Lessons learned while watching paint dry: film formation, drying fronts, and cracking William B. Russel Department of Chemical Engineering Princeton University Procter and Gamble July 2007 OUTLINE one-dimensional film formation regimes temperature dependence observations on a cantilever capillary stresses drying fronts cracking: theory & experiment critical capillary pressure patterns and spacing critical thickness Collaborators P.R. Sperry Rohm & Haas A.F. Routh *97 Cambridge Univ. M. Tirumkudulu IIT Bombay Weining Man *05
early time Film Formation, Non-Uniformities, and Cracking Driven by Capillary Pressure transparent film water flows stresses crack
Driving Forces for Homogeneous Deformation 1. Wet sintering (Vanderhoff, 1966) polymer/water surface tension! pw a 2. Dry sintering (Dillon, 1951) polymer/air surface tension 2! pa a 3. Capillary Deformation (Brown, 1956) pressure at air-water interface! p cap =!"# wa $ 10!15 # wa a In all three negative pressures put the film in tension. The substrate prevents lateral deformation, while free surface allows compression in normal direction.
MODEL FOR ONE-DIMENSIONAL FILM FORMATION n!"! n = #R 2a Hertz 1882, Matthews 1980 ------------------------------------------------------------------------------------------------------------------------------------------------------------------------ close-packed spheres E γ wa % d F nm =! 16 " dt 3 a2 & G contact 2 1! $ force (! 33 = " p cap + 0.69 #N$ a " #NG 6% 1" & ( ) H p cap a scaling!et t φ t = rcp H 1! " o! = 1 (!n nm i#in nm ) 3/ 2 n nm viscous ( ( )!n i#in nm nm ) 3/ 2 n nm elastic ( ) ( ) ( exp G t" t + * - d. 3/ 2 t 0 dt = 0 "/ ) 2 1" &, dt! = "! 33 = 1" # rcp # -p cap $ 12.5% aw a G = HG %!E& ( ) # t = ap cap N"$ wa evaporation time relaxation time = &a!e viscous collapse time H$ wa evaporation time
viscous closure evaporation Generalized Process Map dry sintering interfacial tension driving force particle-air receding water front capillary deformation wet sintering water-air/ particle-air water-air ratios of time scales evaporation relaxation particle-water
time to close pores t film wet sintering 0.21!( T )a " pw capillary deformation 0.36 Ḣ E EFFECT OF TEMPERATURE 500 nm dry 50 nm P.R. Sperry, et al., Langmuir 10 2619 (1994) radii [nm] 350 dry sintering 282 233 175 dry sintering 0.21!( T )a " pa receding front capillary wet
Cantilever Experiment for Measuring Stresses Chiu et al. J. Am. Ceramic Soc. (1993); Peterson, et al. Langmuir (1999); Martinez, et al. Langmuir (2000)! xx = h s 3 G" ( ) 6 L h h + h s laser position detector h s : substrate thickness h : film thickness L : length of film G : bending modulus latex dispersion z x copper substrate mirror α
Film forming latices T g < T amb
Low T g Film Forming Latex: WCFA φ ο =0.32 0.35 2a = 290 nm T mft =16 o C! a 2" wa evaporation complete and film turns clear ˆt = t H ( 1! " o )!E decrease in stress with film thickness => viscous response
Capillary pressure required to form film ε = 0.36 elastic limit p cap film ~φng/2(1-ν) viscous limit p film cap ~!N"!E H GH 2( 1! ")#!E
T amb < T g => non-film forming latices large particles => high permeability packing
Film Cracking: Large High T g Particles 2a=342 nm, h dry =79.5 µm, E=2.5 µm/min, RH=66%, T=23.5 o C (PS342-062602b)
evaporation complete
Observations from behind the Glass Slide Polystyrene particles, 2a=342 nm, φ=0.46 2 mm
large high T g PPG342 cracking followed by debonding or failure at first layer of particles small high T g PMMA95
non-film forming [elastic] crit ap cap! theory/3 film forming [viscous] When particles deform viscously, capillary stress drives film formation before onset of cracking. N = H! o 2a! rcp
T amb <T g => non-film forming latices small particles => low permeability packings
Film Cracking: Small High T g Particles 2a=95 nm, h dry =262 µm, E=6.7 µm/min, RH=35%, T=24.4 o C (PMMA95-080802b)
Film Cracking: Small High T g Particles 2a=95 nm, h dry =101 µm, E=6.7 µm/min, RH=35%, T=24.4 o C W/2a H/2a simple and reproducible results, but how to understand the mechanism?
Cracking in Thin Elastic Films Griffiths criterion recovery of elastic energy cost of surface energy x 3 one-dimensional base state compression normal to film x 1 G =!NG ( ) 2" 1# $ tension in plane of film! o 33 = " p o " 2 G# 3/ 2 =0 3 o! o 11 =! o 22 = 1 G" 2 2 o3/ perturbed stress fields after cracking without dilation! 11 +! 33 = 0 stress-free air-water interface relaxation in plane! 33 = " p " 5 G# 1/ 2 8 o # 11 =0! 11 = 3 G" 1/ 2 4 o " 11 $ p = " 5 G# 1/ 2 8 o # 11! 13 = 1 G" 1/ 2 2 o " 13
homogeneous linearly elastic film 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) vertically averaged stress balance u 1 u 3! " 11 = " 13!x 1 H z =0 "lubrication" approximation u 1 ) 13! 3 u z =0 1 2H and ) 13 x! 3 u 3 # 1 H 1" x 3 % $ 2H! 3 u 1 4H & ( equation for displacement! 2 u 1 = u 1 2!x 1 H with 2 p cap " o G +! u 1 0!x 1 ( ) = 0 u 1 (#) = 0 stress-free crack uniform film surface
solve boundary value problems evaluate recovery of elastic energy equate to surface energy! Hp crack cap " minimum capillary pressure for opening crack & $NGH ) # 1.3 ( 2( 1! %)" * + 2/ 5 " + 2.9 $N,!E $NGH 2( 1! %)" voids close before cracking cracking favorable
Direct Measurement of Stresses high pressure filter chamber: gas in camera pressure meter Teflon o-ring polycarbonate chamber film porous support membrane water out
Capillary Pressure Required for Cracking pressure at first crack independent of particle radius variables two dimensionless variables ph/γ Gh/(1-v)γ p kg/m-s one must be function of the other h m energy criterion γ kg/s! hp crack 2/ 5 cap % Gh ( # 2.0 G/(1 ν) kg/m-s " & ( 1! $ )" ) * energy criterion lower bound disordered packings notched O-rings packing flaws initiate cracks ordered packings grain boundaries and notches nucleation sites crystal
smooth top surface fracture surface from below junction of two fractures ordered films 91 nm PS
Normalized spacing as function of excess stress prediction debonding spacing W between cracks as function of excess stress crack pcap pcap! W $ W 2! W $+ = (tanh # * 11cosh #" &%, &% " 2H 2H 2H ) Debonding terminates cracking by relaxing stress. Theory for parallel cracks misses phenomenon. Subsequent cracking controlled by distribution of flaws 3/5
Capillary Pressure to Open or Extend Finite Crack 4 3.5 3 crack crack P ( L ) / P ( ) cap cap crack + crack P ( L ) / P ( ) cap cap 2.5 2 1.5 open crack of length L 1 extend flaw of length L 0.5 0 0.5 1 1.5 2 2.5 3 L / H Data implies flaws with L<< h for randomly close packed layers, but L h for ordered dispersion.
Critical Thickness via Spin Coating K.B. Singh & M.S. Tirumkudulu (in preparation) prediction } }Cima, et }al. 1993 } hypothesis: When capillary pressure exceeds 12.5γ/a without causing cracking, the water front simply recedes into film.! p max cap H " % #NGH ( = 1.30 & 2( 1! $ )" ) * 2/ 5 + a = 0.023 % #NGa ( & 2( 1! $ )" ) * H crit 2/ 3
SUMMARY Mode of film formation depends on rate of evaporation relative to rate of viscous deformation driven by capillary pressure. Particles that deform too slowly allow the capillary pressure to cause cracking before the water front recedes into the film. The capillary pressure required for cracking decreases with increasing film thickness and elastic compliance but is independent of particle size. The density of cracks increases with excess pressure. The onset of cracking and additional cracking at higher pressures appears to be flaw/nucleation controlled. Cracking is not favorable below a minimum thickness even for layers of hard particles.