Differential Dynamic Microscopy microrheology of soft materials: a tracking-free determination of the frequency-dependent loss and storage moduli
Paolo Edera, Davide Bergamini, Véronique Trappe, Fabio Giavazzi, Roberto Cerbino
aa r X i v : . [ c ond - m a t . s o f t ] A ug Differential Dynamic Microscopy microrheology of soft materials: a tracking-freedetermination of the frequency-dependent loss and storage moduli
Paolo Edera, Davide Bergamini, Veronique Trappe, Fabio Giavazzi, ∗ and Roberto Cerbino † Dipartimento di Biotecnologie Mediche e Medicina Traslazionale,Università degli Studi di Milano, Via. F.lli Cervi 93, Segrate (MI) I-20090, Italy Department of Physics, University of Fribourg, Chemin du Musée 3, CH-1700, Fribourg, Switzerland (Dated: August 25, 2017)Particle tracking microrheology (PT- µ r) exploits the thermal motion of embedded particles toprobe the local mechanical properties of soft materials. Despite its appealing conceptual simplicity,PT- µ r requires calibration procedures and operating assumptions that constitute a practical barrierto a wider adoption. Here we demonstrate Differential Dynamic Microscopy microrheology (DDM- µ r), a tracking-free approach based on the multi-scale, temporal correlation study of the imageintensity fluctuations that are observed in microscopy experiments as a consequence of the motionof the tracers. We show that the mechanical moduli of an arbitrary sample are determined correctlyin a wide frequency range, provided that the standard DDM analysis is reinforced with a novel,iterative, self-consistent procedure that fully exploits the multi-scale information made available byDDM. Our approach to DDM- µ r does not require any prior calibration, is in agreement with bothtraditional rheology and Diffusing Wave Spectroscopy microrheology, and works in conditions wherePT- µ r fails, providing thus an operationally simple, calibration-free probe of soft materials. I. INTRODUCTION
Rheology is a well established experimental techniquethat probes the response of materials upon application ofa stress or strain [1]. This probe is particularly significantfor soft materials such as paint, starch, mayonnaise andgelatin that defy the sharp rules according to which wetend to classify a substance as an ideal viscous Newtonianliquid or a perfectly elastic Hookean solid. In practice,depending on the time scales probed and the magnitudeof the stress or strain applied, soft materials may behaveas solids or liquids. For instance, a dense colloidal suspen-sion may exhibit a solid-like response upon applicationof small-amplitude, fast deformations whereas it may bemore similar to a liquid upon applying large-amplitude,slow deformations. Converting these important but qual-itative considerations into some quantitatively and repro-ducibly determined mechanical moduli of the materialsis the realm of rheology [2].Traditional rheology makes use of rheometers, in whicha soft material is loaded in the gap between two solid sur-faces and stressed (or strained) in a controlled fashion tomeasure the strain (or stress) response of the material.This response can be entirely [3] described in terms ofa complex modulus G ∗ ( ω ) = G ′ ( ω ) + iG ′′ ( ω ) . G ∗ canbe measured with a rheometer, by imposing for instancean oscillatory strain γ ( t ) = γ sin( ωt ) and measuring thestress σ ( t ) developed by the material. In general, onefinds that σ ( t ) = G ′ γ sin( t ) + G ′′ γ cos( t ) , where G ′ and G ′′ are the storage (or elastic) and loss (or viscous) mod-uli of the material, respectively.This denomination denotes that a Hookean solid ischaracterized only by a stress in phase with the applied ∗ [email protected] † [email protected] strain, with G ′ corresponding to the elastic modulus ofthe solid, whereas the response of a Newtonian liquid isin quadrature, with G ′′ = ηω , where η is the dynamicviscosity. A generic soft material will have an in-phaseresponse that can be associated to its solid-like characterand an in-quadrature response that is due to its liquid-like nature. Naively one can say that if for a given fre-quency G ′ ≫ G ′′ the material is substantially a solid,whereas if G ′ ≪ G ′′ it behaves as a liquid. Inspect-ing the full frequency dependence of G ′ and G ′′ providesthus a fundamental tool to classify materials based ontheir mechanical response or to monitor changes in theirmechanical properties during for instance gelation or ag-gregation processes [4].Despite their powerfulness and immediacy, rheologytests performed with a rheometer are affected by somelimitations: they require a large quantity of material (ofthe order of a few milliliters), they average over possi-ble heterogeneities of the sample, and the accessible fre-quency range is limited at small ω by torque limitationsand at large ω by inertial effects [5].A complementary approach that addresses the aboveissues, is represented by microrheology [6–10]. Originallyintroduced by Mason and Weitz in 1995 [11], the so-calledpassive microrheology consists of seeding the soft mate-rial of interest with tracer particles of radius a and mea-suring the mean-square displacement (MSD) (cid:10) ∆ r ( t ) (cid:11) of the tracers within the material as a function of time t . The MSD can be related to the frequency-dependentcomplex modulus G ∗ ( ω ) by using the generalized Stokes-Einstein equation [8] G ∗ ( ω ) = dk B T πas h ∆˜ r ( s ) i (cid:12)(cid:12)(cid:12)(cid:12) s = iω (1)where d is the number of dimensions tracked in theMSD, k B the Boltzmann constant, T the temperature, i the imaginary unit and (cid:10) ∆˜ r ( s ) (cid:11) the Laplace trans-form of the MSD. In a Newtonian liquid of viscosity η , the MSD of a tracer particle with diffusion coeffi-cient D = k B T / (6 πηa ) is given by (cid:10) ∆ r ( t ) (cid:11) = 2 dD t ,which leads to (cid:10) ∆˜ r ( s ) (cid:11) = 2 dD /s and, in turn, to G ∗ ( ω ) = iG ′′ ( ω ) = iωη . For a solid, instead, the elas-tic modulus G ∗ ( ω ) = G ′ is obtained from estimating themean squared displacement (cid:10) ∆ r ( t ) (cid:11) = dk B T πaG ′ of a parti-cle in an elastic trap with the condition (cid:10) ∆ r ( t ) (cid:11) = 0 for t < . The Laplace transform is then given by (cid:10) ∆˜ r ( s ) (cid:11) = dk B T πaG ′ s .The MSD of the tracer particles can be obtained in avariety of ways [6, 8, 9]. A direct way is to track in real-space the trajectories of the tracer particles, as done inParticle Tracking (PT) experiments [7]. An alternativeway is to extract the MSD from the measurement of theintensity scattered or fluorescently emitted by a dilutecollection of non-interacting tracer particles, as done inDiffusing Wave Spectroscopy (DWS) [11], Dynamic LightScattering (DLS) [12, 13] and Fluorescence CorrelationSpectroscopy (FCS) [14]. Historically, DWS- µ r was thefirst to be developed and, together with PT- µ r, is stillone of the most common approaches. DWS and PT arein principle quite complementary. DWS gives access toshort time scales and small MSD, while PT gives access tolonger time scales and larger MSD. However, while DWScan be used with almost no user intervention, PT involvesa rather tedious and delicate selection of the trajectories.The disadvantage of DWS in turn is to require largertracer particle concentrations that may more easily alterthe mechanical properties of the material itself.Almost ten years ago, the usefulness of a techniquenamed Differential Dynamic Microscopy (DDM) wasdemonstrated for the characterization of the dynamicsof colloidal suspensions of particles [15]. One of the mainfeatures of DDM is that it lies somehow in-between PTand DLS. Similar to PT, it is based on real-space moviescollected in microscopy experiments. These data aretreated via an image processing algorithm [16] or equiv-alent versions of it [17] that combines image differencesand spatial Fourier transformations to obtain as a resultthe intermediate scattering function f ( q , t ) that is typi-cally probed in DLS experiments as a function of the scat-tering wave-vector q and time t [18]. Since its introduc-tion, DDM has been profitably used and extended alsoby several groups [19–28] for a variety of applications.Surprisingly, as of today, there have been no attemptsto use DDM to perform microrheology experiments oncomplex fluids, which would seem a quite obvious andpowerful application.In this work, we show that DDM can indeed be usedas a convenient and reliable tool to probe the mechan-ical properties of complex fluids, which we demonstratewith both Newtonian liquids, obtained by mixing wa-ter and glycerol in variable proportions, and viscoelasticsamples, consisting of aqueous solutions of a high molec-ular weight polymer (polyethylene oxide). To determinethe tracer MSD in DDM, we demonstrate the advantage of a novel fitting-free, optimization-based procedure thatis applicable to an arbitrary sample and does not requireany prior calibration. The obtained results are found tobe in agreement with standard rheology and with bothPT- and DWS-microrheology. In addition, we show thatDDM- µ r operates also with small tracer particles thatare not suitable for tracking experiments; this widens therange of applicability of microrheology.Our results show that optimization-based DDM- µ r is aflexible, calibration-free approach to microrheology thatcan be almost fully automated, thus eliminating the ar-bitrariness, typical of PT experiments, in sorting and se-lecting the suitable trajectories. We expect that DDM- µ rcan be successfully used to measure the rheological prop-erties of a variety of soft materials, also in cases whereDLS, DWS and PT can not be used. A typical exampleis the cell interior [29] where DDM has already been suc-cessfully used to measure the interplay of diffusion andflow during oogenesis. II. MATERIALS AND METHODSA. Samples preparation
We used two different classes of samples: Newtonianfluids with varying viscosity obtained by adding differentamounts of glycerol to water, and viscoelastic fluids con-sisting of aqueous solutions of polyethylene oxide (PEO, M W = 2 × Da). For the PEO solution we chose towork at a concentration of c=2.0 wt% above the overlapconcentration (c*=0.09 wt%) to obtain a sample with ap-preciable viscoelastic properties in the frequency range ofinterest. At these conditions, the mesh size of the poly-mer network is estimated to be ∼ nm [13]. Glycerol-water solutions
The glycerol-water samples were prepared by mixingsuitable amounts of glycerol (Sigma Aldrich), MilliQ wa-ter and an aqueous suspension of latex beads (SigmaAldrich, LB5, nominal diameter 0.45-0.47 µm , solid con-tent 10%) to reach final mass fractions of glycerol equalto %, . %, . % and . %. The mass fractions ofsuspended beads were . %, . %, . % and . %,respectively. All four samples were investigated by usingboth PT and DDM. Polymer solution
The PEO solution for traditional rheology experimentswas prepared from the pure product purchased as pow-der (Sigma Aldrich, prod. code 372803). The powderwas carefully dissolved in MilliQ water previously filteredwith membrane filters (pore size 0.2 µm ). To prevent theformation of clumps of undissolved polymer, water wasgradually added to the polymer while stirring. The solu-tion was then kept in incubation for nine days at about40 ° C. A few drops of a Sodium Azide solution (molar-ity 4 mM) were added to the PEO solution to preventbacterial proliferation.For the DDM, PT and DWS experiments, in whichcolloidal tracer particles were to be added to the PEOsolution, a prescribed amount of pure water was replacedwith the aqueous colloidal suspensions of latex beads dur-ing sample preparation. The beads were purchased bySigma Aldrich with the part numbers LB1 (nominal di-ameter 0.10-0.12 µm , solid content 10%) and LB5 (nomi-nal diameter 0.45-0.47 µm , solid content 10%). The beadsizes were also tested with DDM and were found to beequal to . ± . µm and . ± . µm for theLB1 and LB5 samples, respectively. After dilution thefinal concentration of PEO was 2% wt/wt and the finalconcentration of tracer beads was (1 . ± . × − wt/wt. To ensure multiple scattering for the DWS ex-periment we used LB5 particles at (1 . ± . × − wt/wt. B. Rheology
Since for the Newtonian glycerol-water mixtures re-liable literature data are available, standard rheologyexperiments were only performed for the PEO sample.We used a commercial rheometer (Anton Paar MCR502)equipped with cone and plate geometry (radius= mm,cone angle = o ) to apply an oscillatory shear strain withstrain amplitude of 5%, and angular frequency in therange [0 . , s − . Our experiments were perforemedin the temperature range T = 20 − °C. To avoid evap-oration during measurement we used a solvent trap. C. Particle tracking (PT)
PT experiments were performed by tracking LB5 par-ticles dispersed in the four glycerol-water samples andin a polymer solution with the PEO concentration alsoused in II B. The samples were loaded in a capillary(Vitrocom) with rectangular cross section and internaldimensions × × . mm . Microscopy experimentswere performed in bright field with an optical microscope(Nikon Ti-E), equipped with a digital camera (Hama-matsu Orca Flash 4.0 v2), and a 40x objective. The re-sulting pixel size was d pix = 162 . nm. Image sequencesmade of , images (512x256 pixels) were acquiredat two different frame rates (777 Hz and 10 Hz). In allacquired images, the sample appeared transparent andthe colloidal particles were clearly visible.Particle tracking analysis was conducted by using acustomized version of the MATLAB code script madefreely available by the group of Maria Kilfoil at UMass(people.umass.edu/kilfoil/). This software reconstructsthe individual trajectories of several particles in parallel, calculating their MSD as a function of time. Comparedto the original code, we added some custom features,mainly to adapt our analysis to bright-field time-lapsemovies and to estimate error bars and experimental un-certainties.Once the MSD was obtained, the data needed to becorrected by subtracting the additive contribution due tothe intrinsic localization uncertainty that becomes domi-nant for small times and particle displacements [30]. Thisstep, which lies at the core of PT- µ r, requires an indepen-dent calibration of the particle localization error. In ourexperiments, the static localization error was determinedas that which minimizes the deviation from a purely lin-ear behavior in the MSD measured in Newtonian samples[9].Once the corrected MSD was obtained, we followed dif-ferent procedures for the two classes of samples. Resultsfor MSD of the Newtonian fluids were simply fitted toa straight line and the sample viscosity η was obtainedfrom the slope, k B T / (6 πηa ) . For the PEO sample, weused the Kilfoil-group software to extract the frequency-dependent elastic and loss moduli, G ′ and G ′′ , respec-tively. The software implements an algebraical inversionprocedure based on the work of T.G. Mason et al. [7]. D. Diffusing Wave Spectroscopy (DWS)
DWS microrheology experiments were performed on apolymer solution with the PEO concentration also usedin II B. The tracer particles concentration (1%) was cho-sen to ensure multiple scattering [11]. In the limit ofmultiple scattering the autocorrelation function of thescattering intensity is given by g ( τ ) = Z ∞ P ( s ) e − k h ∆ r ( τ ) i sl ∗ ds (2)where k = 2 πn/λ is the wave-vector of light withwavelength λ (in our experiment nm) incident on amedium with refractive index n . P ( s ) is the scattering-geometry-dependent relative probability distribution ofphoton path lengths s inside the medium and l ∗ is thetransport mean free path, which quantifies the distancethat a photon has to travel inside the sample before loos-ing memory of its original direction. For our sample, wefound l ∗ = 256 ± µ m. The MSD can be extracted byinverting Eq. 2 [31].For our experiments we used the commercial instru-ment DWS Rheolab (LSInstruments, Fribourg, Switzer-land), a compact stand-alone optical microrheometerthat is based on DWS. The sample was hosted in a cu-vette of thickness L = 2 mm. Measurements were per-formed in transmission geometry with a duration of 3000s each. The MSD of the tracer particles was obtained inthe time range . µ s − . s and subsequently analyzedto extract the moduli G ′ and G ′′ . Figure 1. Schematic representation of the fitting-based procedure used to determine MSD from the image structure function.For a given wave-vector q , the image structure function D ( q, t ) is fitted to a model to obtain the noise baseline B ( q ) and thesignal amplitude A ( q ) . Using Eq. 6, an estimate for the MSD is obtained. The procedure is then repeated for all q -values inthe selected q -range and the best estimate for the MSD is obtained as the average of the curves obtained for all q -values. E. Differential Dynamic Microscopy (DDM)
We performed DDM measurements on all samples.LB5 particles were used for the four Newtonian sam-ples and for the PEO solution. The latter was alsostudied with LB1 particles; for this sample PT is notfeasible. Standard DDM analysis was based on a re-peated sequence of image subtractions and image Fouriertransforms [15, 17, 18]. In more detail, the image struc-ture function for all the accessible two-dimensional wave-vectors q = ( q x , q y ) was calculated for a set of time delays t according to D ( q , t ) = D | I ( q , t + t ) − I ( q , t ) | E t (3)where I ( q , t ) is the Fourier transform of the image I ( x , t ) acquired at time t in a fixed plane in which the horizontalposition is labeled by x = ( x, y ) . It has been recentlyshown that multiplying the images with a windowingfunction before performing the Fourier transform oper-ation removes the artifacts due to the finite image sizeand improves the determination of D ( q , t ) , especially forthose q for which the signal is comparable or smaller thanthe noise [32]. We thus apply this algorithm in our anal-ysis.The image structure function is quantitatively re-lated to the normalized intermediate scattering function f ( q , t ) and in most cases of interest the simple relation-ship D ( q , t ) = A ( q ) [1 − f ( q , t )] + B ( q ) (4)holds, where the functions A ( q ) and B ( q ) , usuallytreated as fitting parameters, are set by the spatial inten-sity correlations and the noise of the detection chain, re-spectively. The normalized intermediate scattering func-tion has some general properties such that f ( q ,
0) = 1 and f ( q , t → ∞ ) = 0 if the particles position are fullyuncorrelated for long times [33]. For a dilute collection of non-interacting particles one has f ( q , t ) = e − q h ∆ r ( t ) i (5)which is the two-dimensional equivalent of the main as-sumption on which DLS microrheology is based [7, 13].Thus, under conditions in which Eq. 5 holds, the MSDof tracer particles dispersed in a soft material can be ob-tained as h ∆ r ( t ) i = − q ln (cid:18) − D ( q , t ) − B ( q ) A ( q ) (cid:19) . (6)provided that an accurate fitting of the structure func-tions can be performed, as sketched in Fig. 1. Once theMSD is obtained it can be used to estimate the loss andelastic moduli of the sample, which is done here with thesame tools used to treat PT data.We note that even though the whole procedure to ex-tract the MSD from the DDM analysis of microscopemovies appears at first rather straightforward, a success-ful and accurate output requires the precise knowledgeof A ( q ) and B ( q ) . When an accurate fitting model for h ∆ r ( t ) i is available, as this is the case for freely diffusingparticles where h ∆ r ( t ) i = 4 D t , this can be done also ifthe key experimental parameters (image exposure time,acquisition frame rate, total number of images) do notallow to observe the full relaxation of the intermediatescattering function from one to zero. By contrast, if sucha model is not available any spurious effect altering thedetermination of A and B will impair the determinationof a correct MSD. Clearly, if DDM is to be used as a gen-eral purpose probe of the mechanical properties of softmaterials, suitable precautions need to be taken to guar-antee a model-free determination of the MSD. In SectionIII B, we will show how this task can be accomplished byreplacing the standard fitting-based DDM analysis witha suitable optimization-based DDM analysis. III. RESULTS AND DISCUSSIONA. Newtonian fluids: fitting-based DDM analysis
The DDM experiments presented here are aimed atmeasuring the viscosity of the Newtonian fluids seededwith ≃ . µm latex beads (LB5) described in detail inSection II. The key step of the analysis consists in ex-tracting the MSD of the tracers directly from the imagestructure function by using Eq. 6. To this aim the am-plitude A ( q ) and the baseline B ( q ) need to be knownwith high accuracy for each q , as any systematic error intheir determination would introduce a bias in the MSD.In particular, an overestimate (underestimate) of B ( q ) would lead to a spurious acceleration (deceleration) ofthe reconstructed tracers dynamics for small times.For monodisperse non-interacting colloidal particlesdispersed in a Newtonian fluid, the intermediate scat-tering function decays exponentially f ( q , t ) = e − D q t and A ( q ) and B ( q ) can be simply obtained by fittingthe image structure functions to Eq. 4.In our experiments, we found indeed that for all sam-ples the intermediate scattering functions were very welldescribed in terms of a single exponential relaxation forall the wave-vectors in the range of [2 . , . µm − forpure water, of [2 . , . µm − for . % glycerol in wa-ter, of [3 . , . µm − for . % glycerol in water, andof [3 . , . µm − for . % glycerol in water. In prac-tice, the width of the wave-vector range is set by the q -region in which both A ( q ) and B ( q ) are known accu-rately and Eq. 6 can be used to obtain the MSD from theintermediate scattering functions. For each wave-vectorin the range [3 . , . µm − we thus extracted an esti-mate for the MSD. These estimates were then combinedto obtain a q -averaged estimate of the MSD for all thesamples. These MSD are reported in Fig.2a for the fourNewtonian samples investigated here. All the curves arein excellent agreement with the PT results obtained byanalyzing the same image sequences. For the viscosity weobtain η meas = 0 . ± . , . ± . , ± , ± mPas. These values are in very good agreement with thoseexpected η lit = 0 . ± . , . ± . , . ± . , ± mPa s, [34], as shown in the inset of Fig. 2(a). The ex-perimental uncertainty on the value obtained for . %glycerol in water is due to the uncertainty in the samplecomposition. B. Newtonian fluids: optimization-based DDManalysis
As we will show in the following, the satisfactory re-sults obtained by using the standard DDM analysis withNewtonian samples depended on the fact that a modelfor the intermediate scattering function was readily avail-able. In general, this would not be the case, as the be-havior of a generic soft material is not known a priori .For this reason, we devised a simple, self-consistent pro- t [s]10 -4 -2 h ∆ r i [ µ m ] -6 -4 -2 (a) t [s]10 -4 -2 h ∆ r i [ µ m ] -6 -4 -2 (b) η lit [Pa/s] η m ea s [ P a / s ] Figure 2. MSD of LB5 tracers in different water-glycerol so-lutions. (a) symbols: MSD obtained from DDM with thefitting-based procedure (details in main text); dots: MSD ob-tained from PT; continuous lines: expected MSD from Ref.[34]. The viscosities η meas experimentally determined withDDM are shown in the inset as a function of the expectedvalues η lit (symbols). The error bars are smaller than thesymbols and the continuous line corresponds to the identity η meas = η lit . (b) symbols: MSD obtained from DDM with themodel-free procedure (see main text for details), continuouslines: expected MSD, as in panel a). cedure that exploits the multi- q capability of DDM toextract the MSD of tracer particles for an arbitrary sam-ple in a robust way. The proposed procedure builds onthe automatic determination of A ( q ) and B ( q ) based onan iterated optimization cycle.The general idea is sketched in Fig.3, where we showa block-diagram that depicts our fitting-free procedure.This procedure is based on an optimization cycle ini- Figure 3. Schematic representation of the optimization-based DDM analysis. The procedure is based on an optimization cycle(yellow arrows), fed by the experimental image structure function D ( q, t ) and by an initial set of parameters ( A ( q ) , B ( q )) . Theobject function is the dispersion σ of the reconstructed mean square displacements (see Eq. A3). New values of ( A ( q ) , B ( q )) are iteratively generated in order to minimize the object function. The output of the procedure is the optimal set of parameters ( A ( q ) , B ( q )) leading to the best estimate of MSD( t ). tially fed by a tentative amplitude-baseline parameterpair ( A ( q ) , B ( q )) , for q -values within a given interval [ q , q ] . These parameters are used to invert the corre-sponding image structure functions (Eq. 6), leading to a"bundle" of MSDs. If the considered pair ( A ( q ) , B ( q )) is the correct one for all q s, than the estimates for theMSD given by Eq. 6 are completely q -independent, re-sulting in an almost perfect collapse of all the curves.Any deviation of the parameters from the correct valuesintroduces a q -dependent dispersion. In our optimizationscheme, the dispersion σ of the curves (see Appendix Afor details) plays the role of an objective function: newvalues of ( A ( q ) , B ( q )) are iteratively generated until aminimum of σ is found. This algorithm, implementedin a custom code developed in MATLAB®, was found torapidly and robustly converge to a minimum for a widerange wave-vectors.Results obtained for the tracer MSD with thisoptimization-based procedure (Fig. 2b) are in excel-lent agreement with those obtained with the fitting-basedanalysis (Fig. 2a) over the whole investigated range ofdelay times . × − s < t < s , which validates theprocedure. Also, we note that the q -averaged MSD shownin Fig. 2b were obtained by averaging the MSD in therange . µm − < q < . µm − ; this range is widerthan that probed with the fitting-based procedure. Theusable q -range is larger in the optimization-based proce- dure, because the full relaxation of the image structurefunctions is here not a requirement for the determinationof the MSD, since A ( q ) and B ( q ) can be obtained self-consistently. Let us stress that the optimization-basedprocedure is largely model- and operator-independent.The only required external parameters are the relevant q -range [ q , q ] over which the optimization is performedand the initial values of the parameters ( A ( q ) , B ( q )) .The key importance of all these properties when study-ing arbitrary samples is described in detail in the nextsubsection. C. Viscoelastic fluid
In this subsection, we apply the optimization-basedprocedure to the data obtained with our model viscoelas-tic fluid, an aqueous solution of PEO, that exhibits elasticbehavior at short-times, high frequencies. The expectedshort time elastic plateau in the MSD would contributeto the baseline B ( q ) , requiring an independent determi-nation of the camera noise. Such requirement would besimilarly involved as the calibration procedure needed inPT- µ r experiments to account for the tracer localizationuncertainty. While such calibration is technically fea-sible, the optimization-based DDM analysis permits acalibration-free implementation of DDM- µ r. t [s]10 -4 -2 h ∆ r i [ µ m ] -4 -2 (a) -2 G ′ [ P a ] -2 (b) ω [rad/s]10 -2 G ′′ [ P a ] -2 (c) Figure 4. (a) Two-dimensional MSD of LB1 (orange triangles)and LB5 (blue circles) tracers in a viscoelastic polymer solu-tion (2% PEO2 in water) obtained from DDM analysis. Blackdots: same quantity obtained from PT analysis for the sam-ple with LB5 tracers. The small insets show representativeimages of the two samples: the one loaded with subdiffractionLB1 particles (left upper corner) and the one loaded with LB5tracers (right lower corner), respectively. (b) Comparison ofthe storage moduli G ′ estimated from the DDM-reconstructedMSD of LB1 tracers (orange triangles) and LB5 tracers (bluecircles), respectively. (c) same as in panel (b) for the lossmoduli G ′′ . Application of the optimization-based procedure to thePEO solutions with small ( ∼ nm) and large ( ∼ nm) tracers provides the results shown in Fig. 4a. Theaccessible range of probed timescales is very similar forthe two tracer sizes. For comparison we also show theresults obtained with PT for the sample containing the ω [rad/s]10 -2 G ′ , G ′′ [ P a ] -2 Figure 5. Comparison of the viscoelastic moduli G ′ and G ′′ of a 2% PEO polymer solution in water, obtained with dif-ferent methods. Gray circles (triangles): G ′ ( G ′′ ) obtainedwith traditional rheology; continuous blue (orange) line: G ′ ( G ′′ ) obtained with DWS using LB5 tracers; black continu-ous (dashed) line: G ′ ( G ′′ ) obtained with DDM microrheol-ogy (weighted average of the results of LB1 and LB5 tracers,shown individually in Fig. 4). larger tracers as black points in Fig. 4a; for the smallerparticles tracking is not feasible, as easily appreciatedfrom the images shown as insets.For each tracer size, we extracted from the MSD themechanical moduli G ′ and G ′′ , as shown in Fig. 4(b)and (c). Results obtained for G ′ with the two tracersizes are off by about 10-20% at small frequencies butthe two datasets are compatible within the experimen-tal errors. To obtain a statistically significant estimatefor the moduli, we combine the data obtained with thetwo tracers and show the results as black lines in Fig. 5.These data are in good agreement with the results ob-tained with traditional rheology, shown as open symbols,and also with the results obtained with DWS, shown asclosed symbols. DDM- µ r extends traditional rheology byone decade at high frequency, whereas at low frequencysimilar performances are obtained, at least as far as thestorage modulus is concerned. However, improvements inthe low-frequency region may be expected by increasingthe mechanical stability of the microscope setup.Let us underline that, without additional calibrationsteps, it would be very difficult to extract meaningfulMSD and thus mechanical moduli with a fitting-basedanalysis of the DDM-data. In the limit of short timesthe MSD displays a non-trivial scaling, compatible witha power-law M SD ≃ t γ with an exponent γ close to . . The counterpart of this behavior in the Fourier t [s]10 -2 D ( q ,t ) [ A . U .] (a) t [s]10 -2 h ∆ r i [ µ m ] -4 -2 (b) t [s] -4 -2 D ( q ,t ) [ A . U .] Figure 6. Effect of the model-dependent determination of the noise baseline B ( q ) on the reconstructed MSD. (a) Symbols:image structure function D ( q, t ) (for q = 6 . µm − ) obtained from DDM on LB5 tracers in a viscoelastic solution of PEO inwater. The continuous red line is an exponential fit to the data obtained at large delay times ( t > s); this fit allows for theestimation of the plateau height A ′ ( q ) . Inset: close-up of the short time behavior of D ( q, t ) (symbols). The data is fitted withdifferent functions, leading to different estimates of the baseline B ( q ) : linear fit over the first data points (continuous blueline), linear fit over the first data points (dashed orange line), fit over the first 20 data points with a function of the form y = ax . + b (dashed-dotted yellow line) and fit over the same interval with a function of the form y = ax . + b (dotted greenline). (b) Mean square displacement obtained from Eq. 5 using the amplitude A ( q ) = A ′ ( q ) − B ( q ) and the noise baseline B ( q ) obtained from the different fitting models shown in the inset of panel (a). Curves are color-coded according to the fits, theblack continuous line is the result of the model-free procedure shown in Fig. 4(a) as blue circles. space is an image structure function taking the form of D ( q, t ) ≃ C ( q ) t γ + B ( q ) at short times. Clearly, an expo-nential or a polynomial fit of D ( q, t ) are inadequate to de-scribe this behavior and any estimate of the baseline B ( q ) based on an exponential or a polynomial fit provides a bi-ased, incorrect result, as shown in Fig. 6. Choosing othermodel functions, such as for instance a power-law withdifferent exponents, also fails, even though the data mayseem deceivingly well described at short times. By con-trast, the optimization-based procedure self-consistentlydetermines the MSD without need of fitting the experi-mentally determined image structure functions. IV. CONCLUSIONS
Microrheology is a very powerful complement to tra-ditional, mechanical rheology [6, 8–10]. For the high-frequency range, rheology is usefully complemented byDWS- µ r [11], whereas in the low-frequency limit bothDLS- µ r [12] and PT- µ r [7] have been usefully employedin the past. PT- µ r is technically the less demanding tech-nique, not requiring any laser source or digital correlation board and is also very flexible for biophysical applica-tions, owing to the possibility of employing different sam-ple contrast mechanisms. However, in its practical real-ization one encounters some challenges. Accurate track-ing algorithms require several input parameters, such asa typical value for the particle radius, a score cut-off todiscriminate signals that are not due to presence of aparticle, an intensity threshold to consider bright pixelsas particles, etc. The results of the tracking dependsseverely on the choice of these parameters that, even forexperienced users, may be sometimes more difficult thanexpected [35]. Also, the extraction of the tracer MSDfrom PT trajectories requires the knowledge of the in-trinsic particle localization uncertainty, which is usuallydetermined by calibration with particles that are keptfixed in space or that freely diffuse in a Newtonian fluidwith similar optical properties [9].We have shown here that DDM [15], a technique thatretains the simplicity and flexibility of PT in terms of ex-perimental setup and applications, can be also used foraccurate microrheology experiments. We also show thatDDM- µ r outperforms PT with small particles in brightfield microscopy. Finally, if an optimization-based algo-rithm is used instead of the standard fitting-based ap-proach, DDM- µ r does not require any calibration or userinput, which limits dramatically the degree of arbitrari-ness on the determination of the mechanical moduli ofthe sample. However, particle tracking is expected to besuperior to DDM in the presence of unwanted and movingscatterers that, being potentially discarded by an accu-rate particle tracking, would affect DDM- µ r experiments.It is likely that these and other DDM features, suchas its capability to handle optically dense samples, forwhich tracking becomes extremely challenging if not im-possible, will make DDM- µ r a useful addition to the port-folio of rheo-scientists, both in academic and in industrialresearch laboratories. Appendix A: Optimization-based determination ofMSD
In this appendix we describe the fitting-free optimiza-tion procedure used to extract from the experimental im-age structure function D ( q, t ) the best estimate for thetracers’ mean square displacement. The main steps ofthe procedure are the following:1. Choice of the interval [ q , q ] of wave-vectors overwhich the optimization is performed. The intervalshould be a subset of the accessible q -range with afair signal to noise ratio. This condition can be alsochecked retrospectively at the end of the procedure,when a q -resolved estimate of the amplitude A ( q ) and the noise background B ( q ) is obtained.2. Choice of the initial set of parameters ( A ( q ) , B ( q )) . This can be done, for exam-ple, by fitting, for each q ∈ [ q , q ] , D ( q, t ) witha linear function near the origin and with aexponential function for large delays (as done forexample in Fig.6).3. Calculation of the mean square displacement M SD ( t | q ) using Eq. 5 for each q in the selectedinterval.4. Determination, for each delay time t , of the subset J ( t ) of q -values such that M SD ( t | q ) < q − . Thischoice ensures that, if q ∈ J ( t ) , then D ( q, t ) hasnot completely lost track of the signal correlationfor that value of q and can be thus meaningfullyinverted. Let N ( t ) be the number of elements in J ( t ) . 5. Calculation of the average mean square displace-ment M SD ( t ) = 1 N ( t ) X q ∈ J ( t ) M SD ( t | q ) . (A1)6. Calculation of the t -dependent dispersion σ t ( t ) as σ t ( t ) = 1 N ( t ) − X q ∈ J ( t ) log M SD ( t | q ) M SD ( t ) . (A2)and of the total dispersion σ as σ = X t σ ( t ) . (A3)7. Generation of a new set of parameters and repe-tition of the procedure from step 3 unless a localminimum in σ is reached (or the prescribed max-imum number of iterations is exceeded).8. If the procedure converges to a minimum of σ ,the optimal set of parameters ( A ( q ) , B ( q )) repre-sents the best estimate for the q -dependent ampli-tude and noise baseline, respectively, and the cor-responding average mean square displacement (Eq.A1) is the best estimate for the tracer’s MSD.Many algorithms are available to search the minimum of σ and to guide the generation of new sets of parametersin step 7. In our implementation, the optimization cycle3-7 was realized using the MATLAB function fminsearch ,which is based on the simplex search method of Lagarias et al. [36]. More refined implementation could possiblyinclude suitable weights when computing the averages inthe right-hand sides of Eqs. A1-A3, accounting for thedifferent statistical errors affecting each term. Also, aneffective weighting scheme could provide an efficient wayto reject the contribution of the most noisy wave-vectorsmaking unnecessary the explicit selection of a predeter-mined optimization interval (step 1). ACKNOWLEDGMENTS
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