Cracking in Soft-Hard Latex Blends: Theory and Experiments

Karnail B. Singh, Girish Deoghare, and Mahesh S. Tirumkudulu*

Department of Chemical Engineering, Indian Institute of Technology-Bombay, Mumbai 400076

ReceiVed September 1, 2008. ReVised Manuscript ReceiVed October 13, 2008

Volatile organic compounds (VOCs) in the traditional paint and coating formulations are an important health andenvironmental concern, and current formulations are increasingly moving toward water-based dispersions. However,even within the water-based systems, small quantities of organic solvents are used to promote particle coalescence.One route to achieving this goal has been to use mixtures of soft and hard particles, also known as latex blends. Weinvestigate the drying of colloidal films containing mixtures of silica and acrylic particles. Since both the particlesdeform only slightly at room temperature, this work investigates the cracking behavior of films containing elasticparticles of two different elastic moduli. We extend an existing model for the stress versus strain relation for identicalparticles in a colloidal film to that containing a mixture of equal-sized hard and soft elastic spheres while accountingfor the nonaffine deformation. A transition from soft to rigidlike behavior is observed beyond a critical hard particlevolume fraction ratio that matches with published results obtained from computer simulations. The model predictionsare validated with extensive experimental data on the critical stress and critical cracking thickness for various ratiosof hard and soft particle volume fraction.

1. Introduction

Adverse impacts on health and the environment are forcingpaint and coating manufacturers to replace the volatile organiccompounds in traditional formulations with water-based for-mulations, also referred to as latex. However, even the water-based formulations are not completely organic solvent-free andcontain small amounts of organic compounds to plasticize polymerparticles so that a balance between film forming ability and themechanical properties of the final film can be achieved underambient conditions. For example, polymers with low glasstransition temperatures (Tgs) will easily form a film at roomtemperature but the final film will be tacky with poor mechanicalproperties. On the other hand, polymers with high Tgs have goodelastic properties but the final film suffers with the problem ofcracking.2,3

To achieve both good film forming ability and excellentmechanical properties from aqueous dispersions without the useof volatile compounds several approaches have been explored.One of these is the use of core/shell latex particles where the coreof the particle is a high Tg polymer whereas the shell is a lowTg film-forming polymer. When particles come in contact, theshell coalesces to form a continuous phase whereas the coreprovides the mechanical strength to the final film. The otherapproach, which forms the subject matter of this study, is the useof latex blends containing soft (low Tg) and hard (high Tg) polymerparticles. In these blends, soft particles coalesce to fill the voidsand the hard ones give mechanical strength to the final film.

While most of the existing literature on the film formation andcracking mechanism in drying colloidal films consider mono-disperse spheres with identical mechanical properties,4,5 a fewhave focused on latex blends. Winnik and Feng1 experimentedwith a latex blend containing a high-Tg polymer [poly(methylmethacrylate), PMMA] and a copolymer of butyl methacrylate

and butyl acrylate [P(BMA-co-BA)] with Tge 10 °C and foundthat the films dry at a slower rate than films cast from dispersionscontaining only one type of particle. They observed that the rateof drying passes through a minimum with increasing hard particlevolume fraction with the minimum occurring at a value of R )0.56, where R is the ratio of hard particles volume to the totalsolid volume. As expected, the ratio of hard to total solid volumewas found to strongly affect the film formation behavior of theblend. Martinez and Lewis6 have reported the film formation inblends of latex and silica. In this study, the film shape evolutionand the stress development during film formation were studiedfor a blend of acrylic latex (particle size of 330 nm and Tg of-6 °C) and silica (particle size of 570 nm). It was found thatblends with R < 0.4 behaved like pure latex films. However, a

(1) Winnik, M. A.; Feng, J. J. Coat. Technol. 1996, 68(852), 39–50.(2) Lepizzera, S.; Lhommeau, C.; Dilger, G.; Pith, T.; Lambla, M. J. Poly Sci

(B) 1997, 35, 2093–2101.(3) Tirumkudulu, M. S.; Russel, W. B. Langmuir 2005, 21(11), 4938–4948.(4) Routh, A. F.; Russel, W. B. Langmuir 1999, 15(22), 7762–7773.(5) Singh, K. B.;Tirumkudulu, M. S.;Phys. ReV. Lett. 2007, 98,

218302(1)–218302(4). (6) Martinez, C. J.; Lewis, J. A. Langmuir 2002, 18, 4689–4698.

Figure 1. Particle positions during deformation.

751Langmuir 2009, 25, 751-760

10.1021/la802857q CCC: $40.75 2009 American Chemical SocietyPublished on Web 12/18/2008

marked transition from a deformable to rigidlike response wasobserved as the silica volume fraction increased beyond thisvalue (i.e., R> 0.4). They measured the stress of the drying filmusing the well-known cantilever technique and found that thestress peak became sharper as the silica volume fraction in theblend was increased.

Lepizzera et al.2 have studied the influence of glass transitiontemperature and relative particle loading on film forming abilityand mechanical properties of latex blend films. The hard latexin this study was synthesized from PMMA with particle sizeand particle glass transition temperature as 210 nm and 105°C, respectively. Soft particles were based on methylmethacrylate-butyl acrylate-acrylic acid with a size rangingfrom 55 to 210 nm and a Tg ranging from -1 to 18 °C. It wasfound that for blends with soft particle Tg less than 0 °C, themaximum weight fraction of hard latex for obtaining crack-free transparent films is constant and equal to R ) 0.55irrespective of soft particle Tg. On the other hand, an inverseproportionality between hard particle weight fraction and softparticle Tg was observed for films with soft particle Tg greaterthan 0 °C. Further, irrespective of soft particle Tg, all filmscrack for R > 0.55. Some of the studies have also consideredthe effect of particle size and size ratio on film formationbehavior of latex blends. Colombini et al.7 found that higherthe soft/hard particle size ratio, the film cracks at lower valuesof R. The size of the dispersed hard phase also affects the

viscoelastic film properties with smaller particles givingmechanically strong films. Tzitzinou and Keddie8 have alsoshown the influence of size ratio on film formation behavior.Depending on the concentration of small particles, they canform a connecting network around the larger ones leading to

Figure 2. Scanning electron micrograph of the surface of a blend filmwith R ) 0.5.

Figure 3. Scanning electron micrograph of the cross-section of a blendfilm with R ) 0.5.

Figure 4. Scanning electron micrograph of the cross-section of a blendfilm with R ) 0.5, heated at 70 °C for about half an hour.

Figure 5. Scanning electron micrograph of the film-substrate interfaceof a blend film with R ) 0.5, heated at 70 °C for about half an hour.

Figure 6. Scanning electron micrograph of the surface of a blend filmwith R ) 0.95.

752 Langmuir, Vol. 25, No. 2, 2009 Singh et al.

a continuous phase inversion. Feng et al.9 have reported theeffect of particle size, glass transition temperature, evaporationrate and soft particle volume fraction ratio on morphologyand transparency of blend films. At high evaporation rates,they obtained turbid films which was attributed to theinsufficient time available for the soft particles to deform andclose the void spaces in the film. The turbidity of the film iscaused by light scattering from defects that are comparableor larger than the wavelength of light. Turbid films were alsoobtained when the hard particles aggregated and a nonuniform

distribution of hard and soft particles was obtained. On theother hand, decreasing the Tg or increasing the fraction of softparticles improved the transparency of the film. They concludethat the size of the soft particles plays a minor role indetermining the transparency of the film.

On the theoretical end, Jagota and Scherer10,11 have studiedthe sintering rates and viscosities of two and three-dimensionalrandom packing of hard and soft spheres of identical size usinga numerical model. The particle packing was replaced by anetwork of points (that represent the center of particles) and

Figure 7. Dimensionless stress versus dimensionless time, tj≡ Et/(ho(1 - φo)), for various R, where E is the evaporation rate, ho is the initial averagewet film thickness, R is the radius of the particles, and φo is the initial particle volume fraction. Note that the initial thickness is different in the fourcases.

Figure 8. Variations of effective shear modulus (Geff) with R. Note that the shear moduli for different interactions are GHH ) 31 GPa, GSS ) 0.8GPa, and GHS ) 0.8 GPa.

Cracking in Soft-Hard Latex Blends Langmuir, Vol. 25, No. 2, 2009 753

links (that represent the contacts between the particles). Further,depending on the type of interaction, the links were eitherrepresented by springs or by dashpots. Finally the deformationof this network was studied to investigate the sintering anddeformation of randomly packed spheres. Here, two differentcases were considered. In the first (bonded) case, the contactbetween hard particles resist all the relative velocities as well as

spins. In second (sliding) case, the contacts between hard particlescan slide and bend but resist interpenetration. In their analysis,force and moment balance equations were solved at each networkpoint assuming that the system is in equilibrium. It was foundthat the transition from soft to hardlike behavior occurs at arigidity threshold which coincides with the percolation threshold(R ) 0.32) for the bonded case whereas in sliding case, therigidity threshold was much higher (R ∼ 1) than the percolationthreshold.

We present both the theoretical and experimental investigationsof latex blends containing equal sized elastic spheres of twodifferent shear moduli. The model developed here extends thestress versus strain relation of Routh and Russel14 for a dryingcolloidal film containing identical spherical particles to thatcontaining a latex blend. They considered the viscoelasticdeformation of a pair of identical particles due to contact andinterfacial forces and related the strain at the particle level tothese forces. Next, they volume averaged the forces over allorientations to arrive at the macroscopic stress versus strainrelationship for a drying film. The model has been successful inpredicting not only the stress profile in drying films of both filmforming and non-film-forming dispersions,12 but also in predictingthe cracking mechanism in the latter.3,5

In the present work, a mathematical model for film formation inlatex blends has been developed. Here, we have employed theconcepts developed for monodisperse systems and derived theconstitutive equation for a blend with soft and hard particles. Weaccount for the nonaffine deformation in the film and predict criticalcrackingstressandcriticalcrackingthickness(CCT)thatarevalidatedwith measurements on a blend of acrylic and silica particles.

2. Model

Consider a drying colloidal film containing equal sized sphericalparticles of two different shear moduli. When the particles reachclose packing, three different types of contacts between a pair

(7) Colombini, D.; Hassander, H.; Karlsson, O. J.; Maurer, F. H. J.Macromolecules 2004, 37, 6865–6873.

(8) Tzitzinou, A.; Keddie, J. L. Macromolecules 1999, 33, 2695–2708.(9) Feng, J.; Winnik, M. A.; Richard, R.; Clubb, B. Macromolecules 1995, 28,

7671–7682.(10) Jagota, A.; Scherer, G. W. J. Am. Ceram. Soc. 1993, 76(12), 3123–3135.(11) Jagota, A.; Scherer, G. W. J. Am. Ceram. Soc. 1995, 78(3), 521–528.(12) Tirumkudulu, M. S.; Russel, W. B. Langmuir 2004, 20, 2947–2961.

Figure 9. Variations of dimensionless critical stress with dimensionless film thickness for a blend of silica and acrylic (BX261) for different hardparticle volume fractions. Here, hrcp is the film thickness at random close packing volume fraction.

Figure 10. Comparison of experimental results with the theoreticalpredictions for various values of R. Here, E represents experimentalmeasurements while T refers to the theory.

Figure 11. Comparison of experimental results for CCT with modelpredictions for different blends.

754 Langmuir, Vol. 25, No. 2, 2009 Singh et al.

of particles are possible: namely, hard sphere-hard sphere (HH)contact, hard sphere-soft sphere (HS) contact, and softsphere-soft sphere (SS) contact. The HH and SS contacts forthe deformation of a two-particle system have been solved byRouth and Russel.4 They assumed that the particle pair is subjectedto contact forces, F, that act along the line joining the centersof spheres and the particles deform such that they remain sectionsof spheres (Figure 1c) throughout the deformation. For the HScontact, we consider a pair of contacting hard and soft elasticspherical particles of equal size acted upon by contact forces, F,again, acting along the line joining their centers but we assumethat the hard sphere is rigid while the soft sphere deformselastically under the applied forces. Therefore, on completedeformation, the soft sphere engulfs the hard sphere. Figures1a-c present schematics of the hard-soft particle pair at variousstages of deformation.

The strain at the particle pair level is defined as the ratio ofthe change in the interparticle center to center distance, at anyinstant to that at complete deformation,

εHS )change in interparticle distance (at time t)

change in interparticle distance at complete deformation(1)

For the present case,

εHS )2R0 - dt

2R0 - df(2)

where df ) (21/3) - 1)R0. Here, R0 is the original radius of thetwo particles, Rt is the radius of the soft sphere after the onsetof deformation, and dt is the distance between the centers of thehard and soft sphere during deformation, while df is the distancebetween the spheres at complete deformation. The variable z′denotes the axial distance between the center of the soft sphereand the vertical plane passing through the intersection of the twospheres. The first step toward obtaining the stress-strain relationis to relate the radius of the expanding soft sphere to its originalradius and the particle strain. The calculation of Rt/R0 involvesthe volume conservation equation and is presented in AppendixA. Since, the value of Rt/R0 is known at εHS ) 0, (35) issuccessively differentiated with respect to εHS to obtain a Taylorseries expansion up to O(εHS

2),

Rt

R0) 1+ �εHS

2 +O(εHS3) where � ≡

(3- √3 2)8

2

(3)

As mentioned earlier, the goal is to relate the strain at the particlepair level to the forces acting on the pair. During the deformation,it is assumed that the surface of the soft sphere deforms radiallywhile the surface of the hard sphere translates in the axial direction.Consequently, the surface displacement vector for the soft sphereis given by,

Sb1 ) (Rt -R0)n1 + g(R0, εHS)ez

while that for the hard sphere is given by,

Sb2 )-(2R0 - dt)ez

where, n1 and n2 (see Figure 1c) are unit vectors perpendicularto the surface of the soft and hard sphere, respectively, while ez

is the unit vector in the z-direction. The function, g(R0, εHS), is

introduced to ensure global incompressibility condition. Then,the strain over the volume of the particle pair is given by(Appendix A),

∫VεTHS dV)∫V

12

(∇bSb+ ∇bSbT))∫A

12

(nSb+ Sbn) dA

) 43

πR03�εHS

2( I5- 3ezez) (4)

Since the volume of each sphere is (4/3) πR03, the volume

averaged strain over the two particles is

⟨εTHS⟩ ) �2εHS

2( I5- 3ezez) (5)

Next, we determine the volume averaged stress on the particlepair. The volume integral of stress over the particle pair can betransformed to a surface integral by invoking Gauss’s theorem,

∫VσT dV)∫A

n · σTxb dA (6)

where n is the unit normal from the surfaces of both particlesand xb is the position vector. Since we neglect van der Waalsforce and interfacial tension, only the interparticle contact forcecontributes to stress,

n · σT)-P0n · I5+ F2πRt

δ(z+Rt)ez -

F2πR0

δ(z- dt -R0)ez (7)

where δ(z) is the Dirac-Delta function and P0 is the pressure inthe surrounding fluid. On substituting (7) in (6), we have

∫VσTHS dV)∫A1+A2

nP0xb dA+

∫A1

Fδ(z+Rt)2πRt

ez(rer + zez) dA-ez(rer + zez)dA (8)

∫ A2

Fδ(z- dt -R0)2πR0

)-83

πR03P0 I5- 4FR0

(1-��2εHS)ezez (9)

where A1 and A2 represent surface areas of soft and hard surfacerespectively (see Appendix B). The stress is then averaged overthe two spheres to obtain the volume averaged quantity,

⟨σTHS⟩ ) 1V∫V

σTHS dV)-P0 I5- 3F

2πR02(1-��

2εHS)ezez

(10)

For the particle pair deforming under the contacting force, thedeviatoric stress ⟨σ5HS⟩ and the strain ⟨εTHS⟩ are related to stressrelaxation modulus of the material by

⟨ τTHS⟩ ) 2∫-∞

tG(t- t′ )

d⟨εTHS⟩dt′ dt′ (11)

The deviatoric stress tensor is related to the total stress, ⟨σ5HS⟩ andthe isotropic pressure P as

⟨σTHS⟩ ) ⟨ τTHS ⟩ -⟨P⟩ I5 (12)

where

(13) Landau, L. D.; Lifshitz, E. M. Theory of Elasticity, first ed.; Pergamon:New York, 1986.

(14) Hashin, Z.; Shtrikman, S. J. Mech. Phys. Solids 1963, 11, 127–140.

Cracking in Soft-Hard Latex Blends Langmuir, Vol. 25, No. 2, 2009 755

⟨P⟩ )P0 +F

2πR02(1-��

2εHS), and

⟨ τTHS ⟩ ) F

2πR02(1-��

2εHS)( I5- 3ezez)

On substituting the averaged deviatoric stress and strain tensorin (11), the contact force is related to the particle strain to O(εHS

2),

F)4πR0

2

3�∫-∞

tG(t- t′ ) d

dt′ (εHS2) dt′ (13)

Note that the trace of the volume averaged stress gives the averageisotropic pressure in the particle as well as the equation governingthe viscoelastic deformation of two spheres (12).

2.1. Nonaffine Deformation. Recall that Routh and Russel4

derived the stress-strain relation for a film containing mono-disperse identical spheres by assuming affine deformation, wherethe strain at the particle level is the same as that at the macroscopiclevel. However, for a latex blend containing equal sized hard andsoft spheres, the strain at the particle pair level will depend onthe moduli of the interacting particles and will in general bedifferent from that at macroscopic level. Next, we account forthe nonaffinity by assuming that the strain at the particle pairlevel for the three types of interactions differs from macroscopicstrain by a function that, at lower order, is linear in macroscopicstrain,

⟨εTSS⟩ - ⟨εT⟩ ) a⟨εT⟩ (14)

⟨εTHH⟩ - ⟨εT⟩ ) b⟨εT⟩ (15)

⟨εTHS⟩ - ⟨εT⟩ ) c⟨εT⟩ (16)

Here ⟨ε5⟩ is the macroscopic strain tensor and ⟨ε5SS⟩ , ⟨ε5HH⟩ , and⟨ε5HS⟩ are respectively the strain tensors at the particle pair levelfor soft-soft, hard-hard, and hard-soft interactions. If thefraction of hard particles (of total particles) is R, then theprobability of finding a pair of hard-hard particles in arepresentative volume is R2, that of a pair of soft-soft particlesis (1 - R)2, and that of a pair of hard-soft particles is 2R(1 -R),

R2⟨εTHH⟩ + (1-R)2⟨εTSS⟩ + 2R(1-R)⟨εTHS⟩ ) ⟨εT⟩ (17)

The constraint equation is therefore

a(1-R)2 + bR2 + 2cR(1-R)) 0 (18)

2.2. Orientational Average Stress-Strain Relation. Wefocus on a single particle in the packing and integrate over theconfiguration of its neighbors to obtain the volume averagedmacroscopic stress,4

⟨σT⟩ ) 1V∫V

σT dV){ 1V ∑

contacts

FbRb-∫VP0 I5 dV} (19)

where, ⟨σ5⟩ is the macroscopic stress in a representative volumecontaining the particles, Σ represents the contribution from allits contacting neighbors, and Rb is the position vector for thecontacting forces (Figure 1d). The averaged stress (in indicalnotation) can be rewritten to highlight the contribution from thetwo phases,

Since the stress generated in the solid phase arises from all possibleinteraction, we have

σij )-(1- φ)P0 - { φP0δij +1V∑

n[R2(Fi

nRjn)HH +

(1-R)2(FinRj

n)SS + 2R(1-R)(FinRj

n)HS]} (21)

The product of the contact force and its position vector for eachof the interactions to the lowest order in respective strain is givenby

(FiRj)SS )πR0

3

2 ∫-∞

tGSS(t- t′ ) d

dt′ [(rmεmn(SS)rn

r2 )2] rirj

r2dt′

(22)

(FiRj)HH )πR0

3

2 ∫-∞

tGHH(t- t′ ) d

dt′ [(rmεmn(HH)rn

r2 )2] rirj

r2dt′

(23)

(FiRj)HS )4πR0

3

3 ∫-∞

tGHS(t- t′ ) d

dt′ [(rmεmn(HS)rn

r2 )2] rirj

r2dt′

(24)

where the soft-soft (22) and hard-hard (23) particle contactrelations are taken from Routh and Russel4 and GSS, GHH, andGHS are the relaxation moduli for the three contacts. Note thatGHS becomes equal to GSS when GHH . GSS. Assuming allorientations of the particles to be equally likely, we average overthem to obtain,

σij )-P0δij -Mφ

280×

∫-∞

t { (1-R)2(1+ a)2GSS(t- t′ )+R2(1+ b)2GHH(t- t′ )

+ (83

�)2a(1-R)(1+ c)2GHS(t- t′ ) } × ddt′ [δij(εmm

2 +

2εmmεnm)+ 4εmmεij + 8εimεmj] dt′ (25)

where, φ is the total solid volume fraction and M is the averagenumber of nearest neighbors. Up to this point, we have assumedthe most general form of constitutive relation for the filmdeformation. In order to determine the values of a, b, and c, weshall now consider special case of a packing containing hard andsoft elastic spheres where the above relation becomes,

σij )-P0δij -Mφ

280Geff[δij(εmm

2 + 2εmnεnm)+ 4εmmεij +

8εimεmj] with Geff ) { (1-R)2(1+ a)2GSS

+R2(1+ b)2GHH

+ (83

�)2a(1-R)(1+ c)2GHS } (26)

Landau and Lifshitz13 show that for an isotropic elastic material,the elastic energy density up to third order in strain is given by,

E) µεlkεkl + (κ2 - µ3)εkkεll +

43εlkεkmεml +Bεllεkmεmk +

C3εllεkkεmm (27)

where µ, κ, A, B, and C are material constants. On differentiating(27), with respect to εij and comparing with (26), we find thatin absence of solvent (P0 ) 0),

756 Langmuir, Vol. 25, No. 2, 2009 Singh et al.

µ) 0; κ) 0;A8) B

2)C) Mφ

280Geff

The elastic energy for closed packing of a latex blend (in theabsence of solvent) to the lowest order in strain can be obtainedby substituting values of A, B, and C in (27). Next, we hypothesizethat the nonaffine strains at the particle level will be such thatthey minimize the elastic energy of the network. Here we usethe standard principle of minimum total potential energy, whichstates that of all displacements that satisfy the boundary conditionsof a structural system, those corresponding to configurations ofstable equilibrium make the total potential energy a relativeminimum. This approach is similar to Hashin-Shtrikman’svariational method14 used to determine bounds for the overallshear modulus for composites containing linear elastic phases.

Under a given deformation, minimization of E with respectto nonaffine strains amounts to minimizing the coefficient Geff

with the constraint given in (17), i.e.

minimize Geff for constraint : a(1-R)2 + bR2 +2cR(1-R)) 0 (28)

Using the method of Langrange multipliers, we get

1+ a)1/GSS

[ R2

GHH+

(1-R)2

GSS+ ( 3

8�)2R(1-R)

GHS]

1+ b)1/GHH

[ R2

GHH+

(1-R)2

GSS+ ( 3

8�)2R(1-R)

GHS]

1+ c)(3/8�)(1/GHS)

[ R2

GHH+

(1-R)2

GSS+ ( 3

8�)2R(1-R)

GHS]

so that the effective shear modulus becomes

Geff )1

[ R2

GHH+

(1-R)2

GSS+ ( 3

8�)2R(1-R)

GHS]

Note that the constitutive relation (25) is similar to that derivedby Routh and Russel4 for identical spheres except that the shearmodulus is replaced by the effective modulus.

Tirumkudulu and Russel3 used their constitutive relation tocalculate the stress field in a drying film and determined thecritical stress for the film to crack by adopting the well-knownGriffith energy balance concept. For the case of a drying filmcontaining latex blends, we can write directly the expression forthe critical stress for cracking,

σc,iR0

2γ) 0.1877(2R0

h )2

3(GeffMφrcpR0

2γ )1

3 (29)

Here, γ is the liquid-gas interfacial tension, h is the film thickness,and φrcp is the random close packing volume fraction. Thederivation of the above expression can be understood from asimple scaling analysis. According to the constitutive relation,the in-plane tensile stress generated in the film during drying

scales with the square of the strain, σ ∼ Gε2. On equating theelastic energy recovered per unit length of crack, Ee ∼ h2σε, tothe increased surface energy per unit length, Es ∼ γh, we get thecritical stress for cracking, σc ∼ G1/3(γ/h)2/3. Note that the criticalstress is independent of particle size.

More recently, Singh and Tirumkudulu5 identified two regimesfor obtaining crack free films for a drying colloidal film containingidentical particles. The strain-limited regime pertains to the caseof a drying film containing soft particles where the particlescompletely deform to fill the pores under the capillary pressureexerted by the menisci at the top of the film. This is possible onlywhen the tensile stress at maximum deformation is less than thecritical stress. Thus the critical cracking thickness hmax belowwhich the films are crackfree is obtained from, G1/3(γ/h)2/3 ∼G(1 - φrcp)2 giving hmax ∼ γ/G(1 - φrcp)2. Alternatively, thestress-limited regime occurs when the particles are hard and themenisci adjust their curvature till the maximum capillary pressure(Pmax) is reached, beyond which they recede into the networkresulting in partial deformation. Here the critical thickness isobtained by equating the stress at Pmax to the critical crackingstress, σc ∼ G1/3(γ/h)2/3 ∼-Pmax, giving hmax ∼ γG1/2/(-Pmax)3/2.For latex blends, the corresponding critical cracking thicknessfor the strain-limited regime is given by

hmax )37γ

GeffMφrcp(1- φrcp)3(30)

while that for the stress-limited regime is

hmax ) 0.64[GeffMφrcpR03

2γ ]1⁄2

[ 2γ-PmaxR0]3⁄2

(31)

A single value of the dimensionless capillary pressure,-PmaxR0/2γ ∼ 1.4, resulted in a good match between CCT measurementsand above predictions over a wide range of particle sizes andrigidities. The same value was used in this study for allvalues of R.

3. ExperimentsExperiments were conducted with a blend of silica and acrylic

(BX261) particles of same size. Details of the dispersions used inthis study are given in Table 1. Although the particle size is thesame, there is a significant difference in the particle rigidity. Thecomposition of the blend is described in terms of relative hard particlevolume fraction ratio R. The value of R was varied from 0 to 1 andfor each composition, the critical stress for cracking as a functionof film thickness and critical cracking thickness (CCT) weremeasured.

The CCT measurements were performed over a temperature rangeof 25-28 °C and a relative humidity of 35-40%. Thin films werecast on glass substrates placed on a spin coater (slow rotation rate∼20 rpm) by disbursing small amounts of dispersion. Here, the spincoater was employed to spread the liquid uniformly over a fixed areaso as to obtain a nearly circular film. Further, the thickness of thefilm was varied by disbursing increasing amounts of dispersion overthe same area. After the films were cast, they were left to dry underambient conditions. On complete drying, it was examined under anoptical upright microscope (Olympus, BX-60) for cracks at varyingmagnifications. Finally, the thickness profile of the dry film wasobtained using a surface profilometer (Dektak-150). It should benoted that in some cases the entire film was crack free which gavethe lower limit for CCT.

The classical cantilever technique was employed to measure thestress in a drying film.15 Here, a thin layer of the dispersion wasapplied on a thin silicon wafer of dimensions 13 mm (length) by 5

(15) Petersen, C.; Heldmann, C.; Johannsmann, D. Langmuir 1999, 15, 7745–7751.

Table 1. Details of Dispersions Used in the Present Study

dispersionParticle

CompositionParticle

Diameter (nm)Tg

(°C)Shear Modulus

G (GPa)

colloidal silica silica 330 31acrylic (BX261) acrylate 353 40 0.8

Cracking in Soft-Hard Latex Blends Langmuir, Vol. 25, No. 2, 2009 757

mm (width) by 0.128 mm (thickness) and clamped on one side. Alaser beam is reflected from the free end of the substrate and collectedby a position sensitive detector (On-track). The position of the detectoris set in such a way that the reflected beam falls normally on thesurface of the detector. The entire detector assembly is mounted ona traverse so that the desired positioning can be done with precision.The position data (i.e., the x, y coordinates of the reflected beam)from the detector is directly fed to a computer. As the film dries,the tensile stresses in the drying film bend the substrate thatconsequently results in the movement of reflected beam on detectorsurface. Finally, the displacement of the reflected beam is relatedto the average tensile stress in the film. The details of stressmeasurement technique and CCT measurements can be found in thework of Tirumkudulu and Russel12 and Singh and Tirumkudulu,5

respectively.We also ensured that the evaporation rate for these experiments

was constant throughout the experiment. This was checked by dryingfilms separately under similar conditions on a weigh balance. Wealso noticed that the drying rate remained constant long after thefilm cracked. Hence, we could also measure the evaporation rate ofthe sample used for stress experiments by transferring it to the weighbalance immediately after it cracked and recording its weight change.

In blends, one of the major concerns is the distribution of theconstituent soft and hard particles. As reported by Feng et al.,9 themorphology of blend films is influenced by the distribution of hardand soft particles. Films were examined using scanning electronmicroscopy to see the particle distribution.

4. Results and Discussion

As shown in Table 1, the blends contain particles of similarsize but with a very different shear moduli. In order to see theparticle distribution in the final film, dried films were viewedunder a scanning electron microscope (SEM). Figure 2 showsthe SEM image of the surface of a dried film with R ) 0.5. Thehard particles are uniformly distributed in the matrix of softparticles. Further, the soft acrylic particles seemed to haveundergone significant deformation at the film surface. In orderto see the particle distribution throughout the film, the film wascut and the cross section imaged. Again, the cross section or theinterior of the film has a uniform distribution of both the soft andhard particles (Figure 3). Here also, the soft particles can be seento have deformed significantly. To differentiate between the twotypes of particles and to ascertain their distribution inside thefilm, we exposed one of the films to an ambient temperature ofabout 70 °C in an oven for half an hour. Since the glass transitiontemperature of the soft (acrylic) particles is 40 °C, we expect theparticles to fuse completely on heating at such an elevatedtemperature. Figure 4 shows the SEM image of one such filmwhere the deformed domains confirm that the distribution ofsoft-hard particles in the blend is indeed uniform. Figure 5shows the SEM image of the same film close to the film-substrateinterface. It appears that the film has a slightly larger number ofhard particles at the film-substrate interface compared to thesoft particles but this segregation appears to be restricted closeto the substrate. We next look at the particle distribution forR ) 0.95 (Figure 6). As can be seen, the hard particles in thefilm are packed in well-ordered domains. Further, the coalescedsoft particles can also be seen at a few locations between the hardparticles. This is indeed surprising given that the soft particlefraction is very small which suggests that the hard and soft particlesare dispersed well at all volume fractions.

Figure 7 presents the dimensionless stress evolution as afunction of the dimensionless time for four different values ofR. Once the particle concentration reaches that for randompacking, the stress starts to increase, reaches a maximum at whichthe film cracks, and then decreases. Note that we do not presentthe full stress relaxation profile after cracking as the samples are

removed for measuring the evaporation rate shortly after thecrack initiation. For both R ) 0 and 0.5, the increase and thedecrease in stress is slow suggesting that the film undergoesviscous deformation. This suggests that even with 50% hardparticles, the response is essentially that for soft particles. Thisobservation is in line with the experiments of Martinez and Lewis6

who did not find any significant variation with hard particlefraction in stress profile for films containing less than 40% hardparticles. In contrast to these profiles, those for R ) 0.9 and 1show that the stress reaches a maximum value within ∆tj) 0.01and decreases to a negligible value as quickly. The response hereis entirely elastic as the stress relaxes to a negligible valueimmediately after crack nucleation.

4.1. Comparison with the Model. The dependence ofeffective shear modulus on R, as predicted by the model, ispresented in Figure 8. The effective shear modulus varies littleand is close to the shear modulus of the soft particles for R <0.95. This suggests that the addition of hard particles into thesoft matrix for R < 0.95 does not significantly contribute to themechanical properties of the film and hence the blend essentiallybehaves like the one composed of only soft particles. However,at a value of R close to 0.95, a steep increase in the effectiveshear modulus is seen. It implies that the transition from soft torigidlike behavior takes place at this value ofR. The experimentalresults shown in Figure 6 also point to this type of behaviorwhere the hard particles are packed in a very ordered structureand hence are capable of transmitting the applied load to theentire film. It is interesting to note that the model predictions arein good agreement with the numerical simulations of Jagota andScherer11 who showed that the rigidity threshold in a mixtureof soft-hard particles with sliding contacts between hard particlesis at a site fraction close to 0.95. However, it is important to notethat, for cases where the shear moduli for soft-soft (GSS) andhard-hard (GHH) interactions are comparable, a more gradualtransition from soft to rigidlike behavior will be observed unlikethat seen in Figure 8.

The variations in dimensionless critical stress with thedimensionless film thickness over the entire range of volumefraction ratio (R) are shown in Figure 9. The dependence ofdimensionless critical stress on dimensionless film thickness for0 < R < 1 is similar to the predicted behavior of Geff in that ajump in the critical stress is seen for values of R g 0.95. Thestress measurements also confirm the transition from soft torigidlike behavior at R ∼ 0.95. Figure 10 compares the stressmeasurements with those predicted by the model. Here, thetheoretical plots are scaled in such a way that the line for R )1 matches with the experiments. The predicted critical stressvariation with film thickness matches with all the experimentsin that the exponent of two-thirds is close to the observed trend.This confirms the validity of the proposed approach to lump themodulus variations in Geff. However, at lower values of R,the absolute values are lower than the experimental results. Thediscrepancy can be attributed to the viscous deformation of thesoft particles. The energy dissipation reduces the stored elastic

Figure 12. Sectional volumes, V1 and V2.

758 Langmuir, Vol. 25, No. 2, 2009 Singh et al.

energy in the network thereby requiring higher stresses to crackthe film.

The experimental results for critical cracking thickness (CCT)are compared with the theoretical predictions for different valuesofR in Figure 11. The measured CCT varies almost linearly withR whereas the model shows an abrupt increase in CCT at a valueof R close to 0.95. This is somewhat surprising given that thetrend for the measured critical stress shows a negligible variationfor R < 0.95. The model predictions match well with theory forboth low and high values of R but exhibit large differences forthe intermediate values. One possible reason for the discrepancycould be the following. The critical cracking stress for a thin filmvaries inversely with film thickness (29). For the measurementof critical stress, we typically apply thick films (∼100-200 µm)on our substrates and measure the stress at which the film cracks.Since the stresses are low, we would expect small particledeformations in the film and, for higher values of R, they wouldbe mostly elastic. On the other hand, we apply thin films (∼5-20µm) to determine the critical cracking thickness. Here, the filmexperiences the highest possible stress that corresponds to themaximum capillary pressure. Consequently, at such large stresseswe can expect significant viscous deformation of the particles.It is possible that the simple assumptions made to account forthe nonaffine deformation (14)–(16) may not be valid at suchlarge deformations. Thus the large mismatch may be due to thesignificant viscous deformation of the soft particles although themodel assumes them to undergo small elastic deformation inthe packing. More work is required to resolve these issues.

5. ConclusionA mathematical model for film formation in latex blends has

been developed. The model captures the main aspects of dryingand cracking behavior of the blend films. The model predictionscorrespond well with the numerical simulations reported in theliterature for the transition in soft to rigidlike behavior for blendsof hard and soft particles. Further, the model predictions arevalidated with extensive experimental data. While the predictedcritical stresses match well with the measured values, the criticalthickness shows a trend different from the experiments. Theviscous deformation of the particles in the drying film which hasnot been accounted for in the model may be the cause for thediscrepancy.

Acknowledgment. This work was supported by the Depart-ment of Science and Technology, India. The authors thank Dr.Martin Murray of AkzoNobel for discussions and Dr. S. L. Kamathof Metallurgy and Materials Science Department of IIT-Bombayfor help with SEM images.

Appendix A: Calculation of Rt/Ro Ratio

Consider the following position of hard and soft spheres. Withthe origin of the coordinate system fixed at the center of the softsphere, Rt the radius of the soft sphere, R0 the radius of the rigidsphere at any stage of deformation, and dt defined as distancebetween centers of both sphere, the equation for the soft spherebecomes,

x2 + y2 + z2 )Rt2 (32)

and that for the hard sphere is

x2 + y2 + (z- dt)2 )R02 (33)

The plane of intersection between two spheres is given by solving(32) and (33) and is at distance z ) z′ from the origin,

z′ )Rt

2 -R02 + dt

2

2dt(34)

The sectional volumes are shown in Figure 12. The volume ofthe section of the sphere is given by

Vi )π6

hi(3ri2 + hi

2) i) 1, 2

where V1 is the volume enclosed in section ACBA and V2 is thevolume of the section ADBA. The height h1 and radius r1 forthe sectional volume V1 are, h1 ) (R0 - dt + z′) and r1 ) (R0

2

- (dt - z′)2)1/2. Similarly, for the sectional volume of soft sphere,we have h2 ) (Rt - z′) and r2 ) (Rt

2 - z′2)1/2. The hard and softspheres are assumed to be intially of equal radii. Enforcing thevolume conservation condition, we have

43

πRt3 -V1 -V2 )

43

πR03 (35)

Appendix B: Two Particle Coordinate System

The equation of the soft sphere at any time is given by

x2 + y2 )Rt2 - z2 ) f1(z) (36)

The equation of the surface is given by

F1 ) x2 + y2 - f1(z)

so that the normal unit vector to the surface of the soft particleis given by

n1 )∇bF1

|∇bF1| )2xex + 2yey - f1′ (z)ez

√4(x2 + y2)+ f1′ (z)2

) rRt

er -f1′ (z)

2Rtez (37)

where f1′(z) ) -2z. Similarly, the normal to the surface of theharder particle is given by

n2 )2xex + 2yey - f2′ (z)ez

√4(x2 + y2)+ f2′ (z)2

) rR0

er -f2′ (z)

2R0ez (38)

where the equation of the second sphere is given by

x2 + y2 )R02 - (z- dt)2 ) f2(z) (39)

The area of the surface element for the first sphere is given by(Figure 13)

dA)√f1(z)

cos ψdz dθ)Rt dz dθ

so that the surface integral of a function M(z, θ) over the surfaceof the soft sphere is given by (40)

Figure 13. Calculation of area for the surface element.

Cracking in Soft-Hard Latex Blends Langmuir, Vol. 25, No. 2, 2009 759

(SI)1 )∫0

2π ∫z1

z2 M(z, θ)Rt dz dθ (40)

Integrating the above surface integral for the soft sphere withinlimits, z1 ) -Rt and z2 ) z′, the value of surface deformationis given by

(SI)1 )∫0

2π ∫-Rt

z′ 12[2(Rt -R0)n1n1 + g(R0, εHS)(ezn1 +

n1ez)]Rt (41)

For the hard sphere, the limit of definite integral becomes z1 )z′ and z ) R0 + dt, and the value of the surface integral thenbecomes

(SI)2 )∫0

2π ∫z′R0+dt

[-(2R0 - dt)]R0(n2ez + ezn2) dz dθ

(42)Calculation of the Trace

For an incompressible material, the trace of {∫V εTHS dV} )0. The strain integrated over the volume of the two spheres, canbe written in terms of the surface integral over surfacedisplacements,

∫VεTHS dV)∫V

12

(∇bSb+ ∇bSbT) dV)∫A

12

(nSb+ Sbn) dA

(43)

For the soft sphere, the displacement vector is along the outwarddirection radial from its center,

Sb1 ) (Rt -R0)n1 + g(R0, εHS)ez

The normal n1 to the surface of soft sphere is given by (37), sothat

n1n1 )r2

Rt2erer +

f1′ (z)2

4Rt2

ezez -12

rf1′ (z)

Rt2 (erez + ezer) (44)

The zero trace condition results in two surface integrals, I1 withterms containing exex and eyey, and I2 and I3 with terms containingezez for the soft sphere while I4 is obtained for the hard sphere,

I1 )∫0

2π ∫-Rt

z′ r2

Rt2[(Rt -R0)Rt+(cos θex + sin θey) ×

(cos θex + sin θey) dz dθ

)π∫-Rt

z′ (Rt -R0

Rt)(Rt

2 - z2)(exex + eyey) dz

)π(Rt -R0)[Rtz′ -z′

3

3Rt+ 2

3Rt

2](exex + eyey)

I2 )∫0

2π ∫-Rt

z′ f1′2

4Rt2[(Rt -R0)Rt]exex dz dθ

) 2π∫-Rt

z′ (Rt -R0

Rt)z2ezez dz

) 2π(Rt -R0)[ z′3

3Rt+

Rt2

3 ]ezez

I3 )∫0

2π ∫-Rt

z′ f1′2Rt

g(R0, εHS)Rtezez dz dθ

For the hard sphere, the displacement is along the ez direction,so that the surface displacement vector is

Sb2 )-(2R0 - dt)ez (45)Hence, {∫A

1/2(n2Sb2 + Sb2n2) dA} is given by

I4 )∫0

2π ∫z′dt+R0

f2′ (z)

2R0(2R0 - dt)R0ezez dz dθ

)-2π∫z′dt+R0

(2R0 - dt)(z- dt) dz

)-2π(2R0 - dt)[ z2

2- dtz]z′

dt+R0

)-2π(2R0 - dt)[R02 - dt

2

2- z′

2

2+ dtz ′ ]ezez

The other terms in the trace {∫V ε5HS dV} go to zero because forthe coordinate system defined,

∫0

2πerdθ)∫0

2π(cos θex + sin θey) dθ) 0 (46)

Also, it can be verified that

∫0

2πerer dθ)∫0

2π(cos θex + sin θey)(cos θex +

sin θey) dθ)π(exex + eyey) (47)

LA802857Q

760 Langmuir, Vol. 25, No. 2, 2009 Singh et al.