1

byAnupam Sengupta

Title of Presentation

Doctoral Researcher

Centre of Advanced Fluid MechanicsAm Weichselgarten 34, D-91058 Erlangen, FRG

Fluid Mechanics of

Coating Flows

2 Contents of Presentation

The presentation:

• Conventional and Modern Coating Techniques

• Navier-Stokes equations in relevance to coating flows

• Slot Coating and Coating Windows

• Recent Works on Dip Coating

• Final remarks

3 Current State and Scopes

• Coating Technology is important from an industrial point of viewCoating Technology is important from an industrial point of view

•• Extensively used in: Extensively used in:

a) Papera) Paper

b) Pharmaceuticalsb) Pharmaceuticals

c) Packagingc) Packaging

d) Textiles d) Textiles

e) Electronics etc.e) Electronics etc.

•• Industries still rely upon trial and error methods and techniquIndustries still rely upon trial and error methods and techniques es

•• Considerable scope of innovations exists: applications of fluidConsiderable scope of innovations exists: applications of fluid mechanicsmechanics

•• Understanding the roles of surface and rheological properties fUnderstanding the roles of surface and rheological properties for improving or improving

coating productscoating products

•• Detailed fluid mechanics studies are possible these days using Detailed fluid mechanics studies are possible these days using

numerical/analytical methodsnumerical/analytical methods

4 Classifications

Wet Film(micro)

Vacuum(nano-micro)

Coating

Pre Metered

•Slot Coating

•Slide Coating

Self Metered

•Dip Coating

•Roll Coating

•Knife Coating

Chemical

• CVD

•Etching

•Lithography

Physical

•PVD:Sputtering

•Evaporation

5 Premetered Coating Techniques

• Premetered Coating : Mass flow of the coating liquid pre-decided by the

desired wet film thickness

Slot Coating : Curtain Mode Slide Coating

M=ρ.UW.h

6 Self-metering Coating Techniques

• Self-metered Coating : Mass of coating liquid is not pre-decided by the

desired wet film thickness

• Generally substrate emerges from an infinite bath. Film thickness is

controlled by varying substrate velocity or post treatments

Substrate Web speedwU

1x

2xg

Coating liquid

h

Dip Coating

Knife Coating

Air-Knife Coating

7 Navier-Stokes Equations

0

i

i

xU

t Mass Conservation

ji

ij

ji

ji

j gxx

PxU

Ut

U

23

j i kij ij

i j k

U U Uµ µx x x

Momentum Conservation

i

jij

j

j

i

i

ii x

UxU

Pxq

xeU

te

with

Energy Conservation

ii x

Tq

8 Application of NS equations

• Mass Conservation• Momentum Consrv.• Surface Properties• Rheology

• Mass Conservation• Momentum Consrv.• Energy Consrv.

Substrate Wet Film Dry Film

Coating Liquid

Coating is done in a sequence of processes

Wet film coating takes place at high speeds : 2000m/min

9 Application of NS equations:Slot Coating in Bead Mode

• All coating flows can be analysed by NS equations

• What is that differentiates various coating flows ?

• Boundary Conditions !

• Let us try to look into some basic flows related to coating :

10 Slot Coating in Bead Mode

• Pre metered coating technique• Various modes of operation : Bead, Curtain, Extrusion, Slide

11 Slot Coating and NS Equations

1 2

1 2

0U Ux x

2 21 1 1 1 1

1 2 12 21 2 1 1 2

] [ ]U U U P U UU U µ gt x x x x x

21 2 1 2 2

1

12

dU x C x Cµ dx

2 22 2 2 2 2

1 2 22 21 2 2 1 2

] [ ]U U U P U UU U µ gt x x x x x

1xIn direction :

2xIn direction :

2 2 1P g x x

12 Boundary Conditions in Slot Coating

Boundary Conditions :

13 Resultant Flow Equation

Downstream Flow

where

Upstream Flow

14 Interpretation of Flow

15 Interpretation of Flow

Hagen Poiseuille Term

Couette Flow Term

1 2d dFor the flow equation reduces to :

21 2 2 2

1 2w

LinearTermQuadraticTerm

x UU x x d x dx µ d

16 Pressure Variation within the Die

23

1 1.34oP P Ca h

3 UU

P Pr

2 3 26 wµU lP P

d

1 2 261 2 wh µU lP P

d d

(Ruschak)

17 Concept of Coating Window

• Pressure term arising across the upstream meniscus can have 2 extremes : depending upon curvature

• When liquid-air interface is convex :

• When liquid-air interface is concave :

• Combining :

3 3U UP P P P ve

3 3U UP P P P ve

3 3 3,min ,max

,min ,max

( ) ( ) ( )

( ) ( ) ( )

U U Uconcave convex

concave convexU U U

P P P P P P

r r r

18 Concept of Coating Window

• From geometrical considerations of the meniscus :

,max

,min

1 , for convex surface

1 , for concave surface

UU

UU

Cosrd

Cosrd

0.22 0.33 0.04=541 aU C where (E.B.Guthoff, C.E.Kendrick (1982),AIChE Journal)

Front-pinned convex Front-pinned concave

Back-pinned convex Back-pinned concave

The meniscus has the flexibility to adjust itself in any of the above depending upon the sub pressure applied

19 Concept of Coating WindowPressure vs Web Speed

20 Industrial Relevance

• Typically thickness and coating speeds are adjusted through practical knowledge

• The flows can be analysed using fluid mechanics

• Innovative techniques can be introduced by applying fundamentals of fluid mechanics

• Knowledge of subpressure values applicable and introduction of appropriate pressure chambers at the back meniscus

• Extend the current knowledge for more challenging cases

21 Dip Coating

• One of the earliest methods of coating

• Self metering coating technique : Process defines thickness

• Continuous or discontinuous : Physics is identical

g

Substrate

,,

wU

H

h

hH

Coating liquid

Roller

,,

Substrate

Discontinuous

Continuous

,,

22 Analytical Treatment

2 2

2 2] [ ]x x x x xx y

U U U P U UU U µ gt x y x x y

reduces to 2

2xU Pµ

y x

23 Analytical Treatment

.p nR Young-Laplace Relation :

3

2 2

2 2

2 2

( )11

( ) ( )

xx

Rx x

x x

where

32

2 3 0x xUµy x

Finally :

Boundary Conditions :

1) No slip B.C :

2) Free Surface :

@ 0xU U y

0@xxy

Uµ yy

24 Flow Profile

Integrating the D.E for a constant x wrt y, we get :3 2

3( )2xyU y U y

µ x

For determining ,we conserve the mass over the film thickness x

0 0

w

x fU y dydz U w

where is the width if the plate in the cross direction.

Finally the thickness is obtained as :

w

25 Solution by Landau and Levich

• One of the earliest solutions

• Valid for only low capillary number cases

• Formulation does not hold good for very low and for high capillary numbers

• Solution based on matched asymptotic expansion and lubrication approx.

23

WµUH cg

Valid for (???)210WµUCa

h 0=

( g/U

)1/2

Ca10-4 10-3 10-2 10-1 100 101 102

1.0

0.8

0.6

0.4

0.2

0.0

Landau-Levich (1942)

Data of Gutfinger and Tallmadge (1965)

White and Tallmadge (1965)

Spiers et al.(1974)

Results of Shunk et al.

26 Observed anomalies

• Discrepencies in the measured and computed film thickness

• Discrepancy was prominent not only at higher capillary numbers (as expected) but also at capillary numbers where the relation should have held good

• Need for a fundamental treatment : Non dimensional analysis

• According to Landau-Levich :

• Far from being complete

• Overprediction and underprediction of experimental results

23 2

3.WµUH c H c Cag

27 Dimensional Analysis

thH

1 2, ...H f • Non dimensional thickness should be

• Aim : To identify 1 2, ,...

gUfH Wth ,,,,

H WU g

L

0 1 1 1 0 0

1 -1 -3 0 1 1

0 -1 0 -2 -1 -2

•Buckingham theorem : Dependence of coating thickness expressed by all the parameters that have an influence on

• Number of dimensionless parameters : N = number of influencing parameters – rank of the dimensional matrix

(5) – (3)

28 Dimensional Analysis

Hence, 1 2 3,H f

1 : 0 0 0H g M L T

0

3 1

2 2 0

12

Thus, 2/11 /

Bog

HH

Similarly, and CaUW

/2

Dig

4/1

3

,thH f Ca Di

29 Numerical Simulations

• The dip coating processs was then numerically simulated using a FEM tool

• The capillary number was varied for different set of Dip Numbers

30 Comparison with Landau-Levich

Underprediction

Overprediction

31 Physical Interpretation :Dip Number

•Interplay of three different characteristic velocities :

Coating Velocity

Adjustment velocity of a free surface on a viscous fluid

Adjustment velocity of liquid with free surface after verticaldisplacement

WU

/

4/1

g

32 Interpretation

• : upward moving free surface adjusts with static meniscus

• This is the zone of typical variation of with : for low dip numbers

this takes place at low capillary numbers and vice versa

• : Distortion of free surface faster than readjustment ( which is

controlled by )

• : Substrate-Liquid interactions take place through London-vander Waals forces which diffuse double layers

1WU Caµ

H Ca

1Ca

µ

1Ca

33 Concept of Drainage Number

• Additional physical process present in dip coating process : Drainage

• Drainage : of the liquid film that takes place when substrate comes out

of the liquid bath

• Takes place with a characteristic velocity

• Can be defined as

• For final thickness does not depend on the generating velocity

4/1

g

4/1// gUDr W

1Dr

34 Discussion and Conclusion

• It is most likely that available expression, is a theoretical limit

• Thus dependence on or does not show up in the expression

• Scope for setting a physical limit of coatability from very low to very high capillrary numbers : Coating window for dip coating

• Defining as a closed mathematical form based on thenumerical data obtained

• Compare the numerical results with experimental data

• Incorporation of surface and rheological aspects and their influence

3/2.. CaconstH

H Di Dr

,thH f Ca Di

Thank You !!!