2D Non-Newtonian Slot-Die Coating

Achieving uniform coating quality is important in several different industries: from optical coatings, semiconductor and electronics industry, through technologies utilizing thin membranes, to surface treatment of metals. Bad coating quality will compromise the performance of the products, or lead to complete failure in some cases.

Several different coating processes exist. This tutorial investigates the performance of a slot-die coating process, a so-called premetered coating method. In this process, the coating fluid is suspended from a thin slot die to a moving substrate. The final coating layer thickness is evaluated from the continuity relationship for a coating liquid. Therefore, the thickness of the liquid layer is determined by the slot gap, the coating fluid inlet velocity and the substrate speed.

The final goal of coating processes is to achieve a defect-free film of a desired thickness. However, manufacturing the uniform coating is not a trivial task, various flow instabilities or defects such as bubbles, ribbing, and rivulets are frequently observed in the process. The die geometry, the size of the slot and height above the substrate, together with the non-Newtonian fluid nature of the coating fluid are important to consider.

This tutorial demonstrates how to model the fluid flow in a polymer slot-die coating process using the interface and an inelastic non-Newtonian power law model for the polymer fluid.

Model Definition

Model Geometry

A typical setup of the slot-die coating process is shown in Figure 1.

Figure 1: Typical geometry for a slot-die coating process with the slot die positioned over a substrate.

This model uses a 2D cross section of the die shown in Figure 1, assuming out-of-plane invariance. The inlet for the coating fluid is at the top of the die, as shown in Figure 2, and there are open boundaries at both ends. The bottom boundary is the coating substrate which is moving at the coating velocity.

Figure 2: Model geometry. 2D cross section of a slot die.

The geometrical and material parameters in this model are taken from the Ref. 1.

Domain Equations and Boundary Conditions

The flow in this model is laminar, so a interface will be used together with a interface to track the interface between the air and the polymer fluid. The coupling of these two interfaces is handled by the multiphysics interface. You can select which constitutive relationship to use for each of the fluid phases. The air is specified as a Newtonian fluid, and the coating fluid is a non-Newtonian power law fluid.

The inlet fluid velocity is increases smoothly from 0 m/s to 0.1 m/s. Both the upstream and downstream boundaries of the model are specified as open boundaries. The corresponding inlet and outlet boundary conditions must also be set in the interface together with the initial values for both fluids to correctly define the position of the initial interface. For the moving substrate, a moving wall boundary condition with a Navier-Slip condition is used.

Results and Discussion

The Figure 3 shows the evolution of the coating fluid interface for t = 0.03 s, t = 0.06 s, and t = 0.2 s.

Figure 3: Coating fluid interface at t = 0.03 s, t = 0.06 s, and t = 0.2 s (top to bottom).

The coating film attains a constant thickness downstream of the die at t = 0.2 s. The film forms upstream and downstream meniscii with the upstream and downstream walls of the die. As the substrate speed increases or the inlet velocity decreases, the upstream meniscus is pulled closer to the slot, eventually causing defects in the coating film. The evolution of the film thickness and position of the upstream meniscus as a function of time is shown in Figure 4.

Figure 4: Film thickness and upstream meniscus position as a function of time.

By changing the geometry, the inlet velocity and wall velocity, it is easy to explore the sensitivity of the design parameters toward the film thickness and coating velocity for a variety of fluid properties in a fast and efficient manner.

Notes About the COMSOL Implementation

The default method for averaging the fluid properties across the interface between the two phases is linear with respect to the volume fraction. When working with fluids that have a large difference in viscosities, switching to a different averaging method increases the performance. In this example, the Heaviside averaging method is applied. The averaged viscosity is defined as

where V2 is the volume fraction of fluid 2, μ1 and μ2 are the viscosities for fluid 1 and 2, respectively, and H is a smoothed Heaviside function. The default value of a mixing parameter lμ is 0.8. In this model, the value of the lμ is increase to 0.9. A lower value will sharpen the interface, but also increase the computation time.

where ρ1 and ρ2 are the densities of fluid 1 and Fluid 2, respectively and lρ is a mixing parameter defining the size of the transition zone.

The default surface tension for the phase field model is evenly distributed across the fluid-fluid interface. In cases with a large difference in density between the two phases, significant spurious oscillations in the velocity field can occur for the lighter phase which results in smaller time steps and longer computing time. Thus, it may be advantageous to shift the surface tension toward the heavy phase to avoid such oscillations. This is done by multiplying the surface tension force by

where ds,Fst is a mixing parameter defining the size of the transition zone.

In addition, this example demonstrates how to fit measured rheology data to a selected inelastic non-Newtonian fluid model.

Reference

1. K.L. Bhamidipati, Detection and elimination of defects during manufacture of high-temperature polymer electrolyte membranes, PhD Thesis, Georgia Institute of Technology, 2011.

Polymer_Flow_Module/Tutorials/slot_die_coating_2d

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Load the model parameters from a text file.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file slot_die_coating_2d_parameters.txt.

Follow the steps below to calculate the parameters for the power-law fluid model based on measurement data in this example.

Least-Squares Fit 1 (lsq1_fun1)

In the toolbar, click

and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file slot_die_coating_2d_viscosity_input.txt.

Locate the section. In the table, click to select the cell at row number 1 and column number 1.

In the text field, type gammadot.

In the text field, type 1.

In the table, click to select the cell at row number 2 and column number 1.

In the text field, type mu_app.

In the text field, type m*max(gammadot,0.01)^(n-1).

In the text field, type Pa*s.

Locate the section. In the table, enter the following settings:


Parameter	Value
m	1
n	1

Click

. The computed parameter values from the least-squares fit appear in the table.

Click

, to see how close data points are to the fitted function.

Create a step function to use for ramping up the inlet velocity. To improve convergence, define a smoothing transition zone to gently increase the inlet velocity from zero.

Step 1 (step1)

In the toolbar, click

and choose .

In the window for , type step1 in the text field.

Locate the section. In the text field, type 0.01.

Click to expand the section. In the text field, type 0.02.

Create the geometry by using a rectangle and a polygon.

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the toolbar, click

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type W.

In the text field, type Hc.

Locate the section. In the text field, type -W/2.

In the text field, type H.

Add an additional layer at the bottom of the channel. You will use it later to define the initial domain for the coating fluid.

Click to expand the section. In the table, enter the following settings:


Layer name	Thickness (mm)
Layer 1	0.1[mm]

Click

Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


x (mm)	y (mm)
-W/2	H
-W/2-W_ud	H
-W/2-W_ud-tan(alpha_u)*Hc	Hc+H
-W/2-L_u	Hc+H
-W/2-L_u	0
W/2+L_d	0
W/2+L_d	Hc+H
W/2+W_dd+tan(alpha_d)*Hc	Hc+H
W/2+W_dd	H

Click

Click the

button in the toolbar.

Compare the resulting geometry to Figure 2.

Definitions

Next define integration operators. First define an integration coupling that integrates along the outlet boundary, to calculate the film thickness. Then define a coupling operator that integrates along the upstream die lip. You will use it later for the integration of the volume fraction along the boundary to evaluate the location of the upstream meniscus.

Integration 1 (intop1)

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 16 only.

Integration 2 (intop2)

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Select Boundary 5 only.

Now specify the material properties for the model. The material can be created by clicking the in the -coupling node and new Multiphase material node will be created in the Materials section.

Multiphysics

Two-Phase Flow, Phase Field 1 (tpf1)

In the window, under click .

In the window for , locate the section.

Click

Now, define the two phases for the model: Air and Coating Fluid.

Materials

Phase 1 (mpmat1.phase1)

In the window, under click .

In the window for , locate the section.

Click

. This button is found when expanding the options next to the list.

Add Material to Phase 1 (mpmat1.phase1)

Go to the window.

In the tree, select .

Click .

Materials

Phase 2 (mpmat1.phase2)

In the window, under click .

In the window for , locate the section.

Click

. This button is found when expanding the options next to the list.

Use the parameters from the least-squares fit lsq1.m and lsq1.n to define material coefficients.

Global Definitions

Coating Fluid

In the window, under click .

In the window for , type Coating Fluid in the text field.

Locate the section. In the tree, select .

Right-click and choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Fluid consistency coefficient	m_pow	lsq1.m	Pa·s	Power law
Flow behavior index	n_pow	lsq1.n	1	Power law
Density	rho	1400	kg/m³	Basic

When working with fluids that have large viscosity and density ratios, switching from the default linear volume fraction to a Heaviside-function dependent on the volume fraction for the properties can increase the performance.

Materials

Multiphase Material 1 (mpmat1)

In the window, under click .

In the window for , locate the section.

Click to select row number 1 in the table.

In the dialog box, choose from the list.

In the lmix text field, type 0.9.

Click

From the list, choose .

Click

From the list, choose .

Click .

Now, set up the physics of the problem by defining the domain physics conditions and the boundary conditions.

Laminar Flow (spf)

Wall 2

In the window, under right-click and choose .

Select Boundary 2 only.

In the window for , locate the section.

From the list, choose .

Click to expand the section. Select the check box.

In the Uw text field, type -U_wall.

Inlet 1

In the toolbar, click

and choose .

Select Boundary 10 only.

In the window for , locate the section.

From the list, choose .

Locate the section. In the Uav text field, type step1(t[1/s])*U_in.

Open Boundary 1

In the toolbar, click

and choose .

Select Boundaries 1 and 16 only.

The initial interface between the coating fluid and air is automatically assigned to the boundaries between the two initial value domains. Set up the initial coating fluid domain in the inlet channel.

Phase Field (pf)

Initial Values, Fluid 2

In the window, under click .

Select Domain 3 only.

Wetted Wall 1

In the window, click .

In the window for , locate the section.

In the θw text field, type 68.5[deg].

Inlet 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose ϕ .

Select Boundary 10 only.

Outlet 1

In the toolbar, click

and choose .

Select Boundaries 1 and 16 only.

Wetted Wall 2

In the toolbar, click

and choose .

Select Boundary 2 only.

In the window for , locate the section.

In the θw text field, type 74[deg].

Multiphysics

Two-Phase Flow, Phase Field 1 (tpf1)

For models with a large difference in viscosity and density, spurious oscillations in the surface tension can occur. To reduce this problem, one can shift the surface tension force so that it applies mostly to the dense phase in your model. Proceed to apply this condition.

Click the

button in the toolbar.

In the dialog box, select in the tree.

In the tree, select the check box for the node .

Click .

In the window, under click .

In the window for , click to expand the section.

Select the check box.

Change the surface tension from the default setting to a user defined value.

Locate the section. From the list, choose . In the σ text field, type 0.049.

If you want to inspect the progress of the fluids during the simulation, you can enable the plot while solving option in the node. By calculating the initial values first, the solver sequence and default plots will be generated. In the following section you generate the default plot groups and use one of them for plotting the volume fraction while solving. Note that plot while solving in general will affect the computation time slightly since the plot needs to be updated in each time step.

Study 1

In the toolbar, click