High resolution strain measurements in highly disordered materials

The ability to measure small deformations or strains is useful for understanding many aspects of materials. Here, a new analysis of speckle diffraction peaks is presented in which the systematic shifts of the speckles are analyzed allowing for strain (or flow) patterns to be inferred. This speckle tracking technique measures strain patterns with a accuracy similar to x-ray single crystal measurements but in amorphous or highly disordered materials.

The measurement of small deformations in samples is a useful method to characterize the properties of materials, soft materials in particular. Deformations are typically measured via strain, or the geometrical relative displacements of elements in a body. High resolution strain has been measured using x-rays and neutron scattering ([1], [2], [3]) and visible light ( [4], [5], [6]). However, these measurements were mostly limited to crystalline materials and or macroscopic sections of samples. There are few measurement techniques involving amorphous materials with x-rays. The Nielsen group [3] applied a 3D absorption tomography technique. This technique provides a high resolution 3 dimensional picture of the static strain of materials. However, scan times are long, with at least 2 hours per scan. The long times make it difficult for in-situ measurements, where the strain may occur over short times. In this paper a technique using XPCS is proposed for in-situ measurements of strain in amorphous materials.
For a simple insight into the technique, imagine a deformation of the underlying space, r = M · r. For small deformations, the transformation consists of a possible rigid body displacement and a local distortion, ∇·M . From this it follows that the strain is the difference between ∇·M and the identity matrix. When ∇·M is independent of r a homogeneous deformation results. Consider the effect on the density, ρ( r), and its Fourier transform, ρ( Q). To get the Fourier transform of the deformed material, ρ ( Q): by simple substitution. This calculation assumes a negligible change in the volume of integration.
Speckle measurements use a small diffraction volume and this sets the scale determining when a deformation is small and for which the approximation is to be valid. Another approach is given by solving the convection-advection diffusion equation by the method of characteristics and may be found in Fuller et. al [7]. So, if we can relate features in ρ( q) (or scattering intensity) before and after the deformation, their time dependent shifts in reciprocal space are: and related to the transpose of the deformation. For a simple velocity field The deformation gradient tensor is related to the velocity gradient matrix, Γ = ∇ v( r) for a velocity field The rigid body shift gives a phase factor, exp(i v 0 · Qτ ), due to the shift in r and has been ignored as it will not be seen in the scattered intensity.
The related wave-vector for a deformation can be obtained from the following correlation function: where the average is over a small region in Q centered around Q 0 . Noticing that when the shifts are time invariant, it is possible to also average in time. This correlation function is an extension of the intensity-intensity correlation function used in XPCS, where ∆ Q = 0. When the scattering is sharply peaked, as for Bragg peaks and for speckles in coherent diffraction, one measures ∆ Q by following the local maxima.
The experiments were carried out at beamline 8-ID-I of the Advanced Photon Source. For this setup, the energy was 7.35 Kev(λ = 1.67Å) monochromated by double bounce Ge (111) crystals. The incident flux was ≈ 10 9 photons/sec through a 20×20 µm 2 aperture. Each pixel in the area detector corresponds to 2.0 × 10 −5Å−1 , which is close to the speckle size. The filled rubber samples where held in a vacuum chamber with an in-situ tensile stress-strain cell. The exposure time per frame was .1 second recurring every 2.0 seconds. Further details are given in references [8,9].
The sample consists of a cross-linked elastomer (Ethylene Propylene Diene Monomer, EPDM, rubber) filled with hydoxylated pyrogenic silica (Aerosil 200 [10][11][12]). The volume fraction of silica is close to 0.16. In our measurements, the one millimeter thick sheets are punched out to a classical dumb-bell shape (width = 4 mm). As described in Ref. [8] upon applying a step strain on the sample, the stress jumps and then slowly relaxes as the sample is help at constant strain.
For this article, only one 400 second data set is presented for a sample which was stretched by 60%. The data collection for this run started approximately 1250 seconds after the application of the strain step. This data is part of those in Ref. [8]. A single run is described as this simplifies the exposition of the technique. The description of the rheological aspects of the measurements is partially described in the references [8,9]. The implications of this new analysis on the viscoelastic properties of the various samples is left for a future paper.
To calculate the cross-correlations, the scattering images were decomposed into wedges or bins of ∆Q and ∆φ. The azimuthal angle φ is oriented with increasing horizontal pixels from beam center as 0 • . The orientation of the detector is such that up on an image is up in the sample. The wedges are 20 pixels (.001Å −1 ) wide in | Q| and 10 • in φ. Cross-correlations for all wedges with more than 1000 pixels up to wave-vector .024Å −1 (1200 pixels) were calculated. This gave 376 wedges. Bins are numbered with φ increasing for fixed Q and then Q increases for the next set of φ. Figure 1 shows a typical bin, bin 57 and its cross-correlation. As for each cross-correlation [13,14] it has a peak of size 1 + β (speckle contrast) sitting on a background of one. Each cross-correlation is least squares fit to a 2D Gaussian peak with widths giving the speckle size. The shift of the peak from the center gives the speckle movement. Figure 1c demonstrates that it is easy to see a shift by a fraction of a pixel as reflected by the asymmetric placement of the data around the center. A preliminary version of this analysis is given in Lhermitte's PhD thesis [15].