y |_{X_{s q}} + H_{s}

(10)

G (x) = - \frac{12 f L_{1}^{3} c o s θ}{E h^{3}} \ln (\frac{(x + X_{s q})}{X_{s q}}) + \frac{6 f}{E h^{3}} [2 L_{1}^{2} - {(L - L_{1})}^{2}] \frac{x \cdot c o s \frac{α}{2} \cdot c o s θ}{c o s (θ - \frac{α}{2})} + x \cdot t a n (θ - \frac{α}{2}) + H_{s}

(11)

3.1. Equation for Dynamic Pressure at Squeegee Tip

Region I is the area from the deformed squeegee tip to the adhesive boundary, as shown in Figure 6. The filling process depends on the fluid dynamic pressure and velocity at the tip of the squeegee. The purpose of analyzing region I is to obtain the velocity distribution in front of the deformed squeegee and the dynamic pressure equation at the tip.

The width of the squeegee is greater than the width of the adhesive coating, so it is assumed that the flow in the z-direction is the same in the coating section. Assuming that the adhesive is incompressible, control volume analysis is performed using Navier–Stokes momentum equations and continuity equations. In Figure 6, The momentum equation for the x-direction in the 2D flow field is:

ρ (v_{x} \frac{\partial v_{x}}{\partial x} + v_{y} \frac{{\partial v}_{x}}{\partial y}) = ρ f_{x} - \frac{\partial p}{\partial x} + μ (\frac{\partial^{2} v_{x}}{\partial x^{2}} + \frac{\partial^{2} v_{x}}{\partial y^{2}})

(12)

The flow velocity in region I is low, and the pressure and friction are high, so the influence of gravity can be ignored,

f_{x} = 0

. The flow area passing through the screen before the squeegee is much larger than perpendicular to the squeegee in the direction of screen movement, so the vertical velocity component of the slurry flow on the screen is very small. We will ignore the flow in the vertical direction,

v_{y} = 0

, so

\frac{{\partial v}_{y}}{\partial y} = 0

. According to the continuity equation

\frac{\partial v_{x}}{\partial x} + \frac{{\partial v}_{y}}{\partial y} = 0

(13)

Equation (12) can be simplified as

0 = - \frac{d p}{d x} + μ \frac{d^{2} v_{x}}{d y^{2}}

(14)

For the viscosity model of a power-law fluid,

μ = k {(\frac{{\partial v}_{x}}{\partial y})}^{n - 1}

(15)

where

n

is a power-law index and a measure of non-Newtonian properties, the lower the

n

, the faster the viscosity decreases and the stronger the non-Newtonian properties;

k

is a measure of the viscosity, but not equal to the viscosity value.

Equation (16) changes to the following equation,

\frac{d p}{d x} = k \frac{d [{(\frac{{\partial v}_{x}}{\partial y})}^{n}]}{d y}

(16)

Integrating,

v_{x} = \frac{n}{n + 1} {(\frac{1}{k} \frac{d p}{d x})}^{\frac{1}{n}} y^{\frac{n + 1}{n}} + c_{5} y + c_{6}

(17)

With the boundary conditions

v_{x} |_{y = 0} = 0

v_{x} |_{y = g (x)} = V_{s}

, the flow velocity distribution in front of the squeegee is

v_{x} = v_{x} (y)

v_{x} = \frac{n}{n + 1} {(\frac{1}{k} \frac{d p}{d x})}^{\frac{1}{n}} \{y^{\frac{1 + n}{n}} - y {[g (x)]}^{\frac{1}{n}}\} + \frac{V_{s}}{g (x)} y

(18)

The expression after non-dimensional for (20) is:

\frac{v_{x}}{V_{s}} = ε_{p} (ε_{y} - ε_{y}^{\frac{n + 1}{n}}) + ε_{y}

(19)

where

ε_{p} = - \frac{n}{(n + 1) V_{s}} {(\frac{1}{k} \frac{d p}{d x})}^{\frac{1}{n}} {[g (x)]}^{\frac{- n - 1}{n}} and ε_{y} = \frac{y}{g (x)}

Dimensionless velocity varies with the pressure gradient in Figure 7a. According to Equation (21), the speed is related to the pressure gradient, squeegee speed, and squeegee deformation. When

ε_{p}

= 0, the variation in dimensionless velocity is linear, and there is no viscosity flow at this time. The x-direction velocity distribution curve of the flow field on the vertical section at x = X conforms to the results in Figure 7a, which is similar to the velocity between plates, as shown in Figure 7b.

In Figure 7b, the adhesive flow velocity

V_{p}

below the squeegee is caused by the pressure accumulation at the tip of the squeegee, which is opposite to the coating direction. The linear velocity of the rotary screen at the tip of the squeegee is opposite to the translational velocity of the coating device, and both magnitudes are equal. For the squeegee wedge shape, the continuity equation of finite fixed control volume between x = 0 and x = X is

\int_{0}^{G (x)} (v_{x} - V_{s}) d y + V_{p} H_{s} = 0

(20)

Equation (20) is brought into Equation (22), and we then integrate

\frac{d p}{d x} = k {[\frac{(4 n + 2) (n + 1)}{2 n}]}^{n} \frac{{(2 V_{p} H_{s} - V_{s} G (x))}^{n}}{{[G (x)]}^{2 n + 1}}

(21)

Integrating, with the boundary conditions

p = p_{s q}

x = 0

p = p_{A T M}

x = X_{a q}

. The pressure difference equation in front of the squeegee becomes

p_{s q} - p_{A T M} = k {[\frac{(2 n + 1) (n + 1)}{2 n}]}^{n} \int_{0}^{X_{a q}} \frac{{(2 V_{p} H_{s} - V_{s} G (x))}^{n}}{{[G (x)]}^{2 n + 1}} d x

(22)

It can be seen that the dynamic pressure is related to the flow velocity below the squeegee in region II.

3.2. Equation for Flow Velocity Under Squeegee

The purpose of analyzing region II is to obtain the flow velocity below the squeegee. For a power-law fluid, we have the modified Poiseuille’s equation for the volumetric flow rate (q) through a rectangular channel of height (h), length (L), and width (W):

q = \frac{2 n}{2 n + 1} {(\frac{h ∆ P}{2 L k})}^{\frac{1}{n}} {(\frac{h}{2})}^{2} W

(23)

As shown in Figure 8, when the rotary screen makes contact with the substrate, for an unopened area caused by deformation at the tip of the squeegee, the flow channel under the squeegee is not entirely rectangular. Based on lubrication theory, a thin film is formed between the squeegee and the rotary screen due to relative motion [49]. The fluid flows within the rotary screen and flows through the gaps between the squeegee and the rotary screen. Therefore, the area below the squeegee is approximated as a rectangular flow channel,

h = H_{s}

L = X_{s q}

. The flow rate of adhesive passing through the outlet below the squeegee is

q = V_{p} H_{s} W

(24)

By combining Equations (25) and (26), it can be concluded that:

p_{s q} - p_{A T M} = \frac{2 k X_{s q}}{H_{s}} {[\frac{2 V_{p}}{H_{s}} \frac{(2 n + 1)}{n}]}^{n}

(25)

The dynamic pressure near the tip of the squeegee is the same, so the equation is transformed into:

\frac{2 X_{s q}}{H_{s}} {[\frac{{2 V}_{p}}{H_{s}}]}^{n} = {(n + 1)}^{n} \int_{0}^{X_{a q}} \frac{{(2 V_{p} H_{s} - V_{s} G (x))}^{n}}{{[G (x)]}^{2 n + 1}} d x

(26)

When n =1, we obtain the following equation:

V_{p} = \frac{\int_{0}^{X_{a q}} {[G (x)]}^{- 2} d x}{2 H_{s} \int_{0}^{X_{a q}} {[G (x)]}^{- 3} d x - \frac{2 X_{s q}}{{H_{s}}^{2}}} V_{s}

(27)

The first approximation of

V_{p}

from (28) is

V_{p} = \frac{{n (2 H_{s})}^{n - 1} \int_{0}^{X_{a q}} {[G (x)]}^{- 2 n} d x}{2 {H_{s}}^{n} \int_{0}^{X_{a q}} {[G (x)]}^{- 2 n - 1} d x - \frac{2^{n + 1} X_{s q}}{{H_{s}}^{n + 1} {(n + 1)}^{n}}} V_{s}

(28)

We can obtain an approximate expression for

p_{s q} - p_{A T M}

from (30). The entire gluing process is carried out under atmospheric pressure, so the atmospheric pressure is set to

p_{A T M}

= 0. The variation in dynamic pressure

p_{s q}

at the squeegee tip under different conditions is shown in Figure 9. Under the same power-law index, the initial contact angle has the greatest impact on the variation in dynamic pressure. The dynamic pressure at the tip of the squeegee increases with the increase in adhesive force and rotary screen speed and decreases with the increase in the initial contact angle and squeegee elastic modulus, which is consistent with the results in previous literature. The dynamic pressure at the tip of the squeegee increases with the increase in the power-law index. The larger the power-law index, the faster the dynamic pressure at the tip of the squeegee decreases or rises.

3.3. Equation for Theoretical Thickness of Adhesive Layer

Figure 10 shows the relationship between the thickness

H_{d}

of the deposited adhesive and the thickness

H_{s}

of the rotary screen under the squeegee. Assuming that all the adhesive on the rotary screen is transferred to the substrate, the time continuity equation for the adhesive flow per unit squeegee width reads:

V_{s} H_{d} ∆ t = V_{p} H_{s} ∆ t

(29)

The equation for the theoretical thickness of the adhesive layer,

H_{d}

, can be summarized as follows:

H_{d} = \frac{n H_{s}^{2 n} \int_{0}^{X_{a q}} {[G (x)]}^{- 2 n} d x}{H_{s}^{2 n + 1} \int_{0}^{X_{a q}} {[G (x)]}^{- 2 n - 1} d x - \frac{4 X_{s q}}{{(n + 1)}^{n}}} H_{s}

(30)

X_{s q} = \frac{12 f L_{1}^{3}}{6 f [L_{1}^{2} - L^{2} + 2 L L_{1}] + E h^{3} (θ - \frac{α}{2})}

(31)

G (x) = - \frac{12 f L_{1}^{3} c o s θ}{E h^{3}} \ln (\frac{(x + X_{s q})}{X_{s q}}) + \frac{6 f}{E h^{3}} [2 L_{1}^{2} - {(L - L_{1})}^{2}] \frac{x \cdot c o s \frac{α}{2} \cdot c o s θ}{c o s (θ - \frac{α}{2})} + x \cdot t a n (θ - \frac{α}{2}) + H_{s}

(32)

In the expression, it can be seen that the theoretical thickness of the non-contact rotary screen coating for the variable cross-section polyurethane squeegee is mainly determined by the thickness

H_{s}

of the rotary screen, the power-law index

n

, the shape

G (x)

of the deformed squeegee, and the amount of adhesive

X_{a q}

in front of the squeegee. The contact length

X_{s q}

between the squeegee and the wire mesh is not an independent variable. It is determined by the squeegee material, adhesive force, and initial contact angle. The influence of various main parameters on the coating thickness is shown in Figure 11.

In Figure 11, it can be seen that the coating thickness of the same fluid decreases with the increase in the initial contact angle and elastic modulus of the squeegee but increases with the increase in the coating force and mesh thickness. The larger the power-law index, the larger the range of change for the adhesive thickness, which is basically consistent with the dynamic pressure change at the squeegee tip in Figure 9. However, in Figure 11a, during the process of increasing the initial contact angle, the coating thickness does not increase with the increase in the power-law index, as in the case of dynamic pressure. When the initial contact angle is 85°, it can be seen from Figure 11b–d that the larger the power-law index, the smaller the coating thickness. When the initial contact angle is 75°, the larger the power-law index, the greater the coating thickness, which corresponds to the results in Figure 11a. The initial contact angle and mesh thickness have a significant impact on the coating thickness, and the thickness can be roughly adjusted. The influence of the squeegee pressure and squeegee elastic modulus on the coating thickness is relatively small, and the thickness can be carefully adjusted. The device can control the servo motor and sensor to adjust the adhesive pressure and initial contact angle to achieve the adjustment of adhesive thickness.

4. Simulation and Testing of Adhesive Coating Process

In this paper, the ARES-G2 rotational rheometer (TA Instruments, New Castle, DE, USA) was used to test the rheology of adhesives used in spacecraft structural plates and collect test data for simulations and tests [50,51,52]. The viscosity of the adhesive conforms to the characteristics of power-law fluid, as shown in Figure 12.

This section simulates the rotary screen coating process in Fluent, and the CFD model is shown in Figure 13a,b. Rough partitioning is performed on non-key domains (non-adhesive regions) to reduce the computational time, while fine partitioning is performed on key domains (adhesive regions) to improve computational accuracy and convergence, as shown in Figure 13b. In order to visually describe the whole process, Smoothing and Remeshing are used to calculate the dynamic mesh. The profile is translated in the opposite velocity direction to the rotation of the rotary screen and translation of the entire outer boundary of the computational domain. The transient calculation adopts the VOF model in Fluent, which includes two phases: air and adhesive. The k-epsilon turbulence model is used for the viscous flow calculation. The right boundary is set as the pressure inlet of the flow field, and the left and upper boundaries are set as the pressure outlets. The rotary screen is a thin, porous zone, which is simulated by a porous jump. The squeegee curve is generated by expression (13) and set as a non-sliding wall; the lower boundary is also set as a non-sliding wall. The solution process adopts the pressure-based type and considers the effect of gravity.

Figure 13c,d shows the phase contours of coating thickness at different times with a rotary screen thickness of 0.1 mm, rotary screen speed of 0.6 rad/s, boundary motion of 33 mm/s, squeegee pressure of 40 N, and initial contact angle of 85

°

. The height of the volume fraction on the substrate in the contour is taken as the thickness of the adhesive. Figure 13e shows the distribution of the total pressure in the flow field of region I. Dynamic pressure at the squeegee tip increases significantly and reaches the maximum value at the contact gap between the squeegee and rotary screen. Figure 13f shows the flow pattern of the squeegee wedge part, which confirms the typical rolling motion of the fluid power pump and verifies Riemer’s analysis results.

This article conducts experiments on spacecraft structural plates using rotary screen coating equipment and measures the coating thickness using a line laser scanner. The screen and squeegee need to be removed and reinstalled. Some adhesives need to add a catalyst. These factors affect the continuity of the experiment and increase the experimental error. The squeegee pressure and initial contact angle can be easily adjusted. Therefore, the coating tests take the initial contact angle and squeegee pressure as control variables, while the other variables are set as constants as follows: rotary screen thickness of 0.1 mm, rotary screen speed of 0.6 rad/s, and coating equipment of 33 mm/s. The same group of experiments was conducted three times under the same experimental conditions, and the photos and scanning results of the experiments are shown in Figure 14 and Table 2. In order to better demonstrate the availability of the theoretical thickness equation, the comparison of simulation, experiment, and theoretical results is presented more vividly in Figure 15; all have similar trends of change. In Figure 16, the experimental and simulation errors compared to theoretical values are less than 5%, which proves the rationality of the theoretical expression of the non-contact rotary screen coating thickness under the condition of considering the actual squeegee deformation. The expression can provide theoretical guidance for the coating process and equipment design. The shear rate below the squeegee may be very high, so it can be considered that the power-law exponent is constant at high shear rates [38]. Different fluids can use the power-law index to determine the theoretical coating thickness of the fluid.

5. Conclusions

In this paper, combining rheology, fluid mechanics, and material mechanics, the theoretical expression of the coating thickness is obtained by applying the continuity of the non-contact rotary screen coating process, and its rationality is verified through theoretical analysis, CFD simulations, and a coating experiment on a spacecraft structural plate. Firstly, deformation analysis was conducted on the true geometric shape of the squeegee to obtain the deformation angle and vertical displacement of the lower surface of the wedge-shaped part of the squeegee. The deformation of the squeegee is a non-linear change, which increases with the increase in adhesive force and decreases with the increase in the adhesive angle. The contact length between the squeegee and the screen is obtained through the deformation angle equation, which is proportional to the squeegee angle and the gluing force, which has not given an accurate analytical expression in previous studies. Compared with the deformation analytical expression through ANSYS simulation, the consistency is good. Using the squeegee deformation equation as the boundary of the flow field in region I, the velocity distribution and squeegee tip pressure in region I are obtained. The flow velocity in region I conforms to the distribution of flow velocity between plates, which is influenced by the deformation of the squeegee, the power-law index of the adhesive, and the squeegee speed. The tip pressure, as a key factor in adhesive thickness, increases with the increase in adhesive force and adhesive speed and decreases with the increase in the squeegee angle. The average velocity of adhesive under the squeegee is calculated from the generated fluid power pressure, contact length, and net thickness, and the theoretical expression of the adhesive layer thickness is finally derived. The theoretical expression has a similar trend of variation with the results of the CFD simulation and adhesive testing. Experimental and simulation errors compared to theoretical values are less than 5%, which proves the rationality of the theoretical expression of the non-contact rotary screen coating thickness. The theoretical expression provides a design basis for the automatic coating equipment of spacecraft structural plates. Changing the model parameters to obtain adhesive layers of different thicknesses enables large-scale manufacturing. However, this paper assumes that the operating environment temperature is constant. It is necessary to introduce a temperature field to obtain more realistic analysis results in the following research.

Author Contributions

Methodology, P.L.; investigation, Y.C.; data curation, Y.C.; writing—original draft, Y.G.; writing—review and editing, P.L. and Y.S.; supervision, P.L. and Y.S.; funding acquisition, X.C. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the National Key R&D Program of China (Grant No. 2017YFB0309800) and Fundamental Research Funds for the Central Universities (Grant No. 21D110326).

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. The actuator of the non-contact rotary screen coating process with measurement function: (b) shows the internal structure of the actuator in (a).

Figure 2. The geometric model of the squeegee.

Figure 3. The contours with FEM for deformed configurations of the squeegee. Initial and deformation states are presented by the solid border and color area, respectively.

Figure 4. Deformation curve of the lower edge of the wedge part for the FEM and the analytical solution. (a) 40 N, 75

°

; (b) 60 N, 75

°

; (c) 40 N, 85

°

; (d) 60 N, 85

°

Figure 4. Deformation curve of the lower edge of the wedge part for the FEM and the analytical solution. (a) 40 N, 75

°

; (b) 60 N, 75

°

; (c) 40 N, 85

°

; (d) 60 N, 85

°

Figure 5. Schematic of flow field division for the rotary screen printing process.

X_{a q}

is the accumulated length of adhesive in front of the squeegee.

V_{p}

is the adhesive flow speed under the squeegee.

V_{s}

is the printing speed of the squeegee and rotary screen.

H_{d}

is the adhesive layer thickness.

H_{s}

is the rotary screen thickness.

Figure 5. Schematic of flow field division for the rotary screen printing process.

X_{a q}

is the accumulated length of adhesive in front of the squeegee.

V_{p}

is the adhesive flow speed under the squeegee.

V_{s}

is the printing speed of the squeegee and rotary screen.

H_{d}

is the adhesive layer thickness.

H_{s}

is the rotary screen thickness.

Figure 6. Flow field model in front of the squeegee.

Figure 7. Velocity distribution in the flow field: (a) Non-dimensional x-velocity of a vertical section under different non-dimensional pressure gradients; (b) velocity distribution of adhesive in the vertical section at x = X relative to the substrate.

Figure 8. Partial schematic diagram (dashed square) and flow area of the rotary screen. a is the projection direction. The blue area is the mesh. the dashed square is the sampling area of the rotary screen.

Figure 9. The dynamic pressure at the squeegee tip under different conditions. The pressure of the squeegee with (a) force, (b) angle, (c) elastic modulus, and (d) coating speed. The black line represents the given parameters, and the x-axis represents the changing parameters in the figure.

Figure 10. Deposited thickness in relation to the thickness under the squeegee.

Figure 11. Variations in thickness

H_{d}

under different parameters. (a) Variations in thickness with angle for different power-law indices. Variations in thickness with (b) force F, (c) screen thickness

H_{s}

, and (d) elastic modulus E at θ = 75°and 85° for different power-law indices n.

Figure 11. Variations in thickness

H_{d}

under different parameters. (a) Variations in thickness with angle for different power-law indices. Variations in thickness with (b) force F, (c) screen thickness

H_{s}

, and (d) elastic modulus E at θ = 75°and 85° for different power-law indices n.

Figure 12. Variation and fit equation in viscosity with shear rate.

Figure 13. CFD simulation of rotary screen printing: (a) Boundary conditions of CFD model; (b) grid division of CFD model; volume fraction of the phase contours in (c) t = 0.2 and (d) t = 0.6 cases; (e) total pressure field; and (f) streamline pattern of velocity.

Figure 14. Photo and scanning contours of rotary screen printing: (a) Coating test; (b) scanning image of the adhesive layer. Red is the thicker area, while blue is the thinner area.

Figure 15. Comparison of the thickness

H_{d}

is obtained by the approximate analytical solution (red solid line), the experimental solution (black triangle), and the CFD solution (Blue Square). (a) Variation in thickness with force. (b) Variation in thickness with angle.

Figure 15. Comparison of the thickness

H_{d}

Figure 16. Experimental and simulation errors compared to theoretical values.

Table 1. Parameter values of the squeegee.

Parameter	$α (°)$	L (mm)	$L_{1}$ (mm)	h (mm)
Value	69.42	35	6.5	9

Table 2. Coating test date.

	Group	1	2	3	Average
Name		1	2	3	Average
F	20 N	1.043	1.055	1.050	1.049
	30 N	1.102	1.096	1.093	1.097
	40 N	1.145	1.141	1.138	1.141
$θ$	$75 °$	1.345	1.312	1.333	1.330
	$80 °$	1.29	1.278	1.278	1.282
	$85 °$	1.134	1.150	1.138	1.141

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Guo, Y.; Li, P.; Chen, Y.; Chi, X.; Sun, Y. Thickness Model of the Adhesive on Spacecraft Structural Plate. Aerospace 2025, 12, 159. https://doi.org/10.3390/aerospace12020159

AMA Style

Guo Y, Li P, Chen Y, Chi X, Sun Y. Thickness Model of the Adhesive on Spacecraft Structural Plate. Aerospace. 2025; 12(2):159. https://doi.org/10.3390/aerospace12020159

Chicago/Turabian Style

Guo, Yanhui, Peibo Li, Yanpeng Chen, Xinfu Chi, and Yize Sun. 2025. "Thickness Model of the Adhesive on Spacecraft Structural Plate" Aerospace 12, no. 2: 159. https://doi.org/10.3390/aerospace12020159

APA Style

Guo, Y., Li, P., Chen, Y., Chi, X., & Sun, Y. (2025). Thickness Model of the Adhesive on Spacecraft Structural Plate. Aerospace, 12(2), 159. https://doi.org/10.3390/aerospace12020159

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Thickness Model of the Adhesive on Spacecraft Structural Plate

Abstract

1. Introduction

2. Analysis of Squeegee Deformation

3. Modeling of Adhesive Layer Thickness

3.1. Equation for Dynamic Pressure at Squeegee Tip

3.2. Equation for Flow Velocity Under Squeegee

3.3. Equation for Theoretical Thickness of Adhesive Layer

4. Simulation and Testing of Adhesive Coating Process

5. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

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