Complex optical transport, dynamics, and rheology of intermediately attractive emulsions

Introducing short-range attractions in Brownian systems of monodisperse colloidal spheres can substantially impact their structures and consequently their optical transport and rheological properties. Here, for size-fractionated colloidal emulsions, we show that imposing an intermediate strength of attraction, well above but not much larger than thermal energy (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 5.6$$\end{document}≈5.6 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{\textrm{B}}T)$$\end{document}kBT), through micellar depletion leads to a striking notch in the measured inverse mean free path of optical transport, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\ell ^*$$\end{document}1/ℓ∗, as a function of droplet volume fraction, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document}ϕ. This notch, which appears between the hard-sphere glass transition, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{\textrm{g}}$$\end{document}ϕg, and maximal random jamming, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi _{\textrm{MRJ}}$$\end{document}ϕMRJ, implies the existence of a greater population of compact dense clusters of droplets, as compared to tenuous networks of droplets in strongly attractive emulsion gels. We extend a prior decorated core-shell network model for strongly attractive colloidal systems to include dense non-percolating clusters that do not contribute to shear rigidity. By constraining this extended model using the measured \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/\ell ^*(\phi )$$\end{document}1/ℓ∗(ϕ), we improve and expand the microrheological interpretation of diffusing wave spectroscopy (DWS) experiments made on attractive colloidal systems. Our measurements and modeling demonstrate richness and complexity in optical transport and shear rheological properties of dense, disordered colloidal systems having short-range intermediate attractions between moderately attractive glasses and strongly attractive gels.

mixing, each diluted sample is split into two portions: 1.5 mL for optical transport and DWS measurements and 0.5 mL for mechanical rheometry. This guarantees that exactly the same sample with identical φ has been used for both DWS and mechanical rheometry measurements, which is necessary in order to make accurate microrheological comparisons.

Optical transport and diffusing wave spectroscopy measurements
Optical transport and DWS measurements are performed with a Rheolab 3 light scattering instrument (LS Instruments, Fribourg CH), equipped with backscattering option. Laser light (wavelength λ = 685 nm), transmitted from a rotating ground-glass diffuser, is subsequently collimated to illuminate the sample. At a given φ , each emulsion is loaded into a clean glass optical cuvette with a width of 10 mm and a thickness (i.e. pathlength) of L = 5 mm. This pathlength is large enough to ensure that the ˚i s always at least a factor of 4 smaller than L for all φ corresponding to the MSDs that we report. To avoid artifacts that could occur because of an inadequate sample volume, we ensure that the upper surface of the loaded emulsion is at least 15 mm high relative to the bottom surface of the cuvette.
The protocols that we have developed for loading emulsion samples into the cuvette and for setting the waiting time, which starts at the completion of sample loading and ends at the beginning of measurements, depend on how viscous or elastic each sample is and therefore on φ . For dilute viscous samples having φ ď 0.3, we only measure and report optical transport ˚, not DWS MSDs, using the following protocol that has been designed to not be excessively influenced by gravitational compaction leading to phase separation, which is known for dilute attractive emulsions 62 . After dilution and mixing, we pour the sample into the cuvette over one wall, cap and seal the cuvette with Parafilm, and allow the sample to rest for 24 hours at 20.0˝C. Then, the cuvette is gently inverted for 5 times and finally righted. After placing the cuvette into the Rheolab 3, we wait 1,200 s before measuring ˚. This waiting time of 1,200 s is significantly larger than the longest doubling time «1.7 s for the lowest φ = 0.052, calculated on basis of colloidal diffusion of droplets in water. Therefore, local dense clusters of droplets have adequate time to form through S-DLCA 15, 16 for all presented 0.052 ď φ ď 0.30. At each φ , the reported 1{ ˚i s averaged from 11 trials and the trial-to-trial standard deviation is less than 4%. No noticeable creaming, which would create a systematic trend in the measured ˚, is observed as a function of trial number.
We measure both ˚a nd DWS g 2 ptq´1 for all samples having φ ě 0.401 after loading the sample in the following manner to avoid introducing air bubbles into the cuvette. For 0.401 ď φ ď 0.541, the yield stress of the emulsion is still low enough that we can simply pour the emulsion into the cuvette over one wall. For 0.571 ď φ ď 0.630, we use a syringe with a stainless-steel needle (inner diameter I.D. = 0.84 mm) to load the samples. We begin by completely inserting the syringe tip into the bottom of the cuvette, then slowly withdrawing the syringe while injecting the emulsion and ensuring that the tip stays below the emulsion's surface. For φ ě 0.64, we transfer the emulsion with a small spatula into the cuvette as close as possible to its bottom; then, we use low speed (< 1,500 rpm) centrifugation for a total duration less than 60 s to eliminate stray air bubbles in loaded cuvettes without generating gradients in φ . After loading, the capped optical cuvettes are sealed with Parafilm, and all of the emulsion samples are stored in a temperature-controlled chamber for 24 hours to equilibrate before measurements. The sample temperature in the Rheolab 3 is maintained at T " 20.0˘0.1˝C for all measurements. For each φ ě 0.401, we measure ˚f ollowed by DWS g 2 ptq´1 in transmission geometry for 11 trials; and then we measure DWS g 2 ptq´1 in backscattering geometry for 11 trials. Each trial of g 2 ptq´1 measurements contains 300 s of multi-tau duration and 60 s of echo duration. The reported 1{ ˚a nd g 2 ptq´1 at each φ are obtained by averaging. The standard deviations of 1{ ˚f or all φ explored over φ ě 0.4 and also the standard deviations of the long-time g 2´1 for all φ explored over φ ě 0.61 are less than 2% of the corresponding average values.
At each φ , using the measured g 2 ptq´1 and ˚, we extract the apparent ∆r 2 a ptq by solving the classic transcendental equation of DWS 47, 58, 64 , and then each apparent MSD is converted into the probe self-motion ∆r 2 ptq to correct for collective light scattering. The apparent MSD is multiplied by a dimensionless ratio, less than unity, given by the actual measured scattering strength at that φ , reflected by 1{ ˚p φ q, divided by the Mie scattering strength that ignores collective scattering at low-φ : r1{ ˚p φ qs{rp1{ I SA,Mie qφ s, where 1{ I SA,Mie " 0.0207 µm´1 is the slope of the calculated 1{ ˚v ersus φ in the dilute limit as φ Ñ 0, based on independent scattering approximation. As a result, the overall magnitude of the apparent MSDs at different φ are reduced by various amounts to obtain the probe self-motion MSDs, as this dimensionless ratio is φ -dependent.
To ensure that the number of droplets per scattering volume is time-invariant within during our DWS measurements of concentrateed IA emulsions at higher φ , we have also performed Rheolab 3 measurements after only a 10-minute waiting time, which serves as a reference for the reported results at a much longer 24-h waiting time. For all φ ě 0.401, changes in the DWS count rate are less than 3%, and changes in the measured 1{ ˚a re less than 4%. Moreover, no distinct layer of cream, which would scatter light more strongly at the top of the cuvette, has been observed visually after the 24-h waiting time. We report dynamic DWS MSD measurements only in the effectively time-invariant range of φ ě 0.541, over which aging and gravity-induced droplet compaction do not significantly influence our measurements and over which the measured DWS g 2 ptq´1 do not fully decay to baseline in the long-time limit. For φ ě 0.64, a slight decrease in the decay rate of long-time g 2 ptq´1 beyond t « 2ˆ10 -1 s is observed in measurements after the 24-h waiting time, which is an indication of very slow aging. This slow aging is not the subject of our study, and our microrheological comparisons are based on plateau MSDs that are at intermediate correlation times, not at long correlation times which show some evidence of slight aging through changes in relaxation. Thus, the primary plateau MSDs, used in the GSER of passive microrheology, occur at shorter times t À 2ˆ10 -2 s and are time-invariant over at least 24 h for all φ ě 0.541 that we report. Microrheological comparisons are facilitated by this effective time-invariance of the plateau feature in the MSD over 24-h long waiting time for the elastic dense emulsions. Our purpose in the present study is to make microrheological comparisons of elastic plateau moduli, not to study long-term aging.

Mechanical shear rheometry
We use a 25 mm diameter cone-and-plate geometry (stainless steel) in a controlled-strain mechanical shear rheometer (RFS-II, Rheometric Scientific, equipped with a vapor trap) to measure the plateau elastic shear moduli, G 1 p,mech , at low strains corresponding to the linear viscoelastic regime. After the same 24-h waiting time also used for DWS, we pre-shear the sample at 50 s´1 shear rate for 30 s by stirring with a spatula and load the sample into the rheometer. Measurements are commenced two minutes after lowering the cone to the appropriate pre-set gap with respect to the plate and adjusting the vapor trap. All measurements are performed at T " 20˝C. At each φ , we perform a small-strain oscillatory frequency sweep from ω " 20 rad/s down to 0.02 rad/s at a small shear strain amplitude of γ " 0.005. The plateau storage modulus becomes noticeably frequency dependent for φ ď 0.620 at this γ " 0.005. We next conduct a strain sweep at each φ and ω " 1 rad/s, yielding the linear and non-linear shear storage modulus, represented as G 1 pγq. This frequency is within the range of time scales that correspond to the plateau associated with DWS MSD measurements. We probe down to shear strains as low as « 1ˆ10´4, which is limited by the resolution of the RFS-II's motor. Under small strains, a dominant linear storage modulus can be detected down to φ " 0.610. To obtain the small-strain G 1 p,mech of the IA emulsion at each φ , we fit the measured G 1 pγq to a function that has a low-γ-plateau 60 (inset in Fig. 6): G 1 pγq " G 1 p,mech {rpγ{γ y q κ`1 s, where γ y is the yield strain associated with the log slope change in G 1 pγq, and κ is a power law exponent related to the non-linear response of G 1 pγq to larger strains. A two-step yielding strain-response, which has been reported previously in strain sweeps on a similar O/W emulsion system but which had a much stronger attractive strength (|U d | « 21 k B T ) 8 , is not apparent in the measured strain sweeps of these IA emulsions.

Regularized fitting using the extended decorated core shell network model
In the prior DCSN model, developed for the SA emulsion, an effective probe-size factor α SA " 2.0, corresponding to a local dense cluster that is approximately tetrahedral on average, has been introduced based on structural concepts for attractive gels of emulsion droplets. Consequently, for attractive colloidal systems, identifying the appropriate probe for interpreting a DWS correlation function as a MSD is complex. The prior study of SA emulsions showed that passive microrheology on these systems can be performed quantitatively via the GSER if the effective size of the DWS scattering probes is taken into account: G 1 p,GSER 91{p a pr ∆r 2 p q " 1{pα a ∆r 2 p q, where α is the dimensionless ratio between the effective average radius of the DWS scattering probes, xa pr y, and the average hydrodynamic radius of an isolated droplet, xay. By contrast, the effective DWS probe-size is comparable to a in the MA system: α MA " 1.0 59 . The IA system, studied herein, has an attractive potential depth that is in between the above-mentioned MA and SA regimes.
To set the magnitude of φ core,perc before introducing the fourth principal component of non-percolating droplets for the IA emulsion, we hypothesize an average droplet volume fraction φ net,core = 0.793 within the percolating core regions, which is the maximum φ that we obtained experimentally with the emulsion having this droplet size distribution. We determine G 1 p within the regions that only have percolating core droplets using the EEI model 27 : G 1 p,EEI pφ net,core q. The magnitude of φ core,perc pφ q is determined based on the effective medium assumption: φ core,perc pφ q " φ net,core rG 1 p,GSER pφ q{G 1 p,EEI pφ net,core qs, where G 1 p,GSER pφ q is determined from the DWS plateau MSD measurements with the assumption of α " 2 in the GSER: G 1 p,GSER pφ q " k B T {rπαxay ∆r 2 pφ q p s. Initially, we exclude non-percolating core droplets from consideration, and we minimize χ 2 of the nonlinear least-squares fit for 1{ I A pφ q by varying the model's parameters [Eq.
(1) with φ core,nonperc temporarily set to zero], in a manner similar to what has been previously done for the SA emulsion. This effectively ignores the notch initially but provides a comparable overall shape for the φ -dependent functional forms of SDD, shell, and percolating core components. Then, we take into account of φ core,nonperc by transferring weights from φ shell at φ ď 0.64 and from φ SDD within the lower end of notch region. This weight-transfer reflects the reorganization of outer droplets between different clusters as a consequence of the applied shear stress while diluting and mixing. We do so in a manner that preserves the smoothness of all four principal components, even as there are some rapid variations in the notch region itself as SDDs are converted to shell droplets, shell droplets into non-percolating core droplets, and shell and non-percolating core droplets into percolating core droplets. After iterations of minimizing χ 2 of the 1{ I A pφ q fit over the entire φ range, having all key features considered, we obtain a regularized curve fit of 1{ I A pφ q with smooth inter-conversions between all component droplet volume fractions.

3/4 Probability density functions of local coordination number
We assume that the distributions for all four principal components in the E-DSCN model are Gaussian; we have also assumed that the same standard deviation of σ N = 2.5 is suitable for all of these distributions in order to provide total distributions that are smooth as a function of N for all φ considered. Enforcing this smoothness, thus, effectively amounts to a regularization assumption. The peaks of p SDD pNq, p shell pNq, p core,nonperc pNq, and p core,perc pNq are located at N = 3, 6, 9, and 12, respectively. We emphasize here that these plotted distributions have been inferred, not directly measured. Yet, these distributions show how increasing the osmotic pressure applied to an attractive emulsion can lead to very substantial changes in the local coordination number that are consistent with the measured trends in the optical transport properties of such emulsions over a wide range of φ .
Here, we note that p N pN ď 2) has been folded to larger N, consistent with slippery diffusion-limited cluster aggregation in the dilute limit 16 . In our experimental system, N = 2 is theoretically possible, but highly unlikely. Droplets trapped in a mobile bridging configurations, for instance spanning between clusters, can have N = 2 in a very unusual situation when clusters are re-established following the cessation of loading and shear disruption. SDDs can also have N = 2 in a transient sense if a very strong Brownian excitation breaks one of the three bonds of the SDD and the droplet shifts into a new configuration with only two bonds still present. By contrast, N ě 3 represents a relatively stable configuration that is only seldom destabilized by Brownian excitations in IA emulsions, resulting in transient droplet motion on the surface of a cluster rather than complete unbinding and liberation as an isolated droplet. Such transient bound droplet motion is one of the potential sources that could lead to excess DWS MSDs that become particularly noticeable toward lower φ .