transport phenomena in drying paint films · transport phenomena in drying paint films ......

Transport Phenomena in Drying Paint Films

by

Nazli Saranjam

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Department of Mechanical and Industrial Engineering University of Toronto

© Copyright by Nazli Saranjam 2016

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Abstract Transport Phenomena in Drying Paint Films

Nazli Saranjam

Doctor of Philosophy

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

2016

Paint films with uniform thicknesses ranging from 150 to 820 µm were applied on stainless steel

substrates using a model paint consisting of a resin dissolved in butanol. Test samples were cured

in a natural convection oven at a temperature of 140°C. Photographs of the paint surface were

taken during drying and the weight loss was measured. Cellular structures appeared on the paint

surface, induced by surface tension-driven flows due to solvent concentration variations. For thin

films (<500 µm), the patterns disappeared in a few minutes and the dried paint surface was smooth,

while for thicker paint films, wave-like structures remained on the hardened paint layer, creating

an uneven surface. An analytical solution of the mass-diffusion equation was used to model solvent

evaporation from the paint film and to calculate the concentration gradient and surface tension

variations in the paint films. In thin films, all the solvent was depleted, and surface tension

gradients disappeared before curing was complete, allowing the surface to become smooth. In

thicker films, concentration gradients that drove cellular flows persisted until the paint dried,

leaving orange peel on the surface. Small air bubbles were introduced into the liquid and test

samples cured in a natural convection oven at temperatures varying from of 100–140 °C.

Relatively thick (1 mm) paint layers were used in experiments to allow large bubbles to grow that

were easy to observe. Small bubbles were transported from the centers to the edges of the cells

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formed in the liquid layer. Bubbles grew larger as the evaporating solvent diffused into them. The

upward curving liquid meniscus around large bubbles created buoyancy forces that drew bubbles

towards each other, making them form clusters.

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Acknowledgement I would like to express my sincere gratitude to my mentor Prof. Sanjeev Chandra for his continuous

support and guidance throughout my research. I acknowledge, with appreciation, my debt of

thanks to Prof. Mostaghimi for his aid and foresight. Without their indispensable advice, this

accomplishment could not have been possible.

My sincere thanks are extended to my thesis committee member, Professor Hani Naguib from the

Department of Mechanical and Industrial Engineering at University of Toronto.

I am also grateful to Professor Chul Park from the Department of Mechanical and Industrial

Engineering at University of Toronto and Professor Daniel Attinger from the Department of

Mechanical Engineering at Iowa State University for participating in my SGS Final Oral Exam.

I addition I would like to thank all my colleagues at the Center for Advanced Coating

Technologies. Their assistance and cooperation were essential for the completion of this work.

It is with great appreciation and thanks that I acknowledge the valuable support and encouragement

of my parents, family and friends.

I am indebted to my husband whose patience and enthusiastic support have been a real contribution

to my study. I gladly dedicate this thesis to you.

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Table of Contents

Abstract .................................................................................................................... ii

Acknowledgement .................................................................................................. iv

Table of Contents ..................................................................................................... v

List of Figures ....................................................................................................... viii

Nomenclature ....................................................................................................... xiii

Chapter 1

Introduction .............................................................................................................. 1

1. Automotive Paint Application .......................................................................... 1

1.1 Motivation and Background ........................................................................................... 1

1.2 Previous Research on Paint Defect ................................................................................. 4

1.2.1 Craters ....................................................................................................................... 5

1.2.2 Bénard-Marangoni Convection, Orange Peel and Wrinkling ................................... 7

1.2.3 Paint Drying and Leveling ...................................................................................... 12

1.2.4 Bubble Migration, Bubble Entrapment, and Blistering .......................................... 16

1.3 Thesis Objectives .......................................................................................................... 19

1.4 Thesis Organization ...................................................................................................... 19

Chapter 2

Experimental Methodology ................................................................................... 21

2 Experimental system ....................................................................................... 21

2.1 Model Paint Formulation and Physical Properties ........................................................ 21

2.2 Experimental Apparatus................................................................................................ 25

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Chapter 3

Film Instability and Defect formation in Drying Paint films ............................ 31

3 Defect Formation in Drying Paint films ........................................................ 31

3.1 Introduction ................................................................................................................... 31

3.2 Experimental System .................................................................................................... 31

3.3 Results and Discussion ................................................................................................. 32

3.3.1 Craters ..................................................................................................................... 32

3.3.2 Bubble Formation ................................................................................................... 35

3.3.3 Wrinkle Formation .................................................................................................. 37

3.3.4 "Orange Peel" defect and Self-Organizing patterns ................................................ 39

3.4 Conclusion .................................................................................................................... 43

Chapter 4

Orange Peel and Marangoni Convection ............................................................. 44

4 Orange Peel Formation due to Surface-Tension-Driven Flows within Drying Paint Films ................................................................................................. 44

4.1 Introduction ................................................................................................................... 44



4.3.1 Surface-Tension-Driven Flows in Drying Paint Films ........................................... 46

4.3.2 Orange Peel Formation on Dried Paint Films ......................................................... 51

4.3.3 Mathematical Modeling of Paint Film Drying ........................................................ 54

4.3.4 Evaluation of the Dimensionless Marangoni Number ............................................ 66

4.3.5 Convective Velocities in Drying Paint Layers ........................................................ 68

4.4 Conclusion .................................................................................................................... 73

Chapter 5

Bubble Growth and Movement ............................................................................ 74

5 Bubble Growth and Movement in Drying Paint Films ............................... 74

5.1 Introduction ................................................................................................................... 74


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5.3.1 Bubble Formation and Growth ............................................................................... 76

5.3.2 Mathematical Model of Bubble Growth ................................................................. 83

5.4 Bubble Motion .............................................................................................................. 89

5.5 Conclusion .................................................................................................................... 99

Chapter 6

Interaction of Growing Bubbles .........................................................................100

6 Interaction of Growing Bubbles in Glycerin and Drying Paint Films .....100

6.1 Introduction ................................................................................................................. 100

6.2 Experimental System .................................................................................................. 100

6.3 Results and Discussion ............................................................................................... 101

6.3.1 The Dynamic of Floating Bubbles ........................................................................ 112

6.4 Conclusion .................................................................................................................. 121

Chapter 7

Closure………………………………………………………………………………...123

7 Summary and Conclusion ............................................................................123

7.1 Contributions............................................................................................................... 124

7.2 Future Work ................................................................................................................ 125

8 References ......................................................................................................126

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List of Figures

Figure 1-1 Paint defects (a) Cratering, (b) Orange Peel, (c) Wrinkling, (d) Bubble formation/

Solvent Boil [15, 16] ....................................................................................................................... 5

Figure 2-1 Thermal analysis on model paint sample ................................................................... 22

Figure 2-2 Surface tension variation with concentration ............................................................. 24

Figure 2-3 Viscosity variation with concentration ....................................................................... 24

Figure 2-4 Schematic arrangement of coating system ................................................................... 26

Figure 2-5 Schematic of the experimental set-up used for conduction curing ............................... 27

Figure 2-6 Schematic of the experimental set-up used for convection curing ............................. 28

Figure 2-7 Schematic of paint drying system ................................................................................ 29

Figure 3-1 Crater formation in 2 mm paint layer using conduction heating at approximately 130

± 5° C. Time t = 0 corresponds to 10 seconds after start of curing. De-wetting initiated at

t = 960 s; One more site grew and can be seen at t = 1180 s. Craters are frozen in place at

t > 1294 s ....................................................................................................................................... 34

Figure 3-2 Bubble formation/ Blistering in 600 μm paint layer using convection heating at

approximately 130 ± 5° C. Time t = 0 corresponds to 10 seconds after start of curing ............... 36

Figure 3-3 Wrinkling in an approximately 1 mm paint layer using convection heating at 130 ± 5°

C. Time t = 0 corresponds to 10 seconds after start of curing. At t= 13.8 min an elastic skin

capable of supporting stress is formed. Wrinkles start propagating at t= 13.9 min ...................... 38

ix

Figure 3-4 Patterns in 1 mm paint layer created by locally decreasing surface-tension on the

surface ........................................................................................................................................... 40

Figure 3-5 Temporal Evolution of Marangoni convective cells initiated by curing of 1000 μm

paint layer at 150 ± 5° C. Time t = 0 corresponds to less than 15 seconds after start of curing

when Marangoni cells appear. At t= 3.1 min initial levelling occurs due to mixing. At t= 5 min

self-organizing structures are fully established ............................................................................. 42

Figure 4-1 Marangoni cell formation in (a) 150 μm, (b) 320 μm, (c) 500 μm, and (d) 820 μm paint

films, t<5 min ................................................................................................................................ 48

Figure 4-2 Self-organizing roll-like patterns in (a) 500 μm, (b) 820 μm paint films, t≥5 minutes .. 50

Figure 4-3 Completely dried paint layers with (a) 150 μm and (b) 820 μm initial thickness .......... 51

Figure 4-4 Typical profilometry traces of the wrinkled patterns completely dried samples with

(a) 150 μm & 300 μm, (b) 500 μm, and (c) 820 μm initial thickness .............................................. 53

Figure 4-5 Self-sustaining Marangoni flow .................................................................................. 55

Figure 4-6 Variation of mass transfer coefficient with time at 100° C .......................................... 58

Figure 4-7 Mass transfer coefficient for various film thicknesses.................................................. 59

Figure 4-8 Reduced desorption curves for paint films of varying thickness ............................... 61

Figure 4-9 Kinetics of drying for paint films for (a) 150 µm and (b) 320 µm (c) 500 µm (d) 820

µm ................................................................................................................................................. 63

Figure 4-10 Concentration profiles for: (a) 150 μm (b) 320 μm (c) 500 μm (d) 820 μm paint

films .............................................................................................................................................. 65

x

Figure 4-11 Marangoni number as a function of time for paint films of varying thickness, using

mean values of diffusivity coefficient and viscosity ..................................................................... 67

Figure 4-12 Change in convective velocities over time for paint films of varying thickness ..... 69

Figure 4-13 Velocity field for 820 µm film at various time steps ............................................... 72

Figure 5-1 Stainless steel substrates, 75 mm in diameter, spray painted with an automotive clear

coat paint and baked in an oven after a flash-off time of (a) 10 min and (b) 2 min [104]. ........... 75

Figure 5-2 Bubble growth and migration in 1000 µm paint film curing at 140° C ..................... 77

Figure 5-3 Bubble growth and migration in 1000 µm paint film curing at 100° C ..................... 78

Figure 5-4 Bubble density variation ............................................................................................. 79

Figure 5-5 Sauter Mean Diameter variation ................................................................................ 80

Figure 5-6 Bubble agglomeration and escape in 1000 μm Glycerin-butanol solution on glass

substrates at T= 100° C, t 4 min .................................................................................................. 82

Figure 5-7 Evaporation curves for glycerin butanol solution and model paint at 120° C ........... 83

Figure 5-8 Reduced desorption curves for (a) glycerin butanol mixtures and (b) mode paint .... 87

Figure 5-9 Bubble growth rate in paint films curing 100 °C, 120 °C, 140 °C ............................. 88

Figure 5-10 Particle agglomeration in 1000 μm paint film containing hollow glass particles

deposited on glass substrates t<2 min ........................................................................................... 90

Figure 5-11 Marangoni cell formation in (a) 1000 μm model paint films curing at 140°C and

(b) 1000 μm Glycerin-butanol solution heated to 100°C ................................................................ 92

xi

Figure 5-12 Bubble agglomeration and escape in 1000 μm Glycerin-butanol solution on steel

substrates at T= 80°C .................................................................................................................... 94

Figure 5-13 Bubble cluster formation in 1000 μm paint film curing at 140° C, t > 8 min .......... 96

Figure 5-14 Corresponding velocity variation for individual bubbles in Figure 5-13 ................ 98

Figure 6-1 Dynamic of a floating bubbles in the vicinity of perturbed meniscus a second bubble

or a cluster of bubbles ................................................................................................................. 102

Figure 6-2 Bubble growth and attraction in 1 mm paint film curing at T= 120 ± 5°C. The solid

and dashed white circles are respectively 9.5 and 13 mm in diameter and identify the individual

bubbles selected for center-to-center separation measurements ................................................. 104

Figure 6-3 Bubble agglomeration in approximately 5 mm glycerin film at room temperature . 106

Figure 6-4 Schematic of the interface curvature by the presence of the wall when (a) the liquid

wets the wall (θ < π/2) and (b) liquid does not wet the wall (θ > π/2) ........................................ 107

Figure 6-5 Bubble growth and attraction to PTFE wall in 1 mm model paint layer curing at room

T= 120 ± 5°C ............................................................................................................................... 109

Figure 6-6 Bubble clustering due to interfacial curvature in 5 mm glycerin layer with floating

PTFE ring on the surface at room temperature ........................................................................... 111

Figure 6-7 Geometry of a bubble floating at a liquid-gas interface with a ring of contact of

radius b. Zc is the height of fluid at the ring of contact, φ the semi-angle subtended at the centre

of sphere by the circle of contact and ψ the liquid-bubble contact angle. The free surface is

inclined at an angle θ to the horizontal plane ............................................................................. 112

Figure 6-8 Variation of β as a function of α plotted from Table 1 in [81]. ................................ 114

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Figure 6-9 Experimental data of Figure 6-3 (bubbles being drawn to the clusters in the area

confined by the white circle) compared to the asymptotic solution for center-center distance of

two identical spheres of radius 0.6 mm and 0.55 with center-to-center distance of L0=4 mm, in

glycerin with σ=63.4 mN/m, μ=1.3 N-s/m2. .............................................................................. 118

Figure 6-10 Experimental data of Figure 6-2 (bubbles clusters in the area confined by the solid

and dashed white circles) compared to the asymptotic solution for center-center distance of two

identical spheres of radius 0.35 mm and with center-to-center distance of L0=3 mm in paint with

σ=26 mN/m, μ=10 N-s/m2 .......................................................................................................... 120

Figure 6-11 Time for cluster formation in paint as a function of constant bubble radius ......... 121

xiii

Nomenclature

Variable Unit

Ap Surface area m2

B Universal gas constant l kPa/K mol

b Bubble contact line radius m

C Concentration mol/lit

C Concentration -

Ci Initial volatile concentration -

Cw Saturation concentration at the bubble interface

mol/lit

C∞ Saturation concentration in liquid bulk

mol/lit

C∞ Volatile concentration above paint layer

-

∆C Concentration difference across the paint film

-

D Diameter m

Dv Volatile diffusivity in paint m2/s

Da Volatile diffusivity in air m2/s

d32 Sauter Mean Diameter m

E Energy Nm

F Force Kg.m/s2

Surface tension force N/m

Viscous shear force N/m

g Gravitational acceleration m/s2

J Volatile mass flux kg/m2s

k Thermal diffusivity m2/s

Kn Modified Bessel function of the first kind and of order n

-

L Bubble center-to-center distance

m

L Paint film thickness m

Lc Capillary length

m

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M Mass kg

P Pressure kPa

PG Pressure inside bubble kPa

P∞ Saturation pressure above liquid

kPa

R Bubble radius m

r Radial coordinate m

r Distance from the vertical axis

m

T Temperature °C

time s

U Velocity m/s

Vp Volume m3

x Vertical coordinate m

Z Height of the surface in the neighborhood of one bubble

m

Greek Letters

α Coefficient in mass transfer equation at paint surface

kg/m2 s

α dimensionless bubble radius -

β Dimensionless length -

δ Dimensionless length -

θ Surface inclination angle °

λn Eigenvalues -

µ Dynamic viscosity N.s/m2

Model paint density kg/m3

Volatile partial density kg/m3

σ Surface tension N/m

φ Semi-angle at bubble center

°

ψ Bubble contact angle °

xv

Dimensionless Numbers

Bi Biot number -

Fo Fourier number -

Marangoni number -

Hc Henry’s constant -

1

Chapter 1

Introduction

1. Automotive Paint Application

1.1 Motivation and Background

The paint applied on the body of an automobile not only has to look attractive but also serve as

a functional coating that resists corrosion and abrasion over a life of many years. Balancing both

appearance and functionality can present unique engineering challenges. Paint, which is typically a

polymer dissolved in a solvent, is sprayed on automotive components and then baked in an oven

where the solvent evaporates while the polymer forms cross-links and cures, forming a hard layer.

Though more than 50% of the total cost of an automotive assembly plant is dedicated to the paint

shop [1], only in recent years has painting begun to be given the same importance as other

engineering areas, with in-depth studies devoted to understanding painting technology.

Paint application is a very carefully controlled, multi-step process, since it is extremely

important to minimize repair-work on painted parts, which increases costs and processing time, and

also creates environmental concerns. Automotive manufacturing companies are among the largest

producers of toxic chemicals and volatile organic compounds (VOC) in the world, and spend large

amounts of money in treating waste, such as paint sludge that needs to be disposed legally [1] [2]

[3]. More than 60% of the pollution control costs for such industries are devoted to air emissions

control [4].

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Toda [2] has summarized the automotive painting process. The BiW (Body in White) arriving

from the welding line undergoes several cleaning steps prior to the application of coating layers

because surface contaminants such as steel mill oils, stamping lubricants, and welding sludge affect

the surface energy of the panel and consequently may mar the final appearance of the applied paint

films [1] [3]. Some of these steps are:

Aqueous cleaning using water and cleaning detergent to remove contaminants. This could

be done through spray and/or immersion cleaning.

After rinsing off the cleaning solution residue the panel goes through an iron (FePO4 and

Fe3(PO4)2) or zinc (Zn3(PO4)) phosphating process. The main purpose of this is to prevent

corrosion on the panel surface. It also increases paint adhesion by providing a rougher

surface, reduces the thermal expansion coefficient and therefore minimizes the build-up of

stresses at the paint/substrate interface in the event of expansion or contraction, and

neutralizes any alkaline residues.

Electro-coating or E-coat immersion is used to efficiently coat irregular geometries. An

electrical current converts the soluble polar ionic resin into a neutral non-polar, i.e.

insoluble, on the panels' surfaces, thus depositing a 25 µm thick E-Coat film on them.

Multiple paint layers are applied in the spray painting booths. The first film is a primer that

is applied to enhance the adhesion of base and clear coats. The base coat is the color coat

and the clear coat is a protective layer that prevents UV damage.

After baking the primer, the top-coat (which consists of both the base coat and the clear

coat) is applied providing the final paint film build-up.

3

Airless, air-spray or rotary gun methods could be used to atomize the paint. The transfer

efficiency of the paint, which is the fraction of sprayed paint that is deposited on the surface, is an

important parameter as it affects the amount of paint wasted and the quantity of paint sludge that

has to be treated. Deposition efficiency is sometimes improved by electro-statically charging the

paint particles as they leave the paint applicator. There has been extensive work in recent years to

optimize the movement of the robot to improve the efficiency of this step [5, 6].

The final thickness of the cured finish is approximately 100-150 µm. The combined layers of

the base-coat with a clear-coat over it offers protection against the weather, sunlight and chemicals

in the air. It also determines the appearance of the automobile, and therefore extra attention needs

to be paid at this stage to ensure that there are no defects in the paint layer.

There are three main categories of coating systems: water-borne, solvent-borne, and powder-

coating systems. In recent years the automotive industry has preferred to use water-based coatings

to meet ever-tighter environmental regulations. Destroying VOCs requires incineration at

approximately 750° C. However, water-based fluidizing medium imposes limitations on the

operation in the painting booth and powder coatings require higher curing temperature and a higher

film build to achieve the same quality as those of liquid coatings [1] [3].

After creating a uniform paint layer in the painting booths, spray painted components are

allowed to dry for 5-10 minutes (known as the “flash-off” time) before being placed in the oven to

dry, to reduce paint blistering [7]. During the flash-off stage the paint film can still flow and level

while air bubbles can escape, minimizing their density. The paint layer will completely set after

being cured in a convection chamber. The component needs to be exposed for at least 30 minutes

to a temperature of 120-125°C for complete drying/curing. This process consumes both time and

4

energy, so typically infra-red radiation is used for 5 minutes after allowing 2 minutes of flash off,

followed by 20 minutes of curing in a forced convection oven.

Due to the large number of controlling factors, defects may be traced back to different coating

steps: paint application, levelling, flash-off, and curing. If the spraying process is not done properly

the paint layer obtained may be unacceptable, typically because the surface is uneven or the paint

does not adhere well. Surface defects either arise because of non-uniform paint application or are

created by gravitational or capillary forces acting on the paint film after it has been deposited on

the surface.Many defects are found to be related to a number of factors including the interfacial

properties of the paint and substrate, and surface and volume forces that arise during solvent

evaporation and curing [8, 9, 10, 11]. Therefore, the stability of the film during curing is of utmost

importance.

1.2 Previous Research on Paint Defect

Interfacial properties become very important in thin films because of their large surface-to-

volume ratio. Surface tension variations along the surface will create tangential forces, setting the

surface into motion and causing film instabilities. Surface tension variation arises from

concentration and/or temperature gradient and in some cases from electrical potentials [12]. The

resulting flows are called solutocapillary, thermocapillary, and electrocapillary respectively and

are sustained as long as sufficiently large temperature and/or composition gradient exist.

When paint is sprayed on a surface, a large number of air bubbles may be entrained by

impacting droplets and trapped in the deposited layer, creating serious defects during drying.

Bubbles act as nucleation sites into which evaporating solvent diffuses, making them grow until

they burst through the paint surface and create visible blisters and pinholes [13]. In a curing liquid

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if the surface energies do not reach equilibrium, or bubbles do not all escape before solidification,

these structures would become defects. Some of the most common defects arising from film

instability are listed in

Figure 1-1 [14, 15].

(a) (b) (c) (d)

Figure 1-1 Paint defects (a) Cratering, (b) Orange Peel, (c) Wrinkling, (d) Bubble formation/ Solvent Boil [14, 15]

Curing of a thin layer of thermosetting paint involves solvent evaporation and polymer cross-

linking, both of which causing spatial temperature and composition gradients. Evaporation causes

surface cooling and solvent concentration variation, while solvent loss and cross-linking both

promote concentration gradients. Non-uniform evaporation and/or cross-linking create regions

with lower temperature and lower solvent concentration. These regions have higher surface-

tension, higher density and viscosity compared to their surroundings. Surface tension and density

gradients generate oscillatory flows across and normal to the paint surface, opposed by viscous

shear forces, causing film instabilities such as substrate de-wetting, bubble motion and self-

organizing patterns.

1.2.1 Craters

Craters (

Figure 1-1 (a)) are depressions in the paint layer, mainly created as a result of surface tension

gradients or de-wetting of the substrate by the paint when the viscosity is sufficiently low to allow

6

fluid motion. Low surface tension areas could be linked to contaminations such as dirt and resin

gel particles, hydrocarbon and fluorocarbon oils and lubricants, or silicon residue [16, 17]. Craters

are also initiated by convective cells (Marangoni-Bénard cells) or bubbles trying to escape the

paint layer and break the surface in the process. When fresh paint with lower surface tension is

brought to the surface due to convection, areas with lower surface energy are created. Because

liquid always flows away from these regions, depressions will occur on the surface. A back-flow

that can fill this depression may not start until later in the curing process, by which time the high

viscosity of the paint will inhibit leveling and leave a crater in the paint layer. Crater diameters

vary from 0.1 to 5 mm and they form on a time scale of 0.1 to 3 s.

Evans et al. [16] developed a mathematical model for two-component paints to predict the size

and structure of craters created by a surfactant being released on a region of the surface. In an

earlier study [18], Bierwagen qualitatively described the time-dependence spreading of a lens of

fluid on a liquid surface.

There have been a number of experimental studies on the formation of craters. Torkar [19]

traced the crater and pinhole defects on a galvanic zinc coated car body to gas trapped in a crack

in the zinc layer. Gaver and Grotberg [20] studied the behavior of glycerin film rupture in the

presence of a surfactant (oleic acid) by using dye markers to visualize flow. Film thickness was

varied and different flow patterns observed. In the case of low film thickness a constant outflow

due to surface tension gradients created by the low surface tension substance was detected. Fraajie

and Cazabat [21] [22] observed the effect of surface tension gradient in spreading of mono-layers

on precursor films. Schwartz et al. [22] presented experimental and mathematical results for de-

wetting patterns in a drying thermosetting paint layer, using a disjoining-conjoining pressure

model that took into account substrate energy. Weidner et al. [23] developed a numerical model

7

taking into account the effect of evaporation, convection, diffusion of solvent, and the resulting

surface tension variations on the time-dependent evolution of the coating thickness. In a recent

study by Zuev [24], solutocapillary flow was created by depositing surface-active soluble droplets

on liquid substrates. The resulting substrate drainage was investigated and the critical thickness at

which film rupture occurred was determined for various liquid pairs. The results showed that the

critical thickness was independent of volume of the surfactant, or viscosity and density of the liquid

substrates; the key influence was the surface tension gradient imposed on the liquid surface.

Therefore both thermocapillary and solutocapillary flows would create film instabilities causing

dewetting that could lead to craters.

1.2.2 Bénard-Marangoni Convection, Orange Peel and Wrinkling

One important aspect of paint finish is the waviness of the surface and what is known as “orange

peel” (

Figure 1-1 (b) and (c)), defined by an ASTM standard as “the appearance of irregularity of a

surface resembling the skin of an orange” [25]. The mechanism by which orange peel appears on

a dried paint surface is not very well understood, since it may be either due to incomplete leveling

of the paint surface before curing, or because of flows created in the paint during the drying process

that make the flat paint surface uneven. The wavelengths of the regular undulations vary from 0.1

to 30 mm and their amplitude is between 0.5-5 μm. Both light reflection optical and mechanical

sensing methods can be used to evaluate the surface profile and determine and orange peel rating

and determine if the finish is acceptable. By appropriately filtering the profilometer data the same

results could be obtained [26, 27]. Visual comparison with standard panels that are commercially

available (for example, P/N 34134 by ACT Laboratories Inc., USA) is the common method of

inspecting cured automotive components.

8

The simplest of the self-organizing patterns (Marangoni-Bénard cells) were observed over a

century ago. Since 1900s, these patterns, observed in a thin fluid layer heated from below and

cooled from above, has been the subject of many studies. Due to the nature of these phenomena it

is an interesting field of research whose results have been applied in many industrial applications.

In coating applications, the goal is to achieve a smooth final finish; however one of the most

common defects is orange peel which is found to be connected to surface irregularities induced by

surface-tension-gradient (STG) which are observed as soon as the curing/drying process begins.

The first systematic investigation was performed by Bénard [28]. He introduced a temperature

difference of about 80° C across a shallow pool of liquid with large aspect ratio of approximately

200. The fluid motion was visualized using fine particles and the regular stable hexagonal cells

were observed by shadowgraph technique. Although the origin of these patterns was not related to

surface tension, Bénard's experiments provided a basis for future work on this subject. Rayleigh

examined Bénard's experiments by approaching the problem as a stability problem, but as the

vertical velocity component and temperature disturbance vanish at the top and bottom boundaries,

this analysis only referred to convection caused by buoyancy [29]. Bénard's problem was reworked

by Pearson [30] who concluded that surface tension variations due to temperature variations were

responsible for these patterns. Nield [31] validated through numerical results that buoyancy and

surface tension are closely linked in forming convection cells, which are approximately the same

size whether formed due to surface tension or buoyancy. There have been many studies on this

subject concerning thermocapillary flow and the onset of convection [32, 33, 34, 35].

Surface tension forces must overcome viscous forces for flow instabilities to be initiated, for

which a critical surface tension gradient, caused by temperature or concentration variations, must

9

be reached. The flow pattern that is created has a unique wavelength (λ), defined by Bénard as the

centre-to-centre distance of two neighbouring cells, and is independent of the fluid depth (L).

Flows caused by a temperature difference ∆T imposed across a liquid film of thickness L are

characterized by the Rayleigh (Ra) and Marangoni (Ma) numbers, which are functions of liquid

layer thickness L, kinematic viscosity ν, thermal diffusivity k, surface tension σ, temperature

difference (∆T), liquid thermal expansion coefficient β, and gravitational acceleration g [34]:

≡ 1-1

≡/

1-2

The dynamic Bond number, which is the ratio of Ra to Ma determines the strength of buoyancy

to thermocapillary forces:

≡ 1-3

When the thickness of the paint film is reduced and L becomes smaller, Ra diminishes much

faster than Ma, leaving surface tension forces dominant.

Past studies have mainly focused on flow visualization and temperature gradient as the driving

means of surface-tension driven flows. Koschmieder and Prahl investigated the effect of aspect

ratio on the shape of typically hexagonal cells, in small circular and square containers [36]. They

reported an increase in the number of cells as the fluid depth was decreased and a steep increase

in Marangoni number with decreasing aspect ratio. Dutton et al. [37] used crystal thermography

to acquire digital images of convective cells and determine the temperature field across the silicon

10

oil film. Critical Marangoni numbers were evaluated and compared with values predicted by

uniform-flux and uniform-temperature cases.

In a recent study by Cerisier et al. [38], surface deformation was visualized via interferometry,

and IR thermography was used to record the temperature field. To simplify the theoretical

modeling, air is frequently chosen as the gas layer above the liquid, which makes it possible to

neglect the properties of the gas. However, in this study, the authors used both air and helium and

examined the effect of physical properties such as vertical temperature gradient, aspect ratio, Biot

number, and Prandtl number on pattern dynamics. Through these complementary measuring

techniques the authors confirmed the findings of previous studies for pattern transition with

different temperature gradients. A more dynamic pattern was observed as the Biot number

increased, but the reverse was observed with increasing the Prandtl number.

Weh [39] created radial temperature gradient using a point heat source below a polymer layer

and complicated patterns were visualized. Temperature gradient takes place during evaporation

due to heat loss from the exposed surface, but few studies have taken into account evaporation

effect as the driving mechanism for Marangoni cells.

Chai and Zhang [40] created a negative temperature gradient in 12 different evaporating liquids

using an aluminum cooling base. Two thermocouple groups were used to monitor the temperature

profile inside the liquid layer and the upper air layer. They derived a modified form of Ma from

Navier-Stokes dimensionless small perturbation equations to characterize the onset of Bénard

instabilities in pure liquids.

∗.

1-4

11

In the equation above, V. denotes volume evaporation rate, Δhv is the enthalpy of vaporization,

and Cp is the specific heat. The authors claimed that the modified Marangoni number to be an

adequate indication of Bénard convection. Abbasian et al. [41] investigated the effect of

evaporation on convective cell formation in various solvents. They reported critical Marangoni

number for cell formation combining the traditional Marangoni number and the modified

Marangoni number from Chai's work [40].

∗ .

1-5

An experimental analysis by Vinnichenko et al. [42] shows different flow regimes for

evaporating water and ethanol layers. They used a Background Oriented Schlieren technique to

determine the temperature field below liquid-gas interface and Infrared Thermal Imaging

techniques to determine liquid surface temperature. They reported a "cool skin" regime for water

and Marangoni convection for ethanol. The dissimilarity was due to ethanol being highly volatile

and having low surface tension. The results of the numerical simulation corresponded to the non-

slip boundary condition for water surface, whereas a vertical velocity gradient due to surface

tension variation relates to the surface of evaporating ethanol.

Surface tension variations can also arise from concentration gradients in a thin evaporating

multi-component film if one component evaporates faster than the others. Random variations in

the surface evaporation rate can create surface tension variations that drive liquid flow from

regions of low to high surface tension. Hansen and Pierce [43] attributed the existence of undesired

phenomena such as pigment segregation in polymer coatings to formation of cellular convective

cells. Kollner et al. [44] observed the evaporation of cyclohexanol/water mixtures and

12

photographed the appearance of cellular structures. Uguz and Narayanan [45] used an analytical

model of fluid instability to show that the occurrence of solutally-driven Marangoni convection

depends on the direction of heating and that the resulting patterns can be made to disappear by

adjusting the liquid and vapor heights. Zhang et al. [46] observed various patterns forming in

butanol-water mixture by varying the initial volatile mass content. Harris and Lewis [47] produced

a variety of patterns in evaporating colloidal films by changing the initial volume fraction of

colloid. Chen et al. [48] reviewed recent progress in solute-driven Marangoni convection in liquid-

liquid systems and discussed how it influences mass transfer. Schwarzenberger et al. [49]

identified three types of convection patterns and showed that interactions between these can create

complex, unsteady behavior.

Despite the large number of experimental and analytical studies, there is little work on surface-

tension-driven flows that are created due to concentration gradients in drying paint layers, which

are important in coating and painting applications.

1.2.3 Paint Drying and Leveling

Paints typically consist of a resin dissolved in a solvent. As the paint dries in an oven the solvent

evaporates, creating strong concentration gradients in the paint layer. Since surface tension

changes with solvent concentration, Marangoni flows are created in the paint film. Then, as the

resin cures, irregularities on the paint surface are frozen in place, creating the orange peel effect.

The relative magnitudes of the timescales for complete diffusion of volatiles (tD) out of the paint

film and the time for paint to cure (tC) play an important role in determining if orange peel occurs.

Additionally, in thin paint films, the substrate roughness or the texture of under layers may

contribute to orange peel as the paint shrinks and conforms to substrate texture. In

13

basecoat/clearcoat systems, the different curing rates of coatings may create residual strains during

curing and could influence the final surface profile [50] [51].

The film drying process consists of different stages as summarized by Yoshida [52]. Based on

this study the rate-determining steps in pure solvents are diffusion through boundary layer and

latent heat of vaporization. But in drying paint films, in addition to previous steps, the change in

solvent composition plays an important part.

In an experimental investigation, Brinckmann and Stephan [53] used flat and z-shaped

electrophoretic coated steel as proxy for different car body parts. The experimental apparatus

consisted of two main sections: a conditioning unit which controlled humidity and temperature; a

drying chamber with various air flow configurations in which weight loss and component

temperature were measured. The range of temperatures and online weight measurements of

samples suggests more rapid weight loss and temperature increase with higher air velocity. The

authors also specify an inflection point, which indicates a two-stage drying process: evaporation

followed by diffusion.

In industrial coatings, it is important to have a flat, smooth surface after paint application and

curing. It is known that even adjusting the coating viscosity cannot prevent the existence of ripples.

Accordingly, previous studies of the levelling of the paint layer have attempted to predict a time-

dependent relationship for the decay of these instabilities. However, the theoretical models

developed were applicable only to solutions with no volatile components, with Newtonian

characteristics, and homogeneous surface tension throughout drying.

Kojima et al. [54] prepared waterborne paint coatings with and without co-solvents applied by

a block-shaped knife edge to tin-free steel surfaces. The block shape made it possible to create an

14

initial rippled layer. A slit beam was reflected from the observation plane and projected on a screen

to analyse the levelling process. The ripple amplitudes of different coatings were recorded, and a

negative (reappearance) ripple amplitude was observed for a waterborne coating containing n-

butanol and solvent-borne counterpart. The authors tested these results with three different

theories, but none could predict the reappearance of ripples, although the amplitude decay of

solvent-borne coatings agreed with the findings of the previous theory. Only one theory takes into

account the effect of evaporation on paint leveling, but even this theory cannot predict the decaying

behaviour of water-borne coatings because the curing of coatings involves polymerization and

cross-linking, which are controlled by numerous chemical and physical parameters.

Basu et al. [55] proposed a mechanism to understand the chemical and mechanical phenomena

associated with curing process. Five different paint systems were examined, having various

chemical formulations, delivery methods, and curing procedures. An automated drawdown

machine was used to create a uniform thickness of powder or liquid coating on a glass substrate,

and the coated surfaces were baked in a convection oven or cured with a UV lamp at room

temperature under a nitrogen atmosphere. Some of the samples were removed from the oven in

the intermediate stages of curing to study the procedure of wrinkle formation via video

microscopy, and the one-dimensional topology of the cured surface was recorded utilizing a stylus-

based mechanical profilometer. Each paint system had a different wrinkle pattern, wavelength, and

amplitude. The curing chemistry of all the systems was found to be highly cross-linked and to have

a depth-wise gradient in terms of solidification, although the cause of the gradient was different

for each system. Based on these similarities, a hypothesis for wrinkle formation was developed

and experimentally validated. Partially but uniformly cross-linked samples were soaked in an

oligomer bath to test the theory proposing oligomer absorption as the cause of surface swelling;

15

pressed elastic films on viscous sub-layers were used to support the theory of wrinkling by

compressive stress.

Weh [56] photographed the appearance of diverse surface structures such as wrinkles, spirals

of hyperbolic type and fractals due to solvent evaporation from thin organic layers. He found that

adding different amounts of surfactant affect their structure. Another study by Weh and Venthur

[57] shows the evolution of fractal-like surface disturbances in a PAN membrane as it hardens.

They learned that thickness differences, convection in the gas phase and high air humidity

promotes these structures.

In the past few years there has been increased interest in studies on polymerization kinetics to

obtain and control certain surface patterns. Vessot et al. [58] used DSC, TGA and FTIR devices to

characterize and analyze the solvent loss and polymerization kinetics of model car paint. However,

these kinetics has not been included in the studies corresponding paint defects. In a study by

Chandra [59] the wrinkle amplitude has been changed by controlling the oxygen content of curing

condition. Basu et al. [60] have developed a theoretical model that predicts the generation of the

surface skin, taking into account the in depth concentration gradients due to evaporation and cross-

linking reaction. However, this model was not able to predict wrinkle wavelength as a function of

solidification time.

Surface tension gradients play a significant role in creating different types of patterns,

especially in curing thermosetting polymer coatings. Small changes in temperature and

concentration will give rise to formation of diverse surface patterns.

16

1.2.4 Bubble Migration, Bubble Entrapment, and Blistering

Decreasing viscosity and allowing a longer flash-off time will help with levelling the paint film

and will allow bubbles to escape. This will reduce a number of common defects such as pin-holes,

blisters and entrapped bubbles (

Figure 1-1 (d)). On the other hand, this could cause another defect called sagging, which is

more prominent on inclined surfaces. Although film thickness is found to be of minor importance

in cratering, thin films reduce formation of bubbles while encouraging Marangoni convection

cells.

Numerous experimental studies have been carried out that report the formation and entrapment

of an air bubble under an impacting droplet both on solid and liquid surfaces [61, 62, 63, 64].

Chandra and Avedisian [61] photographed the impact of a liquid drop on a solid metallic surface

and a liquid film. They observed the formation of a single bubble within the droplet during the

impact. Mehdi-Nejad et al. [62] numerically investigated the effect of viscosity, velocity and

contact angle on bubble entrapment by simulating the droplet impact for water, n-heptane and

molten nickel droplets. Researchers in [63] used ultrafast x-ray phase contrast imaging to visualize

the evolution process of the air film into a bubble. Thoroddsen et al. [64], experimentally

investigated the evolution of an air disc trapped under an impacting drop onto a solid surface, using

a high speed camera. They measured the contraction speed of the disk as it shrank into a bubble at

the droplet center.

Over the past decades there have been well-documented experimental and analytical studies

on thermocapillary flow and bubble motion [65, 66, 67, 68]. Young et al. [65] proved that

imposing a sufficiently strong temperature gradient in a vertical column of liquid would prevent

17

the bubble from rising. They derived an expression to predict the magnitude of temperature

gradient to hold the bubble stationary. Theoretical studies of this system has also been carried out

[65, 66]. Some direct measurements of bubble forces have also been reported by McGrew [67]

and general agreement with theory was indicated.

According to Zuev et al. and Birikh et al. [69, 70], solutocapillary convection around a bubble,

although similar to thermocapillary convection, shows some dissimilarities. This is due to

differences in mass and thermal diffusion time scales, causing concentration gradients to exist

longer than temperature differences. The growth of vapour bubbles in liquids has been studied by

many researchers [71] [72], but there are very few studies of this phenomena in relation to bubbles

in drying paint layers.

Numerous analytical studies have looked at the problem of diffusion-controlled bubble growth

in the absence of mass transfer [73, 74, 75]. The collapse or growth of the bubble is correlated to

the pressure difference across the bubble boundary. Epstein and Plesset [72] first presented

approximate solutions for the rate of gas bubble growth in a liquid-gas solution neglecting the

translational motion of the bubble and hydrodynamic effects. Analysis of the bubble growth by

mass-diffusion in a viscous liquid was first studied by Barlow and Langlois [76] and later studied

by several other investigators [77, 78, 79]. Barlow et al. [76] coupled the diffusion equation to the

equations of viscous hydrodynamics using thin shell approximation around the bubble to

determine the bubble size. Thin shell approximations were used by Plesset and Zwick [74] in the

case of vapor bubble growth by thermal diffusion. Barlow and Langlois [76] obtained solutions

for two extreme cases of bubble growth behaviour at early times when the bubble grows slowly,

and also long after growth has begun. They also developed a numerical solution in the range where

neither the initial solution nor the asymptotic solution apply. Han and Yoo [77] developed a model

18

assuming a thin boundary layer and infinitely dilute solute to predict bubble growth. Exact

solutions were derived for diffusion-controlled growth of spheres using constant property models

using effective diffusivity values in [78]. Venerus et al. [79] formulated a rigorous model of

diffusion-induced bubble growth and showed how previously published models can be derived

from this. The mutual attraction of bubbles or particles floating on the surface of a liquid is well

known, leading to the formation of “bubble rafts” [80] [81]. An analytic model of bubble motion

[80] shows that the force of attraction increases as bubbles get closer.

There are very few studies on the effect of bubble migration and growth in relation to bubbles

escaping or stabilizing in drying paint layers. Domnick et al. [13] developed a physical model

incorporating the mass loss and heat transfer of a painted car body by water based coating and the

results were compared with experimental data. They developed an equation to estimate pinhole

density. In another study [82] the effect of paint application parameters, film build and dehydration

temperature have been considered to investigate solvent boil. The authors identified the most

significant of these parameters but have not been able to suggest conditions for a defect free final

coat.

Bragg [83] developed a model of bubble rafts with uniform bubble size to visualize simulate

imperfections found in crystal and polycrystal lattices. Nicolson [80] derived a dynamic model to

estimate the potential energy of two similar bubbles next to each other. Shi and Argon [84] extend

the results of Nicolson to obtain the attractive force between two bubbles of different radii to

evaluate the energies of transforming cluster configurations. Formation of three dimensional arrays

of air bubbles in a polymer film through evaporation cooling and thermocapillary templating

mechanism was reported by Srinivasarao et al. [85]. Kralchevsky and Nagayama [86] have given

19

a comprehensive review on theoretical and experimental results about lateral capillary forces and

particle structuring.

1.3 Thesis Objectives

The objectives of the research carried out were as follows:

Understanding of the flow dynamics during curing of paint films.

Investigate the effect of film thickness on the onset and evolution of surface-tension-driven

flows (Marangoni convective cells) and severity of “orange peel defect”.

Determine the origin of bubbles in a paint layer and investigate bubble clustering in binary

mixtures in various stage of curing.

Develop an analytical solution of the mass-diffusion equation to model solvent evaporation

from the paint film, calculate the magnitude of concentration gradient and surface tension

variations, and estimate bubble growth rates.

Study mutual attraction of bubbles and predict bubble center-center distance using an

analytical model of bubble motion, showing the force of attraction.

1.4 Thesis Organization

The thesis is organized as follows. Chapter 2 explains the details of the experimental

methodology and illustrates the schematic arrangement of the experimental system. The material

used in the model paint formulation is reported and the physical properties of the material is

described.

20

Chapter 3 reports the results of experimental investigation in which the formation of some of

the most common paint defects in the automotive industry were visualized. The effect of curing

parameters, surface roughness, flash-off time and film thickness were demonstrated. The results

will help us to determine the underlying mechanisms in film instability and defect formation.

Chapter 4 focuses on visualization of orange peel defect formation and its connection to

Marangoni convection is studied. The onset of Bénard-Marangoni cells is photographed and

weight loss of solvent during curing is measured. An analytical solution of the mass-diffusion

equation is used to model solvent evaporation from the paint film surface and to calculate the mass

diffusivity of the solvent. The model allows the prediction of the concentration gradient and

therefore surface tension variations in the paint films. The waviness of the dried paint surface is

quantitatively studied by means of mechanical profilometry.

Chapter 5 experimentally and analytically studies the relative motion of bubbles driven by

Marangoni convection and buoyancy, and their growing rate due to diffusion of gaseous products

of polymerization and evaporated volatiles. Analytical solutions of the mass-diffusion equation were

used to model solvent evaporation from the paint film surface, calculate the magnitude of

concentration gradients, and estimate bubble growth rates.

Chapter 6 experimentally and analytically investigates the mutual attraction of bubbles,

formation of bubble rafts, and attraction of bubbles to a wall in a drying paint layer. The growing

rate of bubbles were taken into account while the force of attraction was analytically studied using

a model of bubble motion.

21

Chapter 2

Experimental Methodology

2 Experimental system

2.1 Model Paint Formulation and Physical Properties

Initial experiments were conducted with a melamine based industrial paint (Model UREGLOSS

CW R10CG062A, BASF, Canada) with a density of 1002 kg/m3, viscosity of 240 cP, and surface

tension of 26 mN/m. However, as the complex mixture of solvents and resins posed difficulties in

characterizing the phenomena observed, a model paint with known components was developed to

overcome this problem. The primary components used to formulate the model paint were:

CYMEL 1159 from Cytec: butylated melamine P/W formaldehyde resin

PARALOID AT-400 from DOW chemicals: hydroxyl-functional thermosetting acrylic

Normal Butanol from CALEDON Laboratory Chemicals

A model paint formulation was developed for experiments that consisted of 85 wt% resin and 15

wt% solvent (normal butanol, CALEDON Laboratory Chemicals, ON, Canada). The resin

composition contains 70 wt% butylated melamine P/W formaldehyde (CYMEL® 1159 Resin,

CYTEC, NJ, USA) and 30 wt% hydroxyl-functional thermosetting acrylic (PARALOIDTM AT400

Resin, DOW Chemicals, PA, USA). According to Sharmin et al. [87], melamine resin has poor

solubility in water, dimethyl sulphoxide, glycols and glycerine. But synthesis of acrylic melamine

overcomes the solubility problem and the solubility tests showed up to 40% solubility in butanol,

methanol, and ethanol. n-butanol is a suitable solvent for many finishes including melamine resins

22

and is widely used in the coating industry. The properties of the paint at room temperature were:

density (ρ) 988 kg/m3, viscosity 240 cP, and surface tension 26 mN/m. The resin also contained

butanol and small amounts of methyl n-amyl ketone solvent, so a thermogravimetric analysis

instrument (Model SDT Q600, TA Instruments, New Castle, USA) was used to determine the total

solvent content of the model paint. Paint samples, weighing less than 15 mg, were heated with a

10°C/min ramp from room temperature to 250°C. This heating cycle was repeated after cooling

the sample to ensure evaporation of all the volatiles and the results showed approximately 45%

solvent content in the model paint composition (see Figure 2-1).

Figure 2-1 Thermal analysis on model paint sample

Therefore, the initial concentration of butanol in the model paint, Ci, used in calculations was

0.45. The model paint is formulated such that it replicates the physicochemical properties of single-

23

component, solvent-based, pigment-free automotive clear coat intended for application over a

cured base and primer coat. This paint contains 30 wt% butylated Melamine Formaldehyde resin

and a complex mixture of volatiles including n-butanol, trimethylbenzene, and many others. The

model paint was developed to have viscosity, surface tension, and curing temperature similar to

that of a commercial clear coat [88] that was analyzed using Fourier transform infrared

spectroscopy (FTIR), thermo-gravimetric analysis (TGA), and differential scanning calorimetry

(DSC).

To measure the variation of surface tension and viscosity with concentration [88], solvent content

was incrementally increased by adding 2% wt n-butanol to the pure resin composition (70% wt

Melamine and 30% wt Acrylic resin). Following each step of solvent increase the viscosity of the

model paint was measured using a digital viscometer (Model DV-I Prime, Brookfield Engineering

Laboratories, Middleboro, MA, USA). A force tensiometer (Model SIGMA 700/701, KSV

Instruments Ltd, Helsinki, Finland) was used to measure the surface tension of the model paint.

Figure 2-2 and Figure 2-3 illustrates the experimental data obtained. The surface tension decreased

with increasing solvent concentration, from 0.0275 N/m for the undiluted resin to 0.0235 N/m for

pure butanol. The viscosity also showed a sharp decrease, from 3700 cP for the resin to 2.6 cP at 25°

C for pure butanol [89].

24

Figure 2-2 Surface tension variation with concentration

Figure 2-3 Viscosity variation with concentration

25

Glycerin-butanol solutions were also used as a viscous non-solidifying liquid, containing 70 wt%

Glycerol (Glycerol, CALEDON Laboratory Chemicals, ON, Canada) and 30 wt% solvent (normal

butanol, CALEDON Laboratory Chemicals, ON, Canada).

To study bubble motion, transparent glass vials (66011-085, VWR International, USA) were

filled three quarters full with model paint or glycerin-butanol mixture and shaken in order to

produce bubbles within the liquid prior to spreading the thin paint layer on test substrates. Particle

migration within thin liquid layers was further investigated by mixing glass tracer particles

(Hollow Glass Microspheres 0.06 g/cc 150-180 um, Cospheric, USA). Convective velocities in

the paint during curing were determined by conducting Particle Image Velocimetry experiments,

in which fine polymer spheres (Orange Polyethylene Microspheres 1.00g/cc 45-53 µm, Cospheric,

USA) were distributed in the paint.

2.2 Experimental Apparatus

Figure 2-4 shows the schematic arrangement of the apparatus used to apply uniform paint

films. Mirror-polished stainless steel discs (51 mm diameter) with roughness less than 0.3 μm, or

heat-resistant borosilicate glass substrates (Model 8477K78, Mc-MASTER-CARR, USA),

63.5 mm in diameter with 3.2 mm thickness, were used as test surfaces. They were cleaned with

acetone to remove dust particles and any oil or silicon residue. The substrates were placed

horizontally on a metal plate resting on a 1-D motion stage. A film applicator (Model Multicator

411, Enrichsen GmbH & Co, Hemer, Germany) was used to create uniform paint layers on

substrates. It consists of a knife edge whose height above the substrate can be adjusted in the range

of 0-1000 µm by means of a micrometer screw with an accuracy of ±1 µm. Substrates were placed

on a motion stage and a line of paint deposited on them with a syringe. The substrates were then

26

drawn under the blade of the applicator by moving the stage to spread the pain and create a uniform

paint film. The paint thickness was then verified through measuring the weight and calculating the

height using the density of paint at room temperature and surface area.

Figure 2-4 Schematic arrangement of coating system

The primary experiments were conducted using both conduction and convection heating methods.

The conduction heating system consists of a metal heater plate used for directly heating the

substrate. The temperature of the heater was controlled by a bench-top temperature controller (Model

MCS-2110K-R, OMEGA, Canada) at 130° C and the paint layer was cooled by the ambient air.

Figure 2-5 shows the system devised for this method.

27

Figure 2-5 Schematic of the experimental set-up used for conduction curing

Figure 2-6 shows the schematic arrangement of the convection drying system. Heated air was

circulated in a metal chamber with a glass top to cure paint samples. Air flowing through a copper

tube was heated by wraparound heating cords (Model HTC-030, OMEGA Engineering, Canada).

A pipe-plug thermocouple probe with ¼” NPT fitting (Model TC-K-NPT-G-72, OMEGA

Engineering, Canada) connected to a bench-top temperature controller (Model MCS-2110K-R,

OMEGA, Canada) was used to control the air temperature at the inlet to the metal chamber. The

air temperature inside the metal chamber was maintained at approximately 130° C and was

constantly monitored using a handheld thermometer (Model HHM 290, OMEGA, Canada) to

avoid large temperature fluctuations.

28

Figure 2-6 Schematic of the experimental set-up used for convection curing

The convection curing system was modified to monitor sample weight variation while curing was

taking place. Coated substrates were placed inside a metal chamber (with less than 15 seconds

delay after applying the coating) that was used as a convection oven to cure paint samples (see

Figure 2-7). A 750 W band heater (Model HB-5075/240V, OMEGA Engineering, Quebec,

Canada) regulated by a bench-top temperature controller (Model MCS-2110K-R, OMEGA

Engineering, Quebec, Canada) was placed inside the chamber, surrounding the substrate, and used

to elevate the temperature of the air to up to 200°C. Air circulation in the chamber was kept at

minimum with a velocity less than 2 m/s at the inlet to the metal chamber to help carry the fumes

out to the exhaust fume hood and maintain only natural convection in the vicinity of the drying

sample. The heater surrounding the test surface prevented any forced convection flows over the

paint. Substrates were placed on a support rod that passed through the bottom of the chamber and

rested on an analytical balance (Model E01140, OHAUS Corporation, Parsipanny, USA) which

recorded their weight with a resolution of 0.1 mg every 60 s. The chamber had a glass top through

29

which the substrate could be viewed and still images of the paint surface, with a resolution of

1280 x 1024 pixels, were taken at 2-4.2 seconds intervals using a camera system (Model SensiCam

Optikon PCO, Cooke Corporation, Germany).

Figure 2-7 Schematic of paint drying system

Pictures of liquid layers with growing bubbles were analyzed using the threshold function in

image analysis software (ImageJ, National Institute of Health) to count the number of bubbles in

each image, the cross-sectional area of each bubble, the location of individual bubbles, and the

distance between bubbles as they formed clusters.

The pictures were also analyzed using the iterative PIV function in image analysis software

(ImageJ, National Institute of Health) to obtain a velocity vector field by cross-correlating two

successive images, approximately 2 s apart, and measuring the displacement of individual

30

particles. At each time step the magnitude of individual vectors was divided by the time elapsed

to give the velocity.

31

Chapter 3

Film Instability and Defect formation in Drying Paint films

3 Defect Formation in Drying Paint films

3.1 Introduction

Paints are applied on surfaces not only because they are attractive, but also because they serve

to protect the coated part from the environment. Therefore, the stability and wetting properties of

the liquid film while it is being applied and cured are of utmost importance to prevent defects that

leave the substrate exposed.

This study was conducted to observe the formation and evolution of some common defects

such as craters, entrapped bubbles, wrinkles, and orange peel effect. The paint provided for this

part of the study by General Motors, was Model UREGLOSS CW R10CG062A, BASF, Canada.

Various steel substrates with different surface roughness were used.

3.2 Experimental System

A melamine based industrial paint with the physical properties in section 2.1, was used to coat 63

mm diameter, 2 mm thick, steel substrates with roughness varying from 0.3 to 1.3 µm. Target

substrates were cured using conduction or convection heating methods (as explained in section 2.2)

at approximately 130 ± 5° C while still images at 1280 x 1024 pixel resolution and 4.2 seconds time

interval, were taken using a camera system. The camera (Model SensiCam Optikon PCO, Cooke

Corporation,Germany) was operated at 7.6 frames per second (fps) and recording images with

1280 x 1024 pixel resolution by averaging 8-32 frames per image. Varying the averaging value

32

provided various time delays between successive images. The average initial thicknesses were

assumed to be constant over the whole substrate area and calculated by simply measuring the

volume of liquid on the substrate and dividing by the substrate area.

3.3 Results and Discussion

As the paint dries in an oven the solvent evaporates, creating strong concentration gradients,

viscosity and density variations in the paint layer. Since surface tension changes with solvent

concentration, Marangoni flows are created in the paint film. Then, as the resin cures and viscous

shear forces become stronger, irregularities on the paint surface are frozen in place, creating various

defects.

3.3.1 Craters

Figure 3-1 shows an image sequence of the surface of a 2 mm paint film curing using conducted

heat to the outer surface of the substrate as per the experimental method shown in Figure 2-5. The

average film thickness was calculated by dividing the dispensed paint volume (determined from the

increase in weight of the coated substrate) by the substrate surface area. The low surface roughness

of 0.3 μm and several minutes of flash-off time minimized air entrapment in the liquid-solid

interface. Time t=0 corresponds to about 10 seconds after the start of curing. The film surface

appeared smooth for the first 5-7 minutes of curing, but as drying continued bubbles, which were

not attached to the substrate, started to form and rise to the surface. The appearance of these bubbles

may be explained by the sharp temperature increase of the substrate when it was placed on the

heater plate, which caused solvent boiling so that additional bubbles nucleated and broke through

the liquid surface.

33

As solvent evaporates from the surface of the paint there is a concentration gradient created,

with high solvent concentration at the bottom of the paint film near the substrate, and low

concentration at the surface exposed to air. When fresh paint with lower surface tension is brought

to the surface due to convection, areas with lower surface energy are created. Because liquid always

flows away from these regions, depression will occur on the surface. This can be seen clearly at

t > 960 s in the two locations marked by arrows. A slight back flow recoated some of the de-wetted

area, but due to rapid drying and/or polymer cross-linking in the region, the ruptured surface was

set in place after 40 seconds for the first crater and only 14 seconds for the second crater.

34

0 s 988 s

560 s 1000 s

960 s 1180 s

965 s 1185 s

980 s 1294 s

Figure 3-1 Crater formation in 2 mm paint layer using conduction heating at approximately 130 ± 5° C. Time t = 0 corresponds to 10 seconds after start of curing. De-wetting initiated at t = 960 s; One more site grew and can be seen at t = 1180 s. Craters are frozen in place at t > 1294 s

35

3.3.2 Bubble Formation

Figure 3-2 shows the formation and expansion of bubbles underneath the glassy skin on the

surface of a 600 μm paint layer applied on a mild steel substrate with a roughness of 1.3 μm. The

paint layer was cured by a heated air current circulating above the surface (see Figure 2-6). No

bubbles were initially introduced into the paint layer. However, the relatively high roughness

caused air entrapment in the substrate cavities and established nucleation sites that initiate bubble

formation as seen at t= 7.6 min. The gaseous products of polymerization and evaporated volatiles

trigger new air pockets (Figure 3-2, t > 14.5 min) that form adjacent to the existing bubbles. They

continue to swell the topmost glassy skin and change shape to that seen at t=18.75 min, when

imaging stopped. Bubbles in an upwards facing liquid layer rise to the surface and escape, but, as

the solvent evaporates from a thermosetting paint layer polymer cross-linking is enhanced and a

skin forms on the surface that traps bubbles. This skin and increased viscosity as solvent

concentration decreases minimizes the escape of bubbles from the surface as seen in Figure 3-2.

In addition to surface roughness, small bubbles that may be entrained under the impacting drops

can act as nucleation sites if they become trapped underneath the skin barrier on the surface. This

phenomenon is discussed in detail chapter 5.

36

t=0 s 14.5 min

7.6 min 15.4 min

8.6 min 16.8 min

10.8 min 17.5 min

13.6 min 18.75 min

Figure 3-2 Bubble formation/ Blistering in 600 μm paint layer using convection heating at approximately 130 ± 5° C. Time t = 0 corresponds to 10 seconds after start of curing

37

3.3.3 Wrinkle Formation

The image sequence in Figure 3-3 shows wrinkling in an approximately 1 mm paint layer

deposited on stainless steel substrate with 0.3 μm roughness and cured using the experimental

system shown in Figure 2-6. Hot air flows over the paint surface and causes rapid solvent depletion

in the top most paint layers. As a direct consequence a higher degree of solidification occurs at the

surface, forming an elastic skin on top of a viscous under-layer. The perimeter of the coating dries

first, anchoring the glassy skin to the substrate. This mechanical skin is capable of supporting stress

and therefore started to deform as seen in figure 2.3 at t= 13.8 min. Basu et al. [55] hypothesized

that the skin suffers from both in-plane tensile and compressive stresses. Tensile stress was

generated by skin shrinkage while compressive stress was utilized by absorption of unreacted

oligomers to the skin, swelling of the topmost surface, and out of plane wrinkling. The physical

deformation of skin due to tensile stresses was studied by Cerda [90]. Once the in-plane tensile

stress exceeds a critical value and the skin is unable to contract locally, it will buckle to

accommodate the strain. As see in Figure 3-3 further shrinkage of the skin and the under-layers

due to drying, caused the skin to continue to deform under the action of various stresses.

38

t=0 s 14.5 min

13.8 min 15.3 min

13.9 min 25.2 min

Figure 3-3 Wrinkling in an approximately 1 mm paint layer using convection heating at 130 ± 5° C. Time t = 0 corresponds to 10 seconds after start of curing. At t= 13.8 min an elastic skin capable of supporting stress is formed. Wrinkles start propagating at t= 13.9 min

39

3.3.4 "Orange Peel" defect and Self-Organizing patterns

One of the most common defects in paint and other coatings is what is known as the “orange

peel effect”, where regular undulations appear on the surface. As discussed before, surface tension

gradients play a significant role in creating different types of patterns that are observed in curing

thermosetting polymer coatings. Small changes in temperature and concentration will give rise to

formation of diverse surface patterns.

Figure 3-4 shows the surface of a 1 mm paint layer and the evolution of organized structures

across the surface. t= 0 corresponds to less than 15 seconds after transferring the sample to the

oven. To create a sharp surface tension gradient a drop of ethanol, with a surface tension of

22.1 mN/m at 20° C, was just touched to the centre of the film and at t= 1.3 min, and roll-like

structures started to appear. As a result of localized surface tension decrease, the self-organizing

patterns start to initiate in an approximately 3mm radius from the point where the ethanol drop

was placed on the surface. In the beginning, the average spreading velocity amounted to

5x10-3 cm/s, up to t =2.6 min. The propagation progress continued undisturbed radially (Figure 3-4

t= 5.5 min), at an average speed of 1x10-4 cm/s, even after it reached the edge zone. After this time

the outward velocity decreased and the structures transitioned to cell-like disturbances.

40

t= 0 s t= 5.5 min

t= 1.3 min t= 6.3 min

t= 2.6 min t= 7.1 min

t= 4 min t= 8 min

Figure 3-4 Patterns in 1 mm paint layer created by locally decreasing surface-tension on the surface

41

Surface tension variation can also arise from concentration gradients in a thin evaporating film

if one component evaporates faster than the others. Random variations in the surface evaporation

rate can create surface tension variations that drive liquid flow from regions of low to high surface

tension. This can be observed in the image sequence in Figure 3-5 . Each row shows successive

states during curing of a 1 mm paint film at 150°C using the experimental system of Figure 2-7.

Time t=0 corresponds to less than 15 seconds after the sample was placed in the oven. Cellular

structures were observed to form immediately on the surface of the film. After the initial levelling

at t= 3.1 min, secondary roll-like structures began to form at t= 3.6 min. At t= 5 min, these

structures fully evolved and spread throughout the paint surface. Little work has been done to

understand the mechanism by which surface-tension-driven flows that are created due to

concentration gradients, lead to the "orange-peel" effect. This is important in coating and painting

applications and will be discussed in more detail in chapter 4.

42

t= 0 s t= 3.1 min

t= 35 s t= 3.6 min

t= 2.1 min t = 5 min

Figure 3-5 Temporal Evolution of Marangoni convective cells initiated by curing of 1000 μm paint layer at 150 ± 5° C. Time t = 0 corresponds to less than 15 seconds after start of curing when Marangoni cells appear. At t= 3.1 min initial levelling occurs due to mixing. At t= 5 min self-organizing structures are fully established

43

3.4 Conclusion

Uniform layers of melamine-based thermosetting paint were photographed during curing.

While paint is drying its composition is constantly changing due to polymerization and solvent loss.

The depth-wise gradient in solvent concentration resulted in variations of physical properties such

as viscosity and surface tension. This caused convective flows within the paint layer, creating

different types of defects.

It was concluded that decreasing viscosity will help levelling the paint and will allow bubbles

to escape. However, increased viscosity could inhibit multiple breakups leading to crater formation.

Film thickness is found to be of minor importance in cratering.

Thin films reduce formation of bubbles but encourage Marangoni convection cells. Also evident

was the dependence of final paint quality on the curing rate and variations of its physical properties.

44

Chapter 4

Orange Peel and Marangoni Convection

4 Orange Peel Formation due to Surface-Tension-Driven Flows

within Drying Paint Films

4.1 Introduction

The appearance of an automobile body is a very important part of its appeal. The exterior paint

should be glossy and smooth since any unevenness of the surface is immediately visible to an

observer as a blurring of images reflected in the paint [91]. Appearance is difficult to quantify in

a single measurement since a viewer’s perception of a painted surface is based on many factors.

One important aspect of paint finish is the waviness of the surface and what is known as “orange

peel”, defined by an ASTM standard as “the appearance of irregularity of a surface resembling the

skin of an orange” [25]. The mechanism by which orange peel appears on a dried paint surface is

not very well understood, since it may be either due to incomplete leveling of the paint surface

before curing, or because of flows created in the paint during the drying process that make the flat

paint surface uneven.

Paints typically consist of a resin dissolved in a solvent. As the paint dries in an oven the solvent

evaporates, creating strong concentration gradients in the paint layer. Non-uniform evaporation

and polymer cross-linking create regions with lower temperature and lower solvent concentration,

which have higher surface-tension than their surroundings. Surface tension gradients generate an

oscillatory flow across and normal to the paint surface and create self-organizing convective cells.

45

Then, as the resin cures, irregularities on the paint surface are frozen in place, creating the orange

peel effect.

Solvent evaporation and chemical curing rates in drying paint films are strongly coupled

phenomena, which proceed in two stages [92]. In the first phase solvent evaporates at a rate

controlled by the paint temperature, solvent concentration and conditions at the paint film surface.

In the second stage resistance to solvent diffusion increases as vitrification (solidification) sets in

and the diffusion rate of solvent decreases sharply. The glass transition temperature, at which a

liquid to solid transformation occurs, is a function of solvent concentration and density of polymer

crosslinking and is the main mechanism that limits the mobility of solvent molecules and polymer

chains [93]. Therefore, the relative magnitudes of the timescales for complete diffusion of volatiles

(tD) out of the paint film and the time for paint to cure (tC) play an important role in determining if

orange peel occurs. Additionally, in thin paint films, the substrate roughness or the texture of under

layers may contribute to orange peel as the paint shrinks and conforms to substrate texture. In

basecoat/clearcoat systems, the different curing rates of coatings may create residual strains during

curing and could influence the final surface profile [50] [51].

This chapter reports the results of an experimental study on curing paint films in a natural

convection oven. A model paint was developed (see section 2.1), with known solvent and resin

composition, to mimic the physical properties of commercial paint. The onset of Bénard-

Marangoni cells was photographed and paint weight loss was measured to determine the

concentration gradient during curing. The objective was to determine the effect of film thickness

on the onset of surface-tension-driven flows in curing paint layers.

46


Figure 2-4 shows a schematic diagram of the apparatus used to apply uniform paint films. Mirror-

polished stainless steel discs (51 mm diameter) with roughness less than 0.3 μm were used as test

surfaces. They were cleaned with acetone to remove dust particles and any oil or silicon residue and

then coated with the method as detailed in section 2.2. Coated substrates were placed inside a metal

chamber (with less than 15 seconds delay after applying the coating) that was used as a convection

oven to cure paint samples (see Figure 2-7).


4.3.1 Surface-Tension-Driven Flows in Drying Paint Films

Figure 4-1 shows images of the surface of paint films curing at a temperature of 140±5° C.

Four different film thicknesses are shown, (a) 150 µm, (b) 320 µm, (c) 500 µm and (d) 820 µm.

Each row shows successive states during the early stages of curing. At the start (t=0) all the paint

surfaces appear to be smooth. Within a few seconds after the start of heating cellular structures

were observed to form on the surface of all the films with the cell size increasing with film

thickness (see Figure 4-1, t=36 s). Schwarzenberger et al [49] identified three different basic

structures that can be observed in systems subject to surface tension driven instabilities;

interactions between these structures can create a wide variety of cellular patterns. The quasi-

stationary dense network of polygonal cells, free of any substructures, seen on the 150 µm and 320

µm thick films would be classified as roll cells of the first order according to their nomenclature

[49], since they do not have any substructures in them (see t=54 s and t=2.25 min for 150 µm and

320 µm films). On the surface of the 500 µm film, several isolated regions containing convective

cells appear each containing substructures, which have been identified [49] as relaxation

47

oscillation cells that are chaotic and appear when the container size is much larger than the

individual cells, which was true in these experiments. The borders of these are round rather than

polygonal and they spread along the surface. The competition for growth of neighboring cells leads

to some expanding while others being compressed (see Figure 4-1c, t=54 s). On the thinnest film

(150 µm) the structures grew more blurred and disappeared after approximately 3 min (see Figure

4-1a, t=3.4 min). As the film thickness increased to 320 µm the surface took a longer time to

become level (see Figure 4-1b, t=4.5 min). The patterns did not reappear on the thin films and their

surfaces remained smooth. However, on the thicker films (L=500 µm and 820 µm) the cells grew

larger by conveying fluid from the bulk with higher concentration of solvent, causing the

expansion of individual cells. Strong oscillatory flows, due to convection from the bulk, deforms

the concentration distribution on the interface. This leads to isolated regions of re-established

concentration gradient and hence ladder-like structure elongated along the flow (see Figure 4-1c,

t= 2.25 min and t=4.5 min). After this stage wave-like structures began to appear on the paint

surface (Figure 4-1c, t=2.25 min).

48

t=0 s

t=36 s

t= 54 s

t=2.25 min

t=3.4 min

t=4.5 min Figure 4-1 Marangoni cell formation in (a) 150 μm, (b) 320 μm, (c) 500 μm, and (d) 820 μm paint films, t<5 min

23 mm

(a) 150 μm (b) 320 μm (c) 500 μm (d) 820 μm

49

No further changes were seen in the thin films (150 and 320 µm) after 5 min: their surfaces

remained smooth until the paint hardened due to polymer cross-linking and no more motion could

be seen, which took approximately 4-5 min. The surfaces of the thicker films continued to evolve

beyond this time.

Figure 4-2 shows images of the 500 and 820 µm thick films for t>5 min. Large-scale stationary

wave-like structures can be seen on the surface of the 500 µm paint layer at t=9 min which began

to level out and become less distinct by t=14.25 min. However, by this time the paint layer had

hardened and these undulations had set in the paint surface. The amplitude of waves on the thickest

film (820 µm) continued to increase until it had completely hardened (t=17.85 min). In general,

the first phase of quasi-stationary cellular convection is followed by a chaotic relaxation oscillation

regime [49]. For thick films a secondary quasi-stationary period was seen to follow resulting in

large-scale wave-like structures.

Figure 4-3 shows images of completely dried samples photographed several days after they had

been completely cured. Figure 4-3 (a) shows the smooth surface of the thinner paint layer with 150

μm initial thickness. Figure 4-3 (b) illustrates the wrinkled surface of a paint layer with

approximately 820 μm initial thickness.

50

(a) 500 μm (b) 820 µm

t=5.25 m t=5.25 m

t=6 min t=6 m

t=9 min t=12 m

t=14.25 m t=17.85 m

Figure 4-2 Self-organizing roll-like patterns in (a) 500 μm, (b) 820 μm paint films, t≥5 minutes

23 mm

51

(a) 150 μm (b) 820 μm

Figure 4-3 Completely dried paint layers with (a) 150 μm and (b) 820 μm initial thickness

4.3.2 Orange Peel Formation on Dried Paint Films

A surface profiler with a low-force mechanical stylus (Model Alpha-Step D-120, KLA-Tencor

Corporation, Milpitas, USA) was used to measure the amplitude and wavelength of surface

features on the dried paint samples. The profiler drew the tip of the stylus across the surface in a

straight line with a speed of 0.1 mm/s and a constant force of 0.1 mg to avoid scratching the dried

paint. Figure 4-5 shows the surface profiles of dried paint films with initial thickness varying from

150 µm to 300 µm. Both the wavelength and amplitude of the surface undulations increased with

initial film thickness. The average amplitude was only 0.06 μm for a 150 μm thick film,

corresponding to a very smooth finish (see Figure 4-5 (a)) and increased to 3 μm for the thickest

film (820 µm) which had very visible orange peel on the surface (Figure 4-5 (c)). The average

wavelength of the surface waviness also increased from 150 μm for thin films (Figure 4-5 (a)) to

1500 μm for the 820 µm film (Figure 4-5 (c)). The average roughness (Ra), defined as the average

deviation of the surface from a hypothetical perfectly flat plane, increasing from approximately

0.02 µm for a 150 μm film to 2 μm for an 820 μm film.

48 mm

52

(a) 150 μm & 300 μm initial thickness

(b) 500 μm initial thickness

Figure 4-4 Typical profilometry traces of the wrinkled patterns completely dried samples with (a) 150 μm & 300 μm, (b) 500 μm, and (c) 820 μm initial thickness

Pro

file

Hei

ght

Scan Length

Scan Length

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53

(c) 820 μm initial thickness

Figure 4-5 Typical profilometry traces of the wrinkled patterns completely dried samples with (a) 150 μm & 300 μm, (b) 500 μm, and (c) 820 μm initial thickness

In the automotive industry paint surface irregularities with wavelengths between 0.1 to 30

millimeters and amplitudes between 0.5 to 5 microns are quantified by "orange peel ratings" that

can be obtained using various techniques. One traditional method is for a human inspector to

visually compare the painted surface with a set of standard panels of varying roughness that are

painted black and labeled with an orange peel rating from 1 to 10. In automated inspections a

scanning instrument (for example, Wavescan Plus, Byk-Gardner, Columbia, MD) is moved over

the surface using a laser light source to illuminate the painted surface at an angle of 60° while a

detector records the reflected light. Distortions of the reflected beam can be correlated with the

wavelength and amplitude of features on the reflecting surface. The optical signal is analyzed to

produce a list of numbers that represent the magnitude of structures in each of five wavelength

windows (0.1-0.3 mm, 0.3-1 mm, 1-3 mm, 3-10 mm, and 10-30 mm). In principle it should be

Scan Length

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ght

54

possible to relate orange peel numbers to profilometer measurements [26, 27], for example by

considering the ratio of amplitude to wavelength [27]. However, there is yet no simple way of

doing this.

4.3.3 Mathematical Modeling of Paint Film Drying

The growth of cellular structure in thin liquid layers is usually due to either temperature or

surface tension gradients that drive convective flow. In these experiments the paint films were

placed in a uniformly heated chamber without any significant temperature gradients. Temperature

measurements using fine-wire thermocouples (Model 5SRTC-TT-K-40-36, OMEGA

Engineering, Quebec, Canada), showed that the horizontal temperature gradient across the width

of paint films was less than 0.2°C/mm. In previous studies [41] horizontal temperature variations

caused by evaporation from pure liquids were reported to be at most 1°C, which would be too

small to produce significant density or surface tension gradients. Observation of the paint surface

with an infra-red camera confirmed that there were no detectable temperature gradients. It is much

more likely that the flows were caused by surface tension gradients due to variations in solvent

concentration during drying of the paint [44] [45] [46] [47].

Figure 4-6 shows the possible mechanism driving Marangoni flows. As solvent evaporates there

will be a lower concentration of butanol on the paint surface than that at the bottom near the

substrate. If there is a random variation in the solvent concentration at two different places on the

paint surface, the surface tension (σ) will be greater at the location with low solvent concentration,

triggering a flow along the surface from the region of low σ to that of high σ. The paint on the

surface will be replenished by liquid drawn from below, which will have a higher solvent

concentration, amplifying the surface concentration difference. This will create a self-sustaining

55

flow that will last as long as there is a concentration difference, beyond the critical value, across

the thickness of the paint layer. The average wavelength on dried samples (see Figure 4-5) correlate

to an estimate of the Marangoni cell sizes (center-center distance of two neighboring cells) and

possibly the length scale over which the surface tension gradient exists. The average wavelength

obtained from the profilometry traces corresponds quite well with the initial film thickness and

can also be used as the length scale in calculating Marangoni number.

Figure 4-6 Self-sustaining Marangoni flow

56

We can estimate the magnitude of surface tension forces due to solvent concentration gradients

by:

4-1

where the film thickness L is used as a length scale. Surface tension driven flows are opposed by

viscous shear forces that can be estimated from:

4-2

There is no imposed flow velocity in this problem but for scaling purposes we can define a

characteristic velocity due to diffusion driven flows that are of the order u ~ Dv /L. Substituting

this in Eqn. 4-2 and assuming x~L we get:

4-3

The ratio of surface tension to viscous shear forces gives the dimensionless Marangoni number:

4-4

In Eqn. 4-4 the viscosity (µ), film thickness (L) and variation of surface tension with concentration

( / are known (see Figure 2-2). We need to determine the mass diffusivity of solvent in the

paint (Dv) and the concentration gradient of the solvent ( / which we can do using a one-

dimensional model of mass transfer from the paint layer [94]. The paint layer was modelled as a

binary mixture with uniform initial volatile concentration (Ci , mass fraction of solute) below the free

57

surface. The volatiles were assumed to only diffuse in a direction normal to the paint surface and the

film thickness (L) was assumed to be constant. Assuming negligible advection in the film and

diffusive transport of volatiles, which is true for the quasi-stationary phase, the governing equation

for the 1-D transient mass transfer with no species generation is [94, 95]:

, ,

, ; 0 4-5

where C denotes concentration, Dv is the solvent mass diffusion coefficient, t is time, and x is location

in the paint layer measured from the solid substrate The substrate is impermeable so the boundary

conditions at the lower surface of the paint is:

0,

0 at 0 4-6

The rate of convection from the upper surface is assumed proportional to the concentration of solvent

at the upper surface of the paint [94] [95] [96]:

, ∞ at 4-7

where the constant of proportionality, α, is assumed to be constant throughout the drying process.

Note that α is not a standard mass transfer coefficient since it is based on the solvent concentration in

the liquid and not the vapor above the evaporating paint. It was assumed that the ambient solvent

concentration, C∞, required to maintain equilibrium with the surrounding atmosphere, is very close to

zero [95] [97], as there is negligible external mass transfer resistance. The time scale for vapor

diffusion in air is much smaller than in the paint layer: The diffusivity of butanol in air is of order the

10-6 m2/s while typical values for the diffusivity of organic vapors in a polymer layer vary between

10-13 m2/s to 10-11 m2/s, depending on the amount of dissolved vapor in the polymer [97].

58

To estimate the magnitude of α experiments were done in which the weight loss of a 60 mm

diameter glass dish in which a 3 mm layer of pure butanol was placed and maintained at a

temperature of 100°C. The mass loss of liquid over a 15 - minute period was recorded at intervals of

30 seconds. Dividing the mass loss by the surface area of the liquid and the elapsed time gave the

mass flux, J, and α was obtained from:

4-8

Where Ci =1 is the initial volatile concentrations of pure butanol. Figure 4-7 illustrates the variation

of mass transfer coefficient with time for pure butanol immediately after the samples were transferred

to the oven. The increasing values of α in the first 3 minutes for pure butanol, reflects the

temperature rise in the liquid pool. An average steady-state value obtained from three experiments

gave α=1.8x10-3 kg/m2s.

Figure 4-7 Variation of mass transfer coefficient with time at 100° C

59

To calculate a value of α for paint films samples of varying thickness were placed in an oven at

140°C and the weight loss recorded at 60 seconds intervals. The value of α was calculated from the

mass loss of solvent during the first 3 minutes, assuming that the surface concentration was constant

at Ci=0.45. Figure 4-8 shows the value of α as a function of film thickness, demonstrating a linear

increase with L. For L≤820 µm, the range of interest for our experiments, the average value of

α=1x10-3 kg/m2s, which was the same order of magnitude as that for pure butane. This value was

therefore used in all subsequent calculations.

Figure 4-8 Mass transfer coefficient for various film thicknesses

Eqn. 4-5 with the boundary conditions given by Eqns. 4-6 and 4-7 is analogous to the problem of

transient conduction heat transfer in a plane wall with one face insulated while there is convection on

the other [98] and the solution to is given by (Crank 1975 page 60) [94] [99]:

60

, ∞

∞

2∞

4-9

Where,

Eigenvalues λn: 4-10

Fourier

number: 4-11

Biot number: 4-12

At time t the cumulative mass of solvent that has evaporated from the paint film (Mt) as a fraction

of the total mass of solvent (M∞) is (Crank 1975 page 60) [94]:

1 e2

4-13

The time dependence on the right hand side of Eqn. 4-13 is only through Fo, which is proportional

to t/L2. Figure 4-9 shows the experimentally measured mass of solvent lost from the paint, divided by

the total solvent mass, as a function of √ / . The evaporative mass flux was highest at the start of the

drying process and then decreased as the surface solvent concentration was reduced. In principle if

Dv and α are constant all the curves in Figure 4-9 should collapse into one. However, as

polymerization of the paint progresses and a glassy film forms on its surface the effective diffusivity

may change [94] [100]. Paint drying time decreases with film thickness because the solvent takes

less time to diffuse out of thinner paint layers. Since the effective paint diffusivity decreases as the

61

paint solidifies the duration for which we can reasonably assume that diffusivity is constant

becomes shorter as film thickness decreases. If we assume that the curves in Figure 4-9 are

reasonably coincident for √ 5 10 s0.5/m, this corresponds to approximately 1 min for a 150 µm

thick paint film, 4.3 min for a 320 µm film and 10.4 min for a 500 µm film. Since orange peel

formation takes place in the first few minutes we assume that Dv is constant, allowing us to use the

analytical solution, though this may not be entirely accurate for the 150 µm paint film for which the

reduced rate of mass transfer is noticeably lower than the others, as seen in Figure 4-9.

Figure 4-9 Reduced desorption curves for paint films of varying thickness

If we assume that the concentration of solvent at the surface is constant, which is a reasonable

approximation for short intervals, the mass loss from the paint film for small times is (Crank 1975

page 48):

62

2

2 1 ierfc√

4-14

In the early stages, this reduces to (Crank 1975 page 244):

2

, → 0 4-15

The average diffusivity was estimated by substituting the measured mass loss over the first 3 min of

paint curing in Eqn. 4-15. Averaging over all the calculated values we obtained Dv ~10-10 m2/s. This

value was used to calculate Fo from Eqn. 4-11, Bi from Eqn. 4-12 and from Eqn. 4-10. The total

amount of diffusing substance was analytically evaluated using Eqn. 4-13. Figure 4-10 shows a

comparison of the predicted and measured mass loss from the 150 µm (Figure 4-10 (a)), 320 µm

(Figure 4-10 (b)), 500 µm (Figure 4-10 (c)), and 820 µm (Figure 4-10 (d)) thick paint films. The

reported mass loss values for each film thickness, are the mean values calculated from at least

three trials with a standard deviation of less than 5%. For thinner films (150 µm and 320 µm),

there is reasonable agreement between the predicted and measured values for t<4min, though after

that the measured value is less than that obtained from calculation showing that the effective

diffusivity had decreased. For larger thicknesses, the diffusivity value should give us a reasonable

estimate for longer periods of time, as the effective diffusivity does not decrease as rapidly. For

500 µm and 820 µm films, when 4 min < t < 25 min, the experimental measured mass loss is

slightly higher than the predicted value. At this time the transport of volatiles is not only by

diffusion, but also assisted by Marangoni convection and constant circulation of fluid which brings

fresh paint to the surface. However, the diffusivity value should give us reasonable estimates of

concentration gradients for t<4 min.

63

(a) 150 μm (b) 320 μm

(c) 500 μm (d) 820 μm

Figure 4-10 Kinetics of drying for paint films for (a) 150 µm and (b) 320 µm (c) 500 µm (d) 820 µm

The solvent concentration variation in the paint layer was calculated from Eqn. 4-9 as a

function of time. Figure 4-11 shows the solvent concentration profiles for four different paint layer

thicknesses at various times.

64

(a) 150 μm

(b) 320 μm

65

(c) 500 μm

(d) 820 μm

Figure 4-11 Concentration profiles for: (a) 150 μm (b) 320 μm (c) 500 μm (d) 820 μm paint films

66

In the thinnest film (L=150 µm) the surface solvent concentration drops to almost zero in the

first minute. The concentration in the bulk of the paint layer then decreases until after 5 min it is

less than 70% of the initial concentration near the substrate. The concentration drops to less than

10% of the initial concentration everywhere after approximately 20 min. In the thickest layer

(L=820 µm), the solvent concentration at the surface rapidly drops to zero and stays there for the

entire evaporation process. There was a significant concentration gradient in the paint layer even

after 30 min, with the concentration near the wall being 70% of its initial value.

The Biot number is large (>1) during our experiments since the diffusivity of the solvent is

much greater in the air than in the paint. This is the reason that the solvent concentration at the

paint surface rapidly drops to zero (see Figure 4-11). Bi increases with paint film thickness, so for

the thicker films C(L,t)=0 for virtually the entire evaporation process (see Figure 4-11 d). This

implies that our results are not very sensitive to the exact value of α used for calculations, as long

as the order of magnitude is correct.

4.3.4 Evaluation of the Dimensionless Marangoni Number

The Marangoni number Ma was calculated from Eqn. 4-4 by assuming ~∆ where ∆C is

the concentration difference across the film thickness calculated from the one-dimensional

diffusion model. The viscosity value at room temperature µ=240 cP was used in calculations. A

mean value of mass diffusivity over the range of concentrations in experiments was used [94].

Figure 4-12 shows the variation of Marangoni number, Mac, with time for varying film

thickness. For the thinnest film Ma increased immediately after heating, reached a maximum

before t =1 min, and then decreased. The time for Ma to reach a maximum corresponds to the time

when the solvent concentration near the substrate starts to diminish (see Figure 4-11 a) so that the

67

concentration gradient across the film decreases. It also marks the time that the cellular structures

on the surface of the paint film start to disappear (see Figure 4-1a, t=3.4 min). Ma decreases as the

solvent is depleted throughout the paint layer and the surface of the paint becomes smooth again

(Figure 4-1a, t=4.5 min).

Figure 4-12 Marangoni number as a function of time for paint films of varying thickness, using mean values of diffusivity coefficient and viscosity

When the paint thickness was increased it took longer for the solvent to diffuse out. In a paint

layer with L=500 µm, the maximum value of Ma was reached at approximately t=5 min. The

cellular structures are seen to reach a maximum a little after this time (Figure 4-2a, t=9 min) and

then slowly start to disappear (Figure 4-2a, t=14.25 min) as Ma decreases. At the greatest film

thickness, L=820 µm, the solvent concentration near the substrate is close to its original value

(Figure 4-11 d) even after 10 min. The Marangoni number therefore continues to rise and reaches

68

a peak only at approximately t=12 min, after which the concentration of solvent throughout the

paint film begins to decrease. The cellular structures are still prominently visible (Figure 4-2 b,

t=12 min). The paint film has hardened at this stage due to polymerization of the resin so that the

surface structures are set in place and cannot disappear (Figure 4-2 b, t=17.85 min). The wrinkled

surface is therefore visible in the dried paint film, as seen Figure 4-3 b.

The formation of orange peel depends on the relative magnitude of the time (tD) for solvent to

diffuse completely out of the paint film and the time for the paint to cure (tC). This depends on the

curing characteristics and chemistry of each coating formulation, as well as specific spray and

oven parameters at which the production line operates at. If tD<tC most of the solvent escapes from

the film so that concentration gradients are no longer large enough to create surface tension driven

flows in the paint. The cellular structures therefore disappear and the surface of the paint layer

becomes smooth before it hardens. Alternately, if tD>tC there are still significant concentration

gradients in the paint film when the paint cures and the surface waves created by surface tension

driven flows set in place. For larger thicknesses, it generally takes longer for the cross-linking

density to increase, due to the larger volume to surface ratio. The free surface for evaporation of

volatiles is unchanged, resulting in trapping of solvent in the lower parts of the coating, due to skin

formation (solidification) in the top layer. As a result, the difference between tD and tC becomes

larger as the film thickness is increased.

4.3.5 Convective Velocities in Drying Paint Layers

The velocity of convective flows in the paint layer first increases as solvent concentration

gradients increases, and then gradually decreases as the paint cures and the viscosity increases.

Paint velocities were measured by using particle image velocimetry to track particles seeded in the

69

paint and calculate velocity vector fields at each instant. The magnitude of the velocity vectors

was averaged to give a representative paint velocity.

Figure 4-13 Change in convective velocities over time for paint films of varying thickness

Figure 4-13 shows the variation in velocity with time for paint films with varying thickness. In

each case the velocity first increased as solvent evaporated and created surface tension gradients

in the paint film. The peak velocity was approximately 30 µm/s and the time required to reach this

increased with paint thickness. Then, after 2-5 min convective velocities decreased and fell to

almost zero. This corresponds to both a decrease in the Marangoni number and also increased paint

viscosity due to depletion of the solvent and polymerization of the resin. Paint curing has a twofold

effect on viscosity: it initially decreases due to shear heating and thermal effects, but as time

elapses, it increases rapidly (to a theoretical infinity) as a result of polymer network forming across

70

the surface. At this point the film has gelled and inhibits convective flow. However, curing reaction

can still proceed although at a reduced rate due to low diffusion [101].

Figure 4-14 shows successive images of the velocity field on the surface of the 820 μm thick

paint film. At t=36 s there is movement on the entire surface of the paint, corresponding to the

presence of cells distributed all over the paint layer (see Figure 4-1, t=36 s). By t=2.25 min the

velocity had increased with the region of high velocity was concentrated along the edge of the

Marangoni cells. The paint velocity had started to decrease by t=4.5 min, showing the effect of

increased viscosity due to curing and solvent evaporation. By t= 6 min, the velocity has dropped

to under 5 µm/s, corresponding to the re-establishment of concentration gradient by diffusion and

appearance of roll-like structures (see Figure 4-2b, t=6 min). Due to the low accuracy of cross-

correlation function in ImageJ software, the precise values of velocities are not of interest.

However, the overall trend of velocity variations and the locations where the maximum velocities

are concentrated, are good indication of flow regime. This provides insight into the flow dynamics

behind the self-organizing structures and an estimate of curing time scale.

In calculating concentration gradients, we have assumed that the paint film is not moving and

transport of solvent is due to diffusion alone. This assumption is true for short times (t< 1 min)

when convection has not started or at long times (t>5 min) when the curing has advanced enough

to prevent liquid motion. The superposition of advective and diffusive transport of solvent can

change the concentration profile and may alter the appearance of surface waves in thicker paint

layers. In Figure 4-1c, for a paint layer with L=500 µm, cellular structures are observed

immediately after the start of heating (t=36 s) that grow larger as convective flows increase.

Mixing of the paint decreases the magnitude of concentration gradients, and the cellular structures

begin to disappear (t=3.4 min). By t= 5 min the convective velocities have begun to diminish

71

rapidly (see Figure 4-13) and the cells reappear (Figure 4-2a, t=9 min), before disappearing again

as solvent is depleted everywhere in the paint (Figure 4-2a, t=14.25 min).

72

t = 36 s

t = 2.25 min

t = 4.5 min

t = 6 min

Figure 4-14 Velocity field for 820 µm film at various time steps

73

4.4 Conclusion

The drying of paint films with varying initial thickness was studied. The diffusivity of solvent

in the paint was calculated assuming one-dimensional diffusion in a binary solvent-resin system.

The effective diffusivity was found to be a function of paint film thickness and time, decreasing

as the paint cured. Photographs of the paint surface showed that cellular structures appeared almost

immediately after the painted steel substrates were placed in the oven. For thin films (<500 µm)

the patterns disappeared in less than 3 minutes and the hardened paint surface was smooth. For

thicker paint films roll-like structures continued to form and remained on the final hardened paint

layer, creating an orange peel effect. The variation of Marangoni number with time was calculated

for different film thickness. For thin films Ma increased as solvent evaporated and created large

concentration gradients, and then decreased as all the solvent was depleted, explaining why cellular

structures disappear. In thicker films Ma continued to increase until the end of drying, creating roll

patterns that remained in the cured film. By this time the high viscosity of the paint inhibited

levelling, leaving orange peel on the paint surface.

74

Chapter 5

Bubble Growth and Movement

5 Bubble Growth and Movement in Drying Paint Films

5.1 Introduction

When paint is sprayed on a surface a large number of air bubbles may be entrained by

impacting droplets and trapped in the deposited layer, creating serious defects during drying.

Bubbles act as nucleation sites into which evaporating solvent diffuses, making them grow until

they burst through the paint surface and create visible blisters and pinholes [13]. As seen in chapter

section 3.3.2, surface roughness and the quality and texture of the surface onto which the paint can

have an effect on the presence of nucleation sites. Presence of pigment particles and flakes in

coatings may also promote bubble entrapment and increase the possibility of bubble formation.

In automotive plants spray painted components are allowed to dry for 5-10 minutes (known as

the “flash-off” time) before being placed in the oven to dry, to reduce paint blistering [7]. Figure

5-1 shows two steel disks that were spray painted with commercial automotive clear coat paint and

then dried in a convection oven [102]. One (Figure 5-1a) was placed in the oven after a flash-off

time of 10 min, while the other (Figure 5-1b) was given a flash-off time of only 2 min. The test

sample with the reduced flash-off time showed pronounced blistering, with a large number of

bubbles in it.

75

(a) (b)

Figure 5-1 Stainless steel substrates, 75 mm in diameter, spray painted with an automotive clear coat paint and baked in an oven after a flash-off time of (a) 10 min and (b) 2 min [102].

This study in this chapter was undertaken to observe the process by which bubbles grow in

drying paint layers. There were several questions that we wanted to address in particular. What is

the origin of the bubbles that are seen in paint layers? Why does increasing the flash-off time result

in elimination of the bubbles? Why do the bubbles form clusters such as those seen in Figure 5-1?


The model paint, with the formulation described in section 2.1, as well as a non-solidifying,

viscous solution of glycerin-butanol was used in the following experiments. Figure 2-4 shows the

schematic arrangement of the apparatus used to apply uniform paint films. Mirror-polished

stainless steel discs (51 mm diameter) with roughness less than 0.3 μm, or heat-resistant

borosilicate glass substrates (Model 8477K78, Mc-MASTER-CARR, USA), 63.5 mm in diameter

with 3.2 mm thickness, were used as test surfaces. The experimental apparatus shown in

Figure 2-7 was used to cure coated substrates. Particle migration within thin liquid layers was

further investigated by mixing glass tracer particles (Hollow Glass Microspheres 0.06 g/cc 150-

180 um, Cospheric, USA). Pictures of liquid layers with growing bubbles were analyzed using the

76

threshold function in image analysis software (ImageJ, National Institute of Health) to count the

number of bubbles in each image, the cross-sectional area of each bubble, the location of individual

bubbles, and the distance between bubbles as they formed clusters.


5.3.1 Bubble Formation and Growth

Figure 5-2 shows a sequence of images of bubbles in a paint film with an average thickness of

1000 µm on a glass substrate curing at 140±5° C. The bubbles were introduced in the paint before

deposition by agitating transparent glass vials that were three-quarters full with paint. The bubbles

remained in the paint while it was transferred to the substrate using a syringe and spread to the

desired thickness using a blade coater. The paint sample was immediately placed in the oven,

without allowing any time for solvent to evaporate. Initially (t = 0 min in Figure 5-2) there were a

large number of bubbles ranging from 0.04 – 0.4 mm in diameter that appear as white dots in the

image. The bubbles were in constant motion as the paint cured, with average velocity of

approximately 25 µm/s. The bubbles grew larger, with the rate of growth relatively slow for the

first 5 min but becoming much more rapid after that time (see Figure 5-2, t > 5.2 min). At the same

time the number of bubbles decreased as bubbles burst through the paint film surface.

77

t = 0 min t = 8.5 min

t = 2.6 min t = 9.8 min

t = 3.6 min t =12.8 min

t = 5.2 min t = 14.3 min

t = 6.7 min t = 16.6 min

Figure 5-2 Bubble growth and migration in 1000 µm paint film curing at 140° C

33 mm

78

Figure 5-3 illustrates the behavior of bubbles in a similarly coated sample, which was cured at

100±5° C. Although the number of bubbles present was comparable to those in the paint layer of

Figure 5-2 when curing began, the rate of growth remained slow (see Figure 5-3, t> 9 min), and

noticeably fewer bubbles remained in the cured paint layer (Figure 5-3, t=23 min).

t = 0 min t= 9 min

t= 4.3 min t=10.2 min

t= 5 min t=13 min

t= 7.9 min t=23 min

Figure 5-3 Bubble growth and migration in 1000 µm paint film curing at 100° C

33 mm

79

Figure 5-4 shows the variation of bubble density (the number of bubbles per square millimeter),

normalized by the initial bubble density, with time. The bubble density decreased to 35 % of its

initial value after approximately 10 minutes for the paint film cured at 140° C and then remained

constant. For the paint film cured at a lower temperature, it continued to decrease to 15% of the

initial density. The bubbles appeared to be attracted to each other as they grew, forming a number

of clusters (see the bubbles within the circles drawn on the images at t = 8.5, 9.8 and 12.8 min in

Figure 5-2).

Figure 5-4 Bubble density variation

To track the growth in bubble size, the Sauter Mean Diameter (SMD, d32) of the bubbles was

calculated at each time step. Image analysis was used to measure the diameter (D) of bubbles in

each image and, assuming they were spherical, to calculate their volume (Vp) and surface area (Ap).

The SMD is defined as:

80

6∑∑

∑∑

5-1

Figure 5-5 shows the variation of SMD with time for the bubbles shown in Figure 5-2 and

Figure 5-3. The SMD was approximately constant for about 5 minutes at 140° C and for 8 minutes

at 100° C, which was also the time in which the number of bubbles decreased most rapidly (see

Figure 5-4). After that time the number of bubbles did not change significantly, but their size

increased by a large amount.

Figure 5-5 Sauter Mean Diameter variation

When bubbles approach the surface of the paint film the thin liquid film formed between

bubble and the liquid-air interface has to rupture before bubbles can escape through the surface

[103] [104] [105]. The higher the viscosity of the paint, the longer the time taken for the paint film

81

to drain and rupture. In a thermosetting paint layer polymer cross-linking is enhanced when

solvent evaporates from the liquid layer and the glass transition temperature, at which a liquid to

solid transformation occurs, is a function of solvent concentration and density of polymer

crosslinking [106]. Solidification is fastest at the paint surface where solvent depletion is most

pronounced, and as a result a skin forms on the paint surface that traps the remaining bubbles in

the paint [60] [51]. The solvent, prevented from escaping due to the solidifying paint surface,

diffuses into the bubbles and makes them grow rapidly. From the data in Figure 5-4 and

Figure 5-5, the skin appears to have started forming at approximately t = 5 min and t=8 min for

paint films of similar thickness respectively cured at 140° C and 100° C, after which the number

of bubbles remained stable while their size increased rapidly.

To compare bubble growth in a paint layer with that in a layer of a viscous liquid which does

not solidify, bubbles were mixed in glycerin-butanol solution prior to spreading on the surface of

stainless steel or glass substrates. Figure 5-6 shows the image sequence in an approximately 1 mm

liquid layer heated to 100° C. In this case the bubbles again showed movement within the paint

film, and also showed some clustering (see t = 1.25 min). However, the bubbles escaped very

rapidly from the paint film, before they had time to grow very much in size, and by t = 4 min there

were no more bubbles within the paint. The absence of a skin on the surface of the liquid layer

meant that there was no barrier preventing bubble escape or forcing solvent into the bubbles and

making them grow.

82

t = 0 t = 1.5 min

t = 50 s t = 2 min

t = 1 min t = 2.5 min

t = 1.25 min t = 4 min

Figure 5-6 Bubble agglomeration and escape in 1000 μm Glycerin-butanol solution on glass substrates at T= 100° C, t 4 min

Figure 5-7 shows the mass of butanol evaporated (Mt), normalized by the initial mass (M∞), as

a function of time for both the paint and the butanol-glycerin mixture, with both maintained at

120°C. Solvent escape from the paint layer was slow, and even after 30 min only about half the

21 mm

83

initial mass of solvent has been lost. By contrast a surface skin is not formed in a glycerin-butanol

and almost all the solvent is depleted from the thin layer. Since solvent is depleted before it has

time to diffuse into the air bubbles in the glycerin-butanol (see Figure 5-6) the bubbles do not

grow, but instead burst through the surface and escape. It is likely that the formation of a skin on

the surface of the paint prevented escape of solvent from the paint surface, leaving the paint film

super saturated with solvent that was forced into the bubbles, increasing their growth rate.

Figure 5-7 Evaporation curves for glycerin butanol solution and model paint at 120° C

5.3.2 Mathematical Model of Bubble Growth

We can estimate the bubble growth rates at constant temperature and pressure, using Fick’s

law, where the mass flux into the bubble depends on the concentration gradient near the bubble

boundary. The gas bubble radius at t =0 is assumed to be R0 and the concentration of the dissolved

84

gas is uniform and equal to C∞. The concentration, C(r,t), of solvent in the liquid surrounding the

bubble is governed by the mass diffusion equation which, written in spherical coordinates is [107]

[72]:

5-2

where C denotes dissolved gas concentration, Dv is the mass diffusion coefficient, t is time, and r is

the radial position within the liquid from the center of the bubble.

The concentration satisfies the following initial and boundary conditions [72]:

, 0 ,

5-3

lim→

, , 0 5-4

, , 0

5-5

Where C∞ is the initial solvent concentration in the paint and Cw is the saturation solvent

concentration at the bubble surface, evaluated at the partial pressure of the gas in the bubble.

A mass balance across the bubble interface, r=R(t) gives:

4 , 5-6

Epstein and Plesset [72] found an approximate solution to this problem [107]:

1

2

1 43

1 / 5-7

where ρG is the density of gas in the bubble.

85

The term / becomes small when t is large, and the characteristic growth is given

approximately by [107]:

02

5-8

In which vapor density ρG is calculated assuming that butanol vapor follows the ideal gas law. A

plot of bubble radius variation with time is parabolic, which is characteristic of diffusion controlled

growth.

The equation of state for gas bubble of radius R in a liquid at ambient pressure P∞ is:

2

5-9

Surface tension ( ) is assumed to be constant throughout the drying process. The surface

tension of paint decreases with solvent concentration, from 27 mN/m for the undiluted resin to

23 mN/m for pure butanol at room temperature (see section 2.1). Surface tension of pure butanol

decreases from 23 mN/m at room temperature to 17 mN/m at 100°C. The surface tension used in

all subsequent calculations of the pressure inside the bubble was approximated as an average of

20 mN/m for a solution of polymer containing dissolved butanol near the bubble boundary.

According to Henry’s law, the dissolved gas concentration in the liquid bulk is related to the

saturation pressure of butanol vapor at the corresponding curing temperature, and the gas

concentration at the bubble wall is related to the pressure inside the bubble. If Hc is the

dimensionless Henry’s constant, B the universal gas constant, T is temperature, and CGL and CGB

are concentrations of butanol vapor on the surface of the liquid and inside the bubble, C∞ and Cw

in mol/lit can be written as [108]:

86

, 5-10

, 5-11

Henry’s constant decreases substantially with temperature, and is approximated using van’t

Hoff equation for normal butanol in water [109].

ln , 12.141, 5892 5-12

The coefficient of the gas diffusion in the liquid can be estimated from the rate of mass loss of

the paint as explained in section 4.3.3. If we assume that the concentration of solvent at the surface

is constant, which is a reasonable approximation for short intervals, Eqn. 4-14 reduces to Eqn. 4-15

which can be used to predict the gas diffusion coefficient.

Figure 5-8 shows plots of experimentally measured solvent mass loss, normalized by the initial

mass, as a function of √ / for both glycerin-butanol (Figure 5-8a) mixtures and paint (Figure

5-8b). Eqn. 4-15 predicts that the reduced curves should all be coincident if the value of Dv is

constant, which is observed to be the case for the first several minutes. The value of diffusivity

was estimated by fitting Eqn. 4-15 to the curves in Figure 5-8 for the first 4 minutes of evaporation

and it was found to slightly increased with increasing temperature from 2x10–11 m2/s at 100°C to

1x10–10 m2/s at 140°C.

87

(a) Glycerin-butanol solution

(b) Model Paint

Figure 5-8 Reduced desorption curves for (a) glycerin butanol mixtures and (b) mode paint

This value agrees well with typical values for the diffusivity of organic vapors in a polymer layer

which vary between 10-13 m2/s to 10-11 m2/s, depending on the amount of dissolved vapor in the

polymer [97]. The theoretically calculated solvent evaporation rate using a value of Dv = 8x10–11 m2/s

is shown in Figure 5-8 and it agrees well with the measured variation for the first few minutes. Then,

88

as Marangoni convection became well established, transported liquid from the substrate to the surface

of the paint film and the rate of solvent loss became much higher than that predicted by a diffusion

model.

Figure 5-9 shows the experimentally measured variation of bubble radius with time at three

different curing temperatures, compared with the bubble growth rate predicted by Eqn. 5-8 at

T=120° C with Dv =10–11 and Psat=109 kPa. The bubble growth rate was lowest at the start of

curing, indicating that the system was close to equilibrium (C∞/Cw ≈ 1) which agreed reasonably

with the predicted values of C∞ and Cw.

Figure 5-9 Bubble growth rate in paint films curing 100 °C, 120 °C, 140 °C

As bubble radius increases the gas pressure in the bubble reduces, resulting in higher mass flux

into the bubble, as C∞/Cw is now slightly larger than unity. However, the experimental results

showed a more rapid growth, with the transition occurring after 5 min at T=140° C, about 8 min

for T=120° C, and over 10 min for T=100° C. The growth rate was an order of magnitude higher

89

than that predicted by the model. Due to skin formation on the paint surface the solvent is trapped

inside the film, which becomes supersaturated and creates a large forcing pressure and a much

larger concentration gradient near the bubble boundary, driving solvent vapor into the bubble and

making it grow rapidly. Consistent with both experimental and analytical results, the initial

concentration of butanol vapor is 10% of the total solvent content of 6 moles/lit. According to the

analytical calculations this value, along with the concentration gradient, does not change

significantly. Although the butanol content decreased from 6 to 2.3 moles/lit due to butanol

evaporation from the surface, butanol vapor concentration increased from 10 to 30% of the total

butanol content after 30 minutes, resulting in C∞/Cw >1.3. This was obtained using the

experimental growth rate data and assuming constant diffusivity coefficient. Finally, the bubble

growth rate decreased again as the paint began to harden, increasing its viscosity and preventing

any further growth of the bubbles.

5.4 Bubble Motion

Figure 5-2 and Figure 5-6 both show that bubbles move as the paint cures and tend to cluster

together. This clustering is not confined to bubbles, but occurs for other particles present in the

paint film also. Figure 5-10 shows a glass substrate coated with a 1000 μm thick layer of paint that

was seeded with hollow glass tracer particles with a density of 60 kg/m3.

90

t = 0 s

t =36 s

t = 72 s

t = 106 s

t= 5min Figure 5-10 Particle agglomeration in 1000 μm paint film containing hollow glass particles deposited on glass substrates t<2 min

33.5 mm

91

At the start (t=0) the particle distribution is reasonably uniform across the paint layer. Within

a few seconds after heating the particles started to rearrange (see Figure 5-10, t = 36 s) forming

self-organizing structures. The particles formed clumps in the hardened paint layer instead of being

evenly distributed (see Figure 5-10, t=5 min).

The migration of both bubbles and particles can be attributed to the flows in a liquid film, arising

from surface tension gradients. Surface tension variation can arise from temperature or

concentration gradients in a thin evaporating film, if one component evaporates faster than the

others. The appearance of self-organizing patterns in thin fluid layers as a result of temperature

gradient, has been the subject of studies for over a century [110, 30, 32, 33, 34]. In polymer

coatings, concentration gradients that create surface tension gradients are the dominant force

driving Marangoni flows, the details of which discussed in detail in section 4.3 of this thesis [48]

[49].

Figure 5-11 shows the cellular structures forming on the surface of two binary systems. In

Figure 5-11(a) successive images show the early stages of curing for an approximately 1 mm

model paint layer on a stainless steel surface. The paint layer was lighted from above to show the

presence of any waves on the surface. Surface undulations were visible within approximately 30 s

of the paint film being placed in the oven. For thin paint films (~100 µm) the waves disappeared

after a few minutes, by which time almost all the solvent had evaporated while in the case of

thicker films (>500 µm) the surface waves remained until the paint had hardened, leaving an

uneven surface (see Figure 4-1) [111]. In the case of a 1 mm glycerin-butanol layer, Marangoni

cells were also visible within a few seconds of the sample being placed in the oven. However, in this

case they were not as prominent as they were for the paint and disappeared after approximately 10

minutes.

92

(a) Model Paint (b) Glycerin-butanol solution

t=0 s

t=36 s

t= 54 s

t=2.25 min Figure 5-11 Marangoni cell formation in (a) 1000 μm model paint films curing at 140°C and (b) 1000 μm Glycerin-butanol solution heated to 100°C

23 mm 23 mm

93

Figure 4-6 shows the mechanism that drives Marangoni flows. As explained in previous chapter

(see section 4.3), when the solvent evaporates from the surface, a concentration gradient is created

across the film thickness, with a higher concentration near the substrate and a low solvent

concentration on the surface. If there is a random variation in the solvent concentration at two

different places on the paint surface, the surface tension (σ) will be greater at the location with low

solvent concentration, triggering a flow along the surface from the region of low σ to that of high

σ. The paint on the surface will be replenished by liquid drawn from below, which will have a

higher solvent concentration, amplifying the surface concentration difference. This will create a

self-sustaining flow that will last as long as there is a sufficiently large concentration difference,

across the thickness of the paint layer. The flow creates polygonal cells in the liquid film (see

Figure 4-1 and Figure 5-11), with downward flow at the centers of the cells and upward flow along

their boundaries.

Figure 5-12 shows bubble motion in a 1000 µm thick layer of glycerin-butanol solution spread

on a stainless steel surface at a temperature of 80°C, in which both the liquid surface and bubbles

are visible. At t = 0 the bubbles are distributed randomly throughout the liquid layer. After only

30 seconds, self-organizing Marangoni cells start to form and the bubbles began to rearrange

themselves. Even though the shapes of the cells are not yet clearly visible, bubble motion appears

to be directed toward what become the centers of cells. As soon as the polygonal structures are

established (Figure 5-12, t = 1.5 min), the bubbles form clusters, located near the center of each

cell. Then, as cells grew larger and smaller ones merged with larger cells, the bubbles were pushed

to the edges of the cells (t = 4 min).

94

t = 0 min t= 6.3 min

t= 0.5 min t=8.6 min

t= 1.5 min t=11 min

t= 4 min t=13.5 min

Figure 5-12 Bubble agglomeration and escape in 1000 μm Glycerin-butanol solution on steel substrates at T= 80°C

5 mm

5 mm

23.5 mm

5 mm

5 mm

95

Bubble motion from the center and edges is representative of the circulating motion shown in

Figure 4-6, where bubbles are transported along with the liquid. Bubbles tend to accumulate near

the stagnation point of the vortex inside a recirculating flow [112]. As bubbles were brought to the

free surface the motion of the liquid pushed some of them through the liquid-air interface so that

they escaped. The bubbles density decreased, as seen at t= 6.3 min, and continued to decrease until

the liquid film was bubble-free at t= 13.5 min.

Bubbles in paint films did not disappear as rapidly as those in butanol-glycerin mixtures, since

the viscosity of paint increases much faster during drying due to cross-linking. Paint reaches a

rubbery or glassy state after curing takes place, when the shear viscosity at gel point approaches

infinity. Cheever & Ngo [50] observed that the dynamic viscosity of a clear-coat paint can increase

up to 10000 cp after only 10 minute of curing. At this point any bubble that has not escaped with

the aid of buoyancy forces or been pushed out by the Marangoni flows will remain in the paint

layer. As the concentration of solvent on the surface continues to diminish and a glassy skin is

formed, the particles clustered at the surface will cease moving while particles from the viscous

under-layers are brought to gather at the edges of the cells.

Once bubbles in the paint layer began to grow rapidly (t > 5 min) they became larger than the

size of the Marangoni cells and were too big to be transported by circulating flows. However, the

bubbles continue to move, appearing to be attracted to each other and forming clusters. Figure

5-13 shows the magnified inset of the area defined by the white circle in Figure 5-2.

96

t = 9 min

t = 10 min

t = 11.6 min

Figure 5-13 Bubble cluster formation in 1000 μm paint film curing at 140° C, t > 8 min

1

23

6.4 mm

97

The image sequence shows a cluster of bubbles forming, as the neighboring bubbles start to

agglomerate in a time interval of 2.6 minutes. Each image is accompanied by a velocity vector

map, with arrows showing the magnitude of bubble velocity. The pictures were analyzed using the

iterative PIV function in image analysis software (ImageJ, National Institute of Health) to obtain

a velocity vector field by cross-correlating two successive images, approximately 15 s apart, and

measuring the displacement of individual bubbles. The individual bubble velocity and center-to-

center bubble distance was measured. The bubble pair at the top of the image had the lowest

velocity compared to the individual bubbles numbered 1 to 3. The bubbles moved towards each

other and once a bubble collided with others its velocity decreased while the remaining bubbles

accelerated to agglomerate with the nearest cluster. The formation of numerous clusters can be

seen to form after t=8.5 min in Figure 5-2. Even as the bubbles clustered they continued to grow

larger due to diffusion of solvent into them. Eventually all bubble movement and growth stopped

when the paint hardened.

The mutual attraction of bubbles or particles floating on the surface of a liquid is well known,

leading to the formation of “bubble rafts” [80] [81]. A bubble projects above a free liquid surface

and creates an upward incline in the liquid meniscus around itself. Since the free liquid surface

rises upwards any other bubble in the vicinity experiences a buoyancy force that drives it up the

liquid meniscus so that the two bubbles move towards each other. When they touch the

deformation of the meniscus around them is enhanced, in turn drawing other bubbles towards

them. An analytic model of bubble motion [80] shows that the force of attraction increases as

bubbles get closer, which is in accordance with the observation that bubbles accelerated as the

distance between them diminished. Figure 5-14 shows the variation in velocity with time for

98

individual bubbles in Figure 5-13. When the separation is large the initial velocity is of the order

of 5 µm/s, which increases to as much as 25 µm/s as the bubbles get close to each other.

Figure 5-14 Corresponding velocity variation for individual bubbles in Figure 5-13

The paint film thickness used in these experiments was significantly higher than that used in the

automotive coating industry, resulting in longer time-scales, but the same mechanisms govern bubble

growth and agglomeration in both cases. As demonstrated by images in Figure 5-2, Figure 5-3, and

Figure 5-6, delaying the surface skin formation by decreasing the curing temperature reduces the

number of bubbles in the dry paint film. The absence of a barrier on the surface increases the time for

solvent to diffuse out of the paint film and also promotes surface-tension-driven flows that brings

bubbles to the surface from where they can escape.

99

5.5 Conclusion

The movement and growth of small air bubbles entrapped in drying paint films and glycerin-

butanol solutions were studied experimentally. Bubbles grew due to diffusion of solvent vapor into

them from the surrounding vapor. Bubble growth rate was initially slow, and agreed reasonably

well with predictions from an analytical model. Then, as the surface of the paint layer dried out

and polymer crosslinking occurred, a surface skin formed that trapped bubbles and prevented

solvent from escaping. The paint layer was supersaturated with solvent that was driven into the

bubbles, making them grow much more rapidly. In the case of glycerin-butanol mixtures, in which

no surface skin was formed, bubble growth rates remained low while bubbles rapidly burst through

the surface and escaped.

As solvent evaporated concentration gradients were created inside the liquid films, driving

Marangoni flows. Cellular structures were observed to form in both paint and glycerin-butanol

mixtures. Small bubbles initially clustered near the center of the cells and were then pushed to the

boundaries between neighboring cells. The bubbles in the glycerin-butanol mixture escaped quite

rapidly, leaving the film free of any bubbles. Bubbles that remained trapped in the paint layer grew

rapidly and became bigger than the size of the Marangoni cells. The bubbles moved towards each

other, driven by buoyancy forces, and formed clusters. This phenomenon will be discussed further

in the next chapter.

100

Chapter 6

Interaction of Growing Bubbles

6 Interaction of Growing Bubbles in Glycerin and Drying Paint Films

6.1 Introduction

Paint, which is typically a polymer dissolved in a solvent, is sprayed on automotive components

and then baked in an oven where the solvent evaporates while the polymer forms cross-links and

cures, forming a hard layer. When paint is sprayed on a surface a large number of air bubbles may

be entrained by impacting droplets and trapped in the deposited layer. These bubbles are examples

of objects that float on the interface, deform the meniscus leading to the interaction of particles

due to both capillary and buoyancy forces. Use of aluminum flakes, a few microns in diameter,

mixed with paint is being widely used by car manufacturers to develop metallic paint. The control

of color quality relies on the orientation and spacing between metallic particles during paint

application and drying. In the current chapter this study was undertaken to observe the process by

which bubbles grow and form clusters in the later stage of curing of paint layers.


The model paint formulation, with physical and chemical properties as explained in chapter 2

section 2.1, was used to coat heat-resistant borosilicate glass substrates (Model 8477K78, Mc-

MASTER-CARR, USA), 63.5 mm in diameter with 3.2 mm thickness prior to curing. The test

samples were cured using the experimental set up illustrated in Figure 2-7. Bubbles were

introduced in the liquid by agitating glass vials three quarters-filled. Pictures of liquid layers with

101

growing bubbles were analyzed using the threshold function in image analysis software (ImageJ,

National Institute of Health) to count the number of bubbles in each image, the cross-sectional area

of each bubble, the location of individual bubbles, and the distance between bubbles as they formed

clusters.

6.3 Results and Discussion As a result of sufficiently large concentration gradient in binary systems bubbles are transported

along with the self-sustaining flow within the liquid film. This has been discussed in detail in

section 5.4. This type of flow continues to bring the bubbles to the surface where they escape.

The characteristic growth rate predicted in section 5.3.2, was an order of magnitude lower than

the experimental results. Bubble radius variation after the transition time was almost linear with

time with dR/dt (mm/s) ranging from 0.0003-0.0005.

When bubbles became too large to be transported by the circulating flow in Marangoni cells,

they continued to move and were attracted to each other. A bubble that projects above the liquid

surface perturbs the liquid meniscus around itself, causing the bubbles in the vicinity to experience

a net upward force due to buoyancy. Since bubbles are constrained at the interface they tend move

along the meniscus until they touch each other, further increasing the distortion around themselves

[80] [81] (See Figure 6-1). The larger a group of bubbles the greater the buoyancy force created

around it, so bubble movement accelerates until all have been drawn into clusters in the paint film.

102

Figure 6-1 Dynamic of a floating bubbles in the vicinity of perturbed meniscus a second bubble or a cluster of bubbles

Figure 6-2 shows bubble growth and movement in a 1 mm thick paint curing at a temperature

of 120 ± 5°C. The paint was placed in a circular glass disc whose edges are visible around the

boundaries of the images in Figure 6-2. Small bubbles were introduced in the paint by stirring it

before spreading it on the glass surface. As curing progressed the bubbles grew larger as solvent

diffused into them (see t=4.1 min). The largest concentration of bubbles was around the edges of

the glass disc where air, trapped when the liquid was spreading on the surface, acted as nucleation

sites. Bubble growth was relatively slow at first, but then grew faster (t > 11.2 min). Some of the

bubbles disappeared as they burst through the surface of the paint film, but most of them survived

and grew larger. Bubbles in the paint film began to form clusters, moving towards each other. Two

103

circles, 10 mm and 13 mm in diameter respectively, are superimposed in Figure 6-2 on the images

for t ≥ 15.7 min, to identify groups of bubbles that grouped together. The bubbles accelerated as

the distance between them diminished which is an indication of the attraction force increasing as

bubbles get closer.

The length scale over which the interaction occurs may be estimated from the capillary length

(Eqn. 6-1):

6-1

For the properties of paint this length was estimated to be approximately 2 mm, which appears

to be a reasonable estimate of the distance over which bubbles were drawn to neighbouring bubbles

to form clusters. In Figure 6-2 two circles are drawn on images corresponding to t>15.7 min, with

radius 5 and 6.5 mm respectively, to show two regions in which clusters formed. The capillary

length therefore gives a reasonable order-of-magnitude estimate of the range over which bubbles

are attracted towards one another. The bubbles accelerated as the distance between them

diminished which is an indication of the attraction force increasing as bubbles get closer.

104

t = 0 min t = 15.7 min

t = 4.1 min t = 18.3 min

t = 7.6 min t = 20.2 min

t = 11.2 min t = 22.7 min

t = 13 min t = 24.5 min

Figure 6-2 Bubble growth and attraction in 1 mm paint film curing at T= 120 ± 5°C. The solid and dashed white circles are respectively 9.5 and 13 mm in diameter and identify the individual bubbles selected for center-to-center separation measurements

56 mm

105

Clustering of bubbles occurs not only in paint but can be observed in other types of liquids with

long-lived bubbles trapped at the interface. Figure 6-3 shows a sequence of images of an

approximately 5 mm pool of glycerin at room temperature. Bubble were mixed with glycerin by

agitating the container and depositing the content in a 63 mm glass petri dish. The sample was

immediately placed under the camera and still images were taken every 2 seconds. The bubbles

were spread randomly in the film as seen at t = 0 min, but the smaller bubbles were attracted to the

larger bubble (approximately 1 mm in diameter) indicated by the dashed circle in less than 4

minutes.

The capillary length as per Eqn. 6-1 is approximately Lc =2.2 mm for glycerin. The white circles

drawn in Figure 6-3 are 5 mm in diameter and show how bubbles confined within this area were

attracted to each other. Since the glycerin did not solidify many of the bubbles were able to burst

through the surface and disappeared. Eventually two main clusters were formed on the surface at

t = 10 min. After sufficiently long times (approximately 30 min), these islands of bubbles also

migrated to the wall of the container.

106

t = 0 min t = 6.5 min

t = 1.5 min t = 8.6 min

t = 3.2 min t = 10 min

t = 4.5 min

Figure 6-3 Bubble agglomeration in approximately 5 mm glycerin film at room temperature

37.5 mm

107

The presence of a wall deforms the interface just as a bubble does, with the interfacial curvature

depending on the liquid-solid contact angle (θ) (See Figure 6-4).

(a) (b)

Figure 6-4 Schematic of the interface curvature by the presence of the wall when (a) the liquid wets the wall (θ < π/2) and (b) liquid does not wet the wall (θ > π/2)

If the surface is hydrophilic, so that θ < 90°, the liquid meniscus rises upwards near the wall

(see Figure 6-4 (a)). Bubbles experience an upwards buoyancy force and move toward the highest

point of elevation of the liquid surface. Viewed from above it appears that the bubble is moving

towards the surface. Alternately, if the surface is hydrophobic with θ > 90, the liquid meniscus

rises upwards away from the solid wall and bubbles should move away from the surface (Figure

6-4 (b)).

To test this hypothesis paint in a glass container was agitated to entrap bubbles and 32 ml of the

liquid was deposited on a 63.5 mm diameter glass substrate (Model 8477K78, Mc-MASTER-

CARR, USA) to create an approximately 1 mm deep layer. A PTFE ring (Model 9559K62, Mc-

MASTER-CARR, USA) with an inside diameter of 19 mm, outside diameter of 22 mm, and

thickness of 1.8 mm, was immediately positioned on the surface that was then placed in an oven

at 120 ± 5°C. Photographs of bubble motion and growth were taken at 2 s interval.

108

Paint wets PTFE (θ ≈ 50°) so that the meniscus formed by a paint layer is concave upwards.

Bubbles are therefore expected to move towards the solid surface as shown in Figure 6-4 (a).

Figure 6-5 illustrates the behaviour of bubbles as the paint layer was cured. At t=0 the bubbles

were scattered across the paint surface, moving freely due to surface tension-driven flows resulting

from evaporation of solvent and concentration gradient. Bubbles in the neighbourhood of others

experience a net upward buoyancy force, move closer to each other and start to form clusters as

seen at t=13.5 min. At the same time small air bubbles trapped on the surface of the solid ring

when it was immersed in the liquid act as nucleation sites. Solvent evaporating from the paint fills

the paint bubbles, making them grow rapidly. The formation of a skin on the surface of the paint

layer inhibits solvent from leaving the surface and also traps bubbles in the paint layer so that they

cannot escape. The increasing viscosity of the paint also slows down the motion of bubbles. By

t = 20.3 min there are a large number of bubbles attached to the surface of the ring surrounded by

several bubble clusters.

109

t = 0 min t = 11.8 min

t = 1.25 min t = 13.5 min

t = 5 min t = 15.2 min

t = 8.5 min t = 20.3 min

Figure 6-5 Bubble growth and attraction to PTFE wall in 1 mm model paint layer curing at room T= 120 ± 5°C

39 mm

110

Glycerin does not wet PTFE very well (σ= 64 mN/m, θ ≈ 100°) so that the meniscus formed is

convex upwards. Bubbles should therefore move away from the wall as seen in Figure 6-4 (b). To

confirm this hypothesis bubbles were introduced in glycerin by agitating the liquid in a glass

container and depositing the contents in a 63 mm glass petri dish in which a PTFE ring (Model

92150A163, Mc-MASTER-CARR, USA) with an inside diameter of 19 mm, outside diameter of

33 mm, and thickness of 1.5 mm, was placed and time-lapse photos taken every 2 seconds.

Figure 6-6 shows a sequence of images of bubbles confined within the inner wall of a PTFE

ring in an approximately 5 mm deep film of glycerin at room temperature. Time t=0 s corresponds

to less than 10 seconds after the ring was placed on the surface. The bubbles were distributed

randomly throughout the liquid, but after only 20 seconds, buoyancy forces drove the bubbles

towards the highest point of the curved surface, which was at the center of the ring (see t=3 min).

Since the film did not solidify in this case bubbles continued to escape from the free surface until

at t=9.5 min the majority of bubbles had disappeared.

111

t = 0 s t = 4.5 min

t = 20 s t = 6 min

t = 2 min t = 8 min

t = 3 min t = 9.5 min Figure 6-6 Bubble clustering due to interfacial curvature in 5 mm glycerin layer with floating PTFE ring on the surface at room temperature

19 mm

Edge of the PTFE ring

112

6.3.1 The Dynamic of Floating Bubbles

To estimate the typical distance between bubbles as a function of time, following Nicolson’s

[80] approximation, we first consider an isolated bubble of radius R at rest on the surface of a fluid.

The bubble creates a meniscus around itself that forms an angle θ with the level liquid surface (see

Figure 6-7). The bubble radius R can be non-dimensionalized using the capillary length Lc to give

a dimensionless radius

6-2

In our experiments, α can be calculated at each time step using the bubble radius and

appropriate capillary length. The liquid contacts the sphere at a radius b and the angle between the

liquid meniscus and bubble surface is ψ.

Figure 6-7 Geometry of a bubble floating at a liquid-gas interface with a ring of contact of radius b. Zc is the height of fluid at the ring of contact, φ the semi-angle subtended at the centre of sphere by the circle of contact and ψ the liquid-bubble contact angle. The free surface is inclined at an angle θ to the horizontal plane

113

If Z is the height of the surface in the neighbourhood of one bubble, R the bubble radius, and r

distance from the vertical axis, we can introduce the dimensionless variables

, ,

The angle and the dimensionless length β can be related by examining the right-angled

triangle formed by the side with length b and hypotenuse R in Figure 6-7:

tan1

6-3

Nicolson [80] solved the Laplace equation to derive the shape of the free liquid surface around

a bubble and calculate the variation of Z with radial distance r, assuming that the capillary pressure

balanced the hydrostatic pressure difference over the height of the meniscus.

If the slope between the cap of the bubble and the remainder of the liquid meniscus is assumed

to be continuous at the ring where the liquid touches the bubble surface (where δ=β), we obtain

the equation [80]:

4

tan 6-4

The height to which the bubble rises above the level liquid surface is determined by equating the

upward buoyancy force it experiences (due to the mass of liquid displaced by the submerged

bubble volume) to the downward component of the surface tension acting on the circular contact

line with radius b. Solving these equations, Nicolson [80] tabulated values of β (the radius of the

114

liquid meniscus around the bubble) as a function of α (the dimensionless bubble radius), which are

shown in Figure 6-8.

Figure 6-8 Variation of β as a function of α plotted from Table 1 in [80].

Using the slope of the interface at the ring of contact

tan , Vella and Mahadevan [81]

found the following expression:

tan sin 6-5

Where B ≡ R2/ Lc2 is the Bond number and represents the relative magnitude of gravity and

surface tension forces. Eqn. 6-5 is valid for B<< 1, in which case surface tension forces become

significant. The value of B varies between 0.05-0.1 in the current experiments, within the

acceptable range for Eqn. 6-5 to be applicable. The second term on the right hand side of Eqn. 6-5,

115

Σ, is the dimensionless effective weight of the bubble, defined as its weight less the buoyancy

force it experiences [81].

Given the bubble radius, R, and fluid properties, we can calculate the dimensionless bubble

radius α from Eqn. 6-2. The radius at which the liquid meniscus attaches to the bubble, β, is

determined from Figure 6-8. For a given β the angles φ and ψ can be evaluated from Eqn. 6-3 and

Eqn. 6-4 respectively, and then Σ from Eqn. 6-5.

Knowing the shape of the meniscus around a bubble, Vella and Mahdevan [81] calculated the

potential energy (E) of an identical bubble placed next to it with horizontal center-to-center

distance L:

2 6-6

Differentiating the potential energy gives the force (F) on the bubble

2 6-7

Where K0 and K1 are modified Bessel functions of the first kind and of order zero and one

respectively.

The variation of the center-to-center distance of two identical bubbles, assuming that the

attractive force F is opposed by viscous drag, was derived to be

23

6-8

116

where μ is the dynamic viscosity of the liquid and L0 the initial bubble center-to center distance.

The drag force acting on a bubble floating on the liquid surface is less than that it would experience

if it was completely submerged: the drag scaling coefficient, D, is defined as the ratio of the drag

on a bubble partially emerging above the fluid to one that is completely immersed. In principle,

given the immersion depth, D can be calculated [113]. In practice it is difficult to measure,

especially if the bubble is growing since D varies with bubble size. In addition, in a drying paint

layer liquid properties such as viscosity are also changing with time. Therefore, in all subsequent

calculations D is treated as a fitting parameter.

We can find the dimensionless center-to-center distance as a function of dimensionless time

by rearranging Eqn. 6-8 :

123

∗ 6-9

Where t* is the dimensionless time and defined as

∗ 6-10

tc is the characteristic time and depends on the initial separation distance as follows:

6-11

117

For L/L0 to approach a theoretical zero, the second term of the radicand in Eqn. 6-9 must

approach 1. Therefore, assuming the bubbles are located within the distance estimated by the

capillary length (Lc / L0~1):

→ 0 23

∗ → 1

by using the average value of D we can find t* as:

∗~32

6-12

Eqn. 6-9 was used to fit data for bubble movement in our experiments. Measurements were

made of the displacement of two bubbles with radius 0.55 and 0.6 mm, starting when they were

approximately 4 mm from the center of a larger bubble, 1 mm in radius, in the glycerin layer shown

in Figure 6-3, identified by the circle drawn in the upper right hand corner of the frame at time t=0.

Figure 6-9 shows the experimentally measured variation of center-to center distances between

the larger (1 mm radius) and the two smaller bubbles (0.55 and 0.6 mm respectively), plotted on

dimensionless axes of L/L0 as a function of t*. Time t*=0 identifies the instant when the smaller

bubbles were within the length scale over which the meniscus around the larger bubble was

estimated to be curved (L0 ~ Lc + 2 R0), when buoyancy forces would act on the small bubbles.

The value of capillary length for glycerin, with surface tension of 63.4 mN/m and density of 1260

kg/m3, was 2.27 mm. Σ was calculated using the procedure outlined above and substituted along

with the physical properties of the liquid and the initial centre-to-centre distance into Eqn. 6-9.

118

Figure 6-9 Experimental data of Figure 6-3 (bubbles being drawn to the clusters in the area confined by the white circle) compared to the asymptotic solution for center-center distance of two identical spheres of radius 0.6 mm and 0.55 with center-to-center distance of L0=4 mm, in glycerin with σ=63.4 mN/m, μ=1.3 N-s/m2.

If D was constant, all the data should collapse on one curve. In reality, D, which quantifies the

drag force on the bubble, changes as the bubble moves and rises above or below the liquid surface.

We varied the value of D to best fit Eqn. 6-9 to the experimental data. Figure 6-9 shows curves for

bubble separation distance calculated using values of D= 0.09 for the bubble with radius of 0.6

mm and D= 0.045 for the bubble radius of 0.55 mm. The initial separation distance was L0=4 mm

in both cases. The equation accurately predicted the variation in bubble velocity, with both bubbles

moving slowly at first when the separation was large, and then accelerating as they came closer to

another bubble.

119

Whereas the bubbles in a glycerine layer have constant diameter, those in a drying paint layer

are growing in size, which adds another level of complexity to analyzing their motion. The bubble

radius variation with time was measured from photographs, and the value of Σ recalculated at each

time step. Using these values Eqn. 6-9 was used to model bubble movement in paint layers.

Figure 6-10 illustrate bubble separation as a function of time for clusters forming in the areas

demarcated by the solid and dashed white circles of Figure 6-2. The bubbles chosen were all

approximately the same size, varying in radius from 0.3-0.4 mm, with t*=0 indicating the time

when each bubble reached within approximately 3 mm of another bubble or cluster of bubbles.

Separation distance measurements were taken of the bubbles within the solid white circle being

attracted to the pair of bubbles within the same circle, which were approximately the same size as

those chosen for the measurements. As seen in Figure 6-10, the model corresponds quite well with

the experimental measurements. All of the bubble separation distances followed similar variations,

speeding up towards each other as they grew closer. The curves drawn in Figure 6-10 represent

Eqn. 6-9 drawn with values of D equal to either 0.02 or 0.05. Most of the data points lie within the

envelope of these two curves, indicating that the average value of D lies in this range.

120

Figure 6-10 Experimental data of Figure 6-2 (bubbles clusters in the area confined by the solid and dashed white circles) compared to the asymptotic solution for center-center distance of two identical spheres of radius 0.35 mm and with center-to-center distance of L0=3 mm in paint with σ=26 mN/m, μ=10 N-s/m2

It is possible to estimate the time for bubbles to come together and form a cluster using

Eqn. 6-12, if we know the value of D. For example, using an approximate value of D=0.035 from

the results of Figure 6-9, we calculate that it will take 46 s for two identical bubbles in a pool of

glycerin, both with 0.5 mm radius and initially separated by a distance L0=3 mm, to come together

and form a cluster.

Figure 6-11 shows the effect of bubble radius on the time required for two bubbles in a paint

film, with initial radius varying from 0.3 mm to 0.65 mm, to come together after being initially

separated by a distance L0=Lc. The value of D=0.035 was used, bubble diameter was assumed

constant in these calculations and bubble growth neglected. As bubble radius increases the time

121

required for a cluster to form decreases. Bubbles with a radius of 0.3 mm took approximately 9

min to form a cluster, while those 0.5 mm in radius came together in less than 1 min. Bubble

clusters that develop before a skin forms on the surface of the paint, which takes approximately 5

min, are trapped in the paint layer and appear as a defect. Larger bubbles, that move towards each

other most rapidly, will be the most obvious in what appear as cloudy patches in the paint layer.

Figure 6-11 Time for cluster formation in paint as a function of constant bubble radius

6.4 Conclusion

The movement and growth of small air bubbles entrapped in drying paint films and

glycerine layers were studied experimentally. Bubbles in paint initially grew slowly due to

diffusion of solvent vapor into them. Then as a solid skin grew on the surface of the paint the

solvent could no longer escape and was forced into the bubbles, leading to rapid growth.

122

Bubbles that remained trapped in the paint layer grew rapidly and moved towards each other,

driven by buoyancy forces. Hardening of the paint layer finally arrested bubble growth and

movement. Each bubble rising above the free surface of the liquid creates an upward curving

meniscus. Neighbouring bubbles experience buoyancy forces that drive them up the rising liquid

surface and towards the first bubble so that clusters are formed. A hydrophilic solid surface also

creates an upward rising meniscus where the liquid wets it, so that bubbles move towards the

surface, while a hydrophobic surface creates a downwards-curving meniscus, so that bubbles move

away from it. An analytical model of the shape of the meniscus around a bubble was used to

analyze bubble dynamics, and its predictions agreed reasonably well with measurements of bubble

displacement. The time for bubbles to approach each other and form clusters decreases as bubble

diameter increases.

123

Chapter 7

Closure

7 Summary and Conclusion

The mechanism that drives fluid motion in thin films consisting of a binary system with one

component evaporating faster than the other, is widely known as solute-driven Marangoni

convection. As a result of sufficiently large concentration difference across or along the surface, a

self-sustaining flow within polygonal cells is established which was experimentally studied by

observing the curing of uniform layers of melamine-based thermosetting paint. Various defects

were observed to form because of variation in physical properties such as surface tension and

viscosity as a result of solvent evaporation.

In this study the effect of film thickness on the onset and evolution of self-organizing

Marangoni cells, and eventually “orange peel” defect, was investigated. This was quantitatively

characterized by evaluating the dimensionless Marangoni number at different time steps during

curing for various thicknesses. Experimental measurements of the decrease in sample weight and

a one-dimensional evaporation model were used to find the diffusivity of the solvent and determine

concentration gradients and Marangoni numbers.

The effective diffusivity was found to be a function of paint film thickness and time, decreasing

as the paint cured. The variation of Marangoni number was in accordance with the photographs of

the paint surface that showed the onset and evolution of cellular structures. Experimental and

analytical studies confirmed that the formation of orange peel depends on the relative magnitude

of the time (tD) for solvent to diffuse completely out of the paint film and the time for the paint to

124

cure (tC). Therefore, altering process parameters such as curing temperature, paint film thickness,

paint composition, and physical properties to obtain tD<tC can eliminate “orange peel” defect.

Cellular structures were found to be also responsible for bubble migration and formation of

clusters when bubbles were smaller than Marangoni cells. It was experimentally shown that

surface-tension-driven flows initially help to reduce bubble density before solidification begins.

Then, solvent evaporation from non-solidifying glycerin and solidifying paint films were

compared analytically and experimentally to determine the role of skin formation in trapping

bubbles in the paint film. The diffusion model was extended to explain and predict bubble growth

in drying paint films.

Bubbles that remained trapped in the paint layer grew rapidly and became bigger than the size

of the Marangoni cells. They moved towards each other, driven by buoyancy forces, and formed

clusters. The hypothesis that the bubbles in paint tend to move toward each other as a product of

the buoyancy force of one bubble and the elevation of the free surface in the meniscus of the second

bubble, was validated by using an analytical model of bubble dynamics. The model agreed well

for bubbles in glycerin and was extended for verifying the motion of growing bubbles in paint.

Hardening of the paint layer finally arrested bubble growth and movement.

7.1 Contributions

The results of Chapter 4 can provide the criteria for practitioners to predict probability

of “orange peel” formation in the coating process. It is confirmed that the formation of

orange peel depends on the relative magnitude of diffusion time scale and curing time

scale. This information can be obtained for any paint formulation through tests on

125

physical and chemical properties of coatings. Even without the aid of flow

visualization, this approach can help optimize the oven and spray parameters more

efficiently to eliminate possible defects.

In Chapter 5 and 6 two various mechanisms were confirmed for bubble clustering in binary

mixtures. These findings suggest the primary reason for significant color variation when

metallic flakes are mixed with paint and can lead to effectively controlling the color quality.

Applying these results, along with the chemo-physical data, can help using the flash-off

time to successfully control bubble clustering and the problem of blisters and pin holes in

any coating process.

7.2 Future Work

The finding of the study can be used as a basis for future work in the field of transport

phenomena and prevention of defects in drying paint layers.

The variation of rheology with curing, addition of surfactants, and curing temperature

can be investigated and incorporated into calculation of Marangoni number and

elimination of “orange peel” defect.

The study of bubble clustering in the presence of Marangoni flow, can be extended to

manipulate color quality in paints mixed with pigments and metallic flakes.

Surface-tension-driven Bénard-Marangoni convection in binary systems may also be

used for the systematic development of multifunctional surface coatings.

126

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