THE JOURNAL OF CHEMICAL PHYSICS 127, 054707 共2007兲

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

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

共Received 20 April 2007; accepted 12 June 2007; published online 7 August 2007兲
We investigate the adsorption of a spherical Janus particle to a spherically curved liquid-liquidinterface. We show that the equilibrium contact angle is determined by the geometry of the particle,
its wettability, and also the interfacial curvature. In contrast with a homogeneous particle, there isa preferred interfacial curvature 共spontaneous curvature兲 due to the Janus particle when the particle
satisfies certain conditions. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2756828兴
Particles which have two distinct parts are called “Janus
particles” in the name of the dual-faced Janus god.1,2 By
providing two different chemical and/or physical properties
to a single particle, one can produce various kinds of Janus
particles which are useful in many modern technologies.3–6
For example, design and synthesis of bicompartmental particles, dumbbell-like particles, acornlike particles, or snowmanlike particles are reviewed in Ref. 7. In addition to usethem as elementary building blocks for supraparticular assemblies, Janus particles can be very useful in various applications. For example, they can be used for the stabilizationof water-in-oil or oil-in-water emulsions similar tosurfactants.8 When bichromal Janus particles were made tohave charge segregation on the surface, they could be usefulfor electronic displays such as the electric reusable paper.
Other applications of Janus particles are catalysis, sensing,drug delivery, etc.
Here we focus on the Janus particles which are analogous to surfactant molecules having both hydrophilic andhydrophobic parts. Such amphiphilic particles were obtained
by Casagrande et al.1 and Ondarçuhu et al.2 from glass
spherical particles. They studied the adsorption at oil/waterinterfaces, and showed that the Janus particles behave differently from ordinary solid particles. In contrast with a dense
film made of conventional molecular surfactants, a film ofJanus particles at the interface has some interstices between
the particles so that it can “breathe.”9 Moreover, it was theoretically shown that Janus particles are more surface active
compared to homogeneous surface particles.10
Not only Janus particles but also homogeneous surface
particles can stabilize emulsions. Such emulsions are called
Pickering emulsions.11–13 In our previous paper, we investigated the adsorption of such homogeneous spherical particles
to a liquid-liquid interface.14 In particular, we looked at theeffect of the curvature of the interface on the particle adsorption. We showed that although the equilibrium contact angle
is determined by the classical Young’s equation 共being independent of the curvature兲, the adsorption energy is affecteda兲Electronic mail: komura@tmu.ac.jp

0021-9606/2007/127共5兲/054707/5/$23.00

by the interfacial curvature. The partitioning of the particles
among the two liquids and the interface was also discussed.
In the present paper, a similar analysis is performed for
Janus particles. We first discuss their equilibrium position ata liquid-liquid interface, and examine the effect of the interfacial curvature. Due to the inhomogeneous nature of theJanus particles, they exhibit completely different adsorption
behavior compared to the homogeneous particles. Especially,
we show that there is a preferred curvature 共spontaneous curvature兲 in the presence of a Janus particle when certain conditions are satisfied. It is known that the spontaneous curvature of a molecular surfactant plays an essential role inemulsifying two immiscible liquids.15 This property is in
sharp contrast compared with the homogeneous particles forwhich the spontaneous curvature does not exist.14
We first consider the equilibrium position of a Janus particle at an interface between two immiscible liquids. As
shown in Fig. 1共a兲, the isolated particle is assumed to be
perfectly spherical and its radius is denoted by a. The particle surface has two chemically distinct regions, say, “A”
and “B” corresponding to hydrophilic and hydrophobic regions, respectively. The borderline between these two regions is parametrized by an angle ␣, as defined in Fig. 1共a兲.
Then the areas of each region are given by
SA = 2␲a2共1 + cos ␣兲

共1兲SB = 2␲a2共1 − cos ␣兲,

共2兲

and

for A and B, respectively. Extreme cases are ␣ = 0 for which
the particle is homogeneously coated with region A, or␣ = ␲ when region B covers the whole surface.
Next we discuss the adsorption of such a Janus particle
to a spherically curved interface between liquids 1 and 2
which correspond to, for example, water and oil, respectively
关see Fig. 1共b兲兴. Here liquid 2 forms a spherical droplet. We
assume that the line connecting the centers of the droplet andthe particle is perpendicular to the borderline of the particle,
and the three phase contact line is perfectly regular without
any distortion. We define ␪ as the contact angle 共measured

127, 054707-1© 2007 American Institute of Physics

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J. Chem. Phys. 127, 054707 共2007兲

Hirose, Komura, and Nonomura

wB = ␴1ASA + ␴2BS2 + ␴1B共S1 − SA兲 − ␴12S12
共6兲+ p 1V 1 + p 2V 2 ,when 0 艋 ␪ 艋 ␪an, or
wA = ␴1AS1 + ␴2BSB + ␴2A共SA − S1兲 − ␴12S12

共7兲+ p 1V 1 + p 2V 2 ,when ␪an 艋 ␪ 艋 ␲. Here S1 and S2 are the 1-particle and2-particle contact areas, respectively, while S12 is the eliminated area of the 1-2 interface due to the presence of theparticle. Note that SA and SB are given in Eqs. 共1兲 and 共2兲,
respectively. The volumes occupied by the particle in liquids
1 and 2 are denoted by V1 and V2, respectively. After somegeometrical calculations, these areas and volumes are obtained as
FIG. 1. 共a兲 Geometry of a spherical Janus particle whose radius is a. Upperand lower regions are denoted by “A” and “B,” respectively. The angle ␣characterizes the borderline between A and B. 共b兲 Janus particle adsorbed toa spherical interface between liquid 1 and liquid 2. The radius of the interfacial curvature is R. The pressures in liquids 1 and 2 are p1 and p2, respectively. The angles ␪ and ␻ are the contact angle and the central angle,
respectively. 共c兲 Definition of the anchoring angle ␪an. The borderline between the two regions coincides with the liquid-liquid interface.

across liquid 1兲 which characterizes the position of the particle, and R is the radius of the liquid droplet. We introduce
the dimensionless curvature ⑀ = a / R which is taken to bepositive when the interface is curved toward liquid 1 ornegative when it is curved toward liquid 2. The central angle
␻ defined in Fig. 1共b兲 is given by冉␻ = arctan冊

⑀ sin ␪,1 + ⑀ cos ␪

2␴12
.
R

S2 = 2␲a2关1 − cos共␪ − ␻兲兴,共9兲

S12 = 2␲R2共1 − cos ␻兲,共10兲

V1 = ␲a3关 32 + cos共␪ − ␻兲 − 31 cos3共␪ − ␻兲兴− ␲R3关 32 − cos ␻ + 31 cos3 ␻兴 ,

共4兲

共5兲

We neglect external forces and a finite thickness of theinterface. Then the interfacial energy attributed to the particle
is given by either of the following two cases depending on
the particle position with respect to the interface:

共11兲

V2 = ␲a3关 32 − cos共␪ − ␻兲 + 31 cos3共␪ − ␻兲兴+ ␲R3关 32 − cos ␻ + 31 cos3 ␻兴 .

共12兲Hereafter, we define the following dimensionless quantities using the above surface tensions:␴=

As shown in Fig. 1共c兲, we call such a position of the particle
as the anchoring position or the anchoring angle ␪an.
For the adsorption of a Janus particle, there are five different surface tensions characterizing the system. Let ␴12,
␴1A, and ␴2A be the 1-2, 1-A, and 2-A surface tensions. We
use similar notations for ␴1B and ␴2B as well. When theinterface is curved, as in Fig. 1共b兲, 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, the pressure difference is maintained by the 1-2 surface tension ␴12,
p2 − p1 =

共8兲

共3兲

in terms of ⑀ and ␪. From the geometrical relations, the borderline between the two regions on the particle coincides
with the liquid-liquid interface when ␻ = arcsin共⑀ sin ␣兲. In
this case, the contact angle is given by␪ = ␣ + ␻ = ␣ + arcsin共⑀ sin ␣兲 ⬅ ␪an .

S1 = 2␲a2关1 + cos共␪ − ␻兲兴,

␴1A + ␴2B
,
␴12

␶=

␴1A − ␴2A
,
␴12

␥B =

␥A =

␴1A − ␴2B,␴12
␴1B − ␴2B
,
␴12

共13兲

共14兲

which characterize the wettability of the particle. Without
loss of generality, we assume that region A prefers liquid 1 toliquid 2, and region B prefers liquid 2 to liquid 1. This meansthat the orientation of the particle with respect to the interface is such that shown in Fig. 1共b兲, and the case of the
opposite orientation is excluded from our consideration.
Since the conditions ␴1A ⬍ ␴2A and ␴1B ⬎ ␴2B hold in thissituation, we can generally set

␥A ⬍ 0,␥B ⬎ 0.

共15兲This condition means that the Janus particle is amphiphilic.
We now present the interfacial energy wi 共i = A , B兲 in thedimensionless form, and subtract the constant terms which
do not depend on ␪ or ⑀,
g i共 ␪ , ⑀ 兲 =a共p1 + p2兲
wi− ␴ − ␶ cos ␣ −.
2␲a2␴123␴12Then we have

共16兲

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J. Chem. Phys. 127, 054707 共2007兲

Adsorption of Janus particles to curved interfaces

FIG. 2. The dimensionless adsorption energy gi as a function of the contact
angle ␪ for ␣ = ␲ / 2 and 3␲ / 8 when ⑀ = a / R = 0. The other parameters are␥A = −0.25 and ␥B = 0.25. Filled circles correspond to the minimum and thearrows indicate the anchoring positions.

gB共␪, ⑀兲 = ␥B关cos共␪ − ␻兲 − cos ␣兴 −1 − cos3 ␻3⑀2⑀− 关3 cos共␪ − ␻兲 − cos3共␪ − ␻兲兴3

共17兲

for 0 艋 ␪ 艋 ␪an, andgA共␪, ⑀兲 = ␥A关cos共␪ − ␻兲 − cos ␣兴 −1 − cos3 ␻3⑀2⑀− 关3 cos共␪ − ␻兲 − cos3共␪ − ␻兲兴3共18兲

for ␪an 艋 ␪ 艋 ␲. Since ␻ ⬇ ⑀ sin ␪ up to the lowest order in ⑀关see Eq. 共3兲兴, the above expressions reduce to those for theflat interface when ⑀ → 0.2,10 The equilibrium position of theparticle is obtained by minimizing gi with respect to ␪ while⑀ is fixed.
First we consider the case when the liquid-liquid interface is flat, i.e., ⑀ = 0. Then we have ␪an = ␣ from Eq. 共4兲. In
Fig. 2, we plot gi as a function of ␪ for ␣ = ␲ / 2 and 3␲ / 8
when ␥A = −0.25 and ␥B = 0.25. Here the filled circles markthe minimum of gi and the arrows indicate the anchoring
angles. When ␣ = ␲ / 2, the minimum of gi coincides with theanchoring position, whereas they differ for ␣ = 3␲ / 8 although ␥A and ␥B are identical in these two cases. In thelatter case, the minimum of gA at ␪ ⬇ 1.32 gives the smallest
adsorption energy. A similar discussion for the adsorption of
a Janus particle to a flat interface was given before,1,2,10 andour result agrees with theirs.
Next we examine the effect of the interfacial curvatureon the adsorption of a Janus particle. In Fig. 3, we haveplotted gi for the cases ⑀ = a / R = 0, 0.2, and 0.4. The rest of
the parameters are the same as in Fig. 2 and hence the wettability of the particle is not changed here. As marked by thefilled circles, the equilibrium position coincides with the anchoring position 共indicated by an arrow兲 when ⑀ = 0 and 0.2,

FIG. 3. The dimensionless adsorption energy gi as a function of the contact
angle ␪ for ⑀ = 0 共solid line兲, ⑀ = 0.2 共dashed line兲, and ⑀ = 0.4 共dotted line兲.
The other parameters are ␣ = ␲ / 2, ␥A = −0.25, and ␥B = 0.25. The minimum
denoted by the filled circle coincides with the anchoring angle 共indicated byan arrow兲 when ⑀ = 0 and 0.2, while it is given by Young’s equation for ⑀= 0.4.

while they differ when ⑀ = 0.4 due to the curvature effect.
When the curvature is relatively large, the high internal pressure prevents the particle from anchoring at the borderline.
For general curvatures, we can obtain the equilibrium contact
angle ␪eq as冦␪B when ␪B 艋 ␪an␪eq = ␪an when ␪A 艋 ␪an 艋 ␪B
␪A when ␪an 艋 ␪A ,

冧

共19兲

where ␪B ⬅ arccos共−␥B兲 艌 ␲ / 2 and ␪A ⬅ arccos共−␥A兲 艋 ␲ / 2.
Again this result reduces to that for a flat interface when ⑀→ 0.1,2,10 The cases which give Young’s equation 共␪eq = ␪Band ␪A兲 are obtained from the condition 共⳵gi / ⳵␪兲⑀ = 0. An
analogous calculation to obtain Young’s equation was explained in our previous paper.14 To represent the above result, we show in Fig. 4 the equilibrium contact angle ␪eq as afunction of the curvature ⑀ for various values of ␥A and ␥B
共taken as 兩␥A兩 = 兩␥B兩 for simplicity兲 when ␣ is fixed to ␲ / 2. Itshould be noticed that the particle surface is homogeneous
when ␥A = ␥B = 0 for which there is no ⑀ dependence 共solidline兲. For finite values of ␥A and ␥B, the equilibrium angle
␪eq depends on the curvature when the particle is anchored.
The curvature-dependent region of ␪eq becomes larger as 兩␥i兩
is increased.
Whether the equilibrium contact angle ␪eq is determinedby the anchoring angle or Young’s relation depends on ␣, ⑀,␥A, or ␥B, as discussed above. It is important to note that theequilibrium contact angle ␪eq is independent of the interfacial
curvature ⑀ when Young’s equation holds, while it does depend on ⑀ when ␪eq = ␪an. Such a property of the Janus particle is in sharp contrast compared to a homogeneous surface
particle for which the equilibrium contact angle is always
determined by Young’s equation.14 A similar result using a

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J. Chem. Phys. 127, 054707 共2007兲

Hirose, Komura, and Nonomura

FIG. 4. The equilibrium contact angle ␪eq as a function of the interfacial
curvature ⑀ for different values of 兩␥i兩 共i = A , B兲 when ␣ = ␲ / 2. The particle
is homogeneous when 兩␥i兩 = 0, and ␪eq does not depend on ⑀ 共solid line兲.
However, ␪eq depends on ⑀ when 兩␥i兩 ⫽ 0 when the particle is anchored.

different approach was given before by di Meglio andRaphaël.16 However, they did not consider the volume energy which becomes important in the following argument.
We now substitute back the equilibrium contact angle ␪eqinto gi, and obtain the adsorption energy which is a functionof ⑀ only. Then we haveg共⑀兲 = −1
关1 + 3⑀2␥B cos ␣ − 共1 − 2⑀2 + ⑀␥B兲
3⑀2

⫻共1 + ⑀2 − 2⑀␥B兲1/2兴,共20兲

when ␪B 艋 ␪an,g共⑀兲 = −1
关1 − 共1 − ⑀2 sin2 ␣兲3/2兴
3⑀2

⑀
− 共3 cos ␣ − cos3 ␣兲,
3FIG. 5. The minimized adsorption energy g as a function of the interfacial
curvature ⑀ for ␣ = ␲ / 2 共solid line兲, ␣ = 31␲ / 60 共dashed line兲, and ␣= 8␲ / 15 共dotted line兲. The parameters are ␥A = −0.25 and ␥B = 0.25 for all thecases. Filled circles correspond to the minimum points.

From the minimization of g共⑀兲, we have numerically calculated ⑀min as a function of ␣ which is plotted in Fig. 6.
Notice again that the minimum appears only when ␣ is closeto ␲ / 2. The range of ␣ for which ⑀min exists is dependent on
the values of ␥i. When 兩␥i兩 becomes larger, ␣ can deviate
from ␲ / 2 in order that g共⑀兲 has a minimum. The preferredcurvature ⑀min is an odd function of ␣ and vanishes when
␣ = ␲ / 2. Hence the flat interface is stable for a Janus particle
characterized by ␣ = ␲ / 2. Such a behavior is completely different from that of the homogeneous particle for which theinterface is always unstable.14 By assuming ⑀ Ⰶ 1, we expand
g共⑀兲 in terms of ⑀ to obtain

共21兲

when ␪A 艋 ␪an 艋 ␪B, andg共⑀兲 = −1
关1 + 3⑀2␥A cos ␣ − 共1 − 2⑀2 + ⑀␥A兲
3⑀2

⫻共1 + ⑀2 − 2⑀␥A兲1/2兴,共22兲

when ␪an 艋 ␪A. Note that these different cases correspond to
those in Eq. 共19兲. In Fig. 5, we have plotted the minimized
adsorption energy g共⑀兲 as a function of ⑀ for ␣ = ␲ / 2 共solidline兲, ␣ = 31␲ / 60 共dashed line兲, and ␣ = 8␲ / 15 共dotted line兲.
For all the cases plotted here, g共⑀兲 is downside convex when
the particle is anchored. As marked by the filled circles, theminimum of g共⑀兲 appears for ␣ = ␲ / 2 and 31␲ / 60. The curvature ⑀min which gives the minimum of g depends on thesurface geometry ␣. On the other hand, the minimum nolonger exists for ␣ = 8␲ / 15. For larger curvature, g共⑀兲 takes amaximum value and then decreases. This is due to the factthat the equilibrium contact angle ␪eq changes from the anchoring angle to that given by Young’s equation.

FIG. 6. The minimum curvature ⑀min as a function of the angle ␣. The otherparameters are ␥A = −0.25 and ␥B = 0.25. The minimum does not exist for ␣
outside the range of the present plot.

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J. Chem. Phys. 127, 054707 共2007兲

Adsorption of Janus particles to curved interfacesg共⑀兲 ⬇ −冉

冊

sin2 ␣
cos2 ␣
⑀2+ ⑀ cos ␣− 1 + sin4 ␣ + ¯ .
8
2
3
共23兲

Then ⑀min can be approximately obtained as⑀min ⬇

冉冊

4 cos ␣cos2 ␣
1−.
sin4 ␣
3

共24兲

The existence of such a preferred curvature is a unique character of the Janus particle. Notice that ⑀min ⬎ 0 and the interface is curved toward liquid 1 when ␣ ⬍ ␲ / 2.
Quite generally, Helfrich proposed an elastic model applicable for biological membranes, microemulsions, andother self-assembled molecular films.15,17 The corresponding
curvature elasticity energy per unit area of the interface isgiven by␬
f c共c1,c2兲 = 共c1 + c2 − 2c0兲2 + ¯␬c1c2 ,2共25兲

where c1 and c2 are two principle curvatures, ␬ and ¯␬ are thebending and Gaussian moduli, respectively, and c0 is the
spontaneous curvature. Using the expansion Eq. 共23兲, we calculate the sum and the difference of g共⑀兲 and g共−⑀兲 to obtain冉␬c0 = ␲␴12a cos ␣ 1 −2␬ + ¯␬ =冊

cos2 ␣
,3

␲␴12a2 sin4 ␣ .
4

共26兲

共27兲

The sum of the elastic moduli 2␬ + ¯␬ is proportional to ␴12a2
which is in contrast with an interface saturated with amphiphilic molecules. In the latter case, the bending modulus
is proportional to the cube of the molecular size.18
In order to give a value for the derived ␬, we consider asystem in which ␴12 = 5 ⫻ 10−2 J / m2 and a = 10−7 m. We thenhave ␬ ⬇ 4.7⫻ 104kBT for ␣ = ␲ / 2 at room temperature 共wehave neglected ¯␬兲. Hence the bending modulus can be muchlarger than that of the interface made of surfactants, and theeffect of thermal fluctuations can be neglected. This result
suggests that emulsions made of Janus particles are quite
stable against the coalescence of the droplets since the en-ergy barrier to break a droplet is primarily determined by thecurvature energy of the interface. A liquid droplet coated byJanus particles may be rather stable and behaves like a soft
solid.
To summarize, we have discussed the adsorption of a
Janus particle to a spherically curved liquid-liquid interface.
We showed that the equilibrium contact angle depends on thegeometry of the particle, its wettability, and also the interfacial curvature. Furthermore, there exists a preferred interfacial curvature due to the presence of a Janus particle.
We thank T. Kato for useful discussions. This work was
supported by KAKENHI 共Grant-in-Aid for Scientific Research兲 on Priority Areas “Soft Matter Physics” and Grant
No. 18540410 from the Ministry of Education, Culture,
Sports, Science and Technology of Japan.
1

C. Casagrande, P. Fabre, E. Raphaël, and M. Veyssié, Europhys. Lett. 9,
251 共1989兲.
2
T. Ondarçuhu, P. Fabre, E. Raphaël, and M. Veyssié, J. Phys. 共France兲
51, 1527 共1990兲.
3
K.-H. Roh, D. C. Martin, and J. Lahann, Nat. Mater. 4, 759 共2005兲.
4
N. Glaser, D. J. Adams, A. Böker, and G. Krausch, Langmuir 22, 5227
共2006兲.
5
S. M. Anthony, L. Hong, M. Kim, and S. Granick, Langmuir 22, 9812
共2006兲.
6
L. Hong, A. Cacciuto, E. Luijten, and S. Granick, Nano Lett. 6, 2510
共2006兲.
7
A. Perro, S. Reculusa, S. Ravaine, E. Bourgert-Lami, and E. Duguet, J.
Mater. Chem. 15, 3745 共2005兲.
8
Y. K. Takahara, S. Ikeda, S. Ishino, K. Tachi, K. Ikeue, T. Sakata, T.
Hasegawa, H. Mori, M. Matsumura, and B. Ohtani, J. Am. Chem. Soc.
127, 6271 共2005兲.
9
P. G. de Gennes, Rev. Mod. Phys. 64, 645 共1992兲.
10
B. P. Binks and P. D. I. Fletcher, Langmuir 17, 4708 共2001兲.
11
S. U. Pickering, J. Chem. Soc. 9, 2001 共1907兲.
12
B. P. Binks, Curr. Opin. Colloid Interface Sci. 7, 21 共2002兲.
13
Colloidal Particles at Liquid Interfaces, edited by B. P. Binks and T. S.
Horozov 共Cambridge University Press, Cambridge, 2006兲.
14
S. Komura, Y. Hirose, and Y. Nonomura, J. Chem. Phys. 124, 241104
共2006兲.
15
S. A. Safran, Statistical Thermodynamics of Surface, Interface, and Membranes 共Addison-Wesley, New York, 1994兲.
16
J. M. di Meglio and E. Raphaël, J. Colloid Interface Sci. 136, 581
共1990兲.
17
W. Helfrich, Z. Naturforsch. C 28, 693 共1973兲.
18
L. D. Landau and E. M. Lifshitz, Theory of Elasticity 共Pergamon, Oxford,
1986兲.