Nanometer-precision non-local deformation reconstruction using nanodiamond sensing

Spatially resolved information about material deformation upon loading is critical to evaluating mechanical properties of materials, and to understanding mechano-response of live systems. Existing techniques may access local properties of materials at nanoscale, but not at locations away from the force-loading positions. Moreover, interpretation of the local measurement relies on correct modeling, the validation of which is not straightforward. Here we demonstrate an approach to evaluating non-local material deformation based on the integration of nanodiamond orientation sensing and atomic force microscopy nanoindentation. This approach features a 5 nm precision in the loading direction and a sub-hundred nanometer lateral resolution, high enough to disclose the surface/interface effects in the material deformation. The non-local deformation profile can validate the models needed for mechanical property determination. The non-local nanometer-precision sensing of deformation facilitates studying mechanical response of complex material systems ranging from impact transfer in nanocomposites to mechano-response of live systems.

A peak force tapping mode was applied to obtain the topography information of the samples.
Before the indentation experiments, correlating the confocal system and the AFM system was realized in two steps. We first manually overlapped the excitation laser spot with the AFM tip under the top view camera on top of the AFM. Then, an AFM topography image and a fluorescence image were captured and an overlapping between the two was established by comparing the patterns of the NDs distribution in the two images.
In the gelatin indentation experiments, the size (height) of gelatin particles (~10 μm) was much larger than the size of NDs (~ hundreds of nanometers), which made it difficult to resolve the NDs on the gelatin in the AFM image. The correlation between the AFM and the fluorescence images was established by overlapping the fluorescence image and AFM image of NDs on the substrate (cover slide). For the NDs on the gelatin particles, their coordinates were determined from the fluorescence images instead of the AFM ones. Consequently, the lateral spatial resolution of the gelatin deformation measurements was limited by the optical resolution (~300 nm).

Supplementary Note 2. Method of non-local deformation measurement (1) ND orientation and rotation
There are four ODMR-inequivalent NV axes in a diamond lattice, (111), (11 1 ), (1 11 ) and The orientation of an arbitrary ND (ND-I) is specified by the Euler angles ( , , ) ( Supplementary Fig. 2b), the angles of three sequential rotations that transfer the orientation of ND0 (whose crystallographic directions set the laboratory coordinate system) to that of ND-I: first a rotation of about the -axis in laboratory coordinate system (which transfers the -axis to where and are the matrices of the rotations about the -, and -axes, respectively, and = cos and = sin . The corresponding NV orientations in ND-I are related to those of ND0 by = ( , , ) . Figure  ) is in the x-z plane. b, The Euler angles representing the orientation of an arbitrary ND, obtained from the orientation of ND0 by three rotations in sequence (1, 2, and 3). c, The two magnetic fields and the orientation of the ND used in the simulations. d, An rotation with the rotation axis (orange arrow) defined by ( , ) and the rotation angle . d, Illustration of the sample surface with (blue grid) and without (grey grid) indentation. The black and red arrows are the normal vectors of the surfaces. The green arrows are the surface gradient field.
The rotation of an ND can be described by the rotation axis ( , ) (in the spherical coordinates) and the rotation angle (see Supplementary Fig. 2d). The rotation matrix R is defined where is an arbitrary vector, and = (sin cos , sin cos , cos ) is the unit vector of the rotation axis. The orientation of an ND after the rotation is characterized by the Euler angles ( ′, ′, ′). The rotation ( , , ) of the ND was obtained by comparing its Euler angles before and after the operation, that is ( , , ) ( , , ) = ( ′, ′, ′).

(2) Determination of the NV orientations
In the presence of known magnetic field(s) (calibration of the magnetic fields is described in Supplementary Note 3), the orientations of NV centres can be determined from the experimentally measured frequency shift of the ODMR spectra 3 . The ground state of an NV center is a spin triplet state ( = 1) with a zero-field splitting ≈ 2.87 GHz between the = 0 and = ±1 states defined along the NV axis. The Hamiltonian of the NV center is written as 4 in the presence of a magnetic field B, where = ( , , ) is the spin-1 operator and ≈ 28 MHz mT is the electron gyromagnetic ratio. By diagonalizing the Hamiltonian, the transition frequencies ± , up to the second order perturbation, are determined as 3 where = cos ( ⋅ / ) is the angle between the NV axis and the external magnetic field . For a given magnetic field, the orientation of the (and hence that of the ND) is determined from the angles . However, if the ND is rotated by any angle about the magnetic field , the ODMR frequencies would not be changed. This ambiguity can be removed by introducing another magnetic field ′ along a different direction from B. Then another set of angles relative to ′ can be obtained, which enables the unambiguous determination of the orientation of the ND. By the least-square fitting of the respective transition frequencies ± (with ) and ± (with '), with the Euler angles ( , , ) and the zero-field splitting as the fitting parameters, we obtained the orientation of the ND. Once the ND orientation is determined, the comparison between the orientations before and after the rotation gives the rotation data ( , , ).
In practice, instead of fitting the transition frequencies, we directly fitted the four ODMR spectra under magnetic field or ' before and after the rotation to deduce the rotation of the ND.
The normalized ODMR spectrum is written as 4 where is the baseline, represents the ODMR contrast of the NV centres along the th direction and Δ is the linewidth (FWHM). In the least-square fitting, the initial orientation of the ND were firstly deduced from the two ODMR spectra before the rotation by employing the Euler angles ( , , ), the baselines, the contrasts, the FWHM, and the shift of the zero-field splitting Δ (slightly varied among NDs) as the fitting parameters. Then, the rotation of the ND were determined by fitting the two ODMR spectra after the rotation with the rotation axis and angle ( , , ), the contrasts, and the baselines as fitting parameters. The Euler angles, the FWHM, and Δ were fixed to be the results obtained by fitting the spectra before the rotation. It was found that for each ND, the contrasts and the baselines obtained by fitting the data before and after the rotation were approximately identical. The slight variation of the parameters was probably due to the laser and the MW power fluctuations between the ODMR measurements.

(3) Reconstruction of deformation from ND rotation data
The surface of the material before indentation is assumed to be horizonal ( In turn, the gradient field ( ) is ( ) = − , − . The deformed surface ( ) was reconstructed from ( ) at a discrete set of locations { }, by minimizing the global least-square cost function 5

(4) Deformation and ODMR simulation
The axisymmetric deformation ( ) of a homogenous material caused by the indentation of an AFM tip (cone-shaped with tip radius 100 nm and half-angle 12°) was numerically simulated with a linear elastic model (the Hertz-Sneddon model) 6,7 . The result is shown as the blue line in

Supplementary Note 3. Method validation
Feasibility of the proposal was first evaluated by measuring the rotation of a bulk diamond (containing NV ensembles) of known orientations. Figure S4a shows an optical image of the bulk diamond. We set the ( 01 1) 1) and (011) crystallographic axes along the x-, and y-axes of the laboratory frame, respectively. A 21° clockwise rotation is applied to the image after rotation in order to regain its original orientation. b, Schematic showing the configuration of the magnetic fields and ', and the orientations of the NV centers inside the bulk diamond. The grey arrow marks the rotation axis. c, Optically detected magnetic resonance (ODMR) spectra of the bulk diamond before (black dots) and after (red dots) rotation under the magnetic field (upper) or ′ (bottom). The grey and red lines are the best fitting. consistent with the rotation angle (~21°) measured from the optical images before and after the sample rotation ( Supplementary Fig. 4a).

Supplementary Note 4. Reconstructing the deformation of a PDMS thin film (1) Fabrication of PDMS films and their characterizations
PDMS was synthesized by mixing the precursor and cross linker with a ratio of 5:1. A microwave antenna was amounted a glass slide. The PDMS was then spin-coated onto the glass slide. After degassing, the surface of the PDMS film was modified by O2 plasma that generated a surface oxidized layer 8,9 . At last, the NDs were spin coated on the PDMS film. The thickness of the PDMS film was measured to be ~50 μm (from optical measurement). To estimate the thickness of the oxidized surface layers, we prepared a cross-sectional PDMS sample by putting together two PDMS films with oxidized surfaces face to face. Figure S5a shows the AFM stiffness mapping of the cross-sectional PDMS. The dark region represents the bulk PDMS, and the bright region is the surface layer (with larger stiffness). A ~500 nm thickness of the surface oxidation layer was estimated from the thickness between the two dashed lines. AFM nanoindentation was performed on the bulk PDMS far from the oxidized surface, and apparent Young's modulus of the bulk PDMS was obtained as ~0.78 MPa with the Hertz-Sneddon model 6,7 .
A silicon nitride (with Young's modulus ~1 GPa) cantilever (DNP10-A Bruker) was applied to image and indent the PDMS film. After calibrating the spring constant and deflection sensitivity of this tip, the PDMS film was imaged in a peak force tapping mode and then indented in a ramp mode. Figure S5b shows a few typical load-depth profiles obtained by nanoindentation performed at the spots labeled in the inset. The almost identical load-depth profiles of indentations carried out at different location on the sample suggest the homogeneity in the x-y plane of the PDMS film.

(2) Methods of orientation measurements of surface anchored NDs under an AFM tip indentation
In this protocol, dilute ND solution (2 μg mL ) was employed to avoid NDs aggregation.
Dynamic light scattering (DLS) measurement of the ND solution suggested an average hydrodynamic diameter of 150 nm ( Supplementary Fig. 6a). We also checked the sample using transmission electron microscopy (TEM) ( Supplementary Fig. 6b), when reasonably dispersed NDs were found. The diameters of ND were in the range of a few tens of nanometers to ~200 nm, consistent with the DLS measurement. NDs were dispersed on the surface of the PDMS film by spin coating. Figure S6c shows the AFM topography image with NDs located on a 40 × 40 μm PDMS surface, correlated with the confocal image ( Supplementary Fig. 6d) taken from the same Using the methods described in Supplementary Note 2-2, the initial orientation of the ND (without indentation) was deduced from its ODMR spectra under and ′ and the corresponding fittings (black dots and lines in Supplementary Fig. 7). The characteristic Euler angles were  Supplementary Fig. 7).

(3) Deformation reconstruction
For the x-y plane homogeneous medium, we adopted the single-ND method to reconstruct The same method as for ND 1 was adopted. The initial Euler angles of ND2 and ND3 were AFM image and the ODMR spectra before and after the indentation experiments were compared for ND1 ( Supplementary Fig. 11). Complete recovery of the PDMS surface after releasing the load is suggested by the absence of surface dent marks (Supplementary Fig. 11b).
Identical patterns on the AFM images before and after the indentations ( Supplementary Fig. 11a Supplementary Fig. 10 | The probability distribution of the deviation of a, the rotation angle and b, the -direction deformation. and b) indicate little relative motion of ND particles on the surface before and after the indentation experiments. Moreover, little change in the orientation of ND1 before and after the indentation was observed, as suggested by the identical resonance frequencies of the corresponding ODMR spectra ( Supplementary Fig. 11c and d). These results suggest that the ND had little relative motion and rotation against the PDMS surface throughout the indentation experiments.

Supplementary Note 5. Simulation of the PDMS deformation.
The indentation induced deformation of PDMS film was simulated using a linear elastic model of a layer/bulk system under an axisymmetric loading, as illustrated in Supplementary Fig.   12a. The PDMS film (thickness ~50 μm) was modeled as a half-infinite substrate with a surface layer (formed due to oxygen plasma treatment as illustrated in Supplementary Note 4-1). Literature reports that the oxidized surface layer is stiffer than the bulk PDMS with approximately one-order of magnitude larger elastic modulus 9 . For simplicity, both the surface layer (with thickness ,

Young's modulus , and Poisson's ratio ) and the bulk PDMS (Young's modulus and
Poisson's ratio ) were assumed to be isotropic. We adopted the same method as in dots in Supplementary Fig. 12b) obtained by the AFM tip indentation on the PDMS sample. As a comparison, the dashed line shows the simulation force-depth profile obtained by the Hertz-Sneddon model 6,7 , with the indented material assumed to be a half-infinity homogenous solid with an apparent Young's modulus of = 3 MPa (Supplementary Fig. 12b).

Supplementary Note 6. Validation of specific contact models
The inadequate information obtained from the local deformation profile causes ambiguity in the choice of contact models to determine mechanical parameters, such as Young's modulus. In other word, more than one models may fit well the same local deformation profile. But they may lead to different non-local deformation responses. In the gelatin experiments (see Supplementary Note 7 and the main text), both the models with and without the effect of surface tension can be adopted to fit the local deformation profile ( Supplementary Fig. 18), but only the model including the surface tension fits well the non-local deformation profile (Fig. 4 in the main text). The nonlocal deformation obtained by our method can be used to constrain different contact models.
As an additional example, the suitability of commonly used Johnson-Kendall-Roberts (JKR) the contrary, when non-local deformations were considered, a significant difference was observed between the JKR and DMT models ( Supplementary Fig. 13b), where the indentation force was chosen as = 3 nN (indicated by the red spot in Supplementary Fig. 13a). The difference between the rotation angles of the surface ND (800 nm away from the indentation point) for the two models was ~3.3° (see Supplementary Fig. 13c), which was sufficiently large to be measured by the ODMR. Therefore, the indentation profile determination of homogeneous materials with high precision by our method provides additional information that can be used to validate the contact models.  Supplementary Fig. 14a).
We imaged gelatin particles in peak force tapping mode with peak force of 300 pN (Fig. 4b in the main text). Indentations were applied on the gelatin samples at different locations using a PFQNM-LC-A cantilever in aqueous solution with holding force setpoint of 15 nN and indent and withdraw speed of 1 μm s in total ramp size of 3 μm. Figure S14b shows the force-depth plots obtained by indentation performed at different location of the gelatin particle. Similar results were obtained, suggesting homogeneity in x-y plane of the gelatin particle.

(2) Orientation measurements of surface anchored NDs under an AFM indentation and deformation reconstruction
Deformation of gelatin was investigated using multiple NDs dispersed on the sample surfaces.
Gelatin particles of 30 µm (diameter) were firstly dispersed on to a culture dish, and then 20 μL  Figure S16 shows as examples the ODMR spectra taken from ND a (marked in Supplementary Fig. 15a)  (red dots and lines in Supplementary Fig. 16). The deduced rotation axes and angles of the NDs in each of the indentation experiments are marked in the corresponding locations in Supplementary   Fig. 15. Figure S17a Supplementary Fig. 15a) and b, NDb (in Set 6 as marked in Supplementary Fig. 15f). Black and red solid lines are the fitting results of the ODMR spectra. Grey lines indicate the positions of the resonance peaks before indentation.
reasonable agreement exists between the experimental results and = − 90° (grey line in Supplementary Fig. 17b), suggesting that the NDs rotated toward the indentation location.
By assuming the axisymmetric condition, the gradient field can be written in polar coordinate as , = (tan ( ) , 0). Then the deformed surface is reconstructed by integration, as ( ) = ∫ tan ( ′) .

Supplementary Note 8. Simulation of the gelatin deformation.
Surface tension can play a major role in the mechanics of soft solids 15  is much smaller than the elastocapillary length scale, surface tension dominates and flattens the surface. A linear-elastic model including the surface tension effect was employed to simulate the deformation of the gelatin upon AFM tip indentation (schematic shown in Supplementary Fig.   18a). In this model, the surface tension was introduced by adding a thin membrane of the same material ideally adhered to the bulk with negligibly thickness 16 . The system was simulated following the method in Supplementary Ref. 17. Young's modulus = 6.5 kPa, Poisson's ratio ν = 0.5 and surface tension = 6.5 mN m were employed as the simulation parameters. The AFM tip was assumed a cone shape with tip radius 150 nm and half-angle 8° (according to the SEM imaging). The simulated indentation force-depth profile agrees with the experiment results ( Supplementary Fig. 18b). The simulated result of force-depth profile using the Hertz-Sneddon

Supplementary Note 9. Sensitivity of the rotation measurement of the surface ND
We estimated the sensitivity of our sensing methods as follows. For optics based sensing, a fundamental limitation to sensitivity is the shot-noise of photon counts 4 . To estimate the sensitivity of our methods, we measured the rotations of the ND using various data acquisition time. Typical data was chosen as the ODMR spectra of the first indentation points around the ND3 on the PDMS film (see Supplementary Note 4). Figure S19  Supplementary References: