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Virtual Journal for Biomedical Optics

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  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 8, Iss. 10 — Nov. 8, 2013
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Simultaneous measurement of the microscopic dynamics and the mesoscopic displacement field in soft systems by speckle imaging

L. Cipelletti, G. Brambilla, S. Maccarrone, and S. Caroff  »View Author Affiliations


Optics Express, Vol. 21, Issue 19, pp. 22353-22366 (2013)
http://dx.doi.org/10.1364/OE.21.022353


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Abstract

The constituents of soft matter systems such as colloidal suspensions, emulsions, polymers, and biological tissues undergo microscopic random motion, due to thermal energy. They may also experience drift motion correlated over mesoscopic or macroscopic length scales, e.g. in response to an internal or applied stress or during flow. We present a new method for measuring simultaneously both the microscopic motion and the mesoscopic or macroscopic drift. The method is based on the analysis of spatio-temporal cross-correlation functions of speckle patterns taken in an imaging configuration. The method is tested on a translating Brownian suspension and a sheared colloidal glass.

© 2013 Optical Society of America

1. Introduction

Rough surfaces and scattering media generate a characteristic speckle pattern [1

1. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications(Roberts and Company, Englewood, 2007).

] when illuminated by coherent light, e.g. from a laser. By analyzing a time sequence of speckle patterns, valuable information can be retrieved on the sample evolution. Broadly speaking, one may distinguish between “static” speckle patterns generated by solid objects and “dynamic” speckles formed by soft matter systems (e.g. colloidal suspensions, emulsions, polymer solutions, biological tissues), whose components undergo Brownian motion, thereby continuously reconfiguring the scattered speckle pattern. In the former case, relevant to metrology and interferometry [2

2. R. K. Erf, Speckle Metrology(Academic, 1978).

], a rigid displacement or a long wavelength deformation is often measured, e.g. in response to vibrations, applied load, or a change of temperature; speckle patterns are recorded onto a 2D detector such as a CCD or CMOS camera, using an imaging optics. By contrast, the microscopic (e.g. Brownian) dynamics of soft systems is quantified by g2(τ) − 1, the autocorrelation function of the temporal fluctuations of the scattered intensity. In these dynamic light scattering measurements (DLS, a.k.a. photon correlation spectroscopy [3

3. B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, 1976).

]) a point-like detector (e.g. a phototube) is placed in the far field, where it collects light within a few speckles.

Fig. 1 a): schematic representation of an image J that is a shifted version of image I (only four pixels are shown for clarity). The intensity in a given pixel of J may be obtained as a linear combination of (up to) four pixels of I, with weights proportional to the colored areas. b): quadrant detection scheme for locating the direction of the shift. The nine elements closest to the peak of the cross-correlation between I and J are represented here. The shaded areas are (in pixel units) α1 = 0.485869913, α2 = 0.545406041, α3 = 0.25; see Sec. 6 for more details. c): typical speckle image obtained in a PCI experiment.

Fig. 2 a)–c) spatial cross-correlation between speckle images generated by a diluted Brownian suspension that is translated along the y direction by a motor. The speckle patterns are recorded on a CCD using an imaging collection optics (see text for more details). As the delay τ between pair of images is increased, the peak position shifts to larger y and its height decreases, due to the relative motion of the Brownian particles. d) cut of the cross-correlation along the Δx = 0 line. Curves are labeled by τ. Inset: same data, replotted as a function of spatial shift with respect to the peak position. e) same as d), but for a ground glass, a scatterer whose internal dynamics are frozen. Note that the peak height remains essentially constant as the sample is translated.

2. Sub-pixel digital imaging correlation algorithm

The first step in our method is to find the local rigid displacement with sub-pixel resolution. To this end, we use a cross-correlation technique inspired by particle imaging velocimetry [9

9. C. E. Willert and M. Gharib, “Digital particle image velocimetry,” Experiments In Fluids 10, 181193 (1991). [CrossRef]

] and image correlation velocimetry [10

10. P. T. Tokumaru and P. E. Dimotakis, “Image correlation velocimetry,” Experiments In Fluids 19, 115 (1995). [CrossRef]

]. A time series of images of the sample is taken, using a PCI setup (see Fig. 1(c) for an example). Each image is divided into a grid of regions of interest (ROIs), corresponding to square regions of side L in the sample. Under a rigid drift, the speckle pattern in a ROI at time t will appear in a shifted position in a successive image taken at time t + τ. If the scatterers undergo relative motion, in addition to a rigid shift, the shifted ROI will not be an identical copy of the original one. Still, the displacement field can be estimated by calculating by what amount a ROI of the second image has to be back-shifted in order to maximize its resemblance with the corresponding ROI of the first image. In practice, for a given ROI the shift along the horizontal and vertical directions, Δx and Δy, is determined by searching the maximum of corr[I, J], the spatial cross-correlation of the intensity of the two images, defined by
corr[J,I](k,l)=covar[J,I](k,l)var[J]var[I]
(2)
with
covar[J,I](k,l)=N1r,cJr,cIr+k,c+lN2r,cJr,cr,cIr+k,c+l
(3)
var[I]=N1r,cIr,c2(N1r,cIr,c)2.
(4)
In the above equations, Ir,c is the intensity at time t of the pixel at row r and column c, Jr,c is the intensity at the same location but at time t + τ, and k and l are the shifts expressed in number of rows and columns, respectively. Here and in the following, double sums over r and c extend over all rows and columns for which the terms of the sum exist, in this case the N pixels of the overlap region between the full image and the shifted ROI. For computational efficiency, covar[J, I] is usually calculated in Fourier space [20

20. W. H. Press and S. A. Teukolsky, Numerical Recipes in C. (Cambridge University Press, 1992).

]. Note that covar[J, I] → 0 far from the peak, where I and J are uncorrelated.

The position (, ) of the global maximum of corr[J, I] yields the desired displacement along the direction of columns (x axis) and rows (y axis), Δx = and Δy = respectively, with pixel resolution. Several schemes have been proposed to improve this resolution, e.g. by calculating the centroid of corr[J, I], or by fitting its peak to a 2-dimensional analytical function such as a Gaussian. While both methods work well for broad, circularly symmetric peaks, they tend to be less robust when the peak is sharp or it has an asymmetric shape. The shape of the peak is determined by the spatial autocorrelation of the intensity pattern; for our speckle images, it depends on the shape and size of the speckles, which may not be symmetrical, depending on the shape of the illuminated sample volume and the imaging optics [1

1. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications(Roberts and Company, Englewood, 2007).

]. Moreover, the peak usually extends over just a few pixels, because one minimizes the speckle size in order to maximize the information content in the image. To overcome the limitations inherent to peak-based schemes, we use an alternative approach based on a least-square method that allows us to obtain the displacement field with a typical resolution of a few hundredths of a pixel, with no requirements on the shape or broadness of the peak and without using any fitting function.

The optimum shift is obtained by imposing ∂χ2/∂ai = 0, i = 1,..,4. By exchanging the order of the sums in Eq. (6) one recognizes that a is the solution of b = M · a, with:
bi=covar[J,I](ki,li)
(7)
Mi,j=covar[I,I](kikj,lilj).
(8)
Any standard method is suitable to solve the above set of equations; in our implementation, we use singular value decomposition [20

20. W. H. Press and S. A. Teukolsky, Numerical Recipes in C. (Cambridge University Press, 1992).

]. It should be emphasized that b is known, since covar[J, I] has been already calculated to estimate the pixel-resolved displacement (see Eqs. (2) and (3)). Therefore, the only extra cost required for calculating the displacement with sub-pixel resolution is the computation of covar[I, I] and the solution of the set of linear equations b = M · a. This moderate computational extra cost is due to the fact that a linear interpolation scheme has been adopted in Eq. (6): higher-order interpolations, although more precise, would lead to a much more complex, non-linear minimization problem. Finally, we note that in a typical multispeckle DLS experiment, one calculates the correlation functions for a given starting time (i.e. a given image I) and several time delays τ (i.e. several images J). Therefore, the computational cost for calculating covar[I, I] is shared between several lags, further increasing the efficiency of the algorithm.

Once a is computed, the shifts along the x and y directions are calculated with subpixel precision according to
Δx=a2+a4i=14ai+l1
(9)
Δy=a3+a4i=14ai+k1
(10)
(see Eqs. (27) and (29) in Sec. 6 for the definition of k1 and l1).

3. Dynamic light scattering: corrections tog2(τ) − 1 for drifting samples

In order to quantify the internal dynamics, one needs to compute the intensity correlation function g2 − 1 between image J and a shifted version of I, so as to avoid any artifact due to the rigid shift of the speckles. Denoting by I′ the image I shifted by (Δx, Δy), the (un-normalized) intensity correlation function corrected for the shift contribution is
G2(τ)=N1r,cJr,cIr,c.
(11)
The shifted image I′ may be constructed using an interpolation method. Tests on real speckle images show that linear interpolation, although suitable for determining the shift with good accuracy, is not precise enough to reconstruct a shifted version of I suitable for the calculation of g2 − 1. Higher-order interpolation schemes are thus required. As shown in [21

21. T. M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” Ieee Transactions on Medical Imaging 18, 1049–1075 (1999). [CrossRef]

], image shifting by interpolation is equivalent to convolving the original image with a suitable kernel:
Ir,c=k,lh(r+Δyk)h(c+Δxl)Ik,l,
(12)
where we have assumed for simplicity that the kernel is symmetrical and separable, i.e. that 2Dh(x, y)= h(x)h(y). Unfortunately, in our case this approach would be too time-consuming, because it requires a convolution operation, Eq. (12), in addition to the calculation of the correlation function, Eq. (11).

We introduce below an alternative method that leads to a much faster algorithm, where the only computational cost is that of evaluating the kernel for a few points, with no need for the calculation of convolution and correlation functions. It is convenient to consider only nonnegative fractional shifts δx˜, δy˜, given by
Δx=jx+δx˜
(13)
Δy=iy+δy˜,
(14)
with jx = floor(Δx), iy = floor(Δy), where floor(x) is the largest integer ≤ x. As we shall discuss it below, the choice of the kernel is not crucial; a good choice is a truncated, windowed sinc function with an even number, M, of supporting points:
h(x)=w(x)sin(πx)/(πx)for|x|M/2h(x)=0elsewhere,
(15)
where we choose the three-term Blackman-Harris window function [21

21. T. M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” Ieee Transactions on Medical Imaging 18, 1049–1075 (1999). [CrossRef]

, 22

22. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier-transform,” Proc. IEEE 66, 5183 (1978). [CrossRef]

] defined as
w(x)=0.42323+0.49755cos(2πxM)+0.07922cos(4πxM).
(16)
With this choice, the kernel is DC-constant [21

21. T. M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” Ieee Transactions on Medical Imaging 18, 1049–1075 (1999). [CrossRef]

, 22

22. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier-transform,” Proc. IEEE 66, 5183 (1978). [CrossRef]

], i.e.
k,l=M/2+1M/2h(x)=1,
(17)
a property that will be of use in the following.

Using Eqs. (13) and (14), the convolution product (12) may be rewritten as
Ir,c=k,l=M/2+1M/2h(δy˜k)h(δx˜l)Ik+r+iy,l+c+jx.
(18)
By replacing the r.h.s. of Eq. (18) in Eq. (11) and by exchanging the order of the sums, we obtain:
G2(τ)=k,l=M/2+1M/2h(δy˜k)h(δx˜l)[N1r,cJr,cIr+k+iy,c+l+jx].
(19)
Finally, by recalling the definition of the covariance between J and I, Eq. (3), and using the fact that the kernel is DC-constant, Eq. (17), one obtains
G2(t,τ)J¯I¯=k,l=M/2+1M/2h(δy˜k)h(δx˜l)covar[J,I](k+iy,l+jx)
(20)
or, equivalently,
g2(t,τ)1=k,l=M/2+1M/2h(δy˜k)h(δx˜l)covar[J,I](k+iy,l+jx)J¯I¯,
(21)
where Ī = N−1r,c Ir,c and similarly for .

Equation (21) is the central result of our method. It shows that the intensity correlation function corrected for the shift contribution can be simply obtained as a linear combination of a few terms of covar[J, I], weighted by the kernel. Since covar[J, I] has already been calculated to determine the shift, the extra cost is essentially limited to the evaluation of M2 values of the kernel, which is typically negligible. Finally, we note that covar[J, I] vanishes on the length scale of the speckle size as its argument departs from (iy, jx), which is close to the location of the peak of the covariance. Hence, it is sufficient to take M on the order of a few speckle sizes (in units of pixels), because in Eq. (21) the contribution of the kernel for larger lags would be multiplied by a vanishingly small quantity. For example, we find that for images with a speckle size of about 5 pixels, the correction is virtually independent of M for M ≥ 8.

4. Experimental tests

We test our method on two samples: a suspension of Brownian particles loaded in a cell displaced by a motor, and a colloidal glass to which a shear deformation is applied. In both cases, no rigid displacement is detected when the samples are left at rest, indicating that other potential sources of drift, such as convection or sedimentation, are here negligible. The Brownian sample is a dispersion of polystyrene microspheres (radius R = 0.265 μm) in an aqueous solution of fructose at 75% weight fraction. The particle volume fraction is 10−5 and the sample is kept at a temperature T = 9 °C. The setup is described in [23

23. D. El Masri, M. Pierno, L. Berthier, and L. Cipelletti, “Aging and ultra-slow equilibration in concentrated colloidal hard spheres,” J. Phys.: Condens. Matter 17, S3543 (2005). [CrossRef]

]; we use the imaging geometry shown in Fig. SM1(a) of [13

13. A. Duri, D. A. Sessoms, V. Trappe, and L. Cipelletti, “Resolving long-range spatial correlations in jammed colloidal systems using photon correlation imaging,” Phys. Rev. Lett. 102, 085702 (2009). [CrossRef] [PubMed]

], where an image of the sample is formed onto a CCD detector using light scattered at θ = 90 deg, corresponding to a scattering vector q = 2.46 μm−1. The field of view is 1820×364 μm2 and images are acquired at a rate of 10 Hz, with an exposure time of 5 msec. To avoid any spurious increase of the base line of g2 − 1, the speckle images are corrected for non-uniform illumination as detailed in [24

24. A. Duri, H. Bissig, V. Trappe, and L. Cipelletti, “Time-resolved-correlation measurements of temporally heterogeneous dynamics,” Phys. Rev. E 72, 051401 (2005). [CrossRef]

]. The sample cell is attached to a motor that can impose a drift in the y direction at a controlled speed, vy = 10 μms−1.

Figures 2(a)–2(c) show the spatial cross-correlation calculated applying Eq. (2) to pairs of speckle images taken while displacing the sample, for three different time lags. For τ = 0 s, Eq. (2) yields the spatial autocorrelation of the speckle pattern: accordingly, a sharp peak of height one and centered at Δx = Δy = 0 is visible, whose FWHM ≈ 2.9 pixels provides the speckle size. As the lag increases, the peak position drifts in the y direction, due to the translation motion imposed by the motor. Additionally, its height decreases, due to the relative motion of the Brownian particles that reconfigure the speckle pattern. Figure 2(d) shows a cut of the cross-correlation peak along the Δx = 0 direction, for four values of τ. From this plot, it is clear that if g2(τ) − 1 was to be computed from a purely temporal cross-correlation, as in Eq. (1), one would observe a spurious, fast decay, essentially due to the rigid shift only. This would correspond to follow corr[I(t), I(t + τ)] at Δx = Δy = 0, as a function of τ. By contrast, if the relative motion of the Brownian particles is to be obtained, one has to measure the height of the peak as it drifts, as in the method proposed here. The inset of Fig. 2(d) shows the same data, plotted as a function of distance along y with respect to the (subpixel) peak position. It is worth noting that the peak width remains constant, in contrast to what suggested (but not demonstrated, to our knowledge) in patent literature [25

25. G. J. Tearney and E. B. Bouma, “Optical methods and systems for tissue analysis,” Patent US20020183601 A1, 5December2002.

], where it was proposed that the peak would broaden with τ as a result of the internal motion of the scatterers. Thus, the relevant parameter for extracting the relative motion is indeed the peak height, not its width.

In Fig. 3(a), we show for the Brownian particles the displacement of the speckle pattern as a function of τ, obtained from the sub-pixel peak position averaged over 200 pairs of images (i.e. 20 s), taken while translating the sample. The data are very well fitted by a linear law (red line) as expected for motion at constant speed, thus indicating that our algorithm captures very well the drift component of the speckle pattern, from a fraction of a pixel up to tens of pixels. The error bars represent σy, the standard deviation of the temporal fluctuations of the peak position detected for a fixed lag time τ.For τ ≤ 4.2s, σyy < 4% and the error bars are smaller than the symbol size, indicating that the detection of the peak position is consistently reliable, even when the peak height is as low as 0.1 or the displacement is just a fraction of a pixel. For τ = 5.6 s, the error bar is significantly larger, because the peak height is typically of order 0.05 − 0.06, comparable to the noise level. As a consequence, the detected peak position departs erratically from its true value for a significant fraction of the 200 pairs of images, leading to a large value of σy. Beyond τ = 5.6 s, no peak can be reliably found, thus preventing the displacement to be measured. The slope of the linear fit to the data is 3.30 ± 0.03 pixel s−1. Recalling that the nominal speed of the motor is vy = 10 μm s−1, this implies that 1 pixel corresponds to 3.03 ± 0.03 μm in the sample, in excellent agreement with 3.15 ± 0.15 μm/pixel as obtained from the magnification of the imaging system, evaluated using geometrical optics. Figure 3(b) shows the intensity correlation function g2(τ)−1, averaged over 20 s. Data are normalized such that g2(τ → 0)−1 = 1. If the sample is kept at rest during the measurement (black squares), the intensity correlation function exhibits an exponential decay, as expected for diluted Brownian suspensions [3

3. B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, 1976).

], as better seen in the inset that shows the same data in a semilog plot. When the sample is translated at constant speed, the uncorrected g2 − 1 decays on a much shorter time scale (blue circles) and its shape departs from a simple exponential. Clearly, no information on the microscopic dynamics can be obtained from the uncorrected data. The red crosses are the data corrected according to Eq. (21): for τ ≤ 4.2 s the corrected g2 − 1 is very close to that measured for the stationary sample, thereby demonstrating the effectiveness of our correction scheme. For larger lags, the corrected data tend to overestimate g2 − 1: this is consistent with the fact that the displacement cannot be reliably measured, as discussed in relation to Fig. 3(a). Indeed, in this case the peak-finding algorithm spuriously interprets the highest level in the noisy base line of corr[I(t), I(t +τ)] as the (higher-than-expected) degree of correlation.

Fig. 3 a): displacement versus time, measured for a diluted Brownian suspension translated at a constant speed. The line is a linear fit to the data. b) Intensity correlation functions probing the microscopic dynamics. Solid squares: quiescent sample; open circles: raw g2 − 1 measured while translating the sample with a motor; crosses: same data, corrected for the contribution of the rigid motion of the speckle pattern.

Figure 4(a) shows the velocity profiles close to the bead wall, for two times t after starting shearing the sample. In this representation, the slope of the data is the local shear rate γ̇. The solid line shows the velocity profile expected for homogeneous shear, corresponding to an average shear rate across the whole gap of γ̇ = 7.9 × 10−5 s−1. It is clear that already at t = 120 s γ̇ is non-uniform across the gap, with a highly-sheared band close to the moving surface (x ≥−164 μm, γ̇ ≈ 1.4 × 10−4 s−1), followed by a low shear region (x ≤−220 μm, γ̇ ≈ 6.0 × 10−5 s−1). Similar shear banding has been reported for other colloidal glasses [30

30. V. Chikkadi, G. Wegdam, D. Bonn, B. Nienhuis, and P. Schall, “Long-range strain correlations in sheared colloidal glasses,” Phys. Rev. Lett. 107, 198303 (2011). [CrossRef] [PubMed]

]. Interestingly, shear banding is seen to evolve with time. At t = 820 s, the shear rate for the high- and low-shear bands is comparable to that at t = 120 s (γ̇ ≈ 1.3 × 10−4 s−1 and γ̇ ≈ 6.1 × 10−5 s−1, respectively), but the boundary between the two zones has moved from x = −190 μm to x = −240 μm. Additionally, the occurrence of a marked drop of vy close to the moving surface suggests slipping. This behavior is reminiscent of that reported for a variety of jammed or glassy soft materials, see e.g. [31

31. T. Divoux, D. Tamarii, C. Barentin, and S. Manneville, “Transient shear banding in a simple yield stress fluid,” Phys. Rev. Lett. 104, 208301 (2010). [CrossRef] [PubMed]

], which exhibit complex spatio-temporal shear patterns. Figure 4(b) shows the intensity correlation function measured for the ROI at the position indicated by the arrow in a) [data are normalized to unity for the smallest available delay]. For the unsheared sample (crosses) no dynamics is observed on time scales up to 20s, about 2000 times the Brownian time for the same particles in the diluted regime. This is consistent with the notion that the microscopic dynamics of a sample at rest is slowed down by orders of magnitude on approaching the glass transition. The open symbols show the uncorrected g2 − 1: a fast decay is observed, essentially due to the translation of the speckle pattern due to the imposed shear. Once corrected, the data still show a decay of g2 − 1 (albeit a slower one), thus indicating that particles move with respect to each other, in addition to be advected by the shear deformation. We emphasize that the corrected g2 − 1 is sensitive to the component of the particle displacement along the direction of q, which lays in the horizontal plane, perpendicular to the shear direction. Therefore, the decay of g2 − 1 is not due to the affine component of the particle displacement along y, but rather to irreversible rearrangements associated with flow in glassy systems [32

32. P. Schall, D. A. Weitz, and F. Spaepen, “Structural rearrangements that govern flow in colloidal glasses,” Science 318, 1895 (2007).

]. Interestingly, we find that the decay of g2 − 1 is faster at t = 820 s, when both the local γ̇ and its gradient are larger. This suggest a direct relation between (local) shear rate and plastic rearrangements, as proposed for granular materials [16

16. A. Amon, V. B. Nguyen, A. Bruand, J. Crassous, and E. Clement, “Hot spots in an athermal system,” Phys. Rev. Lett. 108, 135502 (2012). [CrossRef] [PubMed]

] and emulsions [33

33. P. Jop, V. Mansard, P. Chaudhuri, L. Bocquet, and A. Colin, “Microscale rheology of a soft glassy material close to yielding,” Phys. Rev. Lett. 108, 148301 (2012) [CrossRef] [PubMed]

].

Fig. 4 a): velocity profiles for a sheared colloidal glass (symbols). Data are labeled by the time after initiating the shear. The dotted line indicates the position of the mobile wall, the solid line is the the velocity profile for uniform shear. The arrow indicates the location for which the data shown in b) have been measured. b) intensity correlation functions for the quiescent glass (crosses) and after applying a constant shear. Open (solid) symbols indicate the raw (corrected for drift) intensity correlation functions.

5. Conclusions

We have introduced a method to obtain the mesoscopic displacement field and the microscopic dynamics in soft materials where the constituents undergo both a drift motion and a relative displacement. The algorithm proposed here is optimized for the typical features of speckle images in PCI experiments, where a small speckle size is highly desirable to maximize the spatial resolution and the statistics of the measurement. The algorithm is highly efficient in that the correction of g2 − 1 does not requires any significant computational extra cost, besides that necessary to determine the displacement field. The method has been successfully tested on a Brownian suspension and a colloidal glass. Although similar information may be in principle obtained using confocal or optical microscopy, our method allows one to investigate samples that are difficult or impossible to visualize in real space, such as the very small particles of our colloidal glass. A generalization to speckle patterns obtained under multiple scattering conditions or for partially coherent illumination, such as in recent microscopy developments [34

34. R. Cerbino and A. Vailati, “Near-field scattering techniques: Novel instrumentation and results from time and spatially resolved investigations of soft matter systems,” Current Opinion in Colloid & Interface Science 14, 416 (2009). [CrossRef]

, 35

35. F. Giavazzi, D. Brogioli, V. Trappe, T. Bellini, and R. Cerbino, “Scattering information obtained by optical microscopy: Differential dynamic microscopy and beyond,” Phys. Rev. E 80, 031403 (2009). [CrossRef]

] is also possible [36

36. S. Buzzaccaro, E. Secchi, and R. Piazza, “Ghost particle velocimetry: accurate 3D flow visualization using standard lab equipment,” Phys. Rev. Lett. 111048101 (2013). [CrossRef] [PubMed]

]. The method presented here should be particularly valuable for soft materials where slow dynamics is coupled to the effects of an external stress, as in rheological experiments or in samples submitted to an external field such as gravity [18

18. G. Brambilla, S. Buzzaccaro, R. Piazza, L. Berthier, and L. Cipelletti, “Highly nonlinear dynamics in a slowly sedimenting colloidal gel,” Phys. Rev. Lett. 106, 118302 (2011). [CrossRef] [PubMed]

], or in disordered jammed materials, where internal stress is known to play a major role [19

19. O. Lieleg, J. Kayser, G. Brambilla, L. Cipelletti, and A. R. Bausch, “Slow dynamics and internal stress relaxation in bundled cytoskeletal networks,” Nature Materials 10, 236–242 (2011). [CrossRef] [PubMed]

, 37

37. J.-P. Bouchaud and E. Pitard, “Anomalous dynamical light scattering in soft glassy gels,” Eur. Phys. J. E 6, 231 (2001). [CrossRef]

] in the sample dynamics.

6. APPENDIX A: Center-of-mass algorithm for determining the direction of shift

In our speckle images the speckle size is comparable to the pixel size. Hence, the peak of corr[J, I] extends over a few pixels at most. Accordingly, we calculate the center of mass of the peak based on the values of corr[J, I] at its maximum, located at (, ), and in the eight neighboring pixels as showed in Fig. 1(b). Our aim is to determine in which of the four quadrants A, B, C, and D shown in Fig. 1(b) lays the center of mass of the correlation peak. To avoid any bias introduced by the square shape and the orientation of the pixels, we adopt a circular symmetry by considering only the contribution of the areas indicated by α1, α2, α3 in Fig. 1(b) (for clarity, only the overlap areas for quadrant B are shown in the figure). This is accomplished by weighting the contribution of each element of corr[J, I] by its overlap with the circle shown in the figure. The weights wA, wB, wC, wD associated with quadrants A, B, C, D respectively are then
wA=α2corr[J,I](k¯1,l¯1)+α1[corr[J,I](k¯1,l¯)+corr[J,I](k¯,l¯1)+α3corr[J,I](k¯,l¯)
(22)
wB=α2corr[J,I](k¯1,l¯+1)+α1[corr[J,I](k¯1,l¯)+corr[J,I](k¯,l¯+1)+α3corr[J,I](k¯,l¯)
(23)
wC=α2corr[J,I](k¯+1,l¯1)+α1[corr[J,I](k¯,l¯1)+corr[J,I](k¯+1,l¯)+α3corr[J,I](k¯,l¯)
(24)
wD=α2corr[J,I](k¯+1,l¯+1)+α1[corr[J,I](k¯,l¯+1)+corr[J,I](k¯+1,l¯)+α3corr[J,I](k¯,l¯),
(25)
with α1 = 0.485869913, α2 = 0.545406041, α3 = 0.25. Once the weights of the four quadrants are determined, the indexes to be used in Eq. (6) and following are calculated from
k1=k2=floor(k¯+δr)
(26)
k3=k4=k1+1
(27)
l1=l3=floor(l¯+δc)
(28)
l2=l4=l1+1,
(29)
where δr and δc are obtained from wA,...,wD:
δr=wC+wDwAwBwA+wB+wC+wD
(30)
δc=wB+wDwAwCwA+wB+wC+wD.
(31)

Acknowledgments

Funding from CNES is gratefully acknowledged. We thank Unilever for partially supporting SM.

References and links

1.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications(Roberts and Company, Englewood, 2007).

2.

R. K. Erf, Speckle Metrology(Academic, 1978).

3.

B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, 1976).

4.

J. D. Briers, “Laser Doppler, speckle and related techniques for blood perfusion mapping and imaging,” Physiological Measurement 22, R35–R66 (2001). [CrossRef]

5.

M. Draijer, E. Hondebrink, T. Leeuwen, and W. Steenbergen, “Review of laser speckle contrast techniques for visualizing tissue perfusion,” Lasers in Medical Science 24, 639 (2009). [CrossRef]

6.

R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: a tool to study time-varying dynamics,” Rev. Sci. Instrum. 76, 093110 (2005). [CrossRef]

7.

R. J. Adrian, “Scattering particle characteristics and their effect on pulsed laser measurements of fluid flow: speckle velocimetry vs particle image velocimetry,” Appl. Opt. 23, 1690 (1984). [CrossRef] [PubMed]

8.

T. D. Dudderar, R. Meynart, and P. G. Simpkins, “Full-field laser metrology for fluid velocity measurement,” Optics and Lasers in Engineering 9, 163 (1988). [CrossRef]

9.

C. E. Willert and M. Gharib, “Digital particle image velocimetry,” Experiments In Fluids 10, 181193 (1991). [CrossRef]

10.

P. T. Tokumaru and P. E. Dimotakis, “Image correlation velocimetry,” Experiments In Fluids 19, 115 (1995). [CrossRef]

11.

A. P. Y. Wong and P. Wiltzius, “Dynamic light-scattering with a CCD camera,” Rev. Sci. Instrum. 64, 2547–2549 (1993). [CrossRef]

12.

S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, and H. Sillescu, “Multispeckle autocorrelation spectroscopy and its application to the investigation of ultraslow dynamical processes,” J. Chem. Phys. 104, 1758–1761 (1996). [CrossRef]

13.

A. Duri, D. A. Sessoms, V. Trappe, and L. Cipelletti, “Resolving long-range spatial correlations in jammed colloidal systems using photon correlation imaging,” Phys. Rev. Lett. 102, 085702 (2009). [CrossRef] [PubMed]

14.

S. Maccarrone, G. Brambilla, O. Pravaz, A. Duri, M. Ciccotti, J. M. Fromental, E. Pashkovski, A. Lips, D. Sessoms, V. Trappe, and L. Cipelletti, “Ultra-long range correlations of the dynamics of jammed soft matter,” Soft Matter 6, 5514–5522 (2010). [CrossRef]

15.

P. Zakharov and F. Scheffold, “Monitoring spatially heterogeneous dynamics in a drying colloidal thin film,” Soft Matter 8, 102–113 (2010). [CrossRef]

16.

A. Amon, V. B. Nguyen, A. Bruand, J. Crassous, and E. Clement, “Hot spots in an athermal system,” Phys. Rev. Lett. 108, 135502 (2012). [CrossRef] [PubMed]

17.

L. Cipelletti, “Method and device for characterizing the internal dynamics of a sample of material in the presence of a rigid displacement,” Patent WO 2012/076826, 14June2012.

18.

G. Brambilla, S. Buzzaccaro, R. Piazza, L. Berthier, and L. Cipelletti, “Highly nonlinear dynamics in a slowly sedimenting colloidal gel,” Phys. Rev. Lett. 106, 118302 (2011). [CrossRef] [PubMed]

19.

O. Lieleg, J. Kayser, G. Brambilla, L. Cipelletti, and A. R. Bausch, “Slow dynamics and internal stress relaxation in bundled cytoskeletal networks,” Nature Materials 10, 236–242 (2011). [CrossRef] [PubMed]

20.

W. H. Press and S. A. Teukolsky, Numerical Recipes in C. (Cambridge University Press, 1992).

21.

T. M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” Ieee Transactions on Medical Imaging 18, 1049–1075 (1999). [CrossRef]

22.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier-transform,” Proc. IEEE 66, 5183 (1978). [CrossRef]

23.

D. El Masri, M. Pierno, L. Berthier, and L. Cipelletti, “Aging and ultra-slow equilibration in concentrated colloidal hard spheres,” J. Phys.: Condens. Matter 17, S3543 (2005). [CrossRef]

24.

A. Duri, H. Bissig, V. Trappe, and L. Cipelletti, “Time-resolved-correlation measurements of temporally heterogeneous dynamics,” Phys. Rev. E 72, 051401 (2005). [CrossRef]

25.

G. J. Tearney and E. B. Bouma, “Optical methods and systems for tissue analysis,” Patent US20020183601 A1, 5December2002.

26.

D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: The method of cumulants,” J. Chem. Phys. 57, 4814–4820 (1972). [CrossRef]

27.

T. Asakura and N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” App. Physics 25, 179–194 (1981). [CrossRef]

28.

G. Brambilla, D. El Masri, M. Pierno, L. Berthier, L. Cipelletti, G. Petekidis, and A. B. Schofield, “Probing the equilibrium dynamics of colloidal hard spheres above the mode-coupling glass transition,” Phys. Rev. Lett. 102, 085703 (2009). [CrossRef] [PubMed]

29.

P. N. Pusey and W. van Megen, “Observation of a glass-transition in suspensions of spherical colloidal particles,” Phys. Rev. Lett. 59, 2083 (1987). [CrossRef] [PubMed]

30.

V. Chikkadi, G. Wegdam, D. Bonn, B. Nienhuis, and P. Schall, “Long-range strain correlations in sheared colloidal glasses,” Phys. Rev. Lett. 107, 198303 (2011). [CrossRef] [PubMed]

31.

T. Divoux, D. Tamarii, C. Barentin, and S. Manneville, “Transient shear banding in a simple yield stress fluid,” Phys. Rev. Lett. 104, 208301 (2010). [CrossRef] [PubMed]

32.

P. Schall, D. A. Weitz, and F. Spaepen, “Structural rearrangements that govern flow in colloidal glasses,” Science 318, 1895 (2007).

33.

P. Jop, V. Mansard, P. Chaudhuri, L. Bocquet, and A. Colin, “Microscale rheology of a soft glassy material close to yielding,” Phys. Rev. Lett. 108, 148301 (2012) [CrossRef] [PubMed]

34.

R. Cerbino and A. Vailati, “Near-field scattering techniques: Novel instrumentation and results from time and spatially resolved investigations of soft matter systems,” Current Opinion in Colloid & Interface Science 14, 416 (2009). [CrossRef]

35.

F. Giavazzi, D. Brogioli, V. Trappe, T. Bellini, and R. Cerbino, “Scattering information obtained by optical microscopy: Differential dynamic microscopy and beyond,” Phys. Rev. E 80, 031403 (2009). [CrossRef]

36.

S. Buzzaccaro, E. Secchi, and R. Piazza, “Ghost particle velocimetry: accurate 3D flow visualization using standard lab equipment,” Phys. Rev. Lett. 111048101 (2013). [CrossRef] [PubMed]

37.

J.-P. Bouchaud and E. Pitard, “Anomalous dynamical light scattering in soft glassy gels,” Eur. Phys. J. E 6, 231 (2001). [CrossRef]

OCIS Codes
(100.0100) Image processing : Image processing
(120.6150) Instrumentation, measurement, and metrology : Speckle imaging

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: June 18, 2013
Revised Manuscript: August 22, 2013
Manuscript Accepted: September 3, 2013
Published: September 16, 2013

Virtual Issues
Vol. 8, Iss. 10 Virtual Journal for Biomedical Optics

Citation
L. Cipelletti, G. Brambilla, S. Maccarrone, and S. Caroff, "Simultaneous measurement of the microscopic dynamics and the mesoscopic displacement field in soft systems by speckle imaging," Opt. Express 21, 22353-22366 (2013)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-19-22353


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References

  1. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications(Roberts and Company, Englewood, 2007).
  2. R. K. Erf, Speckle Metrology(Academic, 1978).
  3. B. J. Berne and R. Pecora, Dynamic Light Scattering (Wiley, 1976).
  4. J. D. Briers, “Laser Doppler, speckle and related techniques for blood perfusion mapping and imaging,” Physiological Measurement22, R35–R66 (2001). [CrossRef]
  5. M. Draijer, E. Hondebrink, T. Leeuwen, and W. Steenbergen, “Review of laser speckle contrast techniques for visualizing tissue perfusion,” Lasers in Medical Science24, 639 (2009). [CrossRef]
  6. R. Bandyopadhyay, A. S. Gittings, S. S. Suh, P. K. Dixon, and D. J. Durian, “Speckle-visibility spectroscopy: a tool to study time-varying dynamics,” Rev. Sci. Instrum.76, 093110 (2005). [CrossRef]
  7. R. J. Adrian, “Scattering particle characteristics and their effect on pulsed laser measurements of fluid flow: speckle velocimetry vs particle image velocimetry,” Appl. Opt.23, 1690 (1984). [CrossRef] [PubMed]
  8. T. D. Dudderar, R. Meynart, and P. G. Simpkins, “Full-field laser metrology for fluid velocity measurement,” Optics and Lasers in Engineering9, 163 (1988). [CrossRef]
  9. C. E. Willert and M. Gharib, “Digital particle image velocimetry,” Experiments In Fluids10, 181193 (1991). [CrossRef]
  10. P. T. Tokumaru and P. E. Dimotakis, “Image correlation velocimetry,” Experiments In Fluids19, 115 (1995). [CrossRef]
  11. A. P. Y. Wong and P. Wiltzius, “Dynamic light-scattering with a CCD camera,” Rev. Sci. Instrum.64, 2547–2549 (1993). [CrossRef]
  12. S. Kirsch, V. Frenz, W. Schartl, E. Bartsch, and H. Sillescu, “Multispeckle autocorrelation spectroscopy and its application to the investigation of ultraslow dynamical processes,” J. Chem. Phys.104, 1758–1761 (1996). [CrossRef]
  13. A. Duri, D. A. Sessoms, V. Trappe, and L. Cipelletti, “Resolving long-range spatial correlations in jammed colloidal systems using photon correlation imaging,” Phys. Rev. Lett.102, 085702 (2009). [CrossRef] [PubMed]
  14. S. Maccarrone, G. Brambilla, O. Pravaz, A. Duri, M. Ciccotti, J. M. Fromental, E. Pashkovski, A. Lips, D. Sessoms, V. Trappe, and L. Cipelletti, “Ultra-long range correlations of the dynamics of jammed soft matter,” Soft Matter6, 5514–5522 (2010). [CrossRef]
  15. P. Zakharov and F. Scheffold, “Monitoring spatially heterogeneous dynamics in a drying colloidal thin film,” Soft Matter8, 102–113 (2010). [CrossRef]
  16. A. Amon, V. B. Nguyen, A. Bruand, J. Crassous, and E. Clement, “Hot spots in an athermal system,” Phys. Rev. Lett.108, 135502 (2012). [CrossRef] [PubMed]
  17. L. Cipelletti, “Method and device for characterizing the internal dynamics of a sample of material in the presence of a rigid displacement,” Patent WO 2012/076826, 14June2012.
  18. G. Brambilla, S. Buzzaccaro, R. Piazza, L. Berthier, and L. Cipelletti, “Highly nonlinear dynamics in a slowly sedimenting colloidal gel,” Phys. Rev. Lett.106, 118302 (2011). [CrossRef] [PubMed]
  19. O. Lieleg, J. Kayser, G. Brambilla, L. Cipelletti, and A. R. Bausch, “Slow dynamics and internal stress relaxation in bundled cytoskeletal networks,” Nature Materials10, 236–242 (2011). [CrossRef] [PubMed]
  20. W. H. Press and S. A. Teukolsky, Numerical Recipes in C. (Cambridge University Press, 1992).
  21. T. M. Lehmann, C. Gonner, and K. Spitzer, “Survey: interpolation methods in medical image processing,” Ieee Transactions on Medical Imaging18, 1049–1075 (1999). [CrossRef]
  22. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier-transform,” Proc. IEEE66, 5183 (1978). [CrossRef]
  23. D. El Masri, M. Pierno, L. Berthier, and L. Cipelletti, “Aging and ultra-slow equilibration in concentrated colloidal hard spheres,” J. Phys.: Condens. Matter17, S3543 (2005). [CrossRef]
  24. A. Duri, H. Bissig, V. Trappe, and L. Cipelletti, “Time-resolved-correlation measurements of temporally heterogeneous dynamics,” Phys. Rev. E72, 051401 (2005). [CrossRef]
  25. G. J. Tearney and E. B. Bouma, “Optical methods and systems for tissue analysis,” Patent US20020183601 A1, 5December2002.
  26. D. E. Koppel, “Analysis of macromolecular polydispersity in intensity correlation spectroscopy: The method of cumulants,” J. Chem. Phys.57, 4814–4820 (1972). [CrossRef]
  27. T. Asakura and N. Takai, “Dynamic laser speckles and their application to velocity measurements of the diffuse object,” App. Physics25, 179–194 (1981). [CrossRef]
  28. G. Brambilla, D. El Masri, M. Pierno, L. Berthier, L. Cipelletti, G. Petekidis, and A. B. Schofield, “Probing the equilibrium dynamics of colloidal hard spheres above the mode-coupling glass transition,” Phys. Rev. Lett.102, 085703 (2009). [CrossRef] [PubMed]
  29. P. N. Pusey and W. van Megen, “Observation of a glass-transition in suspensions of spherical colloidal particles,” Phys. Rev. Lett.59, 2083 (1987). [CrossRef] [PubMed]
  30. V. Chikkadi, G. Wegdam, D. Bonn, B. Nienhuis, and P. Schall, “Long-range strain correlations in sheared colloidal glasses,” Phys. Rev. Lett.107, 198303 (2011). [CrossRef] [PubMed]
  31. T. Divoux, D. Tamarii, C. Barentin, and S. Manneville, “Transient shear banding in a simple yield stress fluid,” Phys. Rev. Lett.104, 208301 (2010). [CrossRef] [PubMed]
  32. P. Schall, D. A. Weitz, and F. Spaepen, “Structural rearrangements that govern flow in colloidal glasses,” Science318, 1895 (2007).
  33. P. Jop, V. Mansard, P. Chaudhuri, L. Bocquet, and A. Colin, “Microscale rheology of a soft glassy material close to yielding,” Phys. Rev. Lett.108, 148301 (2012) [CrossRef] [PubMed]
  34. R. Cerbino and A. Vailati, “Near-field scattering techniques: Novel instrumentation and results from time and spatially resolved investigations of soft matter systems,” Current Opinion in Colloid & Interface Science14, 416 (2009). [CrossRef]
  35. F. Giavazzi, D. Brogioli, V. Trappe, T. Bellini, and R. Cerbino, “Scattering information obtained by optical microscopy: Differential dynamic microscopy and beyond,” Phys. Rev. E80, 031403 (2009). [CrossRef]
  36. S. Buzzaccaro, E. Secchi, and R. Piazza, “Ghost particle velocimetry: accurate 3D flow visualization using standard lab equipment,” Phys. Rev. Lett.111048101 (2013). [CrossRef] [PubMed]
  37. J.-P. Bouchaud and E. Pitard, “Anomalous dynamical light scattering in soft glassy gels,” Eur. Phys. J. E6, 231 (2001). [CrossRef]

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