Lessons learned while watching paint dry: film formation, drying fronts, and cracking

Transcription

1 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

2 early time Film Formation, Non-Uniformities, and Cracking Driven by Capillary Pressure transparent film water flows stresses crack

3 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.

4 MODEL FOR ONE-DIMENSIONAL FILM FORMATION n!"! n = #R 2a Hertz 1882, Matthews close-packed spheres E γ wa % d F nm =! 16 " dt 3 a2 & G contact 2 1! $ force (! 33 = " p cap #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

5 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

6 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 (1994) radii [nm] 350 dry sintering dry sintering 0.21!( T )a " pa receding front capillary wet

7 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

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9 Film forming latices T g < T amb

10 Low T g Film Forming Latex: WCFA φ ο = a = 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

11 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

12 T amb < T g => non-film forming latices large particles => high permeability packing

13 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 (PS b)

14 evaporation complete

15 Observations from behind the Glass Slide Polystyrene particles, 2a=342 nm, φ= mm

16 large high T g PPG342 cracking followed by debonding or failure at first layer of particles small high T g PMMA95

17 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

18 T amb <T g => non-film forming latices small particles => low permeability packings

19 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 (PMMA b)

20 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?

21 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

22 homogeneous linearly elastic film A.G. Evans, M.D. Drory, and M.S. Hu, J. Materials Res (1988) J.L. Beuth, Int. J. Solids Structures (1992) X.C. Xia and J.W. Hutchinson, J. Mech. Phys. Solids (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

23 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 " $N,!E $NGH 2( 1! %)" voids close before cracking cracking favorable

24 Direct Measurement of Stresses high pressure filter chamber: gas in camera pressure meter Teflon o-ring polycarbonate chamber film porous support membrane water out

25 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

26 smooth top surface fracture surface from below junction of two fractures ordered films 91 nm PS

27 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

28 Capillary Pressure to Open or Extend Finite Crack crack crack P ( L ) / P ( ) cap cap crack + crack P ( L ) / P ( ) cap cap open crack of length L 1 extend flaw of length L L / H Data implies flaws with L<< h for randomly close packed layers, but L h for ordered dispersion.

29 Critical Thickness via Spin Coating K.B. Singh & M.S. Tirumkudulu (in preparation) prediction } }Cima, et }al } 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 = % #NGa ( & 2( 1! $ )" ) * H crit 2/ 3

30 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.