Manipulating shear-induced non-equilibrium
transitions in colloidal films by feedback control
Tarlan A. Vezirov,
a
Sascha Gerloff
b
and Sabine H. L. Klapp
b
Using Brownian Dynamics (BD) simulations we investigate non-equilibrium transitions of sheared colloidal
films under controlled shear stress s
xz
. In our approach the shear rate _
gis a dynamical variable, which
relaxes on a time scale s
c
such that the instantaneous, configuration-dependent stress s
xz
(t) approaches
a pre-imposed value. Investigating the dynamics under this “feedback-control”scheme we find unique
behavior in regions where the flow curve s
xz
(_
g) of the uncontrolled system is monotonic. However, in
non-monotonic regions our method allows to select between dynamical states characterized by
different in-plane structure and viscosities. Indeed, the final state strongly depends on s
c
relative to an
intrinsic relaxation time of the uncontrolled system. The critical values of s
c
are estimated on the basis of
a simple model.
1. Introduction
Somatter under shear ow can display rich nonlinear
behaviour involving transitions between different dynamical
states,
1,2
spontaneous spatial symmetry-breaking,
3
shear-band-
ing
4–7
(for recent reviews, see ref. 8–12), rheochaos
13–15
and
heterogeneous local dynamics.
16,17
These intriguing phenomena
oen strongly affect the rheological properties of the system.
Understanding shear-induced effects in, e.g., complex surfac-
tant solutions
18
or liquid crystals,
1,2
colloids,
19–21
so
glasses,
16,17,22
and active uids,
23
is thus a major current topic
connecting non-equilibrium physics and somaterial science.
A quantity of particular interest is the ow (or constitutive)
curve,
7,12
that is, the shear stress sas function of the shear rate
_
g, both of which can serve as experimental control parameters.
In many examples, the curve s(_
g) behaves not only nonlinear
(indicating shear-thinning,
24,25
shear-thickening,
25,26
sometimes
connected irregular (chaotic) rheological response
15,27
), but
becomes also multivalued, i.e. different shear rates lead to the
same stress. In complex uids of e.g. wormlike micelles, this
multivalued property is in fact a universal indicator of a shear-
banding instability, specically, gradient banding, where the
(formerly homogeneous) system separates in gradient direction
into coexisting bands characterized by a smaller and a larger
local shear rate
12
(note that this is different from the more exotic
vorticity banding, i.e., the formation of bands along the vorticity
direction as discussed e.g., in ref. 11, 12 and 28). In so
(colloidal) glasses, multivalued functions s(_
g) occur as transient
phenomena aer a sudden switch-on of shear stress (Bau-
schinger effect),
29
or in the vicinity of the so-called yield stress;
30
in these systems one observes strong dynamical heterogene-
ities.
17
A further intriguing feature is that close to such non-
monotonicities, the system's behaviour strongly depends on
whether one uses the shear stress or the shear rate as a control
parameter.
14,31
In fact, both choices are common in rheological
experiments.
22,32,33
Here we present BD computer simulation results of yet
another system with multivalued stress–shear curve, that is, a
thin colloidal lm sheared from an (equilibrium) state within
in-plane crystalline order in a planar Couette geometry. As
shown in previous experimental
34
and simulation
21,35
studies,
such lms can display successive non-equilibrium transitions
from square crystalline over molten into hexagonal states. Here
we demonstrate that the structural transitions lead again to
non-monotonic ow curves, with a shape reminding that of
materials which perform a solid-to-liquid transition beyond a
critical (yield) stress.
36
Based on this nonlinear behaviour, we then investigate the
lms in presence of controlled shear stress. In fact, so far
most simulations of sheared colloids have been done under
xed shear rate _
g, exceptions being e.g. ref. 17, 30 and 37,
where constant shas been realized by xing the force acting
on the atoms forming the walls. Here we introduce an alter-
native, easy-to-apply scheme to control swhich has been
previously used by us in continuum approaches of sheared
liquid crystals.
31
In that scheme _
g(which directly enters the
BD equations of motion) becomes a dynamical variable whose
time dependence is governed by a relaxation equation
involving a time scale s
c
.Therelaxationisbasedonthe
difference between the instantaneous (conguration-depen-
dent) stress s(t)andapreimposedvalues
0
.Theideaof
a
Institut f¨
ur Theoretische Physik, Technische Universit¨
at Berlin, Hardenbergstr. 36,
b
Institut f¨
ur Theoretische Physik, Technische Universit¨
at Berlin, Hardenbergstr. 36, D-
10623 Berlin, Germany
Cite this: Soft Matter,2015,11,406
Received 28th June 2014
Accepted 5th November 2014
DOI: 10.1039/c4sm01414f
www.rsc.org/softmatter
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adapting the shear rate _
gis inspired by experiments of stress-
controlled systems.
32,38
Our scheme differs from earlier
schemes
17,30,37
where the desired value s
0
is imposed instan-
taneously. Moreover, due to the coupling to the particle
positions our method corresponds to a “feedback”(closed-
loop) control scheme, which is similar in spirit to e.g. a
Berendsen thermostat for temperature control.
39
However,
here the choice for s
c
is found to be crucial for the observed
dynamical behaviour. In particular we demonstrate that, if
our scheme is applied within the multivalued regime of s(_
g),
the nal state strongly depends on the magnitude of s
c
rela-
tive to important intrinsic time scales of the system. Thus, the
stress-control can be used to deliberately select a desired
dynamical state.
2. Model and simulation details
We consider a model colloidal suspension where two particles
with distance r
ij
interact via a combination of a repulsive
Yukawa potential, u
Yuk
(r
ij
)¼Wexp[kr
ij
]/r
ij
and a repulsive so-
sphere potential u
SS
(r
ij
)¼43
SS
(d/r
ij
)
1/2
involving the particle
diameter d.
21
The interaction parameters are set in accordance
to a real suspension of charged silica macroions [for details, see
ref. 40 and 41]. Specically, at the density considered (see
below), the interaction strength W/(k
B
Td)z123 (where k
B
Tthe
thermal energy) and the inverse screening length kz3.2d.
Spatial connement is modeled by two plane parallel, smooth,
uncharged surfaces separated by a distance L
z
along the z
direction and of innite extent in the x–yplane. We employ a
purely repulsive uid-wall decaying as z
9
[see ref. 21]. This is
motivated by previous investigations of the equilibrium layer
structure, where we found a very good agreement with
experiments.
40,41
Our investigations are based on standard BD simulations in
three dimensions, where the position of particle iis advanced
according to
42
riðtþdtÞ¼riðtÞþ D0
kBTFiðtÞdtþdWiþg
:ziðtÞdtex;(1)
where F
i
is the total conservative force acting on particle i.
Within the framework of BD, the inuenceofthesolventon
each colloidal particle is mimicked by a friction constant and
a random Gaussian displacement dW
i
. The friction constant
is set to (D
0
/k
B
T)
1
,whereD
0
is the short-time diffusion
coefficient. The value dW
i
has zero mean and variance 2D
0
dt
for each Cartesian component. The time scale of the system
was set to s¼s
2
/D
0
,whichdenes the so-called Brownian
time. We impose a linear shear prole [see last term in
eqn (1)] representing ow in x- and gradient in z-direction,
characterized by uniform shear rate _
g.Wenotethat,despite
the application of a linear shear prole,thereal,steady-state
ow prole can be nonlinear.
43
This approximation has also
been employed in other recent simulation studies of sheared
colloids;
19,44,45
the same holds for the fact that we neglect
hydrodynamic interactions.
One quantity of prime interest in our study is the x–z
component of the stress tensor,
sxz ¼*1
VX
iX
j.i
Fx;ijzij+:(2)
Thus, we consider only the conguration-dependent contri-
bution to s
xz
; the kinematic contribution (which involves the
velocity components in x- and z-direction) is negligible under
the highly conned conditions here. We note that, apart from
the kinematic contribution, eqn (2) also neglects higher-order
contributions involving gradients.
46,47
In continuum
approaches, one typically includes non-local terms which are
essential for the description of interfaces between shear-
bands.
7,14
The importance of such terms in our highly conned
system, which is characterized by pronounced layer formation,
remains to be investigated.
Based on the shear stress, we introduce a feedback scheme
as follows. Starting from an initial value for _
gwe calculate, in
each time step, the conguration-dependent stress s
xz
from eqn
(2) and adjust _
gvia the relaxation equation
d
dtg
:¼1
sc
s0sxzðtÞ
h0
;(3)
where s
0
is a pre-imposed value of s
xz
, and s
c
determines the
time scale of relaxation. Also, h
0
is the shear viscosity obtained
for _
g/0 (Newtonian regime). This control scheme is inspired
by experiments under xed stress [see, e.g. ref. 32], where the
adaptation of the shear rate to a new stress value always takes a
nite time.
From a more formal point of view, we note that through eqn
(3), _
gbecomes an additional dynamical variable. Therefore, and
since s
xy
(t) depends on the instantaneous conguration {r
i
(t)} of
the particles, simultaneous solution of the N+ 1 equations of
motion (1) and (3) forms a closed feedback loop with global
coupling. Interestingly, this feedback scheme is in accordance
with the common view that, in a stable system, the shear stress
s
xz
should increase with the applied shear rate. This can be
shown (at least for a homogeneous system) by a linear stability
analysis of eqn (3) as outlined in Appendix A.
In our numerical calculations, we focus on systems at high
density, specically rd
3
¼0.85, and strong connement, L
z
¼
2.2d. The corresponding equilibrium system forms a colloidal
bilayer with crystalline in-plane structure.
41
We also show some
data with L
z
¼3.2d, yielding three layers. The values L
z
¼2.2 and
3.2 have been chosen because the equilibrium lattice structure
is square
41
(other values would lead to hexagonal equilibrium
structures which do not show the shear-induced transitions
discussed here). The number of particles was set to N¼1058
and 1587, the width of the shear cell to Lz23.8dand 24.2dfor
L
z
¼2.2dand 3.2d, respectively. Periodic boundary conditions
were applied in ow (x) and vorticity (y) direction. The time step
was set to dt¼10
5
swhere s¼d
2
/D
0
the time unit.
The system was equilibrated for 10 10
6
steps (i.e. 100s).
Then the shear force was switched on. Aer the shearing was
started the simulation was carried out for an additional period
of 100s. During this time the steady state was reached. Only
aer these preparations we started to analyze the considered
systems.
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3. Shear-induced transitions
We start by considering ow curves for systems at constant _
g.
The functions s
xz
(_
g) for both, two- and three layer systems are
plotted in Fig. 1, where we have included data for the viscosities
h¼s
xz
/_
g. Note that we have dened hvia the externally applied
rate _
grather than via the effective shear rate within the system
(characterized by the average velocity of the layers
21
), which can
show signicant deviations from _
g. As seen from Fig. 1, both the
bi- and the trilayer systems are characterized by a non-
monotonic ow curve, accompanied by pronounced shear-
thinning. At small shear rates, the systems display linear stress,
related to an Newtonian response of the square in-plane lattice
structure. An exemplary simulation snapshot for the three-layer
system is shown in Fig. 2a [see ref. 21 for corresponding results
for the bilayer]. In fact, within the Newtonian regime (square
state) the layer velocities are zero, i.e., the particles are
“locked”.
21
In this state the lattice structure persists and the
system reacts to the displacement of the particles in elastical
manner. Further increase of the shear rate then destroys the
square order. In the bilayer, the system then enters a “shear-
molten”state characterized by the absence of translational
order within the layers (as indicated, e.g., by a non-zero long-
time diffusion constant in lateral directions
21
). At the same
time, the entire system starts to ow, that is, the layer velocities
increase from zero to non-zero values.
21
From the function
s
xz
(_
g) plotted in Fig. 1a, the appearance of the shear-molten
state is indicated by a sudden decrease of s
xz
, implying the
onset of shear-thinning. In fact, with the shear-molten regime
the slope of the ow curve is negative. For bulk systems, such a
behavior implies that the system is mechanically unstable.
48,49
Here we are considering a strongly conned system, where the
macroscopic arguments cannot be immediately applied.
Nevertheless, it is an interesting question to which extent the
ow curve pertains to a true steady state in the regime where the
shear rate has values corresponding to shear-molten congu-
rations. Investigating the shear stress as function of strain (see
Appendix B) we nd that s
xz
assumes indeed a constant value on
the time scale of our simulations; however, the relaxation time
is extremely long (see next section). We also mention that our
observation of shear-molten (long-time) congurations in the
regime, where s
xz
decreases with _
g, is consistent with ndings
in an earlier theoretical study of a colloidal suspension under
shear.
50
Somewhat different behaviour is found in the trilayer system
which displays, before melting, an intermediate state [see
Fig. 2b]: this state is characterized by a non-zero layer velocity.
In addition, the middle layer separates into two sublayers, in
which the particles are ordered in “lanes”[see inset of Fig. 2b]
and move with the velocity of the corresponding outer layer (a
more detailed discussion of this “laned”state will be given
elsewhere
51
). Only further increase of _
gthen yields a shear-
molten state characterized by a decreasing ow curve (in
analogy to the bilayer). Finally, both systems transform into a
state with in-plane hexagonal ordering [see Fig. 2c] and low
viscosity. In this hexagonal state the layer velocity is non-zero,
21
that is, the systems ows. As demonstrated earlier by us
21
the
mechanism of relative motion involves collective oscillations of
the particles around lattice sites, consistent with recent exper-
iments of 3D sheared colloidal crystals.
20
Regarding the stress,
we see that the hexagonal regime is (in both systems) charac-
terized by a slight increase of s
xz
with _
g. As a consequence, there
is a parameter range (indicated as region “II”in Fig. 1) where
the ow curve is multivalued, that is, different _
glead to the same
s
xz
. In many contexts (such as for worm-like micelles), multi-
valued ow curves are associated with the phenomenon of
shear-banding, that is, the separation of the system into spatial
regions with different shear rates. In our case, where the system
consists of two or three layers such a separation can not occur.
Instead, the non-monotonic stress curve is a consequence of the
structural transitions of the system induced by the shear.
4. Intrinsic time scales
Before exploring the impact of shear–stress control, which
involves a time scale itself through the parameter s
c
[see eqn
(3)], we take a closer look at the intrinsic time scales
Fig. 1 (Color online) steady state shear stress and shear viscosity
(insets) for bi- and trilayer systems as function of the applied shear rate.
Regions indicated as I, II, III are discussed in the main text.
Fig. 2 (Color online) snapshots of a colloidal trilayer at three different
shear rates corresponding to (a) square, (b) laned, and (c) hexagonal
state. In the main figures, the three colours correspond to particles in
the three layers. In the insets, the two colours indicate the particles of
the middle layer which are positioned at z> 0 (red) and z< 0 (yellow),
respectively (where z¼0 is the middle plane).
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characterizing the uncontrolled systems. We focus on the
bilayer (the same ndings apply qualitatively on the trilayer)
and consider the response of the unsheared equilibrium
system, which is in a square state, to a sudden switch-on at time
s
on
of shear with rate _
g
new
. The resulting time dependence of
the instantaneous stress is plotted in Fig. 3.
If _
g
new
shas a value pertaining to the square state, the shear
stress jumps at s
on
to a non-zero value but then settles quickly to
its steady-state value [see Fig. 1]. At shear rates corresponding to
the shear-molten state we can also observe a relaxation at some
non-zero value. However, it should be emphasized that this
value is transient in character. The true, steady state value is only
achieved at much longer simulation times (see the stress–strain
relations presented in Appendix B). Finally, for shear rates
related to the hexagonal steady state ( _
g
new
s>_
g
hex
sz257, [see
Fig. 1]), we observe a well-pronounced stress overshoot, similar
to what is observed e.g. in soglassy systems,
29
wormlike
micelles
52
and polymer melts.
53
Closer inspection shows that
the actual value of s
1
as well as the functional behavior of s
xz
(t)
strongly depends on the distance between _
g
new
sand the
threshold between shear-molten and hexagonal state, _
g
hex
s: the
smaller this distance is, the larger becomes s
1
, and the more
sensitive it is against small changes of the shear rate. Moreover,
a sudden quench deep into the hexagonal state leads to an
oscillatory relaxation of the stress s
xz
(t) [see curves _
g
new
s¼400,
500], with s
1
(which now corresponds to the relaxation time of
the envelope) being still quite large. Taken together, for _
g
new
s>
_
g
hex
s,s
1
can be tted according to (see inset in Fig. 3)
si¼ai
g
:
newsg
:
is
bi;(4)
where we nd a
1
/s¼0.21, b
1
¼0.52 (setting _
g
1
s¼_
g
hex
s). The
oscillations occurring at large _
g
new
induce yet a different
time scale s
os
, which is smaller than s
1
. Specically, we nd
s
os
/s¼O(10
3
). The physical reason for these oscillations is the
“zig-zag”motion of particles in adjacent layers.
21
This motion is
accompanied by periodic variations of nearest-neighbor
distances, and thus, pair forces, which eventually leads to
oscillations of s
xz
(t).
Furthermore it is interesting to relate the relaxation times
emerging from Fig. 3 to the structural transition from square to
hexagonal state in Fig. 1. To this end we consider the Peclet
number Pe ^
_
gs
Pe
where s
Pe
is a “typical”relaxation time.
54
Identifying s
Pe
with s
1
and considering shear rates _
gclose to the
transition from the quadratic into the shear-molten state, we
nd Pe ¼O(10
0
). In other words, the shear-induced structural
transitions happen at Pe S1, consistent with our
expectations.
54
For comparison we have also investigated the reverse situa-
tion, where the system is initially in a hexagonal steady state at
shear rate _
g
init
¼400. We then suddenly change the shear rate
towards a much smaller value and consider the relaxation
towards the square equilibrium state. The corresponding
behaviour of s
xz
(t) is shown in Fig. 4. Again we nd that, the
smaller the difference _
g
new
s_
g
2
sis, the larger s
2
becomes (and
the more pronounced is the sensitivity to small changes in
_
g
new
). The resulting relaxation times can be also tted via
eqn (4) with a
2
/s¼22.58, b
2
¼1.73 and _
g
2
s¼215, whereby
_
g
2
s¼_
g
sq
sdenotes the threshold between the square and the
molten states. The result for this t is visualized in the inset of
the Fig. 4.
5. Impact of feedback control
We now discuss the impact of our shear stress control scheme
dened in eqn (3). The latter involves the zero-shear viscosity,
h
0
, which is estimated from Fig. 1 to h
0
¼0.086/dsand 0.323/ds
for the bilayer and trilayer, respectively.
The overall dynamical behaviour under feedback control
strongly depends on the value of s
0
(imposed shear stress)
relative to the ow curve of the original system [see Fig. 1]. We
can differentiate between regimes I, II, and III, which are indi-
cated in Fig. 1.
Fig. 3 (Color online) response of s
xz
(t) to a sudden switch-on (at time
s
on
) of shear with different rates _
g
new
s. The simulations were started
from the equilibrium (square) state in a bilayer system. The inset shows
the fit of the relaxation times s
1
according to eqn (4).
Fig. 4 (Color online) response of s
xz
(t) to a sudden change (at time
s
on
) of shear. The new shear rates _
g
new
sresult in a relaxation in the
square state. The simulations were started from the hexagonal steady
state at _
gs¼400 in a bilayer system. The inset shows the fit of the
relaxation times s
2
according to eqn (4).
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For a s
0
chosen in region I, the response of the system is
unique, that is, the nal state is independent of the control time
scale s
c
, as well as of the initial shear rate _
g
init
and the initial
microstructure. Thus, when starting from a square state, with a
corresponding initial shear rate _
g
init
, the system immediately
settles at this state. As a more critical test of the injectivity of the
ow curve in region I, we plot in Fig. 5a and b the functions _
g(t)
and s
xz
(t) for the bilayer system at s
0
ds
2
¼6 and various s
c
,
starting from a hexagonal conguration (and _
g
init
s¼400). In all
cases, the shear rate decreases towards the value _
gsz70 and
the structure relaxes into the square state pertaining to the
value s
xz
ds
2
¼6 in the uncontrolled system. This indicates that
the square state in region I is indeed the only xed point of the
dynamics. We also see from Fig. 5a that the relaxation time into
this steady state increases with s
c
. Fig. 5b additionally shows
that s
xz
(t) displays a pronounced peak. The peak indicates the
time window in which the initial hexagonal ordering transforms
into a square one. In fact, the high values of s
xz
at the peak
reect the large friction characterizing the intermediate molten
state. Similar behaviour occurs in region I of the trilayer system
[see Fig. 5c and d] where, however, uctuations of s
xz
(t) are
generally larger.
We now choose s
0
within region II of the ow curve, where
there are three different shear rates (and thus, three xed
points) pertaining to the same stress [see Fig. 1]. We focus on
systems which are initially in a square congurations, whereas
the initial shear rate _
g
init
has a value pertaining to the hexagonal
state (other initial conditions will be discussed below). The
impact of s
c
on the time dependence of _
g(t) and s
xz
(t) is shown
in Fig. 6. For small values of the control time scale the systems
stays in the initial lattice conguration, i.e.,_
grelaxes towards
the value pertaining to the square state ( _
gsz90). Different
behaviour occurs at larger values of s
c
/s: although the initial
structure is square, the nal state is hexagonal, and the shear
rate essentially remains at its high initial value. We stress that
these ndings crucially depend on the choice of _
g
init
.In
particular, the dependency of the long-time behaviour on s
c
/s
only arises for large values of _
g
init
; for small values the system
remains in the square state irrespective of s
c
. An overview of the
nal dynamical states in the feedback-controlled bilayer at
s
0
ds
2
¼8 and various combinations of _
g
init
and s
c
/s(assuming a
square initial structure) is given in Fig. 7. The colour code
indicates the ratio of local bond-order parameters hJ
6
/J
4
i[for a
denition of the J
n
see e.g., ref. 21]. The restriction to values
hJ
6
/J
4
i#6 is related to the actual values observed in the
simulations. From Fig. 7 one clearly sees that for initial shear
rates _
g
init
s>_
g
hex
sz257, the nal state of the feedback-
controlled system depends on s
c
/s. This is in contrast to the
uncontrolled system which becomes hexagonal for all _
g
init
>
_
g
hex
. For a hexagonal initial conguration the diagram (not
shown here) looks similar from a qualitative point of view;
however, the range of control times where the system retains a
hexagonal state despite of _
g
init
<_
g
quad
(with _
g
quad
being the
threshold between square/molten states) is much smaller.
Fig. 5 (Color online) time dependence of the instantaneous shear rate
and shear stress for a bilayer- [a and b] and a trilayer system [c and d] in
presence of feedback control within region I. The imposed stress was
set to s
0
ds
2
¼6(2) for the bilayer (trilayer) system. Various values of s
c
/s
are considered. The initial configuration is hexagonal.
Fig. 6 (Color online) same as Fig. 5, but for s
0
ds
2
¼8(2.7) for the
bilayer (trilayer) system (region II). The initial configuration is square.
Fig. 7 (Color online) state diagram indicating long-time lattice
structures. All simulations were started from a square initial structure
and the imposed shear stress was set to s
0
ds
2
¼8. The line shows the
result from eqn (8).
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We conclude that, by varying s
c
and the initial structure, we
can “switch”between the two stable, steady-state congurations
arising in the multivalued region of the uncontrolled system.
That these states are stable also under feedback (stress) control
is supported by the linear stability analysis presented in
Appendix A. Indeed, the dynamics under feedback control never
evolves towards the intermediate, shear-molten states, consis-
tent with the view that these states are mechanically unstable.
This holds also in region III of the ow curve of the uncontrolled
system, e.g., for s
0
ds
2
¼16(5) for the bilayer (trilayer): here, a
small value of s
c
yields relaxation towards the square state,
whereas for large s
c
, the system evolves into a hexagonal state.
Finally, we note that completely analogous behaviour is found
in the trilayer system [see Fig. 6b and c] for a s
0
pertaining to the
regime where square, molten and hexagonal states exist.
6. Transition line
The most signicant observation from Fig. 7 is that at high
values of _
g
init
, the feedback-controlled system can achieve either
the hexagonal or the square conguration, provided that we
start from a square conguration and choose s
c
/saccordingly.
We now propose a simple model which allows us to estimate the
transition values of the control time, s
trans
c
.
The physical idea behind our model is that, with the initial
conditions described above, relaxation into the hexagonal state
only occurs if the reorganization time s
reorg
required by the
system to transform from a square into a hexagonal congu-
ration, is smaller than the time s
decay
in which _
gdecays to a
value pertaining to the square state. We can estimate s
decay
from eqn (3) if we assume, for simplicity, a linear relationship
s
xz
(t)¼m_
g(t) (note that such a relationship is indeed nearly
fullled within the square and hexagonal states, see Fig. 1).
Under this assumption eqn (3) can be easily solved, yielding
_
g(t)¼m
1
e
mt/h
0
s
c
(m_
g
init
s
0
+e
mt/h
0
s
c
s
0
). (5)
From eqn (5) we nd that the decay time of _
gto the threshold
value _
g
hex
sz257 (below which the hexagonal state of the
uncontrolled system is unstable) is given by
sdecay ¼sch0
mlnmg
:
init s0
mg
:
hex s0:(6)
To estimate the reorganization time s
reorg
(from the initial
square into a hexagonal conguration), we assume that its
dependence on _
g
init
is analogous to that of the relaxation time
s
1
introduced for the uncontrolled system [see eqn (4)]. Speci-
cally, we make the ansatz
sreorg ¼a0
|g
:
initsg
:
hexs|b0:(7)
As stated above, a crucial assumption of our model is that
the system can only reach the hexagonal state if s
reorg
does not
exceed s
decay
. Note that the latter involves (in fact, is propor-
tional to) the time s
c
. By equating expressions (6) and (7) for
s
decay
and s
reorg
, respectively, we can therefore nd an expres-
sion for the minimal control time, s
trans
c
, above which the system
reaches the hexagonal state, that is
strans
c¼a0m
|g
:
initsg
:
hexs|b0
h0lnmg
:
init s0
mg
:
hex s0:(8)
Due to the square initial conguration, we set m¼h
0
and
s
xz
(t)¼m_
g(t)asdened in our ansatz. The remaining param-
eters a0and b0are determined by tting the numerical results
for s
c
/sat the boundary [see Fig. 7] to expression (8), yielding
a0/s¼54.127 and b0¼1.503. The resulting line s
trans
c
(_
g
init
)is
included in Fig. 7, showing that our estimate describes the
transition between square and hexagonal states very well.
Similar considerations are possible, when we use a hexag-
onal initial lattice structure. Choosing then a small value of
g
init
swe nd that we can switch between hexagonal and square
state. This is illustrated in Fig. 8. To obtain the corresponding
transition values of s
c
, we use the same strategy as before,
but take a different ansatz forthestress.Specially,weset
s
xz
(t)¼n+m_
g(t) which approximately describe the ow curve
in the hexagonal state of the uncontrolled system. From the
results plotted in Fig. 1 we nd n¼7.0477/ds
2
and m¼0.0025.
The analog of the eqn (8) then reads
strans
c¼a0m
|g
:
initsg
:
hexs|b0
h0lnnþmg
:
init s0
nþmg
:
hex s0(9)
with a0/s¼0.012 and b0¼0.237 [see Fig. 8]. The result is visu-
alized in Fig. 8. Comparing the typical control time scales at the
transition with those seen in Fig. 7 we nd that s
c
/swhich is
necessary to switch from square into the hexagonal state [see
Fig. 7] is about two orders of magnitude larger than switching
from hexagonal into the square state. We suspect that this
difference results from the differences of the slope of the shear
stress in the square and hexagonal regimes.
Fig. 8 (Color online) state diagram indicating long-time lattice
structures. All simulations were started from the hexagonal initial
structure and the imposed shear stress was set to s
0
ds
2
¼8.
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7. Conclusions
Using numerical simulation we have studied the complex
dynamical behaviour of sheared colloidal lms under a specic
type of shear–stress control. Our approach involves relaxation of
the shear rate in a nite relaxation time s
c
, until the instanta-
neous stress matches its desired value. This approach is
inspired by rheological experiments
32
where the instantaneous
shear rate as function of time can be measured. Focusing on
systems with multivalued ow curves (resulting from successive
non-equilibrium transitions) we have found that, by tuning s
c
and the initial conditions, it is possible to select a specic
dynamical state. In the present system these are either a state
with square in-plane ordering and high viscosity, or a hexagonal
state with low viscosity. Therefore, our study suggests a way to
stabilize states with desired rheological properties, particularly
shear viscosities. Moreover, we have proposed a model which
relates the transition values of s
c
to relevant intrinsic relaxation
times under sudden change of _
g.
Although most of our results pertain to a colloidal bilayer,
the fact that we found analogous results for trilayers suggests
that the proposed technique can also be applied for systems
with larger number of layers. In fact, we think that this method,
aer some minor adaptations such as consideration of the
kinematic (and, possibly, also the non-local) contributions in
eqn (2), should also be applicable and fruitful in bulk systems.
Indeed, we expect the method to allow for state selection in any
shear-driven system with multivalued ow curve. For example,
in an earlier study we have used an analogous approach (based,
however, on continuum equations) to select states and even
suppress chaos in shear-driven nematic liquid crystals.
31
It
therefore seems safe to assume that the capabilities of the
present scheme are quite wide. For colloidal layers one may
envision, e.g., stabilization of time-dependent structures such as
oscillatory density excitation, which may have profound impli-
cations for lubrication properties.
55
Finally, our ndings are quite intriguing in the broader
context of manipulating nonlinear systems by feedback control.
In our case, the feedback character stems from the fact that the
stress control involves the conguration-dependent instanta-
neous stress. Mathematically, this scheme can be viewed as
feedback control with exponentially distributed time-delay
56
(as
can be seen by formally integrating eqn (3) and inserting it into
eqn (1)). Similar schemes are used to stabilize dynamical
patterns in laser networks,
57
neural systems,
58
and more
generally, coupled oscillator systems.
59
The implications of
these connections are yet to be explored.
Appendix
A. Stability of the feedback controlled system
In this Appendix we investigate the stability of the solutions of
eqn (3). Specically, we consider the impact of small variations
of the shear rate from its steady state value _
g
0
related to the
imposed stress s
0
. Expanding the right side of eqn (3) with
respect to the difference _
g_
g
0
yields
d
dtg
:z
s0sxzg
:
0;t
sch0
vsxzg
:
;t
sch0vg
:g
:
0g
:g
:
0þOg
:2:(10)
For long times we expect the rst term on the right side of
eqn (10) to vanish, since s
xz
(_
g
0
,t)/s
0
. To linear order, eqn (10)
then reduces to
d
dtg
:z1
sch0
vsxzg
:
;t
vg
:g
:
0g
:g
:
0þOg
:2:(11)
Noting that the values of s
c
and h
0
are both positive, we can
follow that the feedback controlled shear rate approaches a
steady-state value only if
vsxz
vg
c
.0:(12)
This corresponds to the usual criterion of mechanical
stability.
48
B. Strain–stress relation under constant shear rate
In this Appendix we present results for stress–strain relations at
different xed values of _
g. These can be obtained from the data
shown in Fig. 3 by rescaling the time axis with the applied shear
rate. Numerical results are shown in Fig. 9. Similar to the stress–
time relations shown in Fig. 3, one observes simple, monotonic
behavior for the case _
g
new
s¼200 (quadratic regime), while the
curves for larger shear rates display pronounced stress over-
shoots. The width of these overshoots is largest at _
g
new
s¼240,
where the system is in the molten state. This is consistent with
the appearance of a particularly large intrinsic relaxation time
as discussed in Section 4.
Acknowledgements
This work was supported by the Deutsche For-
schungsgemeinschathrough SFB 910 (project B2).
Fig. 9 (Color online) stress–strain relations in the colloidal bilayer for
different shear rates, starting from the equilibrium (square)
configuration.
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Soft Matter Paper
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