Progress of Theoretical Physics Supplement No. 175, 2008

81

Adsorption Dynamics in Pickering Emulsions
Yuichi Hirose, Shigeyuki Komura and Tadashi Kato
Department of Chemistry, Graduate School of Science and Engineering,Tokyo Metropolitan University, Tokyo 192-0397, Japan
We investigate the adsorption dynamics of colloidal particles to a liquid/liquid interface.
Such an analysis is important in the formation of Pickering emulsions. Using a free energy
formulation that has been used for adsorption of surfactants, we obtain the time evolutionof the surface coverage and dynamic surface tension. We estimate the characteristic time
for adsorption, and show that it corresponds to a diﬀusion-limited adsorption case. The
asymptotic values of the surface coverage and surface tension with respect to their equilibrium
values is shown to decrease as t−1/2 . We also show that the equilibrium equation of state
relating surface tension to surface coverage holds also in nonequilibrium situations.

§1.

Introduction

Small particles such as lamp black or hydrophobic silica adsorb to an oil-waterinterface and act as stabilizers of emulsions. Such emulsions are called “Pickering
emulsions”1) and have been used in various food and cosmetic products.2) For instance, lipoprotein particles and fatty acid crystals stabilize the emulsion state inmayonnaise or margarine. A modern application of Pickering emulsion is to usethem as templates of functional composite materials. For example, spherical emulsion droplets are used to produce micro-structured hollow spheres3) or ribosome-likenano-capsules (“colloidsomes”).4) In addition to spherical droplets, the particles form
several self-assembling structures such as network structures or non-spherical aggregates.5), 6)
Pickering emulsions are not only useful but also interesting from the fundamental point of view. When a molecule exhibits a surface activity, it is conventionally believed that both hydrophobic and hydrophilic groups should exist in a single
molecule. However, this is not the case for a colloidal particle whose surface is chemically uniform. The particles adsorb only when the relative wettability between theparticle and the two liquids balances each other.7), 8) Moreover, colloidal particles canadjust their position with respect to the interface in order to reduce the interfacial
energy.
One of the important aspects of Pickering emulsions resides in their nonequilibrium properties. This is the reason why it was a surprise when thermodynamicallystable Pickering emulsions have been reported recently.9) For example, the coalescence frequency of droplets is shown to depend on time and particle concentration.10)
Computer simulations of phase separation of a binary solvent containing nanoparticles have been also performed.11) It was shown that colloidal particles with equalaﬃnity for two liquids form a bicontinuous interfacially jammed emulsion gel (“bijel”).12) Soon after this study, microscopy study of Pickering emulsions revealed thatquenching into the demixed region leads to the formation of domains which are sta-

82

Y. Hirose, S. Komura and T. Katoble due to a jammed, semisolid, multilayer of colloids.13), 14) Recently, the kinetics ofnanoparticle self-assembly at liquid-liquid interfaces has been reported.15) They measured changes in the interfacial tension from the early to late stages of nanoparticle
adsorption.
In this paper, we discuss the adsorption dynamics of solid particles to liquidliquid interfaces. We use a theory that has been developed to describe the kinetics ofsurfactant adsorption.16), 17) This theory relies on a free energy formulation and has
been successful in explaining the observed dynamic interfacial tensions of surfactant
solutions. Our primary interest is to understand how the wettability (aﬃnity) of
the colloidal particle with respect to the two liquids would aﬀect the adsorption
dynamics. We discuss the characteristic time for the particle adsorption and willshow that it is a diﬀusion-limited process. We also calculate the dynamic surface
tension which will be compared with the recent experimental result of nanoparticle
adsorption.15) It will be argued that the equilibrium equation of state holds also in
nonequilibrium cases.
In the next section, we ﬁrst deﬁne the wettability of a colloidal particle andderive its adsorption energy. In §3, the free energy of the system is deﬁned andsome equilibrium properties are derived including the adsorption isotherm. In §4,
we present the basic equations describing the adsorption kinetics. Section 5 gives
the results of the numerical calculation together with some discussions. Finally, we
shall conclude in §6.
§2.

Wettability of a spherical particle

We start our discussion by considering a spherical particle of diameter a residing
at a ﬂat interface between liquid 1 and liquid 2, as shown in Fig. 1(a). Let θ be thecontact angle deﬁned in liquid 1, and σ1p , σ2p , σ12 be the 1-particle, 2-particle, 1-2
interfacial tensions, respectively. In this paper, we deﬁne the following dimensionlessquantity to characterize the wettability of the particle:γ=

σ1p − σ2p
.
σ12

(2.1)

Neglecting external forces and a ﬁnite thickness of the interface, the interfacial energy
attributed to the particle is A = σ1p S1p + σ2p S2p − σ12 S12 , where S1p and S2p are
the 1-particle and 2-particle contact areas, respectively, while S12 is the eliminatedarea of the 1-2 interface due to the presence of the particle.7), 18) These areas are
given byS1p =πa2
(1 + cos θ),
2

S2p =

πa2
(1 − cos θ),
2

S12 =

Then the total adsorption energy per particle is expressed as
sin2 θ
πa2 σ12σ + γ cos θ −,A(θ, γ) =22where σ = (σ1p + σ2p )/σ12 .

πa2
sin2 θ.
4

(2.2)

(2.3)

Adsorption Dynamics in Pickering Emulsions

83

Fig. 1. (a) A spherical particle of diameter a adsorbed to a ﬂat interface between liquid 1 and liquid
2. The contact angle taken in liquid 1 is denoted by θ. (b) Colloidal particles adsorbing at theliquid/liquid interface. The particles are contained only in liquid 1 which occupies x > 0. Here
φ0 is the surface coverage, φ(x) the volume fraction at x, and φb the bulk volume fraction at
x → ∞.

The contact angle θ which gives the equilibrium position of the particle is determined by minimizing A(θ, γ) with respect to θ. This leads to the conventional
Young’s equation:7)
θ = arccos(−γ).

(2.4)

By substituting this relation into Eq. (2.3), the minimized interfacial energy or theadsorption energy (as a function of γ) is given by18)
1
πa2 σ122σ − (1 + γ ) ,A(γ) =
22(2.5)

as long as |γ| < 1. In other cases, the particle prefers to stay either in liquid 1 whenγ < −1 or in liquid 2 when γ > 1.
§3.

Adsorption isotherm

Next we discuss the equilibrium adsorption isotherm of the interface. As depicted
in Fig. 1(b), we assume that a ﬂat interface between the two liquids spans the planex = 0, and the particles are contained only in liquid 1 which occupies x > 0. Liquid
1 is in contact with a bulk reservoir of particles where the particle volume fraction
and chemical potential are given by φb and μb , respectively.
The particle volume fraction in the bulk is denoted by φ(x), and its value at theinterface by φ0 . Following Refs. 16) and 17), we start from the excess free energy

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Y. Hirose, S. Komura and T. Kato

per unit area written as ∞ΔΓ [φ(x)] =0Δf [φ(x)] dx + f0 (φ0 ),(3.1)

where Δf is the excess free energy per unit volume in the bulk, and f0 is the freeenergy per unit area at the interface. The former is given by1
kB T [φ ln φ − φ − (φb ln φb − φb )]a3+4π(a/2)2 σ1p (φ − φb ) − μb (φ − φb ) ,Δf [φ(x)] =(3.2)

where kB is the Boltzmann constant and T the temperature. The ﬁrst contribution
is the entropy of mixing with respect to the bulk, and the second contribution comesfrom the contact of the particles with liquid 1 (hence the surface tension σ1p ). It
is assumed here that the bulk volume fraction φ(x) is much smaller than unity. On
the other hand, f0 in Eq. (3.1) has the form
1
f0 (φ0 ) = 2 kB T [φ0 ln φ0 + (1 − φ0 ) ln(1 − φ0 )]
a

B
(3.3)
+A(γ)φ0 − φ20 − μ1 φ0 .
2
In the above, the terms in the square brackets is the full entropy of mixing since
φ0  φ(x). Moreover A(γ) is the particle adsorption energy deﬁned by Eq. (2.5),
and the term proportional to B describes the interactions between particles at theinterface. The overall particle interaction is attractive when B > 0. Finally, thechemical potential is deﬁned as μ1 ≡ μ(x → 0).
Variation of ΔΓ with respect to φ(x) leads to the chemical potential given by
μ(x) = kB T ln φ(x) + πa2 σ1p . Hence we have μ1 = kB T ln φ1 + πa2 σ1p , where φ1 ≡φ(x → 0) is the particle volume fraction at the subsurface layer. Note that, inequilibrium, we have φ(x) = φb for x > 0. On the other hand, minimization ofΔΓ with respect to φ0 (hence μ0 = μ1 , where μ0 is the chemical potential at theinterface) and the condition μ1 = μb gives the equilibrium adsorption isotherm
φ0,eq =

φb,
φb + exp[−α(1 + γ)2 − βφ0,eq ](3.4)

where α ≡ π(a/2)2 σ12 /kB T and β ≡ B/kB T are dimensionless energy parameters.
This equation is called the Frumkin adsorption isotherm which reduces to the Langmuir adsorption isotherm when β = 0.19) In Fig. 2(a), we plot φ0,eq as a functionof φb for various values of β. The other parameters are chosen to be α = 10 and
γ = −0.5. When β < 4, the equilibrium surface coverage φ0,eq increases monotonically. For β > 4, however, φ0,eq changes non-monotonically so that a two-phasecoexistence is implied.19) We have determined the vertical dashes lines by the condition that φb minimizes the surface free energy of Eq. (3.3). Such a discontinuous
jump of the surface coverage in the presence of a lateral interaction is also predicted
for athermal polymer-nanoparticle blends near a substrate.20) In the following, weAdsorption Dynamics in Pickering Emulsions

85

Fig. 2. (a) Equilibrium surface coverage φ0,eq as a function of the bulk volume fraction φb for
diﬀerent values of β (adsorption isotherms). The other parameters are α = 10 and γ = −0.5.
For β > 4, φ0,eq changes discontinuously according to the dashed lines. (b) The equilibrium
interfacial energy f0,eq as a function of φ0 .

shall mainly deal with the case of β < 4 for which the phase transition does notoccur. The case of β > 4 will be brieﬂy discussed later in §5.
From Eqs. (3.3) and (3.4), the equilibrium interfacial energy becomes16), 17)


kB T
β 2
f0,eq = 2 ln(1 − φ0,eq ) + φ0,eq .
a
2

(3.5)

This provides the equilibrium equation of state relating the surface tension to thesurface coverage. Notice that Δf in Eq. (3.2) completely vanishes in equilibrium. In
Fig. 2(b), we show f0,eq as a function of φ0 for various values of β. As a result of
the competition between the entropy and interaction terms, the above equilibrium
surface energy decreases only slightly for small φ0 , but drops oﬀ when φ0 → 1. We
also see that a maximum appears when β > 4.
§4.

Adsorption dynamics

In order to discuss nonequilibrium situations, we follow the formulation thathas been developed for surfactant molecules.16), 17), 21) Assuming the proportionality
between the velocities and potential, we employ an ordinary diﬀusion equation in
the bulk:∂ 2φ∂φ= D 2,
∂t
∂x

(4.1)

where D is the diﬀusion constant of the colloidal particle. We further assume herethat the colloidal particles do not leave the interface toward liquid 2. Then the

86

Y. Hirose, S. Komura and T. Kato

condition of the particle conservation at the subsurface layer can be written as
∂φ1
D ∂φ
∂φ0
=
,
−
∂t
a ∂x x=a∂t(4.2)

where we have assumed that the subsurface layer exists at x = a.
By applying the Laplace transform to Eqs. (4.1) and (4.2), we recover the socalled Ward-Tordai equation:22), 23)


 t
√
1 D
φ1 (τ )√dτ + 2φb ,2φb t −
φ0 (t) + φ1 (t) =
(4.3)
a π
t−τ
0
where we have used φ(x, t = 0) = φb for the initial condition. This equation describes
the time evolution of the surface plus subsurface volume fraction (φ0 + φ1 ) due to adiﬀusive transport of colloidal particles from the bulk to the interface. The second
term in the brackets accounts for the desorption of the particles from the subsurface
layer. The dynamics at the interface itself is proportional to the variation of f0 withrespect to φ0 . This gives the following equation:D
φ1 (1 − φ0 )∂φ02
= 2 φ1 ln+ α(1 + γ) + βφ0 .
(4.4)
∂t
a
φ0
Equations (4.3) and (4.4) provide a complete set of equations to determine φ0 (t)and φ1 (t) independently. Although the diﬀusivities at the interface can be diﬀerent
from those inside the aqueous solution, the two diﬀusion constants D in the above
equations are assumed to be the same. Using the numerical scheme described inRef. 24), we calculate the full solution of the coupled equations numerically.
§5.

Results and discussion

In this section, we present the results obtained from the numerical solutionsof Eqs. (4.3) and (4.4). Our primary interest is to understand how the particle
wettability parameter γ aﬀects the adsorption kinetics. In Fig. 3(a), we have plotted
the time evolution of the particle volume fraction φ0 (t) at the interface as a functionof the scaled time t̃ = Dt/a2 for various values of γ. The other parameters arechosen to be φb = 10−2 , α = 10, and β = 1. For all the cases, φ0 (t) increases
monotonically with time approaching to the respective equilibrium values φ0,eq (seeEq. (3.4)) which are indicated by the open circles on the right axis of Fig. 3(a). We
see that φ0 (t) becomes larger when the value of γ is increased from −0.5 to 0. Thistendency is reasonable because the particles have more aﬃnity to liquid 2 when γ ismade larger. In the intermediate time range, φ0 increases almost as t1/2 at least forlarger γ, which is consistent with the Ward-Tordai equation. In Fig. 3(b), we haveshown φ0,eq − φ0 (t) as a function of t̃ in the large time region to see the asymptotic
adsorption behavior. We see here a clear power-law dependence obeying ∼ t−1/2 .
The physical explanation of this behavior will be discussed soon after.
In order to estimate a typical time which is necessary for the particles to approach the equilibrium state, we determined the points which satisfy the condition

Adsorption Dynamics in Pickering Emulsions

87

Fig. 3. (a) The time evolution of the surface coverage φ0 as a function of the scaled time t̃ = Dt/a2
for diﬀerent values of γ. The other parameters are φb = 10−2 , α = 10, and β = 1. The
open circles on the right axis indicate the respective equilibrium values φ0,eq . The open squares
are the points at which the condition d3 (log φ0 )/d(log t̃)3 = 0 is satisﬁed. (b) The asymptotic
behavior of φ0,eq − φ0 in the large time regions.

d3 (log φ0 )/d(log t̃)3 = 0 for each curve in Fig. 3(a). These points are marked byopen squares in the ﬁgure, and the corresponding characteristic time is denoted byτ̃c . Needless to say, this is not the only possibility to deﬁne the characteristic time,
but other deﬁnitions would not alter the present result qualitatively. In Fig. 4, wesummarized the obtained τ̃c as a function of γ by open squares. The characteristic
time τ̃c increases as γ is made larger from −1, and it almost saturates when γ ≥ −0.3
for the present choice of the parameters.
It was argued before that there can be two limiting cases describing the relative time scales of the two processes, namely, the kinetics inside the bulk solution
described by Eq. (4.3) and the kinetics at the interface described by Eq. (4.4).16), 17)The ﬁrst case is the diﬀusion-limited adsorption (DLA) which occurs when the dynamics inside the solution is much slower than the one at the interface. Here theinterface is in constant equilibrium with the adjacent solution. This means that, ina DLA, the interface is always at the minimum of f0 (φ0 ) although the free energy
curve changes with time. In this situation, the asymptotic time dependence of thesurface coverage is predicted to be25)
φ0,eq − φ0 (t) ∼τd /t.

(5.1)

Here the characteristic time scale of diﬀusion is determined by
τd =

a2
D

φ0,eq
φb

2

.

(5.2)

The second case is the kinetically limited adsorption (KLA) which applies when
the dynamics at the interface is much slower. In a KLA, the shape of f0 (φ0 ) does

88

Y. Hirose, S. Komura and T. Kato

Fig. 4. The characteristic time τ̃c obtained from Fig. 3(a). The solid line represents the scaled
characteristic time of diﬀusion τ̃d = (φ0,eq /φb )2 as given by Eq. (5.2).

not change with time, but the surface coverage increases toward φ0,eq . Therefore
φ0 evolves with time according to Eq. (4.4), and its asymptotic behavior is givenby16), 17)
φ0,eq − φ0 (t) ∼ e−t/τk ,(5.3)
with a diﬀerent characteristic time
τk =

a2
D

φ0,eq
φb

2

exp[−α(1 + γ)2 − βφ0,eq ].

(5.4)

In general, KLA is realized when there is a potential barrier that might lead tokinetic limitations.
Comparing Eqs. (5.2) and (5.4), we ﬁnd that τd > τk holds when the lateralinteraction is attractive (β > 0). Hence we expect that colloidal particles without
any surface charge exhibit a DLA rather than a KLA. In Fig. 4, τd is represented
by a solid line. We see that the characteristic time τc obtained from the numericalcalculation (open squares) is well-described by the diﬀusive characteristic time τd .
We have also checked that τk is much smaller than τd in the present parameter choice.
In Fig. 3(b), we have shown that the asymptotic behavior of φ0,eq −φ0 obeys the t−1/2
behavior as described by Eq. (5.1). All these results indicate that the adsorption ofcolloidal particles in Pickeirng emulsions is governed by a diﬀusion-limited process.
Since most of the experiments measure surface tension rather than surface coverage, we next look at the dynamical surface tension of an interface adsorbed bycolloidal particles. We remind that the total excess free energy per unit area is given
by Eq. (3.1). For simplicity, we neglect here the contribution from the bulk solution(the ﬁrst term) which is considered to be relatively small. Hence we are left with
only the second term, i.e., the contribution from the interface. In Fig. 5(a), we show
the time evolution of f0 (φ0 ) for various values of γ while the other parameters are
the same with those in Fig. 2. One typically sees a slow decrease in the beginningAdsorption Dynamics in Pickering Emulsions

89

Fig. 5. (a) The time evolution of the surface energy f0 as a function of the scaled time t̃ for diﬀerent
values of γ. The open circles on the right axis indicate the respective equilibrium values f0,eq .
(b) The asymptotic behavior of f0 − f0,eq in the large time regions.

and a rapid drop followed by a slow relaxation toward the equilibrium. The decreaseof the surface energy becomes larger as the value of γ is approached from negative tozero. Notice that the rapid drop of f0 occurs around t ≈ τc ≈ τd . To investigate thelong time asymptotic behavior, we present in Fig. 5(b) the diﬀerence f0 (φ0 ) − f0,eq
for large t. This plot implies that the dynamic surface tension also decreases as
t−1/2 similar to the behavior of φ0 (t) in Fig. 3(b). Such a power-law dependence ischaracteristic for DLA and should be contrasted with an exponential decay expected
for KLA.
In the case of DLA, we mentioned that f0 (φ0 ) is all the time at its minimum sothat μ0 = μ1 holds even if the free energy is dependent on time (μ1 = μb ). When
this is true, the surface energy can be further written as


kB T
β
(5.5)
f0 [φ0 (t)] = 2 ln[1 − φ0 (t)] + [φ0 (t)]2 ,a2
so that f0 is now a functional of φ0 (t). This relation means that the equilibrium
equation of state as previously given by Eq. (3.5) also holds for nonequilibrium
situations.17) In order to check the validity of this argument, we have plotted inFig. 6 both Eqs. (3.3) and (5.5) by the solid and dashed lines, respectively. We
present here only the case of γ = 0 and the other parameters are the same as thosein Fig. 3. We can see that the two curves coincide almost perfectly for t ≥ τd . Hencethe extension of the equation of state to the nonequilibrium cases can be justiﬁed
for DLA in the long time limit.
In Refs. 16) and 17), it was pointed out that the adsorption of nonionic surfactants is described by a DLA. This fact has been experimentally conﬁrmed for various
types of nonionic surfactant which exhibit the asymptotic t−1/2 decrease of the dynamical surface tension. For salt-free ionic surfactants, however, the adsorption is

90

Y. Hirose, S. Komura and T. Kato

Fig. 6. The solid line is the time evolution of the surface energy f0 as a function of the scaled timet̃ for γ = 0 (as presented in Fig. 5). The dashed line is the time evolution of Eq. (5.5). Bothcurves coincide in the large time region.

described by a KLA due to the electrostatic interaction. Interestingly, addition of salt
to an ionic surfactant solution recovers a DLA. For colloidal particles, it is natural
to expect that the adsorption deviates from a DLA when the particles are charged.
Recently, the adsorption dynamics of cadmium selenide nanoparticles at an
oil/water interface has been investigated.15) Using a pendant drop tensiometer, theymeasured changes in the interfacial tension for the early and late stages. The observed time dependence of the interfacial tension is very similar to that in Fig. 5(a),and its asymptotic long time behavior is indeed described by the t−1/2 dependence
as in Eq. (5.1). Hence the experimental result can be described as a DLA.
Finally we shall brieﬂy argue the case when the interaction parameter satisﬁes
β > 4 so that the phase transition occurs at the interface. In Fig. 7(a) and (b),we present the time evolutions of φ0 and f0 , respectively, for diﬀerent values of β.
Other parameters are φb = 10−2 , α = 10, and γ = −0.5. When β > 4, we observe asteep increase of φ0 in the late stages. Corresponding to this behavior, the surface
energy f0 increases transiently and then drops oﬀ toward equilibrium value as shownin Fig. 7(b). Such a unique time dependence of the surface energy is expected to beobserved in the experiment.
§6.

Conclusion

In this paper, we have discussed the adsorption dynamics of colloidal particles
to liquid/liquid interfaces. We used a free energy formulation for the adsorption
which has been applied to surfactants. By solving the coupled equations including
the Ward-Tordai equation, we obtained the time evolution of the surface coverageand dynamic surface tension. From the estimation of the characteristic time, we
showed that the adsorption process is a diﬀusion-limited one. In this situation, theAdsorption Dynamics in Pickering Emulsions

91

Fig. 7. (a) The time evolution of the surface coverage φ0 as a function of t̃ for diﬀerent values ofβ. The other parameters are φb = 10−2 , α = 10, and γ = −0.5. (b) The corresponding time
evolution of the surface energy f0 as a function of t̃.

asymptotic time dependencies of the surface coverage and surface tension close to
their equilibrium values obey the t−1/2 law. Our result is in agreement with a recent
experiment using nanoparticles. We also showed that the equilibrium equation ofstate relating the surface tension to the surface coverage also holds for nonequilibrium
situations.
Several extensions of the present work are under progress. For example, we areinvestigating the adsorption of two diﬀerent types of colloidal particles characterized
by diﬀerent values of γ. One might also consider the case in which the adsorption
occurs from both liquids. In this case, we allow the particles to leave the interface
toward the other liquid.
Acknowledgements
We thank D. Andelman for useful discussion. This work was supported by
KAKENHI (Grant-in-Aid for Scientiﬁc Research) on Priority Area “Soft Matter
Physics” and Grant No. 18540410 from the Ministry of Education, Culture, Sports,
Science and Technology of Japan.
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