Nonlinear Microrheology Research Articles

Nonlinear microrheology with time-dependent forces: Application to recoils in viscoelastic fluids

Nonlinear microrheology with time-dependent forces: Application to recoils in viscoelastic fluids

Measuring fluctuating dynamics of sparsely crosslinked actin gels with dual-feedback nonlinear microrheology.

We investigate the fluctuating dynamics of colloidal particles in weakly crosslinked F-actin networks with optical-trap-based microrheology. Using the dual-feedback technology, embedded colloidal particles were stably forced beyond the linear regime in a manner that does not suppress spontaneous fluctuations of particles. Upon forcing, a particle that was stably confined in a cage made of the network's crosslinks started to intermittently jump to the next caging microenvironments. By investigating the statistics of the jump dynamics, we discuss how heterogeneous relaxations observed in equilibrium systems became homogeneous when similar jumps were activated under constant forcing beyond the linear regime.

Adiabatic limit collapse and local interaction effects in non-linear active microrheology molecular simulations of two-dimensional fluids.

Nonlinear active microrheology molecular dynamics simulations of high-density two-dimensional fluids show that the presence of strong confining forces and an external pulling force induces a correlation between the velocity and position dynamics of the tracer particle. This correlation manifests in the form of an effective temperature and an effective mobility of the tracer particle, which is responsible for the breaking of the equilibrium fluctuation-dissipation theorem. This fact is shown by measuring the tracer particle's temperature and mobility directly from the first two moments of the velocity distribution of a tracer particle and by formulating a diffusion theory in which effective thermal and transport properties are decoupled from the velocity dynamics. Furthermore, the flexibility of the attractive and repulsive forces in the tested interaction potentials allowed us to relate the temperature and mobility behaviors to the nature of the interactions and the structure of the surrounding fluid as a function of the pulling force. These results provide a refreshing physical interpretation of the phenomena observed in non-linear active microrheology.

Nonequilibrium interactions between multi-scale colloids regulate the suspension microstructure and rheology.

Understanding nonequilibrium interactions of multi-component colloidal suspensions is critical for many dynamical settings such as self-assembly and material processing. A key question is how the nonequilibrium distributions of individual components influence the effective interparticle interactions and flow behavior. In this work, we develop a first-principle framework to study a bidisperse suspension of colloids and depletants using a Smoluchowski equation and corroborated by Brownian dynamics (BD) simulations. Using nonlinear microrheology as a case study, we demonstrate that effective depletion interactions between driven colloids are sensitive to particle timescales out of equilibrium and cannot be predicted by equilibrium-based pair potentials like Asakura-Oosawa. Furthermore, we show that the interplay between Brownian relaxation timescales of different species plays a critical role in governing the viscosity of multi-component suspensions. Our model highlights the limitations of using equilibrium pair potentials to approximate interparticle interactions in nonequilibrium processes such as hydrodynamic flows and presents a useful framework for studying the transport of driven, interacting suspensions.

Unsteady and lineal translation of a sphere through a viscoelastic fluid

Unsteady translation of a sphere in a viscoelastic fluid is studied analytically. General, fully invertible relationships between an imposed time-dependent velocity and the resultant force are described in terms of a Volterra series expansion. These results are used to analyze particle motion in fluids described by the Johnson-Segalman and Giesekus models and to define a general framework for analyzing weakly nonlinear microrheology measurements.

Motor-Driven Restructuring of Cytoskeleton Composites Leads to Tunable Time-Varying Elasticity.

The composite cytoskeleton, comprising interacting networks of semiflexible actin and rigid microtubules, generates forces and restructures by using motor proteins such as myosins to enable key processes including cell motility and mitosis. Yet, how motor-driven activity alters the mechanics of cytoskeleton composites remains an open challenge. Here, we perform optical tweezers microrheology and confocal imaging of composites with varying actin-tubulin molar percentages (25-75, 50-50, and 75-25), driven by light-activated myosin II motors, to show that motor activity increases the elastic plateau modulus by over 2 orders of magnitude by active restructuring of both actin and microtubules that persists for hours after motor activation has ceased. Nonlinear microrheology measurements show that motor-driven restructuring increases the force response and stiffness and suppresses actin bending. The 50-50 composite exhibits the most dramatic mechanical response to motor activity due to the synergistic effects of added stiffness from the microtubules and sufficient motor substrate for pronounced activity.

Transient nonlinear microrheology in hydrodynamically interacting colloidal dispersions: flow cessation

Relaxation of a colloidal dispersion following flow shutoff occurs over distinct time scales, providing a useful method for identifying microscopic forces that are otherwise difficult to detect. Active microrheology facilitates such interrogation for small-scale systems such as biological cells, where a Brownian probe is driven by an external force through the dispersion and its motion is tracked to infer flow properties. An interplay between external forces, Brownian motion, hydrodynamic interactions and interparticle forces deforms the microstructure surrounding the probe, which alters probe speed. Application of the Stokes drag law relates mean probe speed to the effective viscosity of the embedding medium. We present a micromechanical theoretical model for this relationship during nonlinear transient relaxation upon removal of the external force. Upon cessation, contributions to viscosity linear in the force vanish instantly, but hydrodynamics influences subsequent relaxation, as well as the pre-cessation structure that sets the initial condition. Stronger pre-cessation flow drives faster relaxation of both structure and rheology. Non-equilibrium entropic contributions persist after shutoff, decaying over time. Hydrodynamic interactions slow this relaxation by hindering relative Brownian and interparticle displacements. The dissipation of entropically stored energy produces a deterministic force that drives deterministic probe motion even after the external force is removed, giving a natural connection to the chemical potential. Using this idea, we show that probe back-travel gives a direct measure of the osmotic pressure and the time scale over which entropically stored energy is dissipated. Modelling a range of repulsions provides a platform for particle formulation tuned to desired transient response.

Cell contraction induces long-ranged stress stiffening in the extracellular matrix

Animal cells in tissues are supported by biopolymer matrices, which typically exhibit highly nonlinear mechanical properties. While the linear elasticity of the matrix can significantly impact cell mechanics and functionality, it remains largely unknown how cells, in turn, affect the nonlinear mechanics of their surrounding matrix. Here, we show that living contractile cells are able to generate a massive stiffness gradient in three distinct 3D extracellular matrix model systems: collagen, fibrin, and Matrigel. We decipher this remarkable behavior by introducing nonlinear stress inference microscopy (NSIM), a technique to infer stress fields in a 3D matrix from nonlinear microrheology measurements with optical tweezers. Using NSIM and simulations, we reveal large long-ranged cell-generated stresses capable of buckling filaments in the matrix. These stresses give rise to the large spatial extent of the observed cell-induced matrix stiffness gradient, which can provide a mechanism for mechanical communication between cells.

The impact of hydrodynamics on viscosity evolution in colloidal dispersions: Transient, nonlinear microrheology

Evolution of microstructure and rheology during flow startup, and its connection to microscopic transport processes, is studied theoretically via active microrheology. At steady state, the balance between entropic, hydrodynamic, and other forces changes with flow strength, producing sustained microstructural asymmetry and non‐Newtonian rheology. However, the transition from equilibrium to steady flow is sometimes marked by overshoots in viscosity that suggests a temporally evolving competition between these rate processes. Here, we formulate and solve a Smoluchowski equation for the time‐dependent evolution of particle microstructure induced by the motion of a colloidal probe driven through a bath of colloidal spheres. The structure is then utilized to compute the time‐dependent microviscosity. Brownian diffusion always sets short‐time particle dynamics, which hinders maturation of the boundary layer. The disparity in Brownian and advective transport rates produces a reversal from flow thinning to flow thickening during startup, revealing that non‐Newtonian flow phenomenology is not instantaneously established. © 2018 American Institute of Chemical Engineers AIChE J, 64: 3198–3214, 2018

Passive non-linear microrheology for determining extensional viscosity

Extensional viscosity is a key property of complex fluids that greatly influences the non-equilibrium behavior and processing of polymer solutions, melts, and colloidal suspensions. In this work, we use microfluidics to determine steady extensional viscosity for polymer solutions by directly observing particle migration in planar extensional flow. Tracer particles are suspended in semi-dilute solutions of DNA and polyethylene oxide, and a Stokes trap is used to confine single particles in extensional flows of polymer solutions in a cross-slot device. Particles are observed to migrate in the direction transverse to flow due to normal stresses, and particle migration is tracked and quantified using a piezo-nanopositioning stage during the microfluidic flow experiment. Particle migration trajectories are then analyzed using a second-order fluid model that accurately predicts that migration arises due to normal stress differences. Using this analytical framework, extensional viscosities can be determined from particle migration experiments, and the results are in reasonable agreement with bulk rheological measurements of extensional viscosity based on a dripping-onto-substrate method. Overall, this work demonstrates that non-equilibrium properties of complex fluids can be determined by passive yet non-linear microrheology.

Anomalous transport in heterogeneous media

The diffusion dynamics of particles in heterogeneous media is studied using particle-based simulation techniques. A special focus is placed on systems where the transport of particles at long times exhibits anomalies such as subdiffusive or superdiffusive behavior. First, a two-dimensional model system is considered containing gas particles (tracers) that diffuse through a random arrangement of pinned, disk-shaped particles. This system is similar to a classical Lorentz gas. However, different from the original Lorentz model, soft instead of hard interactions are considered and we also discuss the case where the tracer particles interact with each other. We show that the modification from hard to soft interactions strongly affects anomalous-diffusive transport at high obstacle densities. Second, non-linear active micro-rheology in a glass-forming binary Yukawa mixture is investigated, pulling single particles through a deeply supercooled state by applying a constant force. Here, we observe superdiffusion in force direction and analyze its origin. Finally, we consider the Brownian dynamics of a particle which is pulled through a two-dimensional random force field. We discuss the similarities of this model with the Lorentz gas as well as active micro-rheology in glass-forming systems.

The non-Newtonian rheology of hydrodynamically interacting colloids via active, nonlinear microrheology

The non-Newtonian rheology of a dilute suspension of hydrodynamically interacting colloids is studied theoretically via active, nonlinear microrheology. In this Stokes-flow regime, a Brownian probe is driven through the suspension by a fixed external force; its motion distorts the configuration of background bath particles, which in turn alters probe motion. This interplay was utilized in our recent article to obtain the normal elements of the suspension stress via the nonequilibrium statistical mechanics theory [Chu and Zia, J. Rheol. 60(4), 755–781 (2016)]. In the present article, we focus on the normal stress differences N1 and N2, osmotic pressure Π, and their evolution with the strength of the probe force and strength of hydrodynamic interactions. As hydrodynamic interactions grow from weak to strong, the influence of couplings between the stress and the entrained motion on N1 changes with the strength of flow. When flow is strong, hydrodynamic interactions suppress the magnitude of N1, owing to coll...

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.

Probing linear and nonlinear microrheology of viscoelastic fluids

Bulk rheological properties of viscoelastic fluids have been extensively studied in macroscopic shearing geometries. However, little is known when an active microscopic probe is used to locally perturb them far from the linear-response regime. Using a colloidal particle dragged periodically by scanning optical tweezers through a viscoelastic fluid, we investigate both, its linear and nonlinear microrheological response. With increasing particle velocity, we observe a transition from constant viscosity to a thinning regime, where the drag force on the probe becomes a nonlinear function of the particle velocity. We demonstrate that this transition is only determined by the ratio of the fluid's equilibrium relaxation time and the period of the driving.

Linear and nonlinear microrheology of lysozyme layers forming at the air-water interface.

We report experiments studying the mechanical evolution of layers of the protein lysozyme adsorbing at the air-water interface using passive and active microrheology techniques to investigate the linear and nonlinear rheological response, respectively. Following formation of a new interface, the linear shear rheology, which we interrogate through the Brownian motion of spherical colloids at the interface, becomes viscoelastic with a complex modulus that has approximately power-law frequency dependence. The power-law exponent characterizing this frequency dependence decreases steadily with increasing layer age. Meanwhile, the nonlinear microrheology, probed via the rotational motion of magnetic nanowires at the interface, reveals a layer response characteristic of a shear-thinning power-law fluid with a flow index that decreases with age. We discuss two possible frameworks for understanding this mechanical evolution: gelation and the formation of a soft glass phase.

Computer simulations of structure, dynamics, and phase behavior of colloidal fluids in confined geometry and under shear

Using computer simulations, colloidal systems in different external fields are investigated. Colloid-polymer mixtures, described in terms of the Asakura-Oosawa (AO) model, are considered under strong confinement. Both in cylindrical and spherical confinement, the demixing transition of the three-dimensional AO model is rounded and, using Monte Carlo simulations, we analyze in detail the consequences of this rounding (occurrence of multi-domain states in cylindrical geometry, non-equivalence of conjugate ensembles due to different finite-size corrections in spherical geometry etc.). For the case of the AO model confined between two parallel walls, spinodal decomposition is studied using a combination of molecular dynamics simulation and the multiparticle collision dynamics method. This allows us to investigate the influence of hydrodynamic interactions on the domain growth during spinodal decomposition. For a binary glass-forming Yukawa mixture, non-linear active micro-rheology is considered, i.e. a single particle is pulled through a deeply supercooled liquid. The diffusion dynamics of the pulled particle is analyzed in terms of the van Hove correlation function. Finally, the Yukawa mixture in the glass state, confined between walls, is studied under the imposition of a uniform shear stress. Below and around the yield stress, persistent creep in the form of shear-banded structures is observed.

Stress development, relaxation, and memory in colloidal dispersions: Transient nonlinear microrheology

The motion of a single Brownian particle in a complex fluid can reveal material behavior both at and away from equilibrium. In active microrheology, a probe particle is driven by an external force through a complex medium and its motion studied in order to infer properties of the embedding material. Most work in microrheology has focused on steady behavior and established the relationship between the motion of the probe, the microstructure, and the effective microviscosity of the medium. Transient behavior in the near-equilibrium, linear-response regime has also been studied via its connection to low-amplitude oscillatory probe forcing and the complex modulus; at very weak forcing, the microstructural response that drives viscosity is indistinguishable from equilibrium fluctuations. But important information about the basic physical aspects of structural development and relaxation in a medium is captured by startup and cessation of the imposed deformation in the nonlinear regime, where the structure is driven far from equilibrium. Here, we study theoretically and by dynamic simulation the transient behavior of a colloidal dispersion undergoing nonlinear microrheological forcing. The strength with which the probe is forced, Fext, compared to thermal forces, kT/b, governs the dynamics and defines a Péclet number, Pe=Fext/(kT/b), where kT is the thermal energy and b is the colloidal bath particle size. For large Pe, a boundary layer (in which unsteady advection balances diffusion) forms at particle contact on the time scale of the flow, a/U, where a is the probe size and U its speed, whereas the wake forms over O(Pe) diffusive time steps. Similarly, relaxation following cessation occurs over several time scales corresponding to distinct physical processes. For very short times, the time scale for relaxation is set by a boundary layer of thickness δ∼(a+b)/Pe, and so τ∼δ2/Dr, where Dr is the relative diffusivity between the probe of size a and a bath particle. Nearly all stress relaxation occurs during this time. At longer times, the Brownian diffusion of the bath particles acts to close the wake on a time scale set by how long it takes a bath particle to diffuse laterally across it, τ∼(a+b)2/Dr. Although the majority of the microstructural relaxation occurs during this wake-healing process, it does so with little change in the stress. Also during relaxation, the probe travels backward in the suspension; this recovered strain is proportional to the free energy stored in the compressed particle configuration, an indicator that the stress is proportional to the free energy density stored entropically in the microstructure. Theoretical results are compared with Brownian dynamics simulation where it is found that the dilute theory captures the correct behavior even for concentrated suspensions. Two modes of forcing are studied: Constant force and constant velocity. Results are compared to analogous macrorheology results for suspensions undergoing simple shear flow.

Nonlinear active micro-rheology in a glass-forming soft-sphere mixture

We present extensive molecular dynamics computer simulations of a glass-forming Yukawa mixture, investigating the nonlinear response of a single particle that is pulled through the system by a constant force. Structural changes around the pulled particle are analyzed by pair correlation functions, measured in the deeply supercooled state of the system. A regime of intermediate force strengths is found where the structural changes around the pulled particle are small, although its steady-state velocity shows a strong nonlinear response. This nonlinear response regime is characterized by a force-temperature superposition principle of a Peclet number and anisotropic diffusive behavior. In the direction parallel to the force, mean-square displacements show anomalous superdiffusion in the long time limit. We analyze this superdiffusive behavior by means of the van Hove correlation function of the pulled particle. Perpendicular to the force, the driven particle shows diffusive behavior for all considered force strengths and temperatures. We discuss the dynamics perpendicular and parallel to the force in terms of effective temperatures.

Micro–macro-discrepancies in nonlinear microrheology: I. Quantifying mechanisms in a suspension of Brownian ellipsoids

Active and nonlinear microrheology experiments involve a colloidal probe that is forced to move within a material, with the goal of recovering the nonlinear rheological response properties of the material. Various mechanisms cause discrepancies between the nonlinear rheology measured microrheologically and macroscopically, including direct probe–bath collisions, the Lagrangian unsteadiness experienced by the material elements, and the spatially inhomogeneous and rheologically mixed strain field set up around the probe. Here, we perform computational nonlinear microrheology experiments, in which a colloidal probe translates through a dilute suspension of Brownian ellipsoids, whose results we compare against analogous computational experiments on the macroscopic shear rheology of the same model material. The quantitative impact of each of the mechanisms for micro–macro-discrepancy can thus be computed directly, with additional computational experiments performed where the processes in question are ‘turned off’. We show that all three discrepancy mechanisms impact the microrheological measurement quantitatively, and that none can be neglected. This motivates a search for microrheological probes whose geometry or forcing is optimized to minimize these impacts, which we present in a companion article.

Micro–macro discrepancies in nonlinear microrheology: II. Effect of probe shape

In this second article devoted to ‘computational experiments’ of nonlinear microrheology, we examine the effect that changing the probe shape or motion has upon the three sources of discrepancy that we previously examined for spheres. In particular, prolate ellipsoidal probes have relatively long regions of relatively constant strain rate, giving predominantly shear and relative Lagrangian steadiness. The micro–macro discrepancy is shown not to arise from Lagrangian unsteadiness, but largely from the non-viscometric nature of the flows. Second, an oblate ellipsoidal probe exacerbates the extensional regions in front of and behind the probe. However, the relatively low extensional rates around such ‘disks’ would require them to be pulled at much higher rates through the fluid in order to excite the extensional deformations. Because our model material thickens under uniaxial extension, but thins under biaxial extension, the contribution of each to the total drag is partially negated by the other. Finally, we examine a rotating spherical probe, which is Lagrangian steady and pure shear. We show that the apparent viscosity thus recovered is close to the true shear viscosity, and furthermore that the true shear viscosity can be extracted quantitatively from the apparent microviscosity.