Coordinates and their metric coefficients.

## Abstract

Using scaling arguments and perturbation theory, we derive the lubrication form of the fluid mechanical equations governing the motion of a thin liquid film on an arbitrarily curved, rotating, axisymmetric substrate. The resulting equations are discretized and then solved numerically using an efficient implicit finite difference algorithm. The primary application for this work is to model the spin coating of the interior of two-piece metal beverage cans, and we consider this problem in some depth. Specifically, we show how adjusting several parameters can eliminate one possible defect in the spin coating process: the tendency for droplets to detach from the substrate when the can is spun at high rotation rates.

### Keywords

- thin liquid films
- spin coating
- spray coating
- numerical modeling

## 1. Introduction

When a liquid is applied to a spinning substrate, or if a pre-wetted substrate is spun, centrifugal forces act to drive any irregularities in the film thickness outward, away from the axis of rotation. The result is that the film becomes thinner and more uniform as the rotation proceeds. Consequently spin coating is used in such applications as coating magnetic storage discs, optical devices, and semiconductor wafers to obtain very thin but uniform films on flat substrates.

When the substrate is curved, centrifugal forces will act to produce a uniform layer only in horizontal regions where the substrate is perpendicular to the axis of rotation. But the coating layer may be very irregular in regions where the substrate is highly curved and the normal vector from the surface is not parallel to the axis of rotation.

In this work we will derive the lubrication form of the fluid mechanical equations for a thin liquid film sprayed on an arbitrarily curved, rotating, axisymmetric substrate. Our goal is to predict how the coating thickness changes with time as a function of the substrate geometry, the rotation rate, the rheological properties of the coating liquid, and the geometry and flux of the spray gun. We then discretize the equations and solve the partial differential equations governing the flow. Using an implicit method of solving the finite difference representation of the partial differential equations, we require a minimum of computer resources.

The theory we develop in this work is used to analyze one specific application: the spray/spin coating of the interior of aluminum beverage containers. When a spinning can is spray painted, centrifugal forces help to cause a more uniform coating layer on the can substrate. Indeed this is the purpose of rotating the can a high spin rates while spray coating the interior. But centrifugal forces can also cause such coating irregularities as drop formation. In this work we will demonstrate how such parameters as the spray gun placement and the rotation rate contribute to how long the beverage can may remain in the spin phase of its coating process before this potential defect occurs.

## 2. Derivation of the evolution equation

Consider an arbitrarily curved, axisymmetric substrate with the parameter *s* representing arc length along the substrate. The parameter *n* is the distance normal to the substrate (see Figure 1). *u* is the velocity parallel to the substrate, in the *s* direction; *v* is the velocity normal to the substrate, in the *n* direction; and *w* the velocity in the circumferential direction *z* axis. Note that while the substrate is axisymmetric, the coating need not be. *r* is the distance from the can centerline to a given point

Metric coefficient | Coordinate | Velocity |
---|---|---|

If

Here

If

We define

At the free surface, the substantial derivative of *F* must be zero. Consequently the kinematic condition on the free surface is given by

Employing the substitution

the mean curvature of the free surface for this geometry is given by

at

If the atmospheric pressure is zero, then the tensor equation relating the change in pressure across the free surface due to surface tension is given by

Here

If we sum the three equations found in Eq. (5) over

all evaluated at the free surface

We now scale the dependent and independent variables with various characteristic lengths of the substrate geometry. These include

If *s* and

We can eliminate the pressure from the

The scaled form of the continuity equation is given by

At the substrate,

The scaled form of the kinematic condition is shown to be

at

at

Instead of transforming the equations representing the pressure discontinuity across the liquid interface, given by Eq. (9) and Eq. (10), we will eliminate the pressure term in Eqs. (9) and (10) using the momentum equations and the identities for the partial differentiation of implicit functions. Eq. (7) gives the pressure in the liquid at the free surface

Plugging the pressure at the interface, found by Eq. (7), into the momentum equations yields

where

Here the

We now expand the scaled velocities in a regular perturbation series expansion in powers of the small parameter

We then use boundary conditions to determine the appropriate constants of integration and solve for

where

with the constants

Using the appropriate boundary conditions, the order

The evolution equation, to order

The dimensional evolution equation is found to be given by

## 3. Nondimensionalization

We shall employ the maximum radius of the substrate as our length scale,

Substituting these scale factors in the dimensional evolution equation, Eq. (22), leads to the nondimensional evolution equation

in nondimensional units. Here

The geometry of the substrate is delineated by a schematic drawing which gives us the radius of each circular arc, *s* and *t*.

## 4. Application of coating layer

We assume that the coating layer is laid down over time by a spray gun that emits a fan of gas that is directed toward the can substrate. Thus the accumulation of coating with time due to this fan is a function of *s* and *t*:

If

If the flux does not vary significantly in the

When expressed as a function of

From Figure 4 we have the relation

For a sufficiently thin fan, where the point on the substrate is sufficiently far from the centerline

When we are near the centerline, this formula must be modified to

to account for the fact that here the substrate is constantly being reached by the spray fan.

We also assume that there is a secondary “gas” which is uniform in the interior of the can and results in a constant

where

## 5. Pendant drop of maximum volume

In an effort to determine the maximum value a droplet can form without detaching from the substrate, we will consider an axisymmetric droplet forming on the underside of a ceiling, as illustrated in Figure 5. Here

If we nondimensionalize the problem by scaling

Here

Using a standard Runge–Kutta method, we can numerically integrate Eq. (27) assuming *g* is replaced by a centrifugal force that is a function the rotation rate

If

## 6. Applications

Approximately 400 billion two-piece, all-aluminum cans are produced annually for the purpose of storing beverages for distribution worldwide. The interior of each of these cans must be coated to protect the aluminum from onslaught due to corrosive elements in the contained beverage, and the beverage must be protected from picking up metal ions or other off-flavors from the aluminum substrate. Consequently the coating must be as uniform as possible for thick regions may slough off and thin regions may not offer adequate protection. To achieve a uniform film thickness, spin coating is employed using a spray fan to distribute the coating on the can substrate. But because the can is highly curved due to structural considerations, achieving a uniform final film thickness is much more complicated than for a flat substrate. In this section we will apply the analytical and numerical model we developed in previous sections to determine how the many parameters are influencing the flow of the paint coating. These parameters include the rotation rate, the shape of the can, the coating fluids physiochemical properties, and the geometry and flux of the spray fan, all affecting the final film thickness distribution.

The can body is initially punched from sheet aluminum and then goes through a washing process to produce a substrate suited for the spray coating. The can is then spun at between 2500 and 3500 rotations per minute, and one or two spray guns spray the interior with the liquid paint film. Centrifugal and gravitational forces redistribute this liquid layer as the can continues to spin after the initial spray process. The can is then placed in an oven where the solvent is allowed to evaporate leaving only the hardened resin. Then the can is filled with the beverage and the top of the can is attached in place. This conveyer process can produce as many as 1700 filled cans per minute.

In practice, one or two spray guns are used to coat the interior of the spinning cans. These are oriented at between 5° and 30° with respect to the vertical axis of the can and placed between 0.5 and 1.5 cm vertically from the top of the sidewall [6, 7]. Typically the can is sprayed for between 0.05 and 0.2 s and spun for an additional 0.1–0.5 s [6] so that centrifugal forces can act to redistribute the coating layer.

The industry uses schematic drawings which plot the substrate as a function of circular arcs of radius *Ri*, and subtended angles

We will assume that the spray fan has an elliptical cross section with a ratio of the major axis to the length of the minor axis of 10. The parameters determining the placement and orientation of the spray gun are illustrated in Figure 9. For the simulation considered in this work, the nondimensional parameters are listed in Table 1, the dimensional parameters in Table 2, and typical properties of the coating liquid in Table 3.

Parameter | Symbol | Value |
---|---|---|

Distance of spray gun from centerline: fan #1, fan #2 | −0.15, −0.45 | |

Distance of spray gun above can: fan #1, fan #2 | 0.15, 0.45 | |

Angle of spray gun wrt vertical: fan #1, fan #2 | 28^{°}, 15^{°} | |

Subtended angle of spray fan | 100° | |

Time spray gun acts in fast spin phase | 0.15 | |

Percent of secondary spray | 5% |

Parameter | Symbol | Value |
---|---|---|

Can radius | R | 3.33 cm |

Rotation rate | 2500–3500 RPM | |

Distance of spray gun from centerline: fan #1, fan #2 | A | −0.5 cm, −1.5 cm |

Distance of spray gun above can: fan #1, fan #2 | B | 0.5 cm, 1.5 cm |

Average wet coating thickness | 0.0028 cm | |

Time spray gun acts in fast spin phase | 0.05 s |

From Eq. (26), we see that the coating applied by the spray gun is an inverse function of the radius of the can substrate

The dimensionless parameter

The surface tension, density, and viscosity of the coating liquid are difficult to significantly alter as they depend on the required organic solvent content and surfactant levels in the paint formula. Similarly, *R*, the beverage can radius, is fixed by industry production standards. This leaves the rate of rotation as the only significant production parameter for changing the nondimensional (centrifugal force)/(surface tension force) parameter.

The centrifugal ejection of coating liquid from the inner wall of the moat is also predicted to be a strong function of the position and orientation of the spray gun. In the above example, the gun is placed 0.5 to the left of the centerline, is 0.5 cm above the top of the can, and is angled at

## 7. Conclusion

In this work we have used scaling arguments and perturbation theory to derive the lubrication form of the governing fluid mechanical equations for a thin liquid film coating an arbitrarily curved, axisymmetric, rotating substrate. Our main purpose has been to develop mathematical model that can be employed to numerically simulate the application of a paint film to the interior of beverage cans, though the basic algorithm may be useful in other applications. We have used our algorithm to predict the time of centrifugal ejection of coating from the inner moat wall as a function of several input parameters: the physiochemical properties of the coating liquid, the rotation rate, and the spray gun placement. The model can also be used to predict other coating defects and how the input parameters can be used to avoid them. The effect of solvent evaporation during the drying phase, when gravity and surface tension forces affect the coating distribution as the viscosity increases until only a final, hard, film remains, may also be modeled. With so many parameters regulating the final coating thickness, using experiments to model the coating evolution and measure the final, dry, film thickness is an almost impossible task. Instead we can utilize the power and versatility of computer simulation to predict the coating profile as a function of input parameters. This understanding will be useful in optimizing the current application process. It may also be essential in acquiring a satisfactory coating when environmental regulations require a change to high solids and latex paints.