Generalized Stokes-Einstein Relation Research Articles

Effective temperatures and the breakdown of the Stokes-Einstein relation for particle suspensions.

The short- and long-time breakdown of the classical Stokes-Einstein relation for colloidal suspensions at arbitrary volume fractions is explained here by examining the role that confinement and attractive interactions play in the intra- and inter-cage dynamics executed by the colloidal particles. We show that the measured short-time diffusion coefficient is larger than the one predicted by the classical Stokes-Einstein relation due to a non-equilibrated energy transfer between kinetic and configuration degrees of freedom. This transfer can be incorporated in an effective kinetic temperature that is higher than the temperature of the heat bath. We propose a Generalized Stokes-Einstein relation (GSER) in which the effective temperature replaces the temperature of the heat bath. This relation then allows to obtain the diffusion coefficient once the viscosity and the effective temperature are known. On the other hand, the temporary cluster formation induced by confinement and attractive interactions of hydrodynamic nature makes the long-time diffusion coefficient to be smaller than the corresponding one obtained from the classical Stokes-Einstein relation. Then, the use of the GSER allows to obtain an effective temperature that is smaller than the temperature of the heat bath. Additionally, we provide a simple expression based on a differential effective medium theory that allows to calculate the diffusion coefficient at short and long times. Comparison of our results with experiments and simulations for suspensions of hard and porous spheres shows an excellent agreement in all cases.

Passive Microrheology Simulations Using Stochastic Particle Methods

Passive microrheology is an experimental technique for the characterization of complex polymeric fluids based on the tracking of small embedded colloidal particles which undergo Brownian motion. Key modeling features for this flow problem include viscoelasticity of the suspending medium as well as the possibility to incorporate thermal fluctuations consistently. In this article some numerical methods suitable for the simulation of passive microrheology are reviewed. The so-called micro-macro approaches are based on the coupling between the continuum hydrodynamics description of the complex liquid with microscopic simulations of polymers using Brownian Dynamics. The advantage of using these methods is related to the consistent incorporation of thermal noise as well as the possibility to simulate microscopic polymer dynamics exactly. On the other hand, computational limitations are mainly due to the fact that in order to reproduce realistic conditions a large number of stochastic realizations is generally necessary. An alternative approach that bypasses the use of micro-macro models is represented by a Lagrangian particle method based on Smoothed Particle Hydrodynamics. In this method the viscoelasticity of the solvent is modelled via a continuum Oldroyd- B model whereas thermal fluctuations, inherently present at the microscopic scale, are incorporated into the particle framework by application of the GENERIC formalism, ensuring the strict fulfilment of the Fluctuation-Dissipation theorem. As an application, the particle method is used to simulate a realistic case in passive microrheology. In particular, a rigid structure suspended in the viscoelastic solvent is modeled and the rheological properties of the Oldroyd-B fluid, namely frequency-dependent storage and loss moduli, are extracted through the standard microrheological route, that is, by measuring the probe mean square displacement (MSD) and linking it to the solvent viscoelasticity by assuming the validity of a generalized Stokes-Einstein relation (GSER). Good agreement with the analytical theory for the Oldroyd-B model is observed only up to maximal frequencies corresponding to time scales considerably larger than the Brownian time of the probe particle where the diffusive regime is fully established. At larger investigated frequencies, a crossover between diffusive and ballistic behavior for the MSD of the probe is observed and validity of the GSER is questionable. The model presented here provides an optimal computational framework to complement experimental observations and to analyze quantitatively the basic assumptions involved in the theory of microrheology.

Trajectories of probe spheres in generalized linear viscoelastic complex fluids

We have developed a fast simulation that generates a random walk of an isolated probe sphere in a generalized linear viscoelastic complex fluid over a highly extended dynamic range. We introduce a coupled harmonically bound Brownian particle (c-HBBP) model, in which the relaxation modes of the viscoelastic medium are treated as harmonic wells. These wells are coupled to the probe sphere and perform Brownian motion in bound harmonic potentials corresponding to the next-longer relaxation mode, according to the relaxation spectrum of the viscoelastic material. We implement this c-HBBP model by generating variable temporal step sizes that have a uniform distribution in logarithmic time. We create and analyze trajectories for several different viscoelastic complex fluids: a polymer system at its gel point, a dense emulsion system, a blend of two monodisperse polystyrene polymers for which the relaxation spectrum has been measured, and a model anisotropic soft system that shows dense emulsion-like and gel-point behaviors along two orthogonal directions. Except for unusual viscoelastic materials, such as the polymer system at its gel point, the generated trajectories are neither self-similar nor self-affine. The resulting mean square displacements predicted by the c-HBBP model are consistent with the single-particle generalized Stokes-Einstein relation of linear passive microrheology.

Longitudinal measurement of extracellular matrix rigidity in 3D tumor models using particle-tracking microrheology.

The mechanical microenvironment has been shown to act as a crucial regulator of tumor growth behavior and signaling, which is itself remodeled and modified as part of a set of complex, two-way mechanosensitive interactions. While the development of biologically-relevant 3D tumor models have facilitated mechanistic studies on the impact of matrix rheology on tumor growth, the inverse problem of mapping changes in the mechanical environment induced by tumors remains challenging. Here, we describe the implementation of particle-tracking microrheology (PTM) in conjunction with 3D models of pancreatic cancer as part of a robust and viable approach for longitudinally monitoring physical changes in the tumor microenvironment, in situ. The methodology described here integrates a system of preparing in vitro 3D models embedded in a model extracellular matrix (ECM) scaffold of Type I collagen with fluorescently labeled probes uniformly distributed for position- and time-dependent microrheology measurements throughout the specimen. In vitro tumors are plated and probed in parallel conditions using multiwell imaging plates. Drawing on established methods, videos of tracer probe movements are transformed via the Generalized Stokes Einstein Relation (GSER) to report the complex frequency-dependent viscoelastic shear modulus, G*(ω). Because this approach is imaging-based, mechanical characterization is also mapped onto large transmitted-light spatial fields to simultaneously report qualitative changes in 3D tumor size and phenotype. Representative results showing contrasting mechanical response in sub-regions associated with localized invasion-induced matrix degradation as well as system calibration, validation data are presented. Undesirable outcomes from common experimental errors and troubleshooting of these issues are also presented. The 96-well 3D culture plating format implemented in this protocol is conducive to correlation of microrheology measurements with therapeutic screening assays or molecular imaging to gain new insights into impact of treatments or biochemical stimuli on the mechanical microenvironment.

Longitudinal Measurement of Extracellular Matrix Rigidity in 3D Tumor Models Using Particle-tracking Microrheology

The mechanical microenvironment has been shown to act as a crucial regulator of tumor growth behavior and signaling, which is itself remodeled and modified as part of a set of complex, two-way mechanosensitive interactions. While the development of biologically-relevant 3D tumor models have facilitated mechanistic studies on the impact of matrix rheology on tumor growth, the inverse problem of mapping changes in the mechanical environment induced by tumors remains challenging. Here, we describe the implementation of particle-tracking microrheology (PTM) in conjunction with 3D models of pancreatic cancer as part of a robust and viable approach for longitudinally monitoring physical changes in the tumor microenvironment, in situ. The methodology described here integrates a system of preparing in vitro 3D models embedded in a model extracellular matrix (ECM) scaffold of Type I collagen with fluorescently labeled probes uniformly distributed for position- and time-dependent microrheology measurements throughout the specimen. In vitro tumors are plated and probed in parallel conditions using multiwell imaging plates. Drawing on established methods, videos of tracer probe movements are transformed via the Generalized Stokes Einstein Relation (GSER) to report the complex frequency-dependent viscoelastic shear modulus, G*(ω). Because this approach is imaging-based, mechanical characterization is also mapped onto large transmitted-light spatial fields to simultaneously report qualitative changes in 3D tumor size and phenotype. Representative results showing contrasting mechanical response in sub-regions associated with localized invasion-induced matrix degradation as well as system calibration, validation data are presented. Undesirable outcomes from common experimental errors and troubleshooting of these issues are also presented. The 96-well 3D culture plating format implemented in this protocol is conducive to correlation of microrheology measurements with therapeutic screening assays or molecular imaging to gain new insights into impact of treatments or biochemical stimuli on the mechanical microenvironment.

Microrheology and Microstructure of Fmoc-Derivative Hydrogels

The viscoelasticity of hydrogel networks formed from the low-molecular-weight hydrogelator Fmoc-tyrosine (Fmoc-Y) is probed using particle-tracking microrheology. Gelation is initiated by adding glucono-δ-lactone (GdL), which gradually lowers the pH with time, allowing the dynamic properties of gelation to be examined. Consecutive plots of probe particle mean square displacement (MSD) versus lag time τ are shown to be superimposable, demonstrating the formation of a self-similar hydrogel network through a percolation transition. The analysis of this superposition yields a gel time t(gel) = 43.4 ± 0.05 min and a critical relaxation exponent n(c) = 0.782 ± 0.007, which is close to the predicted value of 3/4 for semiflexible polymer networks. The generalized Stokes-Einstein relation is applied to the master curves to find the viscoelastic moduli of the critical gel over a wide frequency range, showing that the critical gel is structurally and rheologically fragile. The scaling of G'/G″ as ω(0.795±0.099) ≈ ω(3/4) at high frequencies provides further evidence for semiflexible behavior. Cryogenic scanning electron micrographs depict a loosely connected network close to the gel point with a fibrillar persistence length that is longer than the network mesh size, further indications of semiflexible behavior. The system reported here is one of a number of synthetic systems shown to exhibit semiflexible behavior and indicates the opportunity for further rheological study of other Fmoc derivatives.

Polymer chain properties and thermodynamic stability in oriented-platelet nanocomposites

We present a Monte Carlo study of the conformational properties of polymer chains in platelet nanocomposites. We find that high platelet orientation leads to an increase in chain confinement and to a substantial decrease in polymer entropy. The latter may be responsible for the observed aging of these systems in which the platelets gradually lose their orientation with time. As a test of our entropy-driven hypothesis, we suggest that aging could be substantially slowed down by the addition of small spherical nanoparticles. Using a generalized Stokes-Einstein relation, a calculation of the thermal random motion of the platelets predicts a fivefold increase in shear modulus during aging, in agreement with experimental observation on compatibilized layered silicate nanocomposites.

Monitoring degradation of matrix metalloproteinases-cleavable PEG hydrogels via multiple particle tracking microrheology

The design of hydrogel matrices for cell encapsulation and tissue regeneration has become increasingly complex. Oftentimes, researchers seek to recapitulate specific biophysical and biochemical cues critical for the resident cell population and an in depth understanding of changes in the local microstructure and rheological properties of the synthetic matrix during enzymatic degradation would be extremely beneficial. Multiple particle tracking microrheology (MPT) enables simultaneous characterization of rheological properties and visualization of the microstructure in an evolving hydrogel scaffold. MPT measures the Brownian motion of fluorescently labeled probe particles embedded in the material, which is directly related to rheological properties using the Generalized Stokes–Einstein Relation (GSER). Here, we study a hydrogel scaffold consisting of a four-arm poly(ethylene glycol) (PEG) end functionalized with norbornene that is cross-linked with both a nondegradable PEG–dithiol and a matrix metalloproteinase (MMP) degradable peptide sequence (KCGPQG↓IWGQCK) using thiol–ene chemistry. The material degradation is measured as a function of time and extent of degradability, focusing on measuring the gel–sol transition. Using time–cure superposition, we determine the critical degradation time and critical extent of degradability for this specific gel formulation as tc = 1.85 h and pc = 0.589, respectively, and the critical relaxation exponent, n = 0.16. Finally, spatial information gained by MPT measurements quantifies the heterogeneity within the scaffold showing that these hydrogels degrade homogeneously when collagenase is introduced in solution at a concentration of 0.1–0.3 mg mL−1. Understanding the microstructural and rheological properties of a material near the gel–sol transition enables researchers to improve their insight as to how cells remodel their microenvironment when encapsulated in gels, and more precisely design and manipulate this parameter to improve three-dimensional culture systems.

The analytic solution of Stokes for time-dependent creeping flow around a sphere: Application to linear viscoelasticity as an ingredient for the generalized Stokes–Einstein relation and microrheology analysis

Analytic expressions for the transient stream function, transient flow field, and transient pressure field for creeping flow around a sphere are derived. An analytic expression for the total force on the sphere is also found. The approach is essentially that of Stokes from 1856. Aside from the (essentially trivial) generalization to linear viscoelastic fluids, there is nothing novel in the derivation. Our purpose is to (1) point out that Stokes, not Basset or Boussinesq derived it first, (2) show how simple the derivation is, which may be compared to the more famous solution of Landau and Lifshitz, (3) show an application of the correspondence between creeping flow and linear viscoelastic flow solutions, and (4) provide sufficiently detailed notes so that the derivation might be given in a graduate fluid dynamics or transport phenomena lecture.

Modeling the microrheology of inhomogeneous media

We model the restricted diffusion of small tracer particles near a gel transition by performing Monte Carlo simulations in a site-percolation model. From the mean squared displacement of the particles, we calculate the effective viscous and elastic moduli using the microrheological equations derived from the generalized Stokes–Einstein relation by Mason [T.G. Mason, Rheol. Acta 39 (2000) 371]. The results show a transition from viscous to elastic behavior at a site-filling probability that is different from that at the percolation transition, and that occurs at higher site-filling probability for smaller tracer particles. This behavior is due to confinement of the tracers in the inhomogeneous system, and we discuss the inapplicability of the generalized Stokes–Einstein relation in this case. The simulations are in partial agreement with experimental results obtained for a gelling clay suspension and polymer blends, but there are also differences which suggest the importance of microstructure, and specifically elastic effects in the experimental systems. We discuss the implications of our simulations for the interpretation of microrheological experiments on inhomogeneous media.

Competing effects of particle and medium inertia on particle diffusion in viscoelastic materials, and their ramifications for passive microrheology

We analyze the appropriate form for the generalized Stokes-Einstein relation (GSER) for viscoelastic solids and fluids when bead inertia and medium inertia are taken into account, which we call the inertial GSER. It was previously shown for Maxwell fluids that the Basset (or Boussinesq) force arising from medium inertia can act purely dissipatively at high frequencies, where elasticity of the medium is dominant. In order to elucidate the cause of this counterintuitive result, we consider Brownian motion in a purely elastic solid where ordinary Stokes-type dissipation is not possible. The fluctuation-dissipation theorem requires the presence of a dissipative mechanism for the particle to experience fluctuating Brownian forces in a purely elastic solid. We show that the mechanism for such dissipation arises from the radiation of elastic waves toward the system boundaries. The frictional force associated with this mechanism is the Basset force, and it exists only when medium inertia is taken into consideration in the analysis of such a system. We consider first a one-dimensional harmonic lattice where all terms in the generalized Langevin equation--i.e., the elastic term, the memory kernel, and Brownian forces-can be found analytically from projection-operator methods. We show that the dissipation is purely from radiation of elastic waves. A similar analysis is made on a particle in a continuum, three-dimensional purely elastic solid, where the memory kernel is determined from continuum mechanics. Again, dissipation arises only from radiation of elastic shear waves toward infinite boundaries when medium inertia is taken into account. If the medium is a viscoelastic solid, Stokes-type dissipation is possible in addition to radiational dissipation so that the wave decays at the penetration depth. Inertial motion of the bead couples with the elasticity of the viscoelastic material, resulting in a possible resonant oscillation of the mean-square displacement (MSD) of the bead. On the other hand, medium inertia (the Basset force) tends to attenuate the oscillations by the dissipation mechanism described above. Thus competition between bead inertia and medium inertia determines whether or not the MSD oscillates. We find that, if the medium density is larger than 4/7 of the bead density, the Basset damping will suppress oscillations in the MSD; this criterion is sufficient but not necessary to present oscillations.

A SPH-based particle model for computational microrheology

In this article we present a particle method based on smoothed particle hydrodynamics for microrheology simulations of polymeric fluids. The viscoelasticity of the solvent is modelled via a standard Oldroyd-B model and thermal fluctuations, inherently present at the microscopic scale, are incorporated into the particle framework by application of the GENERIC formalism, ensuring the strict fulfilment of the Fluctuation–Dissipation theorem at the discrete level. Rigid structures of arbitrary shape suspended in the viscoelastic solvent are modelled by freezing SPH particles within a given solid domain and letting them interact with the solvent particles. The rheological properties of the Oldroyd-B fluid, namely frequency-dependent storage and loss moduli, are extracted via macroscopic deterministic simulations under small amplitude oscillatory (SAOS) flow and, alternatively, through standard microrheological simulations of a probe particle suspended in the same Brownian viscoelastic medium, by assuming the validity of a generalized Stokes–Einstein relation (GSER). We check that good agreement with the analytical theory for the Oldroyd-B model is found in the deterministic SAOS flow over the entire regime of frequencies investigated. Concerning the microrheological measurements, good agreement is observed only up to a maximal frequencies corresponding to time scales considerably larger than the viscous time of the probe particle where the diffusive regime is fully established. At larger investigated frequencies, a crossover between diffusive and ballistic behaviour for the MSD of the probe is observed and validity of the GSER is questionable. The model presented here provides an optimal computational framework to complement experimental observations and to analyse quantitatively the basic assumptions involved in the theory of microrheology.

Treating inertia in passive microbead rheology

The dynamic modulus G(*) of a viscoelastic medium is often measured by following the trajectory of a small bead subject to Brownian motion in a method called "passive microbead rheology." This equivalence between the positional autocorrelation function of the tracer bead and G(*) is assumed via the generalized Stokes-Einstein relation (GSER). However, inertia of both bead and medium are neglected in the GSER so that the analysis based on the GSER is not valid at high frequency where inertia is important. In this paper we show how to treat both contributions to inertia properly in one-bead passive microrheological analysis. A Maxwell fluid is studied as the simplest example of a viscoelastic fluid to resolve some apparent paradoxes of eliminating inertia. In the original GSER, the mean-square displacement (MSD) of the tracer bead does not satisfy the correct initial condition. If bead inertia is considered, the proper initial condition is realized, thereby indicating an importance of including inertia, but the MSD oscillates at a time regime smaller than the relaxation time of the fluid. This behavior is rather different from the original result of the GSER and what is observed. What is more, the discrepancy from the GSER result becomes worse with decreasing bead mass, and there is an anomalous gap between the MSD derived by naïvely taking the zero-mass limit in the equation of motion and the MSD for finite bead mass as indicated by McKinley et al. [J. Rheol. 53, 1487 (2009)]. In this paper we show what is necessary to take the zero-mass limit of the bead safely and correctly without causing either the inertial oscillation or the anomalous gap, while obtaining the proper initial condition. The presence of a very small purely viscous element can be used to eliminate bead inertia safely once included in the GSER. We also show that if the medium contains relaxation times outside the window where the single-mode Maxwell behavior is observed, the oscillation can be attenuated inside the window. This attenuation is realized even in the absence of a purely viscous element. Finally, fluid inertia also affects the bead autocorrelation through the Basset force and the fluid dragged around with the bead. We show that the Basset force plays the same role as the purely viscous element in high-frequency regime, and the oscillation of MSD is suppressed if fluid density and bead density are comparable.

Viscosity and diffusion: crowding and salt effects in protein solutions

We report on a joint experimental-theoretical study of collective diffusion in, and static shear viscosity of solutions of bovine serum albumin (BSA) proteins, focusing on the dependence on protein and salt concentration. Data obtained from dynamic light scattering and rheometric measurements are compared to theoretical calculations based on an analytically treatable spheroid model of BSA with isotropic screened Coulomb plus hard-sphere interactions. The only input to the dynamics calculations is the static structure factor obtained from a consistent theoretical fit to a concentration series of small-angle X-ray scattering (SAXS) data. This fit is based on an integral equation scheme that combines high accuracy with low computational cost. All experimentally probed dynamic and static properties are reproduced theoretically with an at least semi-quantitative accuracy. For lower protein concentration and low salinity, both theory and experiment show a maximum in the reduced viscosity, caused by the electrostatic repulsion of proteins. The validity range of a generalized Stokes-Einstein (GSE) relation connecting viscosity, collective diffusion coefficient, and osmotic compressibility, proposed by Kholodenko and Douglas [PRE 51, 1081 (1995)] is examined. Significant violation of the GSE relation is found, both in experimental data and in theoretical models, in semi-dilute systems at physiological salinity, and under low-salt conditions for arbitrary protein concentrations.

Short-time rheology and diffusion in suspensions of Yukawa-type colloidal particles

A comprehensive study is presented on the short-time dynamics in suspensions of charged colloidal spheres. The explored parameter space covers the major part of the fluid-state regime, with colloid concentrations extending up to the freezing transition. The particles are assumed to interact directly by a hard-core plus screened Coulomb potential, and indirectly by solvent-mediated hydrodynamic interactions. By comparison with accurate accelerated Stokesian Dynamics (ASD) simulations of the hydrodynamic function H(q), and the high-frequency viscosity η(∞), we investigate the accuracy of two fast and easy-to-implement analytical schemes. The first scheme, referred to as the pairwise additive (PA) scheme, uses exact two-body hydrodynamic mobility tensors. It is in good agreement with the ASD simulations of H(q) and η(∞), for smaller volume fractions up to about 10% and 20%, respectively. The second scheme is a hybrid method combining the virtues of the δγ scheme by Beenakker and Mazur with those of the PA scheme. It leads to predictions in good agreement with the simulation data, for all considered concentrations, combining thus precision with computational efficiency. The hybrid method is used to test the accuracy of a generalized Stokes-Einstein (GSE) relation proposed by Kholodenko and Douglas, showing its severe violation in low salinity systems. For hard spheres, however, this GSE relation applies decently well.

Image-based algorithm for analysis of transient trapping in single-particle trajectories

Particle tracking has become an increasingly useful tool in microfluidics and biophysics, allowing measurement of microrheology, local structure, and flow. We introduce a novel, automated approach to analyze single-particle trajectories with transient elements, based on image-processing approaches and physical analysis of probe motion. In many physical and active biological systems, such as living cells, probe particles experience thermally mediated Brownian motion combined with active transport processes that can lead to transient-trajectories of local diffusion and trapping, punctuated by segments of active transport. Analyzing such a trajectory as a single unit masks the intermittent nature of the motion. Moreover, directly applying the generalized Stokes–Einstein relation in out-of-equilibrium systems is incorrect and returns inaccurate rheological parameters. We present an automated image-processing-based method to identify and segment transient trap-escape trajectories, allowing quantitative analysis of each segment. We define and discuss effects of controlling parameters, such as particle size and camera frame rate. Our algorithm provides a general and automated method to segment and analyze transient elements in trajectories of single particles, which can be applied to many different experiments. Our image-based approach allows identification of trapping segments, unbiased by specific step sizes within those traps or the mechanism driving those steps. As an example, we successfully apply this method to experiments of laser tweezers trapped particles and show that trajectory segmentation allows us to calculate both trap and fluid parameters. We accurately identify a round trap, calculate the trap stiffness at 3.1 pN/μm, and find that significant local heating reduces fluid viscosity.

A theoretical bridge between linear and nonlinear microrheology

Passive microrheology exploits the fluctuation-dissipation theorem to relate thermal fluctuations of a colloidal probe to the near-equilibrium linear response behavior of the material through an assumed generalized Stokes Einstein relation (GSER). Active and nonlinear microrheology, on the other hand, measures the nonlinear response of a strongly driven probe, for which fluctuation-dissipation does not hold. This leaves no clear method for recovering the macroscopic rheological properties from such measurements. Although the two techniques share much in common, there has been little attempt to relate the understanding of one to the other. In passive microrheology, the GSER is generally assumed to hold, without the need for explicit calculation of the microstructural deformation and stress, whereas in nonlinear microrheology, the microstructure must be explicitly determined to obtain the drag force. Here we seek to bridge the gap in understanding between these two techniques, by using a single model system to explicitly explore the gentle-forcing limit, where passive (ω→0) and active (U→0) microrheology are identical. Specifically, we explicitly calculate the microstructural deformations and stresses as a microrheological probe moves within a dilute colloidal suspension. In the gentle-forcing limit, we find the microstructural stresses in the bulk material to be directly proportional to the local strain tensor, independent of the detailed flow, with a prefactor related to the effective shear modulus. A direct consequence is that the probe resistance due to the bulk stresses in passive (linear response) microrheology quantitatively recovers the results of macroscopic oscillatory shear rheology. Direct probe-bath interactions, however, lead to quantitative discrepancies that are unrelated to macroscopic shear rheology. We then examine the microstructural equations for nonlinear microrheology, whose U→0 limit reduces to the ω→0 limit in passive microrheology. Guided by the results from passive microrheology, we show that direct probe-material interactions are unrelated to the macroscopic shear rheology. Moreover, we show that the bulk microstructural deformations (which quantitatively recover macroscopic shear rheology in the linear limit) now obey a governing equation that differs qualitatively from macroscopic rheology, due to the spatially dependent, Lagrangian unsteady mixture of shear and extensional flows. This inherently complicates any quantitative interpretation of nonlinear microrheology.

Tensorial generalized Stokes–Einstein relation for anisotropic probe microrheology

In thermal “passive” microrheology, the random Brownian motion of anisotropically shaped probe particles embedded within an isotropic viscoelastic material can be used to extract the material’s frequency-dependent linear viscoelastic modulus. We unite the existing theoretical frameworks for separately treating translational and rotational probe motion in a viscoelastic material by extending the generalized Stokes–Einstein relation (GSER) into a tensorial form that reflects simultaneous equilibrium translational and rotational fluctuations of one or more anisotropic probe particles experiencing viscoelastic drag. The tensorial GSER provides a formal basis for interpreting the complex Brownian motion of anisotropic probes in a viscoelastic material. Based on known hydrodynamic calculations of the Stokes mobility of highly symmetric shapes in a simple viscous liquid, we show simple examples of the tensorial GSER for spheroids and half-stick, half-slip Janus spheres.

Interfacial microrheology as a tool to study viscoelastic transitions in nanoconfined soft matter

We present a method to perform in situ microrheological measurements on monolayers of soft materials undergoing viscoelastic transitions under compression. Using the combination of a Langmuir trough mounted on the inverted microscope stage of a laser scanning confocal microscope we track the motion of individual fluorescent quantum dots partly dispersed in monolayers spread at the air-water interface. From the calculated mean square displacement of the probe particles and extending a well established scheme of the generalized Stokes-Einstein relation in bulk to the interface we arrive at the viscoelastic modulus for the respective monolayers as a function of surface density. Measurements on monolayers of glassy as well as nonglassy polymers and a standard fatty acid clearly show sensitivity of our technique to subtle variations, in the viscoelastic properties of the highly confined materials under compression. Evidence for possible spatial variations of such viscoelastic properties at a given surface density for the fatty acid monolayer is also provided.

Size and Shape of Micelles Studied by Means of SANS, PCS, and FCS

The hexaethylene glycol monododecyl ether (C(12)E(6)) micelles at concentrations up to 10% have been studied in their isotropic phase (10-48 degrees C) by means of small angle neutron scattering (SANS) and photon correlation spectroscopy (PCS). The SANS data obtained at low temperatures could be unequivocally interpreted as a result of scattering from a suspension of compact globular micelles with the shape of a triaxial ellipsoid or a short end-capped elliptical rod. Different models have been applied to analyze the SANS data obtained at higher temperatures: (i) elongated rod-like micelles with purely sterical interactions, (ii) compact globular micelles with a weak attractive potential, and (iii) globular micelles influenced by the critical phenomena in the whole temperature range studied. The good quality of the experimental data indicated model (i) as the best fit for our data. The diffusion coefficients obtained from the PCS measurements have been compared to the diffusion coefficients calculated for the rod-like micelles--results of the SANS data analysis. A good agreement was achieved using the solvent viscosity, in agreement with the theoretical predictions for sterically interacting globular colloidal particles. Finally, the SANS results obtained at 24 degrees C were compared to the micelle self-diffusion coefficients previously measured by means of fluorescence correlation spectroscopy (FCS) at this temperature. The good agreement obtained after scaling the data with solution viscosity supports the validity of the generalized Stokes-Einstein relation in sterically interacting systems: the product of the colloidal particle self-diffusion coefficient and the macroscopic viscosity remains constant in a broad range of concentrations. It has been concluded that the FCS technique in combination with simple viscosity measurements might serve as a tool for estimating the micellar size and shape.