step 1

• First, we introduce the viscous model, which neglects capillary effects and ignores surface tension. The viscous model is subject to the following constraints, as illustrated in the image: the lengths of the upstream and downstream slot-die sections, the height between the slot-die and the substrate, and 𝑡, which represents the wet film thickness.
• In this approach, the ink is pinned to the downstream lip, as shown. The ink is then free to move in and out of the slot-die, as indicated by the arrow between the curved dashed line and the red ink.

step 2

• For the first contribution to the viscous model, we consider two contributions.
• First, consider two parallel plates with an ink experiencing a pressure difference from one end to the other. The speed across the channel can be expressed by solving the Navier-Stokes equation with the no-slip condition at the edges.
• The solution is as shown in the illustration.
• Naturally, a half Poiseuille contribution will be pointing outwards on the upstream and downstream of the slot-die, as depicted in the last image.

step 3

• For the second contribution, consider two parallel plates with ink in between them. Here, the bottom plates represent the moving substrate. Solving the Navier-Stokes equation for this problem (geometry) results in a Couette contribution.
• The Couette flow is illustrated here with a representation of velocity vectors, where the velocity linearly increases towards the substrate.
• Finally, for the purpose of understanding the meniscus forming, consider the Couette contribution for both the upstream and downstream slot-die depicted.

step 4

• From the two contributions of Poiseuille and Couette, one can apply the lubrication theory to the upstream and downstream sections.
• From this, one can write the pressure difference, Δp, from up- to downstream, as the inequality shown in the figure to the left.
• When the pressure is fulfilling this inequality, the meniscus is considered stable.
• The stability is a balance of the Poiseuille and Couette contributions.

step 5

• With no pumped ink and non-moving substrate. There are no resulting velocity vectors.
• This might be trivial, but it’s a good way to start and move to the next step.

step 6

• With no pumped ink and moving substrate. You observe a higher speed of the ink closer to the substrate than at the slot-die lips.
• Here, the dominant contribution is Couette flow.

step 7

• With pumped ink and non-moving substrate. You observe a higher speed in the middle of the gap.
• Here, the dominant contribution is Poiseuille flow.

step 8

• With both pumped ink and moving substrate. You observe that both the aforementioned contributors have an impact.
• In the first image, one can see both contributions individually; in the second image, they are combined.

step 9

• With both pumped ink and moving substrate, you continue to observe that both the aforementioned contributors have an impact.
• In the first image, one can see both contributions individually; in the second image, they are combined.

Step 10

Visual presentation of the two contributors with varying substrate speed, viscosity, coating gap, and flow rate.

• NB: The course changes the parameters at the bottom of the video. Python script available on request.

step 11

• Finally, the inequality describes the viscous model, as thoroughly depicted in the previous steps.
• The lower and upper red lines spanning the coating window are the right and left-hand sides of the inequality presented in the step image.
• These are only separated by the term presented in the figure.
• An important result is that t(zero) for a viscous coating bead is half the downstream coating gap. As indicated by the red dot at the intersection of the first axis.

step 12

• The capillary model assumes that the viscous effect is small, and the operation limits are dictated by capillary pressure in the coating bead (droplet between the slot-die and the substrate).
• This model has two pinned (static contact) points. The first is the downstream, and the second is the upstream slot-die.
• The dynamic (free-to-move) point is between the substrate and the ink.
• The contact angle between ink and substrate is important for this model.

step 13

• Kenneth J. Ruschak (1974) conducted the first theoretical study on coating operation limits. Building upon the Landau-Levich film coating theory, he derived operational boundaries under the capillary-flow assumption.
• The operation limits can be thought of as a capillary model, and they are given by the two inequalities shown to the left. Here:
• Sigma, with both u and d as subscripts, are surface tensions or the upstream and downstream.
• Delta P represents the vacuum pressure between the upstream and the downstream menisci.
• Ca, is the capillary number of the ink.
• my, is the viscosity of the ink.

step 14

• The Delta P is often not as important as one can apply vacuum methods to the upstream meniscus by expanding the operation window.
• To the left is a conceptual plot of the two inequalities, representing the coating window.
• The two expressions indicated in the figure are the two linear lines intersecting the second axis.

step 15

• t zero, as indicated in the figure, where the blue line intersects the first axis. That is the thinnest film thickness possible, when delta P, is zero.
• The rightmost boundary is where one can extract the minimum wet film (t_min) thickness possible.
• Important note: Both minimum conditions depend on the capillary number, Ca, to the power of 2/3.

step 16

• The viscocapillary model is the combination of the two contributions and expressions from both the viscous model and the capillary model.
• Both the viscous and capillary effects are included in the model description.

step 17

• As a reminder, the first and second plots are the coating windows based on the capillary and viscous models, respectively.
• Then depicted in the same plot.
• Finally, the combination of the two and what is understood as the coating window of the viscocapillary coating windows.

step 18

• Finally, an illustration of the coating window is shown to the left.
• Again, defects will occur outside the coating window. Understanding the theory of the coating window and consulting on “coating errors” can help you optimize your slot-die coating process.