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Nonlinear Microrheology: Bulk Stresses versus Direct Interactions† Todd M. Squires* Department of Chemical Engineering, UniVersity of California, Santa Barbara, California 93106 ReceiVed August 2, 2007. In Final Form: October 24, 2007 In passive microrheology, the linear viscoelastic properties of complex fluids are inferred from the Brownian motion of colloidal tracer particles. Active (but gentle) forcing may also be used to obtain such linear-response information. More significant forcing may drive the material significantly out of equilibrium, thus potentially providing a window into the nonlinear response properties of the material. In leaving the linear-response regime, however, the theoretical underpinning for passive microrheology is lost, and a variety of issues arise. Most generally, what exactly can be measured, and how can such measurements be interpreted? Here we motivate and discuss a variety of theoretical issues facing the interpretation of active microrheology. First, in the continuum limit, the inhomogeneous velocity field around the probe gives rise to rheological inhomogeneities, whereupon an assumed generalized Stokes drag yields a weighted average of the viscosities around the probe rather than the (homogeneous) viscosity measured macroscopically. We then explicitly treat the material microstructure using a model system (a large colloidal probe pulled through a dilute suspension of small bath particles). We examine the different sources of stress upon the probe particle (e.g., direct probe-bath collisions as well as microstructural deformations within the bulk suspension) and discuss their analog (or lack thereof) in the corresponding macrorheological system. We discuss several crucial issues for the interpretation of nonlinear microrheology: (1) how to interpret the inhomogeneous and nonviscometric nature of the deformation field around the probe, (2) the distinction between direct and bulk stresses and their deconvolution, and (3) the (Lagrangian) time-dependent nature of the stress histories experienced by material elements as they advect past the probe. Having identified these issues, we briefly discuss adaptations of the basic technique to recover bulk rheology more faithfully. Whereas we specifically discuss a model colloidal suspension, we ultimately envision a technique capable of measuring the nonlinear rheology of general materials.
Introduction Many soft materials found in industry, personal care products, foods, and biology cannot be classified as either elastic solids or viscous fluids but fall somewhere in between. The complex response of such materials to applied stresses results from the complex nature of their underlying microstructure and often depends strongly on both the type of stress (i.e., shear vs extension) and the time scale over which it is applied. Rheology involves the study and characterization of such materials, using a variety of techniques to measure the material’s response to applied stress.1,2 For example, small-amplitude, oscillatory measurements using a cone-and-plate rheometer ensure that the material experiences a homogeneous shear at a precisely defined frequency. In such a manner, the linear elastic and viscous moduli G′(ω) and G′′(ω) can be measured over many decades in frequency. Such measurements reveal important frequencies, which correspond directly to time scales that are significant for the microstructure of the material (e.g., relaxation, reptation, or rearrangement times). As such, the linear viscoelastic moduli G′(ω) and G′′(ω) allow one to connect the bulk macroscopic behavior to the underlying behavior of the material’s microstructure. Also significant to note is that these measurements are (by design) performed in the small-strain, near-equilbrium, linear-response limit. In addition to providing the simplest and cleanest method of material characterization, the near-equilibrium nature of the technique allows a host of powerful techniques in linear-response theory to be employed. †
Part of the Molecular and Surface Forces special issue. * E-mail:
[email protected].
(1) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: New York, 1999. (2) Macosko, C. W. Rheology: Principles, Measurements, and Applications; John Wiley & Sons: New York, 1994.
However, many interesting and important material properties involve the nonlinear response of a material subjected to largeamplitude strain. Examples include yield stresses, which prevent frosting from flowing off of a cake yet allow it to be spread, viscosities that thin or thicken with increasing shear rate, and normal stress differences, which give rise to a host of dramatic effects in non-Newtonian fluids.1,3 All of these phenomena involve material microstructures that are driven significantly out of equilbrium and thus require large-strain deformations for their measurement. For example, cone-and-plate rheometers can employ steady rotations (rather than oscillations) to measure shear rate-dependent viscosity. Just over a decade ago, Mason and Weitz demonstrated that the frequency-dependent, linear viscoelastic moduli G*(ω) can be obtained from measurements of the thermal fluctuations of micrometer-scale colloidal particles.4 In the years since, interest in microrheology has exploded,5-10 in large part because of the small (microliter) quantities that are required for measurements as compared to milliliters required for conventional (macroscopic) rheometry. This is particularly useful when the material of interest is expensive, difficult, or impossible to acquire in large quantities (i.e., biological materials or in newly/combinatorially synthesized materials before scaling up production). Further advantages over macrorheology include a significantly higher range of frequencies available without time-temperature superposition,11 the capabil(3) Bird, R. B.; Armstrong, R. C.; Hassager, O. Dynamics of Polymeric Liquids; John Wiley & Sons: New York, 1987; Vol. 1. (4) Mason, T. G.; Weitz, D. A. Phys. ReV. Lett. 1995, 74, 1250. (5) MacKintosh, F. C.; Schmidt, C. F. Curr. Opin. Colloid Interface Sci. 1999, 4, 300. (6) Gisler, T.; Weitz, D. A. Curr. Opin. Colloid Interface Sci. 1998, 3, 586. (7) Furst, E. M. Soft Mater. 2003, 1, 167. (8) Gardel, M. L.; Weitz, D. A. In Microscale Diagnostic Techniques; Breuer, K. S., Ed.; Springer: Berlin, 2005; pp 1-50. (9) Waigh, T. A. Rep. Prog. Phys. 2005, 68, 685. (10) Breedveld, V.; Pine, D. J. J. Mater. Sci. 2003, 38, 4461.
10.1021/la7023692 CCC: $40.75 © 2008 American Chemical Society Published on Web 12/22/2007
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ity of measuring material inhomogeneities that are inaccessible by macrorheological methods,12,13 and rapid thermal and chemical homogenization that allow the transient rheology of evolving systems to be studied.14,15 Variants on the basic technique of Brownian microrheology have also been developed and proven useful. Multiprobe measurements improve statistics and provide additional advantages. Mason et al. examined the relative motion of bead pairs (subtracting out the collective motion) to alleviate convection.16 Crocker et al. measured the cross-correlated Brownian motion of bead pairs, thus measuring the coupling diffusivity between the beads rather than the self-diffusivity of each. The coupling diffusivity thus measured depends more strongly upon the material that lies between the particles than on the local environment around each. As such, these two-point measurements are less sensitive to probe/material interactions than one-point techniques and can more faithfully recover the macroscopic viscoelasticity in inhomogeneous materials, as shown experimentally17 and theoretically.18,19 The rotational diffusivity can also be measured and related to macroscopic rheology in an analogous fashion.20-24 Most modern microrheology is passive. Embedded colloids are allowed to fluctuate via Brownian forces, often using collections of probes to improve statistics. Because the system is inherently in equilibrium, linear-response theory and the fluctuation-dissipation theorem hold. The response of a colloidal probe to a weak, oscillatory force is given in terms of a linear response function R(ω)
∆xeiωt ) R(ω)f0eiωt
(1)
that is complex and frequency-dependent. The fluctuationdissipation theorem relates the thermal flucutations of embedded colloids to the actively driven response R(ω) of the same colloid. For example, the mean-squared displacement 〈∆x(t)2〉 of particles can be measured (e.g., using video microscopy, dynamic light scattering, or diffusing wave spectroscopy), whose Laplace transform is related to the Laplace-transformed linear response function R˜ (s)4,11,16 via
R˜ (s) )
s 〈∆r˜2(s)〉 6kBT
(2)
A variety of techniques have been developed to invert the Laplace transform numerically to recover the frequency-dependent R(ω).11,25,26 Alternately, one can measure the autocorrelation of (11) Mason, T. G.; Gang, H.; Weitz, D. A. J. Opt. Soc. Am. A 1997, 14, 139. (12) Valentine, M. T.; Kaplan, P. D.; Thota, D.; Crocker, J. C.; Gisler, T. Prud’homme, R. K.; Beck, M.; Weitz, D. A. Phys. ReV. E 2001, 6406, art. no. 061506. (13) Chen, D. T.; Weeks, E. R.; Crocker, J. C.; Islam, M. F.; Verma, R.; Gruber, J.; Levine, A. J.; Lubensky, T. C.; Yodh, A. G. Phys. ReV. Lett. 2003, 90, art. no. 108301. (14) Sato, J.; Breedveld, V. J. Rheol. 2006, 50, 1. (15) Slopek, R. P.; McKinley, H. K.; Henderson, C. L.; Breedveld, V. Polymer 2006, 47, 2263. (16) Mason, T. G.; Gang, H.; Weitz, D. A. J. Mol. Struct. 1996, 383, 81. (17) Crocker, J. C.; Valentine, M. T.; Weeks, E. R.; Gisler, T.; Kaplan, P. D.; Yodh, A. G.; Weitz, D. A. Phys. ReV. Lett. 2000, 85, 888. (18) Levine, A. J.; Lubensky, T. C. Phys. ReV. Lett. 2000, 85, 1774. (19) Levine, A. J.; Lubensky, T. C. Phys. ReV. E 2001, 6304, art. no. (20) Cheng, Z.; Chaikin, P. M.; Mason, T. G. Phys. ReV. Lett. 2002, 89. (21) Cheng, Z.; Mason, T. G. Phys. ReV. Lett. 2003, 90. (22) Bishop, A. I.; Nieminen, T. A.; Heckenberg, N. R.; Rubinsztein-Dunlop, H. Phys. ReV. A 2003, 68. (23) Bishop, A. I.; Nieminen, T. A.; Heckenberg, N. R.; Rubinsztein-Dunlop, H. Phys. ReV. Lett. 2004, 92. (24) Schmiedeberg, M.; Stark, H. Europhys. Lett. 2005, 69, 629. (25) Mason, T. G.; Gisler, T.; Kroy, K.; Frey, E.; Weitz, D. A. J. Rheol. 2000, 44, 917.
individual trajectories 〈∆x(t0 + t)∆x(t0)〉, either of a single particle as detected by laser scattering or held by optical tweezers, or of an ensemble in video microscopy. The Fourier transform can then be related directly to the imaginary part R′′ of the linear response coefficient
R′′ )
ω 〈∆x(t0 + t)∆x(t0)〉ω 2kBT
(3)
where 〈〉ω denotes the Fourier transform. The in-phase (elastic) coefficient R′ can then be derived from the out-of-phase (viscous) component R′′ using the Kramers-Kronig relations.11,27,28 Each technique has its own advantages and disadvantages, but all fundamentally rely upon the same physical foundation. So far, both results are exact and simply restate the fluctuationdissipation theorem. What remains is to relate the linear response function R(ω) to the (complex) viscoelastic modulus G*(ω) ) G′(ω) + iG′′(ω) of the material itself. To do so, a crucial assumption is made: the generalized Stokes-Einstein relation (GSER) is invoked to interpret the linear response function via
R(ω) )
1 6πaG*(ω)
(4)
where G*(ω) is the linear viscoelastic modulus, as would be measured using standard macrorheological means. The GSER takes the well-known Stokes-Einstein relation between the diffusion constant of a colloid in a Newtonian fluid and its Stokes mobility (or friction) and generalizes it to account for an incompressible, homogeneous, continuum, linear viscoelastic material. Microrheology has proven successful in measuring the viscoelastic moduli of a wide variety of materials, including colloidal suspensions, emulsions, polymer solutions, gels, wormlike micellar solutions, and other soft materials. The small sample volumes required and the wide frequency range afforded by microrheology have led to its widespread use both in industrial and academic research, and one can now buy commercial microrheology devices. In what follows, we will focus on the emerging and much less-developed technique of nonlinear microrheology. We will explore and discuss several issues that complicate the interpretation and implementation of nonlinear microrheology, many of which find directly analogous issues in passive microrheology. We thus briefly discuss several issues that have been identified for passive microrheology in order to illuminate what follows. Several cases have been established where the G* measured macrorheologically does not agree with results from passive microrheology, interpreted using the GSER. First, the probe might not “see” a continuum material just like the macrorheological apparatus does. For the GSER to be valid, the probe must be significantly larger than all relevant microstructural elements of the material itself (e.g., pore or mesh sizes, colloid radii or polymer persistence/entanglement lengths). As a counter example, nanoparticle diffusivities in polymer melts have been measured to be orders of magnitude higher than the GSER would suggest.29 As discussed by Brochard-Wyart and de Gennes,30 macroscopic viscous flows of polymer melts require entanglements to relax, (26) Dasgupta, B. R.; Tee, S. Y.; Crocker, J. C.; Frisken, B. J.; Weitz, D. A. Phys. ReV. E 2002, 65, art. no. 051505. (27) Gittes, F.; Schnurr, B.; Olmsted, P. D.; MacKintosh, F. C.; Schmidt, C. F. Phys. ReV. Lett. 1997, 79, 3286. (28) Schnurr, B.; Gittes, F.; MacKintosh, F. C.; Schmidt, C. F. Macromolecules 1997, 30, 7781. (29) Tuteja, A.; Mackay, M. E.; Narayanan, S.; Asokan, S.; Wong, M. S. Nano Lett. 2007, 7, 1276. (30) Brochard-Wyart, F.; de Gennes, P. G. Eur. Phys. J. E 2000, 1, 93.
Nonlinear Microrheology
whereas nanoparticles of different sizes can diffuse through the material via different mechanisms that do not require entanglements to relax. Second, the probe might interact physically or chemically to alter the local material microstructural environment (e.g., electrostatic or steric interactions may deplete a region around the probe, or chemical interactions may bind the probe to the material or cross link the surrounding network17-19,31). Such measurements are inherently sensitive to the altered material, giving apparent properties that depend on the physical and chemical identity of the probe, unlike macroscopic measurements. Typically, if multiple measurements with probes of different size or composition yield the same results, then one can have confidence that the bulk rheology is being measured. Third, inertial effects arise at extreme frequencies (ω J xµ/a2F in viscous fluids), just as in macrorheology, when oscillatory shear waves are established around the probe instead of quasi-steady deformations.4,18,27,32 In fact, the micrometer length scales relevant for microrheology enable much higher frequency measurements than in macroscopic rheometers.4,11 Fourth, the strain field around an oscillating probe is not viscometric (shear-only) but rather contains both shear and extensional components. At low frequencies in viscoelastic gels, for example, fluid can freely drain from the mesh, effectively decoupling the two18,27 and causing micro/macro disagreement. No analog occurs in (viscometric, shear-only) macroscopic rheometry. We have thus far discussed only passive microrheology. A colloidal probe may also be actively forced within the material, as is particularly useful when Brownian motion is too weak to yield a measurable response (e.g., for large probes or stiff materials). Common techniques involve magnetic field gradients33,34 and optical tweezers35,36 or polarized light20-23 and even the molecular motor kinesin37 to pull or torque probes through materials. We have thus far discussed only linear microrheology and how it can be used to recover linear viscoelastic behavior. Given the importance of nonlinear rheological properties in material processing and quality control, and the tremendous academic and industrial interest in passive microrheology, it is natural to ask whether one might adapt microrheology to measure nonlinear material properties. Of course, passive microrheology is by nature performed in equilibrium and thus is incapable of probing nonlinear, non-equilibrium material properties. Active microrheology, however, need not be done gently and therefore may well be capable of driving the material microstructure out of equilibrium, thus probing its nonlinear response. In general, active microrheology is a small-scale analog of the classic falling-ball (or towed-ball) viscometry technique.2,38-40 Nonlinear force-velocity curves have been measured for probes pulled through colloidal suspensions using optical tweezers36 and magnetic forces.34 Furst and co-workers explicitly interpreted their experiments microrheologically using a generalized (31) Chae, B. S.; Furst, E. M. Langmuir 2005, 21, 3084. (32) Atakhorrami, M.; Koenderink, G. H.; Schmidt, C. F.; MacKintosh, F. C. Phys. ReV. Lett. 2005, 95. (33) Ziemann, F.; Radler, J.; Sackmann, E. Biophys. J. 1994, 66, 2210. (34) Habdas, P.; Schaar, D.; Levitt, A. C.; Weeks, E. R. Europhys. Lett. 2004, 67, 477. (35) Lugowski, R.; Kolodziejczyk, B.; Kawata, Y. Opt. Commun. 2002, 202, 1. (36) Meyer, A. Marshall, A.; Bush, B. G.; Furst, E. M. J. Rheol. 2006, 50, 77. (37) Holzwarth, G.; Bonin, K.; Hill, D. B. Biophys. J. 2002, 82, 1784. (38) Milliken, W. J.; Mondy, L. A.; Gottlieb, M.; Graham, A. L.; Powell, R. L. Physicochem. Hydrodyn. 1989, 11, 341. (39) Brenner, H.; Graham, A. L.; Abbott, J. R.; Mondy, L. A. Int. J. Multiphase Flow 1990, 16, 579. (40) Reardon, P. T.; Graham, A. L.; Abbott, J. R.; Brenner, H. Phys. Fluids 2005, 17.
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Stokes drag (GSD) to relate the applied velocity U* to the measured force F* via
η* )
F* 6πaU*
(5)
by analogy with the GSER assumed in passive microrheology.36 (Starred variables will here refer to microrheological quantities.) Their microrheological data showed shear thinning that agreed fairly well with bulk, macroscopic rheological measurements. Also, colloidal probes can be externally torqued20-23 and interpreted via
η* )
τ* 8πa3Ω*
(6)
where Ω* and τ* are the probe rotation and torque, respectively. Unfortunately, however, the fundamental nature of nonlinear rheology involves a microstructure that may be significantly out of equilibrium, and the fluctuation-dissipation that underpins passive microrheology will no longer hold. We and others have made initial forays into the theory,41,42 specifically focusing upon the retarding force exerted on the probe by direct probe-material collisions in dilute colloidal suspensions. A new theory must thus be developed for nonlinear microrheologyswhether meaningful information can be thus obtained at all and how to interpret it if so. We know from passive microrheology a variety of conditions under which the GSER is violated and micro and macro do not agree. Many of these violations have counterparts in active microrheology. Additional complications arise in nonlinear microrheology that have no passive analog. Our central goal here is to identify and elucidate various issues and challenges that arise in the interpretation of nonlinear microrheology. We focus here on general concepts rather than detailed computations for specific systems. To pursue such an understanding, we pose several “theoretical experiments” in which we take a simple theoretical “material” whose (macro)rheology is known or computable, compute the response of such a material in a theoretical nonlinear microrheology experiment, use the GSD (eq 5) to infer a “microviscosity” η*, and compare this against the (macro)viscosity that we input at the beginning. Whether these models correspond to a real materials is of secondary importance for now; internal consistency is our main goal. As such, we will introduce the simplest model material possible that will illustrate each concept or point of interest. We will present detailed results for specific model systems in future work. We will start with the best case scenario: a model material (homogeneous, continuum, no probe-material interactions, no distinction between shear and extension) that exhibits none of the features that are known to invalidate the GSER for passive microrheology. Even in this scenario, we will see that the simple GSD (eq 5) does not faithfully reproduce the macroscopic rheology. We will identify how and why it fails and how far one can go to fix it. Our initial approach takes the constitutive relation between material stress and strain rate as prespecified and effectively instantaneous. Real materials, however, do not respond instantaneously to an applied stress. We thus move on to discuss a specific systemsa dilute colliodal suspensionswhere we can explicitly treat the microstructure itself, derive the resulting (41) Squires, T. M.; Brady, J. F. Phys. Fluids 2005, 17, 073101. (42) Khair, S.; Brady, J. F. J. Fluid Mech. 2006, 557, 73.
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stresses, and compute the retarding force on the probe. A variety of issues that complicate nonlinear microrheology then arise. Direct probe-material interactions must be distinguished from intrinsic bulk rheological stresses, just as in passive microrheology. Furthermore, material elements in nonlinear microrheology may be subjected to straining fields that are unsteady in the Lagrangian sense. This effect has no analog in passive microrheology and may prevent the material from having enough time to fully develop the (deformed) microstructure that gives rise to the analogous macrorheological response. We close with a brief discussion of active and nonlinear microrheology and with a few thoughts on modifying the basic technique in response to the issues raised here.
assumed to do so instantaneously. We define the leading-order velocity u0 on the probe surface to be given by the probe velocity so that the non-Newtonian correction u1 vanishes on the surface, again by definition. To solve this problem, one needs to choose a functional form (constitutive relation) for σ, solve a set of coupled, nonlinear partial differential equations for the flow field u ) u0 + u1, use this flow to compute the stress tensor σ(u), and then compute the force upon the probe via
F* )
σ ) σ0 + σ1
(7)
(8)
or (equivalently) the dissipation via
Nonlinear Microrheology in the Continuum Limit We begin with the best case scenario, where we deliberatley exclude effects that are known to invalidate the GSER in passive microrheology. We work with a model material wherein the material response is not altered by direct physical (collisions) or chemical interactions between the probe and the material and a probe that is large enough for the material to behave as homogeneous, continuum, and incompressible. Even in this case, we will see that the GSD (eq 5) is not generally appropriate. There is no fundamental reason to expect that it should; conventional falling-ball viscometry is viewed as an “index” for viscosity rather than a precision measurement.2 In this continuum limit, there is nothing inherently micro about the technique; rather, the problem to be solved involves the flow of a non-Newtonian fluid past a solid sphere, for which extensive literature exists. Generally, and not surprisingly, the drag force is not proportional to simply the product of the macroscopic, shear-rate-dependent viscosity and the probe velocity,43-45 as is assumed by the GSD. A variety of complications arise: a spatially inhomogeneous rate-of-strain field gives rise to material elements in a variety of states, anisotropic normal stresses (normal stress differences) exist in addition to viscous shear stresses, and strongly nonlinear non-Newtonian flows exhibit instabilities and flow separation that can match or exceed high Reynolds number flows in their complexity. However, because the nonlinear force/velocity relation arises because of non-Newtonian rheology, one might hope that these properties could somehow be extracted. We demonstrate that one can recover the shear-dependent viscosity from active microrheology, at least in principle, under certain (gentle) conditions. Nonlinear microrheology experiments measure the force F* required to pull a probe with velocity U* through such a material. Here, we run the corresponding thought experiment: starting with a given stress/strain relation σ, we compute the drag force F* on a probe as a function of pulling speed U*, interpet the results in terms of a microviscosity η* via the GSD (eq 5), and then compare with the input constitutive relation σ. To explore some of the complications that arise in a simple, clear, and somewhat general fashion, we consider a probe that is driven through a fluid whose stress tensor
∫ nˆ ‚σ dA
F*‚U* )
∫ σ:e dV
1 2
(9)
This is, in general, a difficult and tedious task. Brunn, however, gives a remarkably general equation for the dissipation due to a sphere moving slowly in such a slightly viscoelastic fluid46
F*‚U* )
∫ (σ0(e0) + σ1(e0)):e0 dV + O(2)
1 2
(10)
which gives the force F* accurate to O() without requiring one to solve for u1 or even to choose a particular σ1. A brief derivation is reproduced in Appendix A. It is straightforward to compute an apparent microviscosity η* using the GSD (eq 5) by dividing by the force required to pull the sphere through the corresponding Newtonian fluid (i.e., where σ1 ) 0), obtaining
F*‚U* ≡ η* ) η0 + 2 2 6πaU
∫ σ1:e0 dV ∫ e0:e0 dV
(11)
In particular, Brunn considered a second-order fluid, which incorporates the first non-Newtonian effects (e.g., normal stress effects) in rheologically slow flows and to which a variety of more complicated constitutive relations reduce in the weak flow limit.47 Brunn showed that non-Newtonian effects do not affect the drag on a purely translating sphere to the level of approximation he considered (although non-Newtonian effects do influence the force and torque on a sphere that simultaneously translates and rotates). Caswell and Schwarz explicitly computed the first normal stress effects upon the drag, which come in at a higher order of approximation.48 Here we consider an extremely simple model material whose viscosity changes with shear rate but does not exhibit appreciable elastic or normal stress effects. How realistic the model material we choose is of secondary importance; our primary motivation and approach is to theoretically “perform” and interpret thought experiments in order to identify and explore issues that will arise in the interpretation of real experiments. We thus consider a slightly generalized Newtonian fluid whose viscosity
η ) η0 + η1(γ˘ )
(12)
is nearly Newtonian (i.e., σ0 ) -pI + η0e0, where e0 ) ∇u0 + (∇u0)T is the shear rate tensor) but contains a small, nonNewtonian component σ1. Because is small, σ1 depends exclusively on the leading-order Newtonian flow e0 and is further
has a small component η1 with arbitrary functional dependence on strain rate γ˘ ) (∑ij eijeij)1/2. We are assuming that the material’s viscosity makes no distinction between shear, extensional, and mixed flows. In this case, the microviscosity η* obtained via eq
(43) Lunsmann, W. J.; Genieser, L.; Armstrong, R. C.; Brown, R. A. J. NonNewton Fluid Mech. 1993, 48, 63. (44) Phan-Thien, N.; Fan, X. J. J. Non-Newtonian Fluid Mech. 2002, 105, 131. (45) Gottlieb, M. J. Non-Newtonian Fluid Mech. 1979, 6, 97.
(46) Brunn, P. Rheol. Acta 1976, 15, 163. (47) Larson, R. G. ConstitutiVe Equations for Polymer Melts and Solutions; Butterworths: Boston, 1988. (48) Caswell, B.; Schwarz, W. H. J. Fluid Mech. 1962, 13, 417.
Nonlinear Microrheology
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Table 1. Prefactors (Equation 16) for Power-Law Viscosity η˜ /1 Obtained from Active Microrheology, Given a Macroviscosity of η˜ 1 ) Γn a exponent n γ˘ max A1 A2 A3 A4
translating sphere (TS)
translating bubble (TB)
rotating sphere (RS)
3U/x2a 0.210 0.092 0.056 0.037
x6U/a 0.250 0.120 0.071 0.048
3x2Ω 0.442 0.267 0.184 0.137
a Three different methods of forcing are presented: a translating sphere and bubble and a rotating sphere.
η* ) η0 +
γ˘ 02 dV
η˜ *(Γ) ) 1 +
∫
∫ γ˜˘ 02 dV˜
(14)
(15)
Because η˜ D is a function of Γ and γ˘ 20 is positive definite, all coefficients of a power series in Γ of η˜ D must vanish, implying η˜ D ≡ 0. We thus conclude that eq 14 is indeed invertible, at least for slightly generalized Newtonian fluids. Were this not the case, different macroviscosities could yield the same microviscosity. We now illustrate with specific material examples of η˜ 1 as well as various probes (translating spheres, translating bubbles, and rotating spheres). The simplest example, a fluid with a slight power-law viscosity η˜ 1 ) Γn, is recovered microrheologically as
η˜ /1
) AnΓ
n
∫ γ˜˘ 02+n dV˜ where An ) ∫ γ˜˘ 02 d V˜
γ˜˘ RS ) r˜ -3 sin θ, γ˘ RS max ) 3x2Ω
(20)
∞
∞
∑BnΓn and η˜ /1 ≈ n)1 ∑BnAnΓn n)1
(21)
1 η˜ 1 ) Θ(γ˘ - γ˘ c) ) Θ γ˜˘ Γ
(
)
(22)
where Θ is a step function. For Γ < 1, nowhere does the material thicken, and one recovers η˜ /1(Γ < 1) ) 0. When Γ > 1, some fraction of the material thickens, as seen in Figure 1. Equation 14 can be evaluated exactly for TB and RS forcing:
)1η˜ /TB 1
) η˜ /RS 1
6 1 + 3 1/2 5Γ 5Γ
(23)
(4Γ2 - 1)xΓ2 - 1 - 3Γ2 cos-1
(Γ1)
4Γ3
(24)
Although all three probes do show the onset of shear thickening at the correct Γ, they approach the “true” value very slowly. The reason is geometric: the fluid thickens until γ˘ 0 ≈ 1/Γ, which occurs at r˜ ≈ Γ1/2 (r˜ ≈ Γ1/3) for translating (rotating) probes. The integrand scales like Γ-2 over a volume of V ≈ Γ3/2 (Γ) for translations (rotations), leaving a Γ-1/2 (Γ-1) decay. Such slow approaches to the true rheology are to be expected in nonlinear microrheology whenever a material has abrupt rheological features. One could imagine, however, that the GSD might do fairly well with materials whose viscosity changes sufficiently slowly. More generally, how different is the (scaled) microviscosity η˜ /1(Γ) from the true viscosity η˜ 1 of the material? Assuming that η˜ 1(Γ) varies somewhat smoothly with Γ, we expand η˜ 1 in a Taylor series around its value at the probe surface:
|
∂η˜ 1 + ... η˜ 1(Γγ˜˘ 0) ≈ η˜ 1(Γ) + (γ˜˘ 0 - 1)Γ ∂Γ Γ Inserting this into eq 14, we obtain
η˜ /1(Γ)
(
∫ γ˜˘ 03 dV˜ |Γ ∫ γ˜˘ 02 dV˜ - 1
∂η˜ 1 ≈ η˜ 1(Γ) + Γ ∂Γ
)
(25)
+ O(Γ2η˜ ′′1(Γ))
(16)
(26)
For the translating sphere, for example, Table 1 gives
The functional forms of η˜ /1 and η˜ 1 are identical (both vary like Γn), but their prefactors differ. Table 1 gives coefficients An for translating sphere (TS), translating bubble (TB), and rotating sphere (RS) probes with strain rates of
γ˜˘ TS ) (r˜ -8 + (2r˜ -8 - 6r˜ -6 + 3r˜ -4)cos θ)1/2
(19)
η˜ 1 ≈
(13)
An obvious question is whether one can recover η˜ 1 uniquely from η˜ /1. If not, then different macroviscosities could yield the same microviscosity η˜ /1. This would imply in turn that their difference η˜ D would obey
∫ η˜ D(Γγ˜˘ 0) γ˜˘ 02 dV˜ ) 0
x6U γ˜˘ TB ) r˜ -2 cos θ, γ˘ TB max ) a
because An can be considered to be known. We now turn to a material with an abrupt change in viscosity,
where γ˘ 0 is the strain rate for the (unperturbed) Stokes flow. Thus the non-Newtonian component of the microviscosity η/1 is given by a weighted average of η1. Only when η1 is constant over the integration volume (e.g., in the linear response limit) does the GSD recover the correct viscosity. One can show, however, that this relation can be inverted uniquelysnumerically in general, and exactly in at least one case (Appendix B). We scale lengths by the probe size a, η1 by η0, and γ˘ 0 by the maximum (Stokes) strain rate γ˘ max and denote dimensionless variables with tildes. We consider γ˘ in η1 to be parametrized by a characteristic strain rate γ˘ c to give a dimensionless version of the pulling speed, Γ ) γ˘ max/γ˘ c, analogous to a Weissenberg number. With this notation, eq 13 becomes
η˜ 1(Γγ˜˘ 0) γ˜˘ 02 dV ˜
(18)
In all cases, An * 1 and η* differs from the true viscosity for power-law fluids. However, general viscosities η˜ 1 can be recovered from η˜ /1 by expanding each as a power series in Γ
11 does not equal the macroscopically measured viscosity η but is rather related to it via
∫ η1(γ˘ 0) γ˘ 02 dV
γ˘ TS max ) 3Ux2a
(17)
η/1(Γ) ≈ η1(Γ) - 0.8Γη′1(Γ) + O(Γ2η˜ ′′1)
(27)
Thus if η˜ 1 changes slowly enough (i.e., Γη˜ ′1 , 1 or d(ln η)/d(ln γ˘ ) , 1), the strain rates sampled around the probe drive a nonNewtonian viscosity η˜ 1(Γγ˜˘ 0) that varies slowly enough around the probe that η˜ /1(Γ) ≈ η˜ 1(Γ).
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track collisions and interactions between particles, introducing significant new physics and a non-Newtonian component to the suspension’s response. This effort was largely pioneered by Batchelor50-52 and developed by many others, as described by Russel et al.53 Significantly, the microstructural state of the suspension develops an anisotropic component in response to flow, which can be described statistically by a pair distribution function gbb(r1 - r2) that gives the probability of finding one “bath” colloid at r2 given that another is at r1. The pair microstructure function gbb(r) is itself a function of how hard the material is sheared. Competition occurs between convection with the shear, which tends to drive the microstructure out of equilibrium, and diffusion, which tends to heal the structure. This competition is reflected in the dimensionless Peclet number
Figure 1. Microviscosities obtained from a fluid with abruptly shear-thickening viscosity η1 using a rotating sphere (RS), translating sphere (TS), and translating bubble (TB). Although shear-thickening begins at the correct strain rate Γ, the microviscosities approach the macroviscosity only slowly. (Inset) The inhomogeneous rate-ofstrain field around the probe.
This raises the issue of whether properties measured in active microrheology are linear or nonlinear when the simple GSD can be trusted for the inversion techniques outlined above. We have shown in general that the simplest and most straightforward GSD (eq 5) is not appropriate to recover macrorheologically measured rate-dependent viscosities from active microrheological measurements. Instead, the microviscosity so interpreted represents a weighted average of the viscosities that are established in the inhomogeneous straining field around the probe. As for falling-ball viscometry, the GSD provides an index for viscosity rather than a precise measurement.2 We demonstrated for weakly non-Newtonian flows that the microviscosity rather fairly represents the macroviscosity as long as the latter does not change too rapidly. If, however, the material rheology changes abruptly, the the microviscosity approaches the macroviscosity very gradually. What has been omitted in the above approach? In addition to the obvious assumptions of homogeneity, incompressibility, and the validity of the continuum approximation, we assumed the stress to be an instantaneous function of strain rate. Of course, the behavior of a true non-Newtonian material depends upon its microstructural state, which in turn depends upon the strain history that it has experienced. Therefore, we now treat a simple model material whose microstructural state can be explicitly calculated and the stress response thus elucidated. Specifically, we will examine a dilute suspension of colloidal spheres of radius b, through which a spherical probe of radius a is driven with velocity U*. We begin with a brief overview of theoretical techniques developed for the rheology of dilute colloidal suspensions because our treatment largely modifies this approach. The (macro)rheology of colloidal suspensions has enjoyed more than a century of study, going back to Einstein’s celebrated calculation of the effective viscosity η ) η0(1 + 5φ/2) of a dilute suspension of rigid spheres,49 where φ is the volume fraction of the suspension. At this level of approximation, colloids act individually: each moves and rotates with the fluid but cannot shear as the fluid does, thus exerting a stress back on the fluid. The additional particle stress is given by the stresslet (point stress) exerted by each colloid times the number density of colloids, leading to an O(φ) correction. To compute the O(φ2) correction requires one to (49) Einstein, A. Ann. Phys. (Leipzig) 1905, 17, 549.
PeB )
γ˘ b2 Db
(28)
where Db and b are the diffusivity and radius of bath particles. Computing gbb(r) is generally quite difficult because it involves nontrivial hydrodynamic interactions and interparticle forces. A crude, but not unreasonable, approximation involves treating interparticle repulsions as hard sphere interactions that are enforced at a distance bHS that is greater than the actual (hydrodynamic) radius of the particle by an interaction distance LF (so that bHS ) b + LF). This is meant to mimic the characteristic range of interaction for electrostatic or steric repulsions without accounting for the detailed interaction.53 A more severe approximation then involves taking the limit b f 0, where the intercolloidal repulsion becomes freely draining, effectively eliminating hydrodynamic interactions between particles. In this case, particles interact only at contact, leading to a particle stress
〈σp〉 ) -4n2b3kBT I rˆ rˆ g(r ) rc) dΩ ≈ φ2FM(Pe) (29) that varies like φ2 because two colloids are required for a collision. While this “no-hydro” limit may seem a dubious and excessive approximation, it recovers the shear-thinning portion of colloidal rheology surprisingly well while being significantly simpler to solve.54,55 With this backdrop, Squires and Brady posed an analogous problem for active microrheology41 and computed the probebath pair distribution function gpb(r) in the case where both the probe and bath particles were freely draining. In this no-hydro limit, the relative flux between bath and probe is given by
j ) D∇gpb - U*gpb
(30)
where the bath particle is advected with the uniform probe velocity U*. Conservation of probability gives a Smoluchowski equation for gpb(r)
∂gpb ) D∇2gpb - U*‚∇gpb ∂t
(31)
subject to a no-flux condition at contact
nˆ ‚j|r)aHS+bHS
(32)
(50) Batchelor, G. K. J. Fluid Mech. 1970, 41, 545. (51) Batchelor, G. K.; Green, J. T. J. Fluid Mech. 1972, 56, 401. (52) Batchelor, G. K. J. Fluid Mech. 1977, 83, 97. (53) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, U.K., 1989. (54) Brady, J. F.; Morris, J. F. J. Fluid Mech. 1997, 348, 103. (55) Bergenholtz, J.; Brady, J. F.; Vicic, M. J. Fluid Mech. 2002, 456, 239.
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The retarding force on the probe is given by an expression analogous to eq 29
∆η/D 3 a2φ ≈ η0 4π b Pe
( ) I cos θg
pb(r
) 1; Pe) dΩ ≈ φFµ(Pe) (33)
and was interpreted in terms of microviscosity via the GSD (eq 5). The scaled viscosity functions FM(Pe) and Fµ(Pe) from eqs 29 and 33 for the micro- and macro-problems were then compared and found to agree semiquantitatively. Khair and Brady then extended these results by incorporating finite hydrodynamic interactions,42 as had been done for macrorheology,55 and recovered shear thickening when hydrodynamic radii could approach closely enough for lubrication forces to become significant. Qualitative features in the (nonequilibrium) microstructure gpb were found to agree with directly visualized microstructures in nonlinear microrheology experiments.36 Carpen and Brady performed Brownian dynamics simulations with nohydro suspensions with volume fractions of up to 55%, verifying the scale-up arguments given in the no-hydro theory.41 All seemed consistent; for this model system, properly scaled microviscosities closely resembled their macroscopic analogs. However, this theory tells far from the whole story. Whereas direct probe-bath interactions will certainly play an important role in the interpretation of nonlinear microrheology, a variety of issues were effectively masked by the problems that were solved and the nature of the comparisons that were made. Most significantly, the non-Newtonian correction in (macro)rheology (eq 29) comes in at O(φ2), whereas its analog enters at O(φ) in the microrheological problem (eq 33). This reflects the fundamental nature of the problems solved: in the macro problem, any pair of colloidal particles may interact to give the nonNewtonian stress; because two are required, their contribution comes in at O(φ2). In the microrheological problem, however, one of two is always the probe and thus holds a privileged position. As such, every collision it has with individual bath particles results in a retarding force, which eq 33 then interprets as an O(φ) microviscosity. Although Fµ(Pe) and FM(Pe) look quite similar, it was only with an intimate knowledge of colloidal physics that such a comparison could even be made.41 No such knowledge will be available for general materials, necessitating a more generally applicable theory. Furthermore, the retarding force due to probe-bath interactions depends on the nature and size of the probe itself so that the corresponding apparent microviscosity would not represent a material property. We have thus far presented two somewhat disjointed approaches to this problemsan explicit calculation of the microstructural deformation around the probe and a continuum approach. In fact, these two approaches represent two general classes of stresses in the system that force the probesone involving direct probe-bath interactions and one involving stresses within the bulk material itself (Figure 2). To capture the macroscopic rheology using microrheological means, we introduce a new model system that includes those processes that are directly analogous to macroscopic rheology: the bulk stresses within the material as the probe is driven through it. This introduces a variety of rich and difficult problems, which we will address in future work in more detail. For now, we will emphasize general considerations for the colloidal problem and how these will influence strategies for nonlinear microrheology more generally.
Figure 2. Sources of stress in viscometry measurements of colloidal suspensions in (A) a (macro)rheometer and (B) in nonlinear microrheology. In both systems, an O(φ) Einstein contribution ∆ηE (i) arises because colloids cannot shear with the fluid and an O(φ2) bulk contribution ∆ηB arises when two bath colloids collide (ii), giving rise to statistical anisotropy in the pair distribution function. The bulk contribution is non-Newtonian, typically reducing from one plateau at high Pe to another at low Pe. A third source of stress arises in nonlinear microrheology when the probe collides directly with a bath colloid (iii), giving an O(φ) direct contribution ∆η/D. No macrorheological analog exists.
Figure 3. Definition sketch for the theoretical model material discussed in the text. A large probe of radius a is pulled with velocity U* through a dilute suspension of bath colloids or radius b. Probe and bath colloids collide via hard-sphere interactions at a distance r ) a(1 + ) + b ≈ a(1 + ), and bath colloids are advected along streamlines defined by the (Stokes) flow around the probe. Three sources of stress are examined: (a) direct probe-bath interactions, (b) the Einstein correction, and (c) bath-bath collisions, which also involve hard-sphere interactions and neglect hydrodynamic interactions altogether.
Nonlinear Microrheology with Large Probes: a New Model System We now seek to blend the two approaches that we have discussed so far: what happens when probes get large yet the microstructure itself is explicitly treated? That is, when instead of taking the stress to be a given function of the local fluid strain rate do we actually compute the response of the material? As before, we will pose the simplest model system that will allow us to address both direct probe-bath interactions as well as bulk stresses. Specifically, we will treat a probe particle with radius a that is significantly larger than bath particles with radius b, as in Figure 3. We will continue to use an excluded annulus model and work in the limit where hydrodynamic interactions are negligible between bath particles (i.e., as has been computed for the macrorheology problem55). That is to say, the interaction length scale LF (as set by the electrostatic screening length) is large compared with bath colloids. However, LF need not be large compared to the probe itself, which behaves as an excluded annulus as
aHS ) a + LF ) a(1 + )
(34)
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Squires
(
because LF , ah. However, the bath particles (radius b) may be small compared to LF
bHS ) b + LF
(35)
and will generally be approximated as freely draining. The material itself (i.e., the colloidal suspension) involves freely draining hard-sphere collisions between bath particles, by analogy with simple models for macro-54,55 and microrheology.41 In general, the mobilities bb, diffusivities Db, and advection velocities of bath particles will depend upon their location relative to the probe and will require involved hydrodynamic calculations. However, we assume that bath particles are small enough that their mobilities and diffusivities are given by their bulk values Db ) kBTb ) kBT/6πηbh. Furthermore, unlike our previous microrheological model, the hydrodynamic flow around the probe now matters, and bath particles are small enough that they are simply advected around the probe with the local advection velocity ua(rb). Finally, we assume the probe to be pulled steadily and with constant velocity U0 (rather than constant force) and simply note that the two modes of forcing do not necessarily give the same result.40,41,56 As with previous work,41 the absence of hydrodynamic interactions and the hard-sphere nature of the interactions give rise to a particularly simple expressions for the force exerted by colliding bath particles upon the probe (eq 33) and the particle stress due to a colliding pair of bath particles (eq 29). We will first discuss direct probe-bath effects and demonstrate that they clearly depend upon the nature of the probe. In passive microrheology, probe-dependent measurements are taken as an indication that direct probe-bath interactions are interfering with true bulk measurements of the material’s rheology. We then consider bulk stresses. We start with the Einstein correction, which is also modified by direct interactions. Finally, we will discuss microstructural deformations in the bulk involving collisions between bath particles, which is the direct analog of the relevant non-Newtonian stress in macrorheology. We begin with the direct probe-material interactions in the large-probe limit, where we solve the steady convection-diffusion equation
∇2gpb ) PeDu‚∇gpb
(36)
Here we have non-dimensionalized lengths with the probe radius a and have introduced the direct Peclet number
U*a PeD ) Db
(37)
relating convective to diffusive transport of the bath particles. The hard-sphere interaction between probe and bath gives rise to a simple no-flux boundary condition at contact, rc ≈ 1 + ,
rˆ ‚(∇gpb)|rc ) PeDrˆ ‚ugpb|rc
(
)
1 3 3 + ≈ 2 2λ 2λ3 2
(56) Almog, Y.; Brenner, H. Phys. Fluids 1997, 9, 16.
(41)
and
λ)1+
(42)
Finally, we enforce a no-disturbance condition
gpb(r f ∞) f 1
(43)
far from the probe. The above problem is straightforward conceptually but not mathematically. As in the case of uniform flow for freely draining probes,41 diffusion dominates at low PeD, whereas boundary layers form at high PeD. However, the inhomogeneous flow field complicates the full solution. Because our focus lies not with the detailed solution for this particular model system but rather how this model system illuminates general considerations, we will examine the low- and high-PeD limits and discuss their main features. In the limit PeD , 1, a regular perturbation expansion for the microstructure in powers of PeD gives
cos θ + O(PeD2) 2 2r
gpb ) 1 + PeDm(1 + )
(44)
for a pair-distribution function at contact
gpb ) 1 +
PeD m(1 + )cos θ + O(PeD2) 2
(45)
and thus a direct microstructural contribution to the retarding force, found using eq 33
∆F φ a 2 ≈ m(1 + ) F0 2b
()
(46)
When the interaction range is small compared to the probe (i.e., as f 0), the retarding force vanishes like
() ( )
LF 2 ∆F 3 a 2 2 ≈ φ ≈ φ F0 4 b b
(47)
The retarding force is smaller than for a freely draining probe41 by a factor of 2, reflecting the reduced number of streamlines that bring bath particles into direct collision. As well, note that the direct interaction at low PeD somewhat surprisingly does not depend on probe size (although PeD itself does). In the high-PeD limit, however, the physics are quite different. The governing equation for the pair correlation function becomes, to leading order,
u‚∇gpb ) (39)
where
m(λ) ) 1 -
)
1 3 3 ≈ 4λ 4λ3 2
(38)
Here the advection velocity u at contact is
u‚ rˆ |r)1+ ) m(1 + )cos θrˆ +n(1 + )sin θθˆ
n(λ) ) 1 -
(40)
1 2 ∇ gpb ≈ O(Pe-1) Pe
(48)
so that gpb is essentially constant along streamlines and is thus 1 almost everywhere. Streamlines that do collide with the probe drive bath particles into a thin boundary layer on the leading edge of the probe (i.e., when rˆ ‚u < 0 at r ) 1) that can be treated as locally planar. No particles contact the downstream face of the probe, by contrast, giving gpb ) 0 downstream. Balancing convection normal to contact with diffusion on the upstream face yields a local equation for the boundary layer
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Langmuir, Vol. 24, No. 4, 2008 1155
where ζ is a coordinate directed radially away from the surface. The pair distribution function is thus given approximately by
macrorheology, and it scales differently with φ than does the macroviscosity. Indeed, in the low-PeD limit active microrheology reduces to the zero-frequency limit of passive microrheology, whereupon direct probe-material interactions are generally viewed as artifacts.
gpb(ζ, θ) ≈ 1 + f(θ) exp[-m(λ)PeDζ cos θ]
Bulk Contribution: Einstein Correction
1 ∂2 g ∂g m(λ) cos θ ) ∂ζ PeD ∂ζ2
(49)
(50)
with a contact value of
gpb(rc, θ) ) 1 + f(θ)
(51)
The boundary layer has thickness δ
δ≈
1 2 ≈ m(λ)PeD cos θ 32PeD cos θ
(52)
in the thin-annulus limit, meaning that PeD must be much greater than -2 for the boundary layer to be thin. To find the pair distribution at contact and thus the direct contribution to the stress on the probe, we solve for f(θ) using the flux-balance approach as in ref 41. Integrating eq 50 across the boundary layer, we obtain a surface probability density
σ(θ) )
f(θ) m(λ)Pe cos θ
(53)
and determine f(θ) by requiring that the total flux from the bulk into each portion of the boundary layer balance the surfaceadvected flux out of that portion of the boundary layer. The flux into the boundary layer between θ ) 0 and θ0 is given by
Qin ) 2π
∫0θ
0
[m(λ)cos θ]sin θ dθ ) πm(λ)sin2 θ0
(54)
whereas the outbound flux is
Qout ) 2πσn(λ)sin θ0
(55)
where m(λ) and n(λ) are the perpendicular and parallel flow velocities, respectively, at contact (eqs 40 and 41). Balancing Qin ) Qout gives
m(λ) σ(θ) ) 2n(λ)
(56)
to give a pair distribution at contact
m(λ)2 PeD cos θ gpb ≈ 1 + n(λ) 2
(57)
along the front half of the probe and gpb ) 0 on the “downwind” side. The direct interaction in the limit PeD . 1 is thus given by
∆F(PeD . 1) m(λ)2 φ a 2 3 3 a 2 L3 ≈ ≈ φ ≈ φ 2 (58) F0 8 b n(λ) 4 b ab
()
()
and is O() smaller than the low-PeD value. To reiterate, the direct probe-material interaction calculated here and previously41,42,57 exists and must be considered in any treatment of nonlinear microrheology. However, it has no macrorheological analog and depends upon the size and nature of the probe, its force thinning is more drastic than in macrorheology, it force thins at a much lower velocity than in (57) Khair, A. S.; Brady, J. F. J. Rheol. 2008, in press.
We have thus far accounted explicitly for the direct collisional interactions and their contribution to the apparent microvisocity as interpreted via the GSD (eq 5), extending our previous calculation41 to treat the large-probe limit in a more physically reasonable manner. As we have emphasized, however, this direct interaction has no analog in standard macroscopic rheology and depends entirely upon probe properties. Only stresses that arise in the bulk of the material itself find macrorheological analogs, and it is here that we now turn our attention. The simplest and most obvious is the microscopic analog of the Einstein correction, an O(φ) contribution that occurs when bath colloids cannot shear as the fluid would and exert a corresponding stress back upon the fluid. We will use eq 11, which gives the (normalized) force on a probe moving through a nearly Newtonian fluid, to compute the Einstein correction. The (dilute) volume fraction φ plays the role of the small parameter , and σ1 is given by the statistical stress exerted by a single bath particle in the local straining flow field e0 times the number density of particles n∞gpb(r). Using Batchelor and Green’s result58 for the stress exerted by a sphere in a linear flow ui ) Aijrj,
Sij ) -
20 πηa3Aij 3
(59)
in eq 11, the retarding force upon the probe is given by
∆η/E 5 )1+ φ η 2
∫ gpb(r)e:e dV ∫ e:e dV
(60)
In the low-PeD limit, gpb(r > 1 + ) ≈ 1 + O(PeD), and eq 60 gives an apparent Einstein contribution to the microviscosity ∞ γ˘ 2 dV 5 ∫|r|)1+ ) φ(1 - + O(2)) ∞ 2 ∫|r|)1 γ˘ dV 2
∆η/E 5 ) φ η 2
(61)
that reduces to the standard Einstein correction in the limit f 0. For finite , however, bath particles are kept from the excluded annulus, thereby reducing the effective Einstein correction. This system thus provides an explicit example where direct physical interactions between the probe and bath particles introduce an artifact that causes disagreement between micro and macro. The Einstein correction at high PeD is more involved computationally and simply changes the prefactor for the O() correction. Because our primary aim here is to elucidate general features and challenges, we will defer our detailed analysis to future work. For now, we simply note that the Einstein correction to the apparent microviscosity η/E is correctly recovered but that direct probe-bath interactions can modify it.
Bulk Microstructural Deformations Last, we examine the microrheological analog of the processes that give rise to the non-Newtonian macrorheology of a dilute colloidal suspension. As described above, the macro (58) Batchelor, G. K.; Green, J. T. J. Fluid Mech. 1972, 56, 375.
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problem requires one to compute the statistical pair distribution function for a pair of colloids in a shearing flow, from which the stress exerted by the pair upon the suspension is given by eq 29. Here, we describe the analogous approach, wherein we compute the statistical pair distribution function gbb(r1, r2) for two bath colloids in the (now inhomogeneous) straining flow around the probe. This problem is generally quite complicated, so for clarity we discuss only the simplest of limits: we now ignore the excluded annulus around the probe ( ) 0) and all probe-bath collisions, taking gpb ) 1 everywhere outside the probe. As in the macro problem, the pair distribution function obeys a Smoluchowski equation. The probability flux for each particle includes both diffusive and convective terms
j1 ) -D∇1gbb + u(r1)gbb
(62)
j2 ) -D∇2gbb + u(r2)gbb
(63)
where ∇1 and ∇2 denote gradients with respect to the position of particles 1 and 2 and u is the (Stokes) flow around the probe. Conservation of probability requires
∂gbb ) -∇1‚j1 - ∇2‚j2 ∂t
(64)
(
)
∂gbb b2 ) ∇r˜ 2 + 2∇c˜ 2 gbb ∂t˜ 4a
[
Notably, two Peclet numbers appear. The first is the direct Peclet number
PeD )
The freely draining hard-sphere bath-bath interactions impose a no-flux condition at contact,
nˆ 1‚(j2 - j1)||r1 - r2|)2b
(66)
We switch to center-of-mass and relative coordinates
c)
r 1 + r2 and r ) r1 - r2 2
1 1 ∇r1 ) ∇c + ∇r, ∇r2 ) ∇c - ∇r 2 2
(67) (68)
and expand u about the pair’s mean location c in a Taylor series
( 2r) ≈ u(c) + 2r‚A r r u(r ) ) u(c - ) ≈ u(c) - ‚A 2 2
u(r1) ) u c + 2
(69) (70)
where
A ) ∇ru|c
(71)
PeB )
∂gbb 1 ) Db ∇c2 + 2∇r2 gbb - [u(c)‚∇c + r‚A‚∇r]gbb ∂t 2
(
)
(72)
Non-dimensionalizing relative position r with b and center of mass position c by the probe size a, ua by U0, A by its magnitude ∑ij(AijAij)1/2, and t by b2/2Db gives
(74)
b2A 2Db
(75)
which is the precise analog of the Peclet number that is relevant for suspension macrorheology, wherein A is typically the macroscopic shear rate γ˘ . Here, by contrast, A is a local rate of strain, which scales as A ≈ U0/a. The two Peclet numbers are thus related via
PeB ≈
U0b2 b2 ≈ Pe aDb a2 D
(76)
In other words, when the probe is significantly larger than the bath particles, the Peclet number for direct interactions, PeD, is much larger than that for the bulk rheological processes PeB. This reflects the different length scales for each microstructural deformation: direct interactions form structure over the probe length a, whereas pair collisions in the bulk material occur over the bath size b. The pair problem (eq 73) for gbb involves a six-dimensional PDE whose boundary conditions are conceptually simple but geometrically complicated. The macrorheology problem is made much simpler by virtue of statistical homogeneity: shear rates A are spatially homogeneous so that all dependence upon c decouples from the problem. In microrheology, however, the strain rate A itself depends upon c, requiring the full sixdimensional PDE to be solved. As we have done throughout this work, we defer detailed computations for future work and focus upon general issues that arise in its solution and their implications for nonlinear microrheology. We start by treating the bath pair behavior for high and low PeB. The simplest is PeB , 1 for either low or high PeD. Our deliberate neglect of direct interactions means that the background single-particle concentration is uniform nearly everywhere outside the probe. Diffusion dominates (∇2r gbb ≈ 0), and the gentle collisions that arise from convection are balanced by slight relative diffusive gradients, as described by the no-flux condition
∂gbb ∂r
Using eq 68, the Smoluchowski equation then becomes
U0a Db
that we have used so far to describe direct probe-bath interactions; here it reflects the importance of advection of a pair along streamlines versus diffusion of the pair, over probe length scales. The second is the bulk Peclet number
giving
∂gbb ) Db(∇12 + ∇22)gbb - [u(r1)‚∇1 + u(r2)‚∇2]gbb ∂t (65)
]
b2 PeDu˜ ‚∇c˜ + PeBr˜‚A ˜ ‚∇r˜ gb (73) 2a2
|
r)2
≈ 2PeBrˆ ‚A‚ rˆ
(77)
The bath-bath pair distribution function is thus given approximately by
gbb ≈ 1 -
rˆ ‚A(c)‚ rˆ 32 PeB 3 r3
(78)
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Langmuir, Vol. 24, No. 4, 2008 1157
Figure 4. Strain rate |γ˘ | ) ∑ij(eijeij)1/2 around a translating sphere (a) is spatially inhomogeneous, being highest near the sphere surface and decaying with distance from the probe. Particle pairs advecting along streamlines (white) see a straining flow that is unsteady in the Lagrangian sense (b). This Lagrangian unsteadiness prevents even a quasi-steady nonlinear microstructure from being established at high Peclet numbers. The strain rate around a translating rod (c) is also spatially inhomogeneous; however, the (Lagrangian unsteady) strain rate experienced by pairs advecting along streamlines is relatively more constant, suggesting more faithful nonlinear microstructures at high Peclet numbers.
Evaluating eq 29 for the hard-sphere particle stress using gbb evaluated at contact (r ) 2) gives a local particle stress
〈σp(c)〉 ) -4n2b3kBT I rˆ rˆ gbb(r ) rc) dΩ
(79)
1 12 ) φ2η (A(c) + AT(c)) 5 2
(80)
[
]
The particle stress is symmetric and agrees exactly with the corresponding macrorheological system, where the shear rate A is homogeneous.55 For nonlinear microrheology, we can compute the retarding force (and the bulk microviscosity increment ∆η/B) using eq 11 to yield ∞ γ˘ 2 dV ∫|r|)1+ ∞ γ˘ 2 dV ∫|r|)1
∆η/B(PeB , 1) 12 2 ≈ φ η 5
(81)
Note that in the absence of direct interactions (i.e., in the limit of f 0), one recovers the macroviscosity increment exactly, as one should expect in the linear response limit. A small correction is introduced when is small but not zero, again arising from the direct probe-bath interactions that are not intrinsic to the material. However, for PeB . 1 the physics changes rather significantly. Rather than approaching the problem within a frame fixed upon the probe, we work in the center-of-mass frame for a particle pair as it is advected along its streamline (so that c(t) is given by solving the mean advection problem). Within this frame, the relative motion of the pair satisfies
∂gbb(r, c) ˜ (c(t))‚∇r˜ gbb ≈ ∇r˜ 2gbb - PeBr˜ ‚A δt˜
(82)
subject to the no-flux condition at contact (eq 77). Although the concentration field is steady in the probe frame, the advecting pair experiences a relative straining flow that is unsteady in the Lagrangian sense (Figure 4a,b). Collisional boundary layers form at high PeB that can be treated with a (significantly more involved) flux balance, as we did for the high-PeD direct interactions above. What can we say about the high-PeB microstructure without detailed computation? First, if the microstructure were steady and responded instantaneously to the local straining flow A(t), then eq 11 would give the retarding force, which could be
numerically inverted to obtain the true viscosity η(γ˘ ). However, the Lagrangian unsteadiness of the pair problem certainly calls into question the assumption that the material stress is an instantaneous function of e. How long will it take for the microstructure to reach its steadystate value? Typically, one requires an O(1) accumulated strain γ in order to develop a fully nonlinear material microstructure. Here, (quasi)steady state requires the probability that is brought into the boundary layer be advected a distance of O(b) along the contact surface to the point where the boundary layer separates and enters the bulk. Advection velocities are O(bA), so advection times are τa ≈ A-1 in order for the pair to respond to a new local straining flow. Quasi-steadiness requires the strain rate to remain essentially constant for times significantly longer than τa. During this advection time τa, the probe moves a distance ∆c ≈ U0/A-1 ≈ O(a). This is significant: by the time the microstructure would reach steady state in the probe’s straining flow, the pair would have moved a good fraction of the way past the probe. In other words, in the high-PeB limit, particle pairs do not have enough time to fully develop the nonlinear microstructure that would be obtained at a comparable PeB in the (steady) macrorheology problem. As such, the pair stresses may differ significantly from their quasi-steady-state counterparts, and the corresponding microviscosity would be lower than its macrorheological analog.
Discussion and Conclusions We have presented here several theoretical hurdles that nonlinear microrheology must face before its results can be confidently (and successfuly) compared with macrorheology. Issues include the inhomogeneous and nonviscometric nature of the straining flow, direct probe-material interactions, and the Lagrangian unsteadiness of the straining flow from the standpoint of material elements advected past the probe. We will now recap the main results and put them into perspective. Assuming that the material behaves as a perfect continuum whose stress is an instantaneous function of the strain rate and that there are no direct probe-material interactions, we derived an expression (eq 11) for the apparent microviscosity η* that one would infer from the generalized Stokes drag (eq 5). This microviscosity is not identical to the (input) macroviscosity but rather is a sort of weighted average of the range of viscosities probed in the inhomogeneous straining flow around the probe. In principle, one could invert this relation to obtain the true (macro)viscosity η from the (measured) microviscosity η*. However, a number of complications arise for real materials, as we explored using a large probe driven through a dilute suspension of small colloids. First, direct interactions between the probe and bath colloids give rise to a retarding force that would be interpreted as making a direct contribution ∆η/D to the apparent microviscosity η*. This is problematic in that this direct contribution depends upon the probe characteristics (e.g., size and interaction range), whereas the material’s true viscosity is a property that is intrinsic to the material and independent of the measurement technique. The closest analogy to these direct interactions is slip at rheometer walls. From the standpoint of developing nonlinear microrheology as a small-sample stand-in for large-strain macrorheology, one needs to minimize and extract the direct contributions. One strategy for doing so involves using large probes, since material elements are increasingly advected around increasingly large probes rather than being brought into direct contact. Second, the Peclet number for direct interactions, PeD, is larger than that for bulk microstructural processes, PeB, by a factor of a2/b2. In the particular colloidal system studied here, two distinct force-thinning steps could be anticipated: one
1158 Langmuir, Vol. 24, No. 4, 2008
around PeD ≈ O(1), where a boundary layer forms around the leading edge of the probe, and another around PeB ≈ O(1), where shear thinning begins to occur in the bulk of the material around the probe. Ideally, when faced with two such features, one could identify the first as arising from direct interactions and thus know that no such rheological feature would appear in a macrorheological setting. Experiments with different-sized probes would help to identify those features that depend upon probe size. We examined two sources of micro-rheological stress within the bulk of the material: the Einstein correction and the pair interactions between two bath colloids. Both have exact analogs in macro-rheology, and both contributions find explicit and quantitative micro-macro agreement in the low-Peclet limit, albeit possibly modified by direct interactions. The same is not true for the non-Newtonian (pair) processes in the high-Peclet limit: each particle pair sees a time-dependent straining field that changes appreciably before the pair microstructure has a chance to develop. Thus, although the rheological features will appear at the appropriate PeB, the high-PeB features will reflect nonlinear microstructure that is not fully developed and thus an apparent microviscosity contribution η/B that is not in agreement with the macroscopic contribution ηB at the equivalent PeB, even after accounting for the inhomogeneous straining field described by eq 11. In addition to raising these issues that complicate the interpetation of nonlinear microrheology, we hope to have broadened the theoretical underpinning of active and nonlinear microrheology. With the groundwork that we have laid here, one can treat a variety of model materials, not just colloids, armed with an idea of the issues to expect. Our intention here has been to raise and explore various issues that arise in active and nonlinear microrheology. On the basis of these results, we can hypothesize as to the conditions under which one might confidently interpret active microrheological experiments. As in passive microrheology, concerns about bulk rheology versus direct probe/material interactions can be addressed by looking at the way measurements depend on probe size if measurements for two different probe sizes agree when plotted against the appropriate shear rate (i.e., γ˘ ≈ U/a). Then one can be confident that one is measuring bulk properties. In particular, direct microstructural deformations around the probe depend upon the direct Peclet number PeD ) Ua/Db so that direct microstructural deformations form around differently sized probes at different velocities U ≈ Db/a. Bulk rheology depends upon a bulk Peclet or Weissenberg number τmU/a, where τm is a characteristic material time scale (here τm ) b2/Db) and thus will occur at the same PeB or WiB, irrespective of the size of the probe. As to when active microrheological measurements push into the nonlinear regime, various rules of thumb are already known. With oscillatory measurements, one can ensure that the response does not contain higher harmonics of the fundamental driving frequency. One can also simply change the amplitude and check that the response varies linearly. Regarding expressly nonlinear microrheology, we have briefly discussed how to invert the measured drag to get the shear-dependent viscosity, although we did so only in the weakly non-Newtonian limit. How general will this approach be? First, it is significant to note that the inhomogeneous flow around the probe has the effect of averaging, and thus smoothing, the measured microviscosity to the extent that even perfectly abrupt transitions are measured as smooth transitions. Conversely, any abrupt change in the experimentally measured retarding force almost certainly indicates an abrupt change in the character of the flow rather than the rheological
Squires
behavior itself, which is akin to boundary-layer separation in inertial flows. Furthermore, Caswell and Schwartz48 and Brunn46 demonstrated cases in which normal stress differences give rise to retarding forces upon the probe, which in the present context would be misinterpreted as shear-dependent viscosities. Brunn suggested additional single-sphere experiments (e.g., spheres that rotate and translate about different axes simultaneously) that are sensitive to the different normal stress functions46 and thus may tease apart these effects. Additionally, experiments with multiple probes (i.e., two-point nonlinear microrheology) will be sensitive to normal stress differences, as has been explored in the non-Newtonian fluids context59,60 and suggested in the active microrheology context.41,61 Finally, we close with a few comments on potentially beneficial modifications to the basic technique, motivated directly by the issues that we have explored above. To most faithfully reproduce the nonlinear macrorheology, one must minimize the effects of direct probe-material interactions and Lagrangian unsteadiness. One way to do so is to pull a needle, rather than a sphere, through the material. The cross section for direct collisions is correspondingly smaller than for a sphere; furthermore, the flow along the sides of the needle is largely viscometric and steady (Figure 4). In future work, we will pursue these ideas in concert with more detailed computations of the model systems described here. Acknowledgment. I gratefully acknowledge partial support of the NSF CBET-0730270, the NSF Mathematical Sciences Postdoctoral Fellowship (DMS-0202550), and the Lee A. Dubridge Prize Postdoctoral Fellowship at Caltech, where this work was initiated. I am pleased to acknowledge helpful discussions with Eric Furst, John Brady, Ryan Deput, and Aditya Khair and am grateful to Carl Meinhart for generating the COMSOL plots in Figure 4.
Appendix A: Retarding Forces as Functions of Stress Fields One can show rather generally that the dissipation due to a body moving through a fluid is given by
F*‚U*(or τ*‚Ω*) )
∫ σ:e dV
1 2
(83)
where σ is the stress tensor, e is the shear rate tensor, and F* and U* (or τ* and Ω*) are the force and velocity (or torque and angular velocity), respectively, of the moving body. Using eq 7, the dissipation is given to O() by
F*‚U* )
∫ (σ0 + σ1):e0 dV + ∫ e1:σ0 dV
1 2
(84)
In the second integral, the isotropic (p0) component of σ0 vanishes upon contraction with (traceless) e1. Expanding e1 and using the divergence theorem gives
∫ e1:σ0 dV ) 2η0 ∫ nˆ ‚e0‚u1 dA - 2η0 ∫ u1‚∇‚e0 dV
(85)
The surface integral vanishes because u1 ) 0 on the body and at infinity. Because η0∇‚e0 ) ∇p0, the volume integral becomes -2∫ ∇‚(p0u1) dV, which also vanishes with the divergence theorem. We thus arrive at (59) Joseph, D. D.; Feng, J. J. Non-Newtonian Fluid Mech. 1996, 64, 299. (60) Phillips, R. J. J. Fluid Mech. 1996, 315, 345. (61) Khair, A. S.; Brady, J. F. Proc. R. Soc. London, Ser. A 2007, 463, 223.
Nonlinear Microrheology
F*‚U* )
Langmuir, Vol. 24, No. 4, 2008 1159
∫ (σ0 + σ1):e0 dV + O(2)
1 2
(86)
where Γa ) (b2/a2)Γb to enforce F/a ) F/b ≡ F*. The limit a f b yields the surface-averaged macroviscosity
Significantly, only the Stokes flow u0 around the probe is required to obtain the dissipation to O(). An analogous equation for second-order fluids was found by Brunn.46
Appendix B: Inversion Equations for Nonlinear Microrheology in the Continuum Limit Rotating probes reproduce the macroviscosity more closely than translating probes because the faster spatial decay of γ˘ weights the average closer to the probe. Taking this strategy to its logical extreme (a strain rate that decays infinitely quickly) would yield a microviscosity averaged strictly over the probe surface. In fact, this can be realized. The TB and RS modes are special in that the flows for differently sized probes are identical if their forces/torques are the same (i.e., aUa ) bUb and a3Ωa ) b3Ωb). Subtracting the dissipation integrals (eq 14) at the appropriate Γa and Γb thus yields an integral over the difference in probe volumes because the integrands are identical outside of the larger probe. In the TB mode,
η˜ /TB 2πη0 η˜ /TB 1 (Γb) 1 (Γa) ) b a (F*)2
|r|)b η1(γ˘ 2)γ˘ 2 dV ∫|r|)a
(87)
∂η˜ /TB 3 1 ) 3 + η˜ /TB 1 ∂Γ Γ
∫0Γ η˜ 1(ξ)ξ2 dξ
2Γ
(88)
where ξ ) Γ cos θ. This can be inverted to give a remarkable relation between the micro- and macroviscosities
η˜ 1(Γ) )
[(
∂η˜ /TB 1 ∂ Γ3 1 2Γ + η˜ /TB 1 ∂Γ Γ2 ∂Γ 3
)]
(89)
One can verify eq 89 for the step-function and power-law viscosities considered above. The analogous surface-averaged viscosity relation for RS forcing
∂(η˜ /RS 3 1 Γ) ) ∂Γ 2
∫0π η˜ 1(Γ sin θ) sin3 θ dθ
does not appear to have such a simple inversion. LA7023692
(90)
