1

Shear-driven activation, aggregation, and rheopexy

in sheared interacting colloids

Alessio Zaccone, Hua Wu, and Massimo Morbidelli

Chemistry and Applied Biosciences, ETH Zurich, CH-8093 Zurich, Switzerland

(Dated: June 2009)

CORRESPONDING AUTHOR

Alessio Zaccone

Email: alessio.zaccone@chem.ethz.ch.

Fax: 0041-44-6321082.

ABSTRACT

The convective diffusion equation for interacting colloids, formally identical to the

Smoluchowski equation with shear, governs the macroscopic rheological behaviour of

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complex and biological fluids, and is known to be analytically intractable. Hence, puzzling

rheological behaviours of complex fluids (shear-thickening, thixotropy, rheopexy etc.) are

poorly understood and their microscopic physics has remained obscure, in spite of their being

ubiquitous (from paints and inks to blood plasmas). Thus, an approximation is proposed here

which allows for analytical solutions in linear flow fields (shear and extensional) and with

arbitrary interaction potential between primary constituents. When an interaction barrier is

present, an explicit (aggregation) rate-equation in Arrhenius form with the shear rate showing

up in the exponential factor is derived (which thus extends Kramers’ rate theory to account

for the effect of shear). The predictive power of the approximation (with no fitting parameters)

is verified against numerics for the case of DLVO-interacting colloids in extensional flow.

The results explain the puzzling features observed in rheopectic complex fluids, including the

majority of protein-based fluids, such as the presence of an induction time (corresponding to

an activation delay in our theory) followed by a sudden rise of the viscosity (explained here

as due to a self-accelerating kinetics involving activated clusters).

\body Complex and biological fluids, whose importance in both natural and technological

contexts cannot be underestimated, display a range of intriguiing rheological properties (1),

depending on the nature and interactions of their microscopic constituents (colloidal and

noncolloidal particles such as cells, biological macromolecules, synthetic polymers, inorganic

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particulates etc). Among these properties, thixotropy (viscosity decreasing with time under

steady shear) and its opposite, rheopexy (viscosity increasing with time under steady shear)

are ubiquitous and necessitate a better understanding, possibly going deeper beyond the

current, generic attribution of the former to structure-breaking and of the latter to structure-

building processes involving the constituents (2,3). In particular, rheopexy, as a consequence

of aggregation under shear, is observed in important biological fluids such as protein

solutions (e.g. the synovial fluid which lubricates mammalian freely moving joints) and blood

plasmas (4,5). Further, rheopexy may even lead, under certain conditions, to a liquid-solid

(jamming) transition in time which attracts considerable attention for the fabrication of new

materials with extraordinary properties. For example, it has been shown that rheopexy, again

as a result of clustering and aggregation under shear, plays a crucial role in the formation of

spider silk within the spider’s spinneret, where conditions of elongational and shear flow are

created which enhance protein aggregation leading to large intermediate aggregates that are

further extruded to form a light material with formidable mechanical properties (5,6).

Nevertheless, the current understanding of these phenomena is merely qualitative and largely

built upon empirical evidence. It is clear in fact that rheopexy arises from clustering of the

primary particles into structures which can further aggregate (thanks to short-range attractive

interactions), at the same time withstanding the hydrodynamic stresses. It is however not

clear why under most conditions the viscosity increases suddenly after an induction period

(during which it is constant and equal to the zero-time viscosity), which is reminiscent of an

explosive behaviour (2,6,7). Moreover, the sudden increase of viscosity at a well-defined

point in time may lead to flow arrest (jamming), with formation of a solid (2,6), whereas in

other cases it levels off at a certain value reaching a plateau at long time and remaining liquid

(7). The induction period observed in systems with a long-range repulsive component of

interaction (e.g. charge-stabilization) is highly suggestive of an activation mechanism for

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aggregation driven by shear, as was speculated in (7) on the basis of experiments on

emulsions in simple shear where the induction period was found to decrease exponentially

with the shear rate (see Fig. 1). It is also unclear the interplay between shear, interparticle

interactions, and Brownian motion in the aggregation process. The latter point, indeed a

fundamentally unresolved issue, is a consequence of the current lack of successful theories to

quantitatively describe the aggregation kinetics of interacting colloids in flowing systems.

In this work we present an approximate, analytical theory based on the Smoluschowski

equation with shear which governs the aggregation kinetics of interacting particles under

shear. The results shed light on the problems mentioned above and point to the role of shear-

driven activation which triggers a self-accelerating kinetics once activated clusters are

formed. Our theory explains the induction time as well as the rapid rise found in the typical

rheopectic behaviour shown in Fig. 1, by means of an Arrhenius-type rate equation with shear,

directly derived from the Smoluchowski equation. This lays the groundwork for a

quantitative understanding of shear-induced structure-formation in complex fluids and of the

related macroscopic rheological properties.

Results and discussion

The Smoluchowski equation with shear: reduction to ODE for the aggregation rate

problem

Let us consider a dispersion of diffusing particles interacting with a certain interaction

potential. The (number-) concentration field will be called ( )c r . The associated normalized

probability density function ( ) ( )g c≡r r for finding a second particle at distance r from a

reference one is then normalized such that 0( ) ( )c c c=r r , where 0c is the bulk concentration.

The evolution equation thus reads

/ div( )c t D c D cβ∂ ∂ = ∇ − K [1]

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where D is the mutual diffusion coefficient of the particles ( 02 ( )D D r= G , 0D being the

diffusion coefficient of an isolated particle and ( )rG the hydrodynamic function for viscous

retardation), 1/ kTβ = is the Boltzmann factor, and K is the external force. According to this

equation, the associated stationary current is given by ( )D D cβ= − ∇J K . Hence, when

steady-state is reached (after a transient of order 20/a D ), ( ) / 0c t∂ ∂ =r and

( )div 0D D cβ − ∇ =K [2]

If the dispersing medium is subjected to an externally applied flow, the arbitrary external

force field for our problem may be decomposed into two terms, one accounting for the drift

caused by the flow velocity r θ φ( ) [v ,v ,v ]=v r , and the term accounting for the force field due

to the two-body (colloidal) interaction between particles ( )U r−∇ . Introducing the viscosity

of the dispersing medium η and the particle radius a, one obtains ( ) ( ) ( )U r b= −∇ +K r v r ,

where 3b aπη= is the hydrodynamic drag. Hence, the general two-particle Smoluchowski

equation in the presence of both convection and a conservative field of force can be written as

( )div 0D U b D cβ −∇ + − ∇ =⎡ ⎤⎣ ⎦v [3]

and the associated current is ( )D U b D cβ= −∇ + − ∇⎡ ⎤⎣ ⎦J v . We can now define the collision

frequency or collision rate across a spherical surface of radius r, concentric with the reference

particle, as

( )2

2 2 +r

0 0

4 v sin

G dS D c D U b c dS

c Ur D c r d c dr r

π π

β

π β μ φ θ θ

= ⋅ = ∇ + ∇ − ⋅⎡ ⎤⎣ ⎦

⎡ ⎤∂⟨ ⟩ ∂⎛ ⎞= + ⟨ ⟩ +⎢ ⎥⎜ ⎟∂ ∂⎝ ⎠⎣ ⎦

∫ ∫

∫ ∫

J n v n

[4]

where dS denotes the element of (spherical) collision surface, while ⟨..⟩ denotes the angular

average. Recall that G is the inward flux of particles through the spherical surface. Therefore,

integration runs strictly over only those orientations (or, equivalently, those pairs of angles θ ,

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φ ) such that rv⋅ = −v n . Hence we will use +rv ( )r to denote the positive part of the radial

component of the fluid velocity, rv ( )r . Thus,

+ r rr r

v ( ) if v ( ) 0v ( ) max(v ( ) , 0)=

0 else⎧ >

= ⎨⎩

r rr r

[5]

Under the approximation that the flow velocity and the concentration profile around the

reference particle are weakly correlated with respect to spatial orientation, which is

equivalent to assuming

+ 2rv ( ) ( ) 0c⟨ − ⟩ =r r , [6]

Eq. 4 becomes

2 +r4 vUG r D b c

r rπ β∂ ∂⎛ ⎞= + + ⟨ ⟩ ⟨ ⟩⎜ ⎟∂ ∂⎝ ⎠

[7]

In the absence of flow ( ( ) 0=v r for any r ), Eq. 7 reduces to the standard (Fuchs) form for

collision rate on a surface of radius r concentric with the stationary particle in a stagnant

medium. In that case, the concentration field is obviously isotropic, ( ) ( )c c r=r (cfr. Eq. 87 in

Verwey and Overbeek (8)).

It is clear that the collision rate given by Eq. 7 is equal to the collision rate one has if the

actual flow field ( )v r were replaced by an effective flow field for aggregation

eff r,eff( ) [v ( ),0,0]r r=v where only the radial component, +r,eff rv ( ) v ( )r ≡ ⟨ ⟩r , is non-zero.

Therefore, the collision rate and colloidal stability of the real system may be described, under

the approximation Eq. 6, by the following effective two-particle Smoluchowski equation

2 2eff2

1 v 0d dU d cr D b c Drr dr dr dr

β ⟨ ⟩⎡ ⎤⎛ ⎞+ ⟨ ⟩ + =⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦ [8]

where the interaction potential is assumed isotropic.

Establishing the boundary condition problem

The boundary conditions for the irreversible aggregation problem are as follows. First, the

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reaction kinetics by which the particle irreversibly sticks (due to van der Waals forces) to the

reference one is taken as infinitely fast at 2r a= , which corresponds to the absorbing

boundary condition 0c⟨ ⟩ = at 2r a= . Second, the bulk concentration ( 0/ 1c c = ) must be

recovered at a certain distance from the reference particle. This condition is often

implemented at large distances, namely at r →∞ , which is always possible for velocity

fields which vanish at infinity. Classic examples such as convective diffusion of a solute to a

free-falling particle can be found in Levich (9), Ch. 2. However, it is well known that in the

case of linear velocity fields, application of the second (far field) boundary condition,

0/ 1c c = at r →∞ , is more complicated, due to a singularity at the domain boundary due to

the term = ⋅v Γ r , where Γ is the velocity gradient tensor. In this case, the Smoluchowski

equation becomes

[ ]div ( ) 0D U D cβ− ∇ + ⋅ − ∇ =Γ r [9]

As first diagnosed by Dhont (10) who used Eq. 9 to study the structure distortion of sheared

nonaggregating suspensions (11), the ⋅Γ r term, being linear in r, overwhelms the other

terms at sufficiently large separations, even for very small shear rates. It was shown (10) that

in the case of hard spheres the extent of separation δ where this occurs decreases with the

Peclet number and is given by 1/ 2/ a Peδ −∼ (10). In other terms, δ defines a boundary-layer

width beyond which convection represents by far the controlling phenomenon (9,10). To

overcome the problem of a singular term at the domain boundary, specific techniques are

required. For example, when the final goal is to determine the structure factor of a suspension,

it is convenient to Fourier-transform the radial domain or equivalently to move to a reciprocal

domain 2 /q r= (see e.g. (12)). Alternatively, within numerical studies in real space, the far-

field boundary condition is usually applied at finite separations, after self-consistently

identifying the location beyond which the concentration profile flattens as a consequence of

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convection becoming predominant (13,14).

At this point, it is useful to consider the boundary-layer structure of Eqs. 8-9. In fact, as is

well known within the theory of convective diffusion (9), due to the boundary-layer

behaviour, convective flux dominates at sufficiently large interparticle separations (which has

been verified numerically e.g. in (13)) since the other terms in the bracket in Eq. 9 become

negligible compared with ⋅Γ r , leaving c const= (see also Batchelor and Green (15)). In fact,

the pure effect of convection in a linear flow-field is to flatten out the concentration profile.

Therefore, it follows, from this point of view, that assuming homogeneity, 0c const c= = , at

separations larger than the boundary-layer is justified. In this work we thus propose using

2r aδ= + instead of r →∞ in the far-field boundary condition. In the language of matched

asymptotic expansions, this would correspond to exactly determining the solution within the

boundary-layer and matching it to the leading order in 1Pe− expansion in the outer layer.

(Note however that the problem of Eq. 9 in real space is substantially more complicated than

standard singularly perturbed equations, due to the aforementioned singularity of ⋅Γ r at the

domain boundary, in addition to the singularly-perturbed behaviour for large Peclet numbers.)

Hence, collision kinetics being uniquely determined by the inner solution, the only

approximation involved is on the location of the far-field boundary-condition, which we

estimate in the following as a function of Peclet number and interaction range.

The width of the boundary-layer δ where the major change in the concentration profile

occurs, and within which diffusion, convection and colloidal interactions are all important,

can be univocally determined from dimensional considerations. As shown in the Supporting

Information, by applying the Π -theorem of dimensional analysis and using the result

1/ 2/ a Peδ −∼ (10), there is a unique power-law (Rayleigh) combination allowed which reads

/ ( / ) /a a Peδ λ∼ . [10]

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When Eq. 10 is compared to the case of non-interacting hard spheres, 1/ 2/ a Peδ −∼ , the

effect of the colloidal interactions on the boundary-layer width enters through the interaction

range λ, which for the screened-Coulomb repulsion is identified as 1λ κ −= , 1κ − being the

Debye length.

Approximate, analytical solution for the aggregation rate

Based on the above boundary-condition problem analysis, let us rewrite the effective two-

particle Smoluchowki equation with shear, Eq. 8, in dimensionless form

2 22 2

2r,eff2

1 1 1 1 ( )( 2) ( 2)( 2) ( 2)

1 ( 2) v 0( 2)

d dC d dU xx x CPe x dx dx Pe x dx dx

d x Cx dx

⎛ ⎞+ + +⎜ ⎟+ + ⎝ ⎠

⎡ ⎤+ + =⎣ ⎦+

[11]

with the boundary conditions

0 at 01 at /

C xC x aδ= == =

[12]

where, for simplicity, we have set ( ) ( )c C x⟨ ⟩ ≡r , since the latter is a function only of the

dimensionless separation ( / ) 2x r a= − . The tilde indicates dimensionless quantities. The

Peclet number is given by 2 3/ 3 /Pe a D a kTγ πηγ= = . Eq. 11 is formally identical to a

stationary one-dimensional Fokker-Planck equation (in spherical geometry) with time-

independent drift and diffusion coefficients (16). The concentration profile in dimensional

form after application of the second boundary condition to Eq. 11 reads

r,eff

/

r,eff0 20 / /

( ) exp ( / v )

exp ( / v )8 ( )( 2)

x

a

x x

a a

c x dx dU dx Pe

G dxc dx dU dx PeaD x x

δ

δ δ

β

βπ

⎧ ⎫= − −⎨ ⎬

⎩ ⎭⎧ ⎫

× + +⎨ ⎬+⎩ ⎭

∫

∫ ∫

G

[13]

The rate is determined from the absorbing boundary condition at contact as

10

0 0/

r,eff20 /

8

2 exp ( / v )( )( 2)

a x

a

D acGdx dx dU dx Pe

x x

δ

δ

π

β=

++∫ ∫G

[14]

Using / ( / ) /a a Peδ λ= , Eq. 14 can be easily integrated numerically and can find direct

application to colloidal systems under laminar flows. For an axisymmetric extensional flow,

the radial component of the velocity field is given by (13,15):

[ ]( )2rv (1/ 2) ( 2) 1 ( ) 3cos 1Ea x A xγ θ= + − − , where AE(x) is the corresponding hydrodynamic

retardation function. With the rescaled effective velocity defined above, we thus obtain

( ) [ ]+r,eff rv ( ) v ( ) 1/ 3 3 ( 2) 1 ( )Ex x A x≡ ⟨ ⟩ = − + −r . Similarly, in the case of simple shear we

have ( ) [ ]+r,eff rv ( ) v ( ) 1/ 3 ( 2) 1 ( )Sx x A xπ≡ ⟨ ⟩ = − + −r , where AS(x) is the hydrodynamic

retardation function for simple shear.

Irreversible aggregation kinetics and colloidal stability in shear

Since we have retained all terms in the governing equation (and constructed the solution

exactly in the inner layer), Eq. 14 is valid for arbitrary thickness of the boundary layer δ . In

particular we observe that in the limit 0Pe → , Eq. 14 reduces to the well-known Fuchs’

formula for the aggregation rate constant (collision rate) in the presence of (conservative)

colloidal interaction forces but in the absence of flow, which reads (8):

20 0

0

8 / 2 exp ( ) / ( )( 2)G D ac dx U x x xπ β∞

= +∫ G . Comparing Eq. 14 to the latter expression leads

to defining a generalized stability coefficient which is valid for arbitrary Pe numbers and

interaction potentials

( / ) /

r,eff20 ( / ) /

2 exp ( / v )( )( 2)

a Pe x

Ga Pe

dxW dx dU dx Pex x

λ

λ

β= ++∫ ∫G

[15]

Thus, the simultaneous presence of fluid motion (convection) and colloidal interactions can

either diminish or augment the rate of coagulation with respect to the case of Brownian hard

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spheres in a stagnant fluid by a factor equal to GW . This represents the most general

description of the aggregative stability of colloids hitherto achieved.

Comparison with previous numerical results

Let us compare the theoretical predictions from Eq. 14 with numerical results where the full

convective diffusion equation, Eq. 3, was solved numerically, by means of a finite difference

method. The colloidal system is composed of particles of radius 100a = nm, interacting via

standard DLVO potential, and convection is induced by laminar axysimmetric extensional

flow (13). The numerically obtained values of ( )c r were then used to determine the

aggregation rate constant from numerical evaluation of Eq. 4 where the collision surface is

the sphere surface of radius 2a. The comparison is shown in Fig. (2) for three different values

of ionic strength and a fixed surface potential equal to -14.7 mV. The colloidal potential U(x)

for the same conditions of the numerical simulations, as well as the hydrodynamic functions

A(x) and ( )xG , have been calculated according to (13). As shown in the figure, the theory is

able to reproduce, with no free parameters, the numerical data for practically all conditions.

In particular, it is seen that the inflection point which marks the transition from a purely-

Brownian like regime at 1Pe < to a shear-dominated regime at 10Pe > is well captured by

the theory. Some underestimation arises in the regime of high Peclet numbers 50 100Pe > − ,

which tends to become more important upon further increasing Pe. Such underestimation is

related to the approximation + 2rv ( ) ( ) 0c⟨ − ⟩ =r r made in the derivation of Eq. 14. In fact, the

spatial correlation between the flow velocity and the concentration field around the stationary

particle would become non-negligible at high Peclet numbers. In this regime, the

randomizing effect of Brownian motion is gradually lost, whereas the angular regions

(relative orientations between particles) where the flow velocity is higher and inwardly-

directed tend to coincide with the regions where the probability of finding incoming particles

is higher. On the whole, the present theory can be employed to estimate the stability and

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initial aggregation kinetics of colloidal systems under arbitrary conditions of ionic strength

and shear flow intensity, up to substantially high Peclet numbers ( 210Pe ∼ ).

Potential barrier crossing under shear is a shear-driven activation process

Under consideration of a potential interaction energy exhibiting a barrier or repulsive

shoulder such as for DLVO-type interactions, it is possible to derive explicit forms for the

two-particle aggregation rate constant and the characteristic time of aggregation. This is done

by approximating the interaction potential to second order at the barrier top (denoted with a

subscript m) and subsequently evaluating the integral in Eq. 14 by means of the steepest-

descent (Laplace’s) method. The detailed derivation is reported in the Supporting Information.

The result for the two-particle aggregation rate constant is

33

2 ( 6 ) /1,1

3m mU Pe U a kTm

ma Uk Pe U e e

kTβ α παηγπαηγα β − + − −′′−′′≈ − = [16]

It is interesting to observe that Eq. 16 is an Arrhenius-type form, with the pre-exponential or

frequency factor 3(3 ) /ma U kTπαηγ ′′− and the activation energy, 36mU aπαηγ− . Note that

mU being a point of maximum, mU ′′ is negative, and it follows that the quantity under the

square-root in the prefactor is always positive. In both parameters the shear rate γ plays a

prominent role. Increasing γ leads to an increase in the collision rate, through the prefactor,

at the same time causing a decrease in the activation energy barrier (thus increasing the

fraction of successful collisions). Further, a critical value of the shear rate can be defined,

which corresponds to a vanishing activation barrier: 3cr / 6mU aγ παη= . When crγ γ<< , the

interaction barrier dominates, and k1,1 increases as Um decreases. When crγ γ>> , instead, the

drive induced by shear overwhelms the barrier, and k1,1 increases as γ increases. Thus, such

critical shear rate marks the transition from a slow aggregation regime, with an activation

delay due to the presence of a potential barrier, to a fast aggregation regime, with no

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activation delay. In fact, if crγ is such that the pre-exponential factor is of order unity, then the

resulting kinetics will be of the same order of purely Brownian diffusion-limited aggregation

in a stagnant fluid at crγ γ= . Any further increase of the shear rate above crγ will then result

in coagulation rates higher than that the diffusion-limited case. The k1,1 value given by Eq. 16

defines a characteristic aggregation time given by (see SI):

3( 6 ) /c 3

1,1

1 1(3 ) /

mU a kT

m

t ek a U kT

παηγ

παηγ−≈ ≈

′′− [17]

An exponential dependence upon γ for the aggregation time has been recently observed in

experiments of shear-induced aggregation of charged suspensions in simple shear (7). Eq. 17

as derived here, provides the theoretical justification to those experimental evidences.

Shear-driven self-accelerating kinetics, rheopexy, and shear-induced gelation (jamming)

The aggregation time and the rate constant under shear display a very strong dependence

upon the colloid size, as is evident from Eqs. 16-17. In the case where the potential is fixed,

the dependence on the colloid radius reads 33 1/ 2 6 /

c (3 / ) a kTt a kT e παηγπαηγ − −≈ . As an example,

with 0.001 Pa sη = ⋅ , 1/ 3α π= and 500γ = s-1, the latter expression amounts to 2.26 if

100a = nm and to 0.14 if 200a = nm. Thus, doubling the colloid radius leads to a reduction

of the characteristic time for aggregation by an order of magnitude. This effect becomes of

paramount importance when considering the long-time evolution of the coagulation process.

For simplicity, let us first consider a system of Brownian drops undergoing complete

coalescence upon aggregation under shear. The differential equation which governs the

dynamic evolution of the population, i.e. the variations with time of ic classes solely

characterized by their size i (where 1,2,3,...,i = ∞ ), is (17)

, ,1

12

ki j i j k j k j

i j k j

dc k c c c k cdt

∞

+ = =

= −∑ ∑ [18]

Based on the above considerations, the rate constants will be approximately

14

[ 6 ( ) / ],

3 ( )m i j i jU a a a a kTi j i j m

i j

a a a a Uk e

kTπηγπαηγ − + +′′+ −

≈ [19]

where the mutual diffusivity of two drops of size i and j respectively is defined as

, (1/ 1/ ) / 6i j i jD kT a a πη= + , leading to 2, ,( ) / 4 3 ( ) /i j i j i j i j i jPe a a D a a a a kTγ πηγ= + = + .

Hence, the rate constants are strongly increasing functions of the sizes of the coalescing drops.

In comparison with the diffusion-limited case in stagnant fluids, where actually the kinetics

slows down as the particles grow by coalescence (because the size dependence is dictated by

diffusion), here we can foresee that, under shear, according to Eq. 19, larger drops will

coalesce much faster, thus leading to a self-accelerating kinetics. In particular, we see that,

for a fixed shear rate, a critical size corresponding to activated drops, cra , can be defined such

that the activation barrier for two drops belonging to this class is zero. Hence, extending these

considerations to non-coalescing (solid) particles, one can predict that at the critical time at

which the average size of the population will reach such critical, activated size, the self-

accelerating kinetics will set in to determine an extremely fast growth. In turn, the growing

structures are responsible for the observed increase of viscosity, i.e. for the observed

rheopexy. An important role, in the growth kinetics, is played by shear-induced breakup of

the drops, or of the clusters in the case of solid particles, which may suppress further growth.

However, breakup phenomena show up only once the size has reached a maximum stable

value which generally depends upon the applied shear rate in a power-law fashion, the

exponent being a function of the structure and mechanics of the growing mesoscopic entities

(drops/clusters) (18,19). In the case of non-coalescing colloids aggregating into (typically

fractal) clusters where the maximum stable value is much larger than the cluster size

necessary to span the entire system, a connected, percolated structure (a gel) will form which

may lead to flow arrest (jamming). In coalescing systems, the growth may eventually result in

phase separation. It is thus suggested, based on Eqs. 18-19, that a dispersion of Brownian

15

particles interacting with a potential barrier and subject to finite, constant shear-rate will first

go through a slow-aggregation regime (induction) characterized by the shear-activated

barrier-hopping process, during which the system remains macroscopically dispersed and

liquid-like. This activation delay, which is roughly given by the characteristic aggregation

time, Eq. 17, will be followed by a very fast growth as soon as the average size of the

population reaches the critical (activation) value to enter the self-accelerating regime,

1/3cr ( / 6 )ma U παηγ= (activated drops/clusters). Clearly, the smaller the size of the activated

clusters cra , the shorter will be the induction time preceding the self-accelerating regime and

thus the transition to the phase-separated or arrested state. These considerations are

summarized in the qualitative diagram for the time evolution of the characteristic size in Fig.

(3). The asymptotic value at long times is due to the onset of breakup at bpa γ∼ , where p is a

breakup exponent (19). A qualitatively similar plot may be inferred for the viscosity, since

the viscosity of a dispersion of aggregates increases with the characteristic size of the latter

through their effective volume fraction.

Conclusions

Starting from the two-body Smoluchowski equation for interacting particles with shear, we

derived an approximate, analytical solution which allows for quantifying the aggregation rate

between colloidal particles interacting with an arbitrary colloidal potential, in an arbitrary

linear (laminar) flow field, at arbitrary Peclet. The predictions, with no fitting parameter, are

in good agreement with previous numerical data, up to high Peclet numbers ( 210∼ ). When an

interaction barrier is present (as for DLVO-interacting systems), a rate-theory for the kinetic

constant of aggregation has been derived which exhibits a typical Arrhenius form and

consists of a frequency factor (proportional to the square root of the shear rate γ ) multiplying

an exponential which gives the probability of successful collisions. The exponential factor

16

reads 3( 6 ) /mU a kTe παηγ− − . The shear rate is thus effective in diminishing the activation barrier Um ,

thus causing a rise in the kinetic constant as large as 5 to 6 orders of magnitude, just above a

critical Peclet value which erases the barrier. This result may generalize Kramers’ rate-theory

(20) to activated barrier-crossing processes driven by shear. Further, it offers the key to

explain the induction period followed by self-accelerating kinetics observed in the rheopectic

behaviour of many complex fluids (1,2) where the microscopic constituents interact via a

potential barrier, including protein-based biological fluids (4-6).

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8. Verwey EJW and Overbeek JTG (1948) Theory of the Stability of Lyophobic Colloids

(Elsevier, New York).

9. Levich VG (1962) Physicochemical Hydrodynamics (Prentice-Hall, Englewood Cliffs, NJ).

17

10. Dhont JKG (1989) On the distortion of the static structure factor of colloidal fluids in

shear flow. J Fluid Mech 204:421-431.

11. Ackerson BJ and Clark NA Sheared colloidal suspensions. (1983) Physica A 118:221-249.

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colloidal suspensions J Fluid Mech 456:239-275.

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on the aggregation of small particles subject to interaction forces. AIChE J 45:1383-1393.

14. Lionberger RA (1998) Shear thinning of colloidal dispersions. J Rheol 42:843-863.

15. Batchelor GK and Green (1972) Determination of bulk stress in a suspension of spherical

particles to order c2. J Fluid Mech 56:401-427.

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15:1-89.

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stresses. Phys Rev E 79:061401.

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chemical reactions. Physica 7:284-304.

FIGURE LEGENDS

Fig 1 Viscosity as a function of time for sheared charge-stabilized emulsions from Ref. [7]. tcr

and tb refer to the onset times of self-accelerating kinetics and breakup, respectively.

Increasing numbers on top of each curve correspond to increasing values of applied shear rate.

As shown in Ref. [7], tcr decreases exponentially with the shear rate, which is suggestive of

18

an activation process.

Fig 2 Comparison between the calculated aggregation rate (lines) based on the proposed

theory (Eq. 14), and the numerical simulations of the full convective diffusion equation (Eq.

3) from (13) for different ionic-strength conditions (symbols). The colloid surface potential

equals -14.7 mV in all the cases.

Fig 3 Qualitative diagram of the time evolution of the characteristic size of growing

mesoscopic structure in a sheared suspension of particles with interaction barrier as suggested

by the rate theory reported here.

19

20

21

22

Supporting Information

Shear-driven activation, aggregation, and rheopexy in sheared

interacting colloids

Alessio Zaccone, Hua Wu, and Massimo Morbidelli

Chemistry and Applied Biosciences, ETH Zurich, CH-8093 Zurich, Switzerland

(Dated: June 2009)

Appendix A: Dependence of the boundary-layer width upon the interparticle repulsion

range

The governing parameters upon which the boundary-layer width δ depends are: λ, a, γ and

D0, where λ denotes the range of (repulsive) interaction. Thus the dimensionless boundary-

layer width / aδ is expressible as a dimensionless combination of the governing parameters

(Rayleigh)

0/ l m n pa a Dδ λ γ∼ [1]

From the boundary-layer behaviour of the Smoluchowski equation for non-interacting hard

spheres, it is known that 1/ 2/ a Peδ −∼ (1), which identifies 1/ 2n = − and 1/ 2p = . Hence

considering that [ ] [ ]a Lλ = = and 1/ 2 1/ 20[ ] [ ]D Lγ − = , it obviously follows that

1 0l m+ + = [2]

According to the Π -theorem of dimensional analysis, / aδ has to be expressed in terms of

two independent dimensionless groups, / aλ and 20/a Dγ (the latter defines the Peclet

number), as 20( / , / )a a a Dδ λ γ= Φ , because a and γ are parameters with independent

dimension (2). This fixes m as

3 / 2m = − [3]

It is therefore concluded that the dimensionless boundary-layer width / aδ is of the order of

23

/ ( / ) /a a Peδ λ∼ . [4]

Appendix B: Derivation of Eq. 16 for the shear-driven aggregation rate with potential

barrier

Let us consider the case of a high potential barrier in the interaction between particles (as for

charge-stabilized colloids, according to standard DLVO theory). Further, we will neglect the

effect of the hydrodynamic retardation on the velocity field [i.e., ( ) 0A x = ] so that the

effective velocity, as defined in the manuscript, reads r,effv ( 2)xα= − + , where α is a

numerical coefficient which depends uniquely upon the type of flow (e.g. 1/ 3α π= for

simple shear).

Then, the integrand in the inner integral in the denominator on the r.h.s. of Eq. 14 in the

manuscript reduces to

2r,eff ( / ) /

( / ) /

( / v ) ( 2)2 2

x

x a Pea Pe

dx dU dx Pe U U Pe xλ

λ

α αβ β β λ=

+ ≈ − − + +∫ [5]

Clearly when Pe is not too high, ( / ) /

0x a Pe

Uλ=

≈ . Thus the denominator on the r.h.s. of Eq.

14 in the manuscript becomes

/

( / ) /

r,eff20 ( / ) /

( / ) // 2 2

20

2 exp ( / v )( 2)

2 exp ( 2) / 2( 2)

x Pe

a Pe x

a Pe

a PeU

dx dx dU dx Pex

dxe U Pe xx

λ

λ

λ

λαλ β

β

β α=−

+ ≈+

⎡ ⎤≈ − +⎣ ⎦+

∫ ∫

∫

G

G

[6]

Since U goes through a potential maximum (barrier) in [0, ( / ) / ]x a Peλ∈ , so does the

function 2( 2) / 2U Pe xβ α− + . The argument of the exponential can thus be expanded near

the maximum up to second order

2 2 2( 2) / 2 ( 2) / 2 ( )( )m m m mU Pe x U Pe x U Pe x xβ α β α α′′− + ≈ − + + − − , [7]

24

where the subscript m indicates quantities evaluated at the point of maximum. We can thus

evaluate the remaining integral

( / ) /2

20

exp ( )( )( )( 2)

a Pe

m mdx U Pe x x

x x

λ

β α′′⎡ ⎤− −⎣ ⎦+∫ G [8]

by the method of steepest descent (Laplace’s method) to finally obtain

( / ) /2

/ 2( 2) / 2

2

2 2( 2) ( )

x a Pe

m m

UU Pe x

Gm m m

eW ePe U x x

λαλ ββ απ

α β

=−− +≈

′′− + G [9]

More precise approximations can be obtained by considering terms of order higher than

quadratic in the expansion Eq. 7 (3).

The kinetic equation for the rate of change of the concentration of primary particles reads

20 0/ (16 / )Gdc dt D a W cπ= − , where GW is the generalized stability coefficient given by Eq. 9.

Integration yields the time evolution of the concentration, 0 c( ) /(1 / )c t c t t= + , where

1c 0 0(16 / )Gt D ac Wπ −= is the characteristic time of aggregation. Its reciprocal value defines

the kinetic constant for the aggregation of primary particles, 1,1 0 016 / Gk D ac Wπ= .

Using Eq. 9, in view of being 2 2mx + ≈ , the explicit form for the two-particle aggregation

rate constant, k1,1, is thus derived as

33

2 ( 6 ) /1,1

3m mU Pe U a kTm

ma Uk Pe U e e

kTβ α παηγπαηγα β − + − −′′−′′≈ − = [10]

which is Eq. 16 in the manuscript.

References 1. Dhont JKG (1989) On the distortion of the static structure factor of colloidal fluids in shear

flow. J Fluid Mech 204:421-431.

2. Barenblatt GI (1996) Scaling, Self-similarity, and Intermediate Asymptotics, pp. 39-43

(Cambridge University Press, Cambridge).

25

3. Risken H (1996) The Fokker-Planck Equation (Springer, Berlin).