A novel setup coupling space-resolved dynamic light scattering and rheometetry unveils the heterogeneous flow field and non-affine dynamics in startup shear of a gel
A. Pommella, A.-M. Philippe, T. Phou, L. Ramos, L. Cipelletti
AA novel setup coupling space-resolved dynamic light scattering and rheometetryunveils the heterogeneous flow field and non-affine dynamics in startup shear of a gel
A. Pommella, ∗ A.-M. Philippe, T. Phou, L. Ramos, and L. Cipelletti
L2C, Univ Montpellier, CNRS, Montpellier, France. (Dated: September 5, 2018)We present a new light scattering setup coupled to a commercial rheometer operated in the plate-plate geometry. The apparatus allows the microscopic dynamics to be measured, discriminatingbetween the contribution due to the affine deformation and additional mechanisms, such as plasticity.Light backscattered by the sample is collected using an imaging optical layout, thereby allowing theaverage flow velocity and the microscopic dynamics to be probed with both spatial and temporalresolution. We successfully test the setup by measuring the Brownian diffusion and flow velocity ofdiluted colloidal suspensions, both at rest and under shear. The potentiality of the apparatus areexplored in the startup shear of a biogel. For small shear deformations, γ ≤ I. INTRODUCTION
Soft matter is characterized by a complex structureon the nanometer to micron scale, which results in awide range of dynamical properties and mechanical be-havior. This richness is of great academic interest andhas far-reaching implications in industrial applications,e.g. in the food, cosmetics, home care, pharmaceuticaland packaging industry. Products such as ice creams,gelatin, toothpaste, skin-care creams, detergents, inksand plastics heavily rely on the remarkable properties offoams, emulsions, suspensions, gels, surfactant solutionsand polymers.Rheology is an important characterization tool for softmatter [1–3] and biological materials [4, 5], extensivelyused not only to quantify the mechanical response ofa system, but also to gain insight on its structure anddynamics. In the last years, there has been a growinginterest in coupling rheology to techniques that probethe material structure and dynamics at a microscopiclevel, such as microscopy and a wide range of scatter-ing methods, from static and dynamic light scattering,diffusing wave spectroscopy, to neutron and X-ray scat-tering, including X photon correlation spectroscopy [6–9].The importance of combining a macroscopic rheologicalinvestigation with microscopic measurements has beendemonstrated in a wide range of problems, from the dy-namics of foams [10, 11], the behavior of colloidal crystalsunder shear [12–16], the heterogeneous flow of wormlikemicelles [17–20], to the non-affine deformation of poly-mer gels and glasses [21, 22] and the creep and yieldingof dense emulsions and colloidal suspensions [4, 23–34].Simultaneous rheological and microscopic measurementsare particularly valuable, since the detailed behavior ofcomplex systems may vary from run to run, especially inthe non-linear regime, which is usually the most inter- ∗ [email protected] esting one. A prototypical example is the wide distribu-tion of breaking times of soft solids loaded at constantstress [35].Although microscopy is unsurpassed in its abilityto follow single-particle trajectories, scattering methodshave several advantages, which explain their lasting pop-ularity. Scattering experiments are not restricted tospecifically tailored particles, as required for real-spaceparticle tracking; they can deal more easily with turbidsamples; they afford a larger sample size and thus betterstatistics; they allow the sample structure and dynamicsto be probed over a wider range of length and time scales.The first experiments coupling scattering and rheologyfocussed on the structure, often probed by small-anglescattering apparatuses that can be relatively easily cou-pled to a rheometer. Visible light, X-ray radiation andneutrons have been used, leading to the well-establishedfields of Rheo-SALS [36–47], Rheo-SAXS [12, 15, 25, 48–55], and Rheo-SANS [14, 17, 18, 20, 56–61], respectively.Dynamic scattering methods coupled to rheology haveincreasingly become important: various scattering setupshave been coupled to commercial rheometers [38, 41, 44,62–64] or custom-made devices [22, 65, 66]. We shalldivide dynamic scattering methods in single-scatteringand multiple-scattering techniques. The former includedynamic light scattering [6] (DLS), for visible light, andX-photon correlation spectroscopy [8] (XPCS), for coher-ent X rays. Diffusing wave spectroscopy [7] (DWS), bycontrast, operates in the opposite limit of strong mul-tiple scattering, using laser light as a source. In thecontext of scattering experiments coupled to rheology,the key difference between single and multiple scatteringmethods is the fact that the former probe the motion ofthe scatterers projected on a well-defined direction, whileDWS probes displacements isotropically. Indeed, singlescattering is only sensitive to the component of the dis-placement along the direction of the scattering vector q = k sc − k in , with k sc , k in the wave vector of the scat-tered and incoming radiation, respectively. Therefore,single scattering experiments can be designed so as to a r X i v : . [ c ond - m a t . s o f t ] S e p probe motion in a specific direction with respect to therelevant direction set by the rheology measurement, e.g.parallel or perpendicular to the flow. This is not possiblewith DWS, because multiply scattered photons undergoa random walk in the sample, thus propagating in alldirections with equal probability.If the sample is deformed during the measurement, thisdistinction is crucial: DWS will be sensitive both to the‘ideal’ part of the displacement field (e.g. the affine de-formation field in an ideal solid or the laminar flow fieldin an ideal fluid), as well as to any additional micro-scopic dynamics, e.g. due to Brownian motion, reversiblenon-affine response [67, 68], or irreversible, plastic rear-rangements. In DWS, disentangling the ideal contribu-tion to that associated to additional (and usually moreinteresting) dynamics is generally quite difficult [22, 65],except in the simplest cases where both the structure ofthe flow field and the origin of the additional dynamicsare known a priori , e.g. for diluted Brownian suspensionsunder laminar shear flow [69]. For this reasons, DWShas been mostly restricted to rheology tests where thesample is macroscopically undeformed, e.g. in stress re-laxation [70, 71], or to ‘echo’ experiments that probe stro-boscopically the sample under a sinusoidal drive, moni-toring the microscopic evolution between states that cor-respond to the same macroscopic deformation [23, 24].In single scattering experiments, by contrast, these dif-ficulties can be avoided by orienting q perpendicular tothe direction of the deformation field, such that the ex-periment is only sensitive to deviations of the microscopicdynamics with respect to the behavior of ideal solids orfluids. This approach has been implemented in bothXPCS and DLS. In XPCS, it has been used to measurethe Brownian dynamics of colloids pumped in a capil-lary [72], a scheme aiming at minimizing the sample ex-posure to the intense X-ray radiation, which often causesradiation damage. XPCS has also been used to investi-gate the yielding transition of colloidal gels submitted toan oscillatory shear deformation [73, 74], where the sin-gle scattering geometry allowed the authors to study thedynamics in the directions parallel and perpendicular tothe flow direction and within each cycle, not just strobo-scopically. The ability of resolving non-affine microscopicdynamics thanks to single scattering is also at the coreof the small-angle DLS setup described in Refs. [66, 75],which allowed the microscopic precursors of the macro-scopic failure of a colloidal gel to be unveiled in creepexperiments [34].Both DWS and single scattering methods are usuallyimplemented in the far field, homodyne geometry, wherethe detector is placed very far from the sample (or, equiv-alently, in the focal plane of a lens collecting the scatteredlight), such that only scattered light is detected. Underthese conditions, dynamic scattering methods are sensi-tive to the relative motion of the scatterers, not to theiraverage displacement. This degeneracy is removed in het-erodyne experiments, where a static reference beam illu-minates the detector, together with the scattered light. The beating between the reference beam and the scat-tered light causes distinctive oscillations in the measuredintensity correlation function, whose time scale is directlyrelated to the scatterers’ average velocity. Heterodynescattering setups coupled to rheology have been imple-mented both in XPCS [76, 77], for studying the relax-ation of nanocomposites, and in DLS [78], to measurethe velocity profile in Couette flow.An alternatively way to measure the (local) flow ve-locity in rheology experiments has been recently demon-strated [63], using space-resolved DLS [79, 80], alsoknown as photon correlation imaging (PCI). In the setupof Ref. [63], a 2D detector makes an image of the sam-ple confined in the gap of a Couette cell driven by acommercial rheometer and illuminated by a laser sheet.The image is formed by light scattered at a well definedangle θ = 90 ◦ , such that the setup combines featuresof a traditional scattering experiment and of imaging.The dynamics are observed in the flow-velocity gradientplane. For a sample undergoing a macroscopic deforma-tion, the flow velocity can be measured. Additionally, therelative motion between scatterers can be quantified, butthe microscopic dynamics are typically dominated by thecontribution of the affine displacement, because this ge-ometry does not allow the scattering vector to be orientedperpendicular to the flow direction. Thus, this apparatusis best suited for characterizing the local flow profile, forsamples undergoing a macroscopic deformation, or, al-ternatively, for measuring the microscopic dynamics forsamples macroscopically at rest, e.g. in echo or stressrelaxation experiments.In this paper, we introduce a novel, custom-made dy-namic light scattering setup coupled to a commercialrheometer, allowing for the investigation of the micro-scopic dynamics of driven samples. The setup works in aback-scattering configuration and uses the PCI methodto image the flow-vorticity plane in a plane-plane rheo-logical configuration. This allows one to obtain time- andspace-resolved information of the sample internal dynam-ics, decoupling the affine and non-affine contributions tothe microscopic dynamics. Additionally, the local flowvelocity, averaged over the rheometer gap, can be mea-sured.The rest of the paper is organized as follows: in Sec. IIwe present the experimental setup. In Sec. III we testit on a simple Newtonian fluid, a diluted Brownian sus-pension, showing that its microscopic dynamics are cor-rectly captured both at rest and under shear. The localshear rate obtained by optical measurements is success-fully compared to that imposed by the rheometer. InSec. IV we investigate the microscopic dynamics and themesoscopic deformation of a biogel during shear startup,as an example of the potentiality of the setup. SectionV concludes the paper with a few final remarks. II. EXPERIMENTAL SETUP AND IMAGEPROCESSINGA. Setup
Figure 1 illustrates the experimental setup. A commer-cial stress-controlled rheometer (Anton Paar MCR 502)is coupled to a custom-made wide-angle light scattering(WALS) apparatus. The rheometer is equipped with aplate-plate geometry with temperature control. The bot-tom plate is fixed and made of float glass, to let pass theilluminating laser beam. The top plate, with radius 25mm, is the rotating one. As discussed in the following,it is vital to minimize any backreflection from the upperplate. To this end, we glue an absorptive neutral density(ND) filter (Absorptive ND Filter by Edmund Optics,optical density 3.0, diameter 50 mm, thickness 3 mm) tothe steel plate (Anton Paar part PP-50) of the rheometer.To allow for an easy replacement of the filter (e.g. whenthe optical quality of its surface is degraded by use), weuse fused saccharose as a glue. The steel plate and theND filter are placed in an oven at T = 100 ◦ C. Thesugar is mixed with a small amount of water and heatedso as to obtain a light brown caramel with low viscosity,at T ≈ ◦ C. The caramel is poured on the hot plateand the ND filter is glued to the plate. Excess caramelis removed by gently rotating the ND filter. After cool-ing, a thin layer of nail polish is applied on the rim ofthe ND-plate assembly, to prevent water to penetrate inthe solidified caramel when cleaning the ND filter. Tounglue the filter, the nail polish is removed using acetoneand the tool is immersed in hot water.In the plane-plane geometry used here, the shear de-formation γ and the shear rate ˙ γ vary linearly from zero(at the plate center) to a maximum value (at its edge).For a point at distance r from the rotation axis, one has γ = Θ r/H and ˙ γ = ωr/H , respectively, with H the gapbetween the plates, Θ the angular rotation of the upperplate, and ω = d Θ /dt its angular velocity. Similarly, theshear stress σ is not uniform, but rather varies radiallyas σ = 2 T /πr , where T is the torque. Note that, forhighly scattering samples (DWS regime), backscatteringfrom the upper tool is not an issue: in this case, one cansafely use a conventional plane (glass or metal), or evena cone and plane geometry, which insures constant γ , ˙ γ and σ over the whole sample.The WALS setup can be divided in two parts. Thefirst part is composed of a laser source and optical el-ements that shape the illuminating beam. The laseris a single frequency CW diode-pumped surce (CoboltSamba T M λ = 532 nm and a max-imum power of 150 mW. The collimated beam exitingthe laser (1 /e diameter = 0.7 mm) is first expanded andthen focused on the first mirror (M in Fig. 1) by a beammodifier module, consisting of two diverging lenses (thefirst plano-concave with focal length -12.5 mm and andthe second bi-concave with focal length -6.3 mm, both with a diameter of 6.35 mm) and a converging plano-convex lens (diameter 25.4 mm, focal length 62.9 mm).The horizontally-propagating beam is reflected towardsthe sample by the broadband dielectric mirror M , tiltedby 35 ◦ with respect to the horizontal plane. Before reach-ing the sample, the beam passes through the convex lensL that allows the size of the illuminated sample to becontrolled. Two different bi-convex lenses are used as L .A lens with focal length f = 38.1 mm and diameter 25.4mm is used to illuminate almost the entire 50 mm diame-ter sample (see the left inset of Fig. 1), while a lens with f = 100 mm and diameter 50.8 mm allows a smallerregion, of diameter 10-30 mm to be illuminated, see theright inset of Fig. 1.The second part of the WALS setup forms an imageof the illuminated sample onto the sensor of a CMOScamera (acA2000-340km by Basler AG), using the back-scattered light. Note that the directions of the incidentand back-scattering light are chosen such as to avoid col-lecting the specular reflection from the rheometer plates.The sensor has a matrix of 2048 × µ m. A PC equipped with a frame grabber(Solios eV-CLF by Matrox) is used to control the cameraand acquire images, through a custom-written software.A laser line filter (F) is placed in front of the camerato reduce the contributions due to ambient light. Thebackscattered light is first collected by a second broad-band dielectric mirror (M ), placed below the sample andoriented at 45 ◦ with respect to the horizontal plane. Theconvex lens L is used to form the image of the sample onthe camera. Two different lenses L are used, dependingon the size of the illuminated region. When the entiresample is illuminated ( f = 38.1 mm), a plano-convexlens with focal length f = 85 mm and diameter 50.8mm is used, yielding a magnification M = 0 . f = 100mm), we use a bi-convex lens with f = 200 mm anddiameter 25.4 mm, corresponding to M = 1 . : its aperture controls the speckle size, which is cho-sen to be on the order of the pixels size [81]. The wholeWALS setup is placed under a box made of black paper,to minimize temperature fluctuations and to protect thesetup from ambient light.In the imaging configuration used here, a given speckleof size s on the sensor results from the interference of lightissued from a small sample volume, of later size s/M anddepth equal to the gap H . By calculating intensity cor-relation functions averaged on small subsets of the image(regions of interest, ROIs), one can then measure thelocal dynamics and check for any spatial heterogeneity.The WALS setup is characterized by an average scat-tering angle θ = 170 ± ◦ corresponding to a scatteringvector q = 4 πnλ − sin( θ/ ≈ µ m − for water-basedsamples, with refractive index n = 1 .
33. Note that q varies slightly with the location in the sample, becauseboth the illuminating beam and the collected backscat-tered light form an x − and y -dependent angle with the z CameraPCLaser Beamshaper xy Topview xy TopviewTop plateBottom plateSample xz y FDPGM M L L ND Filter
FIG. 1. Scheme of the wide-angle light scattering setup coupled to a rheometer. The incoming laser beam (gray lines) isexpanded by a beam shaper. The mirror M sends the beam to the sample, which is confined between the bottom plate ofthe rheometer and a ND filter glued to the top plate. The lens L controls the size of the illuminated sample. The mirror M collects the light back-scattered from the sample (red and blue lines), sending it to the camera. The lens L makes an imageof the sample on the camera sensor. The iris diaphragm DPG, placed in the focal plane of L , controls the speckle size. Thelaser-line filter F cuts ambient light. Images are acquired and processed by the personal computer PC. The optical elementsare not to scale. The two insets show as grey disks the size and location with respect to the rheometer plates (white circle) ofthe largest (left) and smallest (right) illuminated sample region, depending on the choice and positioning of L . Throughoutthis paper, we use a reference system with the ( x, y ) plane at the interface between the bottom plate and the sample and thetool rotation axis as the z axis, oriented upward. axis (see Fig. 1). We shall discuss later this dependence;however, we anticipate that in most experiments it hasno significant impact on the data analysis. B. Intensity correlation functions
Simultaneously to the rheology experiments, a time se-ries of speckle images is acquired either at a constant rate,typically 1 to 25 Hz, or using the time-varying schemeof Ref. [82]. The images are saved on a hard disk forsubsequent processing using the Time Resolved Correla-tion [83] method, yielding two-time intensity correlationfunctions. In brief, the images are corrected for the un-even illumination and dark background as explained inRef. [84] and a local, two-time degree of correlation c I iscalculated according to c I ( t, τ, r ) = B (cid:104) I p ( t ) I p ( t + τ ) (cid:105) r (cid:104) I p ( t ) (cid:105) r (cid:104) I p ( t + τ ) (cid:105) r . (1)Here, B > ∼ c I ( τ →
0) = 1, I p ( t ) is the time-dependent intensitymeasured by the p -th pixel, τ a time delay, and (cid:104)· · · (cid:105) r is an average over the pixels belonging to a ROI cen-tered around the position r = ( x, y ) (see Fig. 1 for thechoice of the reference system). For stationary dynam-ics, the previous expression may be averaged over time to improve statistics, yielding the intensity correlationfunction g − g ( τ, r ) − (cid:28) (cid:104) I p ( t ) I p ( t + τ ) (cid:105) r (cid:104) I p ( t ) (cid:105) r (cid:104) I p ( t + τ ) (cid:105) r (cid:29) t , (2)with (cid:104)· · · (cid:105) t the average over time. The intensity correla-tion function is directly related to the intermediate scat-tering function f ( q , τ ) (ISF) by the Siegert relation [6], g − f . The ISF quantifies the microscopic dynamicsprojected onto the scattering vector: f ( q , τ ) = (cid:68)(cid:80) j,k exp[ − i q · ( r j (0) − r k ( τ ))] (cid:69)(cid:68)(cid:80) j,k exp[ − i q · ( r j (0) − r k (0))] (cid:69) , (3)where the double sum runs over all particles in thescattering volume associated to the analyzed ROI andthe brackets indicate an ensemble average. In order toachieve sufficient statistics, the ROIs must contain a largeenough number of speckles: depending on M , the min-imum lateral size of the associated sample volume typ-ically ranges from 185 µ m to about 5 mm, setting thelevel of coarse graining with which local dynamics aremeasurable. g - (s) Quartz plateStainless steel plate with ND filter τ FIG. 2. Intensity correlation functions for a diluted sus-pension of Brownian particles (2 R = 190 nm) at rest. Blacksquares and red circles are data obtained using as the toprheometer tool a transparent quartz plate or an absorptiveND filter glued to a stainless steel plate, respectively. Thelines are fits to the data with a double exponential decay,Eq. 4. Two distinct modes are seen with the transparentplate, while the expected single exponential decay is recov-ered with the absorptive filter, as discussed in the text. III. NEWTONIAN SAMPLEA. Brownian suspension at rest
We first test our setup on a diluted suspension of col-loidal particles, for which we expect Brownian dynamicsat rest and Newtonian rheological behavior under shear.The suspending solvent is a mixture of 40% (v/v) MilliQwater and 60% (v/v) polyoxyalkylene glycol (EmkaroxHV 45-LQ-CQ, Croda Chocques SAS), for which we mea-sure a Newtonian viscosity η = 5.5 Pa s. Two kinds ofparticles were used, at a volume fraction of 0.004%, soas to be in the single scattering regime. Polystyrene par-ticles with diameter 2 R = 105 ± R = 190 ± g − R = 190 nm, measured using a transparent quartz up-per plate (PP43/GL-HT by Anton Paar). Surprisingly,the correlation function exhibits a two-step decay. Thisis in stark contrast with expectations for a diluted Brow-nian suspension, for which [6] f ( q, τ ) = exp( − Dq τ ),with D = k B T / (6 πηR ) the particle diffusion coefficient, k B Boltzmann’s constant and T the absolute tempera-ture. Tests with different kinds of particles and upperplates suggest that the slower relaxation mode may be t M M n = 1.55Float glass n = 1.46AirAirSample n = 1.41 θ t θ r θ t θ t θ t θ i θ bs θ fs θ r θ FIG. 3. Scheme of the incoming beam impinging on the bot-tom plate, passing through the sample and hitting the upperplate. The mirror M sends upward the laser beam. The mir-ror M collects light backscattered by particles illuminatedby the up-propagating beam (in grey), as well as light scat-tered forward by particles illuminated by the blue and redbeams, which originate from the back reflection of the in-coming beam at the sample-upper plate and upper plate-airinterfaces. The scheme is not to scale: θ fs is essentially thesame for the beams shown in blue and red. The values ofthe various angles shown in the figure are: θ i = 10 ± ◦ , θ t =6.8 ± ◦ , θ t = θ t = 7 . ± ◦ , θ t = 6.4 ± ◦ , θ t = 10 ± ◦ , θ r = 7.1 ± ◦ , θ r = 6.4 ± ◦ , θ bs = 170 ± ◦ , θ fs = 10 ± ◦ . due to the (partial) reflection of the incoming beam atthe sample-upper plate and upper plate-air interfaces (seeFig. 3). These reflected beams propagate back in thesample, illuminating the particles. As a consequence,the CMOS camera receives both light backscattered atan angle θ bs (from particles illuminated by the upwardpropagating incident beam, shown in grey in Fig. 3) andlight forward-scattered at an angle θ fs (from particles il-luminated by the downward propagating back-reflectedred and blue beams of Fig. 3). Since the reflection coef-ficient is of the order of a few percents, the intensity ofthe downward propagating back-reflected beam is muchsmaller than that of the primary incoming beam. Onemight then think that the forward scattering contributionshould be negligible. However, colloidal particles withdiameter of a hundred of nm or more scatter light muchmore efficiently in the forward direction than in backscat-tering, thereby compensating for the smaller power of thedown-propagating illuminating beam.To quantitatively test this hypothesis, we model the in-tensity correlation function as the result of two indepen-dent contributions, associated with backscattering andforward scattering from Brownian particles: g ( τ ) − A bs exp( − Dq bs τ )+ A fs exp( − Dq fs τ )] , (4)where the bs and f s indexes refer to backscattering andforward scattering, respectively, q bs and q fs are the scat-tering vectors associated to θ bs and θ fs , and A bs and A fs are the relative weights of the two contributions, with A bs + A fs = 1. The relative weight A α , with α ∈ { bs, f s } ,is proportional to the scattered intensity of that mode,hence to the intensity of the illuminating beam times theparticle form factor P ( q α ). Accordingly, the ratio of themode weights reads A fs A bs = P ( q fs ) (cid:2) R s − qp + (1 − R s − qp ) R q − ap (1 − R q − sp ) (cid:3) P ( q bs ) , (5)where we indicate by R αp = | n cos θ t − n cos θ i n cos θ t + n cos θ i | the powerreflectivity of light impinging on the interface betweentwo media of refractive indexes n , n , with θ i and θ t theangles of the incident and transmitted rays to the normalof the interface [85]. In Eq. 5, the subscript p indicatesthat in our experiment the polarization is that of the p wave [85], while the superscript α indicates the interfaces: α ≡ s − q refers to the sample-quartz plate interface andsimilarly for the other superscripts, a standing for air.The first term in the brackets of the r.h.s. of Eq. 5 is thecontribution due to the light reflected at the s − q interface(blue beam in Fig. 3), while the product of the threesubsequent factors accounts for light that penetrates inthe quartz, is reflected at the q − a interface, and is finallytransmitted through the q − s interface back into thesample (red beam in Fig. 3).We fit the data taken with the quartz upper plate withthe double exponential decay of Eq. 4, using t bs,fs ≡ /Dq bs,fs and A bs,fs (with A bs + A fs = 1) as fittingparameters. As shown by the red solid line in Fig. 2,this expression reproduces very well the data. Fromthe measured values of the viscosity and T and using2 R = 190 nm as provided by the manufacturer, we calcu-late D = 3 . × − µ m s − . Using the setup geometri-cal parameters and the sample refractive index n s = 1 . q bs = 33 . ± . µ m − , which combined with D yields an expected relaxation time t bs = 2 . ± .
01 s,in very good agreement with t bs = 2 . t fs = 233 . D , we obtain q fs = 3 . µ m − , implying θ fs = 11 . ◦ , in good agreement with θ fs = 10 ◦ ± ◦ ascalculated form the setup geometry. The relative am-plitude of the two modes is also in fair agreement withthe predictions of our simple model: from the fit weobtain A fs /A bs = 0 .
53, to be compared to 0 . ± . P ( q fs ) /P ( q bs ) = 13 .
23, as obtained from Mie scatteringtheory [86] using the free package MiePlot [87]. Note that R s − qp = 0 . (cid:28) R a − qp = 4 . n ND = 1 . R s − NDp = 0 . > ∼ (10 − ) , vir-tually eliminating the dominant contribution to forwardscattering.Figure 2 shows as red circles g − A fs set to zero, yellow line) yields al-most indistinguishable results. For the two-modes fit, weimpose t fs = 233 . t bs = 2 . A fs /A bs = 0.04. Thefitted relaxation time is in excellent agreement with theexpected one, while the ratio of the amplitude modes isin fair agreement with A fs /A bs = 0 . R s − qp by R s − NDp = 0 .
12% and ne-glecting the subsequent terms, due to the strong attenu-ation of the ND filter. Fitting with a single exponentialyields t bs = 2 . B. Sheared Brownian suspension
Having validated the measurement of the microscopicdynamics of a sample at rest, we now discuss the casewhere the sample is sheared at a constant rate ˙ γ , us-ing diluted suspensions of Brownian particles as refer-ence systems to test our theoretical analysis. In general,under shear and in the imaging geometry used here, thespeckle pattern evolves as a result of three different mech-anisms: i) each speckle is advected due to the circular mo-tion imposed by the rheometer; ii) the speckle intensity g - (s) At restROI 1ROI 2ROI 3ROI 4 xyTop view β τ FIG. 4. Symbols: intensity correlation functions measuredin four ROIs, for a sheared suspension of Brownian particleswith D = 5 . × − µ m s − . The ROI positions in the x − y plane are shown in the inset, where the gray circle representsthe portion of the sample illuminated by the laser beam. Inthe main graph, data for the same sample at rest are shownas black squares. The solid lines are the behavior expectedfrom the combination of Brownian motion and affine displace-ments, Eq. 10, with no fit parameters. The dotted lines showthe confidence band of the theoretical g −
1, due to the uncer-tainty on q x and q y . For all ROIs, q = 33 . µ m − , while theparameters used in Eq. 10 to account for affine displacementsare as follows: ROI 1: q x = -3.13 ± µ m − , q y = 0 µ m − , β = 0 ◦ , ˙ γ = 1.36 × − s − . ROI 2: q x = -3.02 ± µ m − , q y = 0.27 ± µ m − , β = 21.5 ◦ , ˙ γ = 0.98 × − s − . ROI3: q x = -3.02 ± µ m − , q y = 0.612 ± µ m − , β = 44 ◦ ,˙ γ = 1.27 × − s − . ROI 4: q x = -3.02 ± µ m − , q y =0.96 ± µ m − , β = 57 ◦ , ˙ γ = 1.68 × − s − . changes due to the relative motion of the scatterers re-sulting from the imposed affine deformation field; iii) thespeckle intensity fluctuates due to any additional sourceof microscopic dynamics, e.g. Brownian motion and anynon-affine displacement induced by the shear. The firstcontribution leads to a decay of g − r from the ro-tation axis, this time scale depends on the speckle size,controlled by the magnification M and the aperture ofthe diaphragm DPG in Fig. 1, and on the average driftvelocity ωr/
2, where the factor of 1 / ω . As we shall discuss itlater, the drift contribution can be corrected for by us-ing mixed spatio-temporal intensity correlation functionsthat probe the dynamics in a reference system co-movingwith the sample midplane [80]. For the sake of simplic-ity, however, we start by considering the case where theaffine deformation and the microscopic dynamics inducea decay of g − R = 105 nm, for four different ROIs, while the sampleis sheared by rotating the upper plate at a constant an-gular velocity ω = 1.26 × − rad/s, with a gap fixed to H = 1 .
05 mm. The case at rest is also reported for com-parison. The location of the ROIs is sketched in the inset,their size is 50 ×
10 pixel , corresponding to 2 . × . in the sample. Clearly, shearing the sample results in afaster decay of g −
1. The acceleration of the dynamicsdepends strongly on the location of the ROI: it is negligi-ble for ROI 1, located on the x axis, while the decay rategrows up to more than a factor of 10 for ROI 4, whichhas the largest y component.In order to understand this behavior, we model thedecay of the intermediate scattering function f , Eq. 3,for a sample under shear. Neglecting for the momentthe contribution due to the average advection [i) above],we assume that the displacements due to the microscopicdynamics and to the affine deformation field are uncor-related, which results in the factorization f ( q , τ ) = f µ ( q , τ ) f aff ( q , τ ) , (6)where the subscripts µ and af f refer to the microscopicdynamics and affine deformation contributions, respec-tively. For a Brownian suspension, the former is thesame as for the unperturbed sample discussed in refer-ence to Fig. 2, f µ ( q , τ ) = exp( − Dq τ ). The latter canbe evaluated following the approach of Ref. [75]. Fora Newtonian suspension, the shear flow is purely affine,with no z component, such that the affine displacementof a particle with coordinates ( x, y, z ) over a time τ is∆ aff r ( τ ) = ωzτ (cid:112) x + y /H .We start by considering particles laying on the x axis;later, we shall generalize our results to particles witharbitrary ( x, y ) coordinates. For particles with y = 0,∆ aff r ( τ ) = ˙ γ ( x, y ) zτ ˆ u y , where we have introduced thelocal shear rate ˙ γ ( x, y ) = ω (cid:112) x + y /H and where ˆ u y isthe unit vector along the y axis. The affine contributionto the intermediate scattering function then reads f aff ( q , τ ) =1 N (cid:42) N (cid:88) j =1 e − i q · ∆ aff r j ( τ ) + N (cid:88) j (cid:54) = l =1 e − i q · [ r j ( t ) − r l ( t + τ )] (cid:43) . (7)If the particle positions are uncorrelated, as in our Brow-nian suspension, the second sum vanishes. By replacingthe average and the first sum by an integral of the expo-nential term weighted by the (flat) probability distribu-tion function of the particle position [6], one finds f aff ( q , τ ) = (cid:90) H e − iq y ˙ γzτ dz = sinc (cid:18) q · ˆ u y ˙ γHτ (cid:19) e i q · ˆ u y ˙ γ H τ . (8)The case of particles with arbitrary ( x, y ) coordinatesis simply obtained from Eq. 8. By replacing ˆ u y by thegeneral expression for the unit vector parallel to the di-rection of the flow, sin β ˆ u x + cos β ˆ u y , where the angle β is defined in Fig. 4, one finds f aff ( q , τ ) = sinc (cid:20) ( q x sin β + q y cos β ) ˙ γHτ (cid:21) × exp (cid:20) i ( q x sin β + q y cos β ) ˙ γ H τ (cid:21) . (9)Equation 9 shows that f aff depends only on the x and y components of q . This is a consequence of the factthat DLS probes particle displacements projected ontothe direction of the scattering vector and that affine dis-placements occur in the ( x, y ) plane. Although in oursetup the z component of q is the largest one (typically, | q z | ≈ µ m − and | q x | , | q y | < ∼ µ m − ), the contribu-tion of motion in the ( x, y ) plane cannot be neglected,due to the coupling with the shear flow.Using the Siegert relationship, Eq. 9, and the factor-ization of Eq. 6, we finally obtain the following expressionfor the intensity correlation function of a sheared Brow-nian suspension: g ( q , ˙ γ ( x, y ) , τ ) − − Dq τ ) × sinc (cid:20) ˙ γ ( x, y ) τ H q x sin β + q y cos β ) (cid:21) . (10)Note that in writing Eq. 10 we have assumed that theshear does not induce any additional dynamics, besidesthat due to the affine flow field. This is justified fordiluted suspensions, but may not hold at higher concen-trations, where hydrodynamic interactions cannot be ne-glected.We now use Eq. 10 to model the experimental correla-tion functions shown as symbols in Fig. 4. To account forthe finite size of the ROIs, Eq. 10 is integrated over the( x, y ) extension of each ROI, using the nominal values ofall parameters. This yields the solid lines in Fig. 4; thedotted lines show the confidence band for the theoretical g − q x and q y . An excellentagreement is found between the data and Eq. 10, withno fitting parameters, thus demonstrating the validity ofour theoretical analysis. For ROI 1, g − q y ≈ β ≈
0, such that theaffine displacement projected onto the scattering vectoralmost vanishes. The slightly faster decay of the data un-der shear with respect to the sample at rest is due to thefinite size of ROI 1 and the finite range of scattering an-gles accepted by the collection optics, which implies that g − q y and sin β values. While for ROI 1 Brown-ian motion is overwhelmingly responsible for the decay of g −
1, for the other ROIs the contribution of affine motionis the dominant one. The data shown in Fig. 4 demon-strate that the contribution of the (usually uninteresting)affine deformation may or may not be relevant, depend-ing on the ROI location and the relative importance of the microscopic dynamics with respect to affine displace-ments. Quite importantly, Eq. 10 allows the decay ratedue to affine motion to be reliably predicted, thus pro-viding a means to identify any additional dynamics, e.g.due to plastic rearrangements. As a final comment, itis worth mentioning that for practical purposes the inte-gration over the ROI may be avoided by using directlyEq. 10 and letting β as a free parameter. The β valuethus obtained represents an ‘effective’ angular positionof the ROI that nicely accounts for its finite size. Forexample, we find that the continuous line calculated byintegration for ROI 1 is virtually indistinguishable (max-imum difference < × − ) from Eq. 10 evaluated forthe center of the ROI and an effective β = 1 . ◦ . Notethat the effective β is intermediate between β = 0 (atthe ROI center) and the maximum value β = 2 . ◦ (atthe ROI top left corner).In Fig. 4 the applied rotation speed was quite low, re-sulting in a modest drift velocity of the speckles, rangingfrom 0.3 pixels/s for ROI 1 to 0.4 pixels/s for ROI 4.Because the affine deformation and Brownian motion in-duce a decay of g − c I ( t, τ, r , ∆ x, ∆ y ) = B (cid:104) I p ( t ) I p, ∆ x, ∆ y ( t + τ ) (cid:105) r (cid:104) I p ( t ) (cid:105) r (cid:104) I p, ∆ x, ∆ y ( t + τ ) (cid:105) r , (11)where the subscripts p, ∆ x, ∆ y indicate a pixel spatiallyshifted by (∆ x, ∆ y ) with respect to the location of pixel p . At fixed τ , the spatio-temporal degree of correla-tion typically exhibits a peak as a function of the spa-tial lag, whose position (∆ x ∗ , ∆ y ∗ ) provides the speckledrift between time t and t + τ . The height of the peak, c ∗ I = c I ( t, τ, r , ∆ x ∗ , ∆ y ∗ ), represents the degree of cor-relation corrected for the effect of drift, e.g. the loss ofcorrelation due only to the relative motion of the scat-terers. For a stationary drift motion, as in the experi-ments reported here, c ∗ I is averaged over time, yieldingan intensity correlation function g − D i s p l a c e m en t ( µ m ) (s) (s) g - No correctionCorrected D i s p l a c e m en t ( p i x e l ) Brownian decay τ τ t (s) ∆ x , ∆ y ( p i x e l ) -0.20.0-0.4 ∆ x* ∆ y* FIG. 5. Main plot: rigid displacement of the speckle pat-tern for a ROI centered on the x axis, for a diluted Brow-nian suspension sheared at a constant rate, as a function ofthe lag τ between pairs of images. The error bars are therms temporal fluctuations of the displacement. The line is alinear fit through the origin, yielding a local drift velocity of40 ± µ m / s. Top-left inset: raw intensity correlation function(black squares) and g − g − x and y components of the ROI displace-ment over a lag τ = 0 .
04 s. The black thick line shows theexpected value for ∆ y ∗ . The bottom-right inset of Fig. 5 shows the x and y components of the speckle drift over a fixed time lag τ = 0 .
04 s, as a function of time t . The sample is a Brow-nian suspension sheared by rotating the upper plate atan angular velocity ω = 6 . × − rad/s, correspond-ing to a tangential velocity of 75 ± µ m/s (20.4 pixel/s)for the ROI used for the data analysis (the ROI posi-tion is similar to that of ROI 1 in Fig. 4). As expectedfor a ROI centered on the x axis, ∆ x ∗ is zero, to withinrandom fluctuations due to the noise of the speckle track-ing algorithm. The y component of the displacement is inexcellent agreement with the value expected from the im-posed rotation speed and the location of the ROI, shownby the horizontal black line. To within the experimen-tal noise, both ∆ x ∗ and ∆ y ∗ are constant, in agreementwith the fact that the imposed shear rate is fixed. Forstationary flow, the displacement can be more accuratelymeasured by averaging over time. The result is shownin the main graph of Fig. 5, for six values of τ . The t -averaged displacement grows linearly with τ , as expectedfor uniform motion. Moreover, a linear fit through theorigin of the data yields a drift velocity of 40 ± µ m / s,in excellent agreement with 38 ± µ m / s, the drift veloc-ity averaged over the gap as measured by the rheometer.This demonstrates the ability of our setup to measurethe local, z -averaged flow velocity, a valuable piece of in-formation in order to detect any deviation from uniform shear, e.g. due to wall slippage or shear banding.The top-left inset of Fig. 5 reports the intensity corre-lation function measured under shear. The black squaresshow g − g − c ∗ I ( t, τ ) over time t . Thedecay of the corrected g − β that is allowed to vary in order to account for theROI finite size in the simple way discussed in reference toFig. 4. Using β = 3 ◦ , an excellent agreement is obtained,showing that the relative motion of the scatterers canbe precisely quantified even when the shear rate is largeenough to make the drift contribution significant. Theinset shows also the theoretical g − D = 1.5 × − µ m s − and q = 33.2 µ m − : consistently withthe large value of the applied shear rate, we find that therelative motion is dominated by the affine contribution,rather than by Brownian diffusion. IV. STARTUP SHEAR OF A POLYMER GEL
To demonstrate the potentiality of our setup, we in-vestigate a more complex system, measuring simulta-neously the microscopic dynamics and the shear stressfor an agarose gel [88] during shear startup. The gel isprepared by mixing the agarose powder (Sigma AldrichA9539-10G, 1% by weight) with MilliQ water at roomtemperature. The solution is then heated and kept at T = 95 ◦ C for 15 minutes to allow for complete agarosedissolution. The solution is poured between the rheome-ter plates, which are pre-heated at 95 ◦ C and spaced by agap H = 1 .
025 mm. Silicon oil is put around the plates toprevent water evaporation and the temperature is cooleddown to 23 ◦ C to form the gel in situ . The gel is weaklyscattering, such that the experiments are performed inthe single scattering regime. We investigate the spon-taneous dynamics of the gel (no applied shear), as wellas the flow-induced dynamics upon imposing a constantshear rate of 5 × − s − .The symbols in Fig. 6a show the intensity correlationfunctions obtained for a ROI with β = 40 ◦ and r =15 . q x = − . µ m − and q y =0 . µ m − . This ROI is chosen on purpose far from the x axis, so as to be able to measure the contribution ofthe affine deformation to the decay of g −
1. We fit the0 g - (s)a)
200 400 6000 t (s) (%) S hea r s t r e ss ( P a ) t < 400 s450 s < t < 600 sAt rest γ τ c I t (s) = 60 s τ b) -1 Affine g -1 Experimental dataAffine decorrelation
FIG. 6. a) Intensity correlation functions for an agarose gelunder various conditions. Black squares: dynamics at rest.Blue triangles and red circles: g − γ = 5 × − s − , and averaged over0 s ≤ t ≤
400 s and 450 s ≤ t ≤
600 s, respectively. The greensolid and dotted lines show the expected decay of g − g − τ = 60 s,as a function of time, with t = 0 the shear startup time. Thegreen lines show the expected value of c I and its confidenceband if only affine displacements were present. The blue solidlines mark the start and end of the linear regime. The dashedlines indicate when the two-time degree of correlation startsto be sensitive to the change of regime indicated by the solidvertical line positioned 60 s later. intensity correlation function with the general form g ( q , ˙ γ, τ ) − (cid:20) ˙ γ ( x, y ) τ H q x sin β + q y cos β ) (cid:21) × exp [ − ( τ /t ) p ] , (12)where the first term of the r.h.s. accounts for the contri- bution of the affine deformation, as in Eq. 10, while thesecond one is a conveniently simple form describing anyother source of microscopic dynamics. The black squaresare the dynamics at rest: g − = 1 (because ˙ γ = 0), and a slow, compressedexponential decay ( t = 1660 s and p = 1 . > p ranging from 1.5 at low q downto p > ∼ q [89–95]. These dynamics have beenattributed to the relaxation of internal stress built up atgelation [89, 96].Upon shear startup, the dynamics accelerate dramat-ically, as shown by the blue and red symbols in Fig. 6a.Interestingly, the dynamics significantly evolve over time,although the imposed shear rate is constant. Indeed, theintensity correlation function averaged over the first 400s of the experiment (blue triangles) exhibits a decay rateabout twice as fast as that averaged over 450 s ≤ t ≤
600 s (red circles). Remarkably, the change of dynamicalregime around 400-450 s corresponds to the end of thelinear regime where the shear stress is proportional to t ,and thus to γ , delimited by the blue vertical lines in theinset of Fig. 6a.In order to understand the nature of the microscopicdynamics under shear, it is important to quantify thecontribution due to a purely affine deformation. We cal-culate this contribution by integrating the sinc term ofEq. 12 over the ROI area, using for all parameters theirnominal values. This yields the green line in Fig. 6a, withthe dotted lines indicating the confidence band due to theuncertainty on q x and q y . In the initial regime, t ≤
400 sand γ ≤ g − t and p as free parameters. The pink line showsthe best fit, with t = 53 . p = 1 .
71. Except for τ < ∼
20 s, where the fit slightly underestimates the de-cay of the correlation function, Eq. 12 reproduces verywell the relaxation of g −
1. The fitted decay time is 30times shorter than for the gel at rest and about two timesshorter than the contribution due to affine deformationalone. The stretching exponent p is larger than one, andis larger than that at rest.One possible explanation of the observed dynamics isthat the external stress accumulated as the sample issheared triggers rearrangement events similar to thoseobserved in the sample at rest, where they originate frominternal stress. Another possibility is that the non-affinedynamics are not due to irreversible plastic restructuring,but rather to deviations in the local response of the gel tothat of a homogeneously elastic material, due to the het-erogeneous structure of the gel. Indeed, shear-induced,non-affine microscopic dynamics observed in the small- γ c I between speckle images taken attime t and t + τ (see Eq. 1), for a fixed time lag τ = 60 sand as a function of t , t = 0 being the shear startuptime. The green solid line is the behavior expected for apurely affine deformation, with no additional dynamics.In this case, c I would be equal to one for t ≤ −
60 s, be-cause both images are taken while the sample is at rest.The degree of correlation would progressively decreasefor −
60 s < t <
0, because the first image of the pair istaken while the sample is at rest, while the second onecorresponds to an increasingly deformed sample configu-ration. Finally, for t ≥ c I would reach a steady-statevalue, since the sample deformation over the time lag sep-arating the two images is constant. The experimental c I is shown by the black line: it strongly deviates from theideal behavior of purely affine dynamics. For t ≤
400 s c I is significantly lower than the value expected for affinedeformation, indicating additional non-affine motion, asalready inferred from the average dynamics. However,the degree of correlation strongly fluctuates, revealingbursts of enhanced dynamics (lower c I ) that last sev-eral tens of seconds. These fluctuations are not due tonoise measurement, whose typical rms amplitude is muchsmaller, of order 0.01, as seen from the c I trace beforeshear startup (Fig. 6b, t ≤ −
60 s). They reveal thediscontinuous nature of the microscopic dynamics undershear and may hint at discrete events similar to thoseresponsible for the ‘avalanches’ seen in simulations andexperiments on driven amorphous, glassy materials [98–100], and polymeric and colloidal gels [101, 102]. Notethat the experiments of Refs. [99, 100] reported signa-tures of a non-monotonic evolution of macroscopic quan-tities (the stress or the deformation). Here, for t ≤
400 sthe stress evolution does not exhibit any deviation from amonotonic, linear behavior, see the inset of Fig. 6a. Mostlikely, this is due to the size of our sample, which is muchlarger than the system size in Refs. [98–100], thereby av-eraging out more effectively the macroscopic quantitiesmeasured by the rheometer. This highlights the interestof space-resolved measurement such as those afforded bydynamic light scattering in the imaging geometry.For 450 s ≤ t ≤
600 s, the degree of correlation issignificantly higher, surprisingly exceeding the value ex-pected for the affine contribution. This indicates that onthe probed time scale the particle displacement is smallerthan the affine component, i.e. the gel is actually de-formed less than what expected from the applied shearrate, at least in the probed ROI. Inspection of the fullcorrelation function averaged over the same time inter-val (red circles in Fig. 6a) corroborates the finding thatthe microscopic dynamics strongly deviate from those ex- pected for affine deformation. The decay of g − term: at short τ < ∼
20 s itis close to that measured at the beginning of the shearstartup, but at larger τ g − above the green line cal-culated for affine motion, thus confirming smaller-than-expected particle displacements.The only possible explanation for the behavior ob-served for t ≥
450 s is the existence of shear localisa-tion, most likely due to wall slip, such that a sizeablefraction of the gel is indeed deformed less than for homo-geneous, affine flow. To test this hypothesis, we comparethe z -averaged ROI drift to that expected for an affinedeformation. This comparison cannot be performed onthe same ROI as in Fig. 6. Indeed, that ROI was cho-sen far from the x axis, in order to make the microscopicdynamics easily measurable. As a result, g − x axis, for which the sinc term in Eq. 12 is one, thedecay of g − D i s p l a c e m en t ( p i x e l ) ∆ x* ∆ y* = 150 s τ t (s) Affine ∆ y* displacement FIG. 7. x and y components of the displacement of a ROIlocated on the x axis, for the same shear startup experimentas in Fig. 6. The blue solid vertical lines indicate the timeat which shear starts and the end of the rheological linearregime, respectively. The dashed vertical lines indicate whenthe displacement calculated between pairs of images takenat time t and t + τ starts to be sensitive to the change ofrheological regime (beginning of the shear and end on linearregime). The black and green thick lines are the expectedbehavior of ∆ x ∗ and ∆ y ∗ for a purely affine deformation. Figure 7 shows the drift for such a ROI, positioned onthe x axis ( β = 0, q y = 0 µ m − ), at the same distance r = 15 . x and y components of thedrift over τ = 150 s, one of the shortest delays for whichthe ROI displacement can be reliably measured. The2solid vertical blue lines mark the beginning and the endof the linear regime, as in Fig. 6. The dotted lines arepositioned 150 s before the solid lines and indicate thetime at which the the two-time displacement starts to besensitive to the change of regime marked by the followingvertical solid line. The green line shows the displacement∆ y ∗ calculated for a purely affine deformation field. For t ≤ −
150 s, one expects ∆ y ∗ = 0, since the displacementis calculated for pairs of images of the sample at rest. For t > −
150 s ∆ y ∗ should evolve linearly towards its steadystate value, since the second image of the pair is takenfor an increasingly deformed sample configuration. Thesteady state should be reached at t = 0 s, beyond whichboth images used to measure the displacement are takenwhile shearing the sample at a fixed rate. Throughoutthe experiment, ∆ x ∗ should be null, because β = q y = 0.At the very beginning of shear startup, ∆ x ∗ and ∆ y ∗ appear to be consistent with affine deformation, as seenby comparing their trend to the expected one in therange −
150 s ≤ t ≤ t ≥ x ∗ and ∆ y ∗ are seen for t > ∼
400 s. In this regime, not onlyis the behavior of the displacement inconsistent with anideal affine deformation, it is also incompatible with sim-ple plug flow, which could be expected, e.g., if a neatfracture was formed in the horizontal plane. Indeed, inthis case the largest measurable displacement would cor-respond to a gel detached from the bottom, static planeand attached to the rotating top plane. For the ROI in-vestigated here, this would correspond to ∆ x ∗ = 0 and∆ y ∗ twice as large as for ideal affine deformation. Thestrongly intermittent behavior seen in Fig. 7, with dis-placement values well in excess to those predicted forsimple plug flow, rather point to a chaotic flow behav-ior, where bursts of motion relax the stress progressivelybuilt by the continuous rotation of the upper plate. Theseevents must involve the correlated motion of large por-tions of the sample, in a process intriguingly reminiscentof stick-slip. Indeed, Fig. 6 shows that in this regimethe relative motion of the gel network is small. Collec-tively, the space- and time-resolved data of Figs. 6, 7show that from the very onset of shear startup the lo-cal deformation and the microscopic dynamics stronglydeviate from affine deformation. The large temporal fluc-tuations of both the microscopic dynamics and the meso-scopic displacement suggest that these deviations are dueto a series of rearrangement events associated with shearlocalization, rather than to the continuous (non-affine)deformation of a pristine sample. The proliferation ofthese events for t > ∼
400 s explains the softening of thegel measured by rheology.As a concluding remark, it is worth mentioning thatthe detection of shear localization would not have beeneasy, based only on the rheology data. Indeed, from a rheological point of view, the change of slope seen in theinset of Fig. 6a could also be due to extensive plasticactivity that softens the gel. The absence of a stressovershoot, usually seen in shear startup of gels [97, 102–104], might suggests wall slip. However, the only way tounambiguously discriminate between the various possibleorigins of the shear vs strain behavior of Fig. 6a would beto repeat the measurement using different gaps H , sinceany gap dependence of the mechanical response wouldpoint to wall slip or banding. By contrast, our setupallows an easy and quick detection of heterogeneous flow,a great advantage especially for the non-linear regime ofcomplex systems, where reproducibility may be an issue. V. CONCLUSION
We have presented a home-made dynamic light scat-tering setup coupled to a commercial rheometer. Thanksto the use of a 2D detector in an imaging geometry, thesetup allows one to measure the microscopic dynamicswith both spatial and temporal resolution. Although thescattering vector is mainly oriented along the vertical, orflow gradient, direction, q has also a small component inthe flow-vorticity plane. As a consequence, the intensitycorrelation function contains a contribution due to theaffine deformation. We have shown that this contribu-tion can be reliably calculated and factored out from theexperimental g −
1. In other apparatuses that measureDLS or XPCS under flow [72, 75] in the far field limit, thedecay of the correlation function is accelerated by a thirdcontribution, stemming from the fact that sample mate-rial continuously enters and leaves the scattering volume,which is therefore completely renewed after some time.In our space-resolved apparatus, this contribution is min-imized by following the average displacement of a ROI,using mixed spatio-temporal correlators. Additionally,these correlators allow the local flow velocity (averagedover the gap) to be measured, a valuable tool for checkingfor flow heterogeneities.In this paper, we have presented measurements inthe DLS regime. Single scattering conditions are moredemanding than the strong multiple scattering regimeprobed by DWS, since stray light reflected by the op-tical interfaces or scattered by any imperfection in theoptics is likely to be comparable to or even more intensethan light scattered by the sample. In this respect, it ismandatory to reduce as much as possible forward scatter-ing originating from the reflection of the incident beamon the upper plate. We have proposed and demonstrateda simple way to address this issue, by gluing a ND filterto the upper plate.The apparatus described here can also be used forspace-resolved DWS in the backscattering geometry. Inthis case, the ND filter is not required, since light propa-gating through a multiply scattering medium is stronglyattenuated: the intensity of the transmitted beam reach-ing the upper plate is then negligible. For the same3reason, the adoption of a plate-plate geometry, which iscompulsory for DLS, is actually not needed for DWS.One could safely trade it for a cone-and-plate geometry,more appealing from a rheological point of view, becauseit insures constant stress and strain throughout the sam-ple. Note that multiple scattering intrinsically blurs thespatial resolution of the setup, because photons emerg-ing in a given point will have propagated through a finiteportion of the sample. However, the lateral size of thisregion is comparable to the sample thickness, which istypically of the order of 1 mm, usually comparable to orsmaller than the size of the ROIs over which g − ACKNOWLEDGMENTS
We thank S. Aime for illuminating discussions andJ.-M. Fromental for help with instrumentation. Thiswork was supported by Agence National de la Recherche(ANR) Grant ANR-14-CE32-0005-01 and Centre Na-tional d’´etudes Spatiales (CNES) [1] R. Larson,
The Structure and Rheology of Complex Flu-ids , Topics in Chemical Engineering (OUP USA, 1999).[2] M. Rubinstein and R. Colby,
Polymer Physics (OUPOxford, 2003).[3] I. Cantat, S. Cohen-Addad, F. Elias, F. Graner,R. H¨ohler, O. Pitois, and R. Flatman,
Foams: Structureand Dynamics (OUP Oxford, 2013).[4] D. T. Chen, Q. Wen, P. A. Janmey, J. C. Crocker,and A. G. Yodh, Annual Review of Condensed MatterPhysics , 301 (2010).[5] Y. Wang and D. Discher, “Cell mechanics,” (ElsevierSciencel, 2007) Chap. 1, pp. 1–27.[6] B. Berne and R. Pecora, Dynamic light scattering: withapplications to chemistry, biology, and physics (Wiley,1976).[7] D. A. Weitz and D. J. Pine, in
Dynamic Light Scat-tering , edited by W. Brown (Clarendon Press, Oxford,1993) pp. 652–720.[8] D. M. Mills,
Third-Generation Hard X-Ray SynchrotronRadiation Sources: Source Properties, Optics, and Ex-perimental Techniques (2002).[9]
Neutron, X-Rays and Light. Scattering Methods Appliedto Soft Condensed Matter , elsevier ed. (Zemb, Thomasand Lindner, Peter, North Holland, 2002).[10] A. D. Gopal and D. J. Durian, Phys. Rev. Lett. , 2610(1995).[11] R. H¨ohler, S. Cohen-Addad, and H. Hoballah, Phys.Rev. Lett. , 1154 (1997).[12] F. Molino, J.-F. Berret, G. Porte, O. Diat, and P. Lind-ner, The European Physical Journal B - CondensedMatter and Complex Systems , 59 (1998).[13] P. Schall, I. Cohen, D. A. Weitz, and F. Spaepen, Sci-ence , 1944 (2004).[14] S. Reinicke, M. Karg, A. Lapp, L. Heymann, T. Hellweg,and H. Schmalz, Macromolecules , 10045 (2010).[15] M. A. Torija, S.-H. Choi, T. P. Lodge, and F. S. Bates,The Journal of Physical Chemistry B , 5840 (2011).[16] E. Tamborini, L. Cipelletti, and L. Ramos, Phys. Rev.Lett. , 078301 (2014).[17] M. E. Helgeson, P. A. Vasquez, E. W. Kaler, and N. J.Wagner, Journal of Rheology , 727 (2009).[18] R. Angelico, C. O. Rossi, L. Ambrosone, G. Palazzo,K. Mortensen, and U. Olsson, Phys. Chem. Chem.Phys. , 8856 (2010). [19] T. J. Ober, J. Soulages, and G. H. McKinley, Journalof Rheology , 1127 (2011).[20] A. K. Gurnon, C. R. Lopez-Barron, A. P. R. Eberle,L. Porcar, and N. J. Wagner, Soft Matter , 2889(2014).[21] A. Basu, Q. Wen, X. Mao, T. C. Lubensky, P. A. Jan-mey, and A. G. Yodh, Macromolecules , 1671 (2011).[22] M.-Y. Nagazi, G. Brambilla, G. Meunier, P. Marguer`es,J.-N. P´eri´e, and L. Cipelletti, Optics and Lasers inEngineering , 5 (2017).[23] P. H´ebraud, F. Lequeux, J. P. Munch, and D. J. Pine,Phys. Rev. Lett. , 4657 (1997).[24] G. Petekidis, A. Moussa¨ıd, and P. N. Pusey, Phys. Rev.E , 051402 (2002).[25] T. Bauer, J. Oberdisse, and L. Ramos, Phys. Rev. Lett. , 258303 (2006).[26] R. Besseling, E. R. Weeks, A. B. Schofield, andW. C. K. Poon, Phys. Rev. Lett. , 028301 (2007).[27] P. Schall, D. A. Weitz, and F. Spaepen, Science ,1895 (2007).[28] J. Zausch, J. Horbach, M. Laurati, S. U. Egelhaaf,J. M. Brader, T. Voigtmann, and M. Fuchs, Journalof Physics: Condensed Matter , 404210 (2008).[29] N. Koumakis, M. Laurati, S. U. Egelhaaf, J. F. Brady,and G. Petekidis, Phys. Rev. Lett. , 098303 (2012).[30] D. Denisov, M. T. Dang, B. Struth, G. Wegdam, andP. Schall, Scientific Reports , 1631 (2013).[31] E. D. Knowlton, D. J. Pine, and L. Cipelletti, SoftMatter , 6931 (2014).[32] T. Sentjabrskaja, P. Chaudhuri, M. Hermes, W. C. K.Poon, J. Horbach, S. U. Egelhaaf, and M. Laurati,Scientific Reports , 11884 (2015).[33] M. B. Gordon, C. J. Kloxin, and N. J. Wagner, Journalof Rheology , 23 (2017).[34] S. Aime, L. Ramos, and L. Cipelletti, PNAS , 3587(2018).[35] D. Bonn, H. Kellay, M. Prochnow, K. Ben-Djemiaa,and J. Meunier, Science , 265 (1998).[36] V. B. Tolstoguzov, A. I. Mzhel’sky, and V. Y. Gulov,Colloid and Polymer Science , 124 (1974).[37] D. Beysens, M. Gbadamassi, and L. Boyer, Phys. Rev.Lett. , 1253 (1979).[38] T. Hashimoto, T. Takebe, and S. Suehiro, PolymerJournal , 123 (1986). [39] B. J. Ackerson, J. van der Werff, and C. G. de Kruif,Phys. Rev. A , 4819 (1988).[40] J. W. van Egmond, D. E. Werner, and G. G. Fuller,The Journal of Chemical Physics , 7742 (1992).[41] J. L¨auger and W. Gronski, Rheologica Acta , 70(1995).[42] T. Kume, K. Asakawa, E. Moses, K. Matsuzaka, andT. Hashimoto, Acta Polymerica , 79 (1995).[43] J. Vermant, P. Van Puyvelde, P. Moldenaers, J. Mewis,and G. G. Fuller, Langmuir , 1612 (1998).[44] J. Vermant, L. Raynaud, J. Mewis, B. Ernst, andG. Fuller, Journal of Colloid and Interface Science ,221 (1999).[45] P. Varadan and M. J. Solomon, Langmuir , 2918(2001).[46] R. Scirocco, J. Vermant, and J. Mewis, Journal of Non-Newtonian Fluid Mechanics , 183 (2004).[47] L. Gentile, C. O. Rossi, and U. Olsson, Journal of Col-loid and Interface Science , 537 (2012).[48] L. Ramos and F. Molino, Phys. Rev. Lett. , 018301(2004).[49] L. Yang, R. H. Somani, I. Sics, B. S. Hsiao, R. Kolb,H. Fruitwala, and C. Ong, Macromolecules , 4845(2004).[50] E. Di Cola, C. Fleury, P. Panine, and M. Cloitre,Macromolecules , 3627 (2008).[51] Y. Kosaka, M. Ito, Y. Kawabata, and T. Kato, Lang-muir , 3835 (2010).[52] J. P. de Silva, D. Petermann, B. Kasmi, M. Imperor-Clerc, P. Davidson, B. Pansu, F. Meneau, J. Perez,E. Paineau, I. Bihannic, L. J. Michot, and C. Bara-vian, Journal of Physics: Conference Series , 012052(2010).[53] L. Ramos, A. Laperrousaz, P. Dieudonn´e, andC. Ligoure, Phys. Rev. Lett. , 148302 (2011).[54] R. Akkal, N. Cohaut, M. Khodja, T. Ahmed-Zaid, andF. Bergaya, Colloids and Surfaces A: Physicochemicaland Engineering Aspects , 751 (2013).[55] D. C. F. Wieland, V. M. Garamus, T. Zander, C. Kry-wka, M. Wang, A. Dedinaite, P. M. Claesson, andR. Willumeit-R¨omer, Journal of Synchrotron Radiation , 480 (2016).[56] L. Porcar, W. A. Hamilton, P. D. Butler, and G. G.Warr, Phys. Rev. Lett. , 168301 (2002).[57] M. E. Helgeson, L. Porcar, C. Lopez-Barron, and N. J.Wagner, Phys. Rev. Lett. , 084501 (2010).[58] A. P. Eberle and L. Porcar, Current Opinion in Colloidand Interface Science , 33 (2012).[59] C. R. L´opez-Barr´on, L. Porcar, A. P. R. Eberle, andN. J. Wagner, Phys. Rev. Lett. , 258301 (2012).[60] L. Gentile, M. A. Behrens, L. Porcar, P. Butler, N. J.Wagner, and U. Olsson, Langmuir , 8316 (2014).[61] J. M. Kim, A. P. R. Eberle, A. K. Gurnon, L. Porcar,and N. J. Wagner, Journal of Rheology , 1301 (2014).[62] D. Rusu, D. Genoe, P. van Puyvelde, E. Peuvrel-Disdier, P. Navard, and G. Fuller, Polymer , 1353(1999).[63] N. Ali, D. Roux, L. Cipelletti, and F. Caton, Measure-ment Science and Technology , 125902 (2016).[64] H. Guo, S. Ramakrishnan, J. L. Harden, and R. L.Leheny, Phys. Rev. E , 050401 (2010).[65] M. Erpelding, A. Amon, and J. Crassous, Phys. Rev.E , 046104 (2008).[66] S. Aime, L. Ramos, J. M. Fromental, G. Prvot, R. Je- linek, and L. Cipelletti, Review of Scientific Instru-ments , 123907 (2016).[67] B. A. DiDonna and T. C. Lubensky, Physical Review E , 066619 (2005).[68] A. Zaccone, Modern Physics Letters B , 1330002(2013).[69] X.-L. Wu, D. J. Pine, P. M. Chaikin, J. S. Huang, andD. A. Weitz, Journal of the Optical Society of AmericaB , 15 (1990).[70] M.-Y. Nagazi, Cartographie de la dynamique micro-scopique dans la mati`ere molle sous sollicitation , Ph.D.thesis, Universit´e de Montpellier, Montpellier (2017).[71] H. M. van der Kooij, S. Dussi, G. T. van de Kerkhof,R. A. M. Frijns, J. van der Gucht, and J. Sprakel,Science Advances , eaar1926 (2018).[72] S. Busch, T. H. Jensen, Y. Chushkin, and A. Fluerasu,Eur. Phys. J. E , 55 (2008).[73] M. C. Rogers, K. Chen, L. Andrzejewski, S. Narayanan,S. Ramakrishnan, R. L. Leheny, and J. L. Harden,Physical Review E , 062310 (2014).[74] R. L. Leheny, M. C. Rogers, K. Chen, S. Narayanan,and J. L. Harden, Current Opinion in Colloid & Inter-face Science , 261 (2015).[75] S. Aime and L. Cipelletti, ArXiv e-prints (2018),arXiv:1802.03737 [cond-mat.soft].[76] F. Livet, F. Bley, F. Ehrburger-Dolle, I. Morfin,E. Geissler, and M. Sutton, J Synchrotron Rad, J Syn-chrotron Radiat , 453 (2006).[77] F. Ehrburger-Dolle, I. Morfin, F. Bley, F. Livet,G. Heinrich, S. Richter, L. Pich´e, and M. Sutton,Macromolecules , 8691 (2012).[78] J.-B. Salmon, S. Manneville, A. Colin, and B. Pouligny,Eur. Phys. J. AP , 143 (2003).[79] A. Duri, D. A. Sessoms, V. Trappe, and L. Cipelletti,Phys. Rev. Lett. , 085702 (2009).[80] L. Cipelletti, G. Brambilla, S. Maccarrone, andS. Caroff, Opt. Express , 22353 (2013).[81] V. Viasnoff, F. Lequeux, and D. J. Pine, Rev. Sci. In-strum. , 2336 (2002).[82] A. Philippe, S. Aime, V. Roger, R. Jelinek, G. Prvot,L. Berthier, and L. Cipelletti, Journal of Physics: Con-densed Matter , 075201 (2016).[83] L. Cipelletti, H. Bissig, V. Trappe, P. Ballesta, andS. Mazoyer, J. Phys.: Condens. Matter , S257 (2003).[84] A. Duri, H. Bissig, V. Trappe, and L. Cipelletti, Phys.Rev. E , 051401 (2005).[85] M. Born and E. Wolf, Principles of Optics: Electromag-netic Theory of Propagation, Interference and Diffrac-tion of Light (Elsevier Science, 2013).[86] M. Kerker,
The Scattering of Light, and Other Electro-magnetic Radiation , 3830 (1994).[89] L. Cipelletti, S. Manley, R. C. Ball, and D. A. Weitz,Phys. Rev. Lett. , 2275 (2000).[90] A. Duri and L. Cipelletti, Europhys. Lett. , 972(2006).[91] H. Guo, S. Ramakrishnan, J. L. Harden, and R. L.Leheny, The Journal of Chemical Physics , 154903(2011).[92] E. Secchi, T. Roversi, S. Buzzaccaro, L. Piazza, and R. Piazza, Soft Matter , 3931 (2013).[93] D. Larobina and L. Cipelletti, Soft Matter , 10005(2013).[94] S. Buzzaccaro, M. D. Alaimo, E. Secchi, and R. Pi-azza, Journal of Physics: Condensed Matter , 194120(2015).[95] A.-M. Philippe, L. Cipelletti, and D. Larobina, Macro-molecules , 8221 (2017).[96] J. P. Bouchaud and E. Pitard, Eur. Phys. J. E , 231(2001).[97] J. Colombo and E. Del Gado, Journal of Rheology ,1089 (2014).[98] A. Lemaˆıtre and C. Caroli, Phys. Rev. Lett. , 065501(2009). [99] J.-O. Krisponeit, S. Pitikaris, K. E. Avila,S. K¨uchemann, A. Kr¨uger, and K. Samwer, Na-ture Communications , 3616 (2014).[100] J. Antonaglia, W. J. Wright, X. Gu, R. R. Byer, T. C.Hufnagel, M. LeBlanc, J. T. Uhl, and K. A. Dahmen,Phys. Rev. Lett. , 155501 (2014).[101] P. Coussot, Q. D. Nguyen, H. T. Huynh, and D. Bonn,Phys. Rev. Lett. , 175501 (2002).[102] A. Kurokawa, V. Vidal, K. Kurita, T. Divoux, andS. Manneville, Soft Matter , 9026 (2015).[103] J. D. Park, K. H. Ahn, and N. J. Wagner, Journal ofRheology , 117 (2017).[104] L. C. Johnson, B. J. Landrum, and R. N. Zia, SoftMatter14