Imaging in-plane and normal stresses near aninterface crack using traction force microscopyYe Xu (徐晔)a, Wilfried C. Engla, Elizabeth R. Jerisonb, Kevin J. Wallensteinc, Callen Hylandd,Larry A. Wilene, and Eric R. Dufresnea,b,f,1

aDepartment of Mechanical Engineering, Yale University, New Haven, CT 06511; bDepartment of Physics, Yale University, New Haven, CT 06511;cDepartment of Chemical Engineering, Princeton University, Princeton, NJ 08544; dDepartment of Molecular, Cellular and Developmental Biology, YaleUniversity, New Haven, CT 06511; eUnilever Research and Development, Trumbull, CT 06611; and fDepartment of Chemical Engineering and Departmentof Cell Biology, Yale University, New Haven, CT 06511

Edited by Katepalli R. Sreenivasan, New York University, New York, NY, and approved July 9, 2010 (received for review April 26, 2010)

Colloidal coatings, such as paint, are all around us. However,we know little about the mechanics of the film-forming processbecause the composition and properties of drying coatings varydramatically in space and time. To surmount this challenge, weextend traction force microscopy to quantify the spatial distribu-tion of all three components of the stress at the interface of twomaterials. We apply this approach to image stress near the tip ofa propagating interface crack in a drying colloidal coating andextract the stress intensity factor.

fracture mechanics ∣ colloids ∣ ceramics

The mechanical failure of coatings plagues electronic, optical,and biological systems (1–3). In many cases, fracture occurs at

the interface of dissimilar materials. These interface cracks havebeen the subject of extensive theoretical and experimental study(4–13). Although much progress has been made in understandingthe fracture of hard materials, relatively little is known about thefracture mechanics of soft matter. An extremely challenging andfamiliar example is the drying of a colloidal coating, such as paint,which starts off as a fluid and fails as a solid (14). We present arobust experimental approach for studying fracture mechanics insystems with complex spatially and temporally varying mechani-cal properties. Our approach is inspired by recent work in themechanics of biological cells, where the spatial distributionand magnitude of forces are inferred by the deformation of acompliant substrate (15–22). We extend this technique, calledtraction force microscopy, to image the stress near the tip of anadvancing interface crack. We compare our results to the antici-pated universal scaling of stress near a crack tip to measure thestress intensity factor directly.

Results and DiscussionWe study interface cracks formed during the drying of a colloidalcoating. Our samples start off as aqueous suspensions of 11-nmradius silica particles (Ludox AS-40) at a volume fraction ϕ ¼ 0.2.Drying transforms this fluid suspension into a brittle solid thatcracks prodigiously. To simplify the geometry, we confine the sus-pension to a rectangular capillary tube, which allows evaporationof solvent from only one edge (23, 24). During the course ofdrying, the composition and material properties are highly het-erogeneous, with coexistence of low volume fraction fluid regionsand relatively high volume fraction brittle regions. The concen-tration gradient is localized to a compaction front that movessteadily into the sample (25, 26). The tip of an interface crack,debonding the colloidal coating from the substrate, follows about500 μm behind the compaction front, as shown in Fig. 1A.

We visualize the flow and deformation of the colloid byimaging fluorescent tracer particles with time-lapse 3D confocalmicroscopy, as shown in Fig. 1 B and C and Movies S1 and S2.The leading edge of the crack is clearly visible in confocal micro-graphs in a plane near the substrate. Using a simple edge detec-tion algorithm, we can readily extract the shape of the crackfront, xoðyÞ, at each time point, as shown by the red line in Fig. 1C.

The mean position of the crack front moves smoothly at a velocityof about 460 nm∕s, as shown in the Fig. S1. At this speed, theinterface crack is expected to be quasistatic.

While these three-dimensional images visualize the deforma-tion of the colloidal coating during fracture, they do not revealthe internal stresses that drive fracture. If the mechanical proper-ties of the colloid were known, then the stress fields could bereadily determined from the deformation field. Indeed, carefulanalysis of the images in Fig. 1 B and C can quantify the localstrain tensor (27). However, the mechanical properties of thisdrying colloid are unknown. Therefore, stress measurementsneed to be calibrated against an external reference with well-defined mechanical properties. In conventional studies of themechanics of coatings and films, this is provided by bondingthe coating to a wafer or cantilever and measuring the curvatureof the substrate to provide the in-plane stresses (28–32). Instead,we image and analyze the three-dimensional deformation fieldof a substrate with well-defined mechanical properties to extractall three components of the stress at an interface between sub-strate and coating. To achieve this, we deposit a film of elastomer(Dow Corning Sylgard 184) of thickness h (≈45 μm) onto onewall of our capillary tube. We quantify the deformation of theelastomer by tracking the three-dimensional displacements oftracer particles at the plane z ¼ zo, about 3 μm below the surfaceof the elastomer at z ¼ h, as shown in Fig. 2A.

The internal stresses that drive the fracture of our colloidalcoating (33) readily deform the adherent elastomer as shownin Fig. 1B and Movie S1. The resulting deformation field,~uðx;y;zoÞ is shown in Fig. 2B at a time point of 201.0 min, whenthe crack front is at the center of the field of view. The deforma-tion is highly heterogeneous and concentrated near the crackfront, with large components in the x and z directions. Notably,because of the long-range nature of elastic forces, there are largedeformations in regions where the coating has debonded from thesubstrate. The time-dependent deformation field is shown inMovie S3.

In order to deconvolve these long-range deformations fromtheir localized forces, we solve the governing equations of elasto-statics. For an isotropic linear elastic medium, the equation ofequilibrium is given by

ð1 − 2νÞ∇2 ~uþ ~∇ð ~∇ · ~uÞ ¼ 0; [1]

Author contributions: Y.X., W.C.E., C.H., and E.R.D. designed research; Y.X., W.C.E., E.R.J.,K.J.W., and C.H. performed research; Y.X., W.C.E., E.R.J., L.A.W., and E.R.D. analyzed data;and Y.X. and E.R.D. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. E-mail: eric.dufresne@yale.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1005537107/-/DCSupplemental.

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where ν is Poisson’s ratio and ~u is the displacement (34). For filmsthat have finite thickness and are bonded on one side to a rigidplane, the following two boundary conditions hold valid. First,good adhesion to the comparatively rigid glass substrate demandsthat ~uðx;y;0Þ ¼ 0. Second, we specify the stress on the freesurface, where σizðx0;y0;hÞ equals the force per unit area on thesurface in the ith direction, as indicated in Fig. 2A. BecauseEq. 1 is linear, the stress and displacement are simply relatedby a tensor product in Fourier space,

σizð ~k;hÞ ¼ Qijð ~k;z0;hÞujð ~k;z0Þ; [2]

where ~k is the wave vector in the xy plane, with summation overrepeated indices. The tensor Q describes the mechanicalresponse of the elastic substrate incorporating its material prop-erties and geometry. We calculate Q using a straightforward ex-tension of the method of del Alamo et al. (18), which accountedfor the finite thickness of the film but assumed no normal stresses.The mathematical form of the full three-dimensional Q tensor isgiven in SI Text.

There are a few subtleties in the application of Eq. 2. First, forincompressible materials, uniform out-of-plane stresses cannotdeform the material, regardless of their magnitude. This resultsin an essential ambiguity in σzz for k ¼ 0 when ν ¼ 1∕2. In orderto specify the stress offset in the z component, it is essential toinclude a region where the stresses are known to be zero withinthe field of view. Second, high spatial-frequency stresses havevery little effect on the displacement. Thus, high spatial-frequency components in the measurement error of the displace-ment field are strongly amplified during the inversion process.

Therefore, it is essential to apply a low-pass filter to the measuredstress fields after the application of Eq. 2.

The observed stress distribution near the crack front is shownat one time point in Fig. 2C. The key feature of this plot is that thestress is highly heterogeneous and predominantly in the z direc-tion. The stress is distributed throughout the region bound to thesubstrate, but is concentrated near the crack front. The timeevolution of the stress distribution is shown in Movie S4. The het-erogeneity of the stress is undetectable with techniques thatmeasure only average curvature. While spatially resolved curva-ture measurements can map in-plane stresses, they cannot detectthe normal component of the stress.

We exploit the geometric and temporal symmetries of oursystem to increase the spatial resolution and field of view ofour data. Since the crack front is relatively straight and thedisplacements in the x and z directions, ux and uz, are muchbigger than that in the y direction, uy, the geometry of our experi-ment is essentially two-dimensional [maxðux;uzÞ ≈ 1.5 μm andmaxðuyÞ ≈ 0.15 μm]. Therefore, we consider only the x and zcoordinates of tracer beads. Projecting all of the bead locationsalong y allows us to more densely sample the deformation of thesystem. We can further improve the spatial resolution and expandthe field of view by exploiting the smooth motion of the crackfront, which, over the time course of our experiment, travelswith a constant velocity, as shown in Fig. S1. Collapsing the datain space and time, we arrive at the densely sampled particlepositions in the colloid (Fig. 3A) and displacement fields inthe elastomer (Fig. 3B). Here, each point corresponds to a singlebead at a single time point. To correct for small deviations of thecrack front from a straight line, the x positions of the tracer par-ticles are measured relative to their distance from the crack front.Specifically, the position of a particle at ½x;y;z� is plotted as

Colloid

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201.0 min

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Colloid

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airair

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Fig. 1. Interface Crack in a Colloidal Coating. (A) Schematic drawing ofthe experimental geometry. Colloid rests in a capillary tube and dries fromthe opening on the left side. (B) Side views of the system at two time points,created by a maximum intensity projection along y of confocal micrographsof fluorescent tracer particles embedded in the colloid and elastomer. Here,the crack front lies about 10 mm from the drying edge. (C) Top views ofdrying colloid at the same time points. Each image is a single confocal slicetaken near the surface of the elastomer. The solid red line indicates theposition of the crack front xoðyÞ in each time point. The dashed line indicatesthe position of the crack front in the previous image.

A

B

C

Fig. 2. Distributions of 3D Displacement and Stress. (A) Schematic drawingspecifying the geometry of the elastomer film. Wemeasure the displacementof tracer particles in the plane of z ¼ zo and calculate the stress on the surfaceof the film where z ¼ h. (B) 3D displacement in the elastomer att ¼ 201.0 min. (C) 3D stress, σiz, on the interface at t ¼ 201.0 min. Thestresses are low-pass filtered with a FWHM cutoff of 24 μm. The red solid linesin B and C show the positions of crack front.

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½x − xoðyÞ;z�. Using Eq. 2, we convert these displacements intostresses as shown in Fig. 3C. Here, the out-of-plane stress, σzz,is zero behind the crack front. Just ahead of the crack front,σzz shoots up rapidly before decaying slowly. At all locationsahead of the crack front, the in-plane stresses are much smallerthan the out-of-plane stresses. This suggests that the crack ismixed mode, with mode I dominating over mode II (8). Justahead of the crack front, the in-plane stress points away fromthe front. From x ≈ 50 to 200 μm ahead of the crack front, thereare small (<5 kPa) but systematic in-plane stresses that point inthe direction of fluid flow.

The classic theory of fracture predicts that the stress around anisolated crack tip should decay as 1∕

ffiffir

p, where r is the distance

from tip. A log-log plot of σzz reveals that stress measured aheadof the crack front satisfies this scaling over about one decadein distance from the tip of the crack, from x − xo ¼ 0.2h to 2h.Fitting this region to the explicit form for linear elastic mode Ifracture, σzz ¼ KI∕

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πðx − xoÞ

p, we find the stress intensity factor

KI ¼ 550� 30 Paffiffiffiffim

p. Far away from the crack front, the mea-

sured stresses systematically deviate from this scaling. This isexpected due to the finite thickness of the elastomer film, asdescribed in ref. 13. Near the crack front, the measured stressesare systematically lower than the 1∕

ffiffir

plaw. This is partially due to

the spatial resolution of our technique. To demonstrate this, welow-pass filter the expected mode I stress distribution using thesame low-pass filter applied to our data. Although this signifi-cantly reduces the stress near the peak, as shown by the greencurve in Fig. 3C, a discrepancy remains that is likely due to non-linear deformation of material near the crack front.

ConclusionsWe measure the stress distribution near an interface crack usinga technique suitable for materials with spatially and temporallyheterogeneous mechanical properties. In the future, we hopeto measure the local constitutive relationships of unknown mate-rials in situ by correlating the stress and strain of the coating atthe interface. Our approach is quite general and can be adaptedto measure interfacial stresses over a wide dynamic range bychanging the elastic modulus of the elastomer film. In particular,our extension of traction force microscopy is well-suited to mea-sure three-dimensional forces generated by motile and adherentcells on substrates.

Materials and MethodsPreparation of Elastomer-Coated Coverslips. First, the buffer solution for fluor-escent beads is made by mixing sodium tetraborate and boric acid withdeionized water to obtain the pH value at 7.4. To this solution, red fluores-cent microspheres (0.1 μm, carboxylate-modified, Molecular Probes) and1 wt % 1-ethyl-3-(3-dimethylaminopropyl)carbodiimide (EDC) solution areadded in a volume ratio of 1∶104 and 1∶100, respectively.

A glass coverslip (Fisherbrand) is chemically modified for bonding fluor-escent beads by vapor-depositing silane (3-aminopropyl triethoxysilan) onthe surface. The first layer of fluorescent beads is then adsorbed on thesurface of the silanated glass coverslip by floating it on the bead suspensionfor 5 min before rinsing in the deionized water (20). The resulting density ofbeads is around 100 per 100 × 100 μm2. After the water drys out, elastomer isprepared by mixing a silicone elastomer base (Sylgard 184, Dow Corning) andthe curing agent in a weight ratio of 15∶1, and then spin-coated on the activesurface of the coverslip at 2,000 rpm for 1 min. The sample is baked at 60 °Cfor 2 h to expedite the cross-linking, resulting in a ∼41-μm-thick elastomerfilm. After the cross-linking, the surface of the elastomer film is also treatedwith silane vapor deposition. A second layer of fluorescent polystyrene beadsis dispersed on the surface of the elastomer film by floating it on the beadssolution for 10 min in order to achieve a higher density (around 300 per100 × 100 μm2). Another layer of elastomer is then spin-coated at10,000 rpm for 2 min, resulting in a ∼3-μm-thick film after baking in the ovenat 60 °C for 2 h. The resulting elastomer film has a Young’s modulus of750 kPa and a Possion’s ratio of 0.5.

Assembly of Rectangular Capillary Tube. The elastomer-coated coverslip and amicroscope slide are attached together, with the elastomer coating facingthe slide, by two strips of double-sided tape that also act as spacers, leavinga 100-μm gap in between. The resulting tube has dimensions of Δx ≈ 40 mm,Δy ≈ 5 mm, and Δz ≈ 0.1 mm.

Preparation of Colloidal Suspension. Yellow-green fluorescent microspheres(0.1 μm, carboxylate-modified, 0.1 vol %, Molecular Probes) are added toan aqueous suspension of monodisperse colloidal silica nanoparticles withradii of a ¼ 11 nm (Ludox AS-40). The mixed colloidal suspension is loadedinto the tube by capillary rise.

Confocal Microscopy and Time-Lapse Imaging. Three-dimensional image stacksare acquired using a spinning disk confocal system (Andor Revolution)mounted on an invertedmicroscope (Nikon Eclipse Ti) with 40× oil-immersionobjective lens (Plan Fluor) with a numerical aperture (NA) of 1.30. Two laserlines, 491 nm and 561 nm, are used to image fluorescent beads in the colloidand elastomer, respectively. Confocal image stacks are acquired every 30 s forapproximately 4 h after the colloidal suspension was loaded. Each confocalstack contains 300 slices of images, covering tracer beads in both colloid andelastomer film at a resolution of 512 × 512 pixel2. The field of view is

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Fig. 3. Projection and Space-Time Superposition Increase Spatial Resolution.(A) Side view of tracer particle locations in the colloidal coating collapsedover time and space as described in the text. (B) Displacements of the tracerparticles in the elastomer collapsed in time and space. ux and uz are plottedas cyan and blue dots, respectively. The dashed and solid lines are thesmoothed displacements in x and z, respectively. (C) Stress at the interfacecalculated using the smoothed displacements shown in B with an additional10-μm FWHM spatial filter. σxz and σzz are plotted as blue open and closedcircles, respectively. The red solid line shows the fit of σzz to the expected1∕

ffiffiffix

pscaling, whereas the green line shows the same with a 10-μm FWHM

spatial filter. The dashed red line indicates zero stress. (Inset) σzz on a log-logscale.

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165 × 165 μm2. The observation window is located approximately 10 mmaway from the filling end.

Calculation of Displacement and Stress. The 3D displacements of tracer beadsare calculated using centroid tracking algorithms (35, 36) from the confocalimage stacks. The displacement field of the z ¼ zo plane is then establishedby interpolating the displacement of randomly dispersed beads into a 2Dspatial grid with a spacing equal to the mean distance between tracer beads.In this study, the grid spacing is approximately 6 μm. The displacement caused

by drift during the experiments is eliminated by subtracting the mean displa-cement of tracer beads on the coverslip at z ¼ 0. Once the 3D displacementfield [ ~uðzoÞ] was established, the stress at the interface of the elastomer andcolloid is calculated in Fourier space using Eq. 2.

ACKNOWLEDGMENTS. We gratefully acknowledge helpful conversations withL. Mahadevan, W. B. Russel, and J. W. Hutchinson, as well as financial supportfrom Unilever and the National Science Foundation (DBI-0619674).

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