Accounting for inertia effects to access the high-frequency microrheology of viscoelastic fluids
P. Domínguez-García, Frédéric Cardinaux, Elena Bertseva, László Forró, Frank Scheffold, Sylvia Jeney
aa r X i v : . [ c ond - m a t . s o f t ] D ec Accounting for inertia effects to access the high-frequency microrheology ofviscoelastic fluids
P. Dom´ınguez-Garc´ıa, Fr´ed´eric Cardinaux,
2, 3
Elena Bertseva, L´aszl´o Forr´o, Frank Scheffold, and Sylvia Jeney Dep. de F´ısica de Materiales, Universidad Nacional de Educaci´on a Distancia (UNED), Madrid 28040, Spain ∗ Department of Physics, University of Fribourg, 1700 Fribourg Perolles, Switzerland LS Instruments AG, Passage du Cardinal 1, CH-1700 Fribourg, Switzerland Laboratory of Physics of Complex Matter, Ecole Polytechnique F´ed´erale de Lausanne (EPFL), 1015 Lausanne, Switzerland (Dated: October 6, 2018)We study the Brownian motion of microbeads immersed in water and in a viscoelastic worm-like micelles solution by optical trapping interferometry and diffusing wave spectroscopy. Throughthe mean-square displacement obtained from both techniques, we deduce the mechanical propertiesof the fluids at high frequencies by explicitly accounting for inertia effects of the particle and thesurrounding fluid at short time scales. For wormlike micelle solutions, we recover the 3/4 scaling ex-ponent for the loss modulus over two decades in frequency as predicted by the theory for semiflexiblepolymers.
PACS numbers: 83.80.Qr, 82.70.-y, 83.85.Ei
The quantitative stochastic description of Brownianmotion [1] of spherical micro- and nanobeads in a com-plex fluids has laid the foundations for the inventionof tracer microrheology [2, 3], a powerful, noninvasivemethod that allows the measurement of mechanical prop-erties over an extended range of frequencies using all op-tical instrumentation. At very short time-scales, or highfrequencies, the stochastic description of Brownian mo-tion fails as pointed out already in the original work ofEinstein [4]. At microsecond time scales the influence ofinertial effects and hydrodynamic memory becomes siz-able [5, 6]. Failing to account for these contributionsleads to substantial errors. Removing these effects [7, 8]at high frequencies ω when calculating the complex mod-ulus, G ∗ ( ω ), may allow one to discern relevant data thatis otherwise difficult or impossible to access. Elastic mod-uli at such high frequencies may contain important infor-mation about living cells [9], biopolymers in pharmaceu-tical applications [10] or fast processes encountered, forexample, in ink-jet printing [11].In this work, we demonstrate how to correct for the in-fluence of inertia in an actual experiment. We study theBrownian motion of microbeads immersed in water andin a viscoelastic wormlike micelle solution by two com-plementary experimental techniques accessing the MHzfrequency range: optical trapping microscopy (OTI) anddiffusing wave spectroscopy (DWS). The combined appli-cation of these two methods is unique since it covers themost relevant approaches to high-frequency microrheol-ogy, while other methods, such as particle tracking, arelimited to frequencies well below 10 kHz [12]. We ac-count for inertia effects quantitatively using two differentapproaches: the self-consistent correction of the mean-square displacement suggested in [13] and a theoreticalexpression derived from the recent work of Schieber andcollaborators [14, 15]. We find that these two differentquantitative methods provide similar results, as shown by a study of the high-frequency scaling of the modulus G ′′ ( ω ) ∼ ω α of a viscoelastic wormlike micelle solution.Taking into account inertia effects we find an exponent of α ∼ = 0 .
75, as predicted by the theory for semiflexible poly-mers [16]. Besides, the proposed methodology allows oneto extract parameters relevant to the intrinsic propertiesof the viscoelastic fluid, such as, the bending modulus,the mesh size and the contour length of the molecularcomponents.In a tracer microrheology experiment, the complexmodulus of a bulk material is calculated from the mea-sured mean-square displacements MSD ≡ (cid:10) [ r ( t ) − r (0)] (cid:11) of microbeads with position r ( t ) [2, 17]. OTI is the moreversatile of the two techniques as it acts on individualbeads and, therefore, does not rely on averaging over themotion of many beads, as is the case for DWS. In OTI,the movement of a single microbead is recorded by meansof an interferometric position detector [18]. The bead istrapped in the center of a sample chamber using opticaltweezers [19], applying the lowest optical force available,3 . k . µ N / m, where k is the spring constant of theoptical restoring force. DWS is an extension of dynamiclight scattering applied to materials with strong multi-ple scattering [12, 20–22] and it allows precise measure-ments of the three-dimensional movement of tracer beadsin the complex fluid. The advantages of DWS are its rel-ative ease of use employing standard spectrophotometercuvettes and the minute-scale measurement times withlittle calibration and post-processing required. DWSalso provides access to a large range of experimental pa-rameters and can be applied to viscoelastic fluids andsolids with low- or high-frequency moduli while coveringan extended range of frequencies from ω ∼ . rad/s [13]. By directly comparing the results obtainedby each of the techniques, we study the Brownian mo-tion of melamine resin microbeads with radii a = 0 .
94 or1 . µ m, with density ρ p = 1570 kg/m at T = 21 C. (b) H O, a = 1.47 µ mk = 6 µ N/m -7 -6 -5 -4 -3 (d) Mic. 4%, a = 1.47 µ m -7 -6 -5 -4 -3 M S D ( µ m ) (a) H O, a = 0.94 µ mk = 8 µ N/m -7 -6 -5 -4 -3 -8 -7 -6 -5 (c) t (s) Mic. 4%, a = 0.94 µ m FIG. 1. (Color online) One-dimensional particle MSD in wa-ter and in a 4% micellar solution for two different bead sizes a .Experimental data: OTI ( ) and DWS ( ). Corrected MSDsare shown as triangles: from OTI ( ) and from DWS ( ).Theoretical predictions for water: Chandrasekhar’s result ig-noring inertia effects ( ) and Hinch’s prediction includingthe influence of inertia ( ). The relatively high refractive index ( n = 1 .
68) provides-good trapping efficiency in OTI experiments. It has beenreported in [23] that the colloid surface chemistry of themicrobeads can affect the results obtained by microrhe-ology. Such effects are particularly relevant for proteinspecific bonding in biomaterials. In our case, however,the resin beads are chemically non-active and thereforewe expect such effects to be negligible. The sample un-der study is an aqueous solution at 4 wt % of surfactantcetylpyridinium chloride (CPy + Cl − ) and sodium salicy-late (Na + Sal − ) that self-assemble into a wormlike micellesolution [24]. At sufficiently high frequencies the cylin-drical micelles are expected to behave as a solution ofsemiflexible polymers in the semi-dilute regime formedby entangled micelles [25]. We measure the steady-stateviscosity using a Rheometer MCR502 (Anton Paar, Aus-tria) at 21 C, which yields to η = 360 mPa · s. In theOTI experiments, a double flow chamber is used: Oneof the chambers contains water and beads at very lowconcentration, while the other one encloses the viscoelas-tic fluid with the same type of beads. After aligning theoptical trapping light path to the position detector andadjusting signal amplification, a calibration experiment isperformed in the pure water solution. The measurementin the viscoelastic solution is done right afterwards. Thisprocedure allows using the known properties of a New-tonian fluid [26] to extract the volts-to-meter conversionfactor β , which is then used to calculate the MSD in theviscoelastic medium [27]. For the DWS experiments, weare using a bead concentration of 2 vol. %. Samples areloaded into standard rectangular spectrometer cuvettes with a path length L = 2 mm or L = 5 mm and a widthof 10 mm. Echo two-cell DWS experiments in transmis-sion geometry are performed as described in Ref. [28].The two experimental techniques provide similar MSDsfor the bead motion in water, as shown in Figs. 1(a)and 1(b). However, for the micelle solution, some differ-ences are observed [Figs. 1(c) and 1(d)], in particular forthe larger bead size. Similar global shifts of the MSDhave been observed previously [29, 30]. They have beenattributed to hydrodynamic effects at the fluid-particleinterface and local perturbations of the equilibrium con-figuration of the complex fluid [31]. The optical forcein a micelle solution might trap some of the polymericstructure, hindering the Brownian motion and, thereby,slightly incrementing the apparent measured viscosity ofthe fluid. DWS in turn is sensitive to depletion of the sur-factant solution around the beads and a possible onset ofdepletion-induced bead attraction which can result in en-hanced motion and larger MSD values [30]. To observethe effects of inertia and hydrodynamics in the MSDs,we compare in Fig. 1(a) and 1(b) the results obtainedfor bead motion in pure water with the Chandrasekharexpression for a classic Newtonian fluid [32], given by (cid:10) ∆ r ( t ) (cid:11) = [1 − exp( − kt/γ )] 2 k B T /k where γ = 6 πηa .For the case of DWS, there is no optical trap and theclassical Stokes-Einstein result (cid:10) ∆ r ( t ) (cid:11) = ( k B T / πηa ) t is recovered in the limit k →
0. In both cases, sub-stantial deviations due to inertial effects are observed attimes shorter than t = 10 − s. For water, the inertia ef-fects in the MSD can be reproduced quantitatively usingthe classical result according to Hinch [33] [Fig. 1(a) and1(b)].The standard formalism to convert the measured MSDto the complex elastic modulus G ∗ ( ω ) is the Mason-Weitz(MW) approach based on the generalized Stokes-Einsteinrelation (GSER) [22] concurrent with Mason’s approxi-mation [34] to obtain: Z ∗ ( ω ) = k B Tiωπa h ∆ r ( ω ) i , (1)where h ∆ r ( ω ) i is the one-side Fourier transform of theMSD. In the absence of inertia effects, G ∗ ( ω ) ≡ Z ∗ ( ω )for equilibrium Brownian motion in a homogeneous vis-coelastic fluid [17]. An alternative methodology to con-nect the MSDs with the rheological properties of the fluidwas reported by Evans et al [35]. Both methods [36] havecommon limitations, such as the omission of optical orexternal forces, the neglect particle and fluid inertia andthe non-consideration of active and heterogeneous ma-terials [37]. The inertia effects appearing in the power-spectral density (PSD) of probe particles using opticaltweezers were studied in Refs. [38, 39], while a system-atic approach to account for inertia effects in the MSDobtained from passive bead microrheology has been de-scribed in Ref. [13]. The principal idea in the latter (OTI) G G’’ min G’ G’’ ω (s -1 ) Storage and loss modulus (Pa) G’ G’’(DWS)
FIG. 2. (Color online) Comparison of the microrheology re-sults using OTI and DWS (Inset) for one bead size a = 0 . µ m in a viscoelastic micelle solution with and without tak-ing into account the effects of inertia. Data without inertiacorrection ( ), corrected using Eq. (2) ( ) and using thecorrected-MSDs shown in Fig. 1 ( ). work is to define an effective viscosity of the mediumby η eff ( t ) = [ k B T / π (cid:10) ∆ r ( t ) (cid:11) a ] t and then calculate,for each measured time point t , the correction factor f ( t, η eff ( t )) ≥ et al. [14] and C´ordoba et al. [14, 15]. The latter work conveniently provides an an-alytical expression relating the actual complex modulusand the bead mean-square displacement, assuming thatthe medium is incompressible [43]: G ∗ ( ω ) = Z ∗ ( ω ) + m ∗ ω πa + a ω × "s ρ − ρ πa (cid:18) πaω Z ∗ ( ω ) + m ∗ (cid:19) − ρ , (2)where m ∗ = m particle + 2 πρa / Z ∗ ( ω ) and then using Eq.(2), we obtain the complex moduli without the influenceof inertia. It is important to note that other possible cor-rections to the high-frequency bead motion, for exampledue deviations from the GSER at ω ≥ Hz, are notincluded by these corrections [17].We apply these correction protocols to the experimen-tal MSDs displayed in Fig. 1 to obtain the complex mod-ulus for the micelle solution. In Fig. 2 we show the resultsderived from DWS and OTI using a bead size a = 0 . µ m [44] for OTI and DWS, with and without corrections.The inertia-corrected results for G ′′ ( ω ) calculated usingboth methods yield to similar results. These results can Mic. 4%, a = 0.94 µ m (b) DWS Mic. 4%, a = 0.94 µ m (a) OTI L o ss m odu l u s ( P a ) Mic. 4%, a = 1.47 µ m ω (s -1 ) (c) OTI Mic. 4%, a = 1.47 µ m (d) DWS FIG. 3. (Color online) High-frequency loss modulus, G ′′ ( ω ) − ωη s compared to theoretical predictions. Inertia-correcteddata using Eq. (2) ( ) and using the corrected-MSDs shownin Fig. 1 ( ). (a) OTI, a = 0 . µ m. (b) DWS, a = 0 . µ m.(c) OTI, a = 1 . µ m. (d) DWS, a = 1 . µ m. Black line( ) is G ′′ GMK ( ω ) − ωη s evaluated using Eq. (3) and datafrom Table I. be compared to the the expected behavior at higher fre-quencies for a solution of semiflexible polymers, which,according to Gittes and MacKintosh [16], is given by: G ∗ GMK ( ω ) = iωη s + 115 ρ m κl p (cid:18) − iξκ (cid:19) / ω / (3)Then, it is expected that both moduli, loss and stor-age, follow the G ∗ ∼ ω / behavior at high frequencies.However, at high frequencies the viscous response of thematerial dominates the elastic response by about an or-der of magnitude. This makes it very difficult to ex-tract meaningful information about G ′ from microrheol-ogy. We thus restrict our discussion to the comparisonof Eq. (2) using only the experimental loss modulus, G ′′ ,for CpyCl/NaSal to study in more detail the accuracyof inertia corrections at high frequencies [45]. Next, wecompare Eq. (3) using only the experimental loss modu-lus for CpyCl/NaSal to study in more detail the accuracyof inertia corrections at high frequencies. To evaluate Eq.(3), we use the standard theory of polymers [46] and theexperimental G , G ′′ min , and ω as input parameters. Thequantity G is the value of G ′ at which G ′′ has a localminimum, denoted as G ′′ min . Both characteristic valuescan be easily obtained from Fig. 2. The third inputparameter is ω , the crossover frequency where the expo-nent α of the power-law behavior G ′′ ∼ ω α changes fromthe Rouse-Zimm behavior to the expected 3/4 at higherfrequencies [47]. The persistence length of the polymer-like micelles is then linked to ω by l p = ( kT / η s ω ) / .The micelles diameter is estimated as d mic = 2 . ρ m = φ/ [( π/ d ] = 8 . × TABLE I. Characteristic magnitudes for the wormlike micelle solution obtained from OTI and DWS using Eq. (3). For alldata: d mic = 2 . ρ m = 8 . × m − . a ( µ m) Tech. ω (s − × ) G (Pa) G ′′ min (Pa) l p (nm) κ (Jm × − ) ξ (nm) ζ (Ns/m × − ) l e (nm) L (nm)0 .
94 OTI 1 . ± . ± ± . ± . . ± . ± . ± . ± ± .
94 DWS 1 . ± . ± ± ± . ± . ± . ± . ± ± .
47 OTI 1 . ± . ± ± ± . ± . ± . ± . ± ± .
47 DWS 3 . ± . ± ± . ± . . ± . ± . ± . ± ± m − , where φ is the volumetric concentration [30]. Then,we calculate the bending modulus, κ = kT l p , the meshsize, ξ = ( kT /G ) / , the lateral drag coefficient, ζ =4 πη s / ln(0 . λ/d mic ) where λ = ξ , the contour length be-tween two entanglements, l e = ξ / /lp / , and the con-tour length of the micelles, L = l e G /G ′′ min . The resultsof the calculations for these quantities are summarized inTable I. The different experimental methods (DWS andOTI) as well as the two bead sizes give similar results.The averaged persistence length, l p ∼ ± ξ ∼
42 nm, is slightly lower (52 nm). To quan-tify the inertia corrections for the viscoelastic fluid atfrequencies 10 -10 Hz, we plot in Fig. 3 the loss mod-ulus obtained from applying the two different correctionprocedures. The results quantitatively agree with theory,except for some deviations due to the limited accuracy ofthe experimental methods at higher frequencies. Equa-tion (1) applied to the inertia-corrected MSDs and to theIndei-Schieber method [Eq. (2)] yield to similar results,matching the theoretical curves and the ω / behavior,especially when using OTI and small beads. Differencesbetween both methods arise when using DWS and biggerbeads, where Eq. (2) provides better results.In conclusion, our experimental data and analysisdemonstrate that, when properly accounting for inertiaeffects, passive microrheology easily provides access tothe mechanical properties of viscoelastic fluids up to fre-quencies in the MHz scale. 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Schmidt, Rev. Sci. Instrum. , 3246 (2003).[39] K. Berg-Sørensen and H. Flyvbjerg, Rev. Sci. Instrum. , 594 (2004).[40] D. Mizuno, D. A. Head, F. C. MacKintosh, and C. F.Schmidt, Macromolecules , 7194 (2008).[41] In that work, the Fourier transform was carried out firston the measured time series of bead positions to obtainthe PSD. The inertia correction is then executed on thePSD using the known expression for the frequency depen-dent particle mobility in a simple liquid using an effectivecomplex viscosity of the medium derived from the exper-imental data.[42] B. U. Felderhof, J. Chem. Phys. , 164904 (2009).[43] Equation (37) of Ref. [14] is the follow-ing G ∗ ( ω ) = k B Tiωπa
MSD + m ∗ ω πa + a ω × (cid:20)r ρ + ρ πa (cid:16) k B T ( iω MSD) − m ∗ (cid:17) − ρ (cid:21) , which can berelated to Eq. (1) since the term in the square root is6 k B T / ( iω MSD) = − (6 πa/ω ) Z ∗ ( ω ).[44] An equivalent figure for a = 1 . µ m is included in theSupplemental Material.[45] Using the GSER at high frequencies can introduce non-expected effects. See Supplemental Material documentfor extra comments about this point.[46] M. Doi and S. F. Edwards, The Theory of Polymer Dy-namics. (Clarendon Press, Oxford, 1986).[47] For DWS experiments we use G and G ′′ min from OTI.We select ω for the experimental G ∗ to match G ∗ GMK ,obtaining ω ∼4