Diffusing Wave Microrheology of Highly Scattering Concentrated Monodisperse Emulsions

Diffusing wave microrheology of highly scattering concentrated monodisperse emulsions

Ha Seong Kima, Nesrin S¸enbilb, Chi Zhangb, Frank Scheffoldb,1, and Thomas G. Masona,c,1

aDepartment of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095; bDepartment of Physics, University of Fribourg, CH-1700 Fribourg, Switzerland; and cDepartment of Physics and Astronomy, University of California, Los Angeles, CA 90095

Edited by Steve Granick, IBS Center for Soft and Living Matter, Ulju-gun, Ulsan, Republic of Korea, and approved February 27, 2019 (received for review October 3, 2018) Motivated by improvements in diffusing wave spectroscopy nating light (4). Because the standard analytical framework of (DWS) for nonergodic, highly optically scattering soft matter and DWS neglects collective scattering of colloidal probes at high by cursory treatment of collective scattering effects in prior DWS densities, DWS MSDs extracted for such dense colloidal sys- microrheology experiments, we investigate the low-frequency tems do not necessarily represent the true self-motion of the 0 plateau elastic shear moduli Gp of concentrated, monodisperse, probes. Consequently, using DWS MSDs for thermal-entropic disordered oil-in-water emulsions as droplets jam. In such exper- passive microrheology (6, 7) of dense dispersions, including con- iments, the droplets play dual roles both as optical probes and centrated emulsions, would likely lead to inaccurate predictions as the jammed objects that impart shear elasticity. Here, we of their linear viscoelastic moduli, since passive microrheol- demonstrate that collective scattering significantly affects DWS ogy requires accurate self-motion MSDs in the generalized mean-square displacements (MSDs) in dense colloidal emulsions. Stokes–Einstein relation (GSER) (6, 8). By measuring and analyzing the scattering mean free path as a Despite these potential issues related to collective scatter- function of droplet volume fraction φ, we obtain a φ-dependent ing, DWS MSDs have been used since the inception of passive average structure factor. We use this to correct DWS MSDs by up microrheology to infer the linear viscoelastic response of numer- 0 to a factor of 4 and then calculate Gp predicted by the generalized ous soft materials, such as hard spheres (6, 8), emulsions (6, Stokes–Einstein relation. We show that DWS-microrheological 8), polymer solutions (6, 9–11), DNA solutions (12–14), actin 0 0 Gp(φ) agrees well with mechanically measured Gp(φ) over about solutions (15–17), microgels (18), and micellar solutions (19– three orders of magnitude when droplets are jammed but only 21). For at least some of these systems, and particularly for weakly deformed. Moreover, both of these measurements are concentrated emulsions, collective scattering could play a sig- consistent with predictions of an entropic–electrostatic–interfacial nificant role. In addition, early DWS studies of nonergodic (EEI) model, based on quasi-equilibrium free-energy minimization soft materials (8), including concentrated emulsions, were taken of disordered, screened-charge–stabilized, deformable droplets, using a simple DWS apparatus that did not force a final which accurately describes prior mechanical measurements of long-time decay (22). As a consequence, for such nonergodic 0 Gp(φ) made on similar disordered monodisperse emulsions materials, the values of the plateaus varied significantly from over a wide range of droplet radii and φ. This very good run to run, precluding accurate comparisons of shear elastic quantitative agreement between DWS microrheology, mechanical rheometry, and the EEI model provides a comprehen- sive and self-consistent view of weakly jammed emulsions. Significance Extensions of this approach may improve DWS microrheology on other systems of dense, jammed colloids that are highly By treating collective light scattering in dense colloidal sys- scattering. tems, we improve the analysis approach for diffusing wave spectroscopy (DWS), a dynamic light-scattering technique developed for white, highly scattering soft materials. Using microrheology | jamming | emulsions | diffusing wave spectroscopy | this improved DWS analysis, we accurately measure the viscoelasticity bounded translational motion of crowded oil droplets within concentrated monodisperse oil-in-water emulsions resem- iffusing wave spectroscopy (DWS) (1, 2) is a dynamic light- bling mayonnaise. We show that the plateau elastic shear Dscattering (DLS) technique that can be used to measure moduli of such emulsions determined by passive microrhe- time-dependent mean-square displacements (MSDs), ∆r2 (t) , ology, based on this true bounded droplet motion and of uniform spherical probe particles in opaque, highly scattering the generalized Stokes–Einstein relation, match very closely colloidal dispersions. In DWS, the transport of light is mod- with mechanical measurements and model predictions as the eled as a random walk having an optical transport mean free droplets become more tightly jammed. This excellent quanti- path `∗. The diffusion equation is then applied to a specific tative agreement represents a major improvement over prior sample-cell geometry, which typically has a thickness far in excess microrheology measurements on similar jammed emulsions of `∗, while taking into account the illumination and detec- that yielded only qualitative trends. tion configuration used. DWS is a powerful approach because it is capable of measuring colloidal dynamics over a wide range Author contributions: F.S. and T.G.M. designed research; H.S.K., N.S., and C.Z. performed research; H.S.K., N.S., C.Z., F.S., and T.G.M. analyzed data; and H.S.K., F.S., and T.G.M. of time scales, and it is also sensitive to very small probe dis- wrote the paper. y placements approaching 1 A˚ (1–5). Provided that the scattering Conflict of interest statement: F.S. is a board member and shareholder of LS Instru- probes are well dispersed and dilute, their self-motion MSDs ments AG.y can be accurately inferred from the measured DWS intensity This article is a PNAS Direct Submission.y autocorrelation function (1–5). Published under the PNAS license.y By contrast, for probes at densities well beyond the dilute limit, 1 To whom correspondence may be addressed. Email: [email protected] or frank. collective light-scattering effects could significantly influence [email protected] decays and plateaus in DWS correlation functions, particularly This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. when probes are at high-volume fractions φ and have diameters 1073/pnas.1817029116/-/DCSupplemental.y comparable to or smaller than the wavelength of the illumi- Published online March 28, 2019.

7766–7771 | PNAS | April 16, 2019 | vol. 116 | no. 16 www.pnas.org/cgi/doi/10.1073/pnas.1817029116 Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 i tal. et Kim moduli Entropic–Electrostatic– shear elastic and plateau sured Modulus Shear Model. Interfacial Plateau Mechanical Discussion and Results of systems DWS-GSER disordered objects. quantitative colloidal jammed scattering improving other highly for of basis approaches microrheology a analytical passive and as experimental serve our could that emulsions, anticipate jammed for we obtained have we that comparison titative microrheological hoc ad an used GSER, the in the used for and corrected factor yield when structure DWS, average using φ-dependent measured long-time that MSDs show making plateau when we 6) Here, (3, scales interpretations. time focus microrheological inertial we than droplets inertia, longer viscous scales by less time introduced on for complications times avoid early to very and, otherwise (28), at can DWS by that detected fluctuations highly be interfacial a entropic has suppress which we of each samples, on controlled emulsion experiments same rheometry the mechanical exactly and in uncertainties DWS small both for even perform regime jamming the in large of function a as DWS emulsions, modern monodisperse microscale and jammed, disordered, rheometry of microrheology, mechanical both using moduli, shear systematic measured elastic a plateau collective the can present of which rectify comparison we experimental systems, microrheology, to colloidal passive need dense affect of the adversely DWS by in effects also scattering and systems order nonergodic the on seconds short, of time tens measurement minutes. of total of scales the time keeping to correlation while MSDs DWS the extracted useful of and range are function temporal signals the echo times extend DWS at they Such collected because decay. be forced also the can every than (27) periodically longer this signals beam echo of illuminating DWS scatterers then the (24– cycle, identical at into decay that rotated move is such forced object scatterer rigid frequency, final rigid angular those the moving fixed slowly than of a the shorter scale if significantly Moreover, time mate- 26). scales the soft time nonergodic with for over associated and least ergodic at for both rials, reproducible run are to software, run and from by hardware obtained correlator’s behavior function, digital correlation early-time DWS the the that of ensures values plateau effectively which and improve- technique This significantly time, times. without in long earlier motion, ment at its certain function of correlation a rate the the at affecting by fully controlled cor- The be decay DWS can material. the into to forces soft surface) function scatterer nonergodic rough moving relation the additional a this includes having of that scat- motion glass rigid path etched moving light slowly (e.g., the a 26) or 24) (25, (23, terer cell second involves a solution introducing effective one materials, nonergodic of functions (8). rheometry mechanical with microrheology DWS-GSER using obtained moduli o nld ietcmaio ihameasured did a with that comparison direct nanoemulsions a include concentrated not on recent beyond work the goes light-scattering study for our account Moreover, rheometry. not mechanical a did only accurate, in demonstrated trend as and qualitative factor, not structure was average that tech- φ-dependent DWS 8) refined less (6, and microrheological nique simpler earlier a using over made study jamming, improvement measurements Our droplet jam. major droplets of a as magnitude model represents of analytical orders three an about as over well as rheometry, oiae yteeipoeet nDStcnqe for techniques DWS in improvements these by Motivated correlation DWS in variability undesirable this alleviate To G p 0 (φ) yuigahg-icst i nieordroplets, our inside oil high-viscosity a using By φ. hc ace hto arsoi mechanical macroscopic of that matches which idpnetcreto atrt ecl the rescale to factor correction φ-independent G ecmaetemcaial mea- mechanically the compare we 1A, Fig. In p 0 G (φ) p 0 (φ) 2) ie h tiigyacrt quan- accurate strikingly the Given (29). ntejmigrgm oprdwith compared regime jamming the in eas netite in uncertainties Because φ. G p,mech 0 ftemonodisperse the of G p 0 (φ) G n that and p 0 a be can we φ, G p 0 , ueet nohrfatoae mlin hthv similar G have that emulsions mea- mechanical fractionated past compositions. other with consistent on highly surements is emulsion this of that find with We agreement here. explored the deforma- have below lies we interfacial regime entropic droplet The dominates). (i.e., tion regime jammed screened interfacially and (i.e., dominates) regime repulsion jammed charge electrostatically the to responds ocnrto 3) ydtriigtedmnn contribution dominant SDS the bulk determining to 10-mM same By the (30). at concentration radii microscale shown and been nanoemulsions having have monodisperse O/W that (PDMS) siloxane those polydimethyl are lar measured model mechanically EEI the describe the that to in values here parameter The use disordered limit. we jamming for weak approximation the reasonable in emulsions a is interfa- quasi- deformation, droplet a and cial repulsions, of entropic electrostatic to minimization screened related crowding, terms that includes which assumes energy, free model equilibrium EEI jamming the The above and point. near monodis- emulsions disordered (O/W) stabilized, oil-in-water perse ionically of model concentrated, values (EEI) for predicted entropic–electrostatic–interfacial (30) to the Methods), by radius droplet and obtained average (Materials an nm has which 459 study, this in emulsion in rdce relative Predicted (B) (yellow). in interfacial uncertainty and (green), electrostatic EEI (blue), the [i.e., in to energy Regimes tions free text]. its main in (30); has terms model that interfacial droplets and stabilized electrostatic, of ionically entropic, prediction concentrated, line: uniform, disordered, solid radius of droplet Red average circles). an solid having (red emulsion monodisperse a of 1. Fig. p 0 φ ic h E oe mohycpue h te iein rise steep the captures smoothly model EEI the Since G euei odmntaeta vnrltvl small relatively even that demonstrate to it use we (φ), ie by given p,EEI 0 esrdmcaia lta lsi ha moduli, shear elastic plateau mechanical Measured (A) G p 0 tdifferent at nteEImdlaeidctdb akrudclr:entropic colors: background by indicated are model EEI the in ∆φ G p 0 ie by given , pe ih Inset Right (Upper G p,EEI 0 PNAS φ (φ) 3) efidthat find we (30), δ G | p 0 /G using pi 6 2019 16, April p 0 soitdwt ifrn uncertainties different with associated , ). φ aigdfeetdmnn contribu- dominant different having a φ 5 m Thus, nm. 459 = ≥ 0.592 G G | G p,mech 0 p,mech 0.562 0 p,EEI 0 o.116 vol. orsod othe to corresponds ae namodel a on based (φ) (φ) (φ) < φ < | si excellent in is fohrsimi- other of o 16 no. φ ag that range 0.592 G a G p,mech 5 nm 459 = 0 p,mech 0 G | p,EEI a cor- 0 7767 (φ) (φ), =

PHYSICS experimental uncertainties in φ of emulsion samples can lead to 0 very large uncertainties in Gp at the onset of jamming. We first 0 numerically calculate the first partial derivative of Gp,EEI with respect to φ. In Fig. 1B, we plot the predicted relative magni- 0 tude of the variation in Gp(φ), which is proportional to this first partial derivative and also proportional to the magnitude of the uncertainty in the droplet volume fraction, ∆φ. Even for small 0 ∆φ significantly less than 1%, measured values of Gp near the onset of jamming could exhibit significant scatter and uncertainty of a factor of 2 or more. This highlights the need to control φ very carefully in all studies of mechanical properties of jammed emulsions, irrespective of whether these measurements are based on mechanical rheometry or on light scattering. Here, we have controlled φ to a high degree and have kept ∆φ very low. Moreover, we have ensured a direct comparison between mechanical and light-scattering measurements on exactly the same emulsion at the same set of φ values, thereby avoiding the large uncertain- 0 ties in Gp highlighted by the peaks in Fig. 1B, which adversely affected a prior comparison near and above the jamming point (8).

Diffusing Wave Spectroscopy. To facilitate comparison with mechanical results, we performed DWS studies on the same emulsion at identical φ (Materials and Methods). The measured, normalized, time-averaged intensity autocorrelation functions, g2(t) − 1, for each different φ are shown for DWS transmission (Fig. 2A) and backscattering (Fig. 2B). In these measured g2(t) − 1, after initial decays, we observe long-time plateaus over at least several orders of magnitude in t, except at the lowest φ. Such plateau behavior indicates that droplets in the emulsion are confined by other surrounding droplets. The mag- nitudes of these plateaus increase systematically toward unity for larger φ, indicating greater droplet confinement. For φ ≥ 0.574, transmission g2(t) − 1 curves remain above the baseline even for long times extending into the echo regime (Materi- als and Methods). By contrast, for φ < 0.574, g2(t) − 1 becomes unresolvable from the baseline at long times; so, plateau behavior, if present, cannot be readily determined using transmission Fig. 2. (A and B) Normalized, averaged temporal DWS intensity correlation DWS for such low φ. To overcome this limitation, we also functions, g2(t) − 1, of a monodisperse emulsion having a = 459 nm for the measure g2(t) − 1 using backscattering DWS. We also test sev- same set of φ (values on right) as in Fig. 1A measured using (A) transmis- eral larger φ to compare backscattering plateaus with clearly sion and (B) backscattering (Materials and Methods). DWS echo data are resolved transmission plateaus. For φ > 0.616, the backscat- indicated by arrows. Lines guide the eye. tering DWS correlation functions do not decay sufficiently to be readily analyzed. For the two lowest φ, g2(t) − 1 are sim- S(q), over all q ∈ [0, 2k], weighted by the scattering power P(q) ilar; this is likely caused by experimental uncertainties when . hS(q)i = R 2k q 3P (q)S (q)dq R 2k q 3P (q)dq setting φ. and normalizing: 0 0 , where P(q) denotes the form factor of the scatterers and k = Extracting Self-Motion of Dense Probes. From DWS g2(t) − 1 we 2πns/λDWS is the wavenumber in the solvent with refractive 2 extract the apparent MSDs of probe droplets ∆ra (t) using index ns = 1.33 for water (31, 35). By comparison, a possible standard procedures (31) (Materials and Methods). For passive contribution by the distinct part of the hydrodynamic function DWS microrheology using tracer probes at low φ ∼ 1−2% (6, is small, so we neglect it (SI Appendix). 2 ∗ 12, 19), collective scattering effects are negligible and ∆ra (t) To quantify hS(q)i, we measure 1/` (φ) for the emulsion with reduces to the true ∆r2 (t) associated with probe self-motion. a = 459 nm over a wide range of φ by comparing time-averaged However, for higher probe concentrations, as in the emulsions transmission intensities from these emulsions with transmission here, the individual scattering processes are modulated by the intensities measured for a set of polystyrene latex reference sam- microstructure. In single scattering, this leads to the well-known ples having different sizes, using the DWS instrument’s software de Gennes narrowing of the intensity-spectrum I (q, ω), where q (5) (Fig. 3A). Collective scattering also influences static light is the magnitude of the scattering wavevector and ω is the fre- scattering and thus `∗. Typically, this leads to an increase of `∗, quency of quasi-elastically scattered coherent radiation, near the although in some particular cases it can also lead a reduction, peak of the structure factor of simple liquids (32). The exper- e.g., close to a photonic pseudogap or for high refractive index ∗ imental collective diffusion coefficient of colloids Dc ∝ 1/S(0) scatterers (36, 37). In Fig. 3A we plot the calculated 1/ÌSA ∝ φ measured by dynamic light scattering increases with concentra- (dashed line) in the absence of collective scattering, also known tion, where the structure factor at low q S(0) drops sharply (32– as the independent scattering approximation (ISA). Using ns = 34). In DWS, collective scattering effects contribute at all scatter- 1.33 for water, we infer that the refractive index of the oil inside ∗ ing wavevectors; consequently, to obtain the self-motion MSD of the droplets is n = 1.401 using the criterion that 1/ÌSA and probes, it is necessary to correct the apparent MSD by multiply- 1/`∗ must merge as φ → 0. This oil refractive index is in excel- ing it with the average structure factor hS(q)i of the emulsion: lent agreement with refractometry measurements that we have 2 2 ∆r (t) = hS(q)i ∆ra (t) . hS(q)i is defined by the integral of made (n = 1.40 at λ ∼ 580 nm) and also the supplier’s reference

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PHYSICS microrheology and mechanical rheometry implies that the GSER does work very well for dense emulsion systems if the DWS MSDs have been properly corrected for collective scattering. 0 In addition, the microrheological Gp(φ) matches the predicted 0 Gp(φ) of the EEI model as droplets become jammed, which enables us to identify that the GSER is applicable when both screened electrostatic repulsions and droplet interfacial defor- 0 mations dominate Gp(φ). Moreover, we have derived a time- domain equation for MSDs of droplets in dense emulsions, based on the GSER and a model that has an s1/2-dependent contribution to the linear viscoelasticity, and have used this equation to fit measured DWS self-motion MSDs, yielding excellent agreement. From these fits, we have determined the φ-dependent time scale τ associated with the s1/2 term in the viscoelastic model. In future theoretical investigations, it would be useful to determine quantitative predictions for τ(φ) and the power- 0 law scaling identified for Gp(τ) that could be compared with our measurements. We anticipate that broader application of the method we have demonstrated for correcting DWS MSDs of jammed emulsions with the φ-dependent hS(q)i could lead to improved quantitative accuracy of passive microrheology using the GSER in other dense colloidal soft materials that are highly scattering.

Materials and Methods Monodisperse Oil-in-Water Emulsions. We make emulsions using SDS (Fisher 1/2 Fig. 5. (A) Characteristic time scale τ of the (sτ) term in the viscoelastic Scientific; electrophoresis grade 99% purity), PDMS (Gelest Inc.; viscos- emulsion model (SI Appendix), obtained from fits to DWS MSDs in Fig. 4, as a ity 350 centi-Stokes), and deionized water (Millipore Milli-Q; resistivity function of φ: transmission (open black triangles), and backscattering (open 18.2 MΩ·cm). SDS-stabilized monodisperse PDMS O/W emulsions were pre- 0 blue squares). (B) Plateau elastic storage modulus Gp,GSER as a function of τ, pared through emulsification, a set of depletion fractionation steps to both obtained as fit parameters of DWS MSDs in Fig. 4. Symbols are as in A. reduce droplet polydispersity (39), and repeated centrifugation to set the 0 −χ Dashed line: fit using a power law, Gp,GSER ∼ τ , yielding an exponent χ = bulk concentration of SDS and to concentrate the droplets (40–43) (SI 0.86 ± 0.02. Appendix ). Samples at lower φ were prepared by diluting the concentrated master emulsion sample with 10 mM SDS solution. The master emulsion is also diluted in 10 mM SDS to φ ∼ 10−4 and then characterized by mul- ◦ ◦ macroscopic mechanical G0 (φ) and the predicted G0 (φ) tiangle DLS (LS Spectrometer; LS Instruments) over 60 −120 scattering p,mech p,EEI = ± of the EEI model. Over the entire range of φ, we find excellent angles, yielding an average hydrodynamic droplet radius a 459 15 nm. 0 The polydispersity is δa/a ' 0.176, where δa is the SD of the radial droplet agreement between DWS microrheological and mechanical Gp size distribution. measurements without applying any arbitrary correction factors, which had been previously applied to dense emulsion systems on Mechanical Shear Rheometry. For each φ, we load the emulsion sam- an ad hoc and φ-independent basis (8, 29). Accounting for col- ple into a 25-mm diameter stainless steel cone-and-plate geometry in a lective scattering effects in DWS MSDs by using φ-dependent hS(q)i, based on the measured `∗(φ), is necessary to achieve such quantitative agreement. Conclusions Since the advent of passive microrheology, apparent DWS MSDs have been used in combination with the GSER to show 0 trends in Gp(φ) of dense glassy and jammed colloidal systems, including emulsions, yet until now a highly accurate quantitative match with macroscopic mechanical measurements has not been obtained over a wide range of φ. In past experiments, the lack of conversion of apparent MSDs into true self-motion MSDs has led to the introduction of various correction factors to rescale microrheological measurements into mechanical measurements as well as theoretical speculations about appropri- ateness of boundary conditions and other assumptions inherent in the GSER. Here, we have shown that invoking such ad hoc correction factors is unnecessary, and we have presented and demonstrated a well-defined empirical method for correcting DWS MSDs in dense, highly scattering systems using φ-

dependent hS(q)i to account for collective scattering. Although 0 3 Fig. 6. Comparison of plateau shear elastic moduli Gp of a monodisperse the q weighting inherent in DWS does favor self-motion in 0 emulsion as a function of φ as disordered droplets jam. Gp,mech(φ) was mea- the extracted MSDs, such extracted MSDs still require sig- 0 sured using mechanical shear rheometry (solid red circles) (Fig. 1). Gp,GSER(φ) niﬁcant φ-dependent corrections for scattering probes at high was measured using the hS(q)i-corrected plateau DWS MSDs at long times densities, up to a factor of 4 or more, as we have demon- t in Fig. 4 through the GSER of passive microrheology: transmission (open strated for jammed emulsions. The excellent agreement we black triangles) and backscattering (open blue squares). Red solid line shows 0 0 ﬁnd between Gp(φ) measured using both modern DWS-GSER Gp,EEI(φ) predicted by the EEI model.

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(44, ACKNOWLEDGMENTS. liquid viscous a in spheres hard concentrated for appropriate be constant time a includes an that and emulsions modulus of shear viscoelasticity linear plateau for low-frequency model time-domain a analytical and an relation Eq. to Appendix, MSDs (SI self-motion equation droplet Parameters. measured Rheological Extract the to ted MSDs Self-Motion Droplet Fitting Appendix measured measured and the apparent From the times. extract longer we at `*, acquired also are data echo Rep suspensions. colloidal suspensions. particles. colloidal Lett emulsions. ionic disordered concentrated in nanoemulsions. emulsions. compressed and dense scheme. Vol detection DC), single-mode a Washington, with Soc, Chemical (Am P Alexandridis 143–160. F, pp 861, Case eds experiments. 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