THE JOURNAL OF CHEMICAL PHYSICS 124, 241104 共2006兲

Adsorption of colloidal particles to curved interfaces
S. Komuraa兲 and Y. Hirose
Department of Chemistry, Faculty of Science, Tokyo Metropolitan University, Tokyo 192-0397, Japan

Y. Nonomura
Tokyo Research Laboratories, Kao Corporation, Tokyo 131-8501, Japan

共Received 22 February 2006; accepted 25 May 2006; published online 27 June 2006兲
As a simple model for a Pickering emulsion droplet, we consider the adsorption of sphericalparticles to a spherical liquid-liquid interface in order to investigate the curvature effect on theparticle adsorption. By taking into account both the surface and the volume energies due to thepresence of a particle, we show that the equilibrium contact angle is determined by the classicalYoung’s equation although the adsorption energy depends on the curvature. We also calculate thepartitioning of the colloidal particles among the two liquids and the interface. The distribution ofcolloidal particles is expressed in terms of the interfacial curvature as well as the relative wettabilityof the particle. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2216697兴
Small particles such as lamp black or hydrophobic silica
adsorb to an oil-water interface and act as stabilizers of
emulsions. Such emulsions are called “Pickering emulsions”1and have been used in various food and cosmetic products.2
For instance, lipoprotein particles and fatty acid crystals stabilize the emulsion state in mayonnaise or margarine. A modern application of Pickering emulsion is to use them as templates of functional composite materials. For example,
spherical emulsion droplets are used to produce microstructured hollow spheres3 or ribosome-like nano-capsules
共“colloidsomes”兲.4 In addition to spherical droplets, the particles form several self-assembling structures such as network structures or ball-like aggregates.5 Quite recently, colloidal particles with equal affinity for two liquids are shown
to form a bicontinuous interfacially jammed emulsion gel
共“bijel”兲.6
Pickering emulsions are not only useful but also very
interesting from the fundamental point of view. When a molecule exhibits a surface activity, it is conventionally recognized that both hydrophobic and hydrophilic groups should
exist in a single molecule. This is not the case for colloidal
particles whose surface is chemically uniform. The particles
adsorb only when the relative wettability between the particle and the two liquids balances each other.7,8 Moreover,the connecting point between the hydrophobic and hydrophilic groups in a surfactant molecule is localized at the interface between the two liquids. Colloidal particles, on theother hand, can adjust their position with respect to the interface in order to reduce the interfacial energy.
In this Communication, we discuss the adsorption of
spherical solid particles to a liquid-liquid interface. Special
attention is paid to the effects of interfacial curvature on theadsorption behavior. For a spherically curved droplet interface, we show that the contact angle satisfies the classicalYoung’s equation although the adsorption energy depends onthe curvature. We also consider the partitioning of the colloia兲

Electronic mail: komura@comp.metro-u.ac.jp

0021-9606/2006/124共24兲/241104/4/$23.00

dal particles among the two liquids and the interface. In contrast to the previous work,9 both the relative wettability andthe interfacial curvature are shown to contribute to the equilibrium distribution of colloidal particles. The curvature effect on the adsorption behavior is important because most of
the structures in Pickering emulsions are characterized byfinite curvatures.
We start our discussion by considering a single sphericalparticle of radius a resting at a flat interface between liquid 1
and liquid 2, as shown in Fig. 1共a兲. Let ␪ be the contact
angle, and ␴1p, ␴2p, ␴12 be the 1-particle, 2-particle, 1-2 interfacial tensions, respectively. Neglecting external forces
and a finite thickness of the interface, the interfacial energy
attributed to the particle is w0 = ␴1pA1p + ␴2pA2p − ␴12A12,
where A1p and A2p are the 1-particle and 2-particle contact
areas, respectively, while A12 is the eliminated area of the 1-2
interface due to the presence of the particle.7 For a flat interface, these areas are given by
A1p = 2␲a2共1 + cos ␪兲,A2p = 2␲a2共1 − cos ␪兲,共1兲

A12 = ␲a2 sin2 ␪ .

Hereafter, it is convenient to introduce the following dimensionless quantities:␴=␴1p + ␴2p
,
␴12

␥=

␴1p − ␴2p
.
␴12

共2兲

Then the dimensionless adsorption energy defined byg0共␪兲 = w0 / 共2␲a2␴12兲 is written as
g0共␪兲 = ␴ + ␥ cos ␪ −sin2 ␪
.
2

共3兲

The equilibrium position 共or the contact angle兲 of the particle
is determined by minimizing g0共␪兲 with respect to ␪, i.e.,
dg0 / d␪ = 0. This results in the conventional Young’s
equation:7

124, 241104-1© 2006 American Institute of Physics

241104-2

J. Chem. Phys. 124, 241104 共2006兲

Komura, Hirose, and Nonomura

FIG. 1. 共a兲 A spherical particle of radius a adsorbed to a flat interface
between liquid 1 and liquid 2. The contact angle is denoted by ␪. 共b兲 A
spherical particle adsorbed to a spherical interface whose radius of curvature
is R12. The pressures in liquids 1 and 2 are p1 and p2, respectively. ␻ is thecentral angle.

␪min = arccos共− ␥兲.

共4兲

By substituting this relation to Eq. 共3兲, the minimized interfacial energy is obtained as g0共␪ = ␪min兲 = ␴ − 共1 + ␥2兲 / 2 when兩␥兩 ⬍ 1. On the other hand, the particle prefers to stay either
in liquid 1 when ␥ ⬍ −1 or in liquid 2 when ␥ ⬎ 1.
Next we discuss the adsorption of a spherical particle toa curved interface between liquids 1 and 2. For simplicity,
we consider a spherical 1-2 interface whose radius is denotedby R12 as depicted in Fig. 1共b兲. Here the dimensionless curvature ⑀ = a / R12 is taken to be positive when the interface iscurved toward liquid 1. In the following, we generally assume that ⑀ Ⰶ 1. When the interface is curved, one has to take
into account the pressure in each liquid denoted by p1 and p2.
According to Laplace’s law for a spherical interface, thepressure difference is maintained by the surface tension;p2 − p1 = 2␴12 / R12. For a given droplet curvature ⑀, we now
calculate the energy per particle by w = ␴1pA1p + ␴2pA2p− ␴12A12 + p1V1 + p2V2, where V1 and V2 are the volumes occupied by the particle in liquids 1 and 2, respectively. Aftersome geometrical considerations, we obtain these areas andvolumes as
A1p = 2␲a2关1 + cos共␪ − ␻兲兴,A2p = 2␲a2关1 − cos共␪ − ␻兲兴,共5兲

2共1 − cos ␻兲,A12 = 2␲R12V1 = ␲a3关 32 + cos共␪ − ␻兲 − 31 cos3共␪ − ␻兲兴
3 2− ␲R12
关 3 − cos ␻ + 31 cos3 ␻兴 ,共6兲V2 = ␲a3关 32 − cos共␪ − ␻兲 + 31 cos3共␪ − ␻兲兴
3 2+ ␲R12
关 3 − cos ␻ + 31 cos3 ␻兴 ,

where

冉

␻ = arctan

⑀ sin ␪1 + ⑀ cos ␪

FIG. 2. The minimized scaled interfacial energy difference ⌬g共⑀兲 共see thetext兲 as a function of ⑀ = a / R12 for ␥ = 0 , ± 0.5. Filled circles on each curve
correspond to the maximum points.

冊

共7兲

共8兲

is the central angle defined in Fig. 1共b兲. Then the dimensionless energy g共␪ , ⑀兲 = w / 共2␲a2␴12兲 which now depends on ␪and ⑀ becomesg共␪, ⑀兲 = ␴ +−a共p1 + p2兲
+ ␥ cos共␪ − ␻兲3␴12

1 − cos3 ␻ ⑀− 关3 cos共␪ − ␻兲 − cos3共␪ − ␻兲兴.
3⑀2
3
共9兲

Since ␻ ⬇ ⑀ sin ␪ up to the lowest order in ⑀, it is easy to seethat Eq. 共9兲 reduces to Eq. 共3兲 in the limit ⑀ → 0 except theconstant term related to the pressure. We note here that asimilar calculation was done before,10 but the volume energies were not considered there.
As we did for a flat interface, the equilibrium position of
the particle is obtained by minimizing g共␪ , ⑀兲 with respect to␪ while ⑀ is fixed, i.e., 共⳵g / ⳵␪兲⑀ = 0. Then we obtain the same
Young’s equation as in Eq. 共4兲. This means that the equilibrium contact angle does not depend on the curvature of theinterface. The present result is consistent with the otherwork.11 Notice that we have fixed the curvature ⑀ and allowed the area and volume to change. In a different
treatment,12 the same Young’s equation was recovered evenwhen the volume is conserved.
We now substitute back the Young’s equation into Eq.
共9兲, and obtain g共⑀兲 = g共␪ = ␪min , ⑀兲 which is a function of ⑀
only. In Fig. 2, ⌬g共⑀兲 ⬅ g共⑀兲 − ␴ − a共p1 + p2兲 / 共3␴12兲 is plottedfor ␥ = 0 and ±0.5. We see that g共⑀兲 takes a maximum value
at a certain value of ⑀ 共denoted by ⑀*兲 for each different
values of ␥. Since the absolute value of g共⑀兲 corresponds tothe work needed to remove a particle from the interface, theparticles are most weakly adsorbed when ⑀ = ⑀*.
When the size of the droplet is large 共⑀ Ⰶ 1兲, one can
expand g共⑀兲 in terms of ⑀ to obtaing共⑀兲 ⬇ ␴ +冉 冊

a共p1 + p2兲 1
␥2− 共1 + ␥2兲 + ␥ 1 −
⑀
2
3
3␴12

3
− 共− 1 + ␥2兲2⑀2 + ¯ .
8共10兲Again, g共⑀兲 reduces to g0共⑀兲 in the limit ⑀ → 0 apart from theconstant term. Within this quadratic approximation we obtain⑀* ⬇ 共4 / 9兲␥共3 − ␥2兲 / 共−1 + ␥2兲2, which is valid when ␥ is close241104-3

J. Chem. Phys. 124, 241104 共2006兲

Adsorption of colloidal particles to curved interfaces

to zero. Being consistent with the symmetry consideration, ⑀*
is an odd function of ␥. It should be noted that even theequilibrium contact angle is independent of the interfacial
curvature, the minimized interfacial energy g共⑀兲 is a functionof ⑀. This is the main result of the paper. Such a curvatureeffect becomes important when we consider, for example, thepartitioning of the particles as discussed below.
In the following, we adopt the formalism to discuss thethermodynamics of the surfactant films which are in equilibrium with surfactant in solution.13 Here the particles play therole of surfactants. Consider a ternary mixture in which thetotal volume fractions of liquid 1 and liquid 2 are ␾1 and ␾2,
respectively, and the total volume fraction of colloidal particles in the system is ␾ p. These volume fractions satisfy thecondition ␾1 + ␾2 + ␾ p = 1. The particles are partitioned
among the liquid 1, liquid 2, and the monolayer films withvolume fractions ␾ p,1, ␾ p,2 and ␾ p,m, respectively. Such apartitioning was indeed observed in the experiment.14 Since
the total amount of the particle should be conserved, theysatisfy␾ p,1 + ␾ p,2 + ␾ p,m = ␾ p .

共11兲Next we denote the local volume fractions of particles inliquid 1 and liquid 2 by ␺1 and ␺2, respectively. When the
segregation between the two liquids is strong enough, thetotal and the local volume fractions are related by

␾ p,i
␺i
=
,
␾i 1 − ␺i

共12兲

where i = 1 , 2. It is assumed here that the local concentrationof particle in the film is unity. This assumption is justified
because the interfaces in Pickering emulsions consistof monolayers of almost hexagonally close-packed
particles.15,16
In the liquid regions containing colloidal particles, eachfree energy per particle is given by
f i = ␺i log ␺i + 共1 − ␺i兲log共1 − ␺i兲 + ␮i␺i +

␹i 2
␺ ,
2 i

共13兲

where the first two terms represents the entropy, ␮i is theenergy of one particle in liquid i, and ␹i is the second virial
coefficient among the particles in liquid i. Here and in thefollowing, all the energies are expressed in units of the thermal energy kBT. We consider the case of ␹i 艌 0 in order toinvestigate the effect of the repulsion between the particles.
For attractive cases 共␹i ⬍ 0兲, one needs to include the higher
order terms in ␺i, which will be reported elsewhere. In thecase of a spherical particle, we simply have4␲a2␴ip 4␲a3 pi
+
,
␮i =
k BT
3kBT

共14兲

and ␹i = 8 for hard spheres.17 Since the two liquid phases
consisting of both liquid and particles occupy fractions of␾1 + ␾ p,1 and ␾2 + ␾ p,2 of the total volume, the total free energy per particle is written as

FIG. 3. Equilibrium local volume fractions of particles in liquid 1 共␺1兲 andin liquid 2 共␺2兲 as a function of ␥. The second virial coefficients are
共␹1 , ␹2兲 = 共0 , 0兲 共solid lines兲 and 共␹1 , ␹2兲 = 共5 , 1兲 共dashed lines兲. The otherparameters are ⑀ = 0 and ␲a2␴12 / kBT = 1.

f total = 共␾1 + ␾ p,1兲f 1 + 共␾2 + ␾ p,2兲f 2 + ␾ p,m f m ,共15兲

where f m = 共2␲a2␴12 / kBT兲g共⑀兲 is the interfacial energy per
particle consisting the monolayer 关see Eq. 共10兲兴.
To obtain the equilibrium partition, we minimize f total
with respect to ␺1, ␺2, and ␾ p,m subject to the constraint ofconservation of the total amount of the particle 关see Eq.
共11兲兴. Such a constraint minimization can be performed using
the method of Lagrange multiplier. The resulting equilibrium
conditions are provided by
2 log ␺i − 2共f m − ␮i兲 + ␹i␺i共2 − ␺i兲 = 0,

共16兲

with i = 1 , 2. Elimination of f m from Eq. 共16兲 yields the relation between ␺1 and ␺2 such that冋冉

␺1 = ␺2 exp ␮2 − ␮1 + ␹2 ␺2 −

冊 冉

␺22
␺2
− ␹1 ␺1 − 1
2
2

冊册

.

共17兲
This relation is reminiscent of the Frumkin adsorption
isotherm.18
In the case of a spherical particle adsorbing to a spherical interface, one can show from Eqs. 共10兲 and 共14兲 thatf m − ␮i in Eq. 共16兲 are given byf m − ␮1 =f m − ␮2 =

冋

冉

冊

册

2␲a2␴121
2
␥3− 共1 + ␥兲2 +⑀+ ¯ ,
+␥−
2
3
3
k BT

冋

冉

冊

册

共18兲

2␲a2␴1212
␥3
⑀+ ¯ ,−␥+− 共1 − ␥兲2 −
2
3
3
k BT
共19兲

up to the lowest order in ⑀. Once we obtain ␺i by solving Eq.
共16兲, ␾ p,i is calculated from Eq. 共12兲. Then ␾ p,m is determined according to Eq. 共11兲 provided ␾i and ␾ p are given.
The partitioning of the particle is determined not only by the
relative wettability ␥ but also the interfacial curvature ⑀.
We first consider the adsorption to a flat interface
共⑀ = 0兲. In the absence of repulsion between the particles inboth liquids 共␹1 = ␹2 = 0兲, Eq. 共16兲 can be easily solved for ␺i:241104-4

J. Chem. Phys. 124, 241104 共2006兲

Komura, Hirose, and Nonomura

FIG. 4. Equilibrium local volume fractions of particles in liquid 1 共␺1兲 andin liquid 2 共␺2兲 as a function of ␥ when ⑀ = 0 共solid lines兲 and ⑀ = 0.1 共dashedlines兲. The other parameters are 共␹1 , ␹2兲 = 共5 , 5兲 and ␲a2␴12 / kBT = 10.

␺i = exp共f m − ␮i兲.

共20兲This result for ␺1 and ␺2 are plotted as solid lines in Fig. 3
when ␲a2␴12 / kBT = 1. As is consistent with the physicalmeaning of the relative wettability, ␺1 decreases and ␺2 increases as ␥ is changed from −1 to 1. When ␥ = 0, the adsorption is purely determined by the ratio ␲a2␴12 / kBT.6
Practically, the fraction of the particles in the bulk liquids are
negligibly small when ␲a2␴12 / kBT ⬎ 10. In the presence ofrepulsive interactions, on the other hand, Eq. 共16兲 needs to besolved numerically. As an example, we show the result of
共␹1 , ␹2兲 = 共5 , 1兲 as dashed lines in Fig. 3. The repulsive interactions dramatically suppress ␺i, which in turn leads to an
enhanced particle adsorption. This suppression is more pronounced in liquid 1.
Next we look at the effect of a finite interfacial curvature. In Fig. 4, we have plotted ␺1 and ␺2 for ⑀ = 0 共solidlines兲 and 0.1 共dashed lines兲. The other parameter values arechosen here as 共␹1 , ␹2兲 = 共5 , 5兲 and ␲a2␴12 / kBT = 10. Due tothe latter choice of the parameter, both ␺1 and ␺2 almost
vanish for ␥ 艌 0 and ␥ 艋 0, respectively 共not visible in Fig.
4兲. For a positive curvature as considered here, ␺1 is largerand ␺2 is smaller than those of the flat interface. This result
is in accordance with the fact that the particles prefer to stayin liquid 1 because the pressure in liquid 2 is larger when⑀ ⬎ 0. The shifts of the distribution curves become larger as
the curvature is increased 共not shown here兲.
Finally, we comment that the interfacial curvature ⑀ isprimarily determined by the total volume fractions ␾1, ␾2,and ␾ p. Assuming strong adsorption of the particles, the cur-vature of a spherical droplet consisting of liquid 2 is roughly
given by ⑀ ⬃ ␾ p / ␾2.17 This means that the partition of theparticles is completely determined provided we know thetotal volume fractions of each component. The macroscopic
phase behavior of a Pickering emulsion is characterized by
the competition between ⑀ and the spontaneous curvature of
the interface saturated with particles.10
In summary, we have discussed the adsorption of spherical particles to a spherically curved interface. We showedthat equilibrium contact angle is given by the Young’s equation, whereas the minimized adsorption energy is a functionof curvature. Such a curvature effect affects the equilibrium
distribution of the particles. Currently, we are investigating
analogous problems for a spherical colloidal particle withhydrophilic and hydrophobic hemispheres 共“Janus bead”兲.19
We thank T. Kato and K. Tsujii for useful discussions.
This research is partly supported by the Grant-in-Aid for
Scientific Research 共C兲, Japan Society for the Promotion of
Science under Grant No. 15540395.
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