Table Of ContentCharged hydrophobic colloids at an oil/aqueous phase interface
Colm P. Kelleher
Department of Physics and Center for Soft Matter Research,
New York University, 4 Washington Place, New York, New York 10003, USA
Anna Wang
Harvard John A. Paulson School of Engineering and Applied Sciences,
Harvard University, Cambridge, Massachusetts 02138, USA
Guillermo Iv´an Guerrero-Garc´ıa
Department of Materials Science and Engineering,
Northwestern University, Evanston, IL, USA, and Instituto de Fisica,
7
Universidad Autonoma de San Luis Potosi, San Luis Potosi, Mexico
1
0
2 Andrew D. Hollingsworth, Rodrigo E. Guerra, Bhaskar Jyoti Krishnatreya, David G. Grier
Department of Physics and Center for Soft Matter Research,
n
New York University, 4 Washington Place, New York, New York 10003, USA
a
J
0 Vinothan N. Manoharan
3 Harvard John A. Paulson School of Engineering and Applied Sciences,
Harvard University and Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
]
t
f Paul M. Chaikin
o
Department of Physics and Center for Soft Matter Research,
s
. New York University, 4 Washington Place, New York, New York 10003, USA
t
a (Dated: February 1, 2017)
m
Hydrophobic PMMA colloidal particles, when dispersed in oil with a relatively high dielectric
- constant, can become highly charged. In the presence of an interface with a conducting aqueous
d
phase,imagechargeeffectsleadtostrongbindingofcolloidalparticlestotheinterface,eventhough
n
theparticlesarewettedverylittlebytheaqueousphase. Inthispaper,westudyboththebehaviorof
o
individual colloidal particles as they approach the interface, and the interactions between particles
c
that are already interfacially bound. We demonstrate that using particles which are minimally
[
wetted by the aqueous phase allows us to isolate and study those interactions which are due solely
1 tochargingoftheparticlesurfaceinoil. Finally,weshowthattheseinteractionscanbeunderstood
v by a simple image-charge model in which the particle charge q is the sole fitting parameter.
1
0
8 I. INTRODUCTION sensitive to the protocol used to prepare the samples
8
[13, 14], are highly non-uniform [15], and are strongly
0
time-dependent [14].
. Understanding the behavior of colloidal particles at
1
0 fluidinterfacesisalong-standing[1]andactivelystudied Toexplainthesecomplicatedinteractions,differentau-
7 problem in soft condensed matter physics [2–5]. Exten- thors have proposed various modifications or extensions
1 sive experimental and theoretical work has been carried of Pieranski’s simple model. These include mechanisms
: out on interactions between particles that are partially for interparticle attraction, such as from inhomogenous
v
i wettedbybothfluids,thatis,systemswheretheequilib- chargedistributionontheparticlesurface[16],andinter-
X rium contact angle θ falls in the range 0◦ <θ <180◦. particlerepulsion,forexamplebychargingoftheparticle
C C
r As noted by Pieranski [6], the presence of a fluid inter- surface in oil [14, 17]. Finite-ion-size effects in the aque-
a
face can lead to a charge asymmetry in the vicinity of ous phase have been proposed to explain the anomalous
eachwettedparticle,andhencetointeractionswhichare dependence of the interparticle force on salt concentra-
dipolar in form. Indeed, the r−4 force law characteris- tion [18], while irregular pinning of the contact line on
tic of dipole-dipole repulsion has been observed in many the colloid surface introduces anisotropic capillary forces
experiments [7–9]. between particles [19, 20]. Moreover, since all these ef-
fects can in principle occur at the same time in the same
However, various aspects of the interactions between
sample, it is difficult to disentangle them.
interfacialparticlesarestillnotwell-understood[10]. For
instance, interfacial colloids may form repulsive crystals In this paper, we report measurements of the interac-
orfractalaggregates[11],ormayself-assembleintomore tions between colloidal spheres at an oil/aqueous phase
complex mesoscopic structures [12]. The interactions re- interface in a system with two useful properties. First,
sponsible for this collective behavior are typically very thespheresareembeddedalmostentirelyintheoilphase
2
the vicinity of a glycerol/water mixture (“the aqueous
phase”).
1. Preparation of Glassware & Sample Chambers
Theglassweusetostoretheparticlesandtoconstruct
sample chambers is sonicated for 10s in 5wt% Contrad
70 detergent, followed by sequential rinsing in de-ionized
water, acetone and isopropanol. The glass is then blown
drywithanN sprayerandplacedinanovenat70◦Cfor
2
atleast15minpriortouse. Wenotethefollowingexcep-
tion: the sample chamber we use in the experiment de-
scribedinSectionVBconsistsofaglasscapillarytubeof
internaldimensions100µm×2mm×5cm(VitroTubes)
whichisultrasonicatedinMilliporewaterfor10s,andfi-
FIG.1. (Coloronline)(A)Onceboundtotheinterface,non- nally dried in an oven at 70◦C for 2h. Where necessary,
wetting colloids repel electrostatically. (B) Optical micro- weuseaglycerolbufferphasetoensurethattheoilnever
graphsofasinglerunofanexperimenttoprobetherepulsive comes into contact with the Norland optical adhesive we
force between pairs of interfacial colloids, each with a diam- use to seal the samples.
eter d = 1.1µm. (i) & (ii) Two interfacial particles, more
than 20µm away from any others, are identified using a par-
ticle tracking algorithm, and (iii) automatically dragged to
a pre-assigned location using laser tweezers. (iv) The par-
ticles are released from the tweezers, and move freely along
2. Fluid Phases
the interface. Their trajectores (shown in red) are recorded
throughout.
Theaqueousphaseconsistsof10mMNaClina70wt%
glycerol solution, while the oil phase consists of a 5:3:2
and are wetted very little, or not at all, by the aqueous v/v mixture of cyclohexyl bromide (CHB), hexane and
phase. Second, the oil has a dielectric constant which is dodecane. To prevent ionic contamination of the oil
large compared to that of typical hydrocarbon oils, and phase, we filter and store it according to the protocols
so can harbor mobile charges. These properties allow described in Refs. [23] and [21]. Using the formula given
us to isolate and explore how the interparticle interac- in [24], we estimate that this oil has a relative dielec-
tions are influenced by electrostatic charges on the par- tric constant (cid:15) = 4.2, which is much lower than water
r
ticles’ surfaces. Similar systems have been studied pre- ((cid:15) ≈ 80), but significantly higher than alkanes such as
r
viously [21], particularly for the insights they offer into decane ((cid:15) ≈2). Theoretical estimates [25] indicate that
r
the proliferation and dynamics of topological defects in an oil with (cid:15) =4, in contact with a water reservoir, will
r
two-dimensional curved spaces [22]. By elucidating the reach an equilibrium ionic concentration with a Debye
natureoftheinteractionsinthissystem, wealsohopeto screeninglengthλ ofapproximately50µm,whichisfar
D
cast new light on these phenomena. greaterthanthelengthscalesprobedinourexperiments.
We study two different aspects of the behavior of col-
loids in this system: the approach and binding of indi-
vidual particles to the oil/aqueous phase interface, and
the repulsive force between interfacially bound colloids.
3. Colloidal Particles
We show that both sets of observations can be quantita-
tively described by a simple electrostatic model in which
the aqueous phase plays the role of a conducting sub- The PMMA microparticles are sterically stabilized
strate, and the particle charge q is the only adjustable with covalently bound poly(12-hydroxystearic acid) [27].
parameter. This model is shown schematically in Fig. Suchparticleshaveasurfacechargethatmightbecaused
1(A). byadsorptionofpositivelychargedspeciesresultingfrom
the decomposition of CHB [23], chemical coupling of an
amine catalyst during particle synthesis [28], or some
II. MATERIALS combinationofthesemechanisms. Insomeofourexperi-
ments, weusespheresthatarefluorescentlylabeledwith
Our experimental system is composed of poly(methyl absorbed rhodamine 6G dye [29]. We find that dyeing
methacrylate) (PMMA) spheres, dispersed in oil, in the particles does not affect their measured interactions.
3
edge-finding routine, is shown in Fig. 2. For our system,
we measure the best-fit contact angle of the particles to
be in the range 171◦ to 180◦, consistent with the results
ofRef.[21]. Allmeasurementsareconsistentwithacon-
tact angle of 180◦.
IV. COLLOID-INTERFACE INTERACTION
Weprobetheinteractionofindividualcolloidalspheres
with a flat, horizontal oil/aqueous phase interface by
measuringtheirtrajectoriesastheymovethroughtheoil
phase toward the interface. Because the particles move
at speeds up to 80µms−1, too fast to track with confo-
FIG.2. (Coloronline)Horizontalandverticalslicesofacon- cal microscopy, we measure their trajectories with dig-
focal micrograph of a single PMMA sphere electrostatically ital holographic microscopy. In this technique, an inci-
bound to the neck of a capillary bridge droplet. Top Left: dent monochromatic plane wave scatters from a spheri-
Schematic of the geometry of the particle and interface. The cal colloidal particle. Using the apparatus described in
curvature of the capillary bridge is exaggerated for clarity. Refs. [26] and [20], we digitally record the image that
Main Figure: The dashed white curves show the result of an results from interference of the scattered light with the
edge-finding routine designed to measure the contact angle.
incident plane wave [30, 31]. Fitting the interference
For this particle, θ =180◦+0◦.
C −9◦ pattern predicted by Lorentz-Mie theory to the recorded
hologram [32] gives the particle’s three-dimensional po-
sition with 3nm precision over a 50µm×50µm×50µm
III. MEASUREMENT OF THE CONTACT volume at time intervals as low as 1ms. Since the oil
ANGLE θ andaqueousphaseshavewell-matchedrefractiveindices,
C
wefitthedatausingfunctionsappropriateforscattering
fromadielectricsphereimmersedinamediumofuniform
To verify that our particles remain entirely immersed
refractive index [33]. To avoid interference from multi-
in the oil and are not wetted by the aqueous phase,
pleparticlesinthesameimage,PMMA-in-oildispersions
we measure their contact angle directly by fluorescent
are prepared at volume fractions below 10−6. As well as
confocal microscopy. We do this by preparing a low-
position data, the holographic measurements yield esti-
concentration dispersion of PMMA particles of mean di-
mates for the diameters of the particles, with nanometer
ameter2.6µminoil,andflowthedispersionintoachan-
precision, and their refractive indices, which can be used
nel containing several capillary bridges of the aqueous
for consistency checks. The colloidal particles we use for
phase. To create these capillary bridges, we first use a
these and all subsequent studies have a mean diameter
sprayer to deposit droplets (of typical diameter 10µm to
d=1.08µm and polydispersity 5%.
100µm) of the aqueous phase on a cover slip. We then
To understand the observed trajectories of the parti-
placethecoverslip,dropletsidedown,onDura-larspac-
ersofthickness25µmwhichhavebeenplacedonamicro- clesastheyapproachtheinterface(Fig.3), weconstruct
an equation of motion involving the electrostatic force
scope slide. The larger droplets come into contact with
and drag. Because the aqueous phase contains dissolved
the microscope slide, and spontaneously form capillary
salt ions that act as free charges, we treat it as a good
bridges. We use a two-channel Leica TCS SP5 II confo-
conductor. Asphereofchargeq whosecenterisatheight
cal microscope, with a 63× NA 1.4 oil-immersion objec-
z aboveaflatconductingsurfaceisattractedtowardsits
tive lens to simultaneously image the particles and the
image charge with a force [34]
aqueous phase. For these studies, the aqueous phase is
fluorescentlylabeledbyreplacingsomeoralloftheNaCl
q2
with fluorescein sodium salt. Since the oil and aqueous
F (z)=− . (1)
phases are refractive-index matched to within approxi- z 16π(cid:15)r(cid:15)0z2
mately 1%, optical artifacts arising from the curvature
of the interface are minimized. Because the motion is overdamped, the speed with
As shown in Fig. 2, some of the particles bind to the which the sphere approaches the interface is given by
neck of a capillary bridge, presumably by electrostatic
forces. Using inbuilt edge-detection algorithms from the v (z)= Fz(z), (2)
commercial software package Mathematica, we identify z γ (z)
⊥
the edges of the capillary bridge and the colloidal par-
ticle. The contact angle is calculated from these data. where γ (z) is the viscous drag coefficient for motions
⊥
A typical confocal slice, overlaid with the results of the perpendicular to the fluid-fluid interface located in the
4
8
A B 0
6 1) --2100 ○ ○■○■ ○■○■○■○■○■○○■○○○■○■○■○■○■○■○■○○○○○○○○○
) -s ○ ■ ■
m 4 m -30 ○■ ○ ■
μ ■
( μ
z ( -40
z
2 v
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● --6500 ■■
0
2 3 4 5 6 7 8
-0.6 -0.4 -0.2 0.0 0.2
z(μm)
t(s)
FIG.3. (Coloronline)(A)Thecoloredtrajectoriesareobtainedbydigitalholographicmicroscopyofsevendifferentd=1.1µm
colloidal particles approaching the oil/aqueous phase interface. For each trajectory, we have set t = 0 as the time of the
attachmentevent,andz=0asthepositionofthefluidinterface. Thechargedparticlesarestronglyattractedtotheinterface,
achievingspeedsofupto80µms−1. Forcomparison,theseparticleshaveasedimentationvelocityof0.04µms−1. Wehavealso
plotted (in black) the trajectory of a d=1.6µm PMMA particle in a system where the oil phase consists of pure decane [26].
IntheabsenceofCHB,theparticleisfarlesscharged,andapproachestheinterfacemuchmoreslowlythaninthepresenceof
CHB. The red and gray disks indicate the sizes of the colloidal particles, to scale on the z-axis. (B) Plot of the velocities of
two of the trajectories, as a function of distance from the interface. The error bars indiate the expected width of the velocity
distribution,calculatedfromthebulkdiffusionconstantD . Foreachdataset,thebestfitsofthemodelgivenbyEqs.(1),(2)
0
and (3) are also shown.
plane z =0. Lee and Leal [35] find that eter in this expanded model yields as an upper bound
|σ| < 0.2e /µm2, but neither improves the quality of
γ (z) 3 d2+3λ (cid:18) 3 d2+3λ(cid:19)2 (cid:26)d3(cid:27) the fits nor significantly affects our estimate of the par-
⊥ =1+ + +O , ticle charge q. We therefore omit σ from the model. We
γ 16z 1+λ 16z 1+λ z3
0 alsoneglectthegravitationalforcebecause,overthemea-
(3)
sured range of z, it is negligible compared to the image-
where λ = µ /µ is the ratio of dynamic viscosities
aq oil
charge interaction: |F |<0.02|F |.
of the two fluid phases and γ = 3πµ d is the Stokes grav z
0 oil
These measurements establish that the force draw-
drag on a sphere far from any boundaries. For our
ing the PMMA spheres to the interface is consistent
system, µ = 23mPas (from tabulated values) and
aq
with image-charge attraction and provide an estimate of
µ = 1.7mPas (from our measurements of the parti-
oil
the single-sphere charge. We next investigate how that
cle diameter and diffusion constant in bulk oil) so that
charge influences the interaction between spheres at the
λ = 14. We estimate D , the 3D diffusion constant
0
interface.
of the particles far from any boundaries, by measuring
D (z), the diffusion coefficient parallel to the interface.
(cid:107)
We calculate D (z) from those parts of the trajecto-
(cid:107)
ries where z > 2µm. From the hydrodynamic theory V. PAIR INTERACTION OF INTERFACIAL
in Ref. [35], we estimate that the error in approximat- COLLOIDS
ing D by D (z > 2µm) is around 6%. We then use
0 (cid:107)
the fluctuation-dissipation relation, γ0 = kBT/D0, to In this section, we describe the results of three inde-
obtain the drag coefficient at the absolute temperature pendentexperimentsformeasuringtheforcebetweenin-
T =293K. A typical value for the spheres in this study terfaciallyboundcolloidalparticlesasafunctionofinter-
is γ0 = 17nNsm−1. particle separation r. The results are all consistent with
Applying the model consisting of Eqs. (1), (2) and (3) an electrostatic model in which the charge q on a single
tothedatainFig.3yieldsgoodagreementwiththemea- sphere is 570±30e, which is in turn consistent with the
sured velocities for a mean sphere charge q =530±30e, result described in the previous section.
where e is the elementary charge, and the uncertainty is We treat the colloidal particles as spheres of uniform
given by the standard error of the mean. surface charge sitting directly above the aqueous phase,
A priori,wecannotexcludethepossibilityofthepres- which, as in the previous section, plays the role of a con-
ence of significant amounts of (positive or negative) sur- ductingsubstrate. AsshowninFig.1(A),pairsofspheres
face charge σ on the oil/aqueous phase interface [21], arerepelledbyeachother’scharges, butareattractedto
which would require including an extra force qσ/(cid:15) (cid:15) on their neighbors’ image charges. All our measurements
r 0
theright-handsideofEq.(1). Treatingσ asafitparam- take place in the regime where interparticle separations
5
A. “Catch-and-Release” Laser Tweezer
x(μm)
Experiments
-4 -2 0 2 4
1.6
0.8 A Our first measurement of the repulsive force between
m) apairofinterfacialparticlesproceedsbyforcingthepar-
μ 0 ticles close together with a pair of optical tweezers and
(
y -0.8 thenreleasingthem. Werecordandanalyzetheresulting
-1.6 trajectories to find the interparticle force, as illustrated
in Fig. 4.
3.0
For these experiments, and also those of Section VC,
0.2 2.5 C
) we prepare samples that contain many small (diameter
m) 0.1 1-s 2.0 100µmto500µm),almost-flatinterfacesthatareisolated
(μr 0.0 mμ1.5 fromeachother. Tomaketheseinterfaces, acoverslipis
Δ (1.0 immeresed in a bath of KOH-saturated isopropanol for
-0.1 B Vr
0.5 1h prior to undergoing the treatment described in Sec-
3 4 5 6 7 8 910 0.0 tion II 1. We use a sprayer to deposit droplets of the
3 4 5 6 7 8 910
r(μm) aqueous phase onto the cover slip, which is then incor-
r(μm)
porated into the construction of a capillary channel. Fi-
400
0.08 nally,thechannelisfilledwiththeparticledispersionand
E
)
2 300 sealedwithNorlandopticaladhesive. Followingthispro-
- 0.06
s N) tocol, each droplet of the aqueous phase forms a roughly
mμ0.04 f(200 sphericalcapontheglasssurface,withacontactangleof
( Fr
Dr0.02 D 100 1◦ or less. The resulting interface is flat enough to allow
bright-field imaging of interfacial particles, which adsorb
0.00 0
3 4 5 6 7 8 910 3 4 5 6 7 8 910 totheinterfacebecauseoftheelectrostaticattractionde-
r(μm) scribedinSectionIV.Effectivelyrandomfactors,suchas
r(μm)
how far a given droplet is from the entrance of the capil-
lary channel, influence how many particles are deposited
FIG. 4. (Color online) Stages in the analysis of the data
on each interface. Thus, within a in a single sample cell,
from the catch-and-release experiment to probe the repul-
we obtain many isolated interfaces, each with a different
sive force between a specific pair of interfacial colloids. (A)
Overlayof180post-releasetrajectoriesofasinglepairofcol- interfacial density.
loidal particles. At each instant, the positions are plotted To measure the interparticle force, we first identify
in the center-of-mass frame. (B) Frame-to-frame radial dis- an interface at sufficently low particle density that only
placement,plottedasafunctionofcenter-to-centerseparation two spheres are in the field of view of the microscope.
r. (C)Radialvelocity,obtainedbybinningandaveragingthe A particle tracking algorithm then locates the spheres
datainB.(D)TheradialdiffusionconstantD (r)isobtained
r [36, 37]. Once located, the spheres are confined in holo-
from the variance of data in each bin. The dashed line is a
graphicopticaltrapsprojectedattheirposition(“catch”)
guidetotheeye,andhighlightsther-dependenceofD ,which
r [38–40]. The holographic trapping system is created
we attribute to hydrodynamic interaction between the parti-
with a 1064nm laser (IPG Photonics YLR-10-1064-
cles. We obtain the radial drag coefficient γ (r) by using the
r
LP) whose wavefronts are modified using a computer-
fluctuation-dissipationrelation,γ (r)=k T/D (r). (E)The
r B r
radial force is found by multiplying the velocity by the drag controlledliquid-crystalspatiallightmodulator(Holoeye
coefficient, F (r)=γ (r)V (r). Pluto). Theresultinglightpatternisrelayedtoanobjec-
r r r
tivelens(NikonPlanApo,NA1.45100×,oilimmersion)
that focuses the traps into the sample. The traps drag
the particles towards one another until they reach a pre-
are large compared to the colloid diameter, but small assignedminimumdistance,atwhichpointthetrapsare
compared to the Debye length in the oil phase, so that instantaneouslydisplacedtensofmicronsinthedirection
d2 (cid:28) r2 (cid:28) λ2 . In this limit, the net interaction force, perpendiculartotheimagingplane,allowingtheparticles
D
F (r),betweenpairsofsphereswithcenter-to-centersep- tomovefreelyalongtheinterface(“release”). Thetrajec-
r
aration r takes a dipolar form, tories of the particles are recorded by video microscopy,
and the coordinates of the centers of the particles, r
1
and r , are measured using publicly available tracking
(cid:32) (cid:33) 2
q2 d 1 1 3B routines [37, 41]. This procedure, shown in Fig. 1(B),
F (r)=− − (cid:39) , (4)
r 4π(cid:15) (cid:15) dr r (r2+d2)1/2 r4 is fully automated, and, for a given pair of particles, is
r 0
repeated as many as several hundred times.
Once a particle is released from the optical traps, its
where the force constant 3B is related to the particle motion results from a combination of interaction with
charge by B =q2d2/8π(cid:15) (cid:15) . the other particle, and diffusion. For two subsequent
r 0
6
thefilledendofthecapillarytubeintoasamplevialcon-
taining the particle dispersion. As the oil flows into the
tube, thin patches of the aqueous phase are left behind
on the top and bottom glass surfaces. These patches,
held in place by pinning of the contact line, are typically
0µm to 2µm thick, and millimeters in diameter. We are
thus able to image regions of uniform density as large as
0.3mm2, and which contain hundreds of particles. We
use a Leica TCS SP5 II confocal microscope, mounted
witha10×airobjectivelens,tocollectmovieswhichare
typically 1h to 2h in length. Using publicly available
software [41], we find the positions of the particles in
each frame and obtain g(r) for each movie. Fig. 5 shows
a snapshot of a typical sample, together with the stages
of the anaysis.
FIG. 5. (Color online) Main Image: Confocal micrograph To estimate the error in finding the force in this man-
ofPMMAparticlesboundtoaglycerol/waterinterface. This
ner, we perform a series of Monte-Carlo simulations of
image is a single snapshot taken from a 1h movie, with an
point particles interacting via a set of known interaction
average particle density of 0.005µm−2. Left Column: A se-
potentials. Wechoosetheinteractionpotentialsandden-
ries of plots showing: (i) g(r), from all frames of the movie;
sities in the simulations so that they produce pair corre-
(ii) the potential of mean force w(r) = −k Tlogg and the
B
pair potential U(r). We use the Ornstein-Zernike equation lation functions similar to those observed in the experi-
toobtainU(r)fromg(r);(iii)theradialforce,obtainedfrom ments. Fromeachsimulation,wecalculateg(r),andthen
U(r) by numerical differentiation. apply the Ornstein-Zernike method described above to
obtain U(r) and F (r). We compare the F (r) obtained
r r
from g(r) to the F (r) curve calculated directly from the
r
frames,thedisplacementoftheparticlesalongtheradial potentialthatweuseinthesimulation. Theerror∆Fr is
direction in the center-of-mass reference frame is given by the difference between these two values. Since
thefractionalerror∆F /F doesnotdependstronglyon
r r
∆ (t)=[r(t+τ)−r(t)]·ˆr(t), (5) r, ρ, or the parameters describing the interaction poten-
r
tial,wetakeittobeconstant,∆F /F =0.5. Thisvalue
r r
where r(t) is the instantaneous separation at time t, is assumed when plotting error bars such as those shown
r(t)=r (t)−r (t), and τ =16.7ms is the time interval in Fig. 5.
2 1
betweenvideofields. Therelativevelocity,v (r)=∆ /τ, Fig. 7(B) shows the results of applying the Ornstein-
r r
as well as the separation-dependent drag coefficient in Zernike inversion procedure to samples of interfacial col-
the radial direction γ (r), is found by combining the loids at two different areal densities. For each sample,
r
data from multiple releases, and plotting ∆ as a func- we repeat the measurement one, two and four days af-
r
tion of interparticle separation r. The data are divided ter preparation to confirm that the results for g(r) are
into bins along the r-axis, as shown in Fig. 4, and the time-independent, and thus reflect equilibrium proper-
interparticle force is obtained by using the overdamped ties. Fitting this data to Eq. (4) gives q = 540±30e,
equation of motion, F (r)=γ (r)v (r) [36, 42, 43]. The which is consistent both with the value obtained from
r r r
results of 10 such experiments, on different pairs of par- the colloid-interface interaction experiments in Section
ticles, are shown in Fig. 7(A). Fitting Eq. (4) to this IVandwithotherpairinteractionexperimentsdescribed
data gives q = 580±30e, which is consistent with the in this section.
value obtained from the colloid-interface experiments in
Section IV.
C. Crystal Elasticity Experiments
B. Pair Correlation Function Experiments Our third approach for measuring the pair interaction
takes advantage of the fact that, when confined to an in-
In a second experiment to measure the pair interac- terface at sufficiently high areal density, colloidal mono-
tion, we measure the pair correlation function g(r) of a layers form a hexagonally-ordered solid phase, which is
system of interfacial particles at low areal density ρ. We stableovertime-scalesofmanyweeks[46]. Thiscolloidal
use the Ornstein-Zernike equation from liquid state the- solid is soft enough that thermal fluctuations of the par-
ory,withthehypernettedchainapproximation,toobtain ticle positions can be measured using video or confocal
thepairpotentialU(r)fromg(r)[44,45],andfinallycal- microscopy. From the resulting trajectories, we can esti-
culateF (r)bynumericaldifferentiation. Tosampleg(r) mate the crystal’s bulk modulus, K, and shear modulus,
r
atthelowdensitiesthismethodrequires,wefirsthalf-fill µ. These elastic constants are related to the crystal’s in-
a capillary tube with the aqueous phase. We then place teraction parameter, Γ, which yields the pair potential
7
B
C ϵxx+ϵyy
A
5600 LLLbbb///LLL===111///248 D 2a()kTB0.04 (μ(Lb)+K(Lb))-1LLb E
) 0.03 □
AA/40 Lb1L
10 μm pdf(Δ123-0000.04-0.02 0.00 0.02 0.04 -elasticconst.() 000...0000012.10□●0.1□5●0.20 0(μ.●2(L5b)0)-.13L0Lb0.35
ΔA/A Lb/L
FIG. 6. (Color online) Finding the elastic moduli of a 2D colloidal crystal from video microscopy. (A) Snapshot taken from a
movie of 100 statistically independent frames. The lattice constant a is 4.3µm. We analyze the fluctuations within a square
box of side length L = 8a = 34µm. (B) The instantaneous displacement field u corresponding to the particle configuration
shown in panel A. For clarity, the magnitude of the displacements is exaggerated by a factor of 5. Also shown is the Voronoi
tesselation,computedfromtheaverageparticlepositions. (C)Traceofthestraintensor,(cid:15) (r,t)+(cid:15) (r,t),calculatedfromthe
xx yy
displacement field shown in panel B. (D) Probability distribution function of the dilatation strain (∆A/A) for three different
sub-region sizes L /L. Solid curves are Gaussian fits to the data. (E) We obtain elastic moduli of K = 40 k Tµm−2 and
b B
µ=6k Tµm−2,which,viaEqs.(6),giveΓ=305±35. Thus,foraninterparticleseparationof4.3µm,theinterparticleforce
B
is F =110±30fN.
r
at the mean interparticle separation a [47, 48]. Unlike and the local rotation is
thepreviouslydiscussedmeasurements,whichyieldfunc-
1 (cid:90)
tionalformsfortheseparation-dependentinteraction,in- ∆θ(t)= θ (r,t)d2r.
teractionmeasurementsbasedonlatticeelasticityrequire A A xy
ustoassumeafunctionalform. However,sincethismea-
If we assume equipartition of energy, the variances in
surementtakesplaceathigharealdensity,itcanconfirm
these quantities in a box of side length L are related to
the pair-wise additivity of the interactions. b
the finite-size bulk and shear moduli K(L ) and µ(L )
b b
To determine the interparticle force and the elastic
by [48]
constants, we first measure the instantaneous strain and
rotation. For a displacement field u(r,t) at position r (cid:26)∆A(t)(cid:27) k T
andtimet,theinstantaneousstrainandrotationtensors Var = B and
A A[K(L )+µ(L )]
are defined as Lb b b
k T
Var{∆θ(t)} = B .
(cid:15) (r,t)= 1[∂ u (r,t)+∂ u (r,t)] and Lb 2Aµ(Lb)
ij 2 i j j i
The thermodynamic limits of the elastic constants are
1
θij(r,t)= 2[∂iuj(r,t)−∂jui(r,t)], obtained by the finite-sized scaling procedure [49] shown
in Fig. 6(E).
The elastic moduli are related to the potential en-
respectively, where i,j ∈ {x,y}. Adapting these defini-
ergy U(r) = Br−3 of the particles’ pair repulsion at the
tions to include a displacement field u defined at a dis-
nearest-neighbor separation, r = a. In terms of the di-
crete set of lattice points and times, we use the lattice
mensionless interaction parameter Γ = B(πρ)3/2/k T,
calculus methods described in the Appendix to calculate B
we expect, in the high-density limit [50],
the strain and rotation tensors from the measured set of
particle positions.
a2 a2
ForaregionofareaA,thedilatationstrainisgivenby Γ= µ and Γ= K. (6)
0.3461k T 3.461k T
B B
∆A(t) 1 (cid:90) Thus, each elastic constant provides a measurement of
= [(cid:15) (r,t)+(cid:15) (r,t)] d2r,
A A xx yy Γ. We take the average of these measurements to be
A
8
our estimate for Γ, and half their difference to be the
1000 A 20 B
3pfcnookremroaWBrtreTeoeedsscΓpctou(o-olπnsnnieρtnfidih)grgi−nrmhot3gbhu/to2eugharhneH−ciaO4emnc.rtOtcoeaulrMeiarncacDtuctyl-iyaob∆rnolufΓdefo.yousrnuFrcaiieitmnmeaFip[lc5lrlsy1e(,ma–sw)5iem3en]=ut.ealsaFttt3ioioimBronnpaaosta−fer4patthemh=ries-- FfN()r 101001 4 FfN()r1025-ρ=0.005μm-2
etervaluessimilartothoseoftheexperiment, ouranaly- 0.1 1 1 -ρ=0.003μm-2
sis of the particle trajectories accurately reproduces the 4 6 8 10 6 8 10 12
interaction parameter Γ and the interparticle force F . r(μm) r(μm)
r
Fig. 7(C) shows the results of applying this analysis
to five different samples. We restrict our data collection 500 C 500 D
to crystals that have lattice constants between a=3µm ✶✶
dEtfaahnoqred-.finc(5eo6hlt)aµd.rmhgaFae.sivstqeutCimn=hrgypigs5tEhti9oaq0nle.s±n(row422u)i0(cid:29)tgtheho.dtad2hAe,e(cid:46)snwpsTihl3toaiylµtbetmlteetodhIdodjoussahestntoawiwof,yitswt,hsetathahfiteiin(cid:38)ssdfuyvs5taehµtluhamoeetf FfN()r12005000 FfN()r10150005 --✶tgw(✶re)ed✶zae✶trada●ta●●●●●●
isconsistentwiththeotherpairinteractionexperiments, 20 1 ✶-crystalelasticitydata
and, within two standard deviations, is also consistent 3.0 3.5 4.0 4.5 5.0 4 6 8 10
with the results of the colloid-interface experiments. r(μm) r(μm)
In our experiments, we have observed the behavior
of specific pairs of particles far away from any others
FIG. 7. Log-log plots of the results of three experiments to
(tweezer experiments), as well as systems of many par- find forces between interfacial colloids. (A) Results of the
ticles in both the low density (g(r) experiment) and the catch-and-releaselasertweezerexperiment,for10pairsofin-
high density (crystal elasticity experiment) limits. The terfacial particles. (B) Results of finding the interparticle
fact that the measured charge is consistent in all these force via the pair potential g(r), for two samples at different
casesimpliesthattheinteractionispairwiseadditiveover areal densities. (C) Results of experiments where we found
therangeofinterparticleseparationsexploredbyourex- the interparticle force via the elastic moduli of 2D colloidal
crystals. Eachdatapointcorrespondstoadifferentvideoofa
periments. Thiscontrastswithothersystemsofcolloidal
fluctuatinglattice. (D)ThedatafrompanelsA,BandCare
particles dispersed in oil [54], and may have implications
divided into bins. For each data set, the mean in each bin is
for understanding the origin of the surface charge on the
plotted. Errorbarsindicatethespreadofthedatainagiven
particles.
bin: standarddeviationfordatasetsAandC,maximumde-
viation from the mean in the case of B. The dashed line is a
fit of all the data to Eq. (4), which gives q=570±30e.
VI. CONCLUSIONS
trostatic interactions in systems of interfacial colloids to
In this work, we study the behavior of a system of
betterunderstandthebehaviorofmorecomplexsystems,
charged colloids in the vicinity of a fluid interface. We
such as those with partial wetting.
show that, in the absence of wetting by the aqueous
phase, this behavior is governed by electrostatics alone:
individual colloids interact with the interface via image-
charge attraction, while particles that are already inter- q (e)
facially bound interact with their neighbors as charge-
colloid-interface experiment 530±30
image charge dipoles. Our model, in which the parti-
laser tweezer experiment 580±30
cle charge q is the only fit parameter, is consistent with
g(r) experiment 540±30
datafromthefourindependentexperimentswehaveper- crystal elasticity experiment 590±20
formed.
In our system, interactions between interfacial parti- TABLE I. Particle charge found in each of the four experi-
cles are pairwise-additive, are constant over time-scales ments described in this paper.
of weeks, and are homogenous enough to allow the for-
mation of defect-free crystals over length-scales of tens This work was supported primarily by the National
of lattice spacings. The system is thus well-suited to Science Foundation under Award No. DMR-1105417.
the study of problems in fundamental condensed mat- Partial support was provided by NASA, through Award
ter physics, for example the phase behavior of repulsive No. NNX13AR67G,andtheMRSECprogramoftheNa-
particles in 2D [55, 56], or the structure and dynam- tional Science Foundation through Award Nos. DMR-
ics of topological defects in curved spaces [22, 57, 58]. 1420073 and DMR-1420570. G. I. G-G. acknowledges
Moreover,wecannowhopetouseourknowledgeofelec- CONACYT for financial support via the program Cate-
9
dras CONACYT para Jovenes Investigadores. The au- In order to do this, we use the Voronoi construction to
thors would like to acknowledge fruitful discussion with partition the field of view into cells associated with each
Dan Evans, Gary L. Hunter, Eric DeGiuli and Aleksan- lattice site. This construction provides a well-defined set
dar Donev at New York University. of nearest neighbors for each particle, which does not
change over the course of the movies. For an arbitrary
vectorfieldv,thematrixofpartialderivatives∂ v (α)of
i j
Appendix: Finding Strain and Rotation Tensors particleαcanbecalculatedbyusingadiscreteversionof
from Particle Trajectories the divergence theorem [60]. This works as follows: for
every particle β that neighbors α, the particles share an
Here we outline how we use a discretized version of edge of a Voronoi cell, which we label (α,β). For each
the divergence theorem from vector calculus to the cal- edge, we define the vector v(α,β) as the average of v(α)
culate the strain and rotation tensor from the particle and v(β), while the normal vector nˆ(α,β) and the edge
trajectories obtained from video microscopy. length (cid:96)(α,β) are given by the geometry of the Voronoi
We first define the particle’s equilibrium position. Af- cell. Using this notation, the divergence of v at particle
ter subtracting uniform drift, we still need to eliminate α is given by
effects which are due to slow expansion, compression or
rotationofthelattice,aswellasthelong-wavelengthfluc-
1 (cid:88)
tations which are characteristic of 2D solids [59]. To do ∇·v(α)= (cid:96)(α,β)(v(α,β)·nˆ(α,β)),
this, we use a moving average of the particle’s position A(α)
β n.n.α
using time window which is typically 20 times the relax-
ationtimeofanindividualparticle,ascomputedfromthe
mean square displacement. At each frame of the movie, where A(α) is the area of the Voronoi cell of particle
the displacement u(α) of particle α is calculated relative α, and the sum is taken over all particles β which are
to this moving average. nearest neighbors to particle α. For appropriate choice
Once we obtain the displacement field u for a given of v, the components of the strain and rotation tensors
frame of the movie, we need to calculate the strain and can be found at each lattice site. For instance, (cid:15) =
xx
rotation tensors, which requires taking derivatives of u. ∂ u =∇·v, where v=(u ,0).
x x x
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