lessons from watching paint dry: film formation…rrg/lectures/film_formation.pdf · lessons from...
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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
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
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)
large high Tg PPG342
⇒ cracking followed by“debonding” or failureat first layer of particles
small high Tg PMMA95
When particles deform viscously, capillary stressdrives film formation before onset of cracking.
non-film forming[elastic]
film forming[viscous]
apcap
crit
!
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.
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.