Imaging in-plane and normal stresses near an interface crack using traction force microscopy

Abstract

Colloidal coatings, such as paint, are all around us. However, we know little about the mechanics of the film-forming process because the composition and properties of drying coatings vary dramatically in space and time. To surmount this challenge, we extend traction force microscopy to quantify the spatial distribution of all three components of the stress at the interface of two materials. We apply this approach to image stress near the tip of a propagating interface crack in a drying colloidal coating and extract the stress intensity factor.

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 have been the subject of extensive theoretical and experimental study (4–13). Although much progress has been made in understanding the fracture of hard materials, relatively little is known about the fracture mechanics of soft matter. An extremely challenging and familiar 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 a robust experimental approach for studying fracture mechanics in systems with complex spatially and temporally varying mechanical properties. Our approach is inspired by recent work in the mechanics of biological cells, where the spatial distribution and magnitude of forces are inferred by the deformation of a compliant substrate (15–22). We extend this technique, called traction force microscopy, to image the stress near the tip of an advancing interface crack. We compare our results to the anticipated universal scaling of stress near a crack tip to measure the stress intensity factor directly.

Results and Discussion

We study interface cracks formed during the drying of a colloidal coating. Our samples start off as aqueous suspensions of 11-nm radius silica particles (Ludox AS-40) at a volume fraction ϕ = 0.2. Drying transforms this fluid suspension into a brittle solid that cracks prodigiously. To simplify the geometry, we confine the suspension to a rectangular capillary tube, which allows evaporation of solvent from only one edge (23, 24). During the course of drying, the composition and material properties are highly heterogeneous, with coexistence of low volume fraction fluid regions and relatively high volume fraction brittle regions. The concentration gradient is localized to a compaction front that moves steadily into the sample (25, 26). The tip of an interface crack, debonding the colloidal coating from the substrate, follows about 500 μm behind the compaction front, as shown in Fig. 1A.

Fig. 1.

Interface Crack in a Colloidal Coating. (A) Schematic drawing of the experimental geometry. Colloid rests in a capillary tube and dries from the 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 micrographs of fluorescent tracer particles embedded in the colloid and elastomer. Here, the crack front lies about 10 mm from the drying edge. (C) Top views of drying colloid at the same time points. Each image is a single confocal slice taken near the surface of the elastomer. The solid red line indicates the position of the crack front x_o(y) in each time point. The dashed line indicates the position of the crack front in the previous image.

We visualize the flow and deformation of the colloid by imaging fluorescent tracer particles with time-lapse 3D confocal microscopy, as shown in Fig. 1 B and C and Movies S1 and S2. The leading edge of the crack is clearly visible in confocal micrographs in a plane near the substrate. Using a simple edge detection algorithm, we can readily extract the shape of the crack front, x_o(y), at each time point, as shown by the red line in Fig. 1C. The mean position of the crack front moves smoothly at a velocity of about 460 nm/s, as shown in the Fig. S1. At this speed, the interface crack is expected to be quasistatic.

While these three-dimensional images visualize the deformation of the colloidal coating during fracture, they do not reveal the internal stresses that drive fracture. If the mechanical properties of the colloid were known, then the stress fields could be readily determined from the deformation field. Indeed, careful analysis of the images in Fig. 1 B and C can quantify the local strain tensor (27). However, the mechanical properties of this drying colloid are unknown. Therefore, stress measurements need to be calibrated against an external reference with well-defined mechanical properties. In conventional studies of the mechanics of coatings and films, this is provided by bonding the coating to a wafer or cantilever and measuring the curvature of the substrate to provide the in-plane stresses (28–32). Instead, we image and analyze the three-dimensional deformation field of a substrate with well-defined mechanical properties to extract all three components of the stress at an interface between substrate and coating. To achieve this, we deposit a film of elastomer (Dow Corning Sylgard 184) of thickness h (≈45 μm) onto one wall of our capillary tube. We quantify the deformation of the elastomer by tracking the three-dimensional displacements of tracer particles at the plane z = z_o, about 3 μm below the surface of the elastomer at z = h, as shown in Fig. 2A.

Fig. 2.

Distributions of 3D Displacement and Stress. (A) Schematic drawing specifying the geometry of the elastomer film. We measure the displacement of tracer particles in the plane of z = z_o and calculate the stress on the surface of the film where z = h. (B) 3D displacement in the elastomer at t = 201.0 min. (C) 3D stress, σ_iz, on the interface at t = 201.0 min. The stresses are low-pass filtered with a FWHM cutoff of 24 μm. The red solid lines in B and C show the positions of crack front.

The internal stresses that drive the fracture of our colloidal coating (33) readily deform the adherent elastomer as shown in Fig. 1B and Movie S1. The resulting deformation field, is shown in Fig. 2B at a time point of 201.0 min, when the crack front is at the center of the field of view. The deformation is highly heterogeneous and concentrated near the crack front, with large components in the x and z directions. Notably, because of the long-range nature of elastic forces, there are large deformations in regions where the coating has debonded from the substrate. The time-dependent deformation field is shown in Movie S3.

In order to deconvolve these long-range deformations from their localized forces, we solve the governing equations of elastostatics. For an isotropic linear elastic medium, the equation of equilibrium is given by

[1]

where ν is Poisson’s ratio and is the displacement (34). For films that have finite thickness and are bonded on one side to a rigid plane, the following two boundary conditions hold valid. First, good adhesion to the comparatively rigid glass substrate demands that . Second, we specify the stress on the free surface, where σ_iz(x^′,y^′,h) equals the force per unit area on the surface in the ith direction, as indicated in Fig. 2A. Because Eq. 1 is linear, the stress and displacement are simply related by a tensor product in Fourier space,

[2]

where is the wave vector in the xy plane, with summation over repeated indices. The tensor Q describes the mechanical response of the elastic substrate incorporating its material properties and geometry. We calculate Q using a straightforward extension of the method of del Alamo et al. (18), which accounted for the finite thickness of the film but assumed no normal stresses. The mathematical form of the full three-dimensional Q tensor is given in SI Text.

There are a few subtleties in the application of Eq. 2. First, for incompressible materials, uniform out-of-plane stresses cannot deform the material, regardless of their magnitude. This results in an essential ambiguity in σ_zz for k = 0 when ν = 1/2. In order to specify the stress offset in the z component, it is essential to include a region where the stresses are known to be zero within the field of view. Second, high spatial-frequency stresses have very little effect on the displacement. Thus, high spatial-frequency components in the measurement error of the displacement field are strongly amplified during the inversion process. Therefore, it is essential to apply a low-pass filter to the measured stress fields after the application of Eq. 2.

The observed stress distribution near the crack front is shown at one time point in Fig. 2C. The key feature of this plot is that the stress is highly heterogeneous and predominantly in the z direction. The stress is distributed throughout the region bound to the substrate, but is concentrated near the crack front. The time evolution of the stress distribution is shown in Movie S4. The heterogeneity of the stress is undetectable with techniques that measure only average curvature. While spatially resolved curvature measurements can map in-plane stresses, they cannot detect the normal component of the stress.

We exploit the geometric and temporal symmetries of our system to increase the spatial resolution and field of view of our data. Since the crack front is relatively straight and the displacements in the x and z directions, u_x and u_z, are much bigger than that in the y direction, u_y, the geometry of our experiment is essentially two-dimensional [max(u_x,u_z) ≈ 1.5 μm and max(u_y) ≈ 0.15 μm]. Therefore, we consider only the x and z coordinates of tracer beads. Projecting all of the bead locations along y allows us to more densely sample the deformation of the system. We can further improve the spatial resolution and expand the field of view by exploiting the smooth motion of the crack front, which, over the time course of our experiment, travels with a constant velocity, as shown in Fig. S1. Collapsing the data in space and time, we arrive at the densely sampled particle positions in the colloid (Fig. 3A) and displacement fields in the elastomer (Fig. 3B). Here, each point corresponds to a single bead at a single time point. To correct for small deviations of the crack front from a straight line, the x positions of the tracer particles are measured relative to their distance from the crack front. Specifically, the position of a particle at [x,y,z] is plotted as [x - x_o(y),z]. Using Eq. 2, we convert these displacements into stresses as shown in Fig. 3C. Here, the out-of-plane stress, σ_zz, is zero behind the crack front. Just ahead of the crack front, σ_zz shoots up rapidly before decaying slowly. At all locations ahead of the crack front, the in-plane stresses are much smaller than the out-of-plane stresses. This suggests that the crack is mixed mode, with mode I dominating over mode II (8). Just ahead of the crack front, the in-plane stress points away from the front. From x ≈ 50 to 200 μm ahead of the crack front, there are small (< 5 kPa) but systematic in-plane stresses that point in the direction of fluid flow.

Fig. 3.

Projection and Space-Time Superposition Increase Spatial Resolution. (A) Side view of tracer particle locations in the colloidal coating collapsed over time and space as described in the text. (B) Displacements of the tracer particles in the elastomer collapsed in time and space. u_x and u_z are plotted as cyan and blue dots, respectively. The dashed and solid lines are the smoothed displacements in x and z, respectively. (C) Stress at the interface calculated using the smoothed displacements shown in B with an additional 10-μm FWHM spatial filter. σ_xz and σ_zz are plotted as blue open and closed circles, respectively. The red solid line shows the fit of σ_zz to the expected scaling, whereas the green line shows the same with a 10-μm FWHM spatial filter. The dashed red line indicates zero stress. (Inset) σ_zz on a log-log scale.

The classic theory of fracture predicts that the stress around an isolated crack tip should decay as , where r is the distance from tip. A log-log plot of σ_zz reveals that stress measured ahead of the crack front satisfies this scaling over about one decade in distance from the tip of the crack, from x - x_o = 0.2h to 2h. Fitting this region to the explicit form for linear elastic mode I fracture, , we find the stress intensity factor . Far away from the crack front, the measured stresses systematically deviate from this scaling. This is expected due to the finite thickness of the elastomer film, as described in ref. 13. Near the crack front, the measured stresses are systematically lower than the law. This is partially due to the spatial resolution of our technique. To demonstrate this, we low-pass filter the expected mode I stress distribution using the same low-pass filter applied to our data. Although this significantly reduces the stress near the peak, as shown by the green curve in Fig. 3C, a discrepancy remains that is likely due to nonlinear deformation of material near the crack front.

Conclusions

We measure the stress distribution near an interface crack using a technique suitable for materials with spatially and temporally heterogeneous mechanical properties. In the future, we hope to measure the local constitutive relationships of unknown materials in situ by correlating the stress and strain of the coating at the interface. Our approach is quite general and can be adapted to measure interfacial stresses over a wide dynamic range by changing the elastic modulus of the elastomer film. In particular, our extension of traction force microscopy is well-suited to measure three-dimensional forces generated by motile and adherent cells on substrates.

Materials and Methods

Preparation of Elastomer-Coated Coverslips.

First, the buffer solution for fluorescent beads is made by mixing sodium tetraborate and boric acid with deionized water to obtain the pH value at 7.4. To this solution, red fluorescent microspheres (0.1 μm, carboxylate-modified, Molecular Probes) and 1 wt % 1-ethyl-3-(3-dimethylaminopropyl)carbodiimide (EDC) solution are added in a volume ratio of 1∶10⁴ and 1∶100, respectively.

A glass coverslip (Fisherbrand) is chemically modified for bonding fluorescent beads by vapor-depositing silane (3-aminopropyl triethoxysilan) on the surface. The first layer of fluorescent beads is then adsorbed on the surface of the silanated glass coverslip by floating it on the bead suspension for 5 min before rinsing in the deionized water (20). The resulting density of beads is around 100 per 100 × 100 μm². After the water drys out, elastomer is prepared by mixing a silicone elastomer base (Sylgard 184, Dow Corning) and the curing agent in a weight ratio of 15∶1, and then spin-coated on the active surface of the coverslip at 2,000 rpm for 1 min. The sample is baked at 60 °C for 2 h to expedite the cross-linking, resulting in a ∼41-μm-thick elastomer film. After the cross-linking, the surface of the elastomer film is also treated with silane vapor deposition. A second layer of fluorescent polystyrene beads is dispersed on the surface of the elastomer film by floating it on the beads solution for 10 min in order to achieve a higher density (around 300 per 100 × 100 μm²). Another layer of elastomer is then spin-coated at 10,000 rpm for 2 min, resulting in a ∼3-μm-thick film after baking in the oven at 60 °C for 2 h. The resulting elastomer film has a Young’s modulus of 750 kPa and a Possion’s ratio of 0.5.

Assembly of Rectangular Capillary Tube.

The elastomer-coated coverslip and a microscope slide are attached together, with the elastomer coating facing the slide, by two strips of double-sided tape that also act as spacers, leaving a 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 to an aqueous suspension of monodisperse colloidal silica nanoparticles with radii of a = 11 nm (Ludox AS-40). The mixed colloidal suspension is loaded into the tube by capillary rise.

Confocal Microscopy and Time-Lapse Imaging.

Three-dimensional image stacks are acquired using a spinning disk confocal system (Andor Revolution) mounted on an inverted microscope (Nikon Eclipse Ti) with 40× oil-immersion objective lens (Plan Fluor) with a numerical aperture (NA) of 1.30. Two laser lines, 491 nm and 561 nm, are used to image fluorescent beads in the colloid and elastomer, respectively. Confocal image stacks are acquired every 30 s for approximately 4 h after the colloidal suspension was loaded. Each confocal stack contains 300 slices of images, covering tracer beads in both colloid and elastomer film at a resolution of 512 × 512 pixel². The field of view is 165 × 165 μm². The observation window is located approximately 10 mm away from the filling end.

Calculation of Displacement and Stress.

The 3D displacements of tracer beads are calculated using centroid tracking algorithms (35, 36) from the confocal image stacks. The displacement field of the z = z_o plane is then established by interpolating the displacement of randomly dispersed beads into a 2D spatial 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 displacement of tracer beads on the coverslip at z = 0. Once the 3D displacement field [] was established, the stress at the interface of the elastomer and colloid is calculated in Fourier space using Eq. 2.