Generalized Stokes-Einstein Relation Research Articles

Nanoscopic Interfacial Hydrogel Viscoelasticity Revealed from Comparison of Macroscopic and Microscopic Rheology.

Deviations between macrorheological and particle-based microrheological measurements are often considered to be a nuisance and neglected. We study aqueous poly(ethylene oxide) (PEO) hydrogels for varying PEO concentrations and chain lengths that contain microscopic tracer particles and show that these deviations reveal the nanoscopic viscoelastic properties of the particle-hydrogel interface. Based on the transient Stokes equation, we first demonstrate that the deviations are not due to finite particle radius, compressibility, or surface-slip effects. Small-angle neutron scattering rules out hydrogel heterogeneities. Instead, we show that a generalized Stokes-Einstein relation, accounting for an interfacial shell around tracers with viscoelastic properties that deviate from bulk, consistently explains our macrorheological and microrheological measurements. The extracted shell diameter is comparable to the PEO end-to-end distance, indicating the importance of dangling chain ends. Our methodology reveals the nanoscopic interfacial rheology of hydrogels and is applicable to different kinds of viscoelastic fluids and particles.

Non-invasive rheological testing overcomes a critical setback

By adjusting the known Generalized Stokes-Einstein Relation, light can measure bulk rheological dynamics.

Two-point microrheology and diffusing wave spectroscopy

The dynamics of charged and hard-sphere silica colloidal suspensions are measured using diffusing wave spectroscopy and are interpreted by the Generalized Stokes–Einstein Relation (GSER). At high concentration, the resulting moduli are in good qualitative agreement with bulk rheology, including the frequency response, but are higher by a quantitative factor. We show that the “two-point” GSER, which derives from the correlated motion of the colloidal particles, provides a better quantitative agreement between bulk and microrheology. The two-point GSER applies generally when the average scattering vector sampled by the multiple scattering events corresponds to a length scale greater than the scatterer diameter. We discuss applications for characterizing colloid interactions by measuring the high-frequency moduli of suspensions and extend the two-point analysis to tracer particle microrheology measurements of a semiflexible biopolymer network.

The confined Generalized Stokes-Einstein relation and its consequence on intracellular two-point microrheology

Two-point microrheology (TPM) is used to infer material properties of complex fluids from the correlated motion of hydrodynamically interacting probes embedded in the medium. The mechanistic connection between probe motion and material properties is propagation of disturbance flows, encoded in current TPM theory for unconfined materials. However, confined media e.g. biological cells and particle-laden droplets, require theory that encodes confinement into the flow propagator (Green’s function). To test this idea, we use Confined Stokesian Dynamics simulations to explicitly represent many-body hydrodynamic couplings between colloids and with the enclosing cavity at arbitrary concentration and cavity size. We find that previous TPM theory breaks down in confinement, and we identify and replace the underlying key elements. We put forth a Confined Generalized Stokes-Einstein Relation and report the viscoelastic spectrum. We find that confinement alters particle dynamics and increases viscosity, owing to hydrodynamic and entropic coupling with the cavity. The new theory produces a master curve for all cavity sizes and concentrations and reveals that for colloids larger than 0.005 times the enclosure size, the new model is required.

Self-motion and heterogeneous droplet dynamics in moderately attractive dense emulsions

We show that diffusing wave spectroscopy (DWS) is sensitive to the presence of a moderate short-range attraction between droplets in uniform fractionated colloidal emulsions near and below the jamming point associated with monodisperse hard spheres. This moderate interdroplet attraction, induced by micellar depletion, has an energy of about ∼2.4 k B T, only somewhat larger than thermal energy. Although changes in the mean free path of optical transport caused by this moderate depletion attraction are small, DWS clearly reveals an additional secondary decay-to-plateau in the intensity autocorrelation function at long times that is not present when droplet interactions are nearly hard. We hypothesize that this secondary decay-to-plateau does not reflect the average self-motion of individual droplets experiencing Brownian excitations, but instead results from heterogeneous dynamics involving a sub-population of droplets that still experience bound motion yet with significantly larger displacements than the average. By effectively removing the contribution of this secondary decay-to-plateau, which is linked to greater local heterogeneity in droplet structure caused by the moderate attraction, we obtain self-motion mean square displacements (MSDs) of droplets that reflect only the initial primary decay-to-plateau. Moreover, we show that droplet self-motion primary plateau MSDs can be interpreted using the generalized Stokes-Einstein relation of passive microrheology, yielding quantitative agreement with plateau elastic shear moduli measured mechanically.

Generalization of the Kubo relation for confined motion and ergodicity breakdown.

A time-dependent generalized Kubo relation is derived by introducing the notion of a diffusion function for a particle confined in a harmonic potential. The relation reduces to the standard Kubo relation as a special case but holds for anomalous diffusion, nonergodic processes, and bounded motion. We analyze in detail the behaviors of the diffusion and memory functions and report a generalized Stokes-Einstein relation concerning anomalous diffusion. Furthermore, we demonstrate that when a high finite-frequency cutoff is imposed on the noise spectral density, a breakdown in ergodicity accompanied by the appearance of nonstationarity in the velocity autocorrelation function occurs in forced systems. This breakdown is taken as explicit evidence for either decay-spring-memory or recovering-force effects leading to nonexponential relaxation kinematics.

The Laplace approach in microrheology.

When coupled with the generalized Stokes-Einstein (GSE) equation, it is often reported that micro-rheology probes the dynamic properties differently than do macroscopic rheological measurements, especially in relatively condensed systems. In the present work, we empirically examine the GSE in its widely used form: following an analytical continuation, the Fourier transformed particle mean-square displacement (MSD) is used to determine the dynamic moduli [G'(ω) and G''(ω)] and we compare the results with those obtained by direct inverse Laplace transform calculation of the relevant viscoelastic functions (either relaxation modulus or creep compliance) from the MSD. The results show that the inverse Laplace approaches can differ from the Fourier approach and give better agreement with macroscopic rheological measurements when this is the case. Some instances of agreement between the Fourier approach and the direct Laplace transform approaches are also shown. It is recommended that micro-rheology MSD data be interpreted using one of the direct Laplace transform based approaches.

Slow Linear Viscoelastic Relaxation of Polymer Nanocomposites: Contribution from Confined Diffusion of Nanoparticles

In this work, we investigated the linear rheology of polymer nanocomposites with the purpose to find the correlation with the confined diffusion of nanoparticles (NPs), whose diameter (2Rp) is larger than the entanglement mesh size ae of the polymer matrix. The diffusion of NPs is confined by the entanglements of polymer chains at shorter time scale and the particle cages at longer time scale. Diffusion out of entanglement networks is subdiffusive and relies on the retarded relaxation of sticky polymer chains, while the diffusion out of particle cage is a hopping process. Linear viscoelasticity predicted from the generalized Stokes–Einstein relation using the particle diffusion dynamics can describe the experimental data quantitatively. It is found that the characteristic parameters of confined diffusion out of entanglement networks depend solely on the effective surface area of NP clusters. For the diffusion out of the particle cages, the size of the cage equals to the mean interparticle surface distance...

Microrheological quantification of viscoelastic properties with photonic force optical coherence elastography.

Photonic force optical coherence elastography (PF-OCE) is a new approach for volumetric characterization of microscopic mechanical properties of three-dimensional viscoelastic medium. It is based on measurements of the complex mechanical response of embedded micro-beads to harmonically modulated radiation-pressure force from a weakly-focused beam. Here, we utilize the Generalized Stokes-Einstein relation to reconstruct local complex shear modulus in polyacrylamide gels by combining PF-OCE measurements of bead mechanical responses and experimentally measured depth-resolved radiation-pressure force profile of our forcing beam. Data exclusion criteria for quantitative PF-OCE based on three noise-related parameters were identified from the analysis of measurement noise at key processing steps. Shear storage modulus measured by quantitative PF-OCE was found to be in good agreement with standard shear rheometry, whereas shear loss modulus was in agreement with previously published atomic force microscopy results. The analysis and results presented here may serve to inform practical, application-specific implementations of PF-OCE, and establish the technique as a viable tool for quantitative mechanical microscopy.

Orthogonal superposition rheometry of model colloidal glasses with short-ranged attractions

Attractive colloidal glasses are unique as their dynamical arrest is a combination of entropic crowding effects and energetic bonds formation. When such systems are subjected to flow, their dynamics are activated in a way which differs from hard-sphere glasses that melt through a “convective cage release mechanism.” Here, we investigate the microscopic dynamics by measuring the relaxation spectrum during flow using orthogonal superposition rheometry. A small amplitude oscillatory strain is imposed perpendicularly to a steady shear flow, and superposition moduli are measured. Brownian dynamic simulations are utilized complementary to extract the moduli from mean-squared displacements using the generalized Stokes–Einstein relation. At low Péclet number, a crossover frequency between elastic and viscous moduli is detected, representing the relaxation time associated with shear-induced particles escape from their frustration (localization) under flow. For the repulsive glass, this is related to shear-induced cage renewal of particles. For attractive glasses, however, when particles escape their localized length (bonds), they move with no further hindrance with the escape time, which is independent of attraction strength and interestingly faster than that in the repulsive glass. This is attributed to particle localization at shorter length scales due to bonding. At high Péclet, a second low frequency crossover is observed and a low frequency elastic dominated response emerges. This elastic response may originate from slow relaxation of hydroclusters or be a consequence of more intricate nature of superposition moduli. At high frequencies, both orthogonal moduli increase relative to quiescent state due to shear-induced cage deformation, which slows down in-cage dynamics.

Structural Changes in Polymeric Gel Scaffolds Around the Overlap Concentration.

Cross-linked polymeric gels are an important class of materials with applications that broadly range from synthetic wound healing scaffolds to materials used in enhanced oil recovery. To effectively design these materials for each unique applications a deeper understanding of the structure and rheological properties as a function of polymeric interactions is required. Increasing the concentration of polymer in each scaffold increases physical interactions between the molecules that can be reflected in the material structure. To characterize the structure and material properties, we use multiple particle tracking microrheology (MPT) to measure scaffolds during gelation. In MPT, fluorescently labeled probe particles are embedded in the material and the Brownian motion of these particles is captured using video microscopy. Particle motion is related to rheological properties using the Generalized Stokes-Einstein Relation. In this work, we characterize gelation of a photopolymerized scaffold composed of a poly(ethylene glycol) (PEG)-acrylate backbone and a PEG-dithiol cross-linker. Scaffolds with backbone concentrations below and above the overlap concentration, concentration where polymer pervaded volume begins to overlap, are characterized. Using time-cure superposition (TCS) we determine the critical relaxation exponent, n, of each scaffold. The critical relaxation exponent is a quantitative measure of the scaffold structure and is similar to a complex modulus, G*, which is a measure of energy storage and dissipation. Our results show that below the overlap concentration the scaffold is a tightly cross-linked network, navg = 0.40 ± 0.03, which stores energy but can also dissipate energy. As polymeric interactions increase, we measure a step change in the critical relaxation exponent above the overlap concentration to navg = 0.20 ± 0.03. After the overlap concentration the scaffold has transitioned to a more tightly cross-linked network that primarily stores energy. Additionally, continuing to increase concentration results in no change in the scaffold structure. Therefore, we determined that the properties of this scaffold can be tuned above and below the overlap concentration by changing the polymer concentration but the structure will remain the same in each concentration regime. This is advantageous for a wide range of applications that require scaffolds with varying stiffness and the same scaffold architecture.

Diffusing wave microrheology of highly scattering concentrated monodisperse emulsions

Motivated by improvements in diffusing wave spectroscopy (DWS) for nonergodic, highly optically scattering soft matter and by cursory treatment of collective scattering effects in prior DWS microrheology experiments, we investigate the low-frequency plateau elastic shear moduli [Formula: see text] of concentrated, monodisperse, disordered oil-in-water emulsions as droplets jam. In such experiments, the droplets play dual roles both as optical probes and as the jammed objects that impart shear elasticity. Here, we demonstrate that collective scattering significantly affects DWS mean-square displacements (MSDs) in dense colloidal emulsions. By measuring and analyzing the scattering mean free path as a function of droplet volume fraction φ, we obtain a φ-dependent average structure factor. We use this to correct DWS MSDs by up to a factor of 4 and then calculate [Formula: see text] predicted by the generalized Stokes-Einstein relation. We show that DWS-microrheological [Formula: see text] agrees well with mechanically measured [Formula: see text] over about three orders of magnitude when droplets are jammed but only weakly deformed. Moreover, both of these measurements are consistent with predictions of an entropic-electrostatic-interfacial (EEI) model, based on quasi-equilibrium free-energy minimization of disordered, screened-charge-stabilized, deformable droplets, which accurately describes prior mechanical measurements of [Formula: see text] made on similar disordered monodisperse emulsions over a wide range of droplet radii and φ. This very good quantitative agreement between DWS microrheology, mechanical rheometry, and the EEI model provides a comprehensive and self-consistent view of weakly jammed emulsions. Extensions of this approach may improve DWS microrheology on other systems of dense, jammed colloids that are highly scattering.

Combining Microfluidics and Microrheology to Determine Rheological Properties of Soft Matter during Repeated Phase Transitions.

The microstructure of soft matter directly impacts macroscopic rheological properties and can be changed by factors including colloidal rearrangement during previous phase changes and applied shear. To determine the extent of these changes, we have developed a microfluidic device that enables repeated phase transitions induced by exchange of the surrounding fluid and microrheological characterization while limiting shear on the sample. This technique is µ2rheology, the combination of microfluidics and microrheology. The microfluidic device is a two-layer design with symmetric inlet streams entering a sample chamber that traps the gel sample in place during fluid exchange. Suction can be applied far away from the sample chamber to pull fluids into the sample chamber. Material rheological properties are characterized using multiple particle tracking microrheology (MPT). In MPT, fluorescent probe particles are embedded into the material and the Brownian motion of the probes is recorded using video microscopy. The movement of the particles is tracked and the mean-squared displacement (MSD) is calculated. The MSD is related to macroscopic rheological properties, using the Generalized Stokes-Einstein Relation. The phase of the material is identified by comparison to the critical relaxation exponent, determined using time-cure superposition. Measurements of a fibrous colloidal gel illustrate the utility of the technique. This gel has a delicate structure that can be irreversibly changed when shear is applied. µ2rheology data shows that the material repeatedly equilibrates to the same rheological properties after each phase transition, indicating that phase transitions do not play a role in microstructural changes. To determine the role of shear, samples can be sheared prior to injection into our microfluidic device. µ2rheology is a widely applicable technique for the characterization of soft matter enabling the determination of rheological properties of delicate microstructures in a single sample during phase transitions in response to repeated changes in the surrounding environmental conditions.

Nanorheology of Entangled Polymer Melts.

We use molecular simulations to probe the local viscoelasticity of an entangled polymer melt by tracking the motion of embedded nonsticky nanoparticles (NPs). As in conventional microrheology, the generalized Stokes-Einstein relation is employed to extract an effective stress relaxation function G_{GSE}(t) from the mean square displacement of NPs. G_{GSE}(t) for different NP diameters d are compared with the stress relaxation function G(t) of a pure polymer melt. The deviation of G_{GSE}(t) from G(t) reflects the incomplete coupling between NPs and the dynamic modes of the melt. For linear polymers, a plateau in G_{GSE}(t) emerges as d exceeds the entanglement mesh size a and approaches the entanglement plateau in G(t) for a pure melt with increasing d. For ring polymers, as d increases towards the spanning size R of ring polymers, G_{GSE}(t) approaches G(t) of the ring melt with no entanglement plateau.

The continuous modeling of charge-stabilized colloidal suspensions in shear flows

Flows of concentrated colloidal suspensions may exhibit a rich set of behaviors due to both hydrodynamic and colloidal interactions between the particles. Colloidal flows are generally modeled with an effective Navier–Stokes equation and a mass balance for the solid phase involving a diffusion coefficient given by the generalized Stokes–Einstein relation. This picture corresponds to a near equilibrium regime in which entropic and colloidal forces dominate over hydrodynamic interactions, the latter being totally ignored. On the other hand, suspension flows far from equilibrium require the modeling of significant hydrodynamic stresses responsible in particular for shear-induced migration, a phenomenon known to occur in some industrial processes involving colloids, such as ultrafiltration. The choice of the proper model ingredients requires a knowledge of the domains in parameter space in which colloidal or hydrodynamic effects are dominant. In this article, such a phase diagram is established for a channel flow of charge-stabilized colloids with a version of the suspension balance model including both colloidal and hydrodynamic effects at the continuous level. It is shown that the classical Peclet number is not sufficient to characterize the flow regime. The phase boundary between the colloidal and hydrodynamic regimes exhibits an original shape explained by the dependence of electrostatic interactions with the colloidal surface charge, and in particular by the phenomenon of ionic condensation. We also show that the phase diagram can be predicted based on the knowledge of a rescaled Peclet number comparing the hydrodynamic stress scale to the bulk modulus of the suspension. The criterion determined here provides important guidelines for the efficient modeling of colloidal flows.

Origin of the effective mobility in non-linear active micro-rheology.

The distinction between the damping coefficient and the effective non-linear mobility of driven particles in active micro-rheology of supercooled liquids is explained in terms of individual and collective dynamics. The effective mobility arises as a collective effect which gives insight into the energy landscape of the system. On the other hand, the damping coefficient is a constant that modulates the effect of external forces over the thermal energy which particles have at their disposition to perform Brownian motion. For long times, these thermal fluctuations become characterized in terms of an effective temperature that is a consequence of the dynamic coupling between kinetic and configurational degrees of freedom induced by the presence of the strong external force. The interplay between collective mobility and effective temperature allows to formulate a generalized Stokes-Einstein relation that may be used to determine the collective diffusion coefficient. The explicit relations we deduce reproduce simulation data remarkably well.

How Far from Equilibrium Is Active Matter?

Active matter systems are driven out of thermal equilibrium by a lack of generalized Stokes-Einstein relation between injection and dissipation of energy at the microscopic scale. We consider such a system of interacting particles, propelled by persistent noises, and show that, at small but finite persistence time, their dynamics still satisfy a time-reversal symmetry. To do so, we compute perturbatively their steady-state measure and show that, for short persistent times, the entropy production rate vanishes. This endows such systems with an effective fluctuation-dissipation theorem akin to that of thermal equilibrium systems. Last, we show how interacting particle systems with viscous drags and correlated noises can be seen as in equilibrium with a viscoelastic bath but driven out of equilibrium by nonconservative forces, hence providing energetic insight into the departure of active systems from equilibrium.

Determination of linear viscoelastic properties of an entangled polymer melt by probe rheology simulations.

Particle rheology is used to extract the linear viscoelastic properties of an entangled polymer melt from molecular dynamics simulations. The motion of a stiff, approximately spherical particle is tracked in both passive and active modes. We demonstrate that the dynamic modulus of the melt can be extracted under certain limitations using this technique. As shown before for unentangled chains [Karim et al., Phys. Rev. E 86, 051501 (2012)PLEEE81539-375510.1103/PhysRevE.86.051501], the frequency range of applicability is substantially expanded when both particle and medium inertia are properly accounted for by using our inertial version of the generalized Stokes-Einstein relation (IGSER). The system used here introduces an entanglement length d_{T}, in addition to those length scales already relevant: monomer bead size d, probe size R, polymer radius of gyration R_{g}, simulation box size L, shear wave penetration length Δ, and wave period Λ. Previously, we demonstrated a number of restrictions necessary to obtain the relevant fluid properties: continuum approximation breaks down when d≳Λ; medium inertia is important and IGSER is required when R≳Λ; and the probe should not experience hydrodynamic interaction with its periodic images, L≳Δ. These restrictions are also observed here. A simple scaling argument for entangled polymers shows that the simulation box size must scale with polymer molecular weight as M_{w}^{3}. Continuum analysis requires the existence of an added mass to the probe particle from the entrained medium but was not observed in the earlier work for unentangled chains. We confirm here that this added mass is necessary only when the thickness L_{S} of the shell around the particle that contains the added mass, L_{S}>d. We also demonstrate that the IGSER can be used to predict particle displacement over a given timescale from knowledge of medium viscoelasticity; such ability will be of interest for designing nanoparticle-based drug delivery.

Transient binding accounts for apparent violation of the generalized Stokes-Einstein relation in crowded protein solutions.

The effect of high concentration, also referred to as crowding conditions, on Brownian motion is of central relevance for the understanding of the physical, chemical and biological properties of proteins in their native environment. Specifically, the simple inverse relationship between the translational diffusion coefficient and the macroscopic solution viscosity as predicted by the generalized Stokes-Einstein (GSE) relation has been the subject of many studies, yet a consensus on its applicability has not been reached. Here, we use isotope-filtered pulsed-field gradient NMR to separately assess the μm-scale diffusivity of two proteins, BSA and an SH3 domain, in mixtures as well as single-protein solutions, and demonstrate that transient binding can account for an apparent violation of the GSE relation. Whereas GSE behavior applies for the single-protein solutions, it does not hold for the protein mixtures. Transient binding behavior in the concentrated mixtures is evidenced by calorimetric experiments and by a significantly increased apparent activation energy of diffusion. In contrast, the temperature dependence of the viscosity, as well as of the diffusivity in single-component solutions, is always dominated by the flow activation energy of pure water. As a practically relevant second result, we further show that, for high protein concentrations, the diffusion of small molecules such as dioxane or water is not generally a suitable probe for the viscosity experienced by the diffusing proteins.

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.