Carbon Chemist

Tag: Structure of solids and liquids

  • X-ray linear dichroic tomography of crystallographic and topological defects

    X-ray linear dichroic tomography of crystallographic and topological defects

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    Materials

    We purchased V2O5 from US Research Nanomaterials and polystyrene latex spheres, 330 nm in diameter, from Thermo Fisher Scientific.

    Sample preparation

    The examined pillar was extracted from a sintered millimetre-sized pellet. This was prepared from a mixture of nanocrystalline V2O5 and polystyrene spheres (85/15 wt%). Mortar and pestle (10 min) was used to homogenize the mixture before the pellet was pressed using a 17 mm die set (3 min, 1.2 t uniaxial load). To increase the V2O5 grain size, sinter and create the desired porous structure, we heated the pellet to 590 °C for 5 h (Supplementary Fig. 1 and Supplementary Note 2). The polycrystalline V2O5 pillar was prepared by mechanically fracturing the sintered pellet, after which a fracture piece was mounted on an OMNY tomography pin51 using epoxy resin. The pillar was then pre-shaped using a microlathe52, before being reduced in diameter to 6 µm using focused ion beam (FIB) milling. This pillar was then transferred onto a second OMNY pin51. The tip of the second OMNY pin was sharpened using FIB milling before transferring the pillar. This was necessary to facilitate tomography measurements with a 30° stage tilt26. See Supplementary Fig. 3 for micrographs of the prepared pillar.

    General material characterization

    Scanning electron microscopy and FIB milling were performed using a Zeiss NVision 40 dual-beam FIB. Powder X-ray diffraction measurements of the sample before and after sintering were acquired using a Cu K-α radiation source with a step size of 0.02° 2θ, (refs. 53,54) (Supplementary Fig. 2). The sintered sample consists of α-V2O5 with a grain size >100 nm.

    Origin of linear dichroism in α-V2O5

    V2O5 has a layered orthorhombic crystal structure consisting of distorted [VO5] pyramids, shown schematically in Fig. 1b. These pyramids tile along the a–b plane and are bound with van der Waals interactions along the c axis. The apical, vanadyl bond of these pyramids, aligned with the crystallographic c axis, is shorter (1.57 Å) compared with the bonds on the base of the pyramid (1.87 Å). This shorter bond breaks the symmetry of an otherwise regular square pyramid55. To examine the spatial orientation of the apical bond and, in turn, the orientation of entire grains and deviations within them, the energy of the incident X-rays was set to that of the vanadium K pre-edge peak55. This peak arises from the V (1 s) to V (4p-3d) transition, more specifically, to V (3d eg + 4p) + O 2pz mixing states, which become accessible as a result of the deviation of the V coordination from the octahedral symmetry. When the apical bond is parallel to the direction of the electric field of the incident X-rays, the interaction is strong, as the transition V (1 s) to V (4p-3d) is allowed. When the apical bond is instead perpendicular to the incident polarization, the interaction is weaker29,55. An illustration of the different absorption strengths that result from the relative orientation between the incident X-ray polarization and the apical bond, known as linear dichroism, is shown in Fig. 1b. The X-ray near-edge absorption and phase spectra of V2O5 measured using LH and LV polarizations are shown in Supplementary Fig. 4 and a schematic of the layered crystal structure is provided in ref. 29.

    In the above-described relationship between the polarization state of the illumination and the examined asymmetry or anisotropy, the linearly polarized light acts as a ‘search light’ for the resonant bond to which the polarization is parallel. This relationship applies, in principle, to all cases of natural linear dichroism16,42. The connection between the investigated anisotropy orientation and the unit-cell orientation of the material can be obtained through the use of reference samples, as showcased in 2D linear dichroic microscopy applications36, and is already available in the literature for numerous materials. It can also be readily determined with previous knowledge of the material’s crystal structure (or molecular arrangement)55.

    Ptychography, PXCT and phase contrast

    Ptychography is a lensless imaging technique in which the phase problem is solved by means of iterative phase-retrieval algorithms27. By applying ptychography to solve the phase problem at different projection angles, its tomographic extension, ptychographic X-ray computed tomography (PXCT)56, is able to retrieve the complex-valued transmissivity of the specimen, providing quantitative tomograms of both phase and amplitude contrast32. Both the individual images—or projections—and resulting tomograms obtained using X-ray ptychography are sensitive to changes in the complex-valued refractive index, η. The real part of the refractive index decrement, δ, corresponds to the phase, whereas the imaginary part of the refractive index corresponds to the amplitude, β. The refractive index is fundamentally an expression of the complex atomic scattering factor, f = f1 + if2. The refractive index is therefore given by:

    $$n=1-\delta -{\rm{i}}\beta =1-\frac{{r}_{{\rm{e}}}}{2{\rm{\pi }}}{\lambda }^{2}\sum _{k}{n}_{{\rm{at}}}^{k}({f}_{1}^{k}\,+{if}_{2}^{k}),$$

    (1)

    with re being the classical electron radius and λ the illumination wavelength57,58. The images and tomograms resulting from measurements performed with incident X-ray energies away from sample-relevant absorption edges can, in the case of tomograms, be converted to quantitative electron-density, ne, and absorption index, µ, tomograms32. Measurements conducted near sample-relevant absorption edges, that is, examining specific electronic transitions and the associated increase in the photoabsorption cross-section, are subject to anomalous scattering effects57,58, including dichroism.

    The angular dependence of the linear dichroism has previously been used in a microscopy context, in particular in X-ray linear dichroism microscopy with secondary imaging modalities such as photoemission electron microscopy, to provide a 2D spatially resolved microstructural characterization tool16,29,30,36,59,60. The reader is directed to the initial work of Ade and Hsiao16 and the more recent works of Gilbert et al.36,61,62,63 and Collins et al.59,60,64. In the present work, we have developed the capability to map the orientation in 3D by combining X-ray linear dichroism microscopy with PXCT (XL-DOT).

    Although XL-DOT can be applied with a range of imaging techniques, such as scanning transmission X-ray microscopy, we have selected X-ray ptychography as the imaging modality, a choice motivated by three factors. (1) PXCT provides quantitative or absolute contrast tomograms, which is ideal for material or component identification and for the detection of marginal signal variations11,30. (2) As a lensless imaging technique, ptychography excels in terms of signal-to-noise ratio (SNR), spatial resolution and dose efficiency (per resolution element) compared with other methods65,66,67,68. Given its superior SNR, it is ideal for measuring the relatively weak linear dichroism signal exhibited by V2O5 (refs. 30,61). (3) Ptychography can access the phase component. Phase changes at the vanadium K-edge are twice as large as changes in the absorption, so that the retrieved phase projections have a higher spatial resolution and superior SNR; see Supplementary Figs. 5 and 6 (ref. 58). We performed all data analysis on the phase component of the projections and tomograms only.

    Ptychographic linear dichroic X-ray tomography

    Data acquisition

    Experiments were carried out at the coherent small-angle X-ray scattering (cSAXS) beamline of the Swiss Light Source. The photon energy was selected using a double-crystal Si(111) monochromator. The horizontal aperture of slits located 22 m upstream of the sample was set to 20 μm, creating a virtual source point that coherently illuminates a 220-μm-diameter Fresnel zone plate with an outermost zone width of 60 nm and with engineered aberrations designed to improve reconstruction contrast and spatial resolution50. Coherent diffraction patterns were acquired using an in-vacuum Eiger 1.5M area detector, with a 75 µm pixel size, placed 5.235 m downstream of the sample inside an evacuated flight tube. Tomography experiments were performed using the positioning instrument described in ref. 69.

    To map the local orientation of the apical bond within the examined sample volume in 3D, we exploited its linear dichroism and acquired eight equiangular ptychographic tomograms over 180° at 5.469 keV for different illumination polarizations and sample tilts. Specifically, ptychographic tomograms were acquired with a LH and LV polarization of the incident illumination at 0° stage tilt and at 30° stage tilt (sample in grey and pink in the top two panels on the right of Fig. 1a). Two further tilts were measured, whereby the sample was first rotated by +90° and −90° about the main axis of the pillar, followed by a 30° stage tilt26. The last two tilts are equivalent to tilting towards and away from the beam by 30° (sample in green and blue in the bottom two panels on the right of Fig. 1a). Examination under different sample tilts and X-ray polarizations is required to have sufficient information for the construction of an orientation tomogram representative of the apical bond orientation in 3D26,47. To change the illumination source native horizontal polarization to vertical, we used a 250-µm-thick diamond crystal phase plate inserted into the illumination path upstream of the zone plate (see Fig. 1a). The phase plate absorbed approximately 65% of the incident photons70. The degree of polarization of the X-rays was determined to be approximately 60% using a polarization analyser set-up. The sample tilt was changed using a sample holder insert26. To minimize the acquisition time, we used an adaptive field of view for each group of ptychographic projections. The maximum field of view, horizontal × vertical, was about 24 × 25 μm2. The scanning followed a Fermat’s spiral pattern71. An average step size of 0.8 µm was used for all tomograms. The exposure time per scanning point was 0.1 s. 280 projections were acquired per tomogram.

    Finally, using the same acquisition parameters, we acquired an off-resonance ptychographic tomogram of the pillar below the absorption edge at 5.4 keV. This tomogram, being insensitive to any dichroic effects, was used for computing the electron-density tomogram and subsequently used for compositional analysis11. It should be noted that the starting angle and angular spacing of projections was kept constant across all tomograms.

    Ptychographic image reconstruction

    Ptychographic images (or tomographic projections) were reconstructed using the PtychoShelves package72. For each reconstruction, a region of 600 × 600 pixels of the detector was used per scanning point, resulting in an image pixel size of 30.91 nm for the pre-edge and 31.29 nm for the below-edge tomogram. Reconstructions were obtained with 200 iterations of the difference map algorithm73, followed by 300 iterations of maximum likelihood refinement74.

    Preprocessing of projections

    Before any tomogram reconstructions, we: (1) resampled all projections to a pixel size of 30.91 nm using Fourier interpolation; (2) extracted the phase from the reconstructed projections, removed constant and linear phase components and spatially aligned the projections using a tomographic consistency approach31; and (3) aligned all projections to a common pillar orientation. As a last step, the different orientations at which projections were measured were characterized by a 3D rotation matrix26, which was input into a specially developed reconstruction code (see the ‘XL-DOT reconstruction’ section below). It should be noted that, owing to the sample tilt and the fixed vertical field of view of the 2D projections, the 3D volume that is commonly sampled in all orientations, and used in the subsequent analysis and visualization, is reduced. (4) Last, to isolate the dichroic component from the isotropic electron-density contribution, the LV projection was subtracted from the LH projection. The resulting set of projections were used in the reconstruction of the XL-DOT dataset, as discussed further below.

    Ptychographic tomogram reconstruction

    The ptychographic tomogram, acquired with the X-ray energy tuned to below the absorption edge, was reconstructed using a modified filtered back-projection algorithm75. This off-resonance phase tomogram was used to derive the electron-density tomogram, which was then used for material component identification11,32.

    XL-DOT reconstruction

    A gradient-based iterative reconstruction algorithm was developed to reconstruct the orientation field in 3D. A schematic of the reconstruction process is shown in Supplementary Fig. 7. The process starts with the creation of a 3D starting, random guess of the sample. Using the sample–illumination interaction relationship in equation (2), a set of projections is simulated. These projections are then compared with the measured set of projections and their difference is used to compute a gradient to iteratively correct the initial guess.

    The interaction between the electric field of the incident linearly polarized X-rays, \(\overrightarrow{E}\), and the orientation of the apical vanadyl bond, \(\overrightarrow{a}\), can be described as:

    $$f={f}_{0}+{f}_{{\rm{lin}}}{(\overrightarrow{E}\cdot \overrightarrow{a})}^{2}$$

    (2)

    Here f is the total scattering factor, which contains the isotropic charge contribution, f0, and the linear dichroism contribution, \({(\overrightarrow{E}\cdot \overrightarrow{a})}^{2}\), with a pre-factor flin that depends on the electronic transition under resonance. Keeping with the experimental geometry (Fig. 1a); using X-rays with a LH polarization parallel to the x axis and denoting an arbitrary polarization angle as φ, in which φ = 0° is LH polarization and φ = 90° is LV polarization, the tomographic rotation and tilting of the sample can be quantitatively represented by the 3D rotation matrix R. In transmission, the measured projection can then be described by the integral given in equation (3). Index summation notation is used to give the rotation of the relevant components of the orientation, aj. The integration is evaluated along the X-ray propagation direction, the z axis.

    $$P(x,y)=\int {f}_{0}({\bf{R}}\overrightarrow{r})+{f}_{{\rm{lin}}}[{R}_{1j}{a}_{j}({\bf{R}}\overrightarrow{r})\cos \varphi +{R}_{2j}{a}_{j}({\bf{R}}\overrightarrow{r})\sin \varphi {]}^{2}{\rm{d}}z$$

    (3)

    Knowing the form of the interaction, the reconstruction algorithm was formulated by generating a guess structure, from which projections were simulated at the same orientations that the sample was measured. These simulated projections, \(\hat{P}\), were then compared with the corresponding measured projections, P. Their square difference was used to define an error metric, ϵ, quantifying how well the guess could reproduce the measured projections, given by

    $${\epsilon }=\sum _{m,x,y}{[{\widehat{P}}^{m}(x,y)-{P}^{m}(x,y)]}^{2}$$

    (4)

    in which m represents the projection index. The error metric was reduced using gradient descent, therefore improving the ability of the guess structure to represent the internal c-axis orientation of the measured sample. By differentiating the error metric in equation (4) with respect to each component, we obtain the following analytical expression for calculating the gradient:

    $$\frac{{\rm{\partial }}{\epsilon }}{{\rm{\partial }}{a}_{k}}={4f}_{{\rm{l}}{\rm{i}}{\rm{n}}}\sum _{x,y}[{\hat{P}}^{m}(x,y)-{P}^{m}(x,y)][{R}_{1j}{a}_{j}\cos {\varphi }+{R}_{2j}{a}_{j}\sin {\varphi }]({R}_{1k}\cos {\varphi }+{R}_{2k}\sin {\varphi })$$

    (5)

    The gradient was evaluated and applied to the guess structure at every iteration. During the reconstruction process, the magnitude of the linear dichroic contrast, corresponding to flin, was not constrained and was therefore also optimized during gradient descent. As a result, it is not necessary to predetermine the flin value. As the iterative gradient descent reconstruction is prone to converging at local minima, 40 individual reconstructions were performed using different random, non-zero initial conditions. The individual reconstructions are combined by averaging all components to obtain a final reconstruction. The difference in the angular orientations between the individual reconstructions and the final, averaged reconstruction was used to evaluate the standard deviation of the orientation, which is an estimate of the uncertainty in orientation.

    Notably, using equation (3), it can be shown that LV polarization (φ = 90°) projection measurements evaluate to

    $$P(x,y)=\int ({f}_{0}({\bf{R}}\overrightarrow{r})+{f}_{{\rm{lin}}}[{{\bf{a}}}_{{\boldsymbol{y}}}({\bf{R}}\overrightarrow{r}){]}^{2}){\rm{d}}z$$

    (6)

    Because there are no vector rotations in this expression, it is equivalent to examining a scalar consisting of two components: the isotropic charge background, f0, and the (out-of-plane) \({a}_{y}^{2}\) component. This can be reconstructed with conventional tomography and gives contrast between grains that are in-plane (x–y plane) and out-of-plane oriented. This contrast was used for further validation of the final reconstruction, as shown in Supplementary Fig. 12.

    Multiaxis tomography

    To obtain a first estimation of how many sample tilts and linear polarization states are necessary for a robust XL-DOT reconstruction, we performed a series of numerical simulations and tomographic reconstructions with fewer sample tilt axes (Supplementary Fig. 14). Preliminary reconstructions can be obtained with as little as two tilt axes using LV and LH polarizations only. Both our simulations (not shown) and recent literature30,47 indicate further that the numerous tilt axes can be replaced by measurements with extra X-ray polarizations76. Similar results can also be achieved using laminography46,48. This offers a route to fewer or even single tilt-axis measurements.

    Dose estimation

    The total deposited dose over the duration of the experiment and the entire volume of the V2O5 pillar was approximately 109 Gy. This estimate is based on the mass density of the sample and the average flux density per projection77. No actions were taken to limit the dose, as V2O5 is not known to degrade under the present experimental conditions11,29. For radiation-sensitive materials, preventative measures can be used to mitigate or account for potential radiation damage78. Dose-limiting options include scanning and projection sparse acquisition schemes11,79 that reduce the total deposited dose, changes to the ptychography acquisition such as using an out-of-focus acquisition with micrometre-sized scanning probes which lead to a reduction of both the total and peak dose per area, as well as the implementation of cryogenic and inert atmosphere measurement conditions80,81.

    Spatial resolution

    Spatial resolution estimates of projections and tomograms were obtained using Fourier ring correlation and Fourier shell correlation, respectively82.

    To evaluate the spatial resolution of the acquired projections, we acquired projections under identical conditions, that is, at the same rotation angle, calculated the correlation between these two images in the Fourier domain and estimated the spatial resolution based on the intersection with a one-bit threshold (see Supplementary Fig. 6). This gives spatial resolutions close to the pixel limit of 30.91 nm and 31.29 nm for the on-resonance (5.469 keV) and off-resonance (5.4 keV) measured projections, respectively.

    To evaluate the spatial resolution of the electron-density tomogram acquired below the absorption edge, we halved the entire dataset and reconstructed two independent tomograms (Supplementary Fig. 10). This gives a 3D spatial resolution of 44 nm.

    To evaluate the spatial resolution of the orientation vector field, the corresponding dataset was similarly split in half and two tomograms of the orientation vector field were calculated. Using Fourier shell correlation, we calculated spatial resolution estimates for each of the orientation scalar components (LDx, LDy, LDz), as shown in Supplementary Fig. 8, providing a lower bound for their spatial resolutions of 84 nm, 45 nm and 89 nm, respectively. Also, we measured edge profiles across sharp features such as 90° grain boundaries, which revealed a maximum edge sharpness of 40 nm, with an average edge sharpness of 73 nm, which we take as the spatial resolution of the orientation tomogram.

    Measurement error estimation

    To estimate the voxel-level electron-density uncertainty, we calculated the standard deviation (σ) of the electron density in a region of air surrounding the imaged pillar. The average electron density in air and uncertainty was calculated as 0.004 ± 0.007 Å−3.

    To estimate the uncertainty in the detected linear dichroism, that is, spatial variations in the pre-edge peak intensity, we independently reconstructed the LV and LH phase tomograms with the sample at a fixed sample tilt and then subtracted them from each other. We then isolated a region of air and calculated the standard deviation in the phase shift associated with the voxels in this region. This standard deviation of the phase associated with the air region corresponds to the uncertainty of the dichroic signal. On the basis of this procedure, the uncertainty of the dichroic signal is found to be 1.3 × 10−4 rad, which corresponds to a refractive index decrement, δ, error of 1.9 × 10−7.

    To estimate the error in the determined orientation, we isolated an elongated grain with a volume of 0.85 µm3 and long-edge length of 3.2 µm that showed the least variance in electron density and V2O5 orientation, that is, which is assumed to be single crystal, and calculated the standard deviation (σ) in orientation to be ±10° for azimuth (x–y-plane angles) and ±8° for elevation (out-of-plane angles) (Supplementary Fig. 11).

    The critical concentration for element detection can be estimated to correspond to a dichroic magnitude (difference between tomograms taken with different polarizations) of at least twice the reconstruction error. The dichroic contrast of the V2O5 is 1.8 × 10−3 and the noise in the reconstruction is an order of magnitude weaker at 1.3 × 10−4. As a result, in V2O5, our dichroic contrast is 12 times the error. We can estimate that, if all other parameters are held constant, the concentration of V can be decreased by a factor of 6 and still be measurable.

    Present XL-DOT acquisition time and future prospects

    The total acquisition time for the XL-DOT dataset used in this work was around 85 h, including sample tilting, changing the polarization and alignment and dead-time overheads. The pure measurement time, however, was only about 24 h. This discrepancy is largely because of the lack of automation. There exist several opportunities to reduce the acquisition time as follows:

    1. 1.

      Reduce oversampling: reconstructions using 50% of the tomograms provide similar results (Supplementary Fig. 14).

    2. 2.

      Automation and imaging geometry: the measurement of intermediate linear X-ray polarization angles30,47,70,76 and/or use of the laminography geometry46,48 will eliminate most of the present acquisition overheads.

    3. 3.

      The increase in coherent flux expected from fourth-generation synchrotron light sources promises to reduce scan times for radiation-hard materials83.

    4. 4.

      Further innovations such as multibeam ptychography and sparse tomography offer routes to even faster data acquisition11,84, providing acquisition times compatible with operando measurements48,85.

    Data analysis

    Analysis of the dichroic tomogram was performed using in-house-developed MATLAB routines, ParaView and Avizo. To account for the damage caused during the FIB milling step of the sample preparation, we defined a mask that excluded the outermost 90 nm of the sample cylinder from orientation and electron-density volume analysis (Supplementary Fig. 9).

    Component identification and isolation

    Materials were identified by comparing the tabulated electron densities of the known sample and reference components, listed in Supplementary Table 1, with the PXCT-measured electron densities. Shown in Supplementary Fig. 9 is a volume rendering and a horizontal cut slice through the electron-density tomogram with the corresponding electron-density histogram. The V2O5 volume was isolated using threshold segmentation with a lower bound of 0.74 Å−3 and an upper bound of 0.90 Å−3.

    Analysis of topological defects

    The topological charge can be determined by considering the winding number associated with a given topological defect. The winding number corresponds to how the crystallographic orientation changes when moving around a circle enclosing the defect in a clockwise manner. For the comet (trefoil) defect, the c axis rotates clockwise (anticlockwise) by +180° (−180°) for one complete revolution. As the crystallographic orientation has completed half a revolution of a full circle (360°), the topological numbers ±1/2 are assigned to them.

    Microstructural analysis of V2O5 domains

    To isolate the V2O5 grains and facilitate a correlation between orientation and electron density, we applied the above-defined threshold mask (electron densities between 0.74 Å−3 and 0.90 Å−3) to the orientation tomogram. To identify and characterize individual V2O5 grains, we downsampled the masked XL-DOT reconstruction by a factor of three (transforming a group of 3 × 3 voxels into 1 voxel with an average intensity value of the same size), thus reducing the sensitivity to intragranular variations. Segmentation was then performed by separating regions along high-angle grain boundaries, showing a c-axis orientation difference larger than 10°. Following segmentation, we then calculated the volume of these grains, their mean diameter and their sphericity86. Shown in Supplementary Fig. 13 are the corresponding distributions and correlations of the segmented grains.

    Sample diameter and photon energy resolution considerations

    As linear dichroic phenomena occur near absorption edges or resonant X-ray energies, the X-ray penetration depth at these energies determines the sample diameter that can be investigated with XL-DOT. For most materials, it is the penetration depth at the X-ray energy of the examined chemical element that sets an upper limit on the sample diameter. Taking pure transition metals as an example, this imposes a typical upper limit to the sample size of around 10 µm. Transition-metal-rich functional materials such as catalyst bodies, cathode materials, ferroelectrics, biominerals and concrete, which are also of interest for XL-DOT measurements, exhibit a substantially larger upper sample size limit owing to their internal porosity or composite nature. For instance, a 100 µm-thick V2O5 sample transmits around 10% of the incident beam in the pre-edge region (https://henke.lbl.gov/). 3D or nanotomography measurements of such sample diameters are increasingly typical for operando measurements48,87,88,89.

    Although XL-DOT measurements should ideally be performed at the X-ray energy of an absorption edge at which linear dichroic contrast is strongest to maximize contrast in the projections, the range of energies at the absorption edge at which dichroism can be measured can be large. For instance, the full width at half maximum of the near-edge peak in our V2O5 spectra used for XL-DOT is approximately 3 eV, which means that even an X-ray energy resolution of up to 3 eV would be sufficient for XL-DOT measurements, albeit at a decreased SNR. There is therefore a degree of flexibility in terms of the required energy position and resolution for XL-DOT measurements.

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  • Work hardening in colloidal crystals

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    The properties of atomic crystals under high strain are well established18; when subjected to stresses beyond the yield stress, they exhibit plastic flow, which causes an irreversible change in their shape. This is mediated by nucleation and motion of topological line defects called dislocations18,23. Increasing plastic deformation requires an increasing flow stress because of the interactions between dislocations. This is work, or strain, hardening. This phenomenon is ubiquitous, yet, owing to the many ways dislocations can interact, our understanding of the mechanism that governs work hardening is still incomplete20,22,24,25,26. In contrast to atomic crystals, colloidal crystals have much simpler interparticle interactions; they consist of solid particles in a fluid and can exhibit purely hard-sphere interactions. Here we show that, despite the simplicity of these interactions, hard-sphere colloidal crystals exhibit work hardening. The micron scale of the particles enables the structure and dynamics of colloidal crystals to be investigated on a particle-by-particle level using optical microscopy5,8,9,12,13,14,27,28,29. Dislocations in these colloidal crystals can, therefore, be directly visualized in three dimensions and in real time. Thus, these measurements provide insight into the general nature of work hardening.

    We disperse 1.55 μm diameter silica spheres in a mixture of water and dimethyl sulfoxide, which closely matches the refractive index of silica. We dissolve a small amount of fluorescein sodium salt to dye the solvent and additional sodium chloride to further screen the surface charge of the particle to produce a nearly hard-sphere interaction between the particles. The colloidal dispersion is put into a cylindrical shear cell, 1 cm in diameter. The bottom of the cell is a template: a coverslip with a square array of dimples of 1.63 μm spacing (Fig. 1a). The template constrains the first layer of the sedimenting particles and imposes the growth of a face-centred cubic (fcc) single crystal along the [001] fcc direction30 (Fig. 1b). We use a spinning-disk confocal microscope to image, at single-particle resolution, five separate regions of 200 × 200 × 60 μm3 volume, containing a total of about 5 million particles (Extended Data Fig. 1 and Supplementary Videos 1–3). By processing the confocal images, we locate the position of the particles in three dimensions, which allows us to determine the local crystalline structure31 and reconstruct the dislocation lines32 with their respective Burgers vectors, b.

    Fig. 1: Plastic shear deformation of colloidal single crystals.
    figure 1

    a, Schematic of the experiment. Colloidal crystals are grown by sedimentation of 1.55 μm particles on templates with a square pattern of dimples. The templates dictate the growth of fcc single crystals along the [001] fcc direction. The crystals are sheared by displacing a grid, embedded in the particles, in the [010] fcc direction. b, The four close-packed {111} planes are marked in the fcc unit cell. c, During the sedimentation process, the hcp stacking faults (orange) in the fcc crystal (green) are formed on the four {111} planes. d, The stacking faults arise because of the motion of Shockley partial dislocations (yellow line in d) that relax the approximately 1% misfit strain due to the mismatch between the template spacing and the crystal lattice constant. e, A snapshot of the von Mises equivalent strain, calculated for γ ≈ 0.04 with respect to a reference frame defined at γ = 0. Plastic flow is mediated by slip on the {111} fcc planes, the classical easy-glide planes, as shown by the high von Mises strain values (Supplementary Video 5). c and e correspond to the same region of the crystal. Scale bar, 20 μm (c,e).

    The as-grown fcc crystals contain multiple stacking faults, characterized by local hexagonal close-packed (hcp) stacking, that lie on the {111} planes, as shown by the orange particles in Fig. 1c. Stacking faults are bounded by Shockley partial dislocations (Fig. 1d) and are formed during the growth of the crystal as a result of the approximately 1% misfit strain between the crystal and the template11 (Supplementary Video 4). After the particles have fully sedimented, we shear the crystal by displacing a square-meshed grid, embedded 50 μm above the template, as shown schematically in Fig. 1a. The grid moves in the [010] fcc direction at a constant speed of 2.5 μm h−1, resulting in a strain rate of 1.4 × 10−5 s−1 (Methods and Extended Data Fig. 2). During the deformation process, three-dimensional (3D) confocal scans are made every 4 min, fast enough to determine the time-dependent trajectories of the individual particles.

    To quantify the deformation of the sheared crystal, we determine the elastic and total strain fields. The elastic strains are obtained for each snapshot of the particle positions by comparing the local particle configuration with the perfect fcc structure12,33,34. By contrast, the total strains are defined with respect to the position of the particles before the deformation33, which requires tracking the particle positions in time (Methods). The average elastic (γE) and total (γ) shear strains are obtained by averaging over all the particles. Here, γ is equivalent to the relative displacement of the particles at the top and bottom surfaces divided by its height (Extended Data Fig. 3).

    We find that the deformation of the crystal is elastic up to γ ≈ 0.005; the total strain is accommodated by elastic strain, γE = γ, as shown in Fig. 2a by the dashed line with a slope of 1. Beyond this yield point, any further increase in the imposed γ causes an increase in the plastic strain, γP = γ − γE. To determine how γP is accommodated by the crystal, we calculate the spatially resolved von Mises equivalent strain, a scalar invariant that quantifies the maximal shear distortion (Methods). Plastic strain is mediated by slip along the oblique {111} planes, as shown by the high von Mises strain values in Fig. 1e. These measurements show that plastic strain is mediated by the classic fcc slip on the close-packed planes.

    Fig. 2: Strain hardening and localization.
    figure 2

    a, The normalized shear stress σ/μ, where μ is the shear modulus, is obtained from the measurements of the elastic strain, γE, and plotted as a function of the total strain, γ. The dashed line with a slope of unity marks elastic deformation, γE = γ. The plastic strain, γP = γ − γE, is denoted by the black double arrow. Inset, strain hardening is demonstrated by replotting σ/μ as a function of γP (black line); increasing stresses are required to sustain plastic flow. Successive unloading and reloading (red line) demonstrate the irreversibility of the plastic flow; the yield stress, the stress at which the slope dσ/dγP suddenly decreases, increases with accumulated γP (A–D). b, Profiles of the average particle displacements (top) in the direction of the shear, y, binned along the crystal height, z, for different values of γ (coloured dots in a). The corresponding profiles of γ (bottom) demonstrate that, at the later stages of the deformation, the strain is localized within a narrow region. c, Strains, averaged over the localized region (red line), the bulk region (yellow line) and over the total thickness (blue line), as defined in b (bottom). During the strain-hardening regime (red background in a,c), the crystals are deformed homogeneously. Saturation of the stress marks the onset of a crossover between the homogenous and localized flow regimes (blue background in a,c). The deformation of the bulk stops entirely at γ ≈ 0.2, and the strain is accommodated in the localized region alone (yellow background in a,c).

    The average elastic strains provide a measure of the normalized stresses σ/μ = γE, where μ is the shear modulus. Remarkably, our experiments show strain hardening in these hard-sphere colloidal crystals: increasing stresses are required to accommodate plastic flow, as seen by plotting σ/μ as a function of γP, shown by the black line in Fig. 2a (inset). To demonstrate the irreversibility of the plastic flow, we reverse the direction of the shear to remove part of the applied shear stress, and then reload. We find that higher stresses are required to reinitiate plastic deformation: the yield stress, the point at which the slope dσ/dγP suddenly decreases, increases with the accumulated γP, as denoted by the capital letters (A–D) in Fig. 2a inset. Although the transition to yield is clear, the unloading and reloading stages are not purely elastic, which would require the loading and unloading curves to be strictly vertical (Fig. 2a, dashed line in the inset).

    The strain-hardening stage ends when the stress reaches σ/μ ≈ 0.02 (Extended Data Fig. 4). Any further flow is maintained at a constant stress level (Fig. 2a). The saturation of stress marks an essential change in the behaviour of the flow. To demonstrate this, we average the particle displacements in the shear direction over the crystalline layers parallel to the substrate. Before the saturation of the stress, the entire crystal undergoes simple shear: the displacement profile is linear, and the strain profile is constant across the thickness of the crystal, as demonstrated by the orange profiles in Fig. 2b. However, during the later stages of the deformation, when the stress saturates, strong displacement gradients near the bottom surface are formed, as shown by the yellow and purple profiles in Fig. 2b (top). The deformation of the crystal is no longer homogeneous, and the strain becomes localized within a 10-μm thick boundary layer at the bottom surface (Fig. 2b, bottom).

    To quantify the crossover between the homogenous and localized flow regimes, we average the strain within two separate regions of the crystal: the region in which localization takes place, 0 < z < 10 μm, and the bulk of the crystal, in which the deformation is homogenous, 10 μm < z < 40 μm. We find that during the strain-hardening stage (red background in Fig. 2a,c), the two strain averages are identical, highlighting again that the deformation is homogeneous over the entire thickness of the crystal. The saturation of the stress, which marks the end of strain hardening, is accompanied by the onset of localization; strain in the localized region exceeds the bulk strain (blue background in Fig. 2a,c). Finally, when the bulk strain saturates, the applied strain is accommodated by the localized region alone (yellow background in Fig. 2a,c).

    To explore the origin of strain hardening, we reconstruct the evolution of the full 3D dislocation network during the deformation process, as shown for γP = 0 (Fig. 3a, left) and γP = 0.035 (Fig. 3a, right). We find that slip is mediated predominantly by two slip systems (Methods and Extended Data Fig. 5): glide of \(\frac{1}{6}[2\bar{1}1]\) and \(\frac{1}{6}[21\bar{1}]\) Shockley dislocations on, respectively, \((\bar{1}\bar{1}1)\) and \((1\bar{1}1)\) planes (Fig. 3a). The two slip systems are symmetric with respect to the applied shear: we calculate the average the density of all the dislocations that pierce through the two slip planes, which defines the forest dislocation density ρf (Methods). We find that late-stage strain hardening in colloidal crystals is described by the Taylor equation, originally derived to explain strain hardening in atomic crystals19: to sustain dislocation glide, the shear stress resolved on a slip system σres (Methods) increases with ρf according to \({\sigma }_{{\rm{r}}{\rm{e}}{\rm{s}}}=\alpha \mu b\sqrt{{\rho }_{{\rm{f}}}}\), where α is a dimensionless constant and b is the magnitude of the Burgers vector22. Moreover, measurements of σres/μ and ρfb2 in our colloidal crystals (blue symbols) are in accordance with experimental and numerical observations in metallic systems20,21,26,35 (grey symbols) and the model predictions (black solid lines for α = 0.2 and 0.5) (ref. 35), as shown in Fig. 3b. This result is particularly interesting given the four orders of magnitude difference in particle size (0.1 nm to 1 μm) and the nine orders of magnitude difference in the shear modulus (GPa to Pa) between metallic and colloidal crystals. Remarkably, the very dense dislocation network of our colloidal crystals leads to a very high σres/μ that exceeds that of most metals and ultimately approaches the theoretical limit of strength (Methods). A more detailed investigation of the data shows that the Taylor equation fails to account for the early stages of the strain-hardening regime. The initial dislocation density is the result of a slight tensile mismatch between the crystal and the template; thus, the Taylor equation overestimates the flow stress. Taylor hardening is established only after a transient evolution of the dislocation network (Fig. 3c).

    Fig. 3: Strain hardening in colloidal crystals.
    figure 3

    a, Snapshots of the dislocation network before (left) and during (right) deformation. The particles displaced by the dominant slip system, \({\bf{b}}=\frac{1}{6}[2\bar{1}1]\) on the \((\bar{1}\bar{1}1)\) plane, are shown in cyan. Another slip system, symmetric with respect to the shear, operates on the \((1\bar{1}1)\) plane (dashed). b, Comparisons of the normalized shear stress resolved on the dominant slip systems, σres/μ (where μ is the shear modulus), as a function of the normalized dislocation density, ρfb2 (where b is the magnitude of the Burgers vector) for a nearly hard-sphere colloidal crystal and various metallic crystals (Methods). The two solid lines are the predictions of the Taylor equation for two values of α (see main text). c, Data in b are on a linear scale. The double arrow denotes the discrepancy between the Taylor model and the colloidal crystals at the early stages of the deformation. d, A 5-μm thick slice of the (\(\bar{1}\bar{1}1\)) plane (solid line in a) is shown for different values of γP. At the early stages (γP = 0.01) crystals flow by nucleation of (half) loops of the dominant dislocations (blue segments. Examples are marked by A and A′). At the later stages (γP = 0.02), dislocations can form immobile junctions (red segments) by interaction with the forest dislocations (green segments, see definition in the text), as shown by examples B, C and D. Thereafter, plastic flow is accommodated by unzipping of the junctions (γP = 0.035), although new dislocations can still be formed (E). Some junctions are very strong and limit slip (F). For the complete time series, see Supplementary Video 6. e, Schematic showing how dislocations can intersect forest dislocations and form immobile dislocation junctions. Scale bars, 50 μm (a); 20 μm (d).

    To gain insight into the mechanism of strain hardening, we examine the evolution of slip on a particular \(\left(\bar{1}\bar{1}1\right)\) plane, marked by the solid line in Fig. 3a. At early stages of the plastic flow (γP = 0.01), slip is mediated by nucleation of mobile dislocations (blue), as marked by A and A′ in Fig. 3d. We find that nucleation takes place at different parts of the crystal, either by formation of half loops at the vicinity of the upper surface of the crystal (examples A and A′) or by homogeneous nucleation of full dislocation loops within the bulk of the crystal (Extended Data Fig. 6). At the later stages of the flow, dislocations expand (γP = 0.02), and the slipped regions coalesce to cover the entire plane (γP = 0.035), as shown in Fig. 3d.

    These observations reveal the underlying mechanism of Taylor hardening. At later stages of strain hardening (γP = 0.02), as the primary dislocations expand, they intersect with Shockley dislocations (Fig. 3e, forest, green) that lie on other glide planes and pierce through the primary plane22, as demonstrated by the schematics in Fig. 3e. We provide here direct experimental evidence that interactions between the primary and forest Shockley dislocations result in the formation of sessile junctions. Most of these junctions are of the Lomer–Cottrell type18 with \({\bf{b}}=\frac{1}{6}[10\bar{1}]\) (Extended Data Table 2), as shown by the red segments in examples B and D in Fig. 3d. However, we also identify Hirth-type junctions with \({\bf{b}}=\frac{1}{3}[0\bar{1}0]\), as occurs, for example, at C. We find that these dislocation junctions form strong obstacles to the motion of the mobile dislocations, as suggested by the high curvature of the primary dislocations pinned by the junctions36 as in example B. Further plastic strain is accommodated by unzipping the immobile junctions (B, C and D). Our measurements show, however, a hierarchy of junction strengths, as some of the junctions remain intact (F), despite the increase in stress. These observations explicitly demonstrate the role of immobile junctions in strain hardening of crystals; to sustain plastic flow, the stress must be sufficiently high to overcome the pinning of dislocations.

    Our measurements also show the origin of the discrepancy between the Taylor predictions and the measured flow stress, as observed at the early stages of the strain-hardening regime (Fig. 3c, double arrow). At low strains (γP = 0.01), the dislocations are short and do not interact with the piercing forest dislocations, so that no dislocation junctions have formed yet. Instead, our measurements suggest that the nucleation of dislocations is the dominant mechanism for strain hardening in this case. We find a hierarchy of barriers to dislocation nucleation; although some regions are susceptible to early nucleation at low stress (A and A′), nucleation can also take place at later stages of the flow and at higher stress (E).

    Strain hardening in our crystals is interrupted by the onset of localization. We find that the specific geometry of the Shockley partial dislocations limits the strain that can accumulate in the bulk. Unlike perfect dislocations, partial dislocations consist of two types23: on applied shear, leading partial dislocations leave behind hcp stacking faults, whereas trailing partial dislocations eliminate the pre-existing stacking faults, as shown in Fig. 4a. We find that the dominant slip systems are of the trailing type. By nucleation and expansion, the \(\frac{1}{6}[2\bar{1}1]\) dislocations on the \((\bar{1}\bar{1}1)\) plane negate the pre-existing hcp stacking faults that were left behind by the misfit dislocations during the epitaxial crystal growth37 (Fig. 4b). As the plastic flow proceeds, the stacking faults are exhausted, as demonstrated by the number of hcp particles on stacking faults on \((\bar{1}\bar{1}1)\) planes; there is a marked decrease between γP = 0 and 0.18 as shown in the two panels of Fig. 4c. We systematically quantify this process by tracking the fraction of the hcp particles on \((\bar{1}\bar{1}1)\) planes, Nhcp/N. These measurements show that the exhaustion of the stacking faults marks the end of the crossover; the deformation of the bulk entirely stops (Fig. 2b, bottom) when the dominant slip systems do not operate because Nhcp/N saturates at a low value (Fig. 4d, yellow region).

    Fig. 4: Exhaustion of the stacking faults terminates bulk deformation.
    figure 4

    a, Glide of leading partial dislocations creates hcp stacking faults, whereas glide of trailing dislocations eliminates the stacking faults. b, A slice of a \((\bar{1}\bar{1}1)\) plane, corresponding to example A in Fig. 3d. The dominant dislocations (blue) are of the trailing type; \(\frac{1}{6}[21\bar{1}]\) dislocations eliminate hcp stacking faults (orange particles) that were left behind by the motion of the Shockley misfit dislocations formed during the epitaxial crystal growth. c, Stacking faults that lie on the dominant \((\bar{1}\bar{1}1)\) planes. Stacking faults that were formed before the deformation process (top) are exhausted by the time plastic flow reaches γP = 0.18 (bottom). The crystal was sliced in the range 10 < z < 40 μm. d, Evolution of the number of hcp particles on \((\bar{1}\bar{1}1)\) plane, Nhcp, shown in c, normalized by the total number of particles, N. The decrease of Nhcp/N to small values marks the end of the crossover to localization (blue background). Thereafter, bulk deformation stops entirely (yellow background). Scale bars, 20 μm (b); 50 μm (c).

    Finally, our measurements reveal the mechanism that drives the localization of flow. In contrast to the strain-hardening regime, during which slip occurs along the classic easy-glide {111} planes, flow localization is mediated by slip on an unconventional (001) slip plane, as demonstrated by the red arrow in Fig. 5a. Slip takes place through glide in the \([110]\) and \([1\bar{1}0]\) directions of Lomer dislocations38,39,40, perfect dislocations of edge character in which \({\bf{b}}=\frac{1}{2}[110]\) and \(\frac{1}{2}[1\bar{1}0]\) (Fig. 5b). The presence of perfect dislocations in near-hard-sphere colloidal crystal is surprising. Perfect dislocations are expected to dissociate into partial dislocations because of the vanishingly small stacking fault energy. Here, as a result of the applied shear stress, mobile partial dislocations merge with the misfit dislocations to form perfect Lomer dislocations near the bottom interface. Lomer dislocations are often considered to be relatively immobile because of the high frictional forces on the (001) plane23. Activation of Lomer dislocations in our experiments is a direct consequence of the severe work hardening of the {111} slip systems, which allows the activation of the less favourable (001) slip system. At this stage, the resolved flow stress is about 0.01μ as shown in Fig. 3b; this is close to the theoretical limit of strength and is reached in very few other materials.

    Fig. 5: Glide of Lomer dislocations on an unconventional (001) slip plane mediates strain localization.
    figure 5

    a, A snapshot of the von Mises equivalent strain, calculated for γ ≈ 0.06, after the onset of the localization of flow. High values show that slip on an unconventional (001) plane has been activated (red arrow), apart from easier slip along the classical {111} planes. b, Cross-sections of the colloidal crystal along the \((\bar{1}10)\) plane (see coordinate system in a) demonstrate glide of a perfect dislocation \({\bf{b}}=\frac{1}{2}\left[110\right]\) in the [110] direction on a (001) plane. The order of the crystal layers is shown by the red and blue particles, located out of and into the view plane, respectively. Black lines show an extra half-plane. Scale bar, 20 μm (a).

    We show here that colloidal crystals exhibit work hardening and that their normalized strength approaches the theoretical limit for materials and exceeds that of most atomic systems. The strength of atomic systems is set by the dislocation density; the maximum density is thought to be limited by the annihilation of nearby dislocations41,42. The probability of this annihilation process strongly decreases with decreasing stacking fault energy43,44. Hard-sphere colloidal crystals have vanishingly small stacking fault energy because of the lack of next-nearest-neighbour interactions. Annihilation of dislocations is not observed in our experiments, which may account for the very high dislocation density and, in turn, very high strength of these crystals. Moreover, we find that, after an initial transient, the relationship between the dislocation density and the strength of the colloidal crystals is in direct agreement with Taylor’s prediction for atomic systems, although hard-sphere interactions lack the complexity of atomic interactions. It is known that the specific details of the interatomic potential can affect, for example, the lattice resistance to the dislocation motion, which, in turn, explains why some atomic crystals break (brittle), whereas others deform plastically (ductile)18,23. However, our work demonstrates that colloidal crystals follow the same universal behaviour of many ductile crystals: the exact interparticle interactions and the lattice resistance are of secondary importance to the interactions among dislocations. These measurements provide a new means to study crystal plasticity, as colloidal systems allow unprecedented real-time observation of dislocation dynamics that is inaccessible in atomic crystals. We provide direct experimental evidence that work hardening can be caused by the formation and destruction of sessile dislocation junctions. Although this is widely accepted, it has so far been supported only by numerical simulations35,45,46 and indirect experimental evidence20,47. Furthermore, the discrepancy between the Taylor equation and the measured flow stress during the transition from the early misfit dislocation configurations to those established by the shear highlights the importance of memory48 and suggests that the deformation history is encoded in the structure of the dislocation network. These insights are essential for understanding the classic latent hardening experiments, during which different slip systems are activated successively47. Our experiments highlight the competition of the different deformation processes—glide of mobile Shockley and glide of Lomer dislocations—at the limit when the flow stresses approach the theoretical limit of strength. Finally, the exhaustion of stacking faults through the motion of trailing partial dislocations is not unique to colloids but should also be important, for example, in the deformation of nanocrystalline materials49, in which the stacking fault and sample sizes are similar. Observation of work hardening in colloidal crystals not only provides insights into this soft matter system but also provides an opportunity to gain important insights into the underlying mechanisms of work hardening itself.

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