Micro/Nano-scale Strain Distribution Measurement from Sampling Moiré Fringes

1. Confirmation of the Micro/Nano-scale Grid on the Specimen

1. Machining of the Specimen
   1. Cut the specimen to the size required by the specific loading device used under a microscope (e.g., 1 x 5 x 30 mm³), making the surface to be observed 1.5x greater than the region of interest.
   2. Polish the specimen surface to be observed (e.g., 1 x 30 mm²), successively using coarse and fine sand paper on an automatic polishing machine (e.g., use SiC foil #320 for 3 min and then #800 for 1 min at 150 rpm and 30 N). Clean the specimen using water after each polishing step.
   3. Polish the same specimen surface, successively using coarse and fine polishing solutions on the automatic polishing machine (e.g., use DP-Spray P 15 µm for 5 min, P 1 µm for 8 min, and P 0.25 µm for 10 min at 150 rpm and 30 N). Clean the specimen using water after each polishing step.
2. Fabrication of the Micro/Nano-scale Grid If No Periodic Pattern Exists on the Specimen
   NOTE: This step can be omitted if a natural periodic pattern exists on the micro/nano scale on the specimen surface. Choose the grid fabrication method from the following: ultraviolet (UV) or heating nanoimprint lithography (NIL)²⁶, electron beam lithography (EBL)², and focused ion beam (FIB) milling⁶.
   NOTE: The grid fabrication process is introduced here, with UV NIL as an example.
   1. Drop 2 mL of UV resist on the specimen surface using a pipette.
   2. Coat the resist on the specimen surface using a spin coater at 1,500 rpm for 60 s.
   3. Press a nanoimprint mold to the resist layer at a pressure of 0.2 MPa. Expose the resist to UV with a wavelength of 375 nm for 30 s.
   4. Separate the nanoimprint mold from the specimen surface.
3. Observation of the Grid on the Specimen Using a Microscope
   1. Coat a platinum or gold layer with a thickness of 3-10 nm on the grid surface using an ion coater (e.g., coating for 30 s at 3 Pa with a sputtering current of 30 mA).
   2. Put the specimen under a laser scanning microscope (LSM)²³.
      NOTE: Other microscopes can also be used, such as a transmission electron microscope (TEM)⁵, an atomic force microscope (AFM)¹², or a scanning electron microscope (SEM)⁷.
   3. Adjust the focus and save one grid image using the microscope by clicking "Capture" and "File | Export | Image File" in the image recording software of the microscope.
4. Calculation of the Grid Pitch (nm or µm) of the Specimen from the Grid Image
   1. Calculate the average value of more than 10 grid pitches in the central area of the grid image to avoid the potential influence of the scanning or lens distortion.
      NOTE: The grid on the specimen can be saved for several days at room temperature.

2. Acquisition of Grid Images in the Loading Test

1. Preparation of the Loading Test Under the Microscope
   1. Fix the specimen to a loading device, such as a tensile, compressive, heating, or electrical loading device, under the microscope.
      NOTE: If the grid pitch is less than 20 nm, a TEM or AFM should be used. If the grid pitch is 20 nm to 10 µm, an SEM can be used. If the grid pitch is greater than 400 nm, an LSM can be used.
   2. Set the load speed (e.g.,0.01 mm/s) and the load or displacement incremental step (e.g., 0.5 N/step or 0.024 mm/step) according to the specific requirements. Preset both the load and displacement to zero.
   3. Make the grid surface in the observation plane. Choose an area of interest under a low magnification by moving or rotating the sample stage of the microscope.
   4. Select an appropriate magnification by making the grid pitch in the image greater than 1.8× one pixel size.
      NOTE: Usually, it is better to make the grid pitch in the image greater than 2 pixels. The more pixels one grid pitch corresponds, the higher the accuracy of the deformation measurement, but the smaller the field of view of the measurement.
2. Collection of Grid Images in the Loading Test
   1. Save a grid image of the area of interest before loading by clicking "Capture" and "File | Export | Image File" in the image recording software of the microscope.
   2. Begin to load the specimen in situ onto the microscope by exerting the first load step (e.g., 0.5 N or 0.024 mm) using the operating software of the loading device.
   3. Record a grid image of the area of interest after the first load step (e.g., at 0.5 N or 0.024 mm) by clicking "Capture" and "File | Export | Image File" in the image recording software of the microscope. Ensure that the magnification and working distance of the microscope remain unchanged.
   4. Continue to load the specimen by exerting each load step using the loading device. Record the grid image after each load step until the specimen is broken or until a certain value is attained (e.g., load 19 times and record 19 grid images at 1 N, 1.5 N, 2.0 N, …, 10 N, at intervals of 0.5 N; or at 0.048 mm, 0.072 mm, 0.096 mm, …, 0.48 mm, at intervals of 0.024 mm). Make sure that the magnification and working distance of the microscope remains unchanged.
      NOTE: The grid images can be saved for an arbitrarily long period of time.

3. Generation of Sampling Moiré Fringes before and after Deformation

1. Estimation of Grid Pitches (pixel) in the Grid Images
   1. Estimate the grid pitch (unit: pixel) in the grid image before loading by measuring the distance between the centers of two adjacent grid dots in an image processing software (e.g., Microsoft Paint).
   2. Estimate the grid pitch in the grid image at the maximum load.
2. Determination of Sampling Pitch (pixel)
   1. Move to step 3.2.2 when the grid pitches before and after deformation are between 1.8 and 2.5 pixels. Skip to step 3.2.3 when the grid pitches before and after deformation are between 2.4 and 3.6 pixels. Skip to step 3.2.4 when the grid pitches before and after deformation are greater than 3.2 pixels.
   2. Set the sampling pitch to T = 2 pixels. Skip to step 3.3.
   3. Set the sampling pitch to T = 3 pixels. Skip to step 3.3.
   4. Set the sampling pitch T to a positive integer within 0.75x and 1.25x the grid pitches before and after deformation, determined from abundant simulation results²².
      NOTE: If there are 2 positive integers that meet the requirements in steps 3.2.1 and 3.2.4, it is better to choose the greater integer as the sampling pitch. If there are 3 or more positive integers that meet the requirements, it is better to choose the middle integer, as long as it is a little bit greater than the sampling pitch.
3. Generation of Sampling Moiré Fringes Before Deformation
   1. Open the grid image before deformation. Assuming that the x direction is horizontally rightward, the y direction is vertically downward, and the coordinate (0, 0) is at the top-left corner, calculate the image width W in the x direction and the image height H in the y direction.
      NOTE: The y direction can also be defined as vertically upward.
   2. Move to step 3.3.3 to generate moiré fringes in the y direction. Skip to step 3.3.7 to generate moiré fringes in the x direction.
   3. Process the grid image to a grating image using a low-pass filter (LPF). For example, use an FT algorithm to suppress the grating, with a principal direction of x, where the principal direction is defined as the direction perpendicular to the grating lines. Set the filter size to be close to the grid pitch.
   4. Thin out the grid image by only extracting the gray values in several horizontal lines, with the spacing of the sampling pitch T (T ≥2) from y = k pixels (k = 0) (Figure 1) (i.e., only keep the gray values in sampling lines of y = k pixels, y = k+T pixels, … , y = k + iT pixels, where i is a positive integer). Make the coordinate of the last sampling line, k + iT, less than the image height H.
   5. Generate a sampling moiré pattern in the y direction by performing full-field intensity interpolation (linear or B-spline) of the image with horizontal sampling lines.
   6. Generate other T-1 sampling moiré patterns in the y direction by repeating steps 3.3.4 and 3.3.5 T-1 times by changing k at the incremental step of 1 pixel (i.e., shifting the starting point of thinning out to y = k pixels; k = 1, …, T-1).
   7. Use the same procedures in steps 3.3.3-3.3.6 to generate T-step spatial phase-shifting sampling moiré patterns in the x direction by changing x to y in step 3.3.3, changing the image height H to the image width W, and changing y to x in steps 3.3.4-3.3.6.
      NOTE: The sampling pitch in the x direction can be different from that in the y direction.
4. Generation of Sampling Moiré Fringes After Deformation
   1. Open all grid images at different loads. Suppose the number of grid images is N.
   2. Generate N groups of T-step spatial phase-shifting moiré fringes in the y direction by repeating steps 3.3.3-3.3.6 N times.
   3. Generate N groups of T-step spatial phase-shifting moiré fringes in the x direction by repeating step 3.3.7 N times.

4. Deformation Measurement of the Specimen in the Loading Test

1. Determination of the Intensities of Moiré Fringes Before and After Deformation
   1. Extract the intensities of the T-step moiré fringes before deformation in the y direction in steps 3.3.5 and 3.3.6; determine those moiré intensities in the x direction in step 3.3.7. Describe the T-step (T ≥2) moiré intensities before deformation in the j (j = x, y) direction using the following equation²³:
      (1)
      where p_j is the grid pitch before deformation in the j (j = x, y) direction, A is the modulated amplitude, and D includes the background and the higher-frequency intensities.
   2. Extract the intensities of the T-step moiré fringes after deformation in the y direction in step 3.4.2 and determine those moiré intensities in the x direction in step 3.4.3. Describe the T-step (T ≥ 2) moiré intensities after deformation in the j (j = x, y) direction using the same equation as above (Equation 1) by changing I_m,j(k), p_j, A, and D to I'_m,j(k), p'_j, A', and D', respectively, where the superscript single quote means after deformation.
      NOTE: If the sampling pitch is T ≥ 3 pixels, ignore this step and skip to step 4.3.
2.  
   1. Reconstruct multiplication moiré fringes from the multiplicative interference between the two-step sampling moiré intensities (Figure 1a) before deformation using the following equation²³
      (2)
      where I_multi,j stands for the intensity of the reconstructed multiplication moiré fringes in the j ( j = x, y) direction before deformation.
   2. Process the reconstructed multiplication moiré fringes before deformation using the fringe centering technique²⁴. Assign consecutive integers and half integers f_j = [1, 1.5, 2, 2.5, …] to the fringe orders at the center lines of the reconstructed multiplication moiré.
      NOTE: If the multiplication moiré fringes are too dense, the fringe orders of the two-step sampling moiré can be determined first (i.e., f_j(0) = [1, 0, 2, 0, 3, 0, …] and f_j(1) = [0, 1.5, 0, 2.5, 0, 3.5, …]). The fringe order of the multiplication moiré fringes will be f_j = f_j(0) + f_j(1) = [1, 1.5, 2, 2.5, 3, 3.5, …]. The rigid-body displacement will not affect the strain result.
   3. Measure the relative strain of the specimen before deformation relative to the sampling pitch using the following equations²³
      (3)
      (4)
      where u_{j_rela} and ε_{j_rela} represent the relative displacement and the relative strain of the specimen before deformation in the j (j = x, y) direction, respectively, and γ_{xy_rela} expresses the relative shear strain before deformation.
   4. Repeat steps 4.2.1-4.2.3 to determine the relative strains of the specimen after deformation in the x and y directions for N times, changing I_multi,j, I_m,j(0), I_m,j(1), p_j, A, D, u_{j_rela} (j = x, y), ε_{j_rela}, and γ_{xy_rela} in equations (2)-(4) to I'_multi,j, I'_m,j(0), I'_m,j(1), p'_j, A', D', u'_{j_rela} (j = x, y), ε'_{j_rela}, and γ'_{xy_rela}, respectively, where the superscript single quote means after deformation.
   5. Determine the actual normal strain ε_j in the j (j = x, y) direction, which is the relative change of the grid pitch and the shear strain, γ_xy, which is the absolute variation of the grid angle of the specimen caused by the load from the relative strains before and after deformation²⁰.
      (5)
      (6)
3. Deformation Measurement When the Sampling Pitch is T ≥ 3 pixels
   1. Calculate the phase of the sampling moiré fringes in the j (j = x, y) direction before deformation when k = 0 (Figure 1b) using the spatial phase-shifting technique²¹
      (7)
   2. Obtain the phase of the sampling moiré fringes in the j (j = x, y) direction after deformation when k =0 by replacing φ_m,j and I_m,j(k) in equation (7) with φ'_m,j and I'_m,j(k), respectively, where the superscript single quote means after deformation. Repeat N times for N loads.
      NOTE: If there is too much random noise in the phase distributions in steps 4.3.1 and 4.3.2, a sin/cos filter²⁵ can be used to smooth the phases.
   3. Determine the phase difference of the sampling moiré fringes in the j (j = x, y) direction before and after deformation (i.e.,Δφ_m,j=φ'_m,j–φ_m,j).
   4. Measure the distributions of displacement u_j, normal strain ε_j in the j (j = x, y) direction, and the shear strain γ_xy of the specimen caused by the load. Use the following equations^6,21
      (8)
      (9)
      (10)
      NOTE: If there is too much noise in the strain distributions, an average smoothing filter can be used, with a filter size smaller than 2 grid pitches.
4. Result Storage
   1. Save the data of the moiré fringes, phases (when the sampling pitch is T ≥3 pixels), displacements, and strains in the forms of images, such as .tif or .bmp files, and text, such as .txt or .csv files.