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: [email protected].
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
1
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
2
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 structurebuilding 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
3
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 sheardriven 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 (r ) ≡ c(r ) for finding a second particle at distance r from a
reference one is then normalized such that c(r ) = c0 c(r ) , where c0 is the bulk concentration.
The evolution equation thus reads
∂c / ∂t = div( D∇c − β DKc)
[1]
4
where D is the mutual diffusion coefficient of the particles ( D = 2 D0G (r ) , D0 being the
diffusion coefficient of an isolated particle and G (r ) the hydrodynamic function for viscous
retardation), β = 1/ kT is the Boltzmann factor, and K is the external force. According to this
equation, the associated stationary current is given by J = ( β DK − D∇ ) c . Hence, when
steady-state is reached (after a transient of order a 2 / D0 ), ∂c(r ) / ∂t = 0 and
div ( β DK − D∇ ) c = 0
[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 v(r ) = [v r ,vθ ,vφ ] , 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 K (r ) = −∇U (r ) + bv (r ) ,
where b = 3πη a 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 ⎡⎣ β D ( −∇U + bv ) − D∇ ⎤⎦ c = 0
[3]
and the associated current is J = ⎡⎣ β D ( −∇U + bv ) − D∇ ⎤⎦ c . We can now define the collision
frequency or collision rate across a spherical surface of radius r, concentric with the reference
particle, as
G=
∫ J ⋅ n dS
=
∫ ⎡⎣ D∇c + β D ( ∇U − bv ) c ⎤⎦ ⋅ n dS
2π
π
⎡⎛ ∂〈c〉
⎤
∂U
⎞
2
= 4π r D ⎢⎜
+β
〈 c〉 ⎟ + μ r ∫ dφ ∫ v r+ c sin θ dθ ⎥
∂r
⎠
0
0
⎣⎝ ∂r
⎦
2
[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 ,
5
φ ) such that v ⋅ n = − v r . Hence we will use v +r (r ) to denote the positive part of the radial
component of the fluid velocity, v r (r ) . Thus,
⎧ v (r )
v +r (r ) = max(v r (r ) , 0)= ⎨ r
⎩0
if v r (r ) > 0
[5]
else
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
〈 v +r (r ) − c(r )〉 2 = 0 ,
[6]
Eq. 4 becomes
∂U
⎛∂
⎞
+ b〈 v +r 〉 ⎟ 〈 c〉
G = 4π r 2 D ⎜ + β
∂r
⎝ ∂r
⎠
[7]
In the absence of flow ( v(r ) = 0 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(r ) = c(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
v eff (r ) = [v r,eff (r ), 0, 0] where only the radial component, v r,eff ( r ) ≡ 〈 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
1 d ⎡ 2 ⎛ dU
d 〈 c〉 ⎤
⎞
r D⎜β
+ bv eff ⎟ 〈 c〉 + Dr 2
=0
2
⎢
r dr ⎣
dr ⎥⎦
⎝ dr
⎠
[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
6
reaction kinetics by which the particle irreversibly sticks (due to van der Waals forces) to the
reference one is taken as infinitely fast at r = 2a , which corresponds to the absorbing
boundary condition 〈 c〉 = 0 at r = 2a . Second, the bulk concentration ( c / c0 = 1 ) 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,
c / c0 = 1 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 [ (− β D∇U + Γ ⋅ r ) − D∇ ] c = 0
[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 δ / a ∼ Pe −1/ 2 (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 q = 2 / r (see e.g. (12)). Alternatively, within numerical studies in real space, the farfield boundary condition is usually applied at finite separations, after self-consistently
identifying the location beyond which the concentration profile flattens as a consequence of
7
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, c = const = c0 , at
separations larger than the boundary-layer is justified. In this work we thus propose using
r = δ + 2a 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 Pe−1 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
δ / a ∼ Pe −1/ 2 (10), there is a unique power-law (Rayleigh) combination allowed which reads
δ / a ∼ (λ / a ) / Pe .
[10]
8
When Eq. 10 is compared to the case of non-interacting hard spheres, δ / a ∼ Pe −1/ 2 , 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 λ = κ −1 , κ −1 being the
Debye length.
Approximate, analytical solution for the aggregation rate
Based on the above boundary-condition problem analysis, let us rewrite the effective twoparticle Smoluchowki equation with shear, Eq. 8, in dimensionless form
⎞
d
dC 1
d ⎛
1
1
1
2 dU ( x )
+
C⎟
( x + 2)2
⎜ ( x + 2)
2
2
Pe ( x + 2) dx
dx Pe ( x + 2) dx ⎝
dx
⎠
d ⎡
1
( x + 2) 2 v r,eff C ⎤ = 0
+
⎦
( x + 2) 2 dx ⎣
[11]
with the boundary conditions
C = 0 at
x=0
C =1
x =δ /a
at
[12]
where, for simplicity, we have set 〈 c(r )〉 ≡ C ( x ) , since the latter is a function only of the
dimensionless separation x = (r / a ) − 2 . The tilde indicates dimensionless quantities. The
Peclet number is given by Pe = γ a 2 / D = 3πηγ a 3 / kT . Eq. 11 is formally identical to a
stationary one-dimensional Fokker-Planck equation (in spherical geometry) with timeindependent drift and diffusion coefficients (16). The concentration profile in dimensional
form after application of the second boundary condition to Eq. 11 reads
x
⎧
⎫
c( x ) = ⎨exp ∫ dx ( − β dU / dx − Pe v r,eff ) ⎬
⎩
⎭
δ /a
x
x
⎧
⎫
G
dx
× ⎨c0 +
exp
dx ( β dU / dx + Pe v r,eff ) ⎬
2
∫
∫
8π aD0 δ / a G ( x )( x + 2)
⎩
⎭
δ /a
The rate is determined from the absorbing boundary condition at contact as
9
[13]
G=
δ /a
2∫
0
8π D0 ac0
[14]
x
dx
exp ∫ dx( β dU / dx + Pe v r,eff )
G ( x)( x + 2) 2
δ /a
Using δ / a = (λ / a ) / Pe , 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):
v r = (1/ 2)γ a ( x + 2) [1 − AE ( x) ] ( 3cos 2 θ − 1) , where AE(x) is the corresponding hydrodynamic
retardation function. With the rescaled effective velocity defined above, we thus obtain
(
)
v r,eff ( x) ≡ 〈 v r (r )〉 = − 1/ 3 3 ( x + 2) [1 − AE ( x)] . Similarly, in the case of simple shear we
+
have v r,eff ( x) ≡ 〈 v r (r )〉 = − (1/ 3π ) ( x + 2) [1 − AS ( x)] , 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 Pe → 0 , 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):
∞
G = 8π D0 ac0 / 2 ∫ dx exp β U ( x) / G ( x)( x + 2) 2 . Comparing Eq. 14 to the latter expression leads
0
to defining a generalized stability coefficient which is valid for arbitrary Pe numbers and
interaction potentials
( λ / a ) / Pe
WG = 2
∫
0
dx
exp
G ( x)( x + 2) 2
x
∫
dx( β dU / dx + Pe v r,eff )
[15]
( λ / a ) / Pe
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
10
spheres in a stagnant fluid by a factor equal to WG . 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 a = 100 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 G ( x) , 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 purelyBrownian like regime at Pe < 1 to a shear-dominated regime at Pe > 10 is well captured by
the theory. Some underestimation arises in the regime of high Peclet numbers Pe > 50 − 100 ,
which tends to become more important upon further increasing Pe. Such underestimation is
related to the approximation 〈 v +r (r ) − c(r )〉 2 = 0 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 inwardlydirected 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
11
initial aggregation kinetics of colloidal systems under arbitrary conditions of ionic strength
and shear flow intensity, up to substantially high Peclet numbers ( Pe ∼ 102 ).
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 steepestdescent (Laplace’s) method. The detailed derivation is reported in the Supporting Information.
The result for the two-particle aggregation rate constant is
k1,1 ≈ α Pe − β U m′′ e− βU m + 2α Pe =
3παηγ a 3 − U m′′ − (U m −6παηγ a3 ) / kT
e
kT
[16]
It is interesting to observe that Eq. 16 is an Arrhenius-type form, with the pre-exponential or
frequency factor
(3παηγ a 3 − U m′′ ) / kT and the activation energy, U m − 6παηγ a 3 . Note that
U m being a point of maximum, U m′′ 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: γ cr = U m / 6παη a 3 . When γ << γ cr , the
interaction barrier dominates, and k1,1 increases as Um decreases. When γ >> γ cr , 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
12
activation delay. In fact, if γ cr 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 γ = γ cr . Any further increase of the shear rate above γ cr 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):
tc ≈
3
1
1
≈
e(U m −6παηγ a ) / kT
k1,1
(3παηγ a 3 − U m′′ ) / kT
[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 tc ≈ (3παηγ a 3 / kT ) −1/ 2 e −6παηγ a
3
/ kT
. As an example,
with η = 0.001 Pa ⋅ s , α = 1/ 3π and γ = 500 s-1, the latter expression amounts to 2.26 if
a = 100 nm and to 0.14 if a = 200 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 ci classes solely
characterized by their size i (where i = 1, 2,3,..., ∞ ), is (17)
∞
dck 1
= ∑ ki , j ci c j − ck ∑ k j , k c j
dt 2 i + j = k
j =1
[18]
Based on the above considerations, the rate constants will be approximately
13
ki , j ≈
3παηγ (ai + a j )ai a j − U m′′
kT
[ −U m + 6πηγ ( ai + a j ) ai a j / kT ]
e
[19]
where the mutual diffusivity of two drops of size i and j respectively is defined as
Di , j = kT (1/ ai + 1/ a j ) / 6πη , leading to Pei , j = γ (ai + a j ) 2 / 4 Di , j = 3πηγ (ai + a j )ai a j / kT .
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, acr , 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 selfaccelerating 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
14
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,
acr = (U m / 6παηγ )1/ 3 (activated drops/clusters). Clearly, the smaller the size of the activated
clusters acr , 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 ab ∼ γ p , 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 ( ∼ 102 ). 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
15
reads e− (U m −6παηγ a
3
) / kT
. 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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thixotropic, yielding fluids J Rheol 46:573-589.
3. Vermant J and Solomon MJ (2005) Flow-induced structure in colloidal suspensions J Phys:
Cond Matter 17:R187-R216.
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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
17
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.
18
19
20
21
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 boundarylayer width δ / a is expressible as a dimensionless combination of the governing parameters
(Rayleigh)
δ / a ∼ λ l a mγ n D0p
[1]
From the boundary-layer behaviour of the Smoluchowski equation for non-interacting hard
spheres, it is known that δ / a ∼ Pe −1/ 2 (1), which identifies n = −1/ 2 and p = 1/ 2 . Hence
considering that [λ ] = [a] = L and [γ ]−1/ 2 [ D0 ]1/ 2 = L , it obviously follows that
l + m +1 = 0
[2]
According to the Π -theorem of dimensional analysis, δ / a has to be expressed in terms of
two independent dimensionless groups, λ / a and γ a 2 / D0 (the latter defines the Peclet
number), as δ = aΦ (λ / a, γ a 2 / D0 ) , because a and γ are parameters with independent
dimension (2). This fixes m as
m = −3 / 2
[3]
It is therefore concluded that the dimensionless boundary-layer width δ / a is of the order of
22
δ / a ∼ (λ / a ) / Pe .
[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., A( x) = 0 ] so that the
effective velocity, as defined in the manuscript, reads v r,eff = −α ( x + 2) , where α is a
numerical coefficient which depends uniquely upon the type of flow (e.g. α = 1/ 3π 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
x
∫
dx( β dU / dx + Pe v r,eff ) ≈ β U − β U
( λ / a ) / Pe
Clearly when Pe is not too high, U
x = ( λ / a ) / Pe
x = ( λ / a ) / Pe
−
α
2
Pe( x + 2) 2 +
α
2
λ
[5]
≈ 0 . Thus the denominator on the r.h.s. of Eq.
14 in the manuscript becomes
( λ / a ) / Pe
2
∫
0
dx
exp
G ( x + 2) 2
αλ / 2 − β U
≈ 2e
x
∫
dx( β dU / dx + Pe v r,eff ) ≈
( λ / a ) / Pe
( λ / a ) / Pe
x= λ / Pe
∫
0
[6]
dx
exp ⎡⎣ β U − α Pe( x + 2) 2 / 2 ⎤⎦
G ( x + 2) 2
Since U goes through a potential maximum (barrier) in x ∈ [0, (λ / a ) / Pe ] , so does the
function β U − α Pe( x + 2) 2 / 2 . The argument of the exponential can thus be expanded near
the maximum up to second order
βU − α Pe( x + 2) 2 / 2 ≈ βU m − α Pe( xm + 2) 2 / 2 + (U m′′ − α Pe)( x − xm ) 2 ,
23
[7]
where the subscript m indicates quantities evaluated at the point of maximum. We can thus
evaluate the remaining integral
( λ / a ) / Pe
∫
0
dx
exp ⎡⎣( βU m′′ − α Pe)( x − xm ) 2 ⎤⎦
G ( x)( x + 2) 2
[8]
by the method of steepest descent (Laplace’s method) to finally obtain
αλ / 2 − β U
WG ≈
x = ( λ / a ) / Pe
2
2π
2e
e βU m −α Pe ( xm + 2) / 2
2
α Pe − β U m′′ ( xm + 2) G ( xm )
[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
dc / dt = −(16π D0 a / WG )c02 , where WG is the generalized stability coefficient given by Eq. 9.
Integration yields the time evolution of the concentration, c(t ) = c0 /(1 + t / tc ) , where
tc = (16π D0 ac0 / WG ) −1 is the characteristic time of aggregation. Its reciprocal value defines
the kinetic constant for the aggregation of primary particles, k1,1 = 16π D0 ac0 / WG .
Using Eq. 9, in view of being xm + 2 ≈ 2 , the explicit form for the two-particle aggregation
rate constant, k1,1, is thus derived as
k1,1 ≈ α Pe − β U m′′ e− βU m + 2α Pe =
3παηγ a 3 − U m′′ − (U m −6παηγ a3 ) / kT
e
kT
[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).
24
3. Risken H (1996) The Fokker-Planck Equation (Springer, Berlin).
25