1
M.G. Nikolaides* ,§ , A.R. Bausch* ,§ , M.F. Hsu * , A.D. Dinsmore * ║ , M.P. Brenner † ,
C. Gay ‡ , and D.A. Weitz *,† .
* Department of Physics, Harvard University, Cambridge, MA 02138, USA, † Division of
Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA,
‡ CRPP-CNRS, Avenue Schweitzer, 3600 Pessac, FRANCE.
$ Present address: Lehrstuhl für Biophysik - E22, Technische Universität München, GERMANY.
║ Present address: . Department of Physics, University of Massachusetts, Amherst, MA 01003-4525.
Nanometer and micrometer sized particles adsorbed at aqueous interfaces are ubiquitous in technological applications, as well as in biological constructs. The particles are typically stabilized by a repulsive Coulomb interaction; if one side of the interface is non polar (air or oil), and cannot sustain a charge, then the repulsive interaction is dipolar in character and can be surprisingly long-ranged
1
.
This can lead to ordering,
2
and even crystallization,
3,4
of the surface particles, but only if their area is constrained. However, there are persistent reports of structuring even in the absence of area constraint;
5
this can only be due to an attractive interaction.
6
Despite many proposed models,
7,8
the origin of this attraction between like-charged particles has heretofore defied quantitative explanation. In this Letter, we present quantitative measurements of the interaction, and suggest that the attraction is caused by a novel mechanism, the distortion of the interface shape due to electrostatic stresses caused by the dipolar field; this induces a capillary attraction between the particles. This picture resolves

2 all reports to date; moreover, such attractive interactions can now better be controlled in interfacial and colloid chemistry.
The repulsion between surface adsorbed colloidal particles results from the dipoledipole interaction caused by the asymmetric charging of the particle surface due to the mismatch in the dielectric constants; only the aqueous side of the particle surface can acquire a charge. This surface charge, combined with the screening ions in the water, produces an effective dipole moment of the particle, which leads to the observed repulsion.
1,2,9 An obvious candidate to account for the origin of the attractive interaction is the capillary force; a deformation of the interface leads to a logarithmic attraction between neighboring particles.
10,11,12 For sufficiently large particles, this deformation is caused by gravity; the weight of a particle is balanced by the surface tension, thereby deforming the interface.
4 This effect results in the common observation of clumping of breakfast cereals at the surface of a bowl of milk, and has been cleverly harnessed to drive the self-assembly of millimeter sized particles at interfaces.
13 However, the buoyancy mismatch of micron sized colloidal particles is too small to significantly deform the interface and the resulting attractive energies are far less than the thermal energy.
14 An alternative candidate for the origin of the attraction is wetting of the particles which also deforms the interface; however, a spherical particle positions itself to exactly achieve the equilibrium contact angle without distorting the interface unless the particle positions are constrained to a thin layer of fluid.
7,14 Another candidate for the origin of this attractive force is surface roughness, which might also cause a deformation of the interface; however, the amount of roughness required is far greater than that which typically exists on colloidal particles.
8 A final candidate for the origin of this attraction is thermal fluctuations; entropic interactions lower the free energy of the interface when two particles approach, resulting in a Casimir-type of effective attraction.
15,16 However, the resulting forces are too small to cause a significant attraction for colloid particles.

3
A typical configuration of particles at the interface of a large water drop in oil is shown in figure 1. The long range of the repulsive interaction is apparent from the large particle separation. Moreover, this repulsion, combined with the geometric confinement due to the finite area of the emulsion drop, which is completely covered with particles, causes a pronounced ordering of the particle positions evidenced in figure 1.
Remarkably however, even when the particles coverage is not complete, similar ordering is still observed. An example is illustrated by the inset in figure 2, which shows a group of seven particles in a hexagonal crystallite. These were the only particles on the surface of a 24 µ m -diameter water drop, and the colloidal crystallite remained stable for more than 15 minutes. The persistence of this structure over long times is dramatic evidence of an additional long ranged attractive interaction (figure3). To extract this interaction potential, we exploit the symmetry of the geometry and measure the probability distribution, P(r)
,
of the center to center distance, r , of the center particle to each of the six outer particles. To obtain the interparticle potential V(r) , we invert the
Boltzmann distribution (figure 2),
P ( r ) ∝ e
−
V ( r k
B
T
)
(1)
The minimum of the potential appears at an interparticle separation of r eq
=5.
7µ m , and can be fit by a harmonic potential with a spring constant of k=25k
B
T/ µ m 2 as shown in figure 2.
The motions of any given particle in the crystallite are influenced by the pair interaction with all of the other particles; however by analyzing only the radial distance of the outer particles to the center particle, these many particle effects are largely cancelled by symmetry i . i To verify that many-particle effects do not bias the results, we performed a molecular dynamics simulation of a crystallite with 7 particles interacting through the measured pair potential. We extracted the potential from these numerical data using the same procedure as done in the experiments. The

4
What is the origin of this attraction? If there is a force F on the particles normal to the interface, the resultant distortion of the interface leads to an interparticle attraction described by the potential
U int erface
( r ) =
F
γ
2 log
æ
ççè r r
0
ö
÷÷ø
(2) where γ is the interfacial surface energy, r is the distance between the particles and r
0
is an arbitrary constant.
12 Decreasing γ or increasing F increases the interfacial distortion, and thus increases the strength of this capillary attraction. The force that the colloidal particles exert on the interface due to gravity is of order 10 -14 N , far too small to account for our observations. There is however another natural cause for F : The same dipolar electric fields that cause the interparticle repulsion create electrical stresses which distort the oil-water interface. The sum total of this stress is the net force F that the particle exerts on the interface.
The electrical stresses arise because oil and water have very different dielectric
and ε water
∝ 80 . When field lines cross the interface, the intensity of the electric field E and the electrostatic energy density
1
2
εε
0
E 2 are thus roughly 40 times smaller in water than in oil. As a result, the free interface tends to move towards the oil so as to lower the total energy of the system. This is equivalent to a pressure being exerted on the interface towards the oil. The colloidal particle therefore behaves as if pulled into the water by an external force F , as shown in figure 4. Thus the particles, which are the source of the electric fields, tend to be surrounded by water, thereby lowering the total electrostatic energy. difference between the measured values of req and k from the many particle data and those for the single particle interaction potential is about 10-20%.

5
The force F can be calculated by the integral of the electric pressure
1
2
ε
0
ε oil
E 2 oil over the free surface. Since the dipolar electric field vanishes like 1/r 3 and hence the electrostatic pressure like 1/r 6 , most of the effect is concentrated in the vicinity of the particle, thus within a distance of order particle radius a from the contact line. Beyond that distance, the interfacial distortion is indistinguishable from any other source of interfacial distortion with the same strength F . A rough estimate of F is obtained by considering that the total electric dipole P of the particle is concentrated at the center of the water-wetted area. The dipolar field in the oil near the interface, at a distance r from the dipole, is
E oil
≅
P
4 πε oil
ε
0 r 3 ε
2 ε oil water
(3)
Here ε
0
is the dielectric constant of vacuum and the damping factor 2 ε oil
/ ε water reflects the image charges required for the interface to remain an equipotential.
Integrating the electrostatic pressure from r=a w
, to r= ∞ where a w
is the radius of the water-wetted region of the particle, we obtain:
F ≅
P 2
16 πε
0 a 4 w
ε
ε oil
2 water
(4)
U =
F 2
2 πγ log
æ
ççè r r
0
ö
÷÷ø
+
P 2
4 πε
0 r 3 ε
2 ε oil
2 water
(5) where the first term is the capillary attraction of Equation (2) and the second term is the dipolar repulsion.
From this energy we can predict both the equilibrium separation r eq
between two particles and the spring constant k = d 2 U / dr 2 : r eq a w
≅
æ
ççè
γ a 2 w
ε
0 a
P 2
3 w 768 π 2
ε 2
ε water oil
ö
÷÷ø
1 / 3
(6)

6
γ k
≅
γ a
1
( )
8 ε
( ) a 3 w
8
8
3
2 43 π 13 ε
ε 8 oil
16 water
÷
1 / 3
(7)
There are two unknown materials parameters in Eqs. (6) and (7). The dipole moment P is determined by the surface charge density of the spheres, this results from the dying process and its value is not precisely determined. The radius of the wetted area is determined by the equilibrium contact angle Θ , through a w
=acos Θ . The equilibrium contact angle is also unknown, and will depend on the surface charge density.
We obtain excellent agreement between the predicted potential and the experimental data, as shown in figure 3, taking a value for a w
which corresponds to an equilibrium contact angle of 57° , and a value for P which corresponds to a total charge on the sphere of 1.8 10 5 unit charges, or a density of 0.4 unit charges per nm 2 . Both of these values are quite reasonable. This fit corresponds to r eq
=5.
7µ m and k ≈ 25k
B
T/ µ m 2 , both in good agreement with the experimentally measured values. The interfacial distortion predicted by the theory is given by the estimate h ≈ F/(2 πγ) log(R/a) , and is h ≈ 10nm , too small to be observed with an optical microscope.
We emphasize that these estimates do not account for the specific geometry of the wetted region, nor do they include a self-consistent determination of the wetted area.
Additionally, our calculations assume that the screening length is much smaller than the size of the sphere. Nevertheless, our prediction adequately captures the essential physics; such other issues will be addressed in detail elsewhere.
17
The results reported here demonstrate that the long range attraction between particles confined to an interface is consistent with electrocapillary attraction. The same electric fields that cause the interparticle repulsion also distort the oil-water interface and lead to the attractive interaction. This may have much broader implications in

7 interfacial and colloid chemistry, since adsorption of charged particles at interfaces between two distinct fluids is a very common phenomenon in foods, drugs, oil recovery, and even biology.

8
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9
10. Chan, D. Y. C., Henry, J. D., White, L. R. The Interaction of Colloidal Particles
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, 79-94 (1992)
12. Morse, D. C., Witten, T. A., Droplet Elasticity in Weakly Compressed Emulsions,
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, 549-555 (1993)
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Mesoscale Objects into Ordered Two-Dimensional Arrays. Science
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14. Kralchevsky, P. A., Nagayama, K. Capillary interactions between particles bound to interfaces, liquid films and biomembranes. Adv. Colloid Interface Sci. 85, 145-192
(2000)
15. Goulian, M., Bruinsma, R., Pincus, P. Long-Range Forces in Heterogeneous Fluid
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17. Brenner, M. P., Gay, C., In Preparation (2002)
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Acknoledgements:
C.G. thanks B. Berge for important early discussions on electrowetting phenomena.
18
The authors gratefully acknowledge support from the NSF, the Materials Research Science and Engineering Center through the auspices of the NSF and the Division of Mathematical Sciences. A.B. acknowledges the support from the Emmy Noether-Program of the DFG.
Correspondence should be sent to D. Weitz (e-mail: weitz@deas.harvard.edu).
11

12
Figure Captions:
Figure 1: Fluorescence microscopy image of an ordered structure of colloidal at an oil-water interface, demonstrating the long range of repulsive interaction. We use poly(methylmethacrylate) (PMMA) particles, sterically stabilized with poly(hydroxystearic acid).
19
The particles have a radius of a=0.75
µ m and are suspended in decahydronapthalene (decaline) at a volume fraction of 0.01% .
The particle cores are labeled with fluorescent dye. An emulsion of water drops in oil is produced by gently shaking ≈ 2 µ l of deionized water in 1ml of the decalin-PMMA mixture, whereupon particles adsorb at the oil-water interface.
They are observed with a CCD camera mounted on an inverted fluorescence microscope, and the images are recorded on videotape. Sequences of interest are digitized and the particle positions are determined with particle tracking routines.
20
Figure 2: Scatter plot showing positions of a seven-particle hexagonal crystallite on a water droplet of
24 µ m
radius. The particles maintained this configuration for more than 15 minutes; 5 min. of data, extracted at time intervals of
1/30 sec are shown. The inset shows a fluorescent microscope image of the crystallite.
Figure 3: Secondary minimum obtained from the particle distribution function of the crystallite shown in figure2. The minimum at a 5.7
µ m separation produces the observed hexagonal crystallite. The inset is a sketch of the full interparticle potential and includes the repulsion barrier which stabilizes the particles, and a deep primary minimum at short range due to van der Waals attraction. These crystallites were observed to collapse and form a gel after several hours, confirming the presence of the primary minimum and the large repulsive barrier.
The accessible range of particle separation allows us to explore the shape of

13 the secondary minimum; there is also a strong primary minimum at particle contact due to van der Waals attraction, as shown in the inset, but the repulsive potential is sufficiently large that we can not explore its shape or magnitude using thermal fluctuations.
Figure 4: Sketch of the equipotential lines at the fluid interface and the resulting distortion of the oil-water interface. The distortion of the interface shape is greatly enhanced for clarity.

Figure1:
10 µ m
14

Figure2:
Y[
µ m]
10
µ m
0
-2
-4
-6
4
2
-6 -4 -2 0 2 4 6 X[ m m]
15

Figure3:
V/k
B
T
2.0
V(r)
1.5
5.7
µ m r[
µ m]
1.0
0.5
measurement
0.0
5.4
5.6
5.8
6.0
r [ µ m]
16

Figure4:
17