Thickness Prediction of Negative Electrodes for Lithium Batteries in the Slot-Die Coating Process

Abstract

Slot-die coating is widely used in the preparation of negative electrodes for lithium batteries. The thickness of the negative electrode has a significant influence on the battery performance and lifespan, and different manufacturers have different requirements for its thickness. In order to reduce the waste caused by trial and error in the electrode preparation process, a prediction model for the negative electrode thickness was established and verified through simulation and experiments. Based on the Landau–Levich film equation and the Ruschak model, a high-precision prediction model was constructed by taking into account the influence of factors’, such as temperature and slurry, spreading characteristics on the coating thickness. The minimum coating thickness and its influencing factors were explored. Meanwhile, the simulation analysis of the coating thickness was performed, and the theoretical values of three common process parameters were compared with the simulation results, showing a deviation of only 2.9%. An experiment on predicting the thickness of the negative electrode of lithium batteries was conducted. Thickness measurements were performed on the samples prepared through the experiment and compared with theoretical values. The accuracy rate of this thickness prediction model can reach 98.75%.

1. Introduction

With the vigorous development of the lithium battery industry, the demand for batteries has become higher and higher. Reducing the thickness of the electrode is an effective way to reduce electronic polarization and ion polarization. The life and safety of lithium batteries may also be potentially enhanced [1]. A key factor to assess die-head coating performance is the parameter, called coating thickness uniformity of slurry on the foil surface [2]. Slot-die coating (SDC) is one of the common coating processes. Slot-die coating is a process in which the coating fluid flows out of the coating die head and spreads evenly onto the substrate through the movement of either the coating die head or the substrate. As a pre-metering coating technology, it is very suitable for the preparation of lithium-battery electrodes and the preparation of thin films of solar cells.

A large number of studies have already been carried out on issues related to thickness prediction for slot-die coating. Electrode manufacturing techniques affect the efficiency, energy density, safety performance, and cycle life of lithium-ion batteries. During the electrode manufacturing process, various defects, such as thicker at the beginning and thinner at the end and thick edges, may occur. These coating defects can cause a severe decline in the performance of the electrode, thereby affecting battery efficiency [3]. The presence of defects leads to a dramatic deterioration in the electrochemical performance of lithium-ion battery electrodes [4,5]. One of the main problems with the slot-die coating process is how to set an appropriate range of operating parameters, such as coating speed, flow rate, coating gap, liquid viscosity, and surface tension which have influence on coating thickness [6,7,8]. Under a given coating width, the thickness of the liquid film can be controlled by the flow rate of the solution flowing to the coating head, the substrate transfer speed, and the concentration of the solution [9]. Considering the influence of Couette flow and Poiseuille flow, the minimum coating thickness achievable under different parameters is predicted based on the limiting conditions of pressure drop [10]. Considering the influence of the inertial effect at the slot exit, the visco-capillary (VC) model was improved, and the inertia-capillary (IC) model was proposed, which predicted the minimum coating thickness during the SDC process at high coating speeds more effectively [11]. The interrelationships among the coating film surface morphology, coating speed, and substrate temperature were studied, and the thickness determination equation for protruding coating was provided [12]. A theoretical model was established for the coating thickness obtained at the initial stage of slot-die coating of the negative electrode slurry of lithium-ion batteries based on the finite element analysis method of fluid mechanics [13]. Using methods such as experiments and statistical analysis, the influencing factors affecting coating thickness and coating width were studied, and it was determined that the substrate movement speed is the most significant influencing factor [14]. Although previous studies explored coating thickness prediction, most did not fully consider the impact of multiple-factor interactions in actual production on thickness. Some only focused on the relationship between one or a few process parameters and coating thickness, without delving into the synergistic effects of factors, like temperature and slurry-spreading characteristics, in a complex environment. The prediction accuracy of some models needs improvement. In different practical production scenarios, their accuracy and reliability are hard to ensure, causing issues, like high trial-and-error costs and low production efficiency, in electrode preparation.

Coating thickness is crucial for lithium-battery capacity and cycle life. Theoretically, a thinner negative-electrode coating leads to a decrease in battery capacity and cycle life. This is because it may not supply enough active materials for the electrochemical reaction, limiting energy storage and charge–discharge cycles. Coating thickness also impacts battery energy and power density. Reasonably adjusting it within a certain range can optimize these densities. A too-thick coating may raise battery internal resistance, increasing ion-transport resistance, reducing charge–discharge efficiency, and thus lowering power density. Meanwhile, the proportion of inactive materials increases, potentially reducing energy density. Conversely, a too-thin coating limits capacity, negatively affecting energy density as well.

In order to reduce the cost of negative electrodes for lithium-battery production, a coating thickness prediction model was established by taking into account the influence of the actual production environment on the electrode preparation, which is used to predict the actual coating thickness under different parameters. This research, based on the Landau–Levich film equation and the Ruschak model, innovatively considers key factors like temperature and slurry-spreading characteristics, constructing a highly comprehensive prediction model for lithium-battery negative-electrode thickness. To verify the model’s validity and reliability, a combined simulation–experiment method was adopted. This model integrates multi-factor influences on coating thickness, better fitting the actual production complexity than previous models. During construction, a complete calculation system was set up through factor-influence analysis and mathematical derivation. From calculating the slurry volume flowing out of the slot channel, to analyzing the diffusion volume on the substrate, and then to calculating the dried-coating thickness by introducing the solution mass fraction and error coefficient considering the drying process, each link is closely connected, forming a unique and efficient thickness-prediction model. It offers a new, more accurate and practical way to predict lithium-battery negative-electrode coating thickness, holding important innovative significance and application value in lithium-battery production.

2. Analysis of Influencing Factors on Coating Thickness

In the actual production process, different manufacturers have different requirements for the electrode thickness, and the thickness of negative electrodes produced under different process coating parameters is also inconsistent. Generally speaking, the thickness of negative electrodes is one of the important factors that affect the performance and lifespan of lithium batteries [15]. When the thickness of the negative electrode is thinner, the capacity and life cycle of the battery will also decrease accordingly. Moreover, the thickness of the negative electrode is also closely related to the energy density and power density of the battery. Therefore, in actual production, it is necessary to determine the optimal thickness of the negative electrode according to different requirements [16]. When the thickness of the negative electrode is determined, a mathematical model for thickness prediction can be established for forecasting, and it can effectively reduce the process of trial and error.

2.1. Theory of Minimum Coating Thickness

The minimum coating thickness of slot-die coating refers to the thinnest coating thickness that can be achieved in the coating process. During the coating process, when the liquid strikes a solid surface, a coating bead is formed, and then gradually spreads to form a liquid film as the coating head moves, as shown in Figure 1. At the downstream meniscus of the coating bead, the flow of the liquid will be affected by the radius of curvature and surface tension σ, thereby causing changes in pressure difference. When the liquid passes through the meniscus, the radius of curvature of the meniscus will cause a change in the pressure difference [17].

At the downstream meniscus, the pressure difference is determined by the radius of curvature of the meniscus and the surface tension of the solution; the equation is expressed as follows:

$$\mathrm{\Delta }\mathrm{p}={\mathrm{p}}_1-{\mathrm{p}}_2={\displaystyle \frac{\mathrm{\sigma }}{\mathrm{r}}}$$

(1)

In Formula (1), $\mathrm{\sigma }$ represents the surface tension of the solution, and r represents the radius of curvature of the meniscus at the downstream position.

When the pressure drop increases, the theoretical minimum thickness of the coating layer will decrease, and the radius of curvature at the meniscus will also become smaller. However, excessive pressure drop will lead to coating damage, and the maximum pressure difference can be obtained at the minimum radius of curvature. The expression for the minimum radius of curvature is as follows:

$$\mathrm{r}={\displaystyle \frac{\mathrm{H}-{\mathrm{h}}_{\mathrm{m}}}{2}}$$

(2)

In Formula (2), H represents a specific reference height (such as the initial distance from the coating head to the substrate), h is a key height value related to the coating thickness under the current state, and $h_m$ is the theoretical minimum film thickness, which can be calculated by the Landau–Levich film equation [18]. The Landau–Levich thin film equation can describe the relationship between the film thickness and the liquid velocity when the liquid forms a thin film on the solid surface. The Landau–Levich equation is as follows:

$${\mathrm{h}}_{\mathrm{m}}=\mathrm{C}\cdot {\displaystyle \frac{{(\mathrm{\mu }\cdot {\mathrm{U}}_{\mathrm{c}})}^{\frac{2}{3}}}{{\mathrm{\sigma }}^{\frac{1}{6}}\cdot {(\mathrm{\rho }\mathrm{g})}^{\frac{1}{2}}}}$$

(3)

In Formula (3), C is a constant, $\mu$ represents the viscosity of the solution, Uc represents the moving speed of the substrate, and g is the gravitational acceleration.

The pressure difference $\Delta p$ can be obtained from the Ruschak model [19]. The Ruschak model can describe the flow of liquid in a small space. The model reveals the relationship between the pressure difference $\Delta p$, the capillary number $Ca$, the solution surface tension $\sigma$, and the theoretical minimum thickness $h_m$. When $Ca$ is less than 0.1, the expression for the pressure difference is as follows:

$$\mathrm{\Delta }\mathrm{p}=1.34\mathrm{C}{\mathrm{a}}^{{\displaystyle \frac{2}{3}}}{\displaystyle \frac{\mathrm{\sigma }}{{\mathrm{h}}_{\mathrm{m}}}}$$

(4)

The capillary number, also known as the interfacial tension number, has a value related to the solution viscosity $\mu$, the characteristic shear rate of the fluid, and the solution surface tension. Its equation is expressed as follows:

$$\mathrm{C}\mathrm{a}={\displaystyle \frac{\mathrm{\mu }\mathrm{v}}{\mathrm{\sigma }}}$$

(5)

In Formula (5), v is the characteristic shear rate of the fluid, and $Ca$ is the capillary number. The capillary number comprehensively reflects the relationship among the solution viscosity, the characteristic shear rate of the fluid, and the surface tension of the solution. It plays a crucial role in the calculation of the pressure difference in the previous Formula (4), which in turn affects the derivation of the theoretical minimum coating thickness. Substituting Formulas (1), (2) and (5) into Formula (4) can derive the theoretical limiting minimum coating thickness; the equation is expressed as follows:

$${\mathrm{h}}_{\mathrm{m}}={\displaystyle \frac{\mathrm{H}}{(1+\frac{2}{1.34\mathrm{C}{\mathrm{a}}^{\frac{2}{3}}})}}$$

(6)

2.2. Analysis of the Effect of Temperature on Coating Thickness

In actual production, temperature is a crucial variable. Temperatures differ among workshops and fluctuate seasonally in the same one. Temperature is vital when building a coating thickness prediction model as it impacts prediction accuracy. Temperature changes alter fluid density, affecting fluid flow during coating, and thus, the coating thickness. When temperature rises, fluid density drops, and according to the mass conservation law, coating thickness may vary under the same coating conditions. Ignoring temperature in the model makes it impossible to accurately predict coating thickness in different temperature settings, which may lead to lithium-battery negative-electrode coatings not meeting requirements and affecting battery performance and lifespan.

When the coating process reaches a stable state, the shape of the coating bead remains stable. Therefore, all the slurry flowing out of the slot channel spreads onto the substrate to form a coating, that is, the mass of the fluid flowing in at the coating head is equal to the mass of the fluid added to the coating layer, which is expressed as ${\rho }_u{V}_u={\rho }_d{V}_d$. During the coating process of the negative electrode, if the temperature of the process environment changes, the temperature and density of the fluid will also change, and the expression of their change relationship is as follows:

$${\mathrm{\rho }}_{\mathrm{d}}={\mathrm{\rho }}_{\mathrm{u}}[1-{\mathrm{a}}_{\mathrm{v}}({\mathrm{T}}_1-{\mathrm{T}}_0)]$$

(7)

In Formula (7), $T$ represents the thermodynamic temperature; $T_1$ and $T_0$ represent the final temperature and initial temperature, respectively; $a_v$ represents the fluid expansion coefficient; ${\rho }_u$ and ${\rho }_d$ represent the solution density when the fluid enters the coating die head at the initial stage of the coating and the solution density of the fluid coated on the substrate, respectively.

2.3. Analysis of the Influence of Slurry Spreading on Coating Thickness

According to rheology theory, particle distribution and interaction in a fluid affect its viscosity and flow. For lithium-battery negative-electrode slurries, like graphite slurry, graphite particle size matters. Smaller graphite particles, with a larger specific surface area, disperse better in the slurry. This evens out intermolecular forces, reducing slurry viscosity and enhancing fluidity and spreading ability. During coating, such slurry flows smoothly from the coating head and spreads evenly on the substrate, forming a thinner and more uniform coating.

Conversely, larger graphite particles tend to agglomerate in the slurry, leading to uneven distribution. Agglomerated particles impede slurry flow, increase local viscosity, and make uniform spreading difficult during coating. When they reach the substrate, they cause coating-thickness variations, resulting in an uneven coating. In extreme cases, large-particle agglomerations may cause overly thick coatings in some areas and thinner coatings nearby, severely affecting battery negative-electrode coating quality.

During the process of the slurry flowing out of the slot channel, the volume of the outflowing slurry $V_u$ can be determined by the inlet velocity of the slurry and the process dimensions of the coating head; the equation is expressed as follows:

$${\mathrm{V}}_{\mathrm{u}}={\mathrm{U}}_{\mathrm{u}}\cdot \mathrm{b}\cdot \mathrm{W}$$

(8)

In Formula (8), $U_u$ represents the fluid inlet velocity at the coating head; $b$ represents the width of the coating head; and $W$ represents the slot width of the coating head. After the slurry comes into contact with the substrate, it will spread out, causing the coating width to diffuse. When coating slurry, it is necessary to control the width of the coating head and the degree of coating diffusion. The expression for the volume of the substrate slurry $V_d$ after coating diffusion is as follows:

$${\mathrm{V}}_{\mathrm{d}}=\mathrm{n}\cdot {\mathrm{b}}^{\prime }\cdot {\mathrm{U}}_{\mathrm{c}}\cdot {\mathrm{h}}_{\mathrm{m}}$$

(9)

In Formula (9), $n$ represents the number of coating strips; $b\prime$ represents the width of the coating after diffusion; $U_c$ represents the coating speed; and $h_m$ represents the predicted coating thickness.

The concentration of the slurry also plays a crucial role in the spreading process. A higher concentration usually leads to an increase in slurry viscosity, which may slow down the spreading speed and make it more difficult for the slurry to spread evenly on the substrate. On the contrary, a lower concentration may result in a decrease in viscosity, causing the slurry to spread too quickly and making it hard to control the coating width. Therefore, an appropriate slurry concentration is essential for obtaining a uniform coating thickness under the three sets of process parameters.

2.4. Research on the Spreading Characteristics of Slurry

The substrate of the negative electrode of power lithium-ion batteries is copper foil. When the slurry comes into contact with the copper foil, a contact angle will be formed on the contact surface. The smaller the contact angle, the better the wettability of the slurry on the copper foil, and it can adhere to the surface of the copper foil more effectively. Therefore, the size of the contact angle is an important indicator of the quality of the slurry spreading. In practical applications, the contact angle between the slurry and the copper foil significantly influences the performance and stability of electronic devices. For instance, when preparing based conductive films, the contact angle between the slurry and the copper foil affects the electrical conductivity and stability of the conductive film. Therefore, studying the spreading characteristics, especially the contact angle, is of great significance for the preparation of negative-electrode sheets.

The contact angle formed by the copper foil substrate and the slurry can be described by the Young–Laplace formula [20]. Let the surface tensions among the slurry, copper foil, and air be $r_{sg}$, $r_{cg}$ and $r_{ca}$, respectively. Then, the contact angle $\theta$ is as follows:

$$\cos \mathrm{\theta }={\displaystyle \frac{({\mathrm{r}}_{\mathrm{c}\mathrm{g}}-{\mathrm{r}}_{\mathrm{s}\mathrm{g}})}{{\mathrm{r}}_{\mathrm{c}\mathrm{a}}}}$$

(10)

This formula only describes the size of the contact angle under a static process. However, in actual production, the slurry may be in a flowing state, so the morphological change, viscosity change, and surface tension change of the slurry also need to be considered. In this regard, the Wenzel model [21] is introduced to adjust the contact angle model. Suppose the contact angle between slurry and the substrate under the Wenzel model is ${\theta }_w$, and the contact angle under Young’s model is ${\theta }_Y$, as shown in Figure 2.

The relationship between the contact angles under the two models is expressed as follows:

$$\cos {\mathrm{\theta }}_{\mathrm{w}}=\mathrm{r}\cos {\mathrm{\theta }}_{\mathrm{Y}}$$

(11)

And the surface energy of the substrate and the fluid and the surface roughness of the substrate will affect the degree of diffusion, so the Young–Wenzel equation is as follows:

$$\mathrm{r}\cdot {\mathrm{E}}_{\mathrm{a}}=\mathrm{r}\cdot {\mathrm{E}}_{\mathrm{b}}+{\mathrm{E}}_{\mathrm{c}}\cdot \cos {\mathrm{\theta }}_{\mathrm{w}}$$

(12)

So, the contact angle ${\theta }_w$ under the Wenzel model is as follows:

$${\mathrm{\theta }}_{\mathrm{w}}=\arccos ({\displaystyle \frac{\mathrm{r}({\mathrm{E}}_{\mathrm{a}}-{\mathrm{E}}_{\mathrm{b}})}{{\mathrm{E}}_{\mathrm{c}}}})$$

(13)

In Formula (13), $E_a$ represents the surface energy of the substrate; $E_b$ represents the interface energy of the fluid; $E_c$ represents the surface energy between the substrate and the fluid; and $r$ is the surface roughness of the substrate, that is, the ratio of the actual surface area of the substrate to its projected surface area, which can be obtained by measuring the root mean square many times.

3. Prediction Model of Coating Thickness

In addition to the characteristics of the droplet itself and the droplet velocity, the droplet falling height will also affect the spreading width. Generally speaking, the greater the droplet falling height is, the greater the droplet spreading width will be. Slot-die coating allows for flexible and convenient adjustment of the coating gap size, which means it can effectively control the height from which the slurry descends. The relationship between the droplet falling height and its spreading width on the substrate is shown in Figure 3.

As the height changes, the radius of the droplet deposited on the substrate will also change. According to the injection height, the Harth model of the ink droplet radius can be obtained [22]; this equation is expressed as follows:

$$\mathrm{r}(\mathrm{t})={\mathrm{r}}_{\mathrm{e}}{[1-\exp ({\displaystyle \frac{2{\mathrm{r}}_{\mathrm{L}}}{{\mathrm{r}}_{\mathrm{e}}^{12}}}+{\displaystyle \frac{\mathrm{\rho }\mathrm{g}}{9{\mathrm{r}}_{\mathrm{e}}^{10}}})\cdot {\displaystyle \frac{24\mathrm{\lambda }{\mathrm{V}}^4(\mathrm{t}+{\mathrm{t}}_0)}{\mathrm{\pi }\mathrm{\eta }}}]}^{{\displaystyle \frac{1}{6}}}$$

(14)

In Formula (14), $\eta$ indicates fluid viscosity; $\lambda$ represents the shape factor; $r_L$ represents the surface tension of the fluid; $V$ represents the volume of injected fluid; $r_e$ indicates the emission radius of the droplet; $t$ indicates the time interval between fluid injection and fluid deposition on the substrate; $t_0$ indicates the experiment delay time; $g$ represents the acceleration of gravity; and $\rho$ represents the density of the fluid.

The time interval is determined by the height at which the droplet falls, the fluid is injected from above the coating head, and the initial velocity $V_0=U_u$. The drop height is equal to the coating gap height H, while t and fall height are derived from the Bernoulli equation of the total flow as follows:

$${\mathrm{Z}}_1+{\displaystyle \frac{{\mathrm{p}}_1}{\mathrm{\rho }\mathrm{g}}}+{\displaystyle \frac{{\mathrm{u}}_1^2}{2\mathrm{g}}}={\mathrm{Z}}_2+{\displaystyle \frac{{\mathrm{p}}_2}{\mathrm{\rho }\mathrm{g}}}+{\displaystyle \frac{{\mathrm{u}}_2^2}{2\mathrm{g}}}$$

(15)

In Formula (15), $Z_1=H$, $Z_2=0$, $p_1=p_2=U_u$. Substituting into Formula (15), this equation is expressed as follows:

$${\mathrm{u}}_2=\sqrt{2\mathrm{g}\mathrm{H}+{\mathrm{u}}_1^2}$$

(16)

The expression of the acceleration component of fluid particle in unsteady flow is as follows:

$$\begin{array}{l}{\mathrm{a}}_{\mathrm{x}}={\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{t}}}+\mathrm{u}{\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{x}}}+\mathrm{v}{\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{y}}}+\mathrm{w}{\displaystyle \frac{\partial \mathrm{u}}{\partial \mathrm{z}}}\\ {\mathrm{a}}_{\mathrm{y}}={\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{t}}}+\mathrm{u}{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{x}}}+\mathrm{v}{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{y}}}+\mathrm{w}{\displaystyle \frac{\partial \mathrm{v}}{\partial \mathrm{z}}}\\ {\mathrm{a}}_{\mathrm{z}}={\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{t}}}+\mathrm{u}{\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{x}}}+\mathrm{v}{\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{y}}}+\mathrm{w}{\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{z}}}\end{array}$$

(17)

And assume that $a_z=g$, only considering that the process of fluid flowing down from above the coating head is only affected by gravity, after introducing the boundary conditions. The equation is expressed as follows:

$$\begin{array}{l}\mathrm{g}={\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{t}}}+\mathrm{w}{\displaystyle \frac{\partial \mathrm{w}}{\partial \mathrm{z}}}\\ \mathrm{t}={\displaystyle \frac{\sqrt{2\mathrm{g}\mathrm{H}+{\mathrm{u}}_1^2}-{\mathrm{U}}_{\mathrm{u}}}{\mathrm{g}}}\end{array}$$

(18)

The expression obtained by substituting Formula (18) into Formula (14) is as follows:

$$\mathrm{r}(\mathrm{t})={\mathrm{r}}_{\mathrm{e}}{[1-\exp ({\displaystyle \frac{2{\mathrm{r}}_{\mathrm{L}}}{{\mathrm{r}}_{\mathrm{e}}^{12}}}+{\displaystyle \frac{\mathrm{\rho }\mathrm{g}}{9{\mathrm{r}}_{\mathrm{e}}^{10}}})\cdot {\displaystyle \frac{24\mathrm{\lambda }{\mathrm{V}}^4(\sqrt{2\mathrm{g}\mathrm{H}+{\mathrm{u}}_1^2}-{\mathrm{U}}_{\mathrm{u}}+\mathrm{g}{\mathrm{t}}_0)}{\mathrm{g}\mathrm{\pi }\mathrm{\eta }}}]}^{{\displaystyle \frac{1}{6}}}$$

(19)

As the coating expands, the spreading range will increase with the increase in the coating gap. A mathematical model can be established to estimate the coating width. Firstly, the width ratio of the coating $E=r(t)/r_e$, so the actual coating width $b^{\prime }=bE$.

Through the above derivation, the wet thickness of the coating can be obtained. For the coating preparation of the lithium-battery negative electrode, the dry thickness of the coating after drying also needs to be considered. Therefore, the solution mass fraction G of its solution needs to be introduced. In addition, there are usually some errors in the coating process [23], and a coefficient of $K(0\le \mathrm{K}\le 1)$ can be introduced. The formula for calculating the coating thickness after drying is as follows:

$${\mathrm{h}}_{\mathrm{k}}={\displaystyle \frac{\mathrm{G}\cdot \mathrm{K}\cdot {\mathrm{U}}_{\mathrm{u}}\cdot \mathrm{w}}{\mathrm{n}\cdot {\mathrm{U}}_{\mathrm{c}}\cdot [1-{\mathrm{a}}_{\mathrm{v}}({\mathrm{T}}_1-{\mathrm{T}}_0)]\cdot \mathrm{E}}}$$

(20)

Substituting the three sets of process parameters in

Table 1. Three sets of process parameters.

Process Name	Belt Speed	Coating Gap	Gasket Thickness	Upper Lip Length	Lower Lip Length	Oven Temperature
1	9 m/min	200 μm	550 μm	275 μm	275 μm	120 °C
2	10 m/min	160 μm	1 mm	5 mm	1.2 mm	120 °C
3	20 m/min	160 μm	0.8 mm	5 mm	1.2 mm	120 °C

into the above formula, the predicted thickness values can be obtained as 127.328 μm, 138 μm, and 100.01 μm, respectively.

Based on the results obtained above, a line graph of coating thickness corresponding to different coating speeds can be plotted, as shown in Figure 4. The figure shows the relationship between the coating thickness of the negative electrode of lithium batteries and the coating speed. It can be seen from the figure that as the coating speed changes, the coating thickness also changes accordingly. A series of data points were obtained by measuring and calculating the coating thickness at different coating speeds, and these points were connected to form a curve, thus visually presenting the relationship between the two.

4. Simulation Analysis

Before conducting this simulation analysis, it is necessary to review the key factors influencing the coating thickness discussed in the previous sections. Firstly, according to the theory of minimum coating thickness, the surface tension $\sigma$ and the radius of curvature r of the downstream meniscus determine the pressure difference through the formula $\Delta p=p_1-p_2={\displaystyle \frac{\sigma }{r}}$, which in turn affects the minimum coating thickness. During the simulation process, we convert these parameters into corresponding settings in the model. For example, based on the actual solution properties and coating conditions, the value of the surface tension $\sigma$ coefficient is determined, and the variation of the radius of curvature and its influence on the pressure difference and coating thickness are observed through the simulation of fluid flow. Secondly, the influence of temperature on the coating thickness cannot be ignored. In the previous analysis, we concluded that the temperature change will lead to the change of fluid density, as shown in the formula ${\rho }_d={\rho }_u[1-a_v(T_1-T_0)]$. In the simulation, different initial temperature conditions are set according to the actual process temperature range, and the influence of the change of fluid density on the coating thickness is observed. In addition, the spreading characteristics of the slurry, including the inlet velocity of the slurry, the width of the coating head and the width of the slot, would affect the width and thickness of the coating in the actual coating process. In the simulation model, we set the values of these variables according to the actual process parameters and observe their influence on the coating thickness through simulation. Through the above methods, we closely combine the theoretical analysis in the previous sections and the actual process factors into the simulation process of the Volume of Fluid (VOF) model, so as to achieve the accurate simulation and analysis of the coating thickness.

4.1. Establishment Model

The VOF model is primarily used for multiphase flow, where two or more fluids do not mix with each other. Adding an extra phase to the model will introduce a variable, that is, calculating the volume fraction of that phase in each cell. The sum of the volume fractions of all phases is one. The volume fraction $a_q$ of the fluid in the cell can be used to determine the phase represented by the variables and attributes within the cell.

In the VOF model, capillary forces are simulated by setting surface-tension-related parameters. Theoretical analysis shows surface tension significantly impacts liquid flow and morphology during coating, especially in coating-bead and liquid-film formation. Based on actual fluid and coating materials, the surface-tension $\sigma$ coefficient is set in the model. This coefficient is involved in two-phase-interface calculation and fluid-flow simulation. For example, when calculating liquid flow at the slot exit and spreading on the substrate, it affects the liquid’s curvature radius and pressure difference, consistent with theoretical analysis. The calculation, based on the Young–Laplace equation and solved numerically, accurately simulates capillary force effects on coating thickness. Thus, capillary forces are incorporated into the VOF model simulation, making the results more reflective of the actual coating process.

Based on the volume fraction, appropriate attributes and variables are assigned to each unit. In the VOF model, the volume fraction is used to track the interface between the two phases by solving the volume fraction continuity equation for the phase [24]. For the term Q, the equation is expressed as follows:

$${\displaystyle \frac{1}{{\mathrm{\rho }}_{\mathrm{q}}}}\left[{\displaystyle \frac{\partial }{\partial \mathrm{t}}}\left({\mathrm{a}}_{\mathrm{q}}{\mathrm{\rho }}_{\mathrm{q}}\right)+\nabla \cdot \left({\mathrm{a}}_{\mathrm{q}}{\mathrm{\rho }}_{\mathrm{q}}{\overrightarrow{\mathrm{v}}}_{\mathrm{q}}\right)\right]={\mathrm{S}}_{{\mathrm{a}}_{\mathrm{q}}}+{\displaystyle \sum _{\mathrm{p}=1}^{\mathrm{n}}\left({\dot{\mathrm{m}}}_{\mathrm{p}\mathrm{q}}-{\dot{\mathrm{m}}}_{\mathrm{q}\mathrm{p}}\right)}$$

(21)

In Formula (21), $m_{ab}$ represents the mass transfer from phase a to phase b, and its magnitude and direction depend on the concentration gradient, temperature gradient and pressure gradient between different phases. By default, the source term on the right side of this equation is zero, but a constant or user-defined mass source can be specified for each phase. The volume fraction of the primary phase is not solved. The volume fraction of Formula (21) is given to the constraint; the equation is expressed as follows:

$${\displaystyle \sum _{\mathrm{q}=1}^{\mathrm{n}}{\mathrm{a}}_{\mathrm{q}}}=1$$

(22)

When using an implicit formula to solve the volume fraction, the volume fraction needs to be discretized into the following forms; the equation is expressed as follows:

$${\displaystyle \frac{{\mathrm{a}}_{\mathrm{q}}^{\mathrm{n}+1}{\mathrm{p}}_{\mathrm{q}}^{\mathrm{n}+1}-{\mathrm{a}}_{\mathrm{q}}^{\mathrm{n}}{\mathrm{p}}_{\mathrm{q}}^{\mathrm{n}}}{\mathrm{\Delta }\mathrm{t}}}\cdot \mathrm{V}+{\displaystyle \sum _{\mathrm{f}}({\mathrm{\rho }}_{\mathrm{q}}^{\mathrm{n}+1}\cdot {\mathrm{U}}_{\mathrm{f}}^{\mathrm{n}+1}\cdot {\mathrm{a}}_{\mathrm{q}.\mathrm{f}}^{\mathrm{n}+1}})=\left[{\mathrm{S}}_{{\mathrm{a}}_{\mathrm{q}}}+{\displaystyle \sum _{\mathrm{p}=1}^{\mathrm{n}}\left({\dot{\mathrm{m}}}_{\mathrm{p}\mathrm{q}}-{\dot{\mathrm{m}}}_{\mathrm{q}\mathrm{p}}\right)}\right]\cdot \mathrm{V}$$

(23)

In Formula (23), $n$ and $n+1$ denote the previous time step and the current time step, respectively; $a_q^n$, $a_q^{n+1}$ represent the unit volume fraction corresponding to step n and step 1 + n step, respectively; $a_{q\cdot f}^{n+1}$ represents the face value of the volume flux passing through the surface at step $n+1$; $U_f^{n+1}$ represents the volume flux passing through the surface at step $n+1$; and $V$ is the volume of the unit. The volume fraction at the current time step is a function of other quantities, and the scalar transport equation for the volume fraction of the secondary phase at each time step is solved iteratively. The surface fluxes are interpolated using the selected spatial discretization scheme.

The simulation and prediction of fluid motion and interaction are realized by defining marker functions in the computational domain to describe fluid locations and shapes. When considering the momentum conservation equation, gravitational acceleration g is an important factor. In the actual coating process, gravity affects the vertical momentum change of the fluid, and there are gravity-related terms in the equation. For example, in the vertical acceleration formula of fluid particles, g is included. During simulation, as the fluid flows from the coating head to the substrate under gravity, its velocity and trajectory change, affecting the final coating thickness. An increase in g accelerates the vertical falling velocity of the fluid, changing its deposition on the substrate, and thus, the coating thickness. This way, the VOF model accounts for the gravitational factor, which is closely linked to coating thickness simulation results.

The numerical simulation of the slurry flow process is all realized based on the VOF model. By solving the mass conservation equation and the momentum conservation equation, physical quantities such as the fluid’s velocity, pressure, and density can be calculated, and the interaction between different fluids can be predicted according to the information of the marker function, such as the interface morphology between liquid and gas, the shape, movement, position of droplets, etc. From this, the flow state of the slurry and the wet coating thickness value under a stable state can be obtained.

4.2. Fluid Simulation Modeling

Since the fluid morphology of each cross-section along the width direction of the slot channel is basically similar, the middle cross-section of the three-dimensional fluid domain is taken, and the external flow channel fluid domain model is simplified into a two-dimensional model. The two-dimensional fluid domain model obtained is shown in Figure 5.

Taking the two-dimensional fluid domain model as the simulation model, common process parameters in production are selected for theoretical numerical calculations, and the VOF model is used for two-phase flow numerical simulation.

When the coating is stable, the two-phase flow cloud chart under the parameters of Process 1 is shown in Figure 6, where the blue area represents the environmental gas phase and the red area represents the slurry phase. The coating thickness at this time is 128.51 μm, and the theoretical predicted value is 127.33 μm. The accuracy of the theoretical calculation results and the numerical simulation results reached 99.07%.

The two-phase flow cloud charts under the parameters of Process 2 and Process 3 are shown in Figure 7. The measured coating thicknesses are 139.32 μm and 102.99 μm, respectively. The theoretical predicted thickness values are 138.00 μm and 100.01 μm, respectively, and the deviations are 0.95% and 0.29%, respectively.

5. Experiment

In order to verify the established thickness prediction model, an experiment was carried out using the slot-die coating device provided by the laboratory of KATOP Company. The slot-die coating equipment, such as the negative graphite slurry, coating head, copper foil, and oven, required for the experiment were all provided by the laboratory. The coating equipment is shown in Figure 8. Among them, the actual dimensions and various parameters of the coating head were basically consistent with the model established by VOF, and the original gasket with a thickness of 0.8 mm was selected for the gasket. An experiment on single-sided continuous coating was conducted using this equipment for thickness detection.

The experimental steps are as follows:

(1)

Clean the material tank and the rubber roller of the steel roller before testing;

(2)

Start the oven heating and adjust the oven temperature and wind frequency;

(3)

Thread and wind up the copper foil normally;

(4)

Adjust the die head, slurry circulation, pressure, etc., to be normal;

(5)

Remove wrinkles by adjusting the tension of the coating mechanism on the electrode;

(6)

Start the equipment, adjust the distance between the steel roller and the rubber roller, and conduct normal trial coating with materials;

(7)

Confirm the basic working conditions, adjust the tape running speed, feeding speed, coating gap, and other process parameters, and adjust the oven temperature to ensure that the electrode is completely dried;

(8)

Randomly take a sample after trial coating and drying for gram weight testing, and adjust the micrometer of the die head based on the gram weight difference to make the die head coating uniform.

In this experiment, some experimental objects were selected as samples for measurement and analysis to infer the characteristics of the whole. From numerous experimental objects, samples were evenly chosen, and the negative electrode under Process 1 coating parameters was analyzed. After coating, the sheets were dried. Instead of winding up, an 800 mm long electrode sheet was cut from the coated product, and 5×8 uniform sampling was performed with a circular quantitative sampler. A circular sample was collected every 110 mm horizontally and 100 mm vertically, obtaining 40 experimental samples in total. Finally, the obtained negative-electrode sample is shown in Figure 9.

The main instruments used in the experiment were as follows: a precision electronic balance with the model number ESJ200-4A, an electronic digital display micrometer with the model number MDE-25PX, and a quantitative sampler with the model number LD-102.

In the experimental verification of the thickness prediction model for the negative-electrode coating of lithium batteries, the weights of the 40 collected samples ranged from 0.168 g to 0.174 g, with a maximum weight difference of 0.006 g and an error of less than 2%, indicating a high degree of uniformity. The thickness value of the copper foil was 0.008 mm. The thickness value of each electrode sheet sample was between 0.106 mm and 0.114 mm, with an average thickness of 109.27 μm. The thickness range excluding the thickness of the copper foil was between 0.098 mm and 0.106 mm, with an average thickness of 101.27 μm.

Under the same conditions, the value calculated using the thickness prediction model was 100.01 μm, and the deviation was 1.25%; the electrode thickness value obtained from the simulation cloud map was 102.99 μm, and the deviation from the experimental value was 1.67%.

6. Conclusions and Prospects

Based on the Landau–Levich thin film equation and the Ruschak model, considering the influence of temperature and slurry-spreading characteristics on the coating thickness, and combined with the influence brought by the actual production environment, a lithium-battery negative-electrode thickness prediction model with high accuracy was established. Through simulation and experiments on a negative electrode, the effectiveness of this model was proven. Compared with existing models that overlook the multi-factor interaction impact in actual production on coating thickness, this model innovatively considers key factors like temperature and slurry spreading characteristics. It integrates multiple-factor influences, better fitting the actual production complexity. During construction, a complete calculation system was built through factor analysis and mathematical derivation, greatly improving prediction accuracy. Experiments verified an accuracy rate of 98.75%, a notable advancement over existing models. Therefore, this model can effectively predict the thickness of a negative electrode produced under different processes and has certain guiding significance for the actual production of lithium-battery negative electrodes.

The model is robust under different conditions. The paper considered temperature’s impact on coating thickness via formula ${\mathrm{\rho }}_{\mathrm{d}}={\mathrm{\rho }}_{\mathrm{u}}\left[1-{\mathrm{a}}_{\mathrm{v}}\left({\mathrm{T}}_1-{\mathrm{T}}_0\right)\right]$. To boost its temperature-related robustness, expand the temperature range in experiments and simulations, collect data, and optimize temperature-related coefficients. For slurry spreading, study different slurries and additives to refine the model.

However, the model has limitations. Its calculations are complex, involving multiple physical quantities and equations, leading to long calculation times, especially in large-scale simulations. Extreme conditions, like abnormal temperatures or coating speeds, can reduce prediction accuracy.

In industrial production, equipment precision affects the model’s application. Regularly calibrate and maintain equipment, and install sensors for real-time monitoring and model correction. Leverage big data, such as using data warehouses and machine learning, to analyze production data for hidden patterns and anomalies.