Active and Nonlinear Microrheology in Dense Colloidal Suspensions
aa r X i v : . [ c ond - m a t . s o f t ] O c t Active and Nonlinear Microrheology in Dense Colloidal Suspensions
I. Gazuz, A. M. Puertas, Th. Voigtmann,
1, 3 and M. Fuchs Fachbereich Physik, Universit¨at Konstanz, 78457 Konstanz, Germany Departamento de F´ısica Aplicada, Universidad de Almer´ıa, 04.120 Almer´ıa, Spain Institut f¨ur Materialphysik im Weltraum, Deutsches Zentrum f¨ur Luft- und Raumfahrt (DLR), 51170 K¨oln, Germany (Dated: October 26, 2018)We present a first-principles theory for the active nonlinear microrheology of colloidal model sys-tems: for constant external force on a spherical probe particle embedded in a dense host dispersion,neglecting hydrodynamic interactions, we derive an exact expression for the friction. Within mode-coupling theory (MCT), we discuss the threshold external force needed to delocalize the probe froma host glass, and its relation to strong nonlinear velocity-force curves in a host fluid. Experimentalmicrorheology data and simulations, which we performed, are explained with a simplified model.
PACS numbers: 83.10.-y, 83.10.Rs, 64.70.pv
Microrheology is a promising technique providing localprobes of the dynamics in a complex fluid [1]. Monitoringthe motion of a singled-out probe particle embedded ina (dense) host fluid or gel, one addresses questions aboutthe microscopic origins of the host’s complex-fluid behav-ior and in particular the link between microscopic mech-anisms and macroscopic properties amenable to conven-tional rheology. This connection subtly depends on thehost, probe-bath interactions, and on the applied forces.Active microrheology turns this into an advantage, at thecost of requiring much better knowledge about the mi-croscopic processes [2]: applying a known forcing to theindividual particle, one explores the nonequilibrium andusually nonlinear regime, providing detailed insight intothe structure-dynamics relationship, e.g., in cellular en-vironments [3] or close to the glass transition [4, 5, 6].Laser tweezers, magnetic or surface-chemistry forces [7]provide experimental realizations achieving large forcing.The external-force-velocity relations obtained in densesuspensions reveal striking nonlinearities, induced by theslow relaxation of the host. Leaving the linear-responseregime, a sudden strong increase in the velocity revealsthe strength required to pull free the probe from the(transient) local neighbor cage, such that force-inducedmotion overrules structural relaxation. Recent theoreti-cal progress [8, 9] notwithstanding, it remains to under-stand the nonlinear friction induced by the slow struc-tural rearrangements of host particles.Here we develop a theory for active nonlinear microrhe-ology in suspensions close to their glass transition, es-tablishing the conceptual connection between micro- andmacro-rheology, when (de)localization of the probe oc-curs, and how the structure of the cage is distorted closeto this yielding point. We start from microscopic equa-tions of motion and relate the force–velocity relation ofthe probe, by virtue of an exact Green-Kubo-like formula,to a microscopic-force autocorrelation function. Thiscan be approximated through non-equilibrium tagged-particle density correlation functions, which in turn arecalculated in the framework of the mode-coupling theory of the glass transition (MCT) [10]. For a hard-sphere(HS) suspension with a pulled probe of same size as thehost particles, we demonstrate that the theory predicts adelocalization threshold force that explains the nonlinearresponse seen in experiment and simulation.We start from the many-body Smoluchowski equationfor the nonequilibrium distribution function Ψ( t ) of a sys-tem of N Brownian particles (positions r i ) and a singleprobe (labeled s ), ∂ t Ψ( t ) = ΩΨ( t ). Subjecting only theprobe particle to a constant, homogeneous force F ex , theSmoluchowski operator Ω = Ω + ∆Ω readsΩ = X i =1 ,...,N,s ∂ i · ( k B T ∂ i − F i ) /ζ i − ( ∂ s · F ex ) /ζ s , (1)where ∆Ω = − ( ∂ s · F ex ) /ζ s is the nonequilibrium termdescribing active forcing, and Ω the equilibrium time-evolution. We neglect solvent-induced hydrodynamic in-teractions and introduce Stokes friction coefficients forhost ( ζ i =1 ,...N ≡ ζ ) and probe ( ζ s ) particles. The F i,s are (potential) interaction forces among the particles.To obtain nonequilibrium averages formed with theforce-dependent Smoluchowski operator, the integration-through-transients (ITT) formalism [11] recasts Eq. (1):Ψ( t ) = Ψ eq − k B Tζ s Z t dt ′ exp[Ω t ′ ]( F ex · F s )Ψ eq , (2)assuming equilibrium at t = 0. In particular, the sta-tionary friction coefficient ζ ( F ex ), defined via the averagestationary velocity at given external force ζ h v s i t →∞ ≡ ζ h v s i ∞ = F ex , (3)is found by using Eq. (2) and equating external and in-teraction forces on the probe: ζ = ζ s + 13 k B T Z ∞ dt h F s exp[Ω † ( F ex ) t ] F s i eq . (4)This formally exact generalized Green-Kubo relation con-nects the far-from-equilibrium response to a transientequilibrium-averaged correlation function. ITT achievesthat all following averages are equilibrium ones, denotedby h·i (suppressing the eq subscript). Ω † is the adjoint ofΩ, and F ex enters non-perturbatively; linear response isrecovered by neglecting this dependence.Following MCT, we assume that force fluctuations aregoverned by collective and probe-particle density fluctu-ations, ̺ q = P Ni =1 exp[ i qr i ] and ̺ s q = exp[ i qr s ]. Wetake that in the thermodynamic limit, the motion of theprobe has negligible impact on the bulk properties of thehost, and restrict wave numbers to a discrete grid ne-glecting anomalous long distance correlations. Insertinga projector P ∝ P kp ̺ s k ̺ p ih ̺ s k ̺ p on both sides of theoperator exponential in Eq. (4), because forces on theprobe relax by host particle rearrangements and probemotion, and splitting four-point density averages intodynamical density correlators, φ k ( t ) = h ̺ − k exp[Ω † t ] ̺ k i and φ s k ( t ) = h ̺ s − k exp[Ω † t ] ̺ s k i , we arrive at h F s exp[Ω † t ] F s i ≈ X k | k B T kS sk | N S k φ s k ( t ) φ − k ( t ) . (5) S k = h ̺ k ̺ − k i and S sk = h ̺ s k ̺ − k i are the equilibriumstructure functions describing interactions among probeand host particles.The probe correlator φ s q ( t ) is complex-valued, as theperturbed operator Ω † is non-Hermitian. This reflectsthat the probe-density distribution is shifted by appli-cation of an external force: while in equilibrium it iscentered around the origin, the average position of theprobe moves, introducing a complex-valued phase factorin φ s q ( t ). Still, Eq. (5) maintains ζ ∈ R due to the sym-metry φ s − q ( t ) = ( φ s q ( t )) ∗ .Equation (5) recasts the problem of calculating theprobe friction as one of calculating collective and tagged-particle density correlation functions. To this end, weemploy Zwanzig-Mori equations of motion [11], ∂ t φ s q ( t ) = − ω s q , q φ s q ( t ) − Z t dt ′ m s q ( t − t ′ ) ∂ t ′ φ s q ( t ′ ) , (6a)closed by the MCT approximation generalizing Eq. (5)to finite wave vectors, m s q ( t ) = k B Tζ s ω s q , q X k + p = q N S p V s qkp V s, † qkp φ s k ( t ) φ p ( t ) . (6b)Again, the physical idea in the MCT approximation isthat the friction kernel m s q ( t ) relaxes by both probeand host density dynamics. The coupling coefficientsare V s qkp = ( qp ) S sp , V s, † qkp = ω s q , p S sp , where ω s q , p =( q k B T − i F ex ) · p /ζ s . An analogous set of equationsholds for φ q ( t ). Since the external force acts on theprobe only, the φ q ( t ) are in fact determined by the un-perturbed Smoluchowski operator, Ω , resulting in thestandard MCT scenario of glassy dynamics [10, 12]. Thisdescribes arrest driven by wave vectors connected with a ϕ F c e x [ k T / a ] FIG. 1: Threshold force F ex c ( ϕ ) needed to delocalize a hard-sphere probe particle in a glass of equally large hard sphereswith packing fraction ϕ above the glass transition (dottedline), calculated from MCT within the Percus-Yevick approx-imation. Blue circles mark F ex values used in Fig. 2. Inset:corresponding schematic-model result (see text). typical host particle radius a . Thus the dimensionlessparameter measuring the effect of the external force is aF ex / ( k B T ), the work required to pull the probe overthat distance in relation to thermal energy.The macroscopic counterpart to the friction ζ is thedispersion viscosity η measured in bulk flow. WithinITT, the analog to Eq. (4) holds for the latter [13]. MCTexpresses this as a functional only of the host correla-tors φ q ( t ), while in Eq. (5), the probe correlators φ s q ( t )enter. In linear response close to the glass transition,identical scaling laws for both closely link micro- andmacro-rheology [14]. For large external forces, this cor-respondence breaks: Equations (6) for the probe corre-lator contain a novel delocalization transition that is ab-sent in φ q ( t ). A probe arrested in a glassy host suspen-sion remains localized in its (deformed) nearest-neighborcage (described by f s q = φ s q ( t → ∞ ) > F ex c . At larger force, the probe is pulled free( f s q = 0) and attains a steady velocity (finite friction) atlong times. In the liquid, cages are transient, and a rem-nant of the threshold survives as a sudden sharp “forcethinning” in ζ ( F ex ).The details of the delocalization transition depend onthe host properties, which we model now as hard spheresusing the known numerical MCT results for the collec-tive density correlators φ q ( t ) within the Percus-Yevick S q -approximation [15]. This model yields a glass transi-tion at packing fraction ϕ c ≈ . ϕ = (4 π/ ̺a with number density ̺ is the only parameter. Fig-ure 1 shows our results for the delocalization thresh-old force, F ex c , for a probe equal to the host particles( S q = 1 + S sq ). This threshold F ex c ( ϕ c ) > ϕ c , and increases further with increasing density. Notethat F ex c = O (50 k B T /a ), much larger than one mightintuitively expect. This reflects the strong caging forceexerted by the set of nearest neighbors that must be over- y /a F ex = − − F ex = − − F ex = − − F ex = − − x/ a FIG. 2: Contour plot of the probability distribution f s ( r ) fora localized hard-sphere probe of radius a in a hard-sphere sys-tem with same radius at ϕ = 0 .
52, for external forces actingto the right with indicated magnitude in units of k B T /a . come before the probe can be delocalized.The inverse Fourier transform of f s q is the t → ∞ prob-ability distribution for the position of a probe starting atthe origin; Figure 2 shows our results at packing fraction ϕ = 0 .
52, slightly above the glass transition, for severalforces below F ex c . F ex is taken to be in (positive) hor-izontal direction, rendering f s ( r ) rotational-symmetricaround this axis. For zero force, the distribution isspherical-symmetric and centered around the origin; itdecays on a length scale of 0 . a , the typical localizationlength for solids dominated by hard-core repulsion. Smallapplied forces mainly shift the center of the distributionto a position x ≈ . a , i.e., they push the probe to the“cage wall” without essentially distorting the cage. Closeto the delocalization threshold, however, f s ( r ) developsa deformed tail extending into the force direction, reduc-ing the spherical symmetry to a merely rotational one.Interestingly, the tail does not extend along the symme-try axis; rather, a “dip” is seen in direction of the appliedforce. For F ex ≥ F ex c , f s q and f s ( r ) vanish, indicating adelocalized state.To discuss probe friction or similar dynamical quanti-ties, we need to solve the time-dependent and spatiallyinhomogeneous Eqs. (6). A first step towards this isto solve a simplified, “schematic” MCT model. Take ζ = 1 + R ∞ φ s ( t ) φ ( t ) dt , with ∂ t φ s ( t ) + ω s φ s ( t ) + Z t m s ( t − t ′ ) ∂ t ′ φ s ( t ′ ) dt ′ = 0 , (7a) m s ( t ) = v s φ s ∗ ( t ) φ ( t ) , (7b) ω s = 1 − iF ex . The host correlator φ ( t ) is set by the“F model” often used to describe glassy dynamics inequilibrium [10, 16], governed by a separation parame-ter ǫ such that ǫ < ǫ ≥ ǫ thus measures the host interactions. v s describesthe strength of probe–host coupling. Its F ex dependenceis ignored, neglecting the subtle interplay arising fromstatic correlations involving more than one wave vec-tor; still, the model recovers the qualitative behavior ofthe force threshold (see inset of Fig. 1). We expect theschematic model to be reasonable for small enough ve-locities and/or forces, where the friction is dominated by F ex [ kT / a ] ζ / ζ ϕ = 0.62ϕ = 0.57ϕ = 0.55ϕ = 0.50ϕ = 0.40ϕ = 0.45 -1 F ex [pN] -6 -5 -4 -3 -2 -1 〈 v 〉 ∞ [ µ m / s ] ϕ = 0.45ϕ = 0.50ϕ = 0.52ϕ = 0.53ϕ = 0.55 (a) (b) FIG. 3: (a) Probe friction ζ as a function of external force forpacking fractions ϕ as indicated, from simulations of a quasi-hard-sphere system (symbols), from Brownian dynamics formonodisperse HS (Ref. [17], open symbols), and from theschematic model (solid lines; see text) with k B T /a = 0 . /ζ = 3 .
44, and parameters ( ǫ, v s ) = (0 , − . , . − . , − . , − . , . − . , ǫ, v s ) = ( − . , − . , − . , − . , v = v = 0 . v s / x and y by 0 .
058 and 296. universal features of the transition at F ex c .To test the simplified model, we performed simulationsof a slightly polydisperse quasi-hard sphere system un-dergoing strongly damped Newtonian dynamics, whichshows a glass transition at ϕ c ≈ .
595 [18]. Particles(mass m = 1, k B T = 1, radii distributed uniformly in[0 . , . ζ = 50) andrandom forces obeying the fluctuation-dissipation theo-rem. One particle is randomly selected to undergo anexternal force F ex until it reaches a distance half the sizeof the simulation box (elongated in the direction of F ex by a factor of 8). The average probe velocity is measuredsampling more than 300 independent trajectories, andthe friction is calculated using Eq. (3). All simulationswere initially equilibrated, except for ϕ = 0 .
62, where thesystem was aged for t w = 25000. At this density, resultsshow little influence of ageing for forces F ex > ∼ k B T /a .A strong decrease in the dynamical friction ζ around F ex c = O (40 k B T /a ) seen in the simulation [symbols inFig. 3(a)] indicates the force threshold. Fitting ǫ and v s per curve, and two shift factors setting the units, theschematic model reproduces this behavior for ϕ < ϕ c .In the (idealized) glass, it predicts a true delocalizationtransition as ζ → ∞ for F ex < F ex c , exemplified by a ǫ = 0 curve in Fig. 3(a). In the simulation, ζ remainsfinite in the accessible window presumably because ofergodicity restoring processes ignored here [10]. Employ-ing larger v s , our model also explains recent experimentson colloidal systems using larger probes [5], as shownin Fig. 3(b). In these velocity-force curves obtained be-low the glass transition, the force-threshold signature is a -0.50.00.51.0 R e φ ( t ) -2 -1 t [( ma / kT ) ] -0.50.00.51.0 I m φ ( t ) -1 t FIG. 4: Probe-particle density correlation function φ s q ( t ) fromcomputer simulation at ϕ = 0 .
55 (left) and from schematicMCT (right; fit as in Fig. 3), real (top) and imaginary (bot-tom) parts. q k F ex corresponds to the position of the mainpeak in the static structure factor S ( q ). For the simulation, aF ex / ( k B T ) = 1, 10, 20, 30, 50, 100, and 250 (right to left). steep increase of h v s i ∞ around F ex c ≈ . ≫ k B T /a ,again reproduced by the model. For too large externalforces, the model fails, as expected above.The virtue of the schematic model is to allow moredetailed qualitative predictions for the slow nonequilib-rium dynamics. This is demonstrated by Fig. 4, wherewe compare the tagged-particle density correlation func-tion φ s q ( t ) obtained from the simulation for a wave vector q k F ex with a magnitude corresponding to the nearest-neighbor peak in S q . The simulation confirms the ex-istence of complex-valued correlation functions (for this q -direction) as a signature of non-equilibrium, naturallyarising in our microscopic framework. For q perpendicu-lar to the external force, φ s q ( t ) = ( φ s − q ( t )) ∗ remains real-valued, owing to the rotational symmetry φ s q ( t ) = φ s − q ( t ),and exhibits the two-step decay typical for glass formerswith an intermediate plateau and a final relaxation spedup by the external force. No such clear plateau is seenin the figure for q parallel to F ex . For large forces, the φ s q ( t ) show pronounced oscillations, quite unexpected fora Brownian system, and even stronger in the simulationdata.To summarize, we have developed a microscopicallyfounded theory for the nonlinear active microrheologyclose to a glass transition. Starting from the Smolu-chowski equation without hydrodynamic interactions,and applying approximations in the spirit of the mode-coupling theory of the glass transition, we predict theprobe friction as a function of the external force and ofthe equilibrium host structure.The theory predics a finite microrheological forcethreshold needed to delocalize a probe from a glassy host, locally melting it. In the dense liquid, this is reflected bya strong nonlinear decrease in friction coefficients differ-entiating the regimes where cages are either broken byslow structural relaxation (for small external force), orby large enough applied force. A schematic model cap-tures these aspects and allows to fit experimental andsimulation data quantitatively for not too large externalforces.The force threshold could be related to the existence ofa yield stress well established for glassy colloidal systems,and predicted by MCT for constant-velocity bulk driving.It will be promising to study more closely this relationand the dynamical behavior of the system close to micro-and macro-yielding.We thank A. Erbe, W.C.K. Poon and J.F. Bradyfor discussions, and Deutsche Forschungsgemeinschaft(SFB 513 project B12), Helmholtz-Gemeinschaft(Hochschul-Nachwuchsgruppe VH-NG 406), M.E.C.project MAT-2006-13646-CO3-02 and Junta de An-daluc´ıa (P06-FQM-01869) for funding. [1] T. A. Waigh, Rep. Prog. Phys. , 685 (2005).[2] T. M. Squires, Langmuir , 1147 (2008).[3] C. Wilhelm, Phys. Rev. Lett. , 028101 (2008).[4] M. B. Hastings, C. J. Olson Reichhardt, and C. Reich-hardt, Phys. Rev. Lett. , 098302 (2003).[5] P. Habdas, D. Schaar, A. C. Levitt, and E. R. Weeks,Europhys. Lett. , 477 (2004).[6] S. R. Williams and D. J. Evans, Phys. Rev. Lett. ,015701 (2006).[7] L. Baraban, A. Erbe, P. Leiderer, and P. K¨uhler, submit-ted to Phys. Rev. Lett., arXiv:0807.1619.[8] T. M. Squires and J. F. Brady, Phys. Fl. , 073101(2005).[9] R. L. Jack, D. Kelsey, J. Garrahan, and D. Chandler,Phys. Rev. E , 011506 (2008).[10] W. G¨otze and L. Sj¨ogren, Rep. Prog. Phys. , 241(1992).[11] M. Fuchs and M. E. Cates, Phys. Rev. Lett. , 248304(2002).[12] W. van Megen and S. M. Underwood, Phys. Rev. Lett. , 2766 (1993).[13] J. M. Brader, Th. Voigtmann, M. E. Cates, andM. Fuchs, Phys. Rev. Lett. , 058301 (2007).[14] M. Fuchs and M. R. Mayr, Phys. Rev. E , 5742 (1999),and references therein.[15] T. Franosch, M. Fuchs, W. G¨otze, M. R. Mayr, and A. P.Singh, Phys. Rev. E , 7153 (1997).[16] We use Eq. (7) with ω = 1 for the correlator φ ( t ) andmemory kernel m ( t ) = v φ ( t ) + v φ ( t ) with v , = v c , (1 + ǫ ) and ( v c , v c ) = (0 . . . . , , 1483 (2005).[18] Th. Voigtmann, A. M. Puertas, and M. Fuchs, Phys. Rev.E70