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Microrheology of colloidal systems

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 243101 (http://iopscience.iop.org/0953-8984/26/24/243101) View the table of contents for this issue, or go to the journal homepage for more

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 243101 (26pp)

doi:10.1088/0953-8984/26/24/243101

Topical Review

Microrheology of colloidal systems A M Puertas1, and T Voigtmann2,3 1

Group of Complex Fluids Physics, Department of Applied Physics, University of Almeria, 04120 Almeria, Spain 2 Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 51170 Köln, Germany 3 Zukunftskolleg and Fachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany E-mail: [email protected] Received 17 January 2014, revised 1 April 2014 Accepted for publication 1 April 2014 Published 21 May 2014 Abstract

Microrheology was proposed almost twenty years ago as a technique to obtain rheological properties in soft matter from the microscopic motion of colloidal tracers used as probes, either freely diffusing in the host medium, or subjected to external forces. The former case is known as passive microrheology, and is based on generalizations of the Stokes–Einstein relation between the friction experienced by the probe and the host-fluid viscosity. The latter is termed active microrheology, and extends the measurement of the friction coefficient to the nonlinear-response regime of strongly driven probes. In this review article, we discuss theoretical models available in the literature for both passive and active microrheology, focusing on the case of single-probe motion in model colloidal host media. A brief overview of the theory of passive microrheology is given, starting from the work of Mason and Weitz. Further developments include refined models of the host suspension beyond that of a Newtonian-fluid continuum, and the investigation of probe-size effects. Active microrheology is described starting from microscopic equations of motion for the whole system including both the host-fluid particles and the tracer; the many-body Smoluchowski equation for the case of colloidal suspensions. At low fluid densities, this can be simplified to a two-particle equation that allows the calculation of the friction coefficient with the input of the density distribution around the tracer, as shown by Brady and coworkers. The results need to be upscaled to agree with simulations at moderate density, in both the case of pulling the tracer with a constant force or dragging it at a constant velocity. The full many-particle equation has been tackled by Fuchs and coworkers, using a mode-coupling approximation and the scheme of integration through transients, valid at high densities. A localization transition is predicted for a probe embedded in a glass-forming host suspension. The nonlinear probefriction coefficient is calculated from the tracer's position correlation function. Computer simulations show qualitative agreement with the theory, but also some unexpected features, such as superdiffusive motion of the probe related to the breaking of nearest-neighbor cages. We conclude with some perspectives and future directions of theoretical models of microrheology. Keywords: colloids, mircrorheology, non-linear response (Some figures may appear in colour only in the online journal)

1. Introduction

stresses in the material to its (rate of) deformation; such constitutive equations are traditionally built on empirical knowledge of the macroscopic behavior of a class of materials. A much more demanding task is to derive constitutive equations starting from the analysis of microscopic structure and

Rheology is the study of the deformation and flow of matter, and in particular complex fluids. The rheological response is usually modeled in terms of constitutive equations, that relate 0953-8984/14/243101+26$33.00

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© 2014 IOP Publishing Ltd Printed in the UK

Topical Review

J. Phys.: Condens. Matter 26 (2014) 243101

dynamics, and its change under global or local d eformations [1, 2]. In the context of linking microscopic dynamics with macroscopic response, microrheology is a promising technique to access the response of a complex system to a localized mechanical perturbation. Microrheology is based on the observation and manipulation of one or a set of small, typically µm-sized tracer particles (probes) embedded in the host medium whose rheological properties are of interest. It has originally been developed with the idea to provide an alternative device for measuring macroscopic material properties. More recently, the fact that microrheology provides local information on the microstructure of the host system that goes beyond macroscopic rheological quantities, has been emphasized as an advantage. Such information is extremely valuable for complex and inhomogeneous media, be they colloidal glasses or gels, active-matter and biological systems, or in microfluidics and nanotechnology applications. Fundamental questions of condensed-matter physics can be addressed, such as the mechanisms of dynamic arrest at high densities, the validity of fluctuation-dissipation theorems (FDT) or the fluctuation theorem (FT) in out-of-equilibrium systems. Two main operational modes of microrheology are typically distinguished: passive microrheology observes the diffusive motion of the probe particles that is induced through thermal fluctuations in the bath (of energy kBT). In the case of an isolated spherical probe whose (hydrodynamic) radius a is much bigger than the relevant length scales of the embedding fluid, the familiar Stokes–Einstein (SE) equation [3], derived independently also by Sutherland [4], relates the probe diffusion coefficient D to the host-fluid shear viscosity η,

are recovered. Active microrheology extends the measurement of the friction coefficient to the nonlinear-response regime, where γ itself becomes a function of the strength of the applied perturbation. Again, measurements beyond the steady state are possible, for example by applying an oscillatory force to the probe, and relating the result to a tracerbased (nonlinear) compliance of the bath. Establishing a connection between the microscopic rheological properties and the macroscopic counterparts of the bath, measured with bulk rheology, was the initial motivation for the development of microrheology, and deserves special attention. The common procedure is to relate the friction coefficient γ to the macroscopic shear viscosity η by extending the Stokes–Einstein–Sutherland expression, equation (1), to the frequency-dependent, non-hydrodynamic case. Questions of applicability and possible origins of such ‘generalized Stokes–Einstein’ (GSE) relations are long-standing issues in condensed-matter physics. We will touch upon this below. It has become clear that establishing the connection between microscopic probe-particle friction and bulk rheological properties of the embedding fluid is far from trivial (and may not be generically possible in complex fluids). Still, to put, where it works, such a connection on a well-understood theoretical basis is an intriguing prospect, since microrheology has the advantage of allowing measurements on much smaller samples than those conventionally used in macrorheology. It has become clear that potential applications of microrheology go well beyond an alternative to bulk rheology in expensive or difficult to obtain systems [6], including particle depinning and microstructure elasticity in arrested systems, such as gels [7] or glasses [8]. Microrheology consequently has been applied to a wide variety of soft-matter systems, from hard-sphere like colloidal suspensions to eukariotic cells, with different levels of modeling. The idea to infer macroscopic rheological properties of fluids and in particular visco-elastic materials by local probes is, in fact, quite old and goes back to Freundlich and Seifriz in the 1920s [9] with first applications on biological systems [10]. In the 1960s, the analysis of the finite displacement u of a forced magnetic bead in an elastic gel was proposed [11] to infer the shear modulus G via the relation F = 6π G a u; this is nothing but Stokes' expression by recognizing the deep connection between Stokes flow and linear elasticity theory [12]. There is still continuing development from theory, simulation, and experiment. In the present review, we focus on the microrheology of colloidal systems, with special attention to the theoretical models developed for this case. A number of review papers already cover different aspects of microrheology: the description of experimental techniques and application to soft-matter systems in general [6, 13–17], application in biophysics [18, 19], or more specific aspects such as the validity of GSE relations [20] or multiple-particle-tracking microrheology for sol–gel transitions [21]. We will first briefly review the topic of passive microrheology, starting with analyses based on the GSE relation pioneered by Mason and Weitz, continuing to more elaborate

kT kT D= B = B , (1) γs 6πηa

assuming no-flux stick boundary conditions at the probe surface. Here, γs is the probe-particle friction coefficient, given by the Stokes drag evaluated from continuum low-Reynoldsnumber fluid mechanics [5]. Passive microrheology extends the classical Stokes experiment of a solid probe moving relative to a viscous fluid, by considering the case where the probe dimension a is comparable to the interaction length scales relevant for the host, and by discussing the frequency-dependent response related to time scales comparable to that of the host fluid's relaxation dynamics. In active microrheology, the probe particle is subject to an external force: sometimes to hold it in some external potential, linking the resulting fluctuations to properties of the host fluid, most often to drag it. In the latter case, the average force exerted on the tracer, 〈F〉, relates to its average velocity, 〈v〉, in the stationary state through a friction coefficient, 〈F〉 = γ〈v〉. Linear-response theory connects active and passive microrheology in the limit of small external forces. In the case of constant external force, the average position of the probe along the force axis obeys 〈 x ( t ) 〉 = ( 〈 x 02 ( t ) 〉 / 2kBT ) F, where 〈 x 02 ( t ) 〉~ 2Dt is the mean-squared displacement of the probe without an external force. Hence the Stokes-drag expression F = 6πη av and the Einstein relation γ = kBT/D 2

Topical Review


models of tracer diffusion in complex systems. We then discuss active microrheology, in particular those cases that explicitly probe the dependence of the applied perturbation strength, or discuss features of strong driving that cannot be mapped onto the passive-microrhelogy case by virtue of the FDT. Considering colloidal suspensions as host media, the theoretical description of active microrheology currently separates into two major branches: the low-density host-fluid case was considered originally by Brady and coworkers, based on the two-particle solutions of the Smoluchowski equation. On the other hand, high-density host fluids close to their glass transition were studied based on a mode-coupling theory devised by Fuchs and coworkers. After discussing the main aspects of these theoretical approaches and their comparison to experiment and simulation, we conclude with some remarks on related topics emerging out of colloidal microrheology.

, which needs to be converted to the Fourier domain to yield the elastic, G′(ω), and viscous, G′′(ω), moduli: for s = −iω+0, G (̂ s ) = G ′ ( ω ) + iG ′′ ( ω ). To improve statistics, a set of noninteracting probe particles can be used, in particular also to average out sample inhomogeneities [21]. In colloidal suspensions, a convenient choice is to employ as probes the colloidal host-fluid particles themselves. This raises the question of the applicability of the GSE relation in the case where no length scale separation between probe and embedding fluid exists. We will discuss some of the issues below. An alternative formulation of the GSE relation, based on the linear-response connection, was given by Gittes et al [24]. The frequency-dependent complex compliance, α(ω), relates the (small) force applied to the probe to its displacement response: rω = α(ω) fω. In the same spirit as above, the compliance is assumed to be related to the shear moduli by α(ω) = 1/ (6 π a G(ω)). Applying the fluctuation-dissipation theorem, the power spectrum of the MSD is obtained as a function of the imaginary part of the compliance [24],

2. Passive microrheology

4k T Im[ α ( ω ) ] 〈 δr 2 ( ω ) 〉 = B . (4) ω

2.1. Theoretical model for microrheology

The focal point for the analysis of passive microrheology experiments is the Stokes–Einstein relation, equation (1). A generalization was proposed in seminal papers by Mason and Weitz and coworkers [22, 23] to extract the complex viscoelastic mechanical modulus of a complex system. One starts from a generalized Langevin equation for the probe position r(t),

The real and imaginary parts of α(ω) are linked by the familiar Kramers–Kronig relations, allowing to reconstruct G(ω) from the observed MSD. The formulation in terms of the probe compliance also provides a direct connection to small-amplitude oscillatory microrheology and it is a convenient way for data analysis in passive microrheology in experiment [14, 24, 25]. Some care has to be taken in order to avoid cutoff artifacts that arise since the experimental data can only be obtained in a finite time window. Originally, it has been proposed to fit experimental time-domain data using simple functions whose Fourier-Laplace transform can be evaluated analytically [22], [26–28]. For example, Mason et al proposed to describe the MSD by local power laws, of exponent β(ω), yielding [26],

∫

t .. . mr ( t ) = − γ ( t ′− t ) r ( t ′ )dt ′ + f ( t ), (2) −∞

which includes a memory kernel to account for retarded friction arising from structured host media, and a fluctuating force f obeying the generalized fluctuation-dissipation theorem: 〈f(t) f(t′)〉 = 6 kBT γ(t′−t), in three spatial dimensions. The inertial term on the left-hand side of equation (2) can usually be neglected for a colloidal tracer (although its effects can be studied, see below). The velocity auto-correlation function can then be obtained by Laplace transformation, γ (̂ s ) =

∫

0

∞

{

}

kBT π (ω) = exp i β ( ω ) , G (5) 2 πa 〈 δr 2 (1 / ω ) 〉Γ [1 + β ( ω )]

where Γ(x) is the Gamma-function and 〈δr2(1/ω)〉 is the timedomain MSD evaluated at the time t = 2π/ω, justified by the assumption that the MSD is a slowly varying function in time. To avoid possible artifacts by introducing ad-hock fitting functions, various improved data analysis schemes have been proposed. These include time series analysis using maximum likelihood estimators [29], correlation methods working with greatly improved statistics and sampling frequency [25], or more sophisticated interpolation schemes [30]. The GSE relation is a proper generalization of the hydrodynamic Stokes problem only at small frequencies. Felderhof [31] has emphasized the importance of the correction for finite frequencies, in terms of the fluid mass density ρ displaced by the probe particle that was already pointed out by Stokes [5]. The frequency-dependent probe friction attains a correction related to the long-time tails in the velocity autocorrelation function,

exp[ − st ] γ ( t )dt, and leads to the following

expression for the probe-particle mean squared displacement (MSD): kBT 1 〈 δr 2̂ ( s ) 〉 = 6 . (3) 6πaη ̂ ( s ) s 2 * Here, ηˆ ( s ) = γˆ( s ) / 6πa is called the frequency-dependent * microviscosity. In an ad-hoc assumption guided by the original Stokes–Einstein relation, one identifies the frequency-dependent micro- and macro-viscosities, ηˆ ( s ) ≈ ηˆ ( s ), recovering * equation (1) in the long-time limit, s → 0. This generalized Stokes–Einstein (GSE) relation hence links the probe-particle MSD in the Laplace domain to the frequency-dependent elastic bulk moduli of the host, Gˆ ( s ) = sηˆ ( s ). The latter is the quantity typically measured in small-amplitude oscillatory bulk rheology. Assuming the GSE relation to be applicable, the experimental procedure is to measure the MSD of a tracer, typically as a function of time, and Laplace transform it to obtain G (̂ s )

⎡ − iωτv ⎤ ⎥ γ ( ω ) = 6πη ( ω ) a ⎢ 1 + (6) ⎢⎣ η ( ω ) / η0 ⎥⎦

3

Topical Review


[33–36]. At even shorter time scales, compressibility effects may in addition become relevant, and these have been studied [37], prompted by recent developments of experiments capable of resolving the very-short-time probe motion [38]. It has to be noted, however, that these studies typically still model the fluid as a viscoelastic continuum, implying its structural length scales to be well separated from the probe size a.

2.2. Experiments

Experimental data for passive microrheology can be obtained by various means, including direct-imaging techniques such as video or confocal microscopy followed by particle-tracking analysis, and scattering methods such as dynamic light scattering or diffusive wave spectroscopy (DWS) [39]. DWS takes advantage of multiple scattering in turbid samples, where conventional dynamic light scattering is unavailable. This includes many dense colloidal suspensions. Assuming that the light path, due to multiple scattering, is a random variable with Gaussian statistics, the resulting intensity autocorrelation function has the shape of the conventional intermediate scattering function, but the length scales probed are much smaller. They are of the order of the mean free path length of the photons, typically 10−2…10−3λ, where λ is the wave length of the scattered light. Enhancements of DWS have been developed to improve ensemble averaging, which becomes crucial in nonergodic samples [40]. The first experimental applications of passive microrheology were given by Mason and Weitz using DWS in different systems [22]. Figure 1 reproduces their results for a colloidal hard-sphere system at high density, where the host-fluid colloidal beads are also used as tracers, and a polyethylene oxide (PEO) solution with colloidal tracers. The shear moduli obtained from microrheology (lines) are compared with conventional rheology measurements (symbols) in the same frequency range, yielding reasonable agreement. The hardsphere results shown in figure 1 refer to a glass-forming host fluid at large number density ρ: the packing fraction ϕ = (4π / 3) ρah3 ≈ 0.56 is close to the commonly accepted glass-transition value ϕg ≈ 0.58 for similar hard-sphere-like colloidal suspensions [41]. The MSD (inset in figure 1) consequently displays linear growth in time only at short times, related to the short-time diffusivity; in the microrheological interpretation, this connects to the high-frequency viscosity of the suspension. At longer times, the MSD features a subdiffusive ‘plateau’ indicating transient localization of particles on a length scale of a few percent of a particle radius. This transiently solid-like motion correlates with the pronounced visco-elastic behaviour of the host fluid: on the time scales probed in the experiment of [22], the system acts essentially solid-like (i.e., the storage modulus dominates the loss modulus, G′ > G′′ in the accessible frequency window). The longtime diffusion regime that is expected in a fluid host is not seen in the hard-sphere MSD measured by Mason and Weitz. The PEO data (lower panel of figure 1) exemplify this crossover to an ultimately fluid regime: at low frequencies, dissipation dominates over elasticity (G′′ > G′).

Figure 1. Comparison of microrheology and bulk rheology. Upper panel: shear moduli from microrheology (symbols) and bulk rheology (lines) for a hard-sphere colloidal suspension with particle size a ≈ 0.21 µm and volume fraction ϕ = 0.56. The open circles show G (̂ ω ), and in the inset the MSD is presented, measured with DWS. Lower panel: shear moduli for a solution of 4 × 106 molecular weight polyethylene oxide in water at 15% weight concentration from microrheology (symbols) and bulk rheology (lines). The inset shows G (̂ s ). Reprinted with permission from [22]. Copyright (1995) by the American Physical Society.

(correcting a factor a omitted in [31]), where τv = ρ a2/η0 is the viscous relaxation time of the problem, and η0 = η (0) ̂ is the zerofrequency viscosity. Taking into account the finite momentum relaxation time of the probe particle, τm, the relation between the frequency-dependent MSD and the elastic moduli becomes ⎡ η (̂ ω ) = η0⎢ ⎢⎣

⎤2 ⎛ 1 ⎞⎟ 1 ⎜ + iω τm − τv − − iωτv ⎥ , ⎝ ⎥⎦ 4 ⎠ 2 6πη0aZ (̂ ω ) 6kT

where Z (̂ ω ) is the Fourier-transformed velocity autocorrelation function (VACF) of the probe. Recalling that the VACF is related to the second derivative of the MSD, one recognizes the GSE relation as the approximation ωτm, ωτv ≪ 1. Felderhof's correction is hence relevant only for rather large frequencies [31]; however, the viscous relaxation time τv can become relevant for large particles trapped in strong optical traps, revealing ‘the colour of Brownian motion’ [32]. Effects of probe inertia and fluid inertia, including possible implications for data analysis, have been discussed in detail elsewhere 4

Topical Review


Sohn and Rajagopalan [42] have studied in more detail the frequency-dependent storage and loss moduli obtained from microrheology measurements of hard-sphere model suspensions (polymethyl methacrylate particles in cycloheptyl alcohol) at various volume fractions up to φ ≈ 0.44, using silica particles with radius a ≈ 1.65ah as probes measured in dynamic light scattering. The qualitative frequency dependence of the dynamical viscosity converted from microrheology using the GSE relation matches that obtained from bulk rheology, in a regime where the low-frequency viscosity that enters the original SE relation varies by about 1.5 decades. However, Sohn and Rajagopalan had to introduce an empirical volume-fraction-dependent correction factor C(ϕ) ⩽ 1 to bring the two measurements into quantitative agreement, setting Ds(ω, ϕ) = (1/C(ϕ))(kBT/6 π a η(ω, ϕ)). The correction factor decreases from unity (in the dilute suspension) to around 0.6 for the largest volume fractions studied. It takes into account deviations from the Stokes–Einstein relation for these hard-sphere suspensions already noted in earlier computer simulations and experiments relating the high-frequency viscosity to the short-time diffusion coefficient [43–45]. Hard-sphere suspensions with added short-range attraction between the colloidal particles are an important model system for studying dynamical arrest phenomena [46]. They can be realized in colloid–polymer mixtures, where nonadsorbing polymer in the solvent provides a depletion-induced attraction among the colloid particles. At high density, varying the attraction strength allows to distinguish two kinds of glasses: the entropically driven hard-sphere-like ‘repulsive’ glasses at weak attraction, and the ‘attractive’ glasses formed when the attraction is strong and sufficiently short-ranged. Approaching the attractive-glass line, the visco-elastic properties of the fluid are dominated by the transient bonding between particles, leading to much higher dynamical shear moduli than in the repulsive glass. Kozina et al [47] compared bulk rheology and microrheology for repulsive and attractive glasses, using internally crosslinked polystyrene microgels as the hard-core particles in a colloid–polymer mixture. Working at a packing fraction close to the repulsive glass transition, the GSE relation was well fulfilled for weak attraction, setting C(ϕ) ≈ 1. Increasing the attraction strength, the empirical correction factor increasingly differs from unity; values for 1/C(ϕ) between 3 and 18 are found in [47]4. These values increase with sample age in this study, indicating that the MSD measured in microrheology shows weaker aging dependence than the bulk rheology moduli. Since an increasing deviation from the SE close to the glass transition is interpreted as a signature of increasing dynamical heterogeneities (see below), the comparison of bulk and microrheology was suggested as a tool to quantify these heterogeneities. Furthermore, Kozina et al [47] find that also the frequency dependence of the quantities probed in the two techniques differs in the attractive glass. Indeed, structural heterogeneities arise in the attractive glass, as confirmed by earlier simulations [48] and experiments [49], that are responsible for the different dynamics probed on different

Figure 2. Microrheology on laponite. Upper panel: trajectories of a tracer particle with different waiting times, as labeled. Lower panel: shear moduli from bulk rheology (filled squares G′, open squares G′′), and microrheology (filled circles G′ and open circles G′′) at ω = 1 rad s−1. Reprinted with permission from [53]. Copyright (2008) by the American Physical Society.

length scales. This effect is in particular important for lower densities (but strong attraction), where gelation occurs in these short-ranged attractive colloids. Passive microrheology nevertheless is a convenient technique to monitor the dynamics of gel formation, since it is nearly non-intrusive compared to mechanical bulk rheology [50–58]. Clays, in particular laponite, are among the most relevant systems here [59]. Oppong et al [53] performed experiments in gel-forming laponite suspensions (whose diskshaped particles are roughly 15 nm in radius and a few nm thick), using flourescent polystyrene spheres with a radius a ≈ 500 nm. The tracer motion becomes more and more hindered with growing waiting time elapsed after sample preparation, due to the irreversible formation of the gel (see figure 2). Oppong et al compared micro- and bulk rheology, finding that both techniques monitor the crossover from fluid-like (where G′′> G′) to solid-like (G′> G′′) behavior as a function of the waiting time. However, the crossover depends on the length scale probed [56], and occurs first on macroscopic scales, and only later on microscopic scales. Thus, the combination of

4

In [47], values C(ϕ) > 1 are quoted, but since the bulk-rheology moduli shown are larger than the values estimated from the GSE relation, those values are to be interpreted as 1/C(ϕ) instead. 5

Topical Review


remains microscopically flat. Schofield and Oppenheim [74] have recovered this result as the leading order in ah/a from a mode-coupling theory involving all N-particle density fluctuations as hydrodynamic slow modes. Note, however, that in colloidal host-fluid suspensions, the intrinsic motion of the host particles is Brownian, instead of the Newtonian dynamics considered in these theories. It is unclear how the resulting difference regarding momentum conservation affects the approach to the SE limit. For finite ah/a, the hydrodynamic probe radius (the point where boundary conditions for the hydrodynamic fields are imposed) is distinct from the actual probe size [75, 76]. One can argue that even for a ≈ ah, the fundamental relation between diffusivity and viscosity should still be D ∼ 1/η, since the hydrodynamic solution is governed by the far field from the probe and thus robust against changes on the microscopic level [77]. Deviations from the SE relation with slip or stick boundary conditions can then be interpreted as an effective hydrodynamic radius; it will depend on temperature, density, and all interaction details [78–80]. By chance, the SE relation is nearly fulfilled with stick boundary conditions for moderately dense hard-sphere fluids when probe and host particle sizes are equal [81, 82]. This is also a result in the mode-coupling theory of the glass transition for hard spheres [83–85]. Close to the glass transition, the product Dη is observed to grow appreciably for a ≈ ah, both in molecular [86–88] and in colloidal glass formers [89–91]. This is referred to as ‘Stokes–Einstein violation’, and thought to reflect the dynamical heterogeneities of the glass-forming system [92]. The correction factor C(φ) discussed above [42, 47] accounts for this. Interestingly, the SE relation continues to be fulfilled in the accessible temperature- and density-range, within experimental and numerical uncertainties in some cases [93], such as in some metallic melts [94, 95], or a subspecies of ‘sedentary’ particles in polydisperse hard spheres [96]. The models for passive microrheology discussed above ignore the microscopic structure of the host fluid, be it colloidal, polymeric, or biological. Although this effectivehydrodynamics approach is good enough for many purposes, a proper theoretical treatment of probe particles whose size is comparable to the relevant length scales in the host liquid, requires microscopic or at least mesoscopic modeling of the interaction. A prominent example are polymer melts, where a decrease of the friction coefficient with decreasing probe size is understood by considering the length scales associated with the polymer coils and the entanglement length scale [97]. Relevant for many biophysical applications is the case of a viscoelastic host medium consisting of an elastic network coupled to an incompressible Newtonian fluid through friction forces. An effective two-fluid model based on continuum mechanics, but implicitly including a relevant network-mesh length scale, was discussed by Levine and Lubensky [98, 99]. It is found with this model that the response of the system, in terms of its compliance, reduces to the simple form of the generalized SE relation for large frequencies, where the longitudinal compression mode of the network is irrelevant. It is further required that the tracer size must be larger than the mesh size of the network, but smaller than the inertial decay length of

microrheological measurements with different probe sizes, in addition to macroscopic rheology, could provide length-scale resolved information on the growth of the gel network [57]. This possibility opens many fascinating applications of microrheology for the study of complex soft matter, and especially biophysical systems. A complete overview over the numerous experimental studies is beyond the present review; see [18, 19, 21] in particular for the biophysical perspective. But let us highlight a few examples. The persistence length of wormlike micelle solutions could first be determined by combining macro- and microrheology, in order to cover a sufficiently large frequency window in the dynamical shear moduli [60]. One of the most studied systems is actin, an essential protein forming the cytoskeleton, that provides the cell structural support. In solution, it forms filaments with persistence length of the order of micrometers, depending on the conditions (pH, salt concentration, etc.) The influence of local stresses on the actin network is decisive for many biological functions of the cell, and is ideally studied by microrheology. Pioneering microrheology experiments in actin solutions were performed by Ziemann et al [61] and Schnurr et al [24, 62], with beads of different sizes, larger than the mesh size. They found that the tracer motion is hindered, showing subdiffusion related to the viscoelastic properties of the medium [63–66]. The absolute results are however extremely dependent on the preparation procedure, as is also the case for the bulk shear moduli. In microrheology, results also depend on the chemistry of the tracers, as the interaction with the actin filaments modifies the properties of the system [67, 68]. The necessity to minimize the influence of tracer-bath coupling details, and to deal with inhomogeneities in the sample, has prompted the development of two-particle microrheology [69] where the motion of two nearby particles is cross-correlated. Passive microrheology has also been performed with other biological macromolecules, such as DNA [26, 70], and also in vivo cells [18, 19]. A commonly observed feature in biological systems is superdiffusion: the probe-particle MSD in some time window growths faster than linear in time, hence faster than expected for diffusive motion. Since this occurs at time scales much larger than probe-momentum relaxation, it is a clear indicator the non-Brownian forces arising from biological activity [71]. 2.3. Probes in structured fluids

The approximate validity of the GSE relation even in the distinctly non-Stokesian limit of small probes, a ≈ ah, has been demonstrated in many instances, starting in the original work by Mason and Weitz [22, 23]. It remains striking nevertheless. According to the standard Green–Kubo formula, the probeparticle self-diffusion is governed by correlation functions of the self-part of the fluctuating density, while the viscosity involves the collective autocorrelation of stress fluctuations. Kinetic theory can be used to deduce the SE relation in the limit of large probe particles, a ≫ ah, and heavy mass. As first correctly pointed out by van Beijeren and Dorfman [72, 73], one gets equation (1) with 4π replacing the 6π; i.e., slip instead of stick boundary conditions since the particle 6

Topical Review


the medium, giving an upper bound for the frequency where the generalized SE relation holds. The model by Levine and Lubensky also suggests that the compliance of a single tracer is affected by the surroundings of the medium. This has been confirmed experimentally using biological macromolecules, as λ-DNA [100] and F-actin [101, 102], where a depletion layer around the tracer is observed. The model by Levine and Lubensky also suggests that the compliance of a single tracer is affected by the surroundings of the medium more dramatically than the cross-correlated response of two hydrodynamically coupled probes. This cross correlation can be measured through video microscopy in two-point (or two-particle) microrheology [69]. It has the advantage of being far less sensitive to the probe morphology and the details of its interaction with the host medium than conventional microrheology. A detailed analysis of two-particle resp. two-trap microrheology in the framework of lowReynolds number hydrodynamics and related data-analysis techniques was presented in [103]. Several models approach the problem of small-probe microrheology by describing the dynamics of the host liquid as modified by the presence of the probe. A mode-coupling model addressing the deviations from the SE relation was developed by Bhattacharyya, Bagchi, and coworkers [104], linking the problem to that of solvent diffusion in physical chemistry [105]. They consider three sources of microscopic friction describing the probe motion: (i) friction from direct collisions with bath particles, ζb (a short-time contribution); (ii) from coupling of the solute's motion to the bath density fluctuations, ζρρ; and (iii) from the transverse current modes in the bath, ζtc. The second is considered dominant close to kinetic arrest in the mode-coupling theory of the glass transition, where the first is usually neglected as an additional but quickly decaying contribution. The third mode describes an alternative relaxation channel for the probe-density fluctuations. In the Laplace domain, one thus adds the corresponding mobilities,

density. The probe particle is considered as a perturbation that renders this fluid density inhomogeneous. The model recovers the SE relation for large probe particles, for roughly a ≳ 7ah. For smaller probes, the tracer-bath interaction dominates the diffusion process. Inayoshi et al [114] present a simplified calculation, based on inhomogeneous linearized-hydrodynamic equations for the steady state, ∇· ρeq ( r ) v ( r ) = 0, (8) δP ( r ) − ρeq ( r ) ∇ + dr′ M ( r , r ′ ) · v ( r ′ ) = 0. (9) ρeq ( r )

∫

The first equation derives from the conservation of density, and the second is the momentum-balance equation involving an inhomogeneous pressure term δP (r) and a stress kernel M (r, r′) determining the velocity fields v(r) around the probe. Here, ρeq(r) is the host-fluid density, related to the probe-hostfluid pair-distribution function. The stress term is approximated by the standard homogeneous Newtonian-fluid form,

M ( r , r ′ ) = δ ( r − r ′ )[ η ∇2 v ( r )+( ηb+ η / 3)∇ ( ∇ · v ( r )) ⎤⎦ , (10)

where η and ηb are the shear and the bulk viscosity, respectively. The flow field around the probe particle can then be calculated from a singular perturbation expansion [114], matching the well-known homogeneous Stokes solution for the far field to an inner solution that keeps the inhomogeneities. Inayoshi et al recover the SE relation with slip boundary conditions, modified by a first-order correction term that is calculated from the radial distribution function of the hostfluid particles, D = kBT/[4πη a (1+ϵ)]. A different generalization of the SE relation has been proposed by Gaskell, Balucani an coworkers [115, 116]. They start from microscopic theory, but introduce a coarse-graining function (called form factor by the authors) f(r) in the microscopic velocity field, v(r, t) = ∑kvk(t) f (|r−rk|), where the sum runs over all particles, and vk is the velocity of particle number k. Here, f(r) replaces the Dirac delta in the definition of a fully microscopic theory, by a smooth function that is constant across a particle diameter, and constructed to obey proper normalization. A wave-vector and frequency-dependent generalized shear viscosity of the host fluid, η(q, z), is introduced via the memory kernel of the transverse-current correlation function, to arrive at

kBT kBT . + D(s) = (7) m [ ζb ( s ) + ζρρ ( s ) ] mζtc ( s )

Both ζb and ζρρ are given in terms of the tracer-fluid structural correlation function. Taking all three contributions into account, the theory showed that when the probe particle is of similar size as the bath particles, its response depends mainly on the tracer-bath interactions, making it a priori useless for the comparison with bulk rheology. Obviously in the limit of very small probe particles, one approaches a fully decoupled regime, where the probe essentially percolates through the matrix set by the host fluid. This delocalization transition has been addressed in the mode-coupling theory by Bhattacharyya and coworkers [106], and also in the standard mode-coupling theory of the glass transition [107–109], where it is linked to the physics of the Lorentz gas model [110] and the ‘anomalous diffusion’ observed in biophysical systems [111], or diffusion of probe particles in quenched heterogeneous media [112]. Yamaguchi et al [79, 113] developed a theoretical model based on the generalized Langevin equation obtained using the standard projection-operator method for the host-fluid

f (q) k Tρ ∞ dq D= B 2 , (11) 0 η ( q, z = 0) 3π

∫

where f(q) is the Fourier-transformed form factor. Neglecting the q-dependence of η(q) recovers the original SE relation with slip boundary conditions. This formulation highlights the expectation that an effective hydrodynamic radius of the probe should enter the SE relation, in this case related to the length over which the velocity field around the probe remains constant. The wave-vector dependent viscosity will in particular pick up nontrivial structural information of the host fluid, as has been demonstrated close to the glass transition [117]. Independent from the microscopic details, the long- wavelength, low-frequency limit of the flow field around 7

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the probe is expected to decay slowly as |v(r)| ∼ 1/r, in the hydrodynamic limit [5, 118]. This also implies the existence of long-time tails, as shown by Cichocki and Felderhof [119], which are also obtained from microscopic models [79, 120]. For probes that are close to interfaces or boundaries, the bulk description hence has to be appropriately modified. The hydrodynamic aspects of the flow field are also of particular concern in computer simulation when periodic boundary conditions are used. In effect, the simulation then corresponds to a (typically) cubic array of probe particles who interact through the hydrodynamic flow field. The Stokes problem for such arrays has been solved by Hasimoto [121], and the hydrodynamic coupling increases the drag force on any particle in the array, compared to the single-probe case. For a simulation with N particles in a cubic box of length L, the diffusion coefficient DN is thus reduced compared to the infinite-systemsize limit D∞: DN = D∞−DSEξ a/L, where DSE = kBT/(6πη a) is the Stokes–Einstein value. The constant ξ≈2.84, for the simple cubic array relevant to describe periodic boundary conditions, can be determined from Ewald summation techniques [121]. Keeping such finite-size corrections in mind, computer simulations have been used to probe the crossover to the classical Stokes–Einstein limit. Shin et al [122] confirmed that the probe motion in 2D molecular-dynamics simulations is correctly described by the generalized Langevin equation (2), i.e., that the distribution of the fluctuating force becomes Gaussian and δ-correlated in time for large enough probes and large mass ratio. The applicability of the SE relation as a function of the tracer size has been studied in various simulation works [76], [123–125]. It is typically found that the SE relation holds for large or massive tracers. Yeh et al [123] and Sokolovskii et al [124] have shown the importance of finite size effects in the simulations, and confirmed the applicability of Hasimoto's formula, DN = D∞−k/L. Exemplary results are shown in figure 3. The 1/L finite-size correction with the predicted slope is typically only found for sufficiently large tracers; otherwise the direct probe-bath interactions are too dominant, in agreement with the theoretical models discussed above [78]. The results of Sokolovskii et al [124] imply a minimum probe size for the interpretation of microrheology data of about a ≈ 3ah in hard spheres. At smaller a, the Stokesian 1/a dependence of the diffusion coefficient crosses over to the 1/a2 dependence expected from Enskog theory.

Figure 3. Upper panel: dependence of the tracer diffusion coefficient in hard spheres, for tracers of the same size as the bath particles. When the correction from Navier–Stokes equation is applied, a constant diffusion coefficient is obtained. Figure from [123]. Lower panel: test of SE relation for stick and slip conditions as a function of the tracer size, in a hard sphere system. The open symbols mark the simulation result for a large system and the closed circles are the extrapolated values. Reprinted with permission from [124]. Copyright (2006), AIP Publishing LLC.

mechanism the osmotic pressure gradient caused by chemical reactions catalyzed on the cap) may become feasible [131, 132]. We will focus here on recent developments considering the theoretical understanding of the nonlinear response regime in model colloidal suspensions and glass formers, for probes that are not extremely bigger than the host-fluid particles. One needs to distinguish two main modes of active microrheology, as first pointed out by Almog and Brenner [133]: force-driven and velocity-driven. In the former, which is in experiment realized through magnetic tracers dragged with an external magnetic field, the probe particle position is allowed to fluctuate around its mean trajectory. This includes excursions perpendicular to the driving direction in response to both thermal fluctuations and collisions with bath particles. The constant-velocity probe on the other hand ‘bulldozes’ through the host fluid, and thus encounters more frequent

3. Active microrheology In active microrheology, external (typically steady or oscillatory) forces or torques act on the probe particles, whose motion is monitored, typically with microscopy techniques. Experimentally, this is most commonly achieved acting with magnetic field gradients on superparamagnetic tracers [56, 61, 126], with optical tweezers or similar optical means on optically active probes in transparent media [16], [127–129], or with magnetic tweezers suitable also for diffusive media [63]. Even molecular motors have been used [130], and probes made of half-capped ‘Janus-particle’ colloids (utilizing as a driving 8

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collisions with particles accumulating near its front. One therefore expects the friction coefficient to be higher in constant-velocity active microrheology, than in the constant-force realization. This is indeed confirmed in experiment [134]. Nonlinear response phenomena are prevalent in active microrheology. Recall that a typical scale of forces induced by thermal motion is kBT/a. For a typical colloid size this implies forces on the pN scale. If the external force becomes comparable to the local rigidity of such a soft-matter system (which will correspond to forces in the range of tens or hundreds of kBT/a), one needs to expect significant nonlinear-response contributions. For this reason, experimental investigations of active microrheology in dense soft matter usually reach far into the nonlinear-response regime. Using particles of size a = 1 µm, suitable for direct-imaging techniques, a typical velocity scale expected in linear response is v ∼ (kBT/a)/γ0, where γ0 = 6πη0a is the solvent-induced friction. With a typical solvent viscosity of η0 ≈ 1 mPa s, the typical velocity scale is v ∼ 0.2 µm s−1. Close to the glass transition, the equilibrium mobility of a probe particle is easily suppressed by a few orders of magnitude. Resolving this mobility in active microrheology would require to observe particle velocities on the scale of µm per week. To quantify the nonlinear response, one introduces a Péclet number quantifying the ratio of the external force to the thermal restoring forces, as a dimensionless measure of the driving force. For Pe ≫ 1, the pulled probe will induce a strong distortion in the microstructure of the host fluid in the vicinity of the probe. The measured microviscosity will in turn itself depend on the strength of the driving. As long as hydrodynamic interaction (HI) effects arising from the solvent in colloidal dispersions do not dominate, one typically finds that this disruption of microstructure leads to ‘force-thinning’: γ (Pe) then drops significantly with increasing Pe. This effect is exemplified in figure 4, where results from a Langevindynamics simulation (for quasi-hard-sphere particles without HI) are shown for both constant-velocity and constant-force active microrheology, employing a tracer of the same size as the bath particles. One notes that the friction obtained from the fixed-force mode is consistently lower than the one from the fixed-velocity mode, in line with the reasoning above. In both cases, one observes a qualitatively similar reduction of the friction coefficient at large Pe. The effect becomes more pronounced with increasing host-fluid density. This is mainly driven by a strong increase in the low-Pe friction coefficient; it reflects the approach towards the glass transition, where the probe mobility is strongly suppressed and the shear viscosity diverges. At low densities, the microscopic friction thinning has been seen in the pioneering simulations by Carpen and Brady [135] and in experiments using laser tweezers [128, 136]. The more pronounced force-thinning behavior close to the glass transition was observed experimentally in magnetic-bead active microrheology [126], described in more detail below. Note that the nonlinear behavior demonstrated in figure 4 is analogous to the well-known non-Newtonian macroscopic rheology of glass-forming fluids: there, the shear viscosity strongly decreases with increasing macroscopic shear rate; an effect referred to as shear thinning.

2

γ / γ0

10

Constant force

1

10

0

10

0

2

1

10

10

3

10

3

10

F

10

Constant velocity φ = 0.55 φ = 0.50

γ / γ0

2

10

φ = 0.40 φ = 0.30 φ = 0.20

1

10

0

10

-4

10

-3

10

-2

10

-1

10

v

0

10

1

10

Figure 4. Effective friction coefficient from simulations, for different densities, as labeled and for the constant pulling of the tracer (upper panel) and constant velocity case (lower panel). The points mark the simulated states, and the lines mark the position of the plateau at low forces or velocities.

In the following, we will discuss in detail two theoretical models for the motion of a probe under steady driving, in a host fluid of hard colloids in the low and high density limits, respectively. The model for low densities is based on the twoparticle Smoluchowski equation, and the friction coefficient is calculated by considering all forces acting on the tracer in the steady state, and has been developed by Brady and coworkers. On the other hand, the model for a high density host requires the full Smoluchowski equation, and describes the dynamics of the bath particles using the mode-coupling approximation; this model was pioneered by Fuchs and coworkers. Oscillatory active microrheology, on the other hand, is often analysed in terms of the theoretical models developed for passive microrheology, using the compliance formula, equation (4). This is certainly valid in the linear-response domain, which however, may require very low applied forces. As detailed above, it also typically implies a simple continuum model for the bath, hence demanding that the oscillation frequency is small compared to the relaxation rate set by the host-fluid dynamics. An improved description based on the low-density expansion of the Smoluchowski equation was given by Khair and Brady [137, 138]. Experiments with hard spheres [139] are analysed with this model, finding good 9

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agreement. Oscillatory microrheology has been also used to test the validity of the fluctuation-dissipation theorem, where the effective temperature to fulfill it, is determined in systems out of equilibrium [140–142]. Controversial results are found in the experiments, however; in gelling laponite, some results indicate that the effective temperature increases waiting time, whereas other find a constant temperature, and equal to the bulk temperature, throughout the experiment. This has motivated extensive theoretical work [143–145], but further studies will be required to fully settle the issue.

be solved with two boundary conditions. At the surface of contact between the tracer and the bath particles the flux must be zero: n · j = D n ·∇ g + n · Ug = 0, (16)

at r = a+ah, and n is the normal to the tracer, in the outward direction. The second condition is that the structure of the bath must be unaffected far from the tracer, i.e. g → 1 for r → ∞. As for the velocity of the tracer, it can be calculated considering all forces that act on it, namely, external driving force, Fext, inter-particle forces, FPj and diffusion [137, 146]:

3.1. Low density bath

N

N

j=1

j=1

U = M11 · F ext + ∑ M1j · FPj − kBT ∑ M1j ∇j ln PN , (17)

Brady and coworkers have developed a detailed theoretical model for active nonlinear microrheology in the limit of low-density colloidal host suspensions [137, 146], rationalizing the previous observations. In the model, the effect of the bath is described in terms of the tracer-bath and bath-bath pair distribution functions [135], [137], [146–148]. The evolution of a system with N Brownian particles (including the tracer, particle 1, of radius a, and N − 1 bath particles, of radius ah) is given by that of the probability density PN(x1, x2, ..., xN, t), where xi denotes the position of particle i, governed by the Smoluchowski equation:

where Mij = Dij/kBT is the mobility tensor. Note that this is a general result; the simplification of Dij due to the neglect of hydrodynamic interactions, and the hard-sphere interactions must be still implemented in this equation. In the linear response regime, the average velocity is proportional to the average external force 〈 U 〉 = M∞s 〈 F ext 〉, where the −1 self mobility at long times M∞s = (6πη a ) is related to the * inverse of the microviscosity introduced in connection with the GSE relation. Squires and Brady [146] calculated the particular cases of applying a constant force to the tracer (the velocity thus fluctuates), or dragging it with a constant speed (the force fluctuates). In the former case:

N

∂ PN + ∑ ∇i · ji = 0. (12) ∂t i=1

The flux of particle i is

⎡ ⎤ 〈 U 〉F = M11 · ⎢ F ext − nkBT ng ( r )dS ⎥ (18) ⎣ ⎦ r = a + ah

∮

N

ji = UiPN − ∑ DijPN ·∇j (ln PN + VN / kBT ) , (13) j=1

The surface integral is over the contact surface between the tracer and bath particles. From this equation, it is straightforward to obtain the constant-force microviscosity as

where Ui is the velocity of particle i, Dij is the relative diffusivity of the of particles i and j, and VN is the interparticle potential. If one restricts oneself to the low-density limit, the equations can be simplified by integrating over the configurational degrees of freedom of N − 2 bath particles. The remaining conditional tracer-bath particle correlation, P1/1(r, t), is given in terms of the pair-distribution function g (r, t): P1/1(r, t) = ng (r, t), and the Smoluchowski equation is written in terms of the radial coordinate r = x1−x2 [137, 146]:

⎡ ⎤−1 ηF * = ⎢ 1 − nkBT nzgdS ⎥ , (19) ext ⎣ ⎦ η F

∮

where it has been assumed that the external force lies along the z-axis. On the other hand, for the case of constant velocity, the average is over the external force, resulting in

∂g + ∇r · ( Ur g ) = ∇r · Dr · ( g ∇r V / kT + ∇r g ) . (14) ∂t

ηU * = 1 + nkBT nzgdS . (20) 6πηaU ext η

where Ur = U2−U1, and Dr is the relative diffusivity. This result is general and exact in the low density limit (since only the tracer-bath particle correlation is considered). In the following we will focus in the system with linear dragging, in a bath of hard-spheres, without hydrodynamic interaction. This simplifies the problem significantly: in the absence of hydrodynamic interactions, Dr is diagonal, and can be written as DI, whereas for hard-sphere interactions, the interaction forces are zero, and only affect the boundaries. Also, in the steady state, the familiar advection-diffusion equation for the tracer is recovered:

It must be further noted that the relative diffusion coefficient appearing in the diffusion-advection equation is different in the case of constant velocity ( DU = Dah ; tracer Brownian motion is suppressed) or constant force ( DF = Da + Dah ), but the equation is unaffected. Note that this difference in the diffusion coefficient implies immediately that η U > η F, as indeed * * observed in the simulations, figure 4. This is rationalized considering the extra work required in the constant velocity case to push bath particles out of the way, while the tracer with a constant force can take advantage of its own diffusion to find its way through the bath. There are also mixed cases between these two limits of forcing, such as dragging a probe that is held in a parabolic potential. This is to a good approximation the case in experiments employing optical tweezers. Here, the

∮

U ·∇ g + D ∇2 g = 0, (15)

where the sub-index 1 for the tracer has been dropped, and the subindex r has been dropped for ∇. This equation can 10

Topical Review


constant-velocity limit is reached only for rather strong traps, depending on the translation speed. Returning to the difference between constant-velocity and constant-force friction coefficients, the calculation by Squires and Brady [146] gives a factor of 1+ah/a between the first and the second case, i.e., a factor of 2 for probes of the same size as the host particles. For small probes, on the other hand, the difference in microviscosity between the two driving modes vanishes. The linear dependence on ah/a was confirmed in experiment [134], and in Brownian-dynamics simulations [135]. Including strong hydrodynamic interactions (HI), the ratio reduces to lower values and starts to depend on Pe [149]. This is related to the fact that the boundary layer around the probe, whose particular importance for the strong-driving case will be discussed below, changes. Behind the probe, a low-density ‘wake’ develops whose shape is strongly influenced by HI. With strong HI, one can also observe a driving-induced increase of the probe-particle friction with increasing Pe, i.e., ‘force thickening’ as the analog of shear thickening in macroscopic rheology [149]. In the high-density theory discussed below, HI are neglected, following a standard argument that these solvent-mediated interactions are subdominant compared to the strong and frequent direct collisions between the colloidal particles in dense systems. Note however, that for active microrheology in a colloidal crystal, HI effects have been studied in simulation and found to be of significant influence [150]. The advection diffusion equation can be written in adimensional form introducing the Péclet number, Pe = U(a+ah)/D and measuring distances in units of (a+ah):

10

Δγ = 4φ g(φ)

γ(Pe→0) / γ0

8 6

Δγ = 2φ g(φ)

4 2

Constant v Constant F

γ(Pe→∞) / γ0

4 3 2

Δγ = 2φ Δγ = φ

1 0

0,1

0,2

0,3

φ

0,4

0,5

0,6

Figure 5. Comparison of the relative effective friction coefficient from the model of Squires and Brady [146] and overdamped Newtonian dynamics simulations in the low-Pe (upper panel) and high-Pe (lower panel) limits. The constant velocity case is presented with the closed black circles (simulations) and black lines, and the constant force case with the open red circles and red lines. The broken lines are the theoretical predictions, as obtained from the model, and the continuous lines are the results corrected with the pair distribution function at contact in equilibrium, calculated with the Carnahan–Starling equation [180].

∂g ∇2 g + Pe = 0, (21) ∂z

where the direction of the external force or imposed velocity has been made explicit. The boundary conditions are now written as: ⎛ ∂g ⎞ ⎜ + Pe cos θ g ⎟ = 0 and g ( r →∞ ) → 1. (22) ⎝ ∂r ⎠r = 1

implying an excess of bath particles in front of the tracer of O ( Pe ), and a wake behind it, of the same order. For the constant force case, this gives a viscosity increment of:

with θ the angle between n and the z-axis. The problem is therefore to solve this diffusion-advection equation for a given Pe, in terms of the pair distribution function, g (r), and use it to calculate the microviscosity. Although this equation can be solved for all Pe analytically, it is instructive to study first the low and high Pe limits, and the viscosity increments in both the constant force and constant velocity cases.

2 ηF * = 1 + φ (1 + α ) (24) 2 η

where α = a/ah is the tracer to bath particles size ratio. In the constant velocity case, the viscosity increment is given by: 3 ηU * = 1 + φ (1 + α ) . (25) 2α η

3.1.1. Low and high-Pe limits. In the low-Pe limit, the bath

Figure 5 presents the test of the predictions of the model against the simulations described in figure 4 for both the constant force and constant velocity cases (upper panel for low-Pe and lower panel for high-Pe). The comparison shows that the model is valid only at rather low volume fractions, and that this disagreement is larger for the case of constant velocity. Squires and Brady proposed that their results could be scaled up to

structure is only slightly modified by the presence of the tracer from its equilibrium state, and diffusion dominates over advection. The pair distribution function up to O ( Pe ) has the dipolar form: cos θ g ( r ) = 1 + Pe 2 (23) 2r

11

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Figure 6. Results from Brownian dynamics simulations for the average bath density around a probe particle pulled with a constant force for φ = 0.35 for different Pe, increasing from left to right and top to bottom. The gray scale is shown in the lower right corner. Reprinted with permission from [135]. Copyright (2005), The Society of Rheology.

higher concentrations by using the pair distribution at contact in equilibrium, accounting for the decrease of the self-diffusion coefficient as ∼1/φ geq(1;φ) [146]. The comparison with this ad-hoc corrected model improves significantly, and describes the simulation results much better, as shown in the graph. The high-Pe limit, on the other hand, is more subtle, as a singular boundary layer develops. The structure of the host medium is strongly modified by the tracer, i.e. advection is dominant, and diffusion is relevant only in a thin layer around the tracer. The friction thus arises from the (quasi-ballistic) collisions of the tracer with bath particles, in addition to the friction with the solvent. It is not obvious, though, that the effective friction coefficient decays asymptotically to a value above the solvent friction. Using again the diffusion-advection equation, the calculated tracer-bath particle pair distribution function decays faster to the equilibrium value in front of the tracer, whereas its contact value is identically zero behind it. The calculation from Squires and Brady [146] gives:

for both the constant force and the constant velocity cases. The comparison with simulations is also presented in figure 5 (lower panel), showing that the model underestimates the friction coefficient, but the correction with g (1;φ) significantly improves the comparison. In comparing with experiments performed on colloidal suspensions, solvent-mediated hydrodynamic interactions (HI) become important, in particular at low host-fluid densities and at large Pe. The two-body Smoluchowski theory of active microrheology has been extended [138, 149] to treat HI approximately using an excluded-annulus model of hard spheres with two interaction diameters (one for the direct excluded-volume interactions, and a second one for the enforcement of no-flux boundary conditions) [151, 152]. Wilson et al [67] have compared experiments and simulations where the tracer, of the same size as the bath particles, is held in an optical trap moving at constant velocity. For the comparison of the high-Pe limit, they corrected for the hydrodynamic interactions using the increase of the short time diffusion coefficient, γhigh PeDss ( φ ) / D0. The so-corrected results give very good agreement between experiments and simulations, up to high volume fractions, indicating that hydrodynamic interactions can be accounted for in this simple manner. They also compared their values with the high shear

Pe g(r)≈1 + cos θ e−Uzcos θ /D for θ < π /2 (26) 2

The viscosity increments in this limit can be calculated using the equations above, yielding Δη*(Pe ≪ 1) = 2 Δη*(Pe ≫ 1) 12

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study active microrheology close to the liquid–gas spinodal of a system with short-ranged attractions. The low-density wake discussed above can lead to cavitation effects, since locally, the density is lowered below that of the critical point of the equilibrium phase transition [156]. From the low-density theory by Brady and coworkers, the final result for the increment of the microviscosity is 3 η * = 1 + (1 + α ) Da ϕV ( Pe ) (28) η D 2

where V (Pe) is an adimensional master function going from 1 at Pe → 0 to 1/2 at Pe → ∞. Figure 7 presents the theoretical increase of the viscosity over the whole Pe range, which shows the shear thinning regime above Pe ∼ 1, and compares it with the early simulation results for a constant force [135] and experiments from Sriram et al [136]. In order to improve the comparison, the simulation and experimental data is normalized with ϕ g (1;ϕ), giving 2 V (Pe). Note that all data (experimental and simulation) collapses indeed onto a master curve, which, however, departs from the calculation of Squires and Brady, particularly at large Pe. The analysis of active microrheology presented here, particularly in the non-linear response regime, has provided interesting results, as for the microstructure of the bath and non trivial plateau at high Pe of the microviscosity. However, it is also worth comparing the obtained microviscosities with their macroscopic counterparts. As discussed previously, the overall shape of the curves shown in figure 7 resembles that of the macroscopic shear viscosity, thus making a direct comparison appealing. However, there are three notable differences between the two quantities that need to be taken into account, as discussed by Squires [147]: (i) the flow field experienced by the host-fluid particles around the probe is inhomogeneous and ‘non-viscometric’, i.e., it does not correspond to the usual simple-shear flow field imposed in macrorheology; (ii) connected to this, the time-dependent stress histories of fluid elements along their Lagrangian path lines around the probe become important in microrheology, while they are absent in macrorheology; (iii) the contributions to the microviscosity come from direct probe-fluid interactions in addition to fluid–fluid particle interactions. As a consequence of the latter point, the microviscosity has a leading low-density contribution of O(ϕ), while the macroviscosity scales as O(ϕ2) in this limit. Still, the low-density microviscosity theory can be compared with its macroscopic counterpart [152]. One finds a slightly larger shear thinning effect in macrorheology than in microrheology, for probe particles of the same size as the host-fluid ones. Increasing the size of the probe is appealing in trying to improve the connection between micro- and macro-response, but this provokes an enhancement of the direct-interaction contribution (iii) which has no counterpart in macroscopic rheology [148]. A similar problem has already been noted in passive microrheology [98, 99]. The non-viscometric nature of the problem remains even if one considers the continuum limit for the host fluid, if the latter is non-Newtonian, since then the classical Stokes-drag calculation must be amended [157]. Assuming a generalized

Figure 7. Increase of the microviscosity, normalized with ϕ g (1;ϕ), from simulations (open symbols) and experiments (closed symbols) at different densities, as labeled, compared with the theoretical predictions for 2 V (Pe). Reprinted with from [136]. Copyright (2010), AIP Publishing LLC.

viscosity from bulk rheology, finding excellent agreement with microviscosity values and corrected simulation values. The correlation function of the tracer position was also measured in this work, showing that the tracer dynamics reflects the collective behaviour of the host fluid even at moderate density. The current model, however, does not provide information about the tracer dynamics, since that requires the full N-particle Smoluchowski equation to be treated. We return to this in the next section discussing the high-density theory. 3.1.2. Arbitrary Pe and connection to bulk rheology. In order

to solve the advection diffusion equation for arbitrary Pe, the substitution g = 1+e−Pe z/2 Pe f is performed, where f fulfills the Helmholtz equation 4∇2 f = Pe2 f. The solution of f is given as an expansion in Legendre polynomials and powers of Pe [146], which shows that particle concentration in front of the tracer increases with Pe, and its wake is also more pronounced. This is confirmed by simulations [135], as shown in figure 6, and experiments [136]. Let us also mention that qualitatively similar results have been obtained using dynamic density functional theory (DDFT). From the Smoluchowski equation describing Brownian particles in potential flow u(r), one arrives at an expression for the density field ρ around the probe. The latter takes the form of a continuity equation where part of the current is determined by a functional derivative of the free energy [153], ⎛ δF[ρ] ⎞ ∂ρ + ∇· ( ρ u ) = ∇· ⎜ ( ρ / γ0 ) ∇ ⎟. (27) ∂t δρ ⎠ ⎝

The density field can then be calculated using well-established expressions for the free energy functional F [ ρ ]. Earlier calculations [154, 155] simply assumed the flow field u(r) to be uniform. Replacing this by the standard Stokes solution improves the calculation [153]. The results agree qualitatively with the features visible in figure 6. DDFT has recently been used to 13

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Newtonian fluid with a small deviation from the Newtonian viscosity, ηh ≈ η0 + ϵη1 ( γ˙ ) with some scalar measure γ˙ of the flow-rate tensor, the relation between the host-fluid viscosity and the microviscosity becomes [147] η1 ( γ˙0 ) γ˙02 dV η = η0 + ϵ , * γ˙02 dV

(a)

2

10

ζ0 〈v 〉∞ [kT/a]

∫

3

10

∫

where γ˙0 ( x ) is the zeroth-order Newtonian Stokes flow field around the probe, and the volume integrals are taken over the whole fluid domain. As discussed by Squires [147], this relation can still be uniquely inverted to give the macroscopic viscosity, but the assumptions leading to it (including an instantaneous connection of the non-Newtonian macroviscosity to the inhomogeneous flow field, ignoring contribution (ii) from above) are unlikely to be fulfilled in a realistic experiment. All three differences pointed out above can be equally important, and non-negligible. This is borne out most clearly if one considers anisotropic particles for the host fluid; changing also the probe-particle shape can then be used to shift the relative weights of the three contributions, e.g., in an attempt to minimise the non-viscometric effects in microrheology [158, 159]. The probe shape in fact matters as it may qualitatively alter the behavior of the microscopic friction coefficient, from ‘thinning’ to ‘thickening’ [160]. Oscillatory active microrheology can also be tackled with the theory, considering an oscillatory force in equation (14), as shown in [137, 138]. This serves to establish a connection between active and passive microrheology, which appears elusive, since passive microrheology is based on the generalized SE relation, and the generalized fluctuation-dissipation theorem, whereas these relations cannot be applied to out of equilibrium systems, i.e. active microrheology. DePuit et al [148] have established a framework to compare both, focusing on the limit of small frequencies in passive microrheology and small driving forces in active. It is found there that a generalized SE relation holds in the linear regime of active microrheology, i.e. low Pe, whereas for finite Pe, the non-linear terms make it fail. It is also shown by DePuit et al that the microstructure is significantly different in micro- and in bulk rheology. The reason for this is that the strain in microrheology is inhomogeneous, and it contains not only a shear component. The final result obtained for the microviscosity thus contains a weighted average of the complete rheological response of the system, and the viscosity does not arise from a prefactor in the stress-strain relationship. Squires [147] showed that this is the case even for a class of generalized Newtonian fluids.

ϕ = 0.45 ϕ = 0.40 ϕ = 0.50 ϕ = 0.55 ϕ = 0.57 ϕ = 0.62

1

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(b)

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ζ0 〈v 〉∞ [kT/a]

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ϕ = 0.45 ϕ = 0.50 ϕ = 0.52 ϕ = 0.53 ϕ = 0.55

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F [kT/a] Figure 8. Velocity–force relations in active microrheology, for a probe particle driven with constant external force through hardsphere-like colloidal systems at various packing fractions ϕ. Upper panel: results from stochastic-dynamics computer simulation (symbols) without hydrodynamic interactions and for a probe of equal size as the host particles; lower panel: results from experiment [126] for a probe with relative size 2.5. Lines are fits using a schematic model of mode-coupling theory. Reproduced from [184] with permission from The Royal Society of Chemistry.

case, at least in a theoretical idealization, cageing becomes permanent, and the structural-relaxation time scale diverges at a critical density. This is the glass transition, identified as a kinetic transition from a slowly relaxation (and hence very viscous) fluid to an amorphous solid. Active microrheology in this case probes the local dynamics around the glass transition. It is a unique tool to provide information on structural relaxation on local length scales, given that the tracer is also caged, and caging other particles, but requires a detailed understanding of the probe-host interactions, as already elaborated above. Typical force–velocity relations obtained in active microrheology of dense colloidal suspensions measured by Habdas and coworkers [126] are reproduced in figure 8. There, a magnetic bead with relative size 2.5 compared to the bath particles was pulled with constant external force. At high densities and the lowest forces (velocities) accessible in experiment, the observed velocity increases superlinearly with increasing force. Only at forces that are very large compared to the thermal-fluctuation force scale, the velocity becomes a linear function of the applied force (again).

3.2. High density bath 3.2.1. Experiment and simulation. At high densities, the structural relaxation dynamics of a colloidal suspension becomes extremely slow. This is attributed to the ‘cage effect’: each particle can only undergo long-range diffusion if it escapes the transient cages that are formed by the neighbours. Since this argument holds for each particle, cage relaxation is a highly collective process, giving rise to a subtle feedback mechanism that depends very sensitively on the density (or other control parameters influencing the dynamics). In the extreme 14

Topical Review


occurs for a driven particle in an ideal-glass host: the (on average) frozen structure of the amorphous solid is characterized by a local rigidity, the microscopic analog of the mechanical moduli quantifying the solid on the macroscopic scale. This ‘cage strength’ results in restoring forces much larger than thermalfluctuation forces. Applying a comparable force to the driven probe particle, one is able to locally melt the glass, and to induce long-range motion for the probe that remains localized at small forces. This corresponds to the transition from an infinitely high friction coefficient at low forces, to a finite one at high forces. The signature of this nonequilibrium state transition in the liquid (or non-ideal glass) is a rapid decrease in the friction coefficient in an external-force window close to the threshold set by the cage strength. One typically gets Fcex = O (100 kT R−1 ), where the exact magnitude depends on the interactions among the host particles and with the probe particle. To model this nonlinear-response phenomenon, one needs to account for the slow structural relaxation dynamics of the host, and its non-perturbative modification due to a strong external force. Since collective processes determine structural relaxation, the simplification to the two-body Smoluchowski equation discussed in the previous section to obtain exact lowdensity results, cannot be used (not even as a starting point for some systematic expansion in density). The many-particle Smoluchowski equation is not amenable to exact analytical treatment, and one needs to resort to approximation schemes. Before going into the detail of the theoretical description in terms of mode-coupling theory, let us briefly comment on the application of lattice models mimicking glassy relaxation in the context of active microrheology. Jack et al [164] have used kinetically constrained models to model the glass-forming host fluid, and found negative differential mobility of the probe for small forces: increasing the applied force first gave rise to the expected linear increase in the probe velocity, attributed to the overcrowding in front of the tracer, but after a maximum the velocity decreased slightly with further increasing force, before it saturated at a large-force value. The latter saturation is not observed in microrheology experiments of colloidal glass formers, but is instead typical for this type of excluded-volume lattice models and the kinetic rules chosen for the driven particle. If one models the host fluid as some sort of exclusion process [164], the external force on the probe is usually modeled by an increasing bias in its jump probabilities,

1000 100 10 1 0.1 0.01 0.001 0.0001 0.01

0.1

1

10

100

1000

10000 100000 1e+06

Figure 9. Force–velocity relation for active force-driven microrheology in a randomly driven 2D dense granular system of inelastic hard disks (coefficient of normal restitution e = 0.9), for different area fractions η as labeled. Symbols are computer simulation results, the dashed line indicates a power-law asymptote for large forces. Reproduced from [162].

Translated into the friction coefficient, ζ = F/v, the superlinear increase of the velocity corresponds to the pronounced ‘force-thinning’ effect discussed above in connection with the low-density models. In the experiment, only the high-Pe linear regime was accessible, due to the very small equilibrium mobility of the probe close to the glass transition. Intuitively, force thinning appears once the driving force is strong enough to disrupt the transient cages that hinder the diffusive motion of the probe particle in the dense bath. Consequently, the phenomenon is quite robust and has been seen in different colloidal experiments [128, 161]. Also, other dense systems are amenable to nonlinear active microrheology. One example of recent interest are driven granular systems. Here, the nonlinear force–velocity relations have been studied in simulations by Fiege et al [162], for a two-dimensional system with dissipative collisions. Their results are reproduced in figure 9. Here, the system is driven by momentum-preserving random kicks applied to particle pairs, implying a typical driving velocity vdr used to non-dimensionalize forces and velocities. Assuming vdr2 to play a role similar to kT in setting the force scale for a thermal system, one recognizes in figure 9 the simulation results as being in the large-force regime predominantly. Indeed, a strong nonlinear rise is seen for the highest density at the lowest forces simulated. This is qualitatively consistent with the colloid results. A difference arises however in the high-force limit, where the granular system exhibits a scaling asymptote v ∼ Fβ, with an exponent β close to 1/2, while in the colloidal suspensions one finds β = 1. The behavior observed in the granular system corresponds to ‘force thickening’, since the effective friction coefficient increases again with increasing force. This has been rationalized in terms of the increased frequency of the dissipative collisions by a simple kinetic argument [163]. A rigorous theoretical treatment of this effect is still lacking. The strongly nonlinear behavior in the friction coefficient is identified as the signature of a delocalization transition that

⎡ F · eα ⎤ ⎡ F · eβ ⎤ pα = exp ⎢ ⎥ ∑ exp ⎢ ⎥, (29) ⎣ 2kBT ⎦ ⎣ 2kBT ⎦ β

where α, β label the spatial directions of the grid, and eα are the corresponding lattice vectors. Then the large-force limit of such models must be governed by the probe moving quasideterministically in the direction of the force whenever the adjacent lattice site becomes available. The latter event occurs with a force-independent rate, since the microstructure distortion of the host fluid by the applied force is not accounted for; consequently, also the probe velocity will attain a force-independent value. In fact, the dynamics can be analyzed exactly to first order in the vacancy density of a nearly occupied lattice, including also fluctuations around the mean particle 15

Topical Review


trajectory [166, 167]. In contrast, the strong modification of the host microstructure around the strongly driven probe in colloidal suspensions (discussed for the low-density theory as a hydrodynamic boundary layer) results in a probe velocity that increases linearly with the applied force.

Following a standard procedure in the treatment of dense colloidal suspensions [178, 179], the Smoluchowski operator Ω is transformed to its one-particle irreducible expression using a Dyson decomposition [170]. One obtains ∞ 1 ζ ( F ex ) = ζs + P dt⟨ Fsexp[Ωirr ( F ex ) t ] Fs ⟩eq . (32) 3kT 0

∫

3.2.2. Mode-coupling theory. One approach to describe strong

friction effects in dense suspensions is based on the modecoupling theory of the glass transition (MCT). This theory was devised to capture the feedback mechanisms behind structural relaxation and gives a good qualitative and often quantitative description of glassy dynamics [168]. MCT has been extended to include constant-force active microrheology [169, 170], capable of treating forces far into the nonlinear response regime. The mode-coupling theory includes external forces through a scheme called integration through transients (ITT). Going back to work by Kawasaki [171] and Evans [172] and coworkers, Fuchs and Cates [173, 174] developed this scheme to describe dense colloidal suspensions under steady shear flow. The microscopic ITT-MCT theory has since then been extended to various non-viscometric and non-steady flows [175–177]. For the description of active microrheology, one starts from the many body Smoluchowski equation ∂tψ (Γ) = Ω(Γ) ψ (Γ, t) for the nonequilibrium distribution function ψ (Γ), where Γ = {ri} is the configuration-space element composed of the particle positions. We assume particles to be spherical and without other relevant degrees of freedom. The Smoluchowski differential operator is then written as

The nonlinear friction coefficient of the driven particle is thus given in terms of a so-called transient correlation function that involves the full nonequilibrium dynamics of the fluctuating forces, but is averaged using the equilibrium distribution function. Equation (32) is still exact, and a suitable starting point for subsequent approximations. MCT proceeds by assuming that slow density fluctuations govern the dynamics of the dense host system. The central quantities of MCT are thus the transient collective density correlation function ϕk ( t ) = ⟨ϱ*kexp[Ω†t ] ϱ k ⟩eq, and the probe-particle density correlation function ϕks ( t ) = ⟨ϱ ks *exp[Ω†t ] ϱ k ⟩eq . In equilibrium, these are the intermediate scattering functions measured in, e.g., dynamic light scattering. Here we have introduced the density fluctuations to a wave vector k with N magnitude k = |k|, ϱ k = ∑ exp[ik · rj ] and ϱ ks = exp[ik · rs ]. j=1

It is also assumed that the system remains homogeneous and translationally invariant in the statistical ensemble average. Also, we assume that in the thermodynamic limit, ϕk(t) will be dominated by the bulk density fluctuations, where the probeparticle motion has negligible influence. Under this assumption, the collective density correlation function ϕk(t) will be independent on the applied force and reflect the spatial isotropy of the bulk system. Note that this neglects a coupling of the probe motion to the locally deformed host-system structure around it. As in the low-density system, one expects the host structure in the vicinity of the probe to reflect the asymmetry induced by the driving direction. As has been studied in computer simulation [180], a force-induced wake behind the probe indeed emerges at large driving forces. However, there is a regime of driving forces Pe ≫ 1, where these structural anisotropies are still small but the dynamical response is already highly nonlinear. For the probe-particle correlations, it is important to recognize the broken symmetry induced by the external force. Hence, the correlation function depends on the full wave vector k, instead of just its magnitude k. Since the external force renders the time-evolution operator non-Hermitian, the autocorrelation function of the probe density fluctuations will in general be complex-valued. At any rate, there holds ϕ ks( t )* = ϕ−sk ( t ). We assume rotational symmetry around the force axis to still hold in the statistical average, so that the correlation function becomes real-valued again for all k⊥Fex. The mode-coupling approximation to equation (32) consists of an expression for the fluctuating-force correlation function,

Ω = ∑ ∂i ·( kT ∂i − Fi ) / ζi − ( ∂s · F ex ) / ζs . (30) i = 1, … N , s

In the last term, δΩ = −(∂s·Fex)/ζs, the external force explicitly enters, acting on a single labeled particle i = s. The Fi are the interaction forces among the particles, assumed to derive from a potential. The ζi are the Stokes friction coefficients of the individual particles. In writing equation (30), hydrodynamic interactions have been neglected. The Smoluchowski equation, with equation (30), can be explicitly, if formally, solved, arriving at an expression that determines the non-stationary distribution function in terms of the known equilibrium one. One gets [170]

∫

t 1 ψ ( t ) = ψeq − d t ′ exp[Ωt ′ ]( F ex · Fs ) ψeq, (31) kTζs 0

assuming that the external force was instantaneously switched on at t = 0. This expression allows in general to reduce nonstationary averages of observables to their equilibrium ones. They involve history integrals over microscopically defined correlation functions, extending the well-known Green–Kubo relations from equilibrium statistical physics to the non-equilibrium situation. Note that the time-evolution operator under the integral still contains the full Smoluchowski operator Ω, including the external force. The linear-response Green– Kubo relations are recovered by applying equation (31) with the additional assumption that inside this integral, Ω can be replaced with the equilibrium operator. The friction coefficient is obtained by averaging the probeparticle velocity using equation (31), using vs = (Fs+Fex)/ζs.

|k TkS s |2 〈 Fsexp[ Ωirr t ] Fs 〉≈ ∑ B k ϕ ks ( t ) ϕk ( t ), (33) NSk k

where Sk and Sks are the equilibrium static structure functions encoding the interactions among the particles in a statistical sense. These structure factors are assumed to be known from 16

Topical Review

J. Phys.: Condens. Matter 26 (2014) 243101 ex

forces and replaces the scalar product p·F that appears in the full theory. The memory kernels in equations (36a) are set to

liquid-state theory [181]. The density correlation functions obey an equation derived by applying a projection-operator scheme to the Smoluchowski equation [182, 183]. Specifically for the probe particle,

∫

m∥s ( t ) = [ υ1sϕ∥s * ( t ) + υ2sϕ⊥s ( t ) ⎤⎦ ϕ ( t ) / (1 − iκ∥F ex ), (37a) m⊥s ( t ) = [ υ1sϕ⊥s ( t ) + υ2sℜϕ∥s ( t ) ⎤⎦ ϕ ( t ) / [1 + ( κ⊥F ex )2 ], (37b)

t

ωq−,1q ∂t ϕqs ( t ) + ϕqs ( t ) + mqs ( t − t ′ ) ∂t ′ ϕqs ( t ′ )dt ′ = 0. (34) 0

Here, ωq, p = (q kBT − iFex)·p/ζs is the first moment of the Smoluchowski operator with respect to the density fluctuations. Equation (34) contains a memory kernel mqs ( t ) that describes the slow structural-relaxation dynamics. Without it, one simply obtains the solution for a freely dissolved particle driven by the external force, viz. an exponential decay due to diffusion modulated by a velocity-dependent phase factor, ϕqs ( t ) ~ exp[ − q 2Dst ] exp[iq · vst ]. This corresponds to a Gaussian van Hove function centered on the deterministic trajectory of the particle. MCT approximates the memory kernel mqs similarly to the correlation function appearing in equation (32). One obtains

m ( t ) = υ1ϕ ( t ) + υ2ϕ( t )2 . (37c)

This model has been discussed in detail in [184–186]. It is constructed on the basis of equation (35) and obeys all the symmetries encoded in the full theory. The adjustable parameters { κα, υ1s , υ2s , υ1, υ2 } mimic the wave-vector dependent coupling coefficients of the full MCT, and are treated as fit parameters. Specifically, v1 and v2 only determine the dynamics of the host suspension; the model for the collective density correlation function ϕ(t) contained here is a well-known model used to study the qualitative features of the MCT glass transition [168]. Equations (36a) and (37a) are supplemented by a schematic version of equation (32),

d3k kBTS s( p )2 ( q · p ) ωq, p mqs ( t ) ≈ ϕp ( t ) ϕ ks ( t ), (35) 8π 3ρ S ( p ) ωq2, q

∫

ζ ( F ex ) = 1 + μ

where ρ is the average density of the host system. In this expression, the density correlation function of the host system enters; it is determined by an equation analogous to the zero-force version of equation (34). The memory kernel mq(t) that appears there needs to describe the collective retardation caused by the cage effect. It is, within MCT, approximated similarly to equation (35), but given as a quadratic expression in the correlation functions ϕk(t). The equations determining ζ(Fex) are then closed, and determined by the equilibrium static structure factors alone. In this sense, MCT is able to make parameterfree predictions about the nonlinear friction coefficient measured in active microrheology given the static structure factors as inputs. A number of qualitative features of the MCT approximation can be studied in a much simplified ‘schematic’ model. Here one ignores the spatial extent of density fluctuations, and retains information only on two distinguished directions, namely parallel and perpendicular to the applied force. Introducing hence two probe-particle correlation functions ϕ∥s ( t ) and ϕ⊥s ( t ), together with a single density correlation function ϕ(t) that is representative of the dynamics of the host system on the cageing length scale, one gets

∫

0

∞

ϕ ( t ) ϕ⊥s ( t )dt + (1 − μ )

∫

0

∞

ϕ ( t ) ℜϕ∥s ( t )dt . (38)

A further adjustable parameter, μ, enters here. If μ = 1/2, the schematic model recovers the expected ratio of friction coefficient increments Δζ = ζ−1, as discussed above, Δζ (0)/Δζ (∞) = 1/2 [146]. Experiments and simulation show that this ratio differs from 1/2 at high densities. The schematic MCT model contains a critical force Fcex separating the close-to-equilibrium low-force response of the driven probe from a nonlinear high-force regime. To see this, recall that close to the host-system glass transition, the density correlation function ϕq(t) decays increasingly slowly. In the idealized-glass limit, it attains a finite long-time limit fq = limt → ∞ ϕq(t) > 0 representing the nonergodic contribution arising from the frozen amorphous structure. In the MCT model, this is contained for sufficiently large coupling coefficients (v1, v2). In equilibrium, a probe particle that is not too small will then also be localized, and fqs = lim t → ∞ ϕq ( t ) ≠ 0 is the Fourier transform of a finite, localized probability distribution for the long-time limit of the probe position. In the schematic model, this is caricaturized by finite long-time limits f, and fαs. At large external forces, the coupling strength entering the memory functions mαs ( t ) of equations (37a) decreases, and above a certain threshold Fcex, the probe-particle motion delocalizes: one obtains fαs = 0. The critical force can be determined by a bifurcation analysis of the equations for the nonergodicity parameters fαs,

t

(1 / ωα ) ∂t ϕαs ( t ) + ϕαs ( t ) + mαs ( t − t ′ ) ∂t ′ ϕαs ( t ′ )dt ′ = 0, (36a) 0

∫

∫

t

(1 / Γ ) ∂t ϕ ( t ) + ϕ ( t ) + m ( t − t ′ ) ∂t ′ ϕ ( t ′ )dt ′ = 0. (36b)

fαs = mαs [ f , f s ] (39) 1 − fαs

0

Here, we label by α ∈ {∥, ⊥} the two relevant spatial directions, and set ω∥ = ω⊥(1 − iκ∥Fex) to capture the low-density features of the probe-particle correlation functions. In the schematic model, Γ and ω⊥ set the short-time relaxation of the probe and host correlation functions and hence set the unit of time, respectively. Setting them unequal allows to adjust the model to the case of a probe particle that is not the same as the host particles. The coefficient κ∥ is introduced to set the unit of

where the right-hand side denotes the memory kernels, equations (37a), with ϕ(t) and ϕαs ( t ) replaced by f and fαs, respectively. For the full MCT model, analogous equations hold for fqs. There, the detailed behavior close to the transition is still open to further research. In the schematic model, asymptotic power laws can be worked out that describe the decrease of the friction coefficient close to Fcex [185]. They 17

Topical Review


shown, one obtains a strong decrease of ζ(F ) with increasing Fex. The oscillations seen in ϕ∥s ( t ) are a signature of the fast high-force motion of the probe. Qualitatively, similar features are seen in the density correlation functions also in a driven granular host medium [163]. The correlation functions for wave vectors perpendicular to the force, ϕ⊥s ( t ) do not contain such oscillations, and instead show a purely real-valued decay ex

are, however, hard to detect in experiment or simulation, since they are quickly masked by non-asymptotic corrections. Adjusting the coupling coefficients of the schematic ITTMCT model, the measured force–velocity relations, or equivalently, the nonlinear friction coefficients can be fit with high precision [184]. This is shown by the lines in figure 8. Solid lines are the model discussed above, while dashed lines are a further simplified model that does not take into account the α = ⊥ direction. Within the model, this influences mainly the large-force limit where the model is expected to work qualitatively at best. Stressing the importance of fluctuations perpendicular to the driving force in this regime is however in line with the exact results obtained by Brady and coworkers in the low-density limit. The local density-fluctuation dynamics of the probe particle can also be studied qualitatively with the schematic-MCT model. Figure 10 shows the most interesting case, ϕ∥s ( t ), compared with simulation results obtained for a wave vector probing the spatial extent of nearest-neighbour cages. In the simulation, transient density correlation functions are obtained by starting from initial configurations that are equilibrated without external force, and ensemble-averaging the subsequent runs with finite driving force over many such configurations. Both, simulation and theory, agree qualitatively. For the smallest forces, the typical two-step decay signaling structural relaxation close to the glass transition is seen: after a shorttime decay due to in-cage motion, a plateau emerges in the density correlation functions, given by the transient-cage nonergodic contribution fqs. At long times, the correlation functions decay to zero since the host system is still a liquid. The external force modifies this final decay and replaces it with a faster, force-induced relaxation. Since the friction coefficient is essentially given by an integral over the correlation function

-2

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10 2

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t [(ma /kT) ]

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t

Figure 10. Transient probe-particle density correlation functions

in active microrheology in a dense host suspension, ϕqs ( t ) for a wave vector q ∼ π/R close to the first peak of the static structure factor, chosen parallel to the external force, q∥Fex. A fixed density is shown, with increasing external force (curves from left to right). Left panels are stochastic-dynamics simulation results for quasi-hard spheres, right panels are obtained from a schematic mode-coupling theory model. Both real and imaginary part of the correlation functions are shown. Reproduced from [184] with permission from the Royal Society of Chemistry.

(b)

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Figure 11. Strain fields around a probe moving in a glass forming host fluid: hard-sphere-like colloidal suspension with ϕ = 0.49 and a

magnetic probe particle with a ≈ 1.45ah; external force applied in cycles with maximum F ex = 0.29nN = O (10 5kBT / a ). The left panel shows the measured bath-particle displacements, the right panel shows residual displacements after subtracting the homogeneous strain field expected from a point force in a linear elastic continuum (magnified by a factor 5). The scale bar indicates a length of 10μ m≈ 6.45ah. Reprinted with permission from [8]. Copyright (2013), AIP Publishing LLC. 18

Topical Review


-1

10

Dorth

10

A particles f = 0.0 f = 0.5 f = 1.0 f = 1.5 f = 2.0 f = 2.5 f = 5.0

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f = 1.5

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10 10 10

~t 1.35

parallel

~t

-1

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orthogonal

-3

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10

10

Figure 12. Force-dependent probe-particle diffusion coefficient for

A particles, T = 0.14 f = 0.0 f = 0.5 f = 1.0 f = 1.5 f = 2.0 f = 2.5

-4

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3[< x 2 (t )>-< x(t )>2]

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Teff /T - 1

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1

t

-1

10

10

2

0

10

1

10

10

3

10

3 10 t

2

10

4

10

5

Figure 13. Mean-squared displacement (MSD) of an actively

motion perpendicular to the applied force in active microrheology, D⊥(Fex), as a function of an inverse effective temperature Teff(f) that depends quadratically on the applied rescaled force f = Fex d/ϵ (where d and ϵ are the diameter and energy scale of the Yukawa potential), as shown in the inset. Reprinted with permission from [190]. Copyright (2012) by the American Physical Society.

driven particle in a glass-forming suspension, for rescaled external forces f as indicated (increasing from left to right), in the direction parallel to the force, obtained from molecular-dynamics simulations of a glass-forming Yuakawa mixture. Lines indicate the longtime asymptotes extracted from the simulation. The inset shows a comparison of the MSD perpendicular and parallel to the force for f = 1.5. Reprinted with permission from [190]. Copyright (2012) by the American Physical Society.

to zero whose relaxation time is shortened by increasing the external force [169, 184]. The microstructural distortion induced by the probe in a dense host fluid has so far not been addressed in the high-density theory. Recent experiments [8] were able to map out the fluid strain field by direct-imaging measurements of colloidal particles. Their results are shown in figure 11. If the host fluid is highly viscous, and can be treated as an effective continuum, it is intriguing to relate the observed distortion to that expected from linear elasticity theory. There, it is a standard exercise to calculate the strain response to a localized point force at the origin [187],

by δrs(t) = rs(t) −rs(0), one introduces the probe-particle mean-squared displacements (MSD) δx2(t) = 〈(ex·δrs(t))2〉 −〈(ex·δrs(t))〉2, δy2(t), and δz2(t) (defined analogously). Choosing Cartesian coordinates such that one direction aligns with the external force (say, Fex∥ex, although different conventions are used in the literature), we again distinguish MSD in the direction of, and perpendicular to the force. We also focus on the transient MSD as it reveals more detailed information on the force-induced breaking of cages. Simulation results were first obtained by Winter et al [190], for a model glass-forming Yukawa mixture. State points in the dense liquid were chosen, so that the probe motion is delocalized for all forces. In the directions perpendicular to the force, ordinary diffusion of the probe particle was obtained at long times. Reminiscent of the speed-up of the relaxation time of density correlations discussed above, the effective diffusion coefficient D⊥(Fex) was found to increase with increasing force. At the same time, as a function of the bath temperature, the diffusion coefficient decreases strongly. Winter et al were able to determine a scaling function for D⊥ as a function of an effective temperature that depends quadratically on the applied force as expected by symmetry. The master curve is shown in figure 12. There, also a comparison with the effective diffusion coefficient obtained under macroscopic shear is shown. Applying a constant velocity gradient γ˙ to the bulk system, the glass-forming fluid shows shear-thinning, i.e., a strong decrease of the shear viscosity with increasing shear rate γ˙. For the single-particle motion, this translates to enhanced diffusion [191]. The comparison highlights the fundamental difference between shear-rate-controlled rheology and applied-force microrheology. Under applied velocity, the system is forced to flow at any arbitrary small shear rate, which induces delocalization of particle motion also in the direction perpendicular to the flow [173, 192]: the glass

1 1+σ u= [(3 − 4σ ) F + er ( er · F ) ] , (40) 8πEr 1 − σ

where er is the radial unit vector, E is Young's modulus, and σ the Poisson ratio of the elastic continuum. Anderson et al [8] were able to fit this expression to their experimental data, uncovering a residual strain field (right part of figure 11) that arises in the non-continuum, thermally fluctuating colloidal glass. It was stressed, that the overall quality of the continuumelasticity fit is remarkable, and that the differences shown in the figure are greatly exaggerated. The non-continuum part of the velocity field appears to have a definite angular signature to it, reminiscent of locally nonaffine displacement fields invoked in the discussion of macroscopic rheology of low-temperature amorphous solids [188]. Pesic et al [189] analyzed the hostfluid structure around a probe dragged in an optical trap in a quasi-two-dimensional suspension; they noted that the deviations from the Stokeslet calculation bear similarities to chainlike stress propagation known from granular media. 3.3. Force-induced diffusion

Further information about the force-induced long-range motion of the probe is given by the fluctuations around the mean path. Denoting the time-dependent probe displacement 19

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the transitions by its first moments, 〈τ〉 and 〈τ 〉, the MSD is asymptotically given by 2

is always shear-melted, and the only relevant time scale for the motion of the otherwise frozen system is set by the external driving. For this reason, the shear-induced diffusion coefficient depends only weakly on temperature close to the glass transition. Under controlled external force, however, the existence of a finite-force threshold below which the glass is kept intact, implies a strong dependence on effective temperature of D⊥(Fex). This aspect of the probe motion is captured qualitatively in the schematic MCT model [186], in the non-linear response regime, beyond the region where SE relation holds. The fluctuations in the motion along the force-induced axis turn out to be qualitatively different, as shown in figure 13. Here, the diffusive long-time asymptote, δx2(t) ∼ t, is not obtained for forces close to the delocalization threshold in the time-span accessed in the simulation. Instead, a super-diffusive growth, δx2(t) ∼ tν, with an exponent ν close to 1.5, is observed. Only at forces higher than the ones shown in the figure, one observes a cross-over to diffusive motion again in the simulation. This leaves open the question whether the true long-time asymptote of the MSD is still diffusive at all forces, but provides an interesting connection to a class of trap models proposed by Bouchaud et al [193, 194]. There, one considers a particle moving in a random force field, in the present application thought to arise due to the random nature of the cageing forces of the amorphous host system. Such a random force field gives rise to a random potential with a mean bias and a fluctuating part. For certain ratios of these quantities, the trap model predicts MSD with a true super-diffusive long-time asymptote. The physical picture behind the trap model is consistent with the appearance of strongly intermittent probe motion close to the delocalization threshold. Individual trajectories show a sequence of localized segments during which the particle remains trapped, interrupted by fast, jump-like escapes from these local traps [180]. The waiting times between these ‘jumps’ extracted from the simulations follows a stretchedexponential distribution [190]. It has, however, been argued that this distribution is not broad enough to warrant the mapping onto Bouchaud’s trap model [167]. Active-microrheology trajectories in a glass-forming Lennard–Jones mixture have also been analyzed in the framework of continuous-time random walks (CTRW) [195, 196] in a potential-energy landscape (PEL) provided by the host system. One can then identify a dominant nonlinear contribution to the average waiting time between transitions from one metabasin of the PEL to another, while nonlinear effects acting on the spatial degrees of freedom are rather weak [196]. The model predicts a transient superdiffusive regime in the MSD [195] (but see [167] for a critical discussion), while the long-time asymptote is diffusive again. This is also expected from MCT, except possibly for a single critical point, since the memory kernels of the theory ultimately decay exponentially fast, allowing for a Markovian approximation on time scales long compared to this decay. The CTRW description starts from the analysis of the dynamics of a dense system as transitions between minima in the energy landscape. Combining those minima between which reversible transitions occur into metabasins, the system explores a number n(t) of metabasins during time t. Characterizing the waiting-time distribution for

⎛ 〈τ 2〉 ⎞ t δx 2 ( t ) ~ 2D∥t + Δx∥2 ⎜ −1⎟ . (41) 2 ⎝ 〈τ〉 ⎠ 〈τ〉

Here Δx∥2 is the average translation of the probe during a metabasin transition, and D∥ is a contribution to the MSD that is equilibrium-like in origin (but still force-dependent). The last term in this expression is related to the variance in the number of metabasin transitions per time, 〈n(t)2〉−〈n(t)〉2, in the long-time limit. At intermediate times, it is responsible for transiently superdiffusive motion. Notably, this variance also appears in the expression for the non-Gaussian parameter (NGP) that quantifies the heterogeneous dynamics in equilibrium. In the CTRW framework, one hence finds an intimate connection between aspects of the nonlinear response and the equilibrium dynamics of the system [195]. While superdiffusive transients appear only at high densities, the force-induced diffusion of a constant-force microrheological response remains anisotropic also in the low-density limit. The theoretical framework of Brady and coworkers starting from the two-particle Smoluchowski equation allows to calculate the different diffusivities parallel and perpendicular to the driving force in this case. At small Pe, Zia and Brady [197] obtain a force-induced microdiffusivity that scales as Pe2, dominated by Brownian motion, while at large Pe the dominant advection terms lead to a scaling as Pe for the enhancement of probe diffusion. The diffusion anisotropy can then furthermore be linked to normal-stress differences in the fluid. Recall that for a non-Newtonian fluid, simple shear forces typically give rise not only to a shear stress in response, but also to shearinduced normal stresses. These can be split into a nonequilibrium pressure contribution, and into normal stress differences N1 = σxx−σyy and N2 = σyy−σzz. Zia and Brady [198] present a simple argument suggesting a link between N1 and the difference in force-induced diffusion coefficients. One recalls that the relation between diffusivity and mobility of a tracer in a dilute bath (of number density n and ideal-gas osmotic pressure Π) can be written as D = M(kBT) = M (∂Π/∂n). Since the osmotic pressure is simply the isotropic part of the colloid-stress tensor, this suggests a generalization to anisotropic pressure, D = M·(∂Σ/∂n), where Σ is the colloid-particle part of the stress tensor. It can be demonstrated that this relation is qualitatively correct, but needs to be completed by terms arising from a more complete force balance between probe and host-fluid particles. Doing so, Zia and Brady [198] arrive at a relation linking −N 1∝(D∥−D⊥), and N2 = 0, which is a reasonable approximation for typical dense colloidal suspensions. A companion method to access normal stress differences using microrheology methods is to employ two forced probes, modeling the host fluid in between as a non-Newtonian continuum [199]. 3.4. Dynamics around the depinning transition

The region of strongly nonlinear increase in the velocity of a forced probe particle in a high-density fluid was attributed to the vicinity of a depinning transition that occurs if the host fluid becomes glassy. The details of this nonequilibrium 20

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transition between localized response at low forces, and delocalized response at high forces, are the subject of ongoing investigations. We first discuss the localized-response regime. In an elastic solid, one expects a certain linear-response regime for small forces, where the probe displacement is related to the applied force by a force-independent spring constant. (Note however the dissenting view that the glass may not possess a linear response regime at all, not even for a force-driven microrheology probe [200].) This allows to estimate the ‘cage strength’ of the host glass. In stochastic-dynamics simulations, a value of ≈80 kBT/a2 was found for hard sphere glasses [201]. This value relates reasonably well to the estimated Young's modulus of the glass [202], establishing a further qualitative connection between microscopic and macroscopic rheology. Comparable values have also been extracted from experiment [8], where the initial displacement of the probe in a macroscopically fluid (but viscoelastic) sample was measured by applying an external force over a short duration. Rich et al [56] studied differently forced magnetic probes in Laponite, and were able to estimate a force threshold separating probes that saturate at a finite displacement from those that move rapidly to the tip of the magnetic tweezers. They converted their measurement into a yield-stress values which compares favorably with the macroscopic yield stress of the system. However, since this conversion has to assume a relevant length scale (over which a force is assumed to be distributed in calculating a stress) that is not a priori known, this agreement has to be interpreted with care. The local melting induced by forces above the threshold was also studied in non-amorphous hosts systems, in particular colloidal crystals [203]. Overall the phenomenology is rather similar, and it was noted that the threshold force is not qualitatively different from the one found in the glassy experiments discussed above [204]. A pioneering simulation study of the force-induced depinning in a glass was performed by Hastings et al [205], finding the probe velocity to asymptotically obey a power law in the distance of the force to the threshold. The decoupling of probe and host-fluid motion was also studied for a system with (additional) quenched disorder, where two regimes of depinning were found: either the probe motion induces a depinning of the bath particles as well, or only the probe decouples from the rest of the system [206]. A number of simulation studies have seen the dynamics to become intermittent, as already mentioned above [180, 190]. A related simulation study was performed by Reichhardt and Olson Reichhardt [207], for a system of charged particles, varying the mismatch between probe and host-fluid charges. These authors noted intermittent dynamics in the probe close to the depinning transition, showing a 1/f noise characteristic. This provides an interesting link to a growing body of literature on ‘intruder’ motion in dense granular matter. There, the intrinsic nonequilibrium nature of the host system and the role of force chains among the granular particles demands first to re-establish the analogous of basic hydrodynamic results from fluid mechanics, such as the Stokes drag coefficient. It is not even clear how far such analogies can go, since, e.g.,

the probe-size or velocity dependence of granular drag force can be distinctly non-Stokesian [208, 209]. Several simulation and experimental studies of forced-probe motion close to the delocalization threshold, in either driven or static granular assemblies, have reported intermittent dynamics [210–213]. The strain respective velocity fields in granular systems around a probe have been measured [214–216], to be compared to the strain field measured in a colloidal glass [8]. Also, the cooperative interactions among multiple intruders has been studied [217].

4. Conclusions Microrheology was developed initially as an alternative to conventional rheology for small samples, as proposed by Mason and Weitz, and has revealed itself to be in addition a powerful technique to access the microscopic properties of complex fluids. An analysis of microrheology data in terms of models that assume the host fluid to be a structureless continuum, is still prevalent in many studies. Refined theoretical models are beginning to emerge, that specifically address the case of equal-sized probe and host-fluid entities. Such models are currently mainly restricted to simple systems, such as hard-sphere suspensions. They already indicate that the response of the tracer, in particular, in active microrheology, is far more complex than expected on the basis of a strict connection between microrheology response and macrorheological quantities. Passive microrheology, where the diffusion of tracers is monitored, has become a popular and economic technique, conventional in many laboratories. The usual interpretation requires the assumption of a generalized Stokes–Einstein relation. However, as shown by the models and some experiments, probe-bath direct interactions and heterogeneities of the host fluid dramatically affect the tracer diffusion, in particular for small probes. Even assuming that a continuum description remains valid for the bath, the analysis of experimental data is hampered by numerical issues arising from the need to perform Laplace or Fourier transforms of this data. Different strategies to simplify and improve these analysis have been devised, although the basic physical assumptions are not modified. Experiments employing passive microrheology have in general reported results that agree qualitatively and quantitatively with bulk rheology in different systems, although a quantitative comparison requires ad-hoc empirical corrections in some cases. These experimental studies have highlighted the microscopic details in the dynamics of complex systems, such as glasses (hard-spheres, laponite, …) or biological ones, and they provide valuable length-scale resolved data to complement standard bulk rheology. Improved modeling for passive microrheology takes into account the non-stationary nature of the motion, inertia effects, and related corrections to the originally proposed generalized SE relation. Understanding probe-size effects gives access to rheological data on many length scales, which is relevant in particular in attractive glasses or gels. But for probes of a size comparable to the relevant length scale in the 21

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host, a better description of the medium is required. In some cases, as in complex network-forming host fluids, an effective modeling based on continuum mechanics is suitable. But if probe size and interaction length scale of the host fluid are comparable, full microscopic theories need to be employed. Several approaches exist, typically starting from a generalized Langevin equation of the microscopic densities. While it is reassuring that the SE relation can be recovered in appropriate limits taken in these models, an interpretation of small-probe microrheology requires a detailed description of the host-fluid dynamics. The inverse problem then, of determining features of this dynamics from the measured probe dynamics, becomes a rather nontrivial one. This is even more true for active microrheology, in particular in the nonlinear-response regime that is relevant for most colloidal fluids. In experiment, probe trajectories have to be monitored down to extremely low average velocities, and small forces have to be precisely controlled. On the theoretical side, two main models that start from the many-body Smoluchowski equation for colloidal suspensions have been developed in recent years: one, developed by Brady and coworkers, is based on exact analyses of the two-particle diffusion equation valid at low host-fluid densities. The other one, by Fuchs and coworkers, employs approximations based on the mode-coupling theory of the glass transition and is thus specifically geared towards high-density, glass-forming host suspensions. Both these theories compare favorably to simulations in their regime of applicability, although especially in the high-density regime, a number of dynamical features— such as the strongly intermittent motion of probe particles and its connection to superdiffusive transient mean-squared displacements—still await explanation. The hallmark of active nonlinear microrheology on model colloidal suspensions, such as hard spheres, is seen in the probe-particle friction coefficient as a function of the driving strength (as quantified through a Péclet number Pe). For small driving, a linear-response regime is seen, with a nearly Pe-independent friction coefficient that increases dramatically with increasing host-fluid density. A pronounced drop in the friction coefficient marks the crossover from this linear response, where thermal motion governs the steady-state dynamics of the probe, to a strong-driving regime, where the force-induced change in the host-fluid structure around the probe dominates the dynamics. For high-density suspensions, this marked nonlinear change in the response is interpreted as the signature of a nonequilibrium delocalization transition occurring in the idealized glass, separating solidlike localized motion from fluid-like motion. In colloidal suspensions, very strong driving again leads to an effective driving-independent friction coefficient that is, based on the two-particle Smoluchowski theory, understood as the effect of a hydrodynamic boundary layer around the probe. First simulations on driven granular systems indicate that the behavior in this regime can qualitatively change with the short-time dynamics governing two-particle collisions. In particular, the inelastic collisions typical of granular systems appear to counter-act the delocalization effect, so that the friction coefficient increases again with Pe at extremely strong driving. Strong

hydrodynamic effects in colloids may have a similar effect, but the details of such a connection remain to be studied. Although both nonlinear features, driving-induced ‘thinning’ and ‘thickening’ of the system, are reminiscent of their macrorheological counterparts, the connection between the microviscosity obtained from active microrheology to the shear viscosity is, in general, not supported by the theoretical models. For length and time scales where the host fluid cannot be treated as a Newtonian continuum fluid, the nonviscometric and non-steady nature of the microrheological flow field destroy such a connection. One encounters a related qualitative breakdown of the Stokes–Einstein relation in polymer solutions or nematic fluids, be it a non-Stokesian probesize dependence [218, 219] or transient effects related to the complex-fluid dynamics [220]. As a matter of fact, microrheology should not be taken as a surrogate for macrorheology; the motion of micron-sized probes in optical traps can be analyzed to yield insight about long-standing issues of complex-matter dynamics. The breaking of the fluctuation-dissipation theorem (FDT) that is the basis of the original SE relation in aging or similar out-ofequilibrium systems can be studied [143, 221, 222]; related is the effective fluctuation-dissipation relation for ‘hot Brownian motion’ [223, 224] that may be relevant for certain nanoparticle trapping setups. Other experimental and simulation studies address the balance between entropy production and consumption expressed by the fluctuation theorem (FT) of nonequilibrium statistical mechanics [225], through fluctuations of a microrheological probe in optical traps that are moving or varied in strength [226–228]. Microrheology has so far been a tool most suitable for the study of soft materials. This owes to the scale of the relevant forces, and the typical length scales to be observed and manipulated, which favor micrometer sized probes. However, extending the length- and force-scales of the traditional microrheology setup, nano-indentation (used in materials science as well in biophysical context [229], and more recently also in colloidal experiment [230, 231]), atomic-force microscopy (AFM) [232, 233], and atomic force acoustic microscopy (AFAM) [234–236] provide interesting developments of local rheology near interfaces. Nanoindentation measurements have been performed with the aim of understanding the generic similarities and differences between molecular, colloidal, and other non-colloidal solids [231]. Another emerging area is the microrheological study of active-particle suspensions and their fascinating nonequilibrium rheology [237]. Foffano et al [238, 239] have performed continuum-mechanics simulations of an active nematic fluid surrounding a probe particle subject to an external force. Again, a non-Stokesian dependence on the probe size was found, along with a region of negative absolute mobility (arising from the boundary condition involving the nematic order parameter on the probe surface). Such studies tie in with the many open issues in understanding the microrheology of biophysical systems (see [21]). Active microrheology borders on a growing field of studying driven colloidal suspensions, and mixtures of driven and non-driven particles, where the interactions among several pulled probe particles provides for rich 22

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Acknowledgments

nonequilibrium induced-interaction effects and kinetic transitions [155, 240–242]. Extending microscopic descriptions to different driving modes is another future direction for the theory, in particular to deal with rotational microrheology of anisotropic probes, in either isotropic or anisotropic host fluids. First studies of anisotropic probe particles subject to external torque go back to Valberg and coworkers [243, 244] (with relatively irregular magnetic-particle aggregates), Wilhelm and coworkers [245, 246], Bishop and coworkers [129, 247], and Cheng and Mason [248]. Analyses are based on the rotational analog of the GSE relation [249], starting (in the simplest case) from the Stokes–Einstein expression for a spherical probe with stick boundary conditions,

ThV acknowledges funding through the Helmholtz-Gemeinschaft (HGF VH-NG 406), and the Zukunftskolleg der Universität Konstanz. AMP acknowledges financial support from the Consejería de Economía, Innovación y Ciencia (Junta de Andalucía) and from the Ministerio de Ciencia e Innovación, and FEDER funds, under projects P09-FQM-4938 and MAT2011-28385, respectively. References [1] Morris J F 2009 Rheol. Acta 48 909 [2] Brader J M 2010 J. Phys.: Condens. Matter 22 363101 [3] Einstein A 1905 Ann. Phys. 19 289 [4] Sutherland W 1905 Phil. Mag. 9 781 [5] Stokes G G 1851 Trans. Camb. Phil. Soc. 9 8 [6] Cicuta P and Donald A M 2007 Soft Matter 3 1449 [7] Lee M H and Furst E M 2008 Phys. Rev. E 77 041408 [8] Anderson D, Schaar D, Hentschel H G E, Hay J, Habdas P and Weeks E R 2013 J. Chem. Phys. 138 12A520 [9] Freundlich H and Seifriz W 1923 Z. Phys. Chem. 104 233 [10] Seifriz W 1924 J. Exp. Biol. 2 1 [11] Gordon M, Hunter S C, Love J A and Ward T C 1968 Nature 217 735 [12] Salençon J 2001 Handbook of Continuum Mechanics (New York: Springer) [13] MacKintosh F C and Schmidt C F 1999 Curr. Opin. Colloid Interface Sci. 4 300 [14] Solomon M J and Lu Q 2001 Curr. Opin. Colloid Inferface Sci. 6 430 [15] Waigh T A 2005 Rep. Prog. Phys. 68 685 [16] Furst E M 2005 Curr. Opin. Colloid Interface Sci. 10 79 [17] Wilson L and Poon W C K 2011 Phys. Chem. Chem. Phys. 13 10617 [18] Weihs D, Mason T G and Teitell M A 2006 Biophys. J. 91 4296 [19] Wirtz D 2009 Annu. Rev. Biophys. 38 301 [20] Squires T M and Mason T G 2010 Annu. Rev. Fluid Mech. 42 413 [21] Schultz K M and Furst E M 2012 Soft Matter 8 6198 [22] Mason T G and Weitz D A 1995 Phys. Rev. Lett. 74 1250 [23] Mason T G, Gang H and Weitz D A 1996 J. Mol. Struct. 383 81 [24] Gittes F, Schnurr B, Olmsted P D and MacKintosh C F C 1997 Phys. Rev. Lett. 79 3286 [25] Yanagishima T, Frenkel D, Kotar J and Eiser E 2011 J. Phys.: Condens. Matt 23 194118 [26] Mason T G, Ganesa K, van Zenten J H, Wirtz D and Kuo S C 1997 Phys. Rev. Lett. 79 3282 [27] Mason T G 2000 Rheol. Acta 39 371 [28] Dasgupta B R, Tee S-Y, Crocker J C, Frisken B J and Weitz D A 2002 Phys. Rev. E 65 051505 [29] Fricks J, Yao L, Elston T C and Forest M G 2009 SIAM J. Appl. Math. 69 1277 [30] Tassieri M, Evans R M L, Warren R L, Bailey N J and Cooper J M 2012 New J. Phys. 14 115032 [31] Felderhof B U 2009 J. Chem. Phys. 131 164904 [32] Franosch T, Grimm M, Belushkin M, Mor F M, Foffi G, Forró L and Jeney S 2011 Nature 478 85 [33] Indei T, Schieber J D, Córdoba A and Pilyugina E 2012 Phys. Rev. E 85 021504 [34] Indei T, Schieber J D and Córdoba A 2012 Phys. Rev. E 85 041504 [35] Córdoba A and Indei T 2012 J. Rheol. 56 185

kT Dr = B 3 . (42) 8πηa

Notably, it displays a stronger probe-size dependence than the translational diffusivity. For anisotropic particles, a tensorial GSE relation including translation-rotation coupling can be derived [250]. Similar to the translational case discussed in this review, above a critical driving frequency, a decoupling of the probe from the host fluid can be observed [251]. Rotational microrheology may help to further increase the range of length scales accessible [252], and to perform optical microrheology also in strongly scattering media [253]. For rotational microrheology in particular, but also for standard translational active microrheology [254], a detailed discussion of transient response will allow to give further insight into the buildup of microscopic stresses in complex host fluids. Surprisingly, the breakdown of the SE relation discussed in glassy systems is less severe for the rotational coupling than for translation [91]. Other issues remain open in the theory of active microrheology. For instance, the relation between fixed-velocity and fixed-force driving of probes is only understood for the low-density models. At high host-fluid density, mainly the fixed-force active microrheology has been studied. Even for the linear-response regime, the results differ when driving the probe by a fixed velocity: for small external forces, the response approaches the equilibrium probe-particle friction coefficient, and hence the diffusivity (assuming FDT). Imposed velocities approaching zero on the other hand correspond to an immobilized particle in a colloidal suspension; similarities to partially pinned systems may then emerge [112]. Future studies will help to refine theoretical models, and turn active, nonlinear-response microrheology into a powerful technique to probe the dynamics of mesostructured complex materials on a wide range of length scales. This will allow to shift the focus of microrheology: while early applications mainly aimed to provide a proxy for macroscopic rheology, microrheology is now increasingly seen as a tool in its own right, harnessing the full detail of the micro-response. An emerging detailed understanding of the nonlinear response in active microrheology opens the possibility to explore in detail dynamical processes that govern the equilibrium and nonequilibrium dynamics of complex fluids. 23

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26

Active microrheology in active matter systems: Mobility, intermittency, and avalanches.

Mean-field microrheology of a very soft colloidal suspension: Inertia induces shear thickening.

Accelerated stability assay (ASA) for colloidal systems.

Frequency modulated microrheology.

The dynamical crossover in attractive colloidal systems.

Colloidal stability of negatively charged cellulose nanocrystalline in aqueous systems.

Design of liposomal colloidal systems for ocular delivery of ciprofloxacin.

Evolution of topological defects in two-dimensional quenched colloidal systems.

DNA-controlled dynamic colloidal nanoparticle systems for mediating cellular interaction.

Active microrheology of driven granular particles.

Microrheology and microstructure of Fmoc-derivative hydrogels.

Magnetic microrheology of block copolymer solutions.

Microrheology of keratin networks in cancer cells.

Active multi-point microrheology of cytoskeletal networks.

Feedback-tracking microrheology in living cells.

Microrheology close to an equilibrium phase transition.

Active microrheology determines scale-dependent material properties of Chaetopterus mucus.

Analytical theory of effective interactions in binary colloidal systems of soft particles.

Particle-Tracking Microrheology Using Micro-Optical Coherence Tomography.

Tracer microrheology study of a hydrophobically modified comblike associative polymer.

Two-Point Microrheology of Phase-Separated Domains in Lipid Bilayers.

Magnetomotive optical coherence elastography for microrheology of biological tissues.

Magnetic wire-based sensors for the microrheology of complex fluids.

Transfection Studies with Colloidal Systems Containing Highly Purified Bipolar Tetraether Lipids from Sulfolobus acidocaldarius.