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LESSONS FROM WATCHING PAINT DRY: film formation, moving fronts, and cracking William B. Russel Department of Chemical Engineering Princeton University ICSCS 2006 Beijing Collaborators P.R. Sperry Rohm & Haas A.F. Routh *97 Cambridge Univ. M. Tirumkudulu IIT Bombay Weining Man *05 OUTLINE one-dimensional film formation regimes temperature dependence measurements with cantilever capillary stresses drying fronts cracking: theory & experiment critical capillary pressure patterns and spacing critical thickness

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LESSONS FROM WATCHING PAINT DRY:film formation, moving fronts, and cracking

William B. RusselDepartment of Chemical Engineering

Princeton UniversityICSCS 2006

Beijing

CollaboratorsP.R. Sperry Rohm & HaasA.F. Routh *97 Cambridge Univ.M. Tirumkudulu IIT BombayWeining Man *05

OUTLINEone-dimensional film formation

regimestemperature dependence

measurements with cantilevercapillary stressesdrying fronts

cracking: theory & experiment critical capillary pressure

patterns and spacingcritical thickness

Driving Forces for Homogeneous Deformation1. Wet sintering (Vanderhoff, 1966)

polymer/water surface tension

2. Dry sintering (Dillon, 1951)polymer/air surface tension

3. Capillary Deformation (Brown, 1956)pressure at air-water interface

2!pa

a

!pw

a

! pcap

= !"#wa

$ 10 !12#wa

a

In all three negative pressure puts the film in tension. The substrate preventslateral deformation, while free surface allows compression in normal direction.

MODEL FOR ONE-DIMENSIONAL FILM FORMATION

contact non-linear viscoelastic response force Hertzian contact (Matthews 1980)

------------------------------------------------------------------------------------------------------------------------------------------------------------------------

initially isotropic thin film of close-packed spheres

γwa scaling H pcap a

t φm

˙ E

Fnm

= !16

3a

2G

2 1!"( )+#

d

dt

$%&

'&

()&

*&!n

nmi+ in

nm( )

3/ 2

nnm

!33= " p

cap"

2

3#$N

G

2 1"%( )+&

d

dt

'()

*)

+,)

-). 3/ 2

= 0

t = !Et / H ! t =128apcap

15N"m2# wa

G =HG$

'

!E%

evaporation time

relaxation time

& =%a !E

H# wa

viscous collapse time

evaporation time

n ! " !n = #R R

! = "!33= 1"#

m#

dry sintering

receding water front

capillary deformation

wet sintering

!Ro

˙ E

"wa

H

G!'H

" ˙ E

1

10

100

1000

10000

0 5 10 15 20 25 30

interfacial tensiondriving force

particle-air

water-air/particle-air

water-air

particle-water

void closureevaporation

ratios oftime scales

evaporation relaxation

Generalized Process Map

REGIMES OF DEFORMATION

key dimensionless groups

dry sintering dry •

• dry receding water front

dry •

! =˙ E R"

H#wa

capillary

wet • partial skin

wet • skinning •

wet sintering • wet skin

Keddie

Lin & Meier

Dobler et al. Pe =6$µRH ˙ E

kT

0.01

1

100

10000

0.01 0.1 1 10 100

! =time for viscous collapse

time for evaporation Pe =

time for evaporation

time for diffusion

0

1

10

100

1000

10000

100000

-5 0 5 10 15

T-T g

time to close pores

wet sintering

capillarydeformation

dry sintering

tcomp

0.26! T( )Ro"pw

0.36H

˙ E

0.26! T( )Ro"pa

500 nm

50 nm

wet

dry

capillary

recedingfront

˙ E tcomp

H

What about experiments?

EFFECT OF TEMPERATURE

T-Tg

P.R. Sperry, B.S. Snyder, M.L. O’Dowd, P.M. Lesko,“Role of water in particle deformation and compaction in latex filmformation” Langmuir 10 2619 (1994)

minimum film formation temperature depends onglass transition temperaturediameter of particlestime

hotdry wet

cold

Tg ->radii [nm]350

282 dry sintering233175

temperaturegradient bar

non-uniform evaporation

pressure driven flow

compression

tension

Film Formation, Non-Uniformities, and CrackingDriven by Capillary Pressure

early time

transparent film

water flows

stresses

Cantilever Experiment for Measuring Stresses Chiu et al. J. Am. Ceramic Soc. (1993); Peterson, et al. Langmuir (1999);

Martinez, et al. Langmuir (2000)

laserpositiondetector

mirror

copper substrate

latex dispersion

α

! xx =hs3G"

6L h h + hs( )

hs : substrate thicknessh : film thicknessL : length of filmG : bending modulus

Film forming latices

Tg < Tamb

Low Tg Film Forming Latex: WCFAφο=0.32−0.35 2a = 290 nm Tmft =16oC

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.0 0.5 1.0 1.5 2.0

N =34

N =61

N =81

N =400

Rel. Humidity = 12-44%

Temperature = 20-23.4 oC

Evap'n Rate = 2.9 - 5.6 µm/min

Dry film thks. = 10 - 116 µm

ˆ ! =! a

2"

ˆ t =t hwet 1! "o( )

˙ E

evaporation completeand film turns clear

decrease in stress with filmthickness => viscous response

Tamb< Tg => non-film forming latices

large particles => high permeability packing

Film Cracking: Large High Tg Particles

2a=342 nm, hdry=79.5 µm, E=2.5 µm/min, RH=66%, T=23.5 oC

(PS342-062602b)

2 mmObservations from behind the Glass SlidePolystyrene particles, 2a=342 nm, φ=0.46

large high Tg PPG342

⇒ cracking followed by“debonding” or failureat first layer of particles

small high Tg PMMA95

evaporationcomplete

When particles deform viscously, capillary stressdrives film formation before onset of cracking.

non-film forming[elastic]

film forming[viscous]

apcap

crit

!

Tamb<Tg => non-film forming latices

small particles => low permeability packings

Film Cracking: Small High Tg Particles 2a=95 nm, hdry=101 µm, E=6.7 µm/min, RH=35%, T=24.4 oC

(PMMA95-080802a)

Film Cracking: Small High Tg Particles 2a=95 nm, hdry=262 µm, E=6.7 µm/min, RH=35%, T=24.4 oC

(PMMA95-080802b)

Crack SpacingPMMA95: 2a=95 nm

N =hwet!o2a!rcp

W

2a

simple and reproducible results, but how to

• measure in more controlled manner

• understand the mechanism

Cracking in Thin Filmshomogeneous linearly elastic films

A.G. Evans, M.D. Drory, and M.S. Hu, J. Materials Res 3 1043 (1988)J.L. Beuth, Int. J. Solids Structures 29 1657 (1992)X.C. Xia and J.W. Hutchinson, J. Mech. Phys. Solids 48 1107 (2000)

Griffiths criterion recovery of elastic energy

||

cost of surface energy

==> critical stress as function of film thickness for isolated crackspacing as function of film thickness and excess stress

!E = h "33d#

33

'+"

22d#

22

'+ 2"

23d#

23

'{ }$$ dx2

||

2h%

!33

o= " p

o"

2

3#$N

G

2 1"%( )&

o

3/ 2=0

!11

o= !

22

o="N

2#

G

2 1$%( )&

o

3/ 2

!11

'=

5

16"#N

G

2 1$%( )&

o

1/ 2&11

'

!13

'=

1

2"#N

G

2 1$%( )&

o

1/ 2&13

'

!33

'= " p'"

3

4#$N

G

2 1"%( )&

o

1/ 2&11

'=0

' p' =3

4#$N

G

2 1"%( )&

o

1/ 2&11

'

• one-dimensional base statecompression

normal to film

tension inplane of film

• two-dimensional stress fields after cracking without dilation

stress-free air-water interface relaxation in plane

!11+ !

33= 0

x3

x1

! "11

'

!x1

=

"13

'

z=0

h

lubrication approximation u1

' ! 3 u1

'x

3

h1"

x3

2h

#

$%&

'(

)13

'

z=0

! 3 u1

'2h and )

13

'! 3 u

1

'4h

!2

u1

'

!x1

2=

6

7

u1

'

h2

with "o+

7

4

! u1

'

!x1

0( ) = 0 u1

'#( ) = 0

• vertically averaged stress balance

• equation for displacement

stress-free crack uniform filmsurface

u1 u3

critical stress for isolated crack

pcapcrita

!= 1.17

a

h

"#$

%&'

3/5GN(oa1)*( )!

"

#$%

&'

2/5

+ pcapcrith

!= 2.0

Gh

1)*( )!"

#$%

&'

2/5

spacing W between parallel cracks as function of excess stress

pcapcrit

pcap=

3

8tanh

6

7

W

2h

"

#$%

&') 5 sinh

6

7

W

h

"

#$%

&')

6

7

W

h

,

-..

/

011

6cosh2 6

7

W

2h

"

#$%

&'234

54

674

84

3/5

minimum thickness for cracking pcapcrit

= 10! a

hcrit

a= 0.068

Ga

1)*( )!"

#$%

&'

2/3

• solve boundary value problems• integrate to evaluate recovery of elastic energy• equate to surface energy

Direct Measurement of Stresses

high pressure filter chamber:

gas in pressure meter

film membraneporous support water out

Teflon o-ring

camera

polycarbonatechamber

Experimental Procedure

• Dispersions consist of polymer latices below Tg or silica.

• Push water out slowly at low pressure (50 kPa) to create close packing.

• Water vapor saturates air inside the chamber, stopping evaporation.

• Gradually increase pressure until the film cracks to determine directlythe critical capillary pressure.

• Continue increasing pressure in steps and photograph cracks todetermine dependence of crack spacing on pressure.

• Remove film and weigh while still wet and then after drying todetermine volume fraction and thickness of film.

Film as onset of cracking

Parametric Dependence of Cracking Stress

crackingpressureindependent ofparticle radius

variablesp kg/m-sh mγ kg/sG/(1−ν) kg/m-s

dimensional analysis two dimensionless variablesone a function of the other

ph

!= f

Gh

1"#( )!$

%&'

()

Dimensional Analysis

Balancing elastic energy recovered by cracking against the additionalsurface energy ( ) provides lower bound on capillary pressurethat produces cracking.

Cracking spacing at pressures above pcrit

A = area of film parallel cracks W=A/LL = total length of cracks orthogonal cracks W=2A/L

triangular cracks W=3A/L

Normalized spacing as function of excess stress

debonding

• Debonding terminates cracking by relaxing stress.• Theory for parallel cracks misses phenomenon.

Subsequent cracking controlled by distribution of flaws?

prediction

Critical Thickness via Spin CoatingK.B. Singh & M.S. Tirumkudulu (in preparation)

hypothesis: When capillary pressure exceeds 10γ/a withoutcausing cracking, the water front simply recedes into film.

10!a

"h

!= 2

Gh

1#$( )!%

&'(

)*

2/5

hcrit

a= 0.068

Ga

1!"( )#$

%&'

()

2/3

}}Cima, et}al. 1993}

prediction

SUMMARY• Mode of film formation depends on rate of evaporation relativeto rate of viscous deformation driven by capillary pressure.

• Edges cause gradients in capillary pressure that draw fluid fromthe center to sustain mass transfer limited evaporation.

• Particles that deform too slowly allow the capillary pressure tocause cracking before the water front recedes into the film.

• The capillary pressure required for cracking decreases withincreasing film thickness and elastic compliance but isindependent of particle size. The density of cracks increases withwith excess pressure but appears to be nucleation controlled.

• There exists a thickness below which films of even hardparticles do not crack.