Dynamic self-assembly of charged colloidal strings and walls in simple fluid flows

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by  Yu Abe

Dynamic self-assembly of charged colloidal strings and walls in simple fluid flows
Yu Abe,1,2 Bo Zhang,1 Leonardo Gordillo,1 Alireza Mohammad Karim,1 Lorraine F. Francis,1 and Xiang Cheng1
1Department of Chemical Engineering and Materials Science,
University of Minnesota, Minneapolis, MN 55455
2Films & Film Products Research Laboratories, Toray Industries,
Inc. 1-1, Sonoyama 1-chome, Otsu, Shiga, 520-8558, Japan
(Dated: January 31, 2017)
Colloidalparticlescanself-assembleintovariousorderedstructuresinfluidflowsthathavepoten-
tial applications in biomedicine, materials synthesis and encryption. These dynamic processes are
7
also of fundamental interest for probing the general principles of self-assembly in non-equilibrium
1
conditions. Here,wereportasimplemicrofluidicexperiment,wherechargedcolloidal particlesself-
0
assemble into flow-aligned 1D strings with regular particle spacing near a solid boundary. Using
2
high-speedconfocalmicroscopy,wesystematicallyinvestigatetheinfluenceofflowrates,electrostat-

n icsandparticlepolydispersityontheobservedstringstructures. Bystudyingthedetaileddynamics
a of stable flow-driven particle pairs, we quantitatively characterize interparticle interactions. Based
J
ontheresults,weconstructasimplemodelthatexplainstheintriguingnon-equilibriumself-assembly

9 process. Our study shows that the colloidal strings arise from a delicate balance between attrac-
2 tive hydrodynamic coupling and repulsive electrostatic interaction between particles. Finally, we
demonstratethat,withtheassistance oftransverseelectricfields,asimilarmechanismalsoleadsto

] theformation of 2D colloidal walls.
t
f
o PACSnumbers:
s
.
t
a I. INTRODUCTION cesses. In equilibrium, colloidal strings have been grown
m
via template-directed colloidal epitaxy [24] and electro-
- magnetic field-assistantself-assembly[5, 25, 26]. In fluid
d Self-assemblyofcolloidalparticlesisaprocessofgrow-
n ing interest because of its broad application in fields flows, the self-assembly of colloidal strings aligned along
o the vorticity direction of shear flows has been reported
ranging from photonics to materials synthesis and to
c when the colloidal particles are subjected to shear in
[ biomedical engineering [1, 2]. Although the mechanisms
of colloidal self-assembly in equilibrium have been well viscoelastic fluids [12, 13, 16] and in Newtonian fluids
1 under strong confinement [15, 17]. Flow-aligned col-
explored with many interesting colloidal structures dis-
v loidalstringsthataremoreusefulinmicrofluidicapplica-
covered[3–6], our understanding of non-equilibriumself-
8
tionshavebeenobservedatfinite Reynoldsnumbers(Re
0 assembly is still undeveloped. Out-of-equilibrium dy-
> 1) when the effect of inertia is significant.[14] Unfor-
4 namicself-assemblyprovidesnotonlyapowerfultoolfor
8 designing novel colloidal materials with unusual meso- tunately,thehigh-Reconditioncannotbeeasilysatisfied
0 inmanymicrofluidicdevices. Asaresult,insteadofrely-
scopic structures, but also a fertile ground for explor-
. ing onthe process of self-assembly,flow-alignedcolloidal
1 ingnon-equilibriumstatisticalmechanics. Amongalldif-
0 strings are normally created manually in microfluidics,
ferent non-equilibrium systems, flow-driven particle self-
7 where particles are injected into microfluidic channels at
assemblyattractsprobablythemostattentionduetothe
1 externally-controlled regular time intervals [27–29]. The
: rapid growthof microfluidic techniques [7–11], where in-
v stringstructuresthusformedaremetastable,susceptible
teractions between fluid flows and colloidal particles are
i to external perturbations [27].
X frequently encountered.
r Particularly, the flow-driven self-assembly of one di- Inthispaper,wereportasimpleexperimentalmethod
a
mensional (1D) colloidal strings with regular particle for constructing 1D colloidal crystals, where, under uni-
spacing shows great potential in microfluidic applica- directionalflows,chargedcolloidalparticlesself-assemble
tions [12–17]. 1D colloidal strings offer a precise control into flow-aligned strings with regular particle spacing
over the positions of colloidal particles such as cells and nearasolidboundary. Usingfastconfocalmicroscopy,we
droplets, which is crucial for flow cytometry, cell sepa- study the dynamicsofthe self-assemblyprocess. Bysys-
ration and diagnostics [18–20], and microfluidic compu- tematically varying key control parameters such as the
tation and encryption [21–23]. Nevertheless, compared flow rate, the strength of electrostatic interactions, the
with a myriad of methods for creating 2D or 3D col- size and polydispersity of particles and the dimension of
loidstructures,fewerapproachesareavailableforassem- microfluidic channels, we delineate the condition for the
bling 1D colloidal strings. Due to the isotropic nature formation of colloidal strings. Based on our experimen-
of interparticle interactions, it is difficult to grow col- talobservation,weproposea simplemodelthatexplains
loidal spheres along a single direction. Thus, 1D col- theintriguingnon-equilibriumself-assemblyprocess. Al-
loidalstringsareusuallyconstructedthroughspecialpro- though our model is minimalistic with crude approxi-
2
mations, it is capable of capturing the essential mech-
anism for the origin of colloidal strings. As such, our
experiments demonstrate the possibility of assembling
flow-aligned 1D ordered colloidal structures at low Re
through the balance of hydrodynamic coupling and elec-
trostatic repulsion. Compared with previous methods,
our study provides a much simpler approach for con-
structing a large number of stable flow-aligned colloidal
strings,potentiallyusefulfordifferentmicrofluidicappli-
cations. Lastly, we show that this approach also leads
to the formation of 2D colloidal walls when a transverse FIG. 1: Formation of colloidal strings in microfluidic flows.
electric field is applied in the same colloidal system. (a) A schematic showing the geometry of our experiments.
A coordinate system is defined in the upper left. (b) An
exampleofcolloidalstringsinthenarrowchannelnexttothe
glassbottom(seealsoSupplementaryVideo1). Flowisalong
II. EXPERIMENTS
thex direction. Theaverage diameterof particles is 1.36 µm
and theflow rate is 3.0 µL/min. The scale bar is 25 µm.
We synthesized poly(methyl methacrylate) (PMMA)
spheres as our colloidal particles [30, 31], which were
fluorescently labeled with Nile red. The particles were
stabilized by covalently bound polymer brushes made of
poly(1,2-hydroxy stearic acid) (PHSA). Most our exper- All channels have a similar height of H = 104.8 9.2
±
iments were conducted using particles of average diam- µm and a length of L 4 cm. We defined a Cartesian
∼
eter d = 1.36 µm, although two other batches of parti- coordinatesystemsuchthatx is the flow directionalong
cles with d = 1.56 µm and 1.86 µm were also used to the channel length, y is along the width direction with
test the effect of polydispersity. The polydispersities of W/2 y W/2 and z is along the height direction
− ≤ ≤
particles in these particle batches are below 11%. Parti- with 0 z H (Fig. 1a). The cell was sealed at the
≤ ≤
cle sizes and polydispersities were determined from dy- bottom by a glass microscope coverslip, allowing for
namic lightscattering(DLS). PMMAparticles weresus- direct confocal imaging. Colloidal suspensions were
pended in a mixture of decahydronaphthalene (decalin, injected into the channels using a syringe pump with a
27.2 wt%) and cis + trans cyclohexyl bromide (CXB, controlled volumetric flow rate ranging from 2.0 to 9.0
72.8 wt%), which matches both the density and refrac- µL/min. The uncertainties on flow rates are less than
tive index ofthe particles. The density andthe viscosity 0.5%. Within this range, the particle Reynolds number,
of the mixture are ρ = 1.20 g/cm3 and η = 1.74 mPas, Re ρvpd/η < 5 10−4, where vp is the velocity of
· ≡ ×
respectively. Its relative dielectric constant is ǫ = 5.6. particles near the bottom wall. Finally, to probe the
r
The volume fraction of suspensions was fixed at φ=1% effect of transverse electric fields on the self-assembly
for most of our experiments. We obtained the surface process, we also fabricated a small microfluidic channel
charge of PMMA particles in the decalin/CXB mixture of 100 15 µm2 (W H), where the bottom and top
× ×
via electrophoresis [32]. The electrophoretic mobility of walls of the channel were made of indium tin oxide
particles in a DC field of 6 V/mm was measured, which (ITO) coated glasses. The details of the ITO channel
can be converted into the zeta potential and the surface can be found in Sec. III.E.
charges of particles based on a theory by Carrique et
al. [33]. The numbers of positive surface charges on the Weusedaninvertedspinning-diskconfocalmicroscope
three batches of PMMA particles with increasing d are to image the dynamics of particles near the center of a
similar with Z = 158 8.2, 152 12 and 160 4.7, re- microfluidic channel away from its inlet and outlet. The
± ± ±
spectively. Note that we used a Debye screening length images were recordedat 100 frames per second. Particle
κ−1 1.3 µm in our calculation, which is derived from dynamics at different heights z above the bottom wall
≈
the measured conductivity of the mixture at 218 pS/cm were taken by varying the focal plane of a 60 micro-
×
[32]. The screening length can be changed when salt is scope lens (NA 1.4). To follow the dynamics of colloidal
added into the solvent. All our measurements were con- particles for the longest possible time, we translated the
ducted at room temperature. sample holder of the microscope in the direction oppo-
We fabricated microfluidic channels from site to the flow of suspensions at a controlled speed in
poly(dimethylsiloxane) (PDMS) using conventional certainexperiments, so that particles at a givenposition
photolithography. The channels have rectangular cross- canbeimagedintheirco-flowingframe. Thepositionsof
sections with two different inner dimensions. A narrow all particles in the field of view were then trackedover a
channel has a width of W = 99.6 1.4 µm, whereas longtimeusingawidely-usedparticletrackingalgorithm
±
a wide channel has a width of W = 299.7 2.1 µm. [34].
±
3
FIG. 2: Colloidal strings at different flowrates in thenarrow channel. Suspensions flowtoward theright. From (a) to(e),the
flow rates are 2.0, 2.2, 2.6, 3.0 and 3.6 µL/min, respectively. The top and the bottom edges of the images correspond to the
two side walls of thechannel. The scale bars are 25 µm.
III. RESULTS AND DISCUSSIONS and the glass surface should be small [35]. The adhesion
mainlyarisesfromtheelectrostaticattractionasweshall
A typical example of colloidal strings formed in our demonstrate below. At this low flow rate, particles are
microfluidic channel is shown in Fig. 1b, where PMMA attracted to the glass surface and form immobile clus-
particlesself-assembleintoflow-aligned1Dstringsatthe ters instead of moving strings (Fig. 2a). At a slightly
boundary next to the glass bottom (see also Supplemen- higher flow rate of 2.2 µL/min, strings start to form on
tary Video 1). The strings move along with the ambient the surface of the glass (Fig. 2b). At even higher flow
flow but with significantly reduced speeds. Above the ratesof2.6µL/minand3.0µL/min,thestringstructure
bottom layer, strings disappear, where particles simply canbe clearlyidentified nearthe side walls (Figs.2c and
move passively with the ambient flow as expected (Fig. d). Short and less regular strings can also be observed
S1a in Supporting Information (SI)). The strings were near the center of the channel. The spacings between
also observed next to the top wall (Fig. S1b in SI). In the strings along the y direction show large variations.
this study, we focus on the dynamics of particles next The average spacing between the strings decreases with
to the glass bottom at z d/2 for the convenience of increasing particle concentrations (Fig. S2 in SI). It is
imaging. Nevertheless, the≈mechanism discussed should worth noting that the strings are very stable and robust
equally apply for strings next to the top wall. To reveal againststrongperturbationssuchasthepresenceofstuck
the origin of colloidal strings, we first investigate the in- particles on the surface and the variation of fluid flows
fluence of flow rates and the strength of electrostatic in- (see Supplementary Videos 1, 2, 4 and 5). However, as
teractions on the string structure. weincreasethe flowratefurther to3.6µL/min,colloidal
stringscollapse(Fig.2e). Atthishighflowrate,particles
move passively with the ambient flow.
A. Flow rate and ionic strength dependence Toquantifythedegreeofthe1Dstringorderacrossthe
widthofthechannel,wemeasurethealignmentfactorat
different y, A (y), defined as [16]
Flow rates show a strong effect on the formation of f
colloidalstrings(Fig.2). Inthe narrowchannel,atalow P∞ N L2
flow rate of 2.0 µL/min, most PMMA particles enter- Af = L∞=1 L , (1)
P N L
ing the channel near the glass bottom stick to the glass L=1 L
bottom resisting the drag from the ambient flow. Since where N is the number of strings that contain L parti-
L
the refractive index of the particles is close to that of cles. A issimilartotheweight-averagemolecularweight
f
the glass bottom, the effective Hamaker constant and, of polymer chains. A 1 and A = 1 when there are
f f
≥
therefore,thevanderWaalsattractionbetweenparticles onlyisolatedparticles. Note thatastringis definedsuch
4
FIG.4: Alignmentfactor,A ,asafunctionofthenormalized
f
velocity difference (vf −vp)/vf. vp is particle velocity. vf is
the velocity of unperturbed ambient flows. Flow rates are
indicated in theplot. The solid line providesa visual guide.
of the microfluidic channels are assumed in these calcu-
lations. We verify our calculation by directly comparing
the calculated velocity profiles with the measured veloc-
ity profiles above the glass bottom (Fig. S5 in SI). The
relative error on v is about 7% (SI). The velocity of
f
particles, v , is measured from experiments via particle
p
tracking velocimetry. Note that v reduces to zero at
p
y slightly different from W/2 and W/2 (Fig. 3a). As
−
showninFig.2,alayerofparticlessticktothesidewalls
of the channel, leading to zero particle velocity near the
FIG. 3: Degree of the string order at different flow rates.
(a) Velocity of particles, vp, across the width of the channel. side wallsnext to the glassbottom. However,awayfrom
From left to right, the flow rates are 2.2 (black squares), 2.6 the glass bottom, fewer particles stick to the side walls
(redcircles),3.0(greenup-pointingtriangles)and3.6µL/min (Fig. S1a in SI).
(blue down-pointing triangles), respectively. The solid lines
As the flow rate increases, the velocity difference be-
are fits with parabolic functions. The dashed lines are the
tween the particles and the unperturbed flow, (v v ),
calculated velocity of the unperturbed flows at z = d/2. (b) f − p
decreases (Fig. 3a). The degree of the string order is
Alignment factor, A , across the width of thechannel. Since
f
the string order is symmetric, the results are shown only for strongest near the side walls of the channel, which in-
the upper half of the channel. The symbols are the same as creaseswiththe flowrateatlowflowratesanddecreases
thoseusedin(a). Theinsetillustratesthedefinitionofstrings. athighflow rates(Fig.3b). More importantly,we find a
Particlesinadashedcirclebelongtoonestring. Neighboring positive correlation between the degree of the string or-
particles in a string should have distances smaller than 3d
der and the velocity difference between the particles and
alongthexdirectionandsmallerthandalongtheydirection.
the ambient flows (Fig. 4), when the velocity difference
is small at (v v )/v < 0.55. The strings disappear
f p f
−
when v v . At larger velocity differences, the degree
f p
thatallthe neighboringparticlesinthe stringarewithin ofthestri≈ngordermaydecrease,althoughthedatashow
three particle diameters away from each other along the strong fluctuations in this limit.
x direction and within one particle diameter away along
It is worth noting that the presence of the side walls
the y direction (Fig. 3b lower inset).
and the shear gradients near the side walls are not nec-
The alignment factor, A (y), the velocity of unper- essary for the formation of the strings. We have con-
f
turbedambientflows,v (y),andthevelocityofparticles, ducted experiments in the wide channel of 300 µm in
f
v (y),atdifferentflowratesareshowninFig.3. Here,v width (Sec. 2). The formation of strings can be clearly
p f
iscalculatedbasedonthe flowprofilesofpressure-driven observedatthe centerofthe channel(Fig.5andSupple-
flowsinrectangularductsofgivencross-sectionsatgiven mentary Video 2). The difference in particle structures
flow rates [36]. No-slip boundary conditions at the walls in the narrow and wide channels can be explained from
5
different velocity profiles in the two channels. With a
fixedflowrateperunitcross-sectionarea,theflowveloc-
ity near the center of the narrow channel is larger than
that near the center of the wide channel. We show the
velocity profiles in the two channels at a flow rate of
3 10−4 µL/(minµm2) (Fig. 5b), which corresponds to
× ·
3 µL/min in the narrow channel (the flow rate used in
Fig.2d)and9µL/mininthewidechannel(the flowrate
used in Fig. 5a). The velocity difference between parti-
cles and the ambient flow necessary for the formation of
colloidalstringsonlyestablishesatintermediateambient
velocities, as illustrated qualitatively by the blue region
inFig.5b. Atlowerambientvelocitiesbelowthebluere-
gion, the drag forces on particles are too weak. Particles
stick to the glass bottom (Fig. 2a). At higher ambient
velocities above the blue region, particles move with the
ambient flow and the velocity difference reduces to zero
(Fig. 2e). Based on the above picture, the strings form
near the wall in the narrow channel (Fig. 5b). There
exist small regions of one or two particle layers next to
bothsidewalls,whereparticlesareimmobileduetoweak
ambient flows (Fig. 3a). Near the center of the narrow
channel,the flowvelocitiesaretoolargeandthe velocity
differencesbetweenparticlesandtheflowaretoosmallto
form colloidal strings (Fig. 2d). In contrast, in the wide
channel, the flow velocities away from the side walls all
fall in the blue region. As such, colloidal strings form
near the center of the wide channel. Particles are im-
mobile in a larger region near the side walls due to the
smallervelocitygradientsnearthewallsofthewidechan-
nel(Fig.5b). Quantitatively,thecorrelationbetweenA
f
and (v v )/v in the wide channel shows the simi-
f p f
−
lar trend as that in the narrow channel (Fig. 4). Thus,
our results indicate that the velocity difference between
the particles and the ambient flows, instead of the shear
gradient, is the key factor dictating the formation of the
strings.
The formation of strings also depends on the electro-
statics of particles. To reveal the effect of electrostatic
interactions, we added 100 µM quaternary ammonium
salt—tetrabutyl ammonium bromide (TBAB)—into the
solvent,whichaccordingtoapreviousstudyreversesthe
FIG.5: (a)Formationofcolloidalstringsinthewidechannel. sign of the surface charge of particles [37]. We also di-
Thesuspensionflowstowardtherightandtheflowrateis9.0
rectlyconfirmthereverseofthesurfacechargeonPMMA
µL/min. Since the width of the channel is larger than the
particles in our experiments using electrophoresis. As a
field of view of our 60× lens, the entire channel is imaged
result, when the salt is added, PMMA particles do not
by tiling three separate images. The top and bottom edges
of the image correspond to thetwo side walls of thechannel. stick to the glass bottom even at low flow rates. Due to
Thescalebarsare25µm. (b)Flowprofilesinthenarrowand the repulsive, instead of attractive, interactions between
wide channels at z = d/2. The flow rates are 3 µL/min in particles and wall,the density of particles at the bottom
thenarrowchanneland9µL/mininthewidechannel,which is dramatically reduced (Fig. 6b). We cannot observe
give the same flow rate per unit cross-section area. The blue
the formation of strings at any flow rates (Figs. 6b and
areaindicatesqualitativelytheregionwheretheambientflow
d). Particle velocity also significantly increases. At 3.0
velocities are in theright range for the formation of colloidal
µL/min, the velocity of particles is almost the same as
strings.
thatoftheunperturbedambientflow(Fig.6c). Thefind-
ingindicatesthatwithoutsalttheelectrostaticattraction
betweenparticlesandtheglassbottominducesfrictional
6
FIG. 6: Colloidal structures at different ionic strength. (a)
Colloidal strings in the solvent without TBAB. (b) The col-
lapse of strings in the solvent with 100 µM TBAB. The flow
rate is 3.0 µL/min. A comparison of particle velocity and
alignmentfactorinthetwocasesareshownin(c)and(d),re-
spectively. EmptysquaresareforthesolventwithoutTBAB.
Red disks are for thesolvent with TBAB. The scale bars are
25 µm.
or lubrication forces on the particles, which slow down
the motion of the particles relative to the ambient flow.
With the reversed electrostatic interaction at high ionic
strength, the repulsion between particles and the glass
bottom eliminates the drag force from the bottom. As
a result, the particles move passively with the flow with
(v v ) 0. The strings do not form (Fig. 6).
f p
− ≈
In summary, our experiments show that the velocity
differences between the ambient flows and particles are
important for the formation of colloidal strings. Such
velocity differences lead to hydrodynamic coupling be-
tweenparticles,crucialforthebondingofparticlesinthe
strings. Moreover, the result with reversed electrostatic
FIG. 7: Dynamics of particle pairs in the wide channel. (a)
interactions suggests that the retarded motion of parti-
Center-to-center distance between two particles in particle
cles is induced by the electrostatic attraction between pairs, r, as a function of time, t. r is normalized by the
particlesandtheglassbottom. Chargesonparticlesnec- average particle diameter d. r and d are defined in the inset
essarilygivesrisetoelectrostaticrepulsionsbetweenpar- of (c). t is normalized by the Brownian relaxation time of
ticles, a factor also needs to be considered in modeling particles inthedilutelimit, τ0 =6πηa3/kBT =2.19 s, where
colloidal strings. η = 1.74 mPa·s is the viscosity of the solvent and a = d/2
is the average radius of particles. The horizontal dashed line
indicates the mean distance between particles. The flow rate
is2.6µL/min. (b)Probabilitydistributionfunction(PDF)of
B. Dynamics of stable particle pairs r. The solid line is a fit of a Gaussian distribution using the
data near the peak r0 and r < r0. The vertical dashed line
indicates the peak of the distribution, r0. The inset shows
To quantify the interparticle interactions, we study
stable particle pairs in our experiments. The scale bar is 25
the dynamics of stable particle pairs. The formation µm. (c) Effective force between particles, F(r). F is normal-
of particle pairs is the precursor and the first step for ized by kBT/d. The black line shows the model result (see
the assembly of longer colloidal strings. To create stable text for details).
particle pairs without interference from longer colloidal
strings, we reduce the volume fractions of suspensions
7
FIG. 8: Average center-to-center distance, r0, between two FIG.9: Autocorrelationofthecenter-to-centerdistance,r(t),
particles in particle pairs of different size ratios, a2/a1. r0 is of a particle pair. The flow rate is 3.0 µL/min. The size
normalized by the mean diameter of particle pairs, a1+a2. ratio of the two particles is a2/a1 = 1.2. Inset shows the
Theblacksolidlineisthenormalizedr0 fromthemodel. The correlationtime,τ,asafunctionofa2/a1. τ isnormalizedby
dashed line shows the normalized [r0 −(a1 +a2)] from the the Brownian relaxation time of particles, τ0. The solid line
model. The red solid line shows the normalized r0 from the is a fit from themodel.
modelwhentheeffectivedistancere=[r0−(a1+a2)]isused
in Eq.6.
tion r =r where F(r =r )=0, the harmonic potential
0 0
to φ = 0.01% in our experiments. Most particles form gives rises to a spring-like force. When r < r , a re-
0
stable particle pairs in the wide microfluidic channel at pulsive force pushes the two particles apart, whereas at
suchalowparticleconcentration(Fig.7binset). Thedy- r > r an attraction pulls the particles together. The
0
namics of stable particle pairs next to the glass bottom force can be as large as several k T/d and, therefore,
B
around y = 0 in the wide channel are then measured, can effectively suppress the thermal motion of particles
where the variation of the speeds of the ambient flow is andmaintainthestabilityoftheparticlepair. Whenthe
small thanks to the small velocity gradients near y = 0. distance between particles is one particle diameter away
We track the center-to-center distance between two par- fromthe equilibriumposition, r >r +d, F(r) 0. The
0
≈
ticles in particle pairs, r, as a function of time (Figs. 7a particles are decoupled.
andcinset). Figure7bshowstheprobabilitydistribution
The results shown in Figs. 7b and c are obtained by
function (PDF) of the distance. The maximal probabil-
averaging tens of particle pairs of different size ratios.
ity occurs at r /d=2.160 0.006. Near r and at short
0 0 A more detailed analysis shows that the equilibrium po-
±
separations with r <r , the distribution can be well ap-
0 sition r depends on the size ratio of the two particles
0
proximated by a Gaussian distribution with a standard
in a particle pair, a /a , where a and a are the radii
2 1 1 2
deviationofσ/d=0.52 0.01. ThePDFshowsalongtail
of the leading and the trailing particles in the particle
±
at large r, indicating higher probabilities to find particle
pair, respectively. Note that when calculating a /a , a
2 1
pairs at large separations.
correction has been made to consider the effect that the
In analogy to equilibrium systems, the Gaussian dis-
centers of small and large particles sit at slightly differ-
tributionnearr allowsustodefineaneffectivepotential
0 ent heights above the glass bottom (Fig. S4 in SI). We
and, accordingly, an effective force between particles in
estimate the errors on a /a to be about 4%. To probe
2 1
this non-equilibrium flow-driven system.
a larger range of size ratios, we mix the d = 1.36 µm
U(r) particle batch with two other batches of particles with
=lnA ln(PDF(r)), d = 1.56 and 1.86 µm (Sec. II). The polydispersity of
k T −
B
these particle batches allowsus to obtainan almostcon-
dU(r) k T dPDF
B
F(r)= = , (2) tinuous range of size ratios. We show r0 as a function
− dr PDF dr
of a /a in Fig. 8. r decreases with increasing a /a .
2 1 0 2 1
where U(r) is the effective potential in the unit of ther- The change is most drastic at small a /a when the size
2 1
mal energy k T and F(r) is the effective force between difference of the two particles in a particle pair is small.
B
particles in a particle pair. A is the normalization con- Note that when we set the average size ratio of particle
stant of the probability distribution. F(r) in the unit of pairs equal to the polydispersity of the particle batch,
k T/d is shown in Fig. 7c. Near the equilibrium posi- i.e., a /a = 1.11, we have r 2.2, consistent with our
B 2 1 0
≈
8
pairs. Our experiments have shown that hydrodynamic
coupling and electrostatic interaction between particles
are the key factors influencing the formation of stable
colloidal pairs and strings. To induce the hydrodynamic
coupling, the glass bottom provides the resistance force
that slows down the motion of particles relative to the
ambient flow.
A full theoretical description of the motion of solid
spheres close to a solid boundary in a semi-infinite fluid
medium cannot be achieved within the limit of low-Re
Stokeshydrodynamics[39,40]. Tofullyaddresstheprob-
lem,complexfactorsincludingtheroughnessofparticles’
surface [39, 41], the permeability and the stiffness of the
layer of grafted PHSA polymer brushes [42], and even
the cavitation of fluids underneath particles need to be
considered [39, 43], which is certainly beyond the scope
FIG. 10: Probability distribution of the size ratio between
of our current work. Instead, we try to build a simple
the trailing and leading particles, P(a2/a1). The red line model, aiming for an order-of-magnitudeestimate of our
is a fit using a log-normal function with the peak value at
observations. Surprisingly,theminimalisticmodelshows
a2/a1 = 1.064. Inset shows the force diagram on a particle
pair. The flow direction is indicated by v . a semi-quantitative agreement with our experiments.
f
In our model, we reduce the complex effects of the
average measurements shown in Fig. 7b. bottom boundary on the motion of particles into one es-
We also calculated the autocorrelationof the particle- sential effect, i.e., it provides a resistance force, f, slow-
pair dynamics (Fig. 9), ing down the motion of particles. The effect enables the
hydrodynamiccouplingbetweenparticles,crucialforthe
(r(t+t0) r )(r(t0) r ) formationofcolloidalstrings. Wefurtherassumef arises
C (t)= h −h i −h i i. (3)
rr r2 r 2 from the friction, instead of lubrication, between parti-
h i−h i
cles and the glass bottom and, therefore, is independent
The average ... is taken over all the initial time t0. ofthevelocityofparticles. Notethatthedensityofparti-
h i
r r0. The vibration of the two particles in a particle clesmatchesthatofthesolvent. Hence,thenormalforce
h i≈
pair around r0 leads to the oscillation of Crr at large t. that results in the friction should arise from the electro-
Thecorrelationtimeoftheparticle-pairdynamics,τ,can static attraction between particles and the wall instead
be defined by fitting Crr at small t with an exponential of the weight of the particles. Since the average sur-
function, exp( t/τ). τ indicates the rigidity of the har- face charge of particles is approximately the same for
−
monic potential in Eq. 2. A smaller τ corresponds to a different batches of particles (Sec. II), f should also be
narrowerandstifferpotentialwell. Wefindthatτ isalso independent of the size of particles. The assumption is
a function of a2/a1 (Fig. 9 inset). A larger a2/a1 gives consistent with our experiments. At low flow rates, the
rise to a smaller τ and, therefore, a stiffer potential. drag on particles from the flow is below the threshold of
The dependence of r0 and τ on a2/a1 indicates that staticfriction. Particlessticktothe glassbottomanddo
particlepolydispersityplaysanimportantroleinthefor- not move (Fig. 2a). At higher flow rates, particles slide
mation of stable particle pairs. Indeed, we measuredthe on the glass bottom. The kinetic friction f follows the
size ratio between the leading particle and the trailing Amonton-Coulomb law f = µN, where µ is the friction
particle, a2/a1, over 220 particle pairs. Figure 10 shows coefficientbetweenPMMAspheresandtheglassbottom
theprobabilitydistributionfunctionofa2/a1. Thedistri- and N is the normal force induced by the electrostatic
bution is clearly biased toward a2/a1 >1 with the max- attraction between the spheres and the glass bottom.
imal probability occurring at a2/a1 = 1.064. In other When salt is added into the solvent, the particle-wall
words, the leading particle is smaller than the trailing interactionbecomesrepulsive,whichremovesthenormal
particle in a particle pair. force. As a result, f 0. Particles move passively with
≈
the flow (Fig. 6b). The hydrodynamic coupling is zero
and the string structures cannot be observed.
C. A minimalistic model
Under the above approximation, the force balance of
Based on our experimental observations, we con- the two particles in a particle pair can be easily written
structed a simple minimalistic model, which provides a (Fig. 10 inset). Note that at low Reynolds number, the
semi-quantitative descriptionof the dynamics of particle total force on each particle is zero. Therefore, for the
9
leading particle (particle 1), we have
F +F =f. (4)
h1 e
For the trailing particle (particle 2), we have
F =F +f. (5)
h2 e
Here, F is the hydrodynamic force onparticle i andF
hi e
is the electrostatic repulsion between the two particles.
i=1 or 2.
For the hydrodynamic forces, we have [44]
F (v v ) [K (v v )/4πr]
h1 f p1 2 f p2
= − − − ,
ηK 1 (K K )/(4πr)2
1 1 2
−
F (v v ) [K (v v )/4πr]
h2 f p2 1 f p1
= − − − , (6)
ηK 1 (K K )/(4πr)2
2 1 2
−
FIG.11: Effectivespring-likeforcesonparticlepairsfromthe
where r is the center-to-center distance between the two model. The main plot shows the ratio of the effective spring
particles, vf is the velocity of the unperturbed ambient constants,k1/k2,ontwoparticlesinparticlepairsofdifferent
flow at the center of particles and vpi is the velocity of sizeratios,a2/a1. Thegrayregionindicatesunstableparticle
particle i. For a stable particle pair, v =v . The two pairs with a2/a1 < 1.062, where k1/k2 > 0. Inset shows the
p1 p2
particles move together with the same speed. K =6πa forcesonthetwoparticlesinaparticlepairwitha2/a1 =1.11
i i
whenthedistancebetweenthetwoparticlesrisslightlyaway
isthecoefficientofStokes’dragofparticlei. Whenv =
f fromtheirequilibriumpositionr0. Theforcesarenormalized
v , the hydrodynamic forces disappear. Furthermore,
pi bykBT/(a1+a2).
as r , F reduces to F = ηK (v v ), i.e.,
hi hi i f pi
→ ∞ −
the Stokes law of isolated particles. The two particles
decouple in this limit.
The prediction of r from Eq. 8 is compared with ex-
0
For the electrostatic repulsion,the DLVO theory gives
perimental results in Fig. 8. Since the leading particles
[35]
arealwayssmallparticlesfromtheparticlebatchwithav-
eragediameterd=1.36µm,wefix a =d/2=0.68µm
Z1Z2e2exp(κa1)exp(κa2)(1+κr) h 1i
F = exp( κr), in our calculation. In addition, we use Z = 157 and
e ǫrǫ0 1+κa1 1+κa2 r2 − the averagevelocity difference betweenhpa1r,t2iicles and the
(7)
ambient flow at v v = 65 µm/s, both of which are
whereZi andai arethenumberofchargesandtheradius h f − pi
measureddirectlyfromexperiments. Themodelqualita-
ofparticle i, e is the elementarycharge,ǫ is the relative
r
tively predicts the decreasing trend of r with increasing
dielectric constant of the solvent, ǫ is the vacuum per- 0
0
mittivity and κ−1 is the screening length of the solvent. a2/a1 (the black solid line in Fig. 8). The quantitative
discrepancy may arise from the crude approximation we
Since Z Z >0, F is always repulsive.
1 2 e
havemadeontheeffectsofthewallandalsofromthefact
The force balance (Eqs. 4 and 5) leads to a simple
the hydrodynamic forces in Eq. 6 is accurate only when
relation,
the distance between particles are large r (a +a )
0 1 2
≫
[44]. Interestingly, when we plot the surface-to-surface
F F =2F . (8)
h2 h1 e
− distance, [r (a + a )], from the model, it quanti-
0 1 2
−
Eq. 8 gives a zero internal force between the particles, tatively agrees with our experiments on r0 (the black
(F F 2F )/2 = 0. For two particles of the same dashed line in Fig. 8). The result indicates that the par-
h2 h1 e
size, F− =−F . In this case, Eq. 8 can be satisfied only ticle distance used in Eq. 6 may be replaced by an effec-
h1 h2
byFe =0whenthedistancebetweenthetwoparticlesis tive distance re when r0 is comparable with (a1 +a2).
significantly larger than the screening length κ−1. With Considering the increasing contribution of the lubrica-
thetwoparticlestravelingatthesamespeed,theparticle tion stress in the gap between particles to the hydrody-
pair thus formed is metastable, susceptible to random namic coupling, re should be smaller than the center-
perturbations. For stable particlepairs with nonzeroFe, to-center distance r0 and close to the surface-to-surface
polydispersity is necessary. Since Fe >0 and Fh1 <Fh2, distance between particles. Indeed, when we replace r0
the leading particle should be smaller than the trailing with re = [r0 (a1 +a2)] in Eq. 6, the result shows a
−
particleinaparticlepair,consistentwithourobservation quantitative agreement with our measurements (the red
(Fig. 9). Eq. 8 determines the equilibrium position r of line in Fig. 8).
0
particle pairs. In addition to the equilibrium position r , the model
0
10
also predicts that the approximately harmonic potential
near r is narrowerand stiffer atlargera /a , leading to
0 2 1
the decreaseofthe correlationtime τ witha /a , consis-
2 1
tent with our observation (Fig. 9 inset). Quantitatively,
the force on the particle i in a particle pair can be ap-
proximated as a spring-like force F = k (r r ) when
i i 0
−
the distance between the two particles is slightly differ-
ent from r , r r /r 1 (Fig. 11 inset), where k
0 0 0 i
| − | ≪
is the effective spring constant. F is balanced by the
i
Langevin-type random force from thermal fluctuations.
The two effective springs on the particles in a particle
pair are in series. Thus, the average spring constant of
the effective force between the two particles can be sim-
ply estimated as k = k k /(k + k ). When we set FIG. 12: Nucleation and growth of 1D colloidal crystals.
1 2 1 2
the average particle ra|tio a|t a| /a| =| 1|.11, the polydis- Fluidflowstowardtheright. (a)Thetimeseriesshowingthe
2 1
formation of a particle pair (see also Supplementary Video
persity of the particle batch used in Fig. 7, the resulting
3). The times of the frames are 0 s, 0.53 s, 0.9 s and 1.91
spring-like force quantitatively matches our experiments
s, respectively. (b) The time series showing the formation of
(the black line in Fig. 7c). The correlationtime can also
a colloidal string of three particles (see also Supplementary
beestimatedasτ 1/√k. Afittingusingthismodelpre- Video 4). The times of the frames are 0 s, 1.85 s, 2.89 s and
∼
diction again agrees well with our experiments at large 5.42s,respectively. (c)Thetimeseriesshowingtheformation
a /a (Fig. 9 inset). The predicted τ, however, shows of a colloidal string of four particles (see also Supplementary
2 1
a sharper increase at small a /a . This discrepancy can Video 5). The times of the frames are 0 s, 0.78 s, 1.58 s and
2 1
2.46 s, respectively. The flow rate is 3.0 µL/min in the wide
be relatedto the limit of the stability ofparticle pairs at
microfluidic channel. The scale bar is 7 µm.
small a /a .
2 1
D. Kinetics and formation of colloidal strings
To have a stable particle pair, the sign of k and k
1 2
must be opposite, so that the forces acting on the parti-
clesareattractivewhenr >r0 andrepulsivewhenr <r0 The size difference between the leading and trailing
(Fig.11inset). Basedonthecriterion,aparticlepairbe- particles also explains the kinetics of the formation of
comes unstable or metastable when the size ratio of the particle pairs. For an isolated particle, the particle ve-
two particles is smaller than 1.062 (Fig. 11). The po- locityisgivenbyv =v f/(6πηa)fromtheStokes’law,
p f
−
tential well becomes infinite soft and τ diverges at this where f is the friction that is independent of the radius
lowerboundofa2/a1. Duetotheapproximationtakenin of the particle a. The larger size of the trailing particle
the model, the lowerbound fails to explainthe existence results in a faster particle speed, which allows the large
of particle pairs of small a2/a1 observed in our experi- particle catch the small particle from behind and forms
ments, which may be metastable in nature. As a result, astableparticlepair(Fig.12aandSupplementaryVideo
the predicted τ shows a sharper increase at small a2/a1 3).
compared with experiments. Interestingly, the estimate
Particle pairs serve as the nuclei for the formation of
lower bound quantitatively matches the size ratio of the
longer colloidal strings at higher particle concentrations.
mostpopulousparticlepairsfoundinourexperimentsat
Duetothehydrodynamiccoupling,thespeedofparticles
a /a =1.064 (Fig. 10).
2 1
in colloidal strings decreases with the length of strings
(Fig. 13). For example, for a pair of equal-sized parti-
cles,thespeedofparticlesisgivenbyv =v cf/(6πηa)
p f
−
Oursimpleminimalisticmodelisabletocapturesemi- with c = 1+3a/(2r) > 1 (Eq. 6), which is smaller than
quantitatively all the essential features of experimental the speed of isolated particles. As a result, a third iso-
observations. Although the slowdown of particle mo- lated particle can approach a particle pair from behind
tions near the boundary is induced by the electrostatic and form a colloidal string of three particles (Fig. 12b
attractions in our experiments, the model shows that and Supplementary Video 4). The same mechanism also
any other attractive interactions such as those induced applies for strings of four particles (Fig. 12c and Supple-
by commensurate polymer brushes attached to particles mentaryVideo5). Astheprocessrepeats,alongcolloidal
andwalls[45]canalsoleadtoasimilarmechanismforthe string eventually forms. The dynamic assembly of long
assembly of colloidal strings. The non-equilibrium self- colloidal strings can be seen as a nucleation and growth
assembly principle explored here should apply in much process of 1D colloidalcrystals in a flow-drivennonequi-
broader contexts. librium system.