Computational Steering of Cluster Formation in Brownian Suspensions

0.52 MB·English

by  Martin Hecht

Computational Steering of Cluster Formation in Brownian Suspensions
Martin Hecht and Jens Harting
Institut fu¨r Computerphysik, Pfaffenwaldring 27, 70569 Stuttgart, Germany
(Dated: February 3, 2008)
WesimulateclusterformationofmodelcolloidalparticlesinteractingviaDLVO(Derjaguin,Lan-
dau, Vervey, Overbeek) potentials. The interaction potentials can be related to experimental con-
ditions, defined by the pH-value, the salt concentration and the volume fraction of solid particles
suspendedinwater. Thesystemshowsdifferentstructuralpropertiesfordifferentconditions,includ-
ingclusterformation,aglass-likerepulsivestructure,oraliquidsuspension. Sincemanysimulations
are needed to explore the whole parameter space, when investigating the properties of the suspen-
siondependingontheexperimentalconditions,wehavedevelopedasteeringapproachtocontrola
running simulation and to detect interesting transitions from one region in the configuration space
8 to another. The advantages of the steering approach and the restrictions of its applicability due to
0 physical constraints are illustrated by several example cases.
0
2
I. INTRODUCTION particles’ interactions. Such a change can be
n
achieved by adding salt to a suspension or by
a
J Soft matter physics is a large field which has changing the pH-value [18, 19, 22, 24]. How-
4 gainedmoreandmoreinimportanceduringthe ever, in mesoscopic systems where many dif-
1 last years. It comprises for example complex ferent effects contribute to the overall behavior
the parameter space is quite large and one has
fluids, biological systems like membranes, so-
] to perform a large number of different simula-
t lutions of large molecules like proteins, or sus-
f tions,eachofthemrequiringalotofcomputing
o pensions of small soft or solid particles, which
resources,togainanunderstandingofallinter-
s are commonly called “colloids”. Since one can
t. find examples for these materials nearly every- relationships.
a
whereineverydaylife(inmedicine,foodindus- Therefore, it is useful to be able to steer
m
try, paintings, glue, blood, ceramics,...), many the simulation on-line. Steering in this context
d- resultsinresearchhaveahighrelevanceforap- can mean to change the interaction potentials
n plications. In this context, especially colloid or other parameters like an externally applied
o science is one of the most active research fields shear rate, but it can also mean to go back in
c of the present time. A considerable effort has the simulation time and follow a different path
[ been invested to describe colloidal suspensions starting from an earlier configuration, both in-
2 from a theoretical point of view and by simula- duced by interaction with the user [25, 26].
v tions [1, 2, 3, 4, 5] as well as to understand the
For our study we have modeled an aqueous
4 particle-particle interactions [6, 7, 8, 9, 10, 11]
5 and the phase behavior [12, 13, 14, 15, 16]. suspension of Al2O3-particles. The particles
9 Typical for soft matter physics, and especially in our simulations are monodisperse spheres of
3 0.37µm in diameter. They interact via DLVO
for colloids, is that the important effects which
. (Derjaguin, Landau, Vervey, Overbeek) poten-
9 govern the material properties are on a meso-
0 scopic length scale, i.e., on lengths much larger tials [6, 7] as well as a repulsive force ensuring
7 excluded volume for the particles. The prop-
thantheatomisticscale,andmuchsmallerthan
0 erties of this model system have been investi-
the macroscopic length scale.
: gated by simulations and experiments in pre-
v This mesoscopic level brings in the possibil-
Xi itytocontroltheparametersofacertainmate- vious works [22, 24, 27, 28, 29, 30]. In this
paper we discuss how steering of a simulation
rialbymanipulatingtheprocessesonthemeso-
r can help to explore the parameter space and
a scopiclengthscale. Forcolloidstherearemany
which problems may occur when using steering
waystocontroltheinteractionsbetweenthein-
techniques.
dividual colloidal particles. As an example, by
changing the particle interactions, it is possi- In the following section we describe our sim-
bletoinitiateaclusteringprocessinacolloidal ulation method and focus on the implementa-
suspension [17, 18, 19, 20, 21, 22, 23]. tion of the steering. Then, we discuss some ex-
However, on the mesoscopic length scale, of- amples of simulations where steering helps to
ten many different effects are in a subtle in- explore the parameter space, but we also give
terplay which makes it difficult to give quan- examples in which a steered simulation might
titative predictions. Computer simulations can lead to different results if compared to an un-
helptostudythesesystemsindetail,e.g.,tosee steered simulation. Finally, we summarize our
the response of the system on a change in the results and draw a conclusion.
2
II. SIMULATION METHOD For the simulation of a fluid solvent, many
different simulation methods have been pro-
Our simulation method is described in detail posed: direct Navier Stokes solvers [31, 32, 33,
inRefs.[24,27,28]andconsistsoftwoparts: a 34], Stokesian Dynamics (SD) [3, 35, 36], Ac-
Molecular Dynamics (MD) code, which treats celerated Stokesian Dynamics (ASD) [37, 38],
the colloidal particles, and a Stochastic Rota- pair drag simulations [4], Brownian Dynam-
tion Dynamics (SRD) simulation for the fluid ics (BD) [17, 39], Lattice Boltzmann (LB) [1,
solvent. In the MD part we include effective 2, 40, 41], and Stochastic Rotation Dynam-
electrostaticinteractionsandvanderWaalsat- ics (SRD) [27, 42, 43]. These mesoscopic
traction,knownasDLVOpotentials[6,7]. The fluid simulation methods have in common that
repulsive term results from the surface charge theymakecertainapproximationstoreducethe
of the suspended particles computational effort. Some of them include
thermalnoiseintrinsically,oritcanbeincluded
(cid:20)
2+κd 4k T consistently. They scale differently with the
V =πε ε · B
Coul r 0 1+κd ze numberofembeddedparticlesandthecomplex-
ity of the algorithm differs largely. In partic-
(cid:18) zeζ (cid:19)(cid:21)2 d2e−κ[r−d]
tanh × , (1) ular, there are big differences in the concepts
4kBT r how to couple the suspended particles to the
surrounding fluid.
where d denotes the particle diameter, r the
We apply the Stochastic Rotation Dynamics
distance between the particle centers, e the el-
method (SRD) introduced by Malevanets and
ementary charge, T the temperature, k the
B Kapral[44,45]. Itintrinsicallycontainsfluctua-
Boltzmann constant, and z is the valency of
tions,iseasytoimplement,andhasbeenshown
the ions of added salt. ε is the permittivity of
0 to be well suitable for simulations of colloidal
thevacuum, ε =81therelativedielectriccon-
r andpolymersuspensions[24,27,42,43,46,47,
stant ofthesolvent, κ theinverseDebyelength
48]. The method is also known as “Real-coded
defined by κ2 = 8π(cid:96) I, with ionic strength I
B LatticeGas”[42]oras“Multi-Particle-Collision
and Bjerrum length (cid:96) =7˚A. We have related
B Dynamics”(MPCD)[49, 50]. Itisbasedonso-
theeffectivesurfacepotentialζ tothepH-value
called fluid particles with continuous positions
and the ionic strength of the solvent by means
and velocities. A streaming step and an inter-
of a charge regulation model in our previous
action step are performed alternately. In the
work [24]. Thus, the particle interaction po-
streamingstep,eachparticleiismovedaccord-
tentials can be related to distinct experimental
ing to
conditions. ThesecondthermoftheDLVOpo-
tentialswhichdoesnotdependonthepH-value r (t+τ)=r (t)+τ v (t), (3)
i i i
or the ionic strength is the attractive van der
where r (t) denotes the position of the parti-
Waals interaction (A = 4.76·10−20J is the i
H cle i at time t and τ is the time step. In the
Hamaker constant) [17]
interaction step the fluid particles are sorted
A (cid:20) d2 d2 (cid:18)r2−d2(cid:19)(cid:21) into cubic cells of a regular lattice and only
V =− H + +2ln . the particles within the same cell interact with
VdW 12 r2−d2 r2 r2
each other according to an artificial interac-
(2)
tion. The interaction step is designed to ex-
The attractive contribution V competes
VdW change momentum among the particles, but at
with the repulsive term and is responsible for
the same time to conserve total energy and to-
the cluster formation one can observe for con-
tal momentum within each cell, and to be very
ditions in which the attraction dominates.
simple, i.e., computationally cheap. Each cell
Since DLVO theory is based on the assump-
j is treated independently: first, the mean ve-
tion of large particle separations, it does not
correctly reproduce the primary minimum in locity uj(t(cid:48)) = Nj1(t(cid:48))(cid:80)iN=j1(t(cid:48))vi(t) in cell j is
the potential, which should appear at particle calculated. N (t(cid:48)) is the number of fluid parti-
j
contact. Therefore, we cut off the DLVO po- clescontainedincellj attimet(cid:48) =t+τ. Then,
tentials and model the minimum by a parabola the velocities of each fluid particle in this cell
as described in Refs. [27, 28]. To ensure ex- are rotated according to
cluded volume of the particles we use a repul-
v (t+τ)=u (t(cid:48))+Ω (t(cid:48))·[v (t)−u (t(cid:48))]. (4)
sive (Hertzian) potential. Below the resolution i j j i j
oftheSRDalgorithmshortrangehydrodynam- Ω (t(cid:48)) is a rotation matrix, which is indepen-
j
icsiscorrectedbyalubricationforcewithinthe dentlychosenatrandomforeachtimestepand
MD framework as explained in Refs. [24, 27, eachcell. Weuserotationsaboutoneoftheco-
28]. ordinate axes by an angle ±α, with α fixed.
3
The coordinate axis as well as the sign of the Let us now shortly sketch the technical re-
rotation are chosen at random, resulting in 6 alization of the steering interface in our sim-
possible rotation matrices. However, there is a ulation code. The program is an object ori-
great freedom to choose the rotation matrices. ented code written in C++. Each object
Any set of rotation matrices satisfying the de- contains virtual routines save() and load(),
tailed balance for the space of velocity vectors which write the whole object data to or load
couldbeusedhere. Toremoveanomaliesintro- it from a buffer, respectively. The buffer con-
ducedbytheregulargrid,onecaneitherchoose tains a plain text description of all variables
the mean free path to be sufficiently large or contained in the object with their values, sim-
shift the whole grid by a random vector once ilar to the C++ source code one would write
per SRD time step [51, 52]. toinitializeaninstanceoftheclass. Thewhole
To couple the SRD and the MD simulation, work flow of a simulation, including the actual
basically three different methods have been in- MD loop as well as data input/output tasks
troduced in the literature. Inoue et al. pro- are described in this manner using specialized
posed a way to implement no slip boundary workstep-classes[59], each providing a special-
conditions on the particle surface [42]. To ized work()-routine which performs different
achieve full slip boundary conditions, Lennard- actions depending on the actual class type of
Jones potentials can be applied for the inter- the respective object. One of the workstep-
action between the fluid particles and the col- classes is designed to change a specified ob-
loidal particles [45, 53]. A more coarse grained ject by using its save() and load() routine.
method was originally designed to couple the First, the current values are stored to a tem-
monomers of a polymer chain to the fluid [54, porary buffer, then one or several variables
55], but in our previous work [22, 24, 27] we may be overwritten by new values, and finally
have demonstrated that it can also be applied the object to be modified is loaded again from
to colloids, as long as no detailed spacial res- the temporary buffer. The description of the
olution of the hydrodynamics is required. We changes may be read from standard input or
usethiscouplingmethodinoursimulationsand from a file, similar to the simulation setup,
describe it shortly in the following. which is read during the initialization of any
simulation. workstep-class objects can also be
For the coupling of the colloidal particles to
included into the work flow at later times, or
the fluid, they are sorted into the SRD cells
may even be disconnected from the usual work
and included in the SRD interaction step. The
flow and bound to standard UNIX system sig-
stochastic rotation is performed in the momen-
nals. By default a workstep writing particle
tum space instead of the velocity space to take
positionstostandardoutputandaworkstepto
into account the difference of inertia between
change objects getting its buffer from standard
light fluid and heavy colloid particles. We have
inputareboundtotheSIGTTOU(“terminalout-
describedthesimulationmethodinmoredetail
put”) and SIGTTIN (“terminal input”) signal,
in Refs. [24, 27, 28].
respectively. This allows to embed the simula-
Inthepresentworkwereportonseveralstud-
tion program into a framework of shell scripts
ies about pressure filtration, cluster formation
which generate the appropriate input, redirect
in a steered simulation, and sedimentation. All
the output to a visualization tool and send the
these studies are proofs of principle, and there-
signals according to the user interaction. This
fore small simulations are performed. Typical
can be realized in a client-server fashion, even
parameters for these simulations are: 60 fluid
on different hosts and platforms using appro-
particles per box, with 0.296µm extension in
priate scripts and TCP/IP connections.
each direction. The volume of the box is set
to be the same as the volume of a colloidal
particle with diameter 0.37µm. The system
size is usually 30 to 60 boxes in each dimen- III. RESULTS
sion. By matching the diffusion constant, den-
sity and viscosity to a real suspension[24], the After the more technical paragraph in the
scaling scheme we have presented in Ref. [27] previous section we now turn to a discussion of
yieldsascaledtemperatureinthesimulationof the advantages of such a steering approach for
14mK and a scaled viscosity of 4.78·10−8Pas. computer simulations and highlight some pit-
Topreservethecorrectdynamics,characterized falls resulting from the physical background in
by the dimensionless numbers (Re, Pe, Kn,...) the context of steering. This section is subdi-
onehastorescalethepotentialsandalldriving vided into several subsections each of them fo-
forces in the MD scheme by the same scaling cusing on a particular example to illustrate the
factor, as well. steering approach in practice.
4
ally low initial densities are used, so that very
1.2
pH = 4
pH = 5 large simulation volumes are needed to obtain
1 pH = 6 large particle numbers for good statistics. On
pH = 7
pH = 8 the other hand, large simulation volumes also
0.8
imply large data files and long transient times
y
nsit 0.6 untilthefiltercakeisformedandaconsiderable
de part of the particles has reached the bottom of
0.4 the system.
Thisisatypicalproblem,inwhichonewould
0.2
liketoobservetherunningsimulationoccasion-
0 ally to check if the filter cake is already formed
0 0.05 0.1 0.15 0.2 0.25 0.3 and when the structure does not change any-
height (in units of system size)
more. One would like to measure pressure pro-
files, local streaming velocities, or simply the
final density profile of the filter cake. Here,
Figure 1: Local density expressed in terms of vol-
ume fraction of the solid particles constituting the steering would mean to initiate data acquisi-
filter cake depending on the pH-value. The ionic tionbyuserinteraction. Thisisofadvantageif
strength is kept constant at I =3mmol/l, and the an action, like starting to average certain data,
volume fraction at Φ=5%. The plots for different shouldbeinitiatedwhenconditionsarefulfilled,
pH-values are shifted against each other vertically whicharedifficulttocheckautomatically. Since
by0.2forbettervisibility. Theshapeoftheprofile the density and structure of the filter cake is a
differs, since the spacing of the particles depends
prioriunknown,anautomatedcheckisdifficult
on their interactions.
to implement.
Additionally, the upper boundary of the fil-
ter cake is diffuse and depends on the condi-
A. Pressure filtration
tions (pH-value and ionic strength) as well. In
Fig.1 the dependence of the density profiles of
Inthiscaseafiltrationofasuspensionissim- filtercakesobtainedinsimulationsfordifferent
ulated. The suspended particles cannot pass pH-valuesareshown. WithincreasingpH-value
the filter, whereas the fluid passes through the the height of the filter cake increases and the
filter without resistance in an idealized filtra- density profile changes slightly. Some voids in
tion process. The suspended particles agglom- thefiltercakediminishthelocaldensityifcom-
erate in front of the filter and form a so-called pared to a dense packing. At the bottom of
filter cake. Since the dynamics of the parti- the filter cake, layers of particles are present,
clesisnotonlygovernedbythehydrodynamics whereas in the upper regions the structure be-
of the fluid, but also by their DLVO interac- comes more irregular. However, the structure
tion, the density and the structure of the filter of the sediment and its height will also depend
cake will depend on the pH-value and the ionic on the pressure exerted on the fluid and proba-
strength. bly on the initial volume fraction. Larger sim-
In our simulation we drive the fluid by an ulations than this proof of principle are needed
applied force pointing downwards acting in a tobeabletoquantifythepH-value-dependence
small region close to the upper boundary of of the height of the filter cake. However, sim-
the system. For the fluid we apply fully peri- ulations without driving force on the fluid, can
odic boundary conditions, whereas the bound- alsohelptounderstandthissystem. Therefore,
aries for the suspended particles are closed in we explore the parameter space in the absence
z-direction. In this way, the fluid is forced to of a driving force in the following subsection.
streaminverticaldirectionandtodragthepar-
ticles to the bottom of the system. In test sim-
ulationswehavechosenconstantionicstrength B. Observation of cluster formation
I = 3mmol/l and a constant overall volume
fraction of Φ = 5%, and simulated the filtra- Since several parameters, namely the vol-
tion process for several pH-values. ume fraction, pH-value, ionic strength, exter-
Depending on the interactions, different in- nal driving forces influence the system, a first
ternal structures of the filter cake are formed step to understand its behavior is to exam-
when the density of the particles increases at ine it in the absence of external forces. Still,
the bottom of the system. To gain a good two parameters–the pH-value and the ionic
statisticsforthestructurelargesimulationsare strength–governthedynamicsofthesystem. If
needed. However, in a filtration process, usu- the potentials are attractive, cluster formation
5
a) b)
c) d)
Figure2: Memoryeffectsinasteeredsimulation: thegrayscaledenotesthecoordinationnumber. Particles
closer than 2.4 radii are considered to be neighbors. Dark particles in this sense are highly coordinated,
bright ones have low coordination numbers. a) right after initialization, homogeneous distribution of the
particles, b) cluster formation after tuning the potentials to be attractive, c) still inhomogeneous regions
aftersteeringthesimulationoutoftheclusteredregion,d)thesystemishomogeneousagainwhendiffusion
hascounterbalancedthedensityinhomogeneities. Thesteeringpathintheparameterspaceaswellasthe
points a)–d) thereon are illustrated in Fig.3.
can be observed [57], provided the attraction has restored a homogeneous system. This time
is strong enough compared to thermal fluctu- depends on the distances between the clusters
ations. In our previous work [22, 28] we have and on the characteristic diffusion time of the
explored this part of the parameter space for particles. The steering path in the configura-
some distinct volume fractions by running nu- tion space is shown in Fig.3. The points at
merous individual simulations. However, with which the snapshots of Fig.2 are taken, are
the newly implemented steering approach one marked by black dots. In the insets in Fig.3
candetectboundariesbetweendifferentregions the total potential is plotted for several cases.
of the stability diagram faster and more sensi- During the simulation we change the pH-value
tively. and the ionic strength and we drive the system
In Fig.2 we illustrate the effects of chang- along a closed trajectory in parameter space.
ing the properties of the interaction potentials. We start at low ionic strength I = 1mmol/l
All the snapshots belong to the same simula- andpH=4(potentialshown)andfirstincrease
tion and represent different points of the tra- thepH-value(potentialshownforI =1mmol/l
jectory the system is steered through param- and pH = 7), then increase the ionic strength,
eter space: a) The initial situation with repul- then reduce the pH-value again while further
sivepotentials,shortlyafterinitializationofthe increasing the ionic strength (potential shown
system, b) after crossing the boundary to the for I = 15mmol/l and pH = 4) and finally de-
clustered region, c) shortly after steering back crease the ionic strength to return to the start-
to the suspended regime, where still some in- ing point. We choose the trajectory such that
homogeneities in the system can be observed, the barrier between the primary and the more
and finally d) after some time when diffusion shallow, secondary minimum of the DLVO po-
6
tential does not go below 5 k T. Since the sec- choice of the steering path (Fig.3).
B
ondary minimum becomes deeper during the In this example the simulation remembers
simulation cluster formation can be observed. theinteractionbytheuser. Evenifthesteering
However, the barrier between the primary and path is selected carefully inside the reversible
the secondary minimum of the DLVO poten- range, the simulated suspension needs a char-
tials prevents irreversible aggregation in the acteristic time to relax after changing the in-
primary minimum. A case in which the bar- teractions. The interaction by the user can be
riervanishesandirreversibleaggregationwould seenasaperturbationinthephysicalsenseand
appear is also shown in the upper right inset the system needs time to adopt to the new sit-
(I = 15mmol/l and pH = 7). In any case, uation. In these cases special care is advised
in our simulation all clusters dissolve again af- when steering a simulation. We illustrate this
ter steering back to a lower pH-value and lower byanotherexampleinthefollowingsubsection.
ionic strength (see Fig.2d)). This confirms
that we have reversible clustering in the sec-
ondary minimum of the DLVO potentials.
C. sedimentation: hydrodynamic
One of the advantages of a steered simula- interaction
tionis,thatonecanmoreaccuratelydetectthe
onset of the cluster formation than by start- Notonlytheparticlepositionsandvelocities,
ing individual simulations for different condi- but also hydrodynamic interactions can influ-
tions. Onemightpredicttheboundarybyeval- ence the behavior of the system. Therefore, in-
uating the depth of the secondary minimum in homogeneous particle distributions induced by
the potential, but since also the hydrodynamic steering the simulation first into a clustered re-
damping forces and an eventually applied driv- gion of the parameter space before simulating
ing pressure influence this process the predic- adifferentsituation,mightinfluencetheresult:
tion of the boundary can become more compli- we show in the following paragraph that a sed-
cated. We have experienced similar difficulties imentation process can happen faster, if clus-
in the context of shear flow simulations of the ter formation by attractive interactions creates
same colloidal suspension[22, 24]. density inhomogeneities in the system.
A quantity to support the visualization of The friction force a single particle feels in a
clusters is the coordination number. We con- sedimentationprocesscanbecalculatedanalyt-
sidertwoparticleswhosecentersarecloserthan ically,butitdiffersifotherparticlesarepresent.
2.4 particle radii as neighbors and define the InRef.[27]wehaveconfirmedthatthesedimen-
coordination number of each particle as the tationvelocityinoursimulationdependsonthe
number of its neighbors in the sense just de- volume fraction, as it is well-known from sed-
scribed. The particles in Fig.2 are drawn in imentation theory [58]. But even at constant
agrayscalecorrespondingtotheircoordination volume fraction the sedimentation velocity can
number, where dark particles are highly co- be different in different configurations. If the
ordinated particles and bright ones are those particlesformclusters,thewholeclustersettles
with a low coordination number. At the begin- down and the fluid streams around the whole
ning of the simulation (top left) the particles cluster. The resistance is much less compared
are distributed homogeneously over the whole to the case when the particles are distributed
simulation volume. After steering the simula- homogeneously and the fluid streams around
tion in the parameter space into the region of each of them separately (compare Ref. [28] and
attractive potentials, cluster formation can be references therein). In Fig.4 we have plotted
observed (top right). When the potentials are the sedimentation velocity evaluated in several
madelessattractiveagain,theclustersdissolve simulations,whichonlydifferbythepotentials.
(bottom left). However, the particles prefer- Thevolumefractionisthesameforallofthem.
ably stay at their positions and thus, the in- However, as one can see in the figure, the sedi-
homogeneities in the system do not disappear mentationvelocityissmallerforlowpH-values.
immediately,whichcanbeseenalsointhedark Thiscanbeexplainedasfollows: Sinceforin-
color denoting high coordination numbers. Af- creasingpH-valuethepotentialsbecomeattrac-
tersometime,diffusionrestoresthehomogene- tive,theparticlesformclustersandsettledown
ity of the system (bottom right). However, if faster. This effect is purely due to the hydro-
the pH-value or the ionic strength is increased dynamics of the system. Since the streaming
too much, so that the barrier between the pri- fielddependsonallparticlepositionsandveloc-
maryandthesecondaryminimumoftheDLVO ities not only at a given time, but also on their
potentials vanishes, the clustering process be- history, steering in this context may be very
comes irreversible, which we avoided by the dangerous. When long range interactions, like
7
total potential / kT 101-1100-0100100-0.5 pd H0is t=a n4c Ie0 = ./5 1p5a rmtic m1leo lr/ald i1u.s5 2 h (mmol/l) 12159 cd)) revierrresvibelers ciblules tcelurisntgering total potential / kT 101-1100-0000110-0.5 pd H0is t=a n7c Ie0 = ./5 1p5a rmtic m1leo lr/ald i1u.s5 2
gt 6
n
e pH = 7 I = 1 mmol/l
total potential / kT 101100-010100 pH = 4 I = 1 mmol/l ionic str 31 4a) pH5 value6 b)7 8 total potential / kT 101-1100-0100010
-10 -0.5 0 0.5 1 1.5 2
-0.5 0 0.5 1 1.5 2 distance / particle radius
distance / particle radius
Figure 3: Steering path of the simulation: we start at the bottom left corner at low pH-value and low
ionic strength and steer the simulation along the dashed path through parameter space. The region of
reversible cluster formation is reached, but the barrier between primary and secondary minimum of the
DLVOpotentialspreventsirreversibleclustering: whenreachingthestartingpointagain,allclustershave
vanished. The boundary of the clustering region is sketched: Reversible clustering is observed when the
depth of secondary DLVO minimum is in the order of k T and irreversible clustering appears when the
B
heightofthebarrierbetweenthetwoDLVOminimaisoftheorderofk T orless. Thelabeledblackdots
B
markthepositionsatwhichthesnapshotsofFig.2aretaken. Theinsetsshowthetotalpotentialatsome
selected extreme cases (see text).
configuration when changing the interactions.
s) 16
d
y n 14
citco
oe 12
vels/s 10 IV. CONCLUSION
n er
atiomet 8 Bearing in mind the underlying physics, one
entdia 6 can roughly estimate the limits of steered sim-
me 4 ulations. Let us summarize the points we have
dicl
searti 2 brought forward in this paper. First of all,
p 0 steering can save computing time when aiming
(
4 5 6 7 8 9 10 ataroughunderstandingoftheinfluencesona
pH value system, especially when looking for the “inter-
esting points” in parameter space, e.g., when
searching for transition lines between different
Figure 4: Sedimentation velocity of a Φ=5% sus-
phases. We have shown this by the onset of
pensioninaclosedvessel. Thesedimentationveloc-
cluster formation due to modifying the interac-
ity of the particles in the upper part of the system
tion potentials in a model suspension. Steering
is averaged over several time steps. The particles
canbeusedaswelltostartdataacquisitionaf-
at the bottom of the system are not encountered.
ter a transient at the beginning of a simulation
Thevelocitydependsontheparticle-particleinter-
action. Clusters settle down faster than individual or to adjust the frequency for the data acquisi-
particles. Symbols denote simulation results, the tion according to the current state of the sim-
line is a guide to the eye. ulation. This is especially interesting for large
simulations with long transient times, as illus-
tratedbytheexampleoffilterflow. Wealsoex-
hydrodynamics, which also depends on the his- plainedwhyinsomecasessteeringmaybeseen
toryofthesystem,isimportant,steeringshould asaperturbationofthesystem,especiallywhen
be avoided. In this case, to obtain reasonable the interaction potentials are changed. Special
results, one has to start again from an initial care is advised when dealing with non ergodic
8
systems, memory effects, or long range inter- percomputing Center Karlsruhe and the Neu-
actions, like the role of hydrodynamics in sedi- mann Institute for Computing in Ju¨lich are
mentation in our last example. highlyacknowledgedforprovidingthecomput-
ing time and the technical support needed for
ourresearch. J.H.thankstheDFGforfinancial
Acknowledgments supportwithinthepriorityprogram“nano-and
microfluidics”. M.H. thanks the ICMMES for
We thank H.J. Herrmann for valuable collabo- travel funds and the DFG for financial support
ration and his support. The High Performance within SFB 716.
Computing Center Stuttgart, the Scientific Su-
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