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Cracking in Drying Colloidal Films of Flocculated Dispersions

Karnail B. Singh, Laxman R. Bhosale, and Mahesh S. Tirumkudulu*

Department of Chemical Engineering, IIT-Bombay, Powai, Mumbai-400076, India

Received December 31, 2008. Revised Manuscript Received March 13, 2009

Understanding the mechanism of cracking during the drying of aqueous colloidal dispersions is important topreventing film failure. While most of the reported work has dealt with stable aqueous dispersions, a few studies havefocused on flocculated systems. The latter especially assumes importance because the role of particle packing in themechanism of cracking is not completely understood. In this work, we study the cracking of colloidal films cast fromflocculated aqueous dispersions of R-alumina. Here, the extent of flocculation is controlled by varying the pH ofthe dispersion and characterized in terms of the final packing volume fraction of the dried film. The influence of varyingthe close-packed volume fraction on the critical cracking thickness and critical cracking stress is measured. Themeasurements are compared with the model predictions based on Griffith’s energy balance, and good agreement isfound between theory and experiments, suggesting that themodel is universal and applies equally well to stable as well asflocculated systems.

Introduction

The cracking of colloidal films has received considerableattention because of its importance in many important applica-tions such as paints, wet clays, ceramics, and so forth.Whena thinfilm of an aqueous colloidal dispersion is applied on an imperme-able substrate, water evaporates, resulting in an increase in thesolid concentration. In stable dispersions, the strong interparticlerepulsive forces disperse the particles against the tendency of theattractive forces to bring them together. Further evaporationcauses particles to concentrate into a closed-packed array,1-6

accompanied by capillary stresses that either completely deformthem to close the pores (soft particles) or resulting in thenucleation and propagation of cracks (hard particles).1,7

The experiments of Chiu et al.8 show that irrespective ofparticle size and rigidity there exists a critical cracking thickness(CCT) below which films do not crack. On the theoretical end,Tirumkudulu and Russel7 proposed a model based on theGriffith’s criterion wherein a balance between elastic energyreleased during crack propagation and the increased surface energydetermines the critical stress at which a drying thin film cracks,

σcR

2γ¼ 0:1877

Mφrcp

N2

� �1=3GR

2γ

� �1=3

ð1Þ

Here, σc is the critical cracking stress,R is the particle radius, φrcp isthe random close-packed volume fraction, γ is the solvent-airinterfacial tension,M is the coordination number,N� (h/2R) is thedimensionless film thickness with h being the film thick-ness, and G is the shear modulus of the particles. Singh andTirumkudulu1 report CCTs for a large number of stable aqueousdispersions and identify two regimes for obtaining crack-free films.

For dispersions containing hard particles (stress-limited regime),particle deformation is negligible, and the CCT increases withparticle size as well as particle rigidity whereas for soft particleswhere the particles completely deform to close the pores (strain-limited regime) the CCT decreases with increasing par-ticle rigidity but is independent of particle size. The CCT (hmax)for the stress-limited regime is obtained by equating the tensiletransverse stress at the maximum attainable capillary pressure(Pmax) to σc,

hmax ¼ 0:64GMφrcpR

3

2γ

" #1=22γ

ð-PmaxÞR� �3=2

ð2Þ

and it agrees well with measurements over a wide range of particleproperties. Furthermore, the model also suggests that the crackingbehavior is influenced by the final packing volume fraction,a prediction yet to be completely verified.

We study the effect of flocculation on both the critical stressand the critical cracking thickness in drying films. Here, theweaker stabilizing interparticle forces are overcome by the attrac-tive van der Waals forces causing the particles to come intocontact with one another well before they reach random closepacking. An electrostatically stabilized dispersion can be floccu-lated either by the addition of electrolytes or by altering the pH ofthe dispersion. In general, the particles agglomerate to form flocs(open fractal structures (Figure 1)).With further evaporation, theflocs connect to form a percolating network that spans the entirethickness of the film. The network has a yield stress, and the filmconsolidates when the capillary pressure exerted by the fluidinterface at the top of the film exceeds the network strength.9

As the film consolidates and the particles rearrange to formstronger networks, the liquid either recedes into the porouspacking when the capillary pressure reaches a maximum whilestill below the yield stress of the network or the film cracks undertransverse tensile stresses generated as a result of the constraintimposed by the substrate on lateral shrinkage. The final packingvolume fraction (φf) of the dried film is less than the randomclose-packing fraction. Recent experiments on lattices containing soft

*Corresponding author. E-mail: mahesh@che.iitb.ac.in.(1) Singh, K. B.; Tirumkudulu, M. S. Phys. Rev. Lett. 2007, 98, 218302

(1)–218302(4).(2) Sheetz, D. P. J. Appl. Polym. Sci. 1965, 9, 3759–3773.(3) Routh, A. F.; Russel, W. B. AIChE J. 1998, 44, 2088–2098.(4) Routh, A. F.; Russel, W. B. Langmuir 1999, 15, 7762–7773.(5) Tirumkudulu, M. S.; Russel, W. B. Langmuir 2004, 20, 2947–2961.(6) Winnik, M. A. Curr. Opin. Colloid Interface Sci. 1997, 2, 192–199.(7) Tirumkudulu, M. S.; Russel, W. B. Langmuir 2005, 21, 4938–4948.(8) Chiu, R. C.; Garino, T. J.; Cima, M. J. J. Am. Ceram. Soc. 1993, 76,

2257–2264. (9) Buscall, R.; White, L. R. J. Chem. Soc., Faraday Trans. 1987, 83, 873–891.

Published on Web 3/26/2009

© 2009 American Chemical Society

DOI: 10.1021/la804331c Langmuir 2009, 25(8),4284–42874284

polymer particles and flocculated by the addition of salt indicateskin formation on top of drying films leading to a significantreduction in evaporation rates.10,11 Chiu et al.8 varied the degreeof flocculation of alumina dispersion by adding salt (NaCl) andfound that the CCT initially increases with salt concentration,passes through a maximum, and then starts to decrease. Theincrease in CCT was attributed to the decrease in stabilizingelectrostatic double layer forces. However, Carreras et al.12 haverecently shown that the measured CCTs for alumina dispersions(pH ∼1.75) were 3 times higher than that obtained from floccu-lated suspensions (pH ∼9), in apparent contradiction to theearlier work of Chiu et al.8 Here, the higher CCT at low pH isattributed to the superior mechanical properties due to the higherpacking fraction.

While the above experiments do suggest a clear influence offlocculation on cracking, the observed trends are contradictory.More importantly, it is not clear what mechanism controlscracking in flocculated systems. Furthermore, if the Griffithcriterion13 is applicable to flocculated systems, how does oneaccount for the network strength? We address these questions byvarying the degree of flocculation of aqueous R-alumina disper-sions over a wide range of pH and measuring the final packingvolume fraction, the critical stress at cracking for varying filmthicknesses, and the CCT. We show that the model for stabledispersions predicts the measurements for flocculated dispersionsequally well if φrcp is replaced by φf while correcting for thenumber of neighbors in contact with one another.

Experimental Section

Dilute dispersions (10-20% by volume) of high-purityR-alumina (AKP-30) (average particle size of 335 nm) wereprepared by dispersing AKP-30 in deionized water. The pH ofthese dispersions was adjusted using analytical-grade HNO3 andKOH. The final packing volume fraction (φf) was determined bydrying the dispersions in glass capillaries with an inner diameterof 500-1000 μm. Here, the original dispersion with differentinitial lengths (l0) was allowed to dry fromone end of the capillarytube, and the final length (lf) of the dried dispersion plug wasmeasured. The final packing volume fraction (φf) was obtainedfromvolumebalance, lfφf= l0φ0. The reported value ofφf for eachpH was averaged over 12 independent experiments with differentinitial lengths. Note that the diameter of the capillary (D ≈ 500-1000 μm) was at least 3 orders of magnitude larger than theparticle size.

For CCTmeasurements, thin films of the dispersions were caston glass substrates by disbursing the liquid using a spin coater(slow rotation rate of∼20 rpm). In the case of highly flocculateddispersions that had very large critical thicknesses, the films weredried in open vials with a square cross section (1.3 cm� 1.3 cm).Finally, the thickness profile of the dry film was obtained usinga surface profilometer (Dektak-150). The classical cantilever

technique was employed to measure the stress in a drying film;the details can be found elsewhere.5,14

Results and Discussion

The zeta potentials for AKP-30 dispersions as reported byJohnson et al.15 are reproduced on the secondary y axis ofFigure 2. The potential is high and positive at low pH anddecreases with increasing pH. At pH ∼9.6, the potential passesthrough the isoelectric point (iep) and decreases further with pHto become large and negative. We quantify the extent of floccula-tion in terms of the final packing volume fraction (φf). Thevariations of final packing volume fraction with pH are plottedon the primary y axis of Figure 2. The measured φf shows directcorrespondence with the reported zeta potential measurement inthat, at low pH, the final packing volume fraction is highestbecause of the higher surface charge. As the pH is increased, thesurface charge decreases, which results in a weaker stabilizingforce and leads to particle flocculation. This is evident from lowervalues of the final packing volume fraction.At the iep, the chargesat the surface are completely neutralized, and there are nostabilizing forces. Consequently, the final packing volume frac-tion is the lowest at pH ∼9.6. With further increase in pH, theparticle surface acquires negative charge, leading to a greaterstability of the dispersion that results in higher φf. The low valuesof φf close to the iep suggest that the system transforms into a gel-like structure with a loose network of particles. The gelationvolume fraction (i.e., the volume fraction at the liquid-solidtransition) at the iep is approximately16 0.14, which is lower thanthemeasured φf. We also include for comparison the final volumefraction at pH 4, 7, and 9 implied by the compressive yield stress(Py) measurements.16,17 While our measured values of φf are inbroad agreement with the yield stress data away from the iep,there is a large discrepancy close to the iep. To resolve this, wecomputed the total network strength ((Py + 4Talf/D) whereTa is the shear yield stress) at φf measured using the capillaryand found it to be 2 orders of magnitude lower than the capillarypressure (eq 3) compressing the network. Although we are unableto explain this discrepancy, drying experiments performed withthe same system but on hydrophobic capillaries also resulted in

Figure 1. Schematic of a drying colloidal film of (i) a stable and(ii) a flocculated dispersion.

Figure 2. Variation of the final packing volume fraction (φf)measured using the capillary tube and that implied by compressiveyield stress and zeta potential as a function of pH for the AKP-30dispersion (26.1 �C and 43%RH). The zeta potential data is takenfrom Johnson et al.15 The solid lines are provided as a guide tothe eye.

(10) Erkselius, S.; Wadso, L.; Karlsson, O. J. Colloid Interface Sci. 2008, 317,83–95.(11) Konig, A. M.; Weerakkody, T. G.; Keddie, J. L.; Johannsmann, D.

Langmuir 2008, 24, 7580–7589.(12) Carreras, C. F.; Dunstan, D. E.; Franks, G. V. J. Colloid Interface Sci. 2007,

313, 160–168.(13) Griffith, A. A. Phil. Trans. R. Soc. 1921, 221, 163.

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

(15) Johnson, S. B.; Russel, A. S.; Scales, P. Colloids Surf. 1998, 141, 119–130.(16) Channell, G. M.; Zukoski, C. F. AIChE 1997, 43, 1700–1708.(17) Zhou, X; Scales, P. J.; Boger, D. V. Chem. Eng. Sci. 2001, 56, 2901–2920.

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similar low values of φf,18 suggesting that the particles are not

sticking to the capillary walls. Whatever be the mechanismcausing the low packing volume fraction in the capillary tube,the same should also be operational in the compressive yield stressmeasurements, with the crucial difference being that it is thecapillary pressure that compresses the network in the capillarytube but it is the centrifugal force in the latter. We believe that theformer technique is more representative of the drying process.However, further investigation is needed to clarify this issue.

In Figure 3, the experimentally measured CCTs are comparedwith those predicted by eq 2. The measured CCT (b) initiallydecreases slightly as the pH is increased from ∼4 to 6. Withfurther increase in pH from 6 to ∼10, there is a sharp increase inthe CCTwith themaximum observed at a pHof∼10. Finally, theCCT is found to decrease as the pH is further increased from 10to∼12. Recall that the critical cracking thickness depends on thevalue of the maximum capillary pressure (-Pmax) (eq 2), which inturn is a function of the final packing volume fraction and thewetting angle1,19

ð-PmaxÞR2γ

¼ 3

2cos θ

φf

1-φf

ð3Þ

Substituting eq 3 into eq 2 gives

hmax ¼ 0:644GMR3ð1-φfÞ327γcos3 θðφf Þ2

" #1=2

ð4Þ

which predicts an increase in CCT with a decrease in φf. This isbecause a decrease in the capillary pressure (due to lower φf)results in lower stored elastic energy and hence requires a thickerfilm to crack so as to equal the increase in surface energy. Themodel prediction compares fairlywell against themeasuredCCTsin that the model captures the peak close to the iep and thesubsequent decrease at higher pH values. The number of con-tacting neighbors (M) is calculated by linearly extrapolating the

data from Uri et al.20 to lower φf. The slight shift in the predictedpeak can be attributed to small errors in pHmeasurements (about(5%). At low pH (<6), however, the experiments exhibit areverse trend compared to the model predictions. One plausiblereason could be the high solubility of alumina at low pH21 thatcould lead to the precipitation of alumina at particle contacts thatin turn would lead to strengthening of the network withoutsignificantly altering the final packing volume fraction. Such aphenomenon is not accounted for in our model and could be thecause of the discrepancy. Chiu et al.8 also report higher CCTsbelow pH 2 and have attributed this to the higher solubility ofalumina at low pH, which explains the difference in the observedtrend with respect to those of Carreras et al.12 Nevertheless, ourmodel captures quite well the measured trend for the remainingpH values. However, the CCT predicted using φf implied by theyield stress data predicts only a modest increase (∼130 μm) closeto the iep. All in all, the above results point to the importance ofcapillary pressure and its role in the cracking of drying films.

Figure 4 plots the dimensionless critical cracking stress againstthe dimensionless characteristic scale (Mφf/ N

2) for dispersionswith pH values of 3.3, 6.5, 7.8, 10.0, and 11.1. Here, the averagethickness of the film (h) is used to calculate the dimensionless filmthickness (N� h/2R). It should be noted that the transverse flow5

generated by capillary pressures at the edges leads to slightinhomogeneties in the thickness over the film. Thus, althoughthe average thicknessmaybe low, the actual thickness over certainregions of the filmwould bewell above theCCT.7 The solid line inFigure 4 represents model predictions (eq 1) with no adjustableparameters.22 The experiments and model predictions are inexcellent agreement, confirming that themodel based on energeticarguments and proposed for stable dispersions is equally capableof predicting the cracking behavior in flocculated systems as well.However, it is important to note that the model is incapable of

Figure 3. Comparison of the measured critical cracking thickness(CCT) with the model predictions (eq 2) at different pH values.Furthermore, the values of φf and M at different pH values are(pH, φf, M) = (3.97, 0.66, 6.96), (4.98, 0.61, 6.18), (6, 0.56, 5.12),(7.25, 0.52, 4.90), (9.3, 0.18, 1.03), (10.46, 0.40, 3.82), and (11.88,0.68, 7.02). A contact angle of 35� was assumed.24

Figure 4. Plot of the measured dimensionless critical crackingstress (σcR/2γ) vs the dimensionless characteristic scale (Mφf/N

2).Thedatapoints are for filmsof theAKP-30dispersionwith varyingpH. The solid line represents eq 1 with no adjustable parameters.The values for φf andM at different pH values are (pH, φf,M) =(3.3, 0.69, 7.2), (6.5, 0.55, 5.23), (7.8, 0.35, 3.3), (10, 0.29, 2.76), and(11.1, 0.58, 5.51). Furthermore, γ=0.072 N/m,R=165 nm, andG= 156 GPa.

(18) Murray, M. Personal communication.(19) White, L. R. J. Colloid Interface Sci. 1982, 90, 536–538.(20) Uri, L.; Walmann, T.; Alberts, L.; Dysthe, D. K.; Feder, J. Phys. Rev. E

2006, 73, 051301(1)–051301(11).

(21) Charles, F. B. Jr.; Robert, E. M. The Hydrolysis of Cations; John Wiley &Sons: New York, 1976.

(22) As in case of Singh and Tirumkudulu,1 the coefficient in eq 1 wasincreased by a factor of 2 while comparing this equation with the experimentalresults.

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predicting the final packing volume fraction at which a givendrying filmwill crack.Thiswould require a knowledge of the forcerequired to compress the network during consolidation, whichin turn depends on the strength of the interparticle bonds andthe microstructure of the network. The models describing suchphenomena are still in a state of early development,23 and thediscrepancy between our measured φf close to the iep and thecorresponding values deduced from the reported compressiveyield stress underlines this fact. However, once the final packingvolume fraction of the dry film is known, the critical stress andthickness variationwith respect to other systemparameters (shearmodulus and particle radius) are captured well by our model.

Conclusions

The cracking behavior of a model aqueous dispersion overa broad range of pH values is investigated. The measured valuesof critical stress and thickness in both the stable and flocculated

regimeswere comparedwith the prediction of the existingmodel.1

Whereas good agreement was found for pH >6, some discre-pancy was observed in the critical thickness at lower pH. Thisstudy highlights the importance of capillary pressure in thecracking of drying films of colloidal dispersions and demonstrateshow merely varying the final packing volume fraction candrastically alter the cracking characteristics. The proposed modelprovides a complete theoretical framework for predicting thecracking characteristics of drying colloidal dispersions and couldform the basis for the efficient design of coating formulations.

Acknowledgment. This work was financially supported bythe Department of Science and Technology, India, and an IITBombay research grant. We thank Dr. Martin Murray ofAkzoNobel for stimulating discussions and for providing theAKP-30 particles.

(23) Channell, G. M.; Miller, K. T.; Zukoski, C. F. AIChE 2000, 46, 72–78.(24) Souheng, W. Polymer Interface and Adhesion;Marcel Dekker: New York,

1982.

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