Toward a nonequilibrium Stokes-Einstein relation via active microrheology of hydrodynamically interacting colloidal dispersions

Accepted Manuscript Toward a nonequilibrium Stokes-Einstein relation via active microrheology of hydrodynamically interacting colloidal dispersions Henry C.W. Chu, Roseanna N. Zia PII: DOI: Reference:

S0021-9797(18)31496-6 https://doi.org/10.1016/j.jcis.2018.12.055 YJCIS 24432

To appear in:

Journal of Colloid and Interface Science

Received Date: Revised Date: Accepted Date:

16 September 2018 13 December 2018 14 December 2018

Please cite this article as: H.C.W. Chu, R.N. Zia, Toward a nonequilibrium Stokes-Einstein relation via active microrheology of hydrodynamically interacting colloidal dispersions, Journal of Colloid and Interface Science (2018), doi: https://doi.org/10.1016/j.jcis.2018.12.055

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Toward a nonequilibrium Stokes-Einstein relation via active microrheology of hydrodynamically interacting colloidal dispersions Henry C. W. Chu Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14850, United States

Roseanna N. Zia∗ Department of Chemical Engineering, Stanford University, Stanford, CA 94302, United States

Abstract We derive a theoretical model for the nonequilibrium stress in a flowing colloidal suspension by tracking the motion of a single embedded probe. While Stokes-Einstein relations connect passive, observable diffusion of a colloidal particle to properties of the suspending medium, they are limited to linear response. Actively forcing a probe through a suspension produces nonequilibrium stress that at steady state can be related directly to observable probe motion utilizing an equation of motion rather than an equation of state, giving a nonequilibrium Stokes-Einstein relation [J. Rheol., 2012, 56, 1175-1208]. Here that freely-draining theory is expanded to account for hydrodynamic interactions. To do so, we construct an effective hydrodynamic resistance tensor, through which the particle flux is projected to give the advective and diffusive components of a Cauchy momentum balance. The resultant phenomenological relation between suspension stress, viscosity and diffusivity is a generalized nonequilibrium Stokes-Einstein relation. The phenomenological model is compared with the statistical mechanics theory for dilute suspensions as well as dynamic simulation at finite concentration which show good agreement, indicating that the suspension stress, viscosity, and force-induced diffusion in a flowing colloidal dispersion can be obtained simply by tracking the motion of a single Brownian probe. Keywords: Colloids; Brownian motion; microrheology; hydrodynamics; Stokes-Einstein; suspensions; stress; osmotic pressure

1. Introduction The fluctuation-dissipation theorem governs various physical processes at thermodynamic equilibrium, e.g. Brownian motion, Johnson-Nyquist noise, thermal radiation: common among them is a balance between the dissipation of spontaneous system fluctuations. An early attempt to relate fluctuation and dissipation ∗ Corresponding

author Email address: [email protected] (Roseanna N. Zia )

Preprint submitted to Journal of LATEX Templates

December 21, 2018

5

dates back to the work of Einstein [1], who considered sedimentation of a dilute suspension of non-interacting hard-spheres. Appealing to the natural balance between sedimentation and the opposing gradient diffusion it produces, he connected observable particle motion — the diffusivity Da — to the viscosity of the fluid, η. The leap in Einstein’s theory was the assumption of equilibrium, which integrated the thermal energy, kT , into the celebrated Stokes-Einstein relation, D a = kT M a , relating the diffusivity Da of a sphere of radius

10

a to its hydrodynamic mobility, M a = I/6πηa, I is the isotropic tensor, k is Boltzmann’s constant, and T is the absolute temperature. Fluctuations of solvent molecules impart an average kinetic energy 3kT /2 to a particle, giving rise to its observable fluctuating motion. As a particle diffuses in a suspension, its fluctuating motion corresponds to temporary gradients in osmotic pressure — short-lived storage of thermal energy in the weakly distorted structure that is dissipated quickly back into the solvent.

15

The primary influence of particle and hydrodynamic interactions on the Stokes-Einstein relation is a change in the timescale over which fluctuations are dissipated. This can be appreciated by recovering the Stokes-Einstein relation via the Langevin equation [2, 3], central to which is the assumption that the random forcing obeys Gaussian statistics on the timescale of solvent-molecule motion. The autocorrelation of the Brownian force is set by the relaxation processes in the material. For a lone particle, decorrelation is

20

instantaneous because Stokes’ drag on a single particle never changes. But in a suspension, hydrodynamic interactions couple particles together and Brownian forces are thus correlated longer in time. While Einstein’s effort was aimed at obtaining Avogadro’s number and proof of the existence of the atom [1, 4], it produced modern-day passive microrheology, including generalizations of Stokes-Einstein relation for suspensions and other complex fluids at equilibrium. A review can be found in [5].

25

In the last decade, Einstein’s model for passive microrheology as been extended to probe the properties of flowing suspensions. Brady and co-workers pioneered this work by constructing the framework of active microrheology, in which a Brownian probe is driven by an external force through a complex medium, and changes in probe velocity are utilized directly to infer the so-called microviscosity via application of Stokes’ drag law [6, 7]. In addition, Zia and Brady utilized mean-square motion to measure force-induced diffusion

30

[8]. Both of these measurements could be obtained by two means. One can evaluating the statistics of probe motion, where the mean and mean-square displacement give the viscosity and diffusivity respectively, as would be measured in experiment or dynamic simulation. Alternatively, one can compute the mean and fluctuating motion by weighting changes in probe velocity with the microstructural distortion (the changes in particle configuration as the probe plows through the bath), as calculated in theoretical modeling or dynamic

35

simulation. This provides a connection between experiment, theory, and dynamic simulation. However, direct measurement of the suspension stress in active microrheology is not possible. Instead, in experiment, theory, and simulation, the stress must be computed from the distorted microstructure, a challenging task in experiments. In experiments, the microstructural distortion can be measured via fast or dynamic confocal

2

microscopy techniques [9, 10, 11]. But to image deep into a suspension, the flow must be stopped long 40

enough to scan the depth, but this allows Brownian motion to relax (smear out) structural distortions of interest. In other techniques, scanning takes place without stopping flow, but imaging is restricted to 10 to 20 particle lengths and channel-flow geometry. Wall-induced structure can then affect statistics. In micromechanical theories, the structure can be determined via solving the Smoluchowski equation for the particle distribution in the pair limit at equilibrium [12, 13, 14] and away from equilibrium [6, 7]. The

45

structure can also be measured directly in dynamic simulations [15, 8, 16, 17]. However, generalization of Einstein’s phenomenological connection of osmotic pressure to diffusion, and thus expansion of the StokesEinstein relation to active flow, required a different approach. Zia and Brady recognized that Einstein’s idea could be generalized to nonequilibrium systems if one modeled the system via an equation of motion rather than an equation of state [18]. The model provided a

50

direct connection between the stress tensor and quantities that can be obtained from simple probe statistics: from measuring just the displacement of the probe over time, one can compute the mean and mean-square displacement. Zia and Brady showed, for a freely draining system, that the two quantities, which represent fluctuation and dissipation, can be combined to give the suspension stress tensor. This approach revealed that gradients in stress drive not only diffusive flux but also advective flux, which they connected constitutively to

55

the force-induced diffusion and microviscosity. Overall, one can view the nonequilibrium stress as a measure of the time required for fluctuations to dissipate in a flowing suspensions. In the freely-draining limit modeled via their “nonequilibrium equation of state”, the configuration-independence of particle mobility gave a simple flow-rate dependence of this balance. Here we seek to understand how nonequilibrium fluctuation and dissipation change when suspension configuration evolves with the strength of hydrodynamic interactions

60

and strength of flow. The focus of this work is to derive a generalized nonequilibrium Stokes-Einstein relation between probe motion and steady-state suspension stress, viscosity, and diffusion, for a dispersion of hydrodynamically interacting colloidal particles with arbitrary ranges of hard-sphere repulsion. To complement the connection of pair interactions modeled in our theory to many-body interactions present in dense suspensions, we

65

conduct in this study Accelerated Stokesian Dynamics [19, 20, 21] simulations and compare to the theory. Accelerated Stokesian Dynamics simulations have long been used to study hydrodynamically interacting colloidal suspensions undergoing bulk rheometric flows. Brownian dynamics has been more recently used to study active microrheology of freely draining suspensions [15]. The present work is the first to report the suspension stress in active microrheology of hydrodynamically interacting particles using Stokesian dynam-

70

ics. While Nazockdast and Morris [17] recently measured the microviscosity via Stokesian dynamics, they could not compute the stress tensor or the osmotic pressure, because their model is traceless [22, 23]: the hydrodynamic functions required to obtain osmotic pressure and the full stress tensor were missing. The

3

goal of their work was instead to develop a microstructural theory to predict the microviscosity for a dense suspension, which required a collective diffusivity which inferred utilizing the phenomenological model of 75

Zia & Brady (2012). In this work, we present an Accelerated Stokesian Dynamics simulations framework to model active microrheology that fully models the hydrodynamic functions that include the trace of the stress couplings and thus enable us to directly measure the suspension stress. In the present study we also expand the survey from dilute to moderate concentrations to allow comparison to statistical mechanics theory and the new theory developed in Section 4

80

The remainder of this paper is organized as follows: in Section 2, the microrheology model system is presented. Next, in Section 3, we derive a phenomenologically modeled, nonequilibrium Stokes-Einstein relation connecting the suspension stress to nonequilibrium fluctuation and dissipation in a suspension undergoing active microrheological flow. The novel contribution here is the detailed accounting of arbitrary strengths of hydrodynamic interactions. This is achieved by development of an anisotropic effective resistance tensor,

85

constructed from the microviscosity [6, 7] and flow-induced diffusivity [8] to model the effect of hindered and fluctuating particle motion in the presence of hydrodynamics which, upon coupling with particle flux, constitutes the advective and diffusive components of a Cauchy momentum balance. Results are presented in Section 4, starting with the normal stresses, the first normal stress difference and osmotic pressure, predicted from the generalized nonequilibrium Stokes-Einstein relation and values obtained from statistical mechanics

90

theory [24, 25], for strong hydrodynamic interactions, varying flow from weak to strong. In Section 5, the predictions from theory are compared with results obtained by Stokesian dynamics simulations. A guide for experimental measurements is presented in Section 6. This study is concluded in Section 7 with a discussion. 2. Model system We consider a suspension of Nb hard-sphere colloidal bath particles of hydrodynamic radius a, dispersed

95

homogeneously in a volume V of incompressible Newtonian solvent of dynamic viscosity η and density ρ. A constant external force, F ext , drives a “probe” particle, also a hard-sphere particle of radius a, through

the suspension. The strength of fluid inertia relative to viscous forces defines a Reynolds number, Re = ρU a/η, where U is the characteristic velocity of the probe. The colloidal particles are small such that Re  1, and thus the fluid motion is governed by Stokes’ equations. The probe number density, na , is much 100

smaller than the number density of bath particles, nb . Probe motion deforms the suspension microstructure while Brownian motion of the bath particles acts to recover an equilibrium configuration. The degree of

microstructural distortion, and its influence of probe motion, is thus set by the strength of the probe forcing, F0 , relative to the Brownian restoring force, 2kT /ath , where k is Boltzmann’s constant, T is the absolute temperature and ath is the thermodynamic size of particles, defining a P´eclet number: P e = F0 /(2kT /ath ). 105

The active microrheology model system has been utilized extensively in prior works [5]. In this work, we

4

(b)

(a) y

a

ath

r

weak hydro

 =1



θ

 = 10-1  = 10-2

z

Fext

 = 103

strong hydro

 = 10-5

Figure 1: (a) The microrheology model system for equally-sized particles of hydrodynamic radius a and hard-sphere radius ath , defining the no-slip and no-flux surfaces, respectively. (b) Minimum approach distance for a range of interparticle repulsion; grey circles of size a are probe and bath particles; dashed circle around each particle is thermodynamic radius ath ; large dashed circle is minimum approach distance rmin .

model only steady-state behavior, corresponding to probe motion well after transient startup and prior to removal of the external force. Modeling the hydrodynamic and non-hydrodynamic physical forces between particles is an essential element of suspension mechanics, and they are interconnected: changing one changes the other. Hydrodynamic interactions can be strengthened or weakened by the introduction of particle surface features such as those utilized to stabilize a suspension, activating some transport processes such as Brownian drift, while suppressing others. Repulsive forces between particles can act over distances that shrink or grow with sample preparation or solvent chemistry, for example, separating the no-slip surface of the particle from its no-flux surface. The former sets its hydrodynamic size, a, and the latter sets its thermodynamic size, ath , as well as defining the closest distance two particles can come to one another. For equally sized particles, this minimum approach distance is twice their thermodynamic size, rmin = 2ath . The ‘excluded shell’ model [26] produces a simple framework to incorporate changes in the strength of hydrodynamic interactions by introducing a ‘contact’ surface at rmin = 2ath . When the thermodynamic surface extends far beyond the hydrodynamic surface, rmin  2a, hydrodynamic disturbance flows decay to zero well before they reach another particle, and the system is freely draining. When rmin = 2a, the two surfaces coincide and particles

can approach closely enough to experience full long-range hydrodynamic and close-range lubrication interactions. Between these two limits, the dimensionless repulsion range κ gives a measure of the strength of hydrodynamic interactions: κ≡

rmin − 2a ath − a = . 2a a

(1)

A conservative interparticle potential V (r) serves as a simplified model for electrostatic or steric repulsion. 5

Its gradient induces an interparticle force where, in the present hard-sphere model, particles exert no force on one another until their no-flux surfaces touch, r = 2ath , at which an infinite potential prevents closer approach: V (r) =

  ∞ if r ≤ rmin  0

(2)

if r > rmin .

For a conservative potential, the interparticle force F P β acting on a particle β can be obtained from the relation F P β = −kT ∇β V , where the derivative is taken with respect to the absolute position of the particle 110

β. 3. Generalized nonequilibrium Stokes-Einstein relation In passive microrheology, thermally driven displacements of a probe particle through a complex fluid can be related to properties of the embedding medium utilizing a generalized equilibrium Stokes-Einstein relation [27], including osmotic pressure [12, 13, 14]. Away from equilibrium, Zia and Brady [18] showed that both

115

advective and diffusive flux are required to expand the connection of osmotic pressure to observable probe motion. Recognizing that an equation of state model is limited to equilibrium, they showed that an equation of motion extends the model to strongly non-equilibrium flow. Utilizing Cauchy’s equation of motion to balance momentum on the probe phase, they argued that bath-particle flux – both advective and diffusive – produces the relevant forces. In their freely-draining model, the influence of hydrodynamic interactions on the

120

nonequilibrium Stokes-Einstein relation was neglected. However, our recent micromechanical study of stress in active microrheology showed that hydrodynamic interactions suppress nonequilibrium stress and osmotic pressure in a flowing suspension, motivating us to generalize the nonequilibrium Stokes-Einstein relation to include the effects of hydrodynamic interactions. The key element to address is the configuration-dependent hydrodynamic resistance tensor, R. The momentum balance in the Stokes flow regime reads [18], 0 = ∇ · hΣi + na hF ext i + na hF int i,

(3)

where the angle brackets denote an ensemble average over all permissible positions of the bath particles relative to the probe, and Σ is the stress tensor associated with the probes. Following experimental approaches, the average over the probe phase corresponds to a material through which many probes are driven by an external force. The probes are dilute and thus do not interact with one another, and are also dilute relative to the bath particles, na  nb , and thus constitute a homogeneous average. A detailed presentation of

the averaging, following Batchelor (1970), is given in Supporting Material A. The averaging procedure was also presented in our previously published work [18, 24]. Comprehensive comparison and reconciliation of 6

active microrheology and to traditional homogeneous flows can be found in [7, 18, 5]. In Eq. (3), F int is an interactive force between the probes and the surrounding material (solvent and bath particles), written constitutively as [18], na hF int i = −na hR · U i + ∇ · hf P i = −hR · ji + hf P i,

(4)

where R is an anisotropic tensor describing the resistance of the medium to the probes’ motion, j = na U is the probe flux, and ∇ · hf P i is an isotropic pressure flux arising from the entropic restriction due to the presence of bath particles. Substituting Eq. (4) into Eq. (3) gives,

0 = ∇ · hΣi + na hF ext i − hR · ji + hf P i.

(5)

Eq. (5) can be written as a linear combination of the equilibrium and nonequilibrium parts, denoted by the superscript eq and neq, respectively, neq ext 0 = ∇ · hΣeq i + ∇ · hΣneq i − h(R · j)eq )i − h(R · j)neq )i + hf eq i. P i + hf P i + na hF

(6)

Simplification of Eq. (6) can be made by considering the momentum balance at and away from equilibrium. In the former case, when external force is absent and particles are driven solely by particle-density-gradientdiffusion, F ext = 0 and ∇ · hΣeq i = 6 0, giving 0 = ∇ · hΣeq i − h(R · j)eq )i + hf eq P i;

(7)

whereas in the latter case, the flow-induced nonequilibrium flux h(R · j neq )i should contain both advective and diffusive contributions

0

0

0

0

h(R · j neq )i = hRneq · j neq i + hRneq · j neq ihRneq i · hj neq i + hRneq · j neq i,

(8)

where we make use of the definition that the average of a product is the sum of the product of the averages and the average of the fluctuations. In Eq. (8), the first term on the right-hand side corresponds to the mean advective motion of the probe, Rneq ≡ Radv and j neq ≡ j adv , and thus allows the dot-product to be 0

taken outside the ensemble average. The second term is the diffusive flux, Rdif f ≡ Rdif f and j neq ≡ j dif f . Combining Eqs. (6)-(8), the momentum balance becomes,

−∇ · hΣneq i = −hRdif f · j dif f i + na hF ext i − hRadv i · hj adv i + hf neq P i.

7

(9)

125

Thus far the momentum balance Eq. (9) is identical in form to the one for a freely draining system [18]. Hydrodynamic interactions affect advective flux [7] and diffusive flux [8, 28, 29] and the resistance tensors in Eq. (9), which differ substantially from the freely draining limit. We expand this model to include the effect of hydrodynamic interactions on the advective and diffusive resistance tensors, starting with the nonequilibrium diffusive flux, hRdif f ·j dif f i. The microstructure hinders

mean probe motion, and probe/bath particle encounters deflect the probe from its mean path. These nonequilibrium fluctuations are characterized by the flow-induced diffusivity D f low [8, 28, 29], which is the O(φb ) forced-induced diffusion of the probes, also known as the microdiffusivity (cf. Supporting Material B.2). The volume fraction of bath particles is defined as φb = 4πnb a3 /3. Here, following [18], we assume a Fickian form for the diffusive flux, j dif f = −D f low · ∇na .

(10)

The force-induced diffusion D neq changes qualitatively as hydrodynamic interactions become important, encoding the influence of hydrodynamic mobility on probe flux. Because diffusive flux is linear in the fluctuating velocity, the configuration dependence of the force arising from this term is captured by D f low : hRdif f · j dif f i = Ra · D f low · ∇na ,

(11)

where Ra ≡ 6πηaI is the characteristic diffusivity.

Next, we model the flux induced by the external force, na hF ext i [cf. Eq. (9)]. The presence of the

microstructure hinders mean probe motion, giving rise to an effective suspension viscosity higher than the

solvent viscosity. The effective viscosity changes with the degree of microstructural distortion which, in turn, depends on the flow strength and the range of entropic repulsion associated with the strength of hydrodynamic interactions [13, 14, 30, 31, 6, 7] (cf. Supporting Material B.1). The advective motion of the probes can be related to the configuration-dependent effective viscosity η ef f of a suspension via Stokes’ drag law [6, 7, 32] ext hF ext i ≡ hF ext = F i=F

and hF ext U i=

ηFef f Ra · hU i = RF · hU i, η

ef f ηU Ra · U = RU · U , η

(12)

(13)

where the subscripts F and U correspond to active microrheology operated under the fixed-force and fixedvelocity mode, respectively. In the fixed-force mode, a probe is driven by a constant external force regardless of the structure it encounters, and thus undergoes fluctuating motion. In contrast, in the fixed-velocity mode, the external force is the fluctuating quantity, and the probe cannot diffuse. In dilute suspensions where pair interactions dominate, the microviscosity η micro is defined as the O(φb ) coefficient of the effective

8

viscosity [6, 7]

η ef f = 1 + η micro φb + O(φ2b ). η

(14)

The direct externally forced advective flux na hF ext i can then be expressed in terms of the microviscosity as,
na hF ext i = na 1 + ηFmicro φb Ra · hU i.

(15)

In addition, the external force induces advective flux hRadv i · hj adv i associated with the mean motion of

the probe [cf. Eq. (9)]. Because the probe can change its speed in the fixed-force mode, we must account for this versus the fixed-velocity mode in the model [18]. Physically, the drag experienced by a probe in the fixed-force mode is lower because the probe is free to fluctuate when it encounters regions of high or low density, whereas in fixed-velocity mode, it must plow through the bath, sometimes demanding higher force to maintain its prescribed velocity. The difference in hydrodynamic resistance of these two modes is denoted as ˜ hRadv i = (RU − R),

(16)

ef f ˜ = R(˜ where RU = Ra (ηU /η) [cf. Eq. (13)], and R η /η) is the additional resistance experienced by a fixed-

velocity probe. The magnitude of the additional resistance must be proportional the the number of particle encounters, i.e. η˜/η = φb . However, the magnitude of particle flux depends not only on the frequency of particle encounters but also on their strength and duration [24]. The latter two factors are accounted for by relating the fixed-velocity and fixed-force microviscosity via the anisotropic tensor α   RU = Ra · I + (I + α)ηFmicro φb , with 

α  ⊥  α= 0  0

0 α⊥ 0

 Dkf low p(κ) f low 0 D⊥     0= 0    αk 0 

0

(17)

0

Dkf low

p(κ) Df low

0

0

a a xa 11 (x11 +y11 )

⊥

   .  

(18)

2

a a Here, xaij = xaij (2ath , κ) and yij = yij (2ath , κ) are the components of the hydrodynamic mobility function that

couples the force on particle j to the induced translational velocity of particle i, and they govern the motion of particle i and j along and transverse to their line of centers, respectively [33]. Because the probe is externally forced in the longitudinal direction, αk is set directly by its mobility during encounters with an upstream a particle, which will include both a longitudinal component, xa11 , and a transverse contribution, y11 , as a bath

particle moves around the probe. Here we utilize their values at thermodynamic contact, as they vary with the strength κ of hydrodynamic interactions. These mobility functions are readily available in the literature [33], requiring no measurement of the evolving structure of a suspension. Because probe displacements in the 9

transverse direction are purely diffusive, α⊥ is set by diffusive anisotropy. Recognizing that a probe driven at a fixed velocity permits no fluctuations, the additional drag in the perpendicular direction on a fixed-force probe relates to the ease with which a probe fluctuates in the direction parallel to the external force (Dkf low )

f low relative to that in the perpendicular direction (D⊥ ). The flow-induced diffusivity [8, 28, 29] parallel and

f low perpendicular to the line of the external force, Dkf low and D⊥ , are obtained by projecting the tensor in f low the corresponding directions, Dkf low = D f low : ez ez and D⊥ = D f low : ey ey , where ez and ey are the

unit vectors in the direction parallel and perpendicular to the external force, respectively (cf. Fig. 1). The

equilibrium diffusivity Deq is present regardless of the strength of flow. A geometrical factor for spheres gives p(κ) = (3/16π)Deq . Plots of αk and α⊥ are presented in Supporting Material B.2. Overall, the advective flux hRadv i · hj adv i is written constitutively as

  hRadv i · hj adv i = na I + (I + α)ηFmicro φb − φb · Ra · hU i.

(19)

The remaining term to be modeled in Eq. (9) is the nonequilibrium particle pressure hf neq P i. We follow Zia

and Brady [18] who recognized that isotropic osmotic pressure arises from diffusive particle fluctuations at equilibrium, similar to the way gas particles collide with the “wall” of a fictitious enclosing container, giving rise to pressure. Colloidal particles also exert a pressure on enclosing boundaries, the osmotic pressure. The thermal energy kT determines the strength of equilibrium fluctuations, thus the osmotic pressure goes up and down with increasing or decreasing temperature. Micromechanical results for the osmotic pressure show that a probe surrounded by bath particles experiences increased osmotic pressure owing to the reduction of accessible volume taken up by the bath particles (Supporting Material B.3 [18, 25]). Zia and Brady modeled nonequilibrium fluctuations as setting the effective suspension “temperature” associated with probe motion — a nonequilibrium flow-induced temperature. They further recognized that this effective temperature is proportional to the average of the squared probe velocity fluctuations, hU 0 U 0 i, and thus is proportional to the

force-induced diffusion, D f low [8, 29]. Finally, the pressure is a force density, f neq P , and as with the advective constitutive models is proportional to the resistance tensor Ra . Following their program, we propose the model hf neq P i=

1 tr(D f low )Ra · ∇na , 3

(20)

where tr(D f low ) is the trace of the force-induced diffusion tensor introduced above. Just as in equilibrium 130

systems, gradients in number density ∇na give rise to osmotic pressure and drive diffusive relaxation of these

temporary gradients; here, the nonequilibrium number-density gradient similarly smooths fluctuations. As noted by Zia and Brady, the assumption of an isotropic pressure (i.e. the trace of the diffusivity) is an approximation, scaling up the ‘temperature’ beyond kT , the strength of the relaxation of number density gradients. This constitutively models changes in probe mobility arising from nonequilibrium distortion of

10

135

the structure. Insertion of this definition, (20), along with the diffusive, force-induced, and advective flux, Eqs. (11), (15), and (19) into the expression relating stress to particle flux (9) gives − ∇ · hΣneq i = Ra · D f low · ∇na −
1 na φb αηFmicro − I · Ra · hU i + tr(D f low )Ra · ∇na . 3

(21)

Eq. (21) can be simplified by approximating the gradient of the probes number density by a Fickian scaling, following Zia and Brady [18]: ∇na ∼

j Df low

∼

na U na P eDa ∼ f low , f low D D a

(22)

where Df low = D f low /φb is the magnitude of the flow-induced diffusivity, and the relation for a dilute suspension [18], ∇ · hΣneq i =

∂hΣneq i hΣneq i · ∇na = · ∇na , ∂na na

(23)

the momentum balance reads,
D f low hΣneq i 1 − = 2I − ηFmicro α · + tr na kT Da 3

D f low Da

! I.

(24)

Rearrangements leads to the Stokes-Einstein form, D f low = −



Σneq +P na



· M neq ,

(25)

where P ≡ Ra · tr(D f low )I/3, and the nonequilibrium mobility tensor is given by, M neq = 2I − ηFmicro α


−1

· M a,

(26)

where M a = I/6πηa. As sought, the form in (25) is identical to that of Zia & Brady and the Stokes-Einstein relation. Taking into account the effect of hydrodynamic interactions, Eq. (25) is a generalized nonequilibrium Stokes-Einstein relation — a nonequilibrium fluctuation-dissipation relation, where the imparted energy to 140

a suspension, represented by Σneq and P , drives fluctuations D f low , and is dissipated back to the solvent precisely by viscous drag (M neq )−1 . In Section 4, we test this relation by comparing its prediction to results obtained from statistical mechanics theory. 4. Results The generalized nonequilibrium Stokes-Einstein relation, Eq. (24), expresses suspension stress as a balance

145

between nonequilibrium fluctuation and dissipation. To test this relation, in this section we compare its 11

(a) 1000

0.1 0.01

b neq

1

⇠ P e2



Stat. Mech. Phen. Model

10

⇠ P e2

0.0001 0.01

1



0.1

Stat. Mech. Phen. Model

102

0.01

10

0.001

10-1

1

103

0.001

10

>/ nakT

⇠ P e0.799

-<

>/ nakT

10

||

neq

100

||

b

100

-<

(b) 1000

⇠ Pe

-5

10-2

0.0001

0.1

1

10

100

0.01

0.1

1

Pe

10

100

Pe

Figure 2: Nonequilibrium parallel normal stress, hΣneq i, scaled by ideal osmotic pressure na kT and volume fraction of bath k particles φb , as a function of flow strength P e for (a) asymptotically weak (κ  1) and strong hydrodynamic interactions (κ  1), and (b) various intermediate strengths of hydrodynamic interactions. Solid lines: statistical mechanics model [18, 24], left-hand side of Eq. (27). Squares: phenomenological model, right-hand side of Eq. (27).

prediction [the right-hand side of Eq. (24)] with the statistical mechanics theory [the left-hand side of Eq. (24)] via four quantities: the suspension stress parallel and perpendicular to the line of the external force, the first normal stress difference, and the particle osmotic pressure, for the full range of flow strength and strength of hydrodynamic interactions, spanning four decades of P e and eight decades of κ. Details of 150

the suspension stress and its statistical mechanics theory are given in Supporting Material B.3. We begin with analyzing the normal stresses. 4.1. Normal stresses The suspension stress is a second-order tensor with six independent elements for an isotropic material. In microrheology, only the normal stresses, Σxx , Σyy and Σzz , are nonzero owing to the axisymmetric microstructure around the probe. The axisymmetric geometry also produces identical normal stresses along the orthogonal axes, Σyy = Σxx . This leaves only two relevant quantities: the normal stresses acting parallel and perpendicular to the direction of the external force (cf. Fig. 1), Σk ≡ Σzz and Σ⊥ ≡ Σyy = Σxx , which are

obtained by projecting the stress tensor onto the corresponding directions. The generalized nonequilibrium Stokes-Einstein relations for the normal stresses thence read
Dk 1 + D a φb 3

f low Dkf low + 2D⊥

f low
D⊥ hΣneq 1 ⊥ i − = 2 − α⊥ ηFmicro + na kT φb Da φb 3

f low Dkf low + 2D⊥

−

hΣneq k i

na kT φb

f low

= 2−

αk ηFmicro

!

D a φb

Da φb

,

(27)

! .

(28)

We test this phenomenological model first by examining the parallel normal stress in the asymptotic limit of weak and strong hydrodynamic interactions, κ  1 and κ  1 respectively. Prediction from the new 12

(a) 1000

⇠ Pe

100

b

b

100

0.01

⇠ P e2



Stat. Mech. Phen. Model

103

0.001

⇠ P e2

0.0001 0.01

1



0.1

neq

⇠ P e0.799

⊥

neq

0.1

10

>/ nakT

1

-<

>/ nakT

10

⊥

-<

(b) 1000

current work

10

0.1

0.01

10

0.001

10-1

1

Zia & Brady (2012)

-5

Stat. Mech. Phen. Model

102

10-2

0.0001

1

10

100

Pe

0.01

0.1

1

10

100

Pe

Figure 3: Nonequilibrium perpendicular normal stress, hΣneq ⊥ i, scaled by ideal osmotic pressure na kT and volume fraction of bath particles φb , as a function of flow strength P e for (a) asymptotically weak (κ  1) and strong hydrodynamic interactions (κ  1), and (b) various intermediate strengths of hydrodynamic interactions. Solid lines: statistical mechanics model [18, 24], left-hand side of Eq. (28). Squares: phenomenological model, right-hand side of Eq. (28) and [18] in the limit of weak hydrodynamics.

155

theory, the right-hand side of Eq. (27), is plotted in Fig. 2(a) as a function of the flow strength, P e, along with the results obtained from statistical mechanics model [18, 24], the left-hand side of Eq. (27). In the asymptotic limit of weak hydrodynamics, the phenomenological model gives excellent prediction of the stress for the entire range of flow strength and, in particular, recovering the low- and high-P e scalings in the asymptotic limits of weak and strong flow. In the opposite limit of strong hydrodynamics, the phenomenological model

160

captures the effects of hydrodynamic interactions accurately: hydrodynamics suppress normal stresses, where the reduction is quantitative and qualitative in the low- and high-P e regimes, respectively [24]. Parallel normal stresses of various intermediate strengths of hydrodynamic interactions are plotted in Fig. 2(b). The prediction by the new theory and statistical mechanics theory matches very well for all values of P e and κ. The comparison for the perpendicular normal stress is plotted in Fig. 3. The results obtained by the

165

present phenomenological model produce accurate predictions for strong and intermediate strength of hydrodynamics, with a small quantitative difference near equilibrium in the freely-draining limit. In that dual limit, we refer to the model by Zia and Brady [18], as shown in Fig. 3(a). The present model captures the monotonic decrease of the nonequilibrium stress with increasing strength of hydrodynamic interactions — a manifestation of the suppressive nature of hydrodynamics [24].

170

4.2. Normal stress differences Owing to the axisymmetry of the microstructure, the second normal stress difference is identically zero regardless of the strength of flow and hydrodynamic interactions (cf. Supporting Material B.3). In this section, we focus on testing the generalized nonequilibrium Stokes-Einstein relation via the first normal stress difference, defined as N1 ≡ Σzz − Σyy . The corresponding expression from the generalized nonequilibrium

13

3

 103 10-5

10

2

10

Zia & Brady (2012)

1

current work

-/ nakT

-/ nakT

b

100

(b) 10

Stat. Mech. Phen. Model

b

(a) 1000

1 0.1 0.01

⇠ P e2

0.001

0

10

-1

10

-2

10



-3

10

⇠ P e4 0.1

1

Stat. Mech. Phen. Model

102

-4

10

10

-5

1

-6

10-1

10

0.0001 0.01

10

10

10-2

-7

10 10

0.01

100

0.1

1

10

100

Pe

Pe

Figure 4: Nonequilibrium first normal stress difference, hN1neq i, scaled by ideal osmotic pressure na kT and volume fraction of bath particles φb , as a function of flow strength P e for (a) asymptotically weak (κ  1) and strong hydrodynamic interactions (κ  1), and (b) various intermediate strengths of hydrodynamic interactions. Solid lines: statistical mechanics model [18, 25], left-hand side of Eq. (29). Squares: phenomenological model, right-hand side of Eq. (29) and [18] in the limit of weak hydrodynamics. Dotted lines and filled squares have been multiplied by −1 to make it visible on the log-log plot.

Stokes-Einstein relation reads f low

−

f low
Dk
D⊥ hN1neq i = 2 − αk ηFmicro − 2 − α⊥ ηFmicro . na kT φb Da φb D a φb

(29)

Fig. 4(a) shows the first normal stress difference in the two asymptotic limits of strong hydrodynamic interactions (present study) and weak hydrodynamic interactions from Zia and Brady [18]. The generalized nonequilibrium Stokes-Einstein relation, the right-hand side of Eq. (29), is compared to statistical mechanics theory [18, 25], the left-hand side of Eq. (29). When forcing is very weak there is a small quantitative offset, 175

a small scalar multiple of O(P e4 ), which is very small: for say, P e = 0.001, this gives P e4 = 10−12 . In this regime of very strong Brownian motion, the noise of thermal fluctuations would be far stronger than this discrepancy. The same comparison was discussed by Zia and Brady [18]. In practice, measurements of smallP e normal stress differences are quite challenging because Brownian motion is so strong that the structure is nearly isotropic, hence the O(P e4 ) normal stress difference. Nonetheless the same scaling is achieved in

180

both the micromechanical and the new phenomenological model, showing that the phenomenological model for strong hydrodynamics recovers excellent agreement with the statistical mechanics theory. Moderate hydrodynamic interactions are also qualitatively well-accounted for in the new model, as shown in Fig. 4(b). The most important physical behavior that must be captured are the strong-flow (high-P e) and weak-flow (low-P e) scaling, and the sign reversal of N1 as P e grows from weak to strong. All three of these

185

qualitative behaviors are recovered for all values of κ, indicating that the phenomenological arguments used to construct the model capture the appropriate physics. Some quantitative difference is observed, however, as hydrodynamic interactions become weak. The sign change predicted by the new model occurs at higher P e than predicted by the statistical mechanics theory. The discrepancy arises because, as hydrodynamic 14

<

0.01

⇠ P e2



⇠ P e2

0.0001 0.01

1



0.1

Stat. Mech. Phen. Model

102

Stat. Mech. Phen. Model

103

0.001

b

⇠ P e0.799

10

neq

0.1

100

10

0.01

10

0.001

10-1

1

Zia & Brady (2012)

-5

0.1

<

b

1

neq

>/ nakT

100 10

(b) 1000

⇠ Pe

>/ nakT

(a) 1000

current work

10-2

0.0001 1

10

100

0.01

Pe

0.1

1

10

100

Pe

Figure 5: Nonequilibrium osmotic pressure, hΠneq i, scaled by ideal osmotic pressure na kT and volume fraction of bath particles φb , as a function of flow strength P e for (a) asymptotically weak (κ  1) and strong hydrodynamic interactions (κ  1), and (b) various intermediate strengths of hydrodynamic interactions. Solid lines: statistical mechanics model [18, 25], left-hand side of Eq. (30). Squares: phenomenological model, right-hand side of Eq. (30) and [18] in the limit of weak hydrodynamics.

interactions weaken, the stress is dominated by the interparticle stresslet, whose longitudinal and transverse 190

contributions cancel precisely. The constitutive modeling produces accurate P e-scaling but small numerical (quantitative) differences which we magnify on the log-log plot. This discrepancy worsens as hydrodynamic interactions become asymptotically weak, but in that limit, the freely-draining theory of Zia and Brady [18] can be utilized. Overall, the excellent qualitative and good quantitative agreement support the idea that single-particle

195

tracking provides an alternative to measuring suspension stress. 4.3. Osmotic pressure The osmotic pressure is defined as the negative one-third of the trace of the stress tensor, Π ≡ −I : Σ/3

and is non-zero away from equilibrium owing to development of asymmetric microstructure [34, 18, 24, 25]. It acts to expand or contract the particle phase to restore symmetry. Here we compare the prediction of the particle pressure from our phenomenological model, hΠneq i 1 = na kT φb 3

f low Dkf low + 2D⊥

Da φb

!

" × 3−

ηFmicro

f low αk Dkf low + 2α⊥ D⊥ f low Dkf low + 2D⊥

!# ,

(30)

with the values predicted by the statistical mechanics theory for a range of flow and hydrodynamic interactions strength. The particle osmotic pressure predicted by the new theory [the right-hand side of Eq. (30)] and the 200

micromechanical model (the left-hand side) is plotted in Fig. 5(a) in the limits of asymptotically weak and strong hydrodynamic interactions. In both limits, the prediction by the phenomenological model matches very well with the statistical mechanics theory. When flow is weak, the osmotic pressure grows quadratically

15

with P e regardless of the strength of hydrodynamic interactions. The pressure increases with flow strength and, when flow is strong, it scales as P e0.799 in the limit of strong hydrodynamics, and as P e when hydro205

dynamic interactions are weak. This scaling of P e emerges directly from the hydrodynamic functions that set how close particles can approach one another, which is captured by the distortion of the arrangement of particles — the non-equilibrium microstructure f around the probe. In the limit of strong hydrodynamics and strong probe forcing where this scaling emerges, the physical picture is as follows: the probe moves through the bath, accumulating particles on its upstream face in the diffusion/advection boundary layer. In

210

the equations governing the density of particles in the boundary layer, the advection term sets the relative motion of bath particles into this boundary layer, and relative motion is set by the hydrodynamic mobility functions. The value of these functions at contact is then carried through in the solution of the equations, giving density of particles in the boundary layer that grows with the flow strength, P eδ , but is slowed by hydrodynamic lubrication interactions, and thus δ is smaller than unity (its value when hydrodynamics do

215

not matter). A detailed derivation of the numerical values can be found in Khair and Brady [7] and Chu and Zia [24]. The predictions for other strengths of hydrodynamics are plotted in panel (b), and excellent agreements are also seen for the full range of P e, showing that the simple model recovers the results of Chu and Zia [25]. The agreement is both useful and satisfying. First, in prior work [6, 7, 8, 28, 29] only the scalar micro-

220

viscosity and flow-induced diffusivity could be inferred from probe motion. The present model broadens the scope of active microrheology such that the full stress tensor can be inferred from the mean (microviscosity) and mean-square motion (flow-induced diffusivity) of a single probe particle. This circumvents the tediousness and expense of imaging and calculating the statistics of all the bath particles for a statistical mechanics computation. The agreement is satisfying because it reinforces the connection between fluctuations that

225

tend to expand the particle phase and viscous drag that tends to hinder it. 5. Accelerated Stokesian Dynamics simulations Beyond the semi-dilute limit, three-body and higher-order hydrodynamic interactions must be modeled, making analytical treatment difficult.

1

The Stokesian Dynamics algorithm is one method for effectively

handling the two primary problems that arise: the infinitely coupled, many-body hydrodynamic interactions 230

that are long-ranged, and singular lubrication interactions near contact [19, 23, 21]. Accelerated Stokesian Dynamics simulations have been employed to investigate hydrodynamically interacting colloidal suspensions for various rheometric flows, including shear [37], extensional [38], and pressuredriven flow [39]. Brownian dynamics has been more recently used to study active microrheology of freely 1 Progress

is being made with concentrated theory. See e.g. [35, 36].

16

draining suspensions [15]. The present work is the first to report the suspension stress in active microrheol235

ogy of hydrodynamically interacting particles using Stokesian dynamics simulations. While Nazockdast and Morris [17] recently measured the microviscosity via Stokesian dynamics, they could not compute the stress tensor or the osmotic pressure, because their model is traceless [22, 23]: the hydrodynamic functions required to obtain osmotic pressure and the full stress tensor were missing. Thus, no results of the suspension stress were presented in [17]. The goal of their work was instead to develop a microstructural theory to predict

240

the microviscosity for a dense suspension for fixed-force and fixed-velocity modes. To do so, they needed a collective diffusivity which could not measured in their simulations, so instead they inferred it utilizing the phenomenological model of Zia & Brady (2012), but using the simplest version that omits the contribution of the viscosity. In this section, we present a modified Accelerated Stokesian Dynamics simulations framework to model active microrheology. Our framework fully models the hydrodynamic functions that include the

245

trace of the stress couplings and thus enabled us to directly measure the suspension stress. In the present study we also expand the survey from dilute to moderate concentrations to allow comparison to statistical mechanics theory and the new theory developed in Section 4. We briefly recapitulate the Accelerated Stokesian Dynamics simulations framework, and present the modifications for computing the suspension stress. Rather than computing a probability distribution describing an ensemble of positions utilized in statistical mechanics theory presented in Section 4, in simulation one computes the trajectory of each particle as it evolves over time, i.e. as particles move due to imposed flow and forces, Brownian motion, hydrodynamic interactions, and other interparticle forces. The N −particle Langevin equation conserves momentum on each of the particles m·

dU = F ext + F H + F B + F P , dt

(31)

where m is a generalized mass/moment of inertia tensor of dimension 6N ×6N and U is a vector of dimension

6N for the translational/rotational particle velocities relative to the solvent. The force/torque vectors F on 250

the right-hand side are of dimension 6N , comprising contributions from external, hydrodynamic, Brownian and interparticle forces. Integration of Eq. (31) over times long compared to the solvent-molecule time scale results in an N -particle displacement equation [37], where the external, hydrodynamic, Brownian and interparticle forces each produce a displacement as follows. The external force, F ext is applied only to the probe particle; for all other particles, F ext = 0. All particles are external-torque free. A hydrodynamic force, F H , arises on the surfaces of all particles due to the translation U and rotation Ω relative to the solvent and other particles. Here, the velocity of particles can arise directly from external forcing or can be induced by disturbance flows from other particles; these

17

interactions are encoded in the N -body resistance tensor RF U : F H = −RF U · U .

(32)

Collisions between the particle and surrounding solvent molecules produce a fluctuating force that, over time long compared to the inertial relaxation time scale of the particle give rise to stochastic displacements, both satisfying Gaussian statistics: F B = 0,

F B (0)F B (t) = 2kT RF U δ(t),

(33)

where the overbar denotes a noise averaging over times much longer than individual solvent collisions and 255

particle momentum relaxation, and δ(t) is the Dirac delta distribution. The N -body coupling produces a configuration-dependent Brownian force. The interparticle force, F P , represents a non-hydrodynamic, deterministic force, e.g. electrostatic repulsion or repulsive force as arises in the presence of surface asperities. We restrict our attention to hard-sphere exclusion, F P = kT rˆ δ(r − 2a) (r is the separation between particles and rˆ is the unit vector connecting

260

their centers) such that particle overlaps are prevented by an infinite potential at contact, and the particles exert no direct force on each other otherwise. However, since κ = 0 (cf. Fig. 1), lubrication forces prohibit particle contact, so the hard-sphere interparticle force has no effect on the evolution of particle configuration or rheology. Evolution of particle positions is obtained by integrating Eq. (31) over a time step ∆t large compared to the inertial relaxation time of the particle, τI , but small compared to the diffusive time of the particle τD = a2 /D over which the configuration changes. We non-dimensionalize the length scale by the particle hydrodynamic radius a. To accurately capture the particle dynamics in different flow regimes, the diffusive time a2 /D and advective time scales a/U are used when flow is weak and strong, respectively. The evolution equation of particle position in the weak-flow regime reads   ˆ ext ∆t + ∆xP + ∇ · R−1 ∆t + X(∆t), ∆x = Pˆe R−1 FU · F FU X = 0,

X(∆t)X(∆t) = 2R−1 F U ∆t;

(34a) (34b)

whereas in the strong forcing regime it is given by   ˆ ext ∆t + ∆xP + 1 ∇ · R−1 ∆t + 1 X(∆t), ∆x = R−1 FU · F FU 1/2 Pˆe Pˆe

(35)

ˆ ext = F ext /F0 is the unit vector pointing in the direction of the external where Pˆe = F0 /(kT /a), and F

18

probe force. In Eqs. (34a) and (35), the first term is the particle displacement induced by the external force; the second term by the interparticle force; the third term is the Brownian drift; and the fourth term is a random Brownian displacement which has a zero mean and covariance given by the inverse of the resistance tensor [cf. Eq. (34b)]. The evolution equations (34a)-(35) update the configuration at each time step. The suspension stress is computed directly from the particle configurations encoded in the resistance tensors, as an average over all permissible bath configurations [12, 14, 24, 25]: hΣi = −na kT I − na hrF P i + hΣiH = −na kT I − na hrF P i + na hSiH,ext + na hSiB + na hSiP,dis ,

(36)

where −na kT I is the ideal osmotic pressure, −na hrF P i is the (non-hydrodynamic) elastic stress, and the

components of the hydrodynamic stress hΣiH associated with the external probe forcing, Brownian motion

and interparticle force can be expressed in terms of the corresponding stresslets via hΣiH,ext = na hSiH,ext , hΣiB = na hSiB , and hΣiP,dis = na hSiP,dis . The angle brackets denote an average of the stress experi-

enced by the probe overall all permissible configurations of the surrounding bath, discussed in detail in [24] and summarized in Supporting Material A and Supporting Material B.3. The configuration-dependent hydrodynamic stresslets can further be expressed as S H,ext = −RSU · U ext − RSΩ · Ωext ,

(37)

S B = −kT ∇ · (RSU · M U F ) − kT ∇ · (RSΩ · M ΩF ) ,

(38)

S P,dis = −RSU · U P − RSΩ · ΩP ,

(39)

where the hydrodynamic resistance and mobility tensors RSU , RSΩ , M U F and M ΩF couple surface tractions on one particle to its own motion and the motion of other particles. As the probe travels through the suspension, it encounters many different configurations of surrounding particles, and corresponding stresslets can be averaged over probe travel (time) and simulations (realizations) because, as the probe travels through the bath, at steady state it continuously encounters new configurations. In the Stokesian Dynamics framework, the primary calculation is thence the grand resistance matrix,  RF U R= RSU

RF E RSE

 ,

(40)

where RSU governs the coupling between stresslet and particle motion, and RF E and RSE describe the 19

265

coupling between force/torque and flow rate of strain, and the coupling between stresslet and flow rate of strain, respectively. We expanded the original Stokesian Dynamics framework [22, 23] by implementing the hydrodynamic functions that give the trace of the stress tensor, viz the resistance tensors RSU and RSE in our model now have a trace, in contrast to prior models of ASD. The trace couples particle motion to the pressure moment on particle surfaces, yielding the sought-after non-equilibrium osmotic pressure [40, 41]. These functions are assembled into the grand resistance matrix that captures hydrodynamic interactions among particles in the entire simulation domain, comprising both near-field lubrication interactions, and infinitely reflected, far-field many-body interactions. For the reader interested in the construction of the grand resistance matrix, the details can be found in [22, 23, 21, 41]. Eq. (31) can now be written in a general form as

270

      0 U F B + F P + F ext   = −R ·   +  , S 0 −rF P

(41)

where straining flow is identically zero as prescribed by the rigidity of particles, and there is no imposed straining flow. At each time step, the particle configuration is utilized to construct the grand resistance matrix as well as the Brownian and interparticle forces. The particle motion U and stresslet S are then obtained upon solving Eq. (41), and the particle motion is further employed to update the configuration via Eqs. (34a)-(35) for the next time step. An average suspension stress hΣi is obtained by averaging over a set

275

of 200 simulations of 2.5 × 104 time steps each for each P e.

Figure 6 shows the results obtained here via dynamic simulation (κ = 0) for the normal stresses, first

normal stress difference, and the osmotic pressure, along with the results from the generalized nonequilibrium Stokes-Einstein relation and statistical mechanics theory, in the limit of strong hydrodynamics, κ  1.

Simulations with bath-particle volume fraction, φb = 0.05, are compared with the dilute phenomenological 280

theory and the statistical mechanics theory, showing very good agreement. The close agreement supports the validity of the nonequilibrium Stokes-Einstein relation, i.e. that suspension stress can be inferred from the mean (microviscosity) and mean-square motion (flow-induced diffusivity) of a single probe particle, without expensive dynamic simulations or detailed imaging of the microstructure. Higher concentrations can also be modeled by the theory. The normal stresses, first normal stress dif-

285

ference and osmotic pressure at volume fractions φb = 0.1 and φb = 0.2 are also presented in Fig. 6. All four quantities increase with increasing particle volume fraction which can be understood as follows. The suspension stress is the entropic energy density stored in a suspension [18, 16, 24]. As particle concentration increases, more frequent collisions produce more pronounced structural distortion, reducing configurational entropy. Interestingly, in the strong-flow regime, the P e-scaling of the stress changes with increasing φb .

290

Since the P e-dependence of the suspension stress in the dilute statistical mechanics results arises from the hydrodynamic functions [18, 24, 25], we predict that hydrodynamic functions recently developed for concen-

20

1000

1000

100

0.1

1

Pe

00

0.001

0.001

Pe

100 0.0001

10

0.01

100 1000 0.01

0.01

SD simulation

100

01 0.01

0.1

0.1

1

Pe

-/ nakT

01

φb = 0.05 = 0.05 φb = 0.1 = 0.1 φb = 0.2 φb = 0.3 = 0.2 φb = 0.4 = 0.3

φb φ φb φb 1φ = 0.4 b

b

01

10 b

1

0.1

Pe

0.01

1 0.1 0.01

10

0.001



Stat. Mech. Phen. Model

10-5 0.01

0.1

1

(d) 1000

0.1

100

SD simulation

SD simulation

100

0.01

0.1

0.1

1

Pe

Stat. Mech. Phen. Model

φb = 0.05 = 0.05 φb = 0.1 = 0.1 φb = 0.2 φb = 0.3 = 0.2 φb = 0.4 = 0.3

φb φ φb φb 1φ = 0.4 b

0.01

0.01

10

Pe

0.0001



100

1

100 0.0001

100

10

0.01

100

0.001

10

100

0.0001

10

0.001

10

10

100

Pe

SD simulation

1

Pe

0.1

1

Pe

0.001

10 1

0.1

1

(c) 1000

-<Σ||neq>/ nakTφb

0.0001

0.1

0.1

100 10-5

00

0.1

1000Mech. Phen. Model Stat.



0.001

10

0.01

neq

0.0001

10

φb = 0.05 = 0.05 φb = 0.1 = 0.1 φb = 0.2 φb = 0.3 = 0.2 φb = 0.4 = 0.3

φb φ φb φb 1φ = 0.4 b

10 b

⊥

0.1

1

0.01

10 b

b

>/ nakT

0.1

-<

0.01

neq

01

||

01

0.01

SD simulation

SD simulation

100

b

φb φ φb φb 1φ = 0.4 b

10 b

b

01

φb = 0.05 = 0.05 φb = 0.1 = 0.1 φb = 0.2 φb = 0.3 = 0.2 φb = 0.4 = 0.3

(b) 1000

0.1

>/ nakT

100

0.1

>/ nakT

SD simulation

1

-<Σ||neq>/ nakTφb

0.1

1

SD simulation

neq

(a) 1000

10

-<

1

10

100

-<Σ||neq>/ nakTφb

10

-<Σ||neq>/ nakTφb

00

1

0.1

Pe

<

00

0.01

10

100

10

100



0.001

10-5

0.0001

Stat. Mech. Phen. Model

10-5

0.0001

0.01

0.1

1

10

100

Pe

0.01

0.1

1

10

100

Pe

neq Figure 6: (a) parallel and (b) perpendicular normal stress, hΣneq i and hΣneq i, and ⊥ i, (c) first normal stress difference hN1 k neq (d) osmotic pressure hΠ i, scaled by ideal osmotic pressure na kT and volume fraction of bath particles φb , as a function of flow strength P e in the asymptotic limit of strong hydrodynamics (κ  1). Solid lines: statistical mechanics theory [24, 25], left-hand side of Eq. (24). Squares: phenomenological model, right-hand side of Eq. (24). Filled symbols: Accelerated Stokesian Dynamics simulations, φb = 0.05 (•), 0.1 (N) and 0.2 (H).

trated suspensions [35, 36] can be used to predict the strong-flow response in a concentrated suspension, and could lead to a scaling theory unifying the stress in dilute and concentrated suspensions. In Section 6, we discuss the experimental measurements required to apply the nonequilibrium Stokes-Einstein relation.

295

6. Experimental measurement The present theory opens a new avenue of experimental measurement. The suspension stress of a hydrodynamically interacting suspension can be obtained simply by tracking the mean (microviscosity) and mean-square motion (flow-induced diffusivity) of a single probe, in contrast to macroscopic rheometry where suspension stress can be measured directly via stress transducers [42, 43, 44, 45] or by measuring parti-

300

cle microstructure via fast or dynamic confocal microscopy [9, 10, 11]. Here we discuss how experimental measurements of these two quantities might be carried out. To obtain the microviscosity, the only measurement required is the total displacement of the probe motion

21

over time, from which one can compute the average speed as hU i = dhxi/dt, which is then utilized to evaluate the microviscosity via the Stokes’ drag law [46, 6, 7]

ηFmicro 6πηa = hU i. η F0

(42)

Here, the angle brackets signify an average over many realizations (many probes in one experiment, or multiple trajectories of a probe through the system), and F0 is the strength of a constant probe forcing. In Eq. (42), the subscript F indicates fixed-force active microrheology where the external force imposed on the 305

probe is fixed and the probe velocity is free to fluctuate. In fixed-velocity active microrheology [47, 48, 32], measurements are taken analogously, with the external velocity being held fixed and the external force being the fluctuating quantity.

2

The average can be taken over time for steady state motion or over repeated

experiments. To obtain the flow-induced diffusivity, the mean squared displacement of the probe is required in addition to the total displacement D f low =

1 d 0 hx (t)x0 (t)i, 2 dt

(43)

where x0 (t) ≡ x(t) − hx(t)i is the displacement from the mean as a function of time. Clearly from Eqs. (42)– 310

(43), one only needs the movement of a single probe particle to obtain the microviscosity, flow-induced

diffusivity, and suspension stress. Detailed knowledge of the embedding material, such as microstructure of the entire suspension, is not required, only computation of statistics. 7. Conclusions We derived a generalized nonequilibrium Stokes-Einstein relation for predicting the steady-state stress in 315

a hydrodynamically interacting suspension via the active microrheology framework. This phenomenological model was derived based on volume-averaging of a pointwise Cauchy equation of motion which takes into account both advective and diffusive flux arising from the relative motion between the probe and surrounding materials. Following Zia and Brady [18], we related the advective flux to the reduction in the mean speed of the probe, the microviscosity, and the diffusive flux to the magnitude of the fluctuating motion, the flow-

320

induced diffusivity. The primary scientific contributions of this work are the phenomenological model that gives the non-equilibrium stress from simply monitoring probe motion, the first measurement of this stress via dynamic simulation of a hydrodynamically interacting suspension, and a non-equilibrium Stokes-Einstein 2 A fixed-force can be closely attained via a uniform magnetic field gradient [49, 46] whereas fixed-velocity mode can be realized with optical tweezers, where the suspending medium is flowed past the optically trapped probe at a fixed rate [50, 47, 48, 51]. In practice, these two modes are always present to some extent in experiments because no trap or field gradient is perfect. This is discussed in e.g. [50, 47, 51]. A pure fixed-force or velocity mode can be made quite accurate by careful accounting of the details of the probe constraint [32].

22

relation between observable particle motion and flow properties. The primary technical contribution of this work was the development of an anisotropic effective resistance tensor, constructed utilizing hydrodynamic 325

mobility functions to constitutively model the difference in the hydrodynamically coupled particle motion between fixed-force and fixed-velocity active microrheology. Similar to other phenomenological models such as the density and viscosity of a liquid, we sacrifice detailed structural information to obtain flow and material properties. The strength of the phenomenology is that it connects observable particle motion directly to the properties of the embedding suspension, without having to have detailed information about the particle-

330

scale motion of that medium — following Einstein’s use of observable motion of a colloidal particle revealing the random motion of solvent molecules that conformed the molecular nature of matter. Thus we pose its heuristic value as a non-equilibrium Stokes-Einstein relation (NESER) — where suspension stress is a balance of fluctuation (diffusion) and dissipation (microviscosity). While microviscosity and force-induced diffusion can be measured in experiments via tracking the mean and mean-square displacement of a probe

335

particle, it is quite challenging to directly measure normal stress. Thus, the NESER provides a means to infer the suspension stress if one has measurements for the mean and mean-square displacement of the probe. We compared the normal stresses, first normal stress difference and particle osmotic pressure predicted by the generalized nonequilibrium Stokes-Einstein relation to statistical mechanics theory and Accelerated Stokesian Dynamics simulations, finding good agreement between the three separate methods. In particular,

340

the suppressive influence of hydrodynamics on the normal stresses and osmotic pressure for the full range of flow strength was recovered by the phenomenological model, and the new theory also accurately captured the change of sign with flow strength in the first normal stress difference in a suspension when the strength of hydrodynamics is moderate. The influence of particle concentration on suspension stress response was also studied, where the normal

345

stresses, first normal stress difference and osmotic pressure all grow in magnitude with increasing particle volume fraction. We expected that a scaling theory based on recently-developed concentrated hydrodynamic functions can be constructed to collapse the response of concentrated suspensions onto the dilute statistical mechanics theory. We referred readers to a separate article for its details [41]. The study of suspension response of various particle volume fractions may inspire experimental studies; to this end, we discussed the

350

laboratory measurements required to apply the generalized nonequilibrium Stokes-Einstein relation. As a matter of practical interest, active microrheology requires only the motion of a single probe, negating the needs for otherwise labor-intensive microstructure measurements or expensive computational simulations. Overall, we showed that a balance of nonequilibrium fluctuation and dissipation produces nonequilibrium stress, reflecting the connection between thermodynamics and hydrodynamics, and generalizing Einstein’s

355

model of fluctuation and dissipation to nonequilibrium. While in this work active microrheology was the platform used to demonstrate the fundamental idea, we expect that such a nonequilibrium relation should

23

be possible for more general soft materials since it is a connection between fluctuation and dissipation. Many interesting questions remain. The formulation of the current phenomenological model is based on a Cauchy momentum balance, and is general for complex materials in which there is a separation of 360

length scale between the dispersed and the suspending phase. In this work, its application is restricted to hard-sphere colloidal suspensions by constitutively modeling the advective and diffusive particle flux using the microviscosity and flow-induced diffusivity taken from studies of hard-sphere suspensions. The model is generalizable to size polydispersity, relevant for many real systems. Further exploration may also focus on an attractive interparticle potential, as well as non-spherical, soft, or slippery (emulsion) particles. Finally, the

365

current framework applies to steady-state flow only. Development of a model for transient behavior would be an interesting contribution to fluctuation dissipation theory and for practical measurements. Acknowledgments This work was supported in part by Office of Naval Research Young Investigator Award (N00014-141-0744). The authors acknowledge many useful conversations with Nicholas Hoh, Yu Su, and Benjamin

370

Dolata.

Supporting Material A. Averaging suspension stress In a statistically homogeneously deformed suspension such as macroscopic shear, one can “stand” on any particle and examine the surrounding structure as a function of solid angle and radial distance. One would 375

get the same statistical response, on average, by standing on any particle in the suspension. In microrheology, one would get a distinctly different response by standing on the probe versus on a bath particle nearby or one far away. Thus, the proper way to carry out an average in active microrheology is to average over many probes traveling through different regions of the same suspension, where the volume fraction of probes is very small relative to the volume fraction of background particles and thus they do not interact with one another.

380

This is the averaging taken in the phenomenological model in the present and a previous work [18], and in previous microstructural models [6, 7, 8, 32, 28, 29, 24, 25]. Chu and Zia [24] presented a detailed account of the averaging procedure, and we reiterate the explanation below due to its relevance to the present work. In the classical framework of suspension mechanics developed by [12], the procedure for computing bulk or average rheological quantities by averaging over an ensemble of microscopic configurations is shown to be

385

equivalent to a volume averaging procedure under certain conditions. This equality of an ensemble average and a volume average is based on the central assumption that the suspension is statistically homogeneous. The use of this averaging technique in the active microrheology framework is clarified as follows.

24

We consider a set of probe particles with number density na in a dilute suspension of bath particles with number density nb . The probe phase is sufficiently dilute that one probe does not interact with another. 390

In addition, the probes are dilute relative to the bath, na  nb . We consider a unit volume of solvent and particles of a characteristic dimension l. In a dilute bath, interactions between the bath particles give rise

to a suspension stress of O(φ2b ) (φb is the volume fraction of bath particles), which is small compared to the O(φb ) contribution arising from the probe-bath interactions [18]. Thus the probe-phase stress dominates the particle stress in any unit volume. Now, we consider the entire suspension, of characteristic dimension L, 395

comprising many of these unit volumes. Since l  L, the variation of the local statistical properties on the microscale l (within each unit volume) is negligible on the macroscale L upon considering the suspension as

a whole. In other words, on the suspension length scale L relevant to bulk rheology, such spatial variations in the suspension stress are negligibly small; thus the suspension can be treated as statistically homogeneous. In justifying the statistical homogeneity of a suspension, another stipulation underlying the averaging 400

process is that all particles are external-force free, implying that in a closed system there is no net migration of particles (no formation of particle-rich and particle-depleted regions, akin to formation of sediment and supernatant in sedimentation), that would yield a statistically inhomogeneous suspension. The model presented in the current work pertains to an unbound suspension which, in practice, corresponds to a large container. Indeed not all particles are external-force free in our system: the probes are driven by a fixed

405

external force, producing net translation of probe particles. However, the probe phase is dilute relative to the bath (na  nb ), and thus the net migration of the probe phase is small compared to the container size.

In a“larg” container, the influence of the accumulation of particles in one region (and depletion in another) on the statistical properties of the suspension is negligible, given that the vast majority of the large domain is statistically homogeneous. The influence of net particle migration can be further minimized in practice, 410

by conducting active microrheology in a system in which one side is connected to a continuous supply of suspension while the other to drainage, i.e. an open system or, more simply, to utilize a container that is large compared to particle size and separation. Supporting Material B. Micromechanical theories and results of the steady-state microviscos-

415

ity, flow-induced diffusivity and suspension stress We present the micromechanical theories and key results of the steady-state microviscosity [6, 7, 32], flowinduced diffusivity [8, 28, 29], and suspension stress [18, 24, 25] from prior studies, along with a summary of their asymptotic behavior in the limits of weak and strong flow strength P e and hydrodynamic interactions κ. Readers interested in transient active microrehology are referred to, e.g., [16, 52].

25

420

Supporting Material B.1. Microviscosity The presence of microstructure hinders the probe’s motion and its deformation of the suspension, giving rise to an effective suspension viscosity higher than the solvent viscosity. This effective viscosity changes with the degree of microstructural distortion which, in turn, depends on the dimensionless force P e, describing the strength of the probe forcing, F0 , relative to the Brownian restoring force, 2kT /ath . Squires and Brady [6] and Khair and Brady [7] related the reduction in the mean velocity of the probe, hU i, to the effective viscosity, η ef f , via the Stokes’ drag law,

F ext = 6πηa

η ef f hU i, η

(A1)

and the effective viscosity is given by η ef f =

F0

ˆ ext 6πahU i · F

(A2)

,

ˆ ext = F ext /F0 is the unit vector pointing in the direction of the external probe force. In the dilute where F limit, φb  1, the effective viscosity can be Taylor-expanded as, η ef f = 1 + η micro φb + O(φ2b ). η

(A3)

The microviscosity comprises contributions from the hydrodynamic, Brownian and interparticle forces, η micro = η H + η B + η P . In fixed-force active microrheology, the probe is driven by a constant external force induced by, e.g. magnetic field [46], and experiences fluctuating motion. Squires and Brady [6] and Khair and Brady [7] derived the microviscosities as ηFH

ηFB

=−

Z

∞ 2

a [xa11 (κr) + 2y11 (κr) − 3] r2 dr

3 ˆ ext ˆ ext − F F : 4π

Z

3 1 ˆ ext = F · 4π P e

Z

ηFP

r≥2

[xa11 (κr)ˆ r rˆ

 r≥2

+

(A4)

a y11 (κr)(I

− rˆ rˆ ) − I] f (r)dr,

 G(κr) − H(κr) 1 dG(κr) + f (r)ˆ r dr, r 2 dr

3 G(2(1 + κ)) ˆ ext = F · 2π Pe

I

f (r)ˆ r dΩ,

(A5)

(A6)

r=2

where the components of the hydrodynamic mobility functions xαβ , yαβ , G, and H are defined following the conventional notations [53, 33, 54, 55]. They depend only on the relative separation between a pair of parti425

cles, r, and the dimensionless repulsion range, κ. The nonequilibrium distortion f (r) of the microstructure is defined as g(r) = g eq (1 + f (r)), where the pair distribution function, g(r) describes the spatial distribution 26

2.5

micro ⌘Fmicro h F

2.0

1.5

1.0 Repulsion range, κ -5

10 (strong hydro) -2 10 -1 10 1 5

0.5

0.0 0.01

10 50 2 10 500 3 10 (weak hydro)

0.1

1

Pe

10

100

micro , as a function of flow strength P e for various strength of Figure 7: Evolution of the total fixed-force microviscosity, ηF hydrodynamic interactions κ.

of bath particles with reference to a probe particle, and g eq is the equilibrium microstructure and is equal to unity in a dilute dispersion. Both g(r) and f (r) have been studied for the full range of flow strength P e and hydrodynamic interactions κ in previous studies [6, 7, 8, 24]. Figure 7 shows the evolution of the total 430

fixed-force microviscosity ηFmicro with P e for various κ. Active microrheology can also be operated in the fixed-velocity mode: the probe is held fixed in an optical trap and moved past the bath at a constant velocity [47, 48]. Fixed-velocity microviscosity can be deduced analogously following Eqs. (A1)-(A3) with the external-velocity being held fixed and the external-force being the fluctuating quantity. The fixed-velocity hydrodynamic, Brownian, and interparticle microviscosities were derived by Swan and Zia [32] as H ηU

B ηU

∞

 1 1 = + a − 3 r2 dr xa11 (κr) 2y11 (κr) 2   Z 3 ˆ ˆ rˆ rˆ I − rˆ rˆ + U1 U1 : + a − I f (r)dr, a 4π y11 (κr) r≥2 x11 (κr) Z

=



   Z 3 1 ˆ 1 G(κr) H(κr) U1 · − a + 8π P e r xa11 (κr) y11 (κr) r≥2   1 d G(κr) f (r)ˆ r dr, 2 dr xa11 (κr)

27

(A7)

(A8)

P ηU =

3 G(2(1 + κ)) 1 ˆ U1 · π xa11 (2(1 + κ)) P e

I

f (r)ˆ r dΩ,

(A9)

r=2

where Uˆ1 = U 1 /U1 is the unit vector pointing in the direction of the imposed probe velocity, and evaluation of the nonequilibrium distortion f (r) was given in [32]. We note that a factor of two was missing in the original results [32], and we corrected them in Eqs. (A8) and (A9) above. Here, we summarize the behavior of the microviscosity in the limits of asymptotically weak and strong probe forcing and hydrodynamic interactions. In both fixed-force and fixed-velocity active microrheology, the deviation of hydrodynamic, Brownian and interparticle microviscosities from their equilibrium values were found to scale as P e2 at asymptotically small P e, P e  1, regardless of the strength of hydrodynamics: H B P (ηF,U − η0HF,U ) , (ηF,U − η0BF,U ) , (ηF,U − η0P F,U ) ∼ P e2 435

for P e  1,

(A10)

where η0 is the equilibrium values of the microviscosities. In the opposite limit of asymptotically large P e, P e  1, fixing external-force or velocity leads to

qualitatively different microstructural deformation, giving rise to distinct scalings of the microviscosities. In the fixed-force mode, the hydrodynamic, Brownian and interparticle microviscosities scale as ηFH ∼ C1 + C2 P eδ−1 , ηFB ∼ P eδ−2 , ηFP ∼ P eδ−1

for P e  1,

(A11)

where 1 ≥ δ ≥ 0.799 as the influence of hydrodynamics evolves from weak (κ  1, δ = 1) to strong (κ  1,

δ = 0.799), and C1 and C2 are obtained by extrapolating the high-P e results to the limit P e  1 [7]. In the fixed-velocity mode, the P e-dependence of the three microviscosities have the same functional form as the fixed-force counterparts, H B P ηU ∼ C3 + C4 P eζ−1 , ηU ∼ P eζ−2 , ηU ∼ P eζ−1

for P e  1,

(A12)

but with new extrapolating parameters C3 and C4 , and δ replaced by ζ, which varies between ζ ∈ [1, 0.825] as the strength of hydrodynamics changes from weak to strong [32]. Supporting Material B.2. Flow-induced diffusivity In addition to causing reduction in the mean velocity of the probe, probe/bath encounters also deflect the probe from its mean path. Flow-induced diffusion (microdiffusivity) is defined as the change in the effective diffusivity of the probe, D ef f , given rise by these interactions [8]. To focus on this nonequilibrium fluctuation, flow-induced diffusivity, D f low , was defined as [29] D ef f = D eq + D f low , 28

(A13)

10 10

I.

1

ABC

0

II.

-1

10

-2

10

-4

1.

-5

0.01

2.

ABC CDE

0.1

3

10 (weak hydro) 500 100 50 10 5 1 -1 10 -2 10 -5 10 (strong hydro)

1 0

-1

10

Repulsion range, κ 3

10 (weak hydro) 500 100 50 10 5 1 -1 10 -2 10 -5 10 (strong hydro)

-2

10

-3

10

-4

10

-5

1

10

Pe

10

100

0.01

0.1

1

10

Pe

100

Acknowledgements

0.7

0.6 10

-5

10

-4

-3

10

10

-2

0

1

10

κ

10

10

2

Repulsion range, κ 3

10 500 2 10 50 10

8

5 1 -1 10 -2 10 -5 10

||

micro

3

10

10

6

^

CDE

0 0.01

2.

2

ABC

4

1.

micro

ABC II.

I.

ABC

ABC.

DeqD,eqp()

(1)

p()

(2)

1.0

1

1.5

0.5

0.0

-1

10

(e)

2.0

p()

0.8

0.1

The authors wish to acknowledge

0.9

Daeq- D, eqp() (D ) / (Dafb)

(d) The authors wish to acknowledge

1.0

Abstract

a

x 11(2ath↵, kκ) , f||(κ)

September 15, 2017

Repulsion range, κ

10

2

What should I do if I want to remove the ’.’ after citing the Sec I’.’ ? Here is a cite Sec. I

-3

10

(c)

10 10

ABC

10

10

(b)

Acknowledgements

micro Dkf low D /Da

|| b

10

2

f low micro D? D/Da^ b

(a)

D / do D 1/↵I ? What should if I want to remove the ’.’ after citing the Sec I’.’ ? Here is a cite Sec. I

ABC.

(2)

(1)

Abstract

1

Sibley School of Mechanical and Aerospace Engineering, Cornell University 2 School of Chemical and Biomolecular Engineering, Cornell University

September 15, 2017

10

-5

-4

10

-3

10

10

-2

10

-1

κ

10

0

1

10

2

10

10

3

1 1

Pe

10

100

f low Figure 8: Evolution of the (a) parallel Dkf low and (b) perpendicular flow-induced diffusivity D⊥ , as a function of flow strength eq P e for various strength of hydrodynamic interactions κ. Evolution of (c) αk and (d) D and p(κ) in Eq. (18), as a function of κ, where p(κ) = (3/16π)Deq . (e) Evolution of α⊥ in Eq. (18), as a function of P e for various κ.

where the first term, D eq is the equilibrium value in the absence of external force, and the second term, 440

D f low , corresponds to the O(φb ) departure from equilibrium arising from the external force. While determination of the microviscosity requires the pair distribution function, g(r), to describe the likelihood finding a bath particle reference to the probe, evaluation of the flow-induced diffusivity requires the fluctuation field, d(r), which details the strength and direction of probe/bath encounters. Akin to micro-

29

viscosity, flow-induced diffusivity comprises contribution from the hydrodynamic, Brownian and interparticle 445

forces. Zia and Brady [8] and Hoh and Zia [29] derived these contributions as D f low,H Da φb

=

Z 3 a Pe [xa11 (κr)ˆ r rˆ + y11 (κr)(I − rˆ rˆ ) − I] · 4π r≥2   1 ˆ ext dneq dr, f (r)I − 2F Pe

D f low,B 3 = D a φb 2π



Z r≥2

 G(κr) − H(κr) 1 dG(κr) + rˆ dneq dr, r 2 dr

I D f low,P 3 = G(2(1 + κ)) Da φb π

rˆ dneq dΩ,

(A14)

(A15)

(A16)

r=2

where Da = kT /6πηa is the bare diffusivity of a colloidal particle of hydrodynamic size a, and dneq (r) is the nonequilibrium part of the fluctuation field associated with the change in the probe’s diffusivity under external force. The flow-induced diffusivity tensor is anisotropic in general. To characterize this anisotropy, flow-induced 450

f low diffusivity parallel and perpendicular to the line of the external force, Dkf low and D⊥ , are obtained by

projecting the tensor in the corresponding directions, Dkf low

= D f low : ez ez ,

(A17)

f low D⊥

= D f low : ey ey ,

(A18)

where ez and ey are the unit vectors in the direction parallel and perpendicular to the external force, respectively. We note that diffusivity in the two perpendicular directions, x and y, are the same owing to f low f low f low axisymmetry of the external flow, and thus D⊥ ≡ Dyy = Dxx . Figure 8 shows the evolution of the 455

f low parallel Dkf low and perpendicular flow-induced diffusivity D⊥ , p(κ), αk and α⊥ in Eq. (18), with P e and

κ. Here, we summarize the behavior of the flow-induced diffusivity in the limits of asymptotically weak and strong probe forcing and hydrodynamic interactions. In the limit of weak hydrodynamic interactions, κ  1, Zia and Brady [8] found that both parallel and perpendicular diffusivities grow as P e2 under weak forcing, f low Dkf low , D⊥ ∼ P e2

for

κ  1 , P e  1,

(A19)

for κ  1 , P e  1.

(A20)

whereas they grow linearly in P e under strong forcing f low Dkf low , D⊥ ∼ Pe

30

In the limit of strong hydrodynamic interactions, κ  1, Hoh and Zia [29] found that the low-P e scalings

of the two diffusivities remain unchanged

f low Dkf low , D⊥ ∼ P e2

for

κ  1 , P e  1.

(A21)

As flow strength increases, the scaling of both diffusivities changes qualitatively Dkf low ∼ P e,

f low D⊥ =0

for κ  1 , P e  1,

(A22)

recovering the result of Hoh and Zia [28] in the strong-flow, pure-hydrodynamic limit. Supporting Material B.3. Suspension stress The particle phase stress, hΣi, can be divided into non-hydrodynamic and hydrodynamic contributions

as [12, 14]

hΣi = −na kT I − na hrF P i + hΣiH ,

(A23)

where the non-hydrodynamic contribution comprises the ideal osmotic pressure, −na kT I, associated with the equilibrium thermal energy of the Brownian particles, and an interparticle stress originated from interparticle elastic collisions, hΣiP,el ≡ −na hrF P i. The remaining term, hΣiH , is the hydrodynamic stress induced by external probe forcing, Brownian motion and interparticle force via hydrodynamic coupling hΣiH = hΣiH,ext + hΣiB + hΣiP,dis ,

(A24)

where the superscript P, dis refers to the dissipative part of the interparticle stress, and the total interparticle 460

stress comprises this hydrodynamic dissipative component and the non-hydrodynamic elastic part, that is hΣiP = hΣiP,dis + hΣiP,el . A detailed derivation of the suspension stress was given by Chu and Zia [24, 25]. As the probe moves through the suspension, distortions to and relaxation of the particle arrangement

give rise to nonequilibrium suspension stress. To focus on the nonequilibrium rheology, the nonequilibrium stress, hΣneq i, was defined as [24, 25]

Σ = Σeq + Σneq ,

(A25)

where Σeq is the equilibrium suspension stress in the absence of external force. Zia and Brady [18] and Chu and Zia [24, 25] derived the nonequilibrium stress arising from external, Brownian and interparticle forces

31

as Z hΣneq iH,ext 1 Pe = − na kT φb π(1 + κ)





r≥2

G X11 (κr)xA 11 (κr)

+

G X12 (κr)xA 12 (κr)



  1 rˆ rˆ rˆ − I rˆ 3

 G  A G A + Y11 (κr)y11 (κr) + Y12 (κr)y12 (κr) (2I rˆ − 2ˆ r rˆ rˆ ) −3 + hΣneq iB = na kT φb



H B Y11 (κr)y11 (κr)

Z 1 Pe 3π(1 + κ)

1 2π(1 + κ)

r≥2



Z

−

r≥2

H B Y12 (κr)y12 (κr)





2I rˆ − 2ˆ r rˆ rˆ

(A26)



ˆ ext

·F

f (r)dr

 P  P A ˆ ext f (r)dr, ˆ ·F X11 (κr)xA 11 (κr) + X12 (κr)x12 (κr) I r


 1 d  2 G G r X11 (κr) − X12 (κr) G(κr) 2 r dr


6 G G − Y11 (κr) − Y12 (κr) H(κr) r  
B
18 1 H H B − Y11 (κr) − Y12 (κr) y11 (κr) − y12 (κr) rˆ rˆ − I f (r)dr r 3   Z
 1 1 d  2 P P + r X (κr) − X (κr) G(κr) If (r)dr, 11 12 6π(1 + κ) r≥2 r2 dr hΣneq iP 3 = − na kT φb π

I

rˆ rˆ f (r)dΩ r=2

 G  2 G − X11 (2(1 + κ)) − X12 (2(1 + κ)) G(2(1 + κ)) π(1 + κ) −

(A27)

2 3π(1 + κ)

  1 rˆ rˆ − I f (r)dΩ 3 r=2 I  P  P X11 (2(1 + κ)) − X12 (2(1 + κ)) G(2(1 + κ)) If (r)dΩ. I

(A28)

r=2

where the components of the hydrodynamic resistance functions Xαβ and Yαβ are defined following the conventional notations [54, 55, 56, 40]. Same as the hydrodynamic mobility functions (cf. Supporting Material B.1), these functions depend only on the relative separation between a pair of particles, r, and the 465

dimensionless repulsion range, κ. The microstructural distortion f (r) in the above expressions is identical to that in the microviscosity expressions. The three-dimensional stress tensor comprises six independent elements in general. In active microrheology, they reduce to three nonzero diagonal elements, the normal stresses Σxx , Σyy and Σzz , owing to the axisymmetry of the microstructure around the probe. This structural axisymmetry also yields identical

470

normal stresses along the orthogonal axes, Σyy = Σxx , leaving only two relevant quantities: the normal stresses acting parallel and perpendicular to the direction of the external force (cf. Fig. 1), Σk and Σ⊥ . They are utilized to characterize non-Newtonian rheology in Section 4, and are obtained by projecting the

32

stress tensor as Σk

= Σ : ez ez ,

(A29)

Σ⊥

= Σ : ey ey .

(A30)

Non-Newtonian rheology also manifests in the normal stress differences and nonequilibrium osmotic pressure, describing the anisotropic and isotropic deformation of a suspension, respectively. The first and the second normal stress differences are defined as N1 ≡ Σzz − Σyy ,

(A31)

N2 ≡ Σyy − Σxx ,

(A32)

and the osmotic pressure is defined as the negative one-third of the trace of the stress tensor: 1 Π ≡ − I : Σ. 3

(A33)

The axisymmetric microstructure around the probe requires the second normal stress difference to be iden475

tically zero regardless of the strength of hydrodynamics. Here, we summarize the behavior of the suspension stress in the limits of asymptotically weak and strong probe forcing and hydrodynamic interactions. In the limit of weak hydrodynamic interactions, κ  1, Zia

and Brady [18] found that, under weak probe forcing, the nonequilibrium parallel and the perpendicular normal stress, and the osmotic pressure grow quadratically in P e, and the first normal stress difference grow with the fourth power in P e, neq neq hΣneq i ∼ P e2 , k i, hΣ⊥ i, hΠ

hN1neq i ∼ P e4

for κ  1 , P e  1,

(A34)

whereas all quantities scale linearly with P e under strong flow, neq neq neq hΣneq i ∼ Pe k i, hΣ⊥ i, hN1 i, hΠ

for κ  1 , P e  1.

(A35)

In the limit of strong hydrodynamic interactions, κ  1, coupled particle motion gives rise to distinct

scalings. When external probe forcing is weak, Chu and Zia [24, 25] found that both normal stress, the first normal stress difference and the osmotic pressure maintain the P e2 -growth, neq neq neq hΣneq i ∼ P e2 , k i, hΣ⊥ i, hN1 i, hΠ

33

for κ  1 , P e  1,

(A36)

where the role played by hydrodynamics in this weak-flow regime is quantitative, except it changes qualitatively the scaling of the first normal stress difference. When external force is strong, all quantities scale as P eδ , neq neq neq hΣneq i ∼ P eδ , k i, hΣ⊥ i, hN1 i, hΠ

for

κ  1 , P e  1,

(A37)

where 1 ≥ δ ≥ 0.799 as the importance of hydrodynamic interactions grows from weak to strong. [1] A. Einstein, On the theory of the Brownian movement, Ann. der Physik 19 (4) (1906) 371–381. [2] P. Langevin, Sur la theorie du mouvement brownien (On the theory of Brownian motion), C. R. Acad. Sci. (Paris) 146 (1908) 530–533. 480

[3] E. J. Hinch, Application of the langevin equation to fluid suspensions, J. Fluid Mech. 72 (1975) 499–511. [4] M. Haw, Middle world - The restless heart of matter and life, Macmillan, 2007. [5] R. N. Zia, Active and passive microrheology: Theory and simulation, Annual Reviews of Fluid Mechanics 50 (2018) 371–405. [6] T. M. Squires, J. F. Brady, A simple paradigm for active and nonlinear microrheology, Phys. Fluids

485

17 (7) (2005) 073101–1–073101–2. [7] A. S. Khair, J. F. Brady, Single particle motion in colloidal dispersions: a simple model for active and nonlinear microrheology, J. Fluid Mech. 557 (2006) 73–117. [8] R. N. Zia, J. F. Brady, Single-particle motion in colloids: force-induced diffusion, J. Fluid Mech. 658 (2010) 188–210.

490

[9] C. Gao, S. D. Kulkarni, J. F. Morris, J. F. Gilchrist, Direct investigation of anisotropic suspension structure in pressure-driven flow, Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 81 (4) (2010) 6–9. [10] N. Y. C. Lin, J. H. McCoy, X. Cheng, B. Leahy, J. N. Israelachvili, I. Cohen, A multi-axis confocal rheoscope for studying shear flow of structured fluids, Rev. Sci. Instrum. 85 (2014) 033905.

495

[11] B. Xu, J. F. Gilchrist, Microstructure of sheared monosized colloidal suspensions resulting from hydrodynamic and electrostatic interactions., The Journal of chemical physics 140 (20) (2014) 204903. [12] G. K. Batchelor, The stress in a suspension of force-free particles, J. Fluid Mech. 41 (3) (1970) 545–570. [13] G. K. Batchelor, J. T. Green, The determination of the bulk stress in a suspension of spherical particles to order c2 , J. Fluid Mech. 56 (1972) 401–427.

34

500

[14] G. K. Batchelor, The effect of Brownian motion on the bulk stress in a suspension of spherical particles, J. Fluid Mech. 83 (1977) 97–117. [15] I. C. Carpen, J. F. Brady, Microrheology of colloidal dispersions by Brownian dynamics simulations, J. Rheol. 49 (6) (2005) 1483–1502. [16] R. N. Zia, J. F. Brady, Stress development, relaxation, and memory in colloidal dispersions: Transient

505

nonlinear microrheology, J. Rheol. 57 (2) (2013) 457–492. [17] E. Nazockdast, J. F. Morris, Active microrheology of colloidal suspensions: Simulation and microstructural theory, J. Rheol. 60 (2016) 733–753. [18] R. N. Zia, J. F. Brady, Microviscosity, microdiffusivity, and normal stresses in colloidal dispersions, J. Rheol. 56 (5) (2012) 1175–1208.

510

[19] L. Durlofsky, J. F. Brady, G. Bossis, Dynamic simulation of hydrodynamically interacting particles, Journal of Fluid Mechanics 180 (1987) 21. [20] A. Sierou, J. Brady, Rheology and microstructure in concentrated noncolloidal suspensions, Journal of Rheology 46 (5) (2002) 1031–1056. [21] A. J. Banchio, J. F. Brady, Accelerated stokesian dynamics: Brownian motion, J. Chem. Phys. 118 (22)

515

(2003) 10323 – 10332. [22] J. F. Brady, Stokesian dynamics, Annu. Rev. Fluid Mech. 20 (1988) 111–157. [23] A. Sierou, J. F. Brady, Accelerated stokesian dynamics simulations, J. Fluid Mech. 448 (2001) 115 – 146. [24] H. C. W. Chu, R. N. Zia, Active microrheology of hydrodynamically interacting colloids: Normal stresses

520

and entropic energy density, Journal of Rheology 60 (2016) 755 – 781. [25] H. C. W. Chu, R. N. Zia, The non-newtonian rheology of hydrodynamically interacting colloids via active, nonlinear microrheology, Journal of Rheology 601 (2017) 551–574. [26] W. B. Russel, The Huggins coefficient as a means for characterizing suspended particles, Journal of the Chemical Society, Faraday Transactions 2 80 (1) (1984) 31.

525

[27] T. G. Mason, D. A. Weitz, Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids, Phys. Rev. Lett. 74 (7) (1995) 1250–1253. [28] N. J. Hoh, R. N. Zia, Hydrodynamic diffusion in active microrheology of non-colloidal suspensions: the role of interparticle forces, J. Fluid Mech. 785 (2015) 189–218. 35

[29] N. J. Hoh, R. N. Zia, Force-induced diffusion in suspensions of hydrodynamically interacting colloids, 530

J. Fluid Mech. 795 (2016) 739 – 783. [30] G. K. Batchelor, Sedimentation in a dilute polydisperse system of interacting spheres. part 1. general theory, J. Fluid Mech. 119 (1982) 379–407. [31] G. K. Batchelor, C.-S. Wen, Sedimentation in a dilute polydisperse system of interacting spheres. Part 2. Numerical results, Journal of Fluid Mechanics 124 (1982) 495 – 528.

535

[32] J. W. Swan, R. N. Zia, Active microrheology – fixed-velocity versus fixed-force, Phys. Fluids. 25 (8) (2013) 083303. [33] D. J. Jeffrey, Y. Onishi, Calculation of the resistance and mobility functions for two unequal rigid spheres in low-reynolds-number flow, J. Fluid Mech. 139 (1984) 261–290. [34] Y. Yurkovetsky, J. F. Morris, Particle pressure in sheared brownian suspensions, Journal of rheology

540

52 (1) (2008) 141–164. [35] R. N. Zia, J. W. Swan, Y. Su, Pair mobility functions for rigid spheres in concentrated colloidal dispersions: Force, torque, translation, and rotation, Journal of Chemical Physics 143 (2015) 224901. [36] Y. Su, J. W. Swan, R. N. Zia, Pair mobility functions for rigid spheres in concentrated colloidal dispersions: stresslet and straining motion couplings, Journal of Chemical Physics.

545

[37] D. R. Foss, J. F. Brady, Structure, diffusion and rheology of Brownian suspensions by stokesian dynamics simulation, J. Fluid Mech. 407 (2000) 167–200. [38] S. Sami, Stokesian dynamics simulation of Brownian suspensions in extensional flow, Ph.D. thesis, California Institute of Technology (1996). [39] P. R. Nott, J. F. Brady, Pressure-driven flow of suspensions: simulation and theory, J. Fluid Mech. 275

550

(1994) 157–199. [40] D. J. Jeffrey, J. F. Morris, J. F. Brady, The pressure moments for two rigid spheres in low-Reynoldsnumber flow, Phys. Fluids 5. [41] Y. Su, H. C. W. Chu, R. N. Zia, Microviscosity, normal stress and osmotic pressure of brownian suspensions by accelerated stokesian dynamics simulation, in preparation, for submission to J. Rheol.

555

(2018). [42] I. E. Zarraga, D. A. Hill, D. T. Leighton, The characterization of the total stress of concentrated suspensions of noncolloidal spheres in newtonian fluids, J. Rheol. 44. 36

[43] A. Singh, P. R. Nott, Experimental measurements of the normal stresses in sheared Stokesian suspensions, J. Fluid Mech. 490 (2003) 293–320. 560

[44] S. Garland, G. Gauthier, J. Martin, J. F. Morris, Normal stress measurements in sheared non-Brownian suspensions, J. Rheol. 57. [45] C. Gamonpilas, J. F. Morris, M. M. Denn, Shear and normal stress measurements in non-Brownian monodisperse and bidisperse suspensions, J. Rheol. 60. [46] P. Habdas, D. Schaar, A. C. Levitt, E. R. Weeks, Forced motion of a probe particle near the colloidal

565

glass transition, Europhys. Lett. 67 (3) (2004) 477–483. [47] A. Meyer, A. Marshall, B. G. Bush, E. M. Furst, Laser tweezer microrheology of a colloidal suspension, J. Rheol. 50 (1) (2006) 77–92. [48] I. Sriram, R. J. DePuit, T. M. Squires, E. M. Furst, Small amplitude active oscillatory microrheology of a colloidal suspension, J. Rheol. 53 (2) (2009) 357–381.

570

[49] A. R. Bausch, W. Moller, E. Sackmann, Measurement of local viscoelasticity and forces in living cells by magnetic tweezers, Biophys. J. 76 (1999) 573–579. [50] E. M. Furst, Applications of laser tweezers in complex fluid rheology, Curr. Opin. Colloid Interface Surf. Sci. 10 (2005) 79–86. [51] L. G. Wilson, A. W. Harrison, A. B. Schofield, J. Arlt, W. C. K. Poon, Passive and active microrheology

575

of hard-sphere colloids, J. Phys. Chem.B 113 (12) (2009) 3806–3812. [52] R. P. Mohanty, R. N. Zia, The Impact of Hydrodynamics on Viscosity Evolution in Colloidal Dispersions: Transient, Nonlinear Microrheology, AIChE Journa 64 (2018) 3198–3214. [53] G. K. Batchelor, Brownian diffusion of particles with hydrodynamic interaction, J. Fluid Mech. 74 (1976) 1–29.

580

[54] S. Kim, R. T. Mifflin, The resistance and mobility functions of two equal spheres in low-Reynolds-number flow, Phys. Fluids 28. [55] S. Kim, S. J. Karrila, Microhydrodynamics:

principles and selected applications, Butterworth-

Heinemann, Boston, 1991. [56] D. J. Jeffrey, The calculation of the low Reynolds number resistance functions for two unequal spheres, 585

Phys. Fluids 4 (1992) 16–29.

37

non-equilibrium osmotic pressure < neq>/ nakT b

1

1000

SD simulation SD simulation particle volume fraction

100

φb φb φb φb φb

10 1

= 0.05 = 0.1 = 0.2 = 0.3 = 0.4

0.1

1 0.01

Pe

10

100 Stat. Mech. New theory

0.001

strong hydrodynamic interactions 0.0001 0.01

0.1

1

Pe flow strength

10

100