Rev


El-Moudny et al – extensive study of cage effect and subdiffusion in Pickering emulsions using molecular dynamics simulations   

 

 

OAJ Materials and Devices, Vol 3 #1, 2404 (2018) – DOI: 10.23647/ca.md20182404 

 

Article type: A-Regular research paper 

 

Extensive study of cage effect and 

subdiffusion in Pickering emulsions using 

molecular dynamics simulations 
S. El-Moudny (1), M. Benhamou (2), M. Badia (3), M. Ossmani (4) 

(1) Physics department, Faculty of sciences, Moulay Ismail University, Morocco, 
elmoudny.soukaina@gmail.com   
(2) Physics department, Faculty of sciences, Moulay Ismail University, Morocco, 
benhamou.mabrouk@gmail.com 
(3) Physics department, Royal air school, Morocco 
(4) ENSAM, Moulay Ismail University, Morocco 
 

Corresponding author : elmoudny.soukaina@gmail.com     

RECEIVED: 30 january 2018 / RECEIVED IN FINAL FORM: 17 april 2018 / ACCEPTED: 24 april 2018 

Abstract : In this work, we aim at an extensive study of the diffusion phenomenon of oil-droplets dispersed in 
water onto which are strongly adsorbed charged point-like particles (Pickering emulsions). This diffusion that 
originates from multiple collisions with the molecules of water, is anomalous, due to the presence of relatively 
strong correlations between the moving oil-droplets. Using Molecular Dynamic simulation, with a pair-potential of 
Sogami-Ise type, we first observe that the random walkers execute a normal diffusion, at intermediate time, 
followed by a slow diffusion (subdiffusion) we attribute to the presence of cages, formed by the nearest neighbors 
(traps). In the cage-regime, we find that the mean-square-displacement increases according to a time-power law, 
with an anomalous diffusion exponent, 𝛼 (between 0 and 1). The existence of a cage effect is shown also by 
computing the velocity auto-correlation function of the random walker. It is found that, in a cage, this function is 
governed by an underdamped (oscillatory) behavior, for strong densities and surface charges, and low-salt 
concentration. In the inverse situation, however, we observe that this correlation-function is rather overdamped 
(non-oscillatory). In the two cases, at large-time, this function fails according to a time-power law, with the 
exponent 𝛼 − 1. 

Keywords : PICKERING EMULSIONS, SOGAMI-ISE POTENTIAL, ANOMALOUS DIFFUSION, CAGE 

EFFECT, MEAN SQUARE DISPLACEMENT 

Introduction 

Pickering emulsions are emulsions that are stabilized by solid 
particles. In fact, the solid particles form a spherical shell and 
impede coalescence when two droplets approach each other. 
Pickering emulsifiers irreversibly adsorb at the oil–water 
interface and require a much higher energy for desorption 
(≈ 106 − 108 𝑘𝐵 𝑇) as compared to the conventional surfactants 
( ≈ 7 𝑘𝐵 𝑇 ). The emulsifier-free character of Pickering 
emulsions makes them attractive regarding applications 

where the surfactants have detrimental effects, in particular, 
when contacted with living matter for health and body care 
applications [1]. 

Pickering emulsions have been the focus of considerable 
research in the past decade due to their properties such as high 
stability with respect to the coalescence, as well as, due to 
advances in nanotechnology that allows us to create and 
characterize the nano-scale structures in new ways. In fact, the 
colloidal assembly of solid particles within Pickering emulsions 
can be used as templates for advanced materials such as Janus 
colloids, composite particles, and colloidosomes. 

mailto:elmoudny.soukaina@gmail.com
mailto:benhamou.mabrouk@gmail.com
mailto:soukaina@gmail


El-Moudny et al – extensive study of cage effect and subdiffusion in Pickering emulsions using molecular dynamics simulations   

 

 

OAJ Materials and Devices, Vol 3 #1, 2404 (2018) – DOI: 10.23647/ca.md20182404 

A Pickering emulsion has a complex structure which presents 
as a dispersion of droplets of a liquid (dispersed phase) in 
another (host liquid). Each droplet is surrounded by discrete 
spherical particles arranged on the surface. If the preparing 
conditions (such as wettability, charge, concentration, shape 
and size of particles, as well as, the pH and salt concentration 
in aqueous phase) are right, the clothed droplets can be 
considered as "soft-colloids". 

Due to the thermal agitation, the clothed droplets experience 
a (lateral) diffusion with the molecules of the continuous 
phase. At the beginning, a single clothed droplet normally 
diffuses, but at large-time, this motion is hampered by the 
presence of the others, and then, the same clothed droplet 
executes rather an anomalous diffusion we are interested in. 
As we shall see below, quantitatively, this anomalous 
phenomenon mainly depends on the pertinent parameters, 
such as the surface charge of the droplets, their number 

density, and the salt-concentration. 

In this work, we quantitatively study the anomalous diffusion 
within Pickering emulsions using the Molecular Dynamics 
(MD) simulations. The dynamic properties are investigated 
through the time-evolution of the mean-square-displacement 
(MSD) combined with the velocity auto-correlation function 
(VACF). For this purpose, we adopted the Sogami-Ise (SI) 
pair-potential [2] between charged clothed droplets. It 
presents as the sum of a repulsive screened Coulomb 
potential and an attractive tail. 

For the dynamics study, our starting point is a proposition of 
a relevant dynamic theory, based on the Langevin equation 
and its generalization (with memory). We then write an 
integro-differential equation for VACF and its relations with 
MSD. The exact solution of this equation, with an appropriate 
choice of the memory-function, enables us to valid the 
obtained results from MD as a computer experiment. 

Pair-potential expression 

Consider a suspension of 𝑁 droplets (soft-colloids) of a liquid 
that are dispersed in another liquid of different chemical 
nature. In this study, we restrict ourselves to oil-in-water 
dispersions, only. The volume of the solution is denoted as 𝑉, 
and the oil-droplet number density as 𝜌 = 𝑁 𝑉⁄ . For simplicity, 
these oil-droplets are assumed to be monodisperse spheres 
of common diameter 𝜎 . The surface of each oil-droplet is 
wetted by 𝑍  irreversibly adsorbed charged particles of small 
diameter in comparison with 𝜎. The strong adsorption of the 
particles rigidifies the surface of the oil-droplets, so they can 
be viewed as soft spherical colloids (as latex particles in 
water, for instance) that carry the same charge 
𝑍𝑒 (macroions). When they are added to the solution, the 
adsorbed particles are ionized and release small ions 
(counterions) in the solution. Generally, the colloids are in 
contact with other free ions resulting from a dissociation of a 
salt or an electrolyte. It is well-known that the surrounding 
mobile ions leads to a screening of the Coulomb forces 
between macroions (oil-droplets). 

Beside the screened Coulomb interactions, the macroions 
experience attractive van der Waals ones, and the 
thermodynamic properties (phase transitions, structure...) of 
the system can be described correctly within the framework 
of the so-called Derjaguin-Landau-Verwey-Overbeek (DLVO) 
theory [3]. In the same context, in his seminal paper [2], I. 
Sogami used a self-consistent theory combined with a 
resolution of the Poisson-Boltzmann equation, satisfied by the 
electric potential created by the macroions (ionized latex 

polyballs), for the determination of their associated effective 
interaction potential. Such a potential involves a short-range 
(screened) Coulomb repulsion, whose origin is self-evident, in 
addition to a long-ranged exponential attractive tail. 

The expression of the SI potential between charged oil-droplets 
reads [2] 

𝑈(𝑟)

𝑘𝐵𝑇
= 𝑙𝐵 [

𝑍𝑠𝑖𝑛ℎ(𝜅𝜎/2)

𝜅𝜎/2
]

2

[
1+(𝜅𝜎/2)coth (𝜅𝜎/2)

𝑟
−

𝜅

2
] 𝑒 −𝜅𝑟                      (1)  

         

for  𝑟 > 𝜎 . In this expression, the screening parameter 𝜅  is 
defined by the standard relation 

                           𝜅2 = 4𝜋𝑙𝐵 ∑ 𝜌𝑖 𝑧𝑖
2

𝑖
                               (2) 

Here, 𝑙𝐵 = 𝑒
2/ 𝑘𝐵 𝑇  denotes the Bjerrum length, which is a 

characteristic length-scale at which the electrostatic interaction 
between a pair of monovalent ions has magnitude 𝑘𝐵 𝑇. The length 
𝑙𝐵 has the value 𝑙𝐵 = 0.7 𝑛𝑚, for water, at room temperature (this 
length is kept fixed to this value). Throughout this work, the 
temperature 𝑇 will be fixed to the value 298 𝐾 (room temperature). 
In the presence of a salt of concentration, 𝐶𝑠, the above formula 
becomes 

𝜅2 = 4𝜋𝑙𝐵 (𝜌𝑍
2 + 𝐶𝑠 )                       (3)  

 

Figure 1 shows the variation of the reduced SI pair-potential 𝑈(𝑟)/
𝑘𝐵 𝑇, with the dimensionless distance, 𝑟/𝜎, for various values of the 
salt-concentration. These curves are drawn with parameters: 𝜌∗ =

0.002, 𝑍 = 500 and 𝜎 = 5000 Å. 

1,0 1,2 1,4 1,6 1,8 2,0 2,2

-0,4

0,0

0,4

0,8

r / 

U(r) / k
B
T

 

 

 C
S
=0.29 M

 C
S
=1.79 M

 C
S
=3.45 M

 C
S
=5.11 M

 

Figure 1: Dimensionless SI pair-potential, 𝑼(𝒓)/𝒌𝑩𝑻 , versus 

the dimensionless distance. 

 

Finally, we note that, using this potential, Tata and coworkers [4] 
performed Monte Carlo (MC) and Brownian Dynamics simulations, 
and the obtained results agree well with certain experimental 
observations [5,6]. Also Kepler and Fraden [7] determined the pair-
potential of the colloidal particles from measurements of the pair 
correlation-function of both dilute and moderately concentrated 
dispersions. Thus, the determined pair-potential can be 
reproduced by SI potential assuming 𝐶𝑠 = 1.75 × 10²¹ 𝑚⁻³ , as 
pointed out by Tata and Arora [8]. The authors argued that the 
results by Kepler and Fraden supported the counterion-mediated 
attraction discussed by Sogami. 



El-Moudny et al – extensive study of cage effect and subdiffusion in Pickering emulsions using molecular dynamics simulations   

 

 

OAJ Materials and Devices, Vol 3 #1, 2404 (2018) – DOI: 10.23647/ca.md20182404 

Results and discussions 

Theory: 
Normal diffusion: Consider a given clothed oil-droplet, 
termed tracer, which experiences a normal diffusion, for early 
times, because of the absence of the correlations. At these 
small-time-scales, the random walker is not yet trapped in a 
cage. As we shall below, such a regime will be put in evidence 
by our computer simulations, for smaller times. 
The Brownian motion of the tracer can be described by the 
following phenomenological Langevin equation 

 

𝑀
𝑑𝒗(𝑡)

𝑑𝑡
= − 𝒗(𝑡) + 𝑭𝑠 (𝑡)                       (4) 

The friction coefficient, , is related to the viscosity of the 

emulsion, , and the oil-droplet radius, 𝑅, by the classical 
Stokes relation: = 6𝜋 𝑅 . Of course, the viscosity of the 
solution depends on the number density of the dispersed oil-
droplets, 𝜌.  
Come back to the stochastic force and note that this is 
considered to be a white noise with 

 

 〈𝑭𝑠 (𝑡)〉 = 0                                 (5) 
 

    〈𝑭𝒔(𝑡). 𝑭𝒔(0)〉 = 6𝑘𝐵 𝑇 𝛿(𝑡)                        (6) 
    

〈𝒗(0). 𝑭𝒔(𝑡)〉 = 0                               (7) 
 

Now, we are interested in two related physical quantities: 
velocity auto-correlation function (VACF), 〈𝒗(𝑡). 𝒗(0)〉 ≡ 𝑐𝑣𝑣 (𝑡), 
and mean-square-displacement (MSD), 〈[𝒓(𝑡) − 𝒓(0)]²〉 ≡ 𝑊(𝑡).  
VACF solves the following differential equation [9] 

 
𝑑𝑐𝑣𝑣(𝑡)

𝑑𝑡
= −𝛾𝑐𝑣𝑣 (𝑡)                          (8) 

 

Its solution is simply 
 

𝑐𝑣𝑣 (𝑡) = 〈𝒗²〉𝑒
−𝛾𝑡 ,      〈𝒗²〉 = 𝑐𝑣𝑣 (0) = 3

𝑘𝐵𝑇

𝑀
             (9) 

 
with the mass weighted friction constant 𝛾 = /𝑀 , called 
relaxation rate. 
On the other hand, VACF and MSD are related by [9] 

 

𝑊(𝑡) = ∫ 𝑑𝑡1 ∫ 𝑑𝑡2𝑐𝑣𝑣(𝑡1, 𝑡2)
𝑡

0

𝑡

0
= 2 ∫ 𝑑𝑡 ′

𝑡

0
(𝑡 − 𝑡 ′)𝑐𝑣𝑣(𝑡′)     (10) 

 
Replacing VACF by its explicit expression (9) in Eq. (10) yields 
the following time-evolution of MSD [9] 
 

𝑊(𝑡) = 6
𝑘𝐵𝑇

𝑀
(

𝑒−𝛾𝑡−1+𝛾𝑡

𝛾²
)                          (11) 

 
For times much longer than the inverse relaxation rate, that 
is for 𝑡 ≫ 𝛾 −1 ≡ 𝑡∗, MSD grows linearly with time, and we have 
 

𝑊(𝑡) = 6𝐷𝑡,                𝑡 ≫ 𝑡 ∗                       (12) 
 

The linear growth of MSD with time is then reached beyond 
the characteristic time  

𝑡 ∗ =
𝑀

6𝜋𝑅𝜂(𝜌)
                                  (13) 

 

This time limit depends only on oil-droplet density through the 
viscosity (𝜌). If, in contrast, 𝑡 << 𝑡∗, we can approximate 𝑒 −𝛾𝑡 
by 1 − 𝛾𝑡 + 𝛾²𝑡²/2, and find 
 

𝑊(𝑡) = 〈𝒗²〉𝑡 2 = 3
𝑘𝐵𝑇

𝑀
𝑡²,         𝑡 ≪ 𝑡 ∗                  (14) 

 
which shows that MSD grows as 𝑡² (ballistic motion).  
 
Anormal diffusion: Now, the raised question is how the tracer 
moves beyond the relaxation rate, 𝑡∗. In this time domain, the 
tracer (target) feels to be trapped in a cage formed by others 
clothed oil-droplets (traps), and cannot escape from this cage 
only after a long-time. As consequence, the presence of the 
traps makes difficult such a diffusion process, and then, the 

random walker executes rather a subdiffusion, characterized by 
an exponent denoted as 𝛼. 
To discuss the cage effect and the subdiffusion laws, the starting 
point is a generalized Langevin equation [9], 

 
𝑑𝒗(𝑡)

𝑑𝑡
= −𝛾𝒗(𝑡) − ∫ 𝑑𝑡 ′𝜅(𝑡 − 𝑡 ′)𝒗(𝑡 ′) + 𝑭𝑠 (𝑡) 

𝑡

0
          (15) 

 
where 𝜅  is the memory-function that expresses the friction 
retardation. We note that this random force 𝐹𝑠 (𝑡) satisfies 
equality (5) and (7), but with the second moment 
(generalization of Eq. (6)) 
 

〈𝑭𝒔(𝑡). 𝑭𝒔(0)〉 = 6𝑀𝑘𝐵 𝑇[𝛾𝛿(𝑡) + 𝜅(𝑡)] ,         𝑡 > 0           (16) 
 

Oscillating regime: In this regime, the random walker is subject 
to oscillations before it undergoes a subdiffusion process. 
Therefore, the tracer moves in a harmonic potential resulting 
from its interactions with the surrounding nearest neighbors. 
This equivalent to take for the memory-function the following 
form 

𝜅(𝑡) = 𝜔0
2 (𝑡)                          (17) 

 
The frequency 𝜔₀ must be considered as a phenomenological 
parameter that depends, of course, on the essential parameters 
of the problem. 
In these conditions, the integro-differential equation (15) writes 
 

𝑑𝑐𝑣𝑣(𝑡)

𝑑𝑡
= −𝛾𝑐𝑣𝑣 (𝑡) − 𝜔0

2 ∫ 𝑑𝑡 ′𝑐𝑣𝑣 (𝑡
′) .

𝑡

0
            (18) 

Using Laplace transform, we find that VACF presents three 

regimes:  

• The underdamped regime (𝛾 < 2𝜔0): 
 

𝑐𝑣𝑣 (𝑡) = 〈𝑣²〉𝑒
−𝛾𝑡/2 {𝑐𝑜𝑠(𝜔 ̃0𝑡) −

𝛾

2𝜔 ̃0
𝑠𝑖𝑛(𝜔 ̃0𝑡)}                  (19) 

 

with the notation: 𝜔 ̃0 =  √𝜔0
2 −

𝛾2

4
 . 

 
• The overdamped regime (𝛾 > 2𝜔0):  

 

 𝑐𝑣𝑣 (𝑡) = 〈𝑣²〉𝑒
−𝛾𝑡/2 {𝑐𝑜𝑠ℎ(𝜔 ̂0𝑡) −

𝛾

2𝜔 ̂0
𝑠𝑖𝑛ℎ(𝜔 ̂0𝑡)}           (20) 

 

with the notation: 𝜔 ̂0 = √
𝛾2

4
− 𝜔0

2  .   

 
• The critical regime (𝛾 = 2𝜔0):  

  

𝑐𝑣𝑣 (𝑡) = 〈𝑣²〉𝑒
−𝛾𝑡/2 (1 −

𝛾𝑡

2
)                    (21) 



El-Moudny et al – extensive study of cage effect and subdiffusion in Pickering emulsions using molecular dynamics simulations   

 

 

OAJ Materials and Devices, Vol 3 #1, 2404 (2018) – DOI: 10.23647/ca.md20182404 

 
Such a regime separates the oscillating regime and the non-
oscillating one. 
 
Using the relation (10) between MSD and VACF, we find that 
MSD also presents three regimes: 
 

• The underdamped regime ( 𝛾 < 2𝜔0): 
 

𝑊(𝑡) = 6
𝑘𝐵𝑇

𝑀𝜔0
2 [1 − 𝑒

−
𝛾𝑡

2 {𝑐𝑜𝑠(�̃�0𝑡) +
𝛾

2�̃�0
𝑠𝑖𝑛(�̃�0𝑡)}]    (22) 

 
• The overdamped regime ( 𝛾 > 2𝜔0): 

 

𝑊(𝑡) = 6
𝑘𝐵𝑇

𝑀𝜔0
2 [1 − 𝑒

−
𝛾𝑡

2 {𝑐𝑜𝑠ℎ(�̂�0𝑡) +
𝛾

2𝜔 ̂0
𝑠𝑖𝑛ℎ(�̂�0𝑡)}]       (23) 

 
• The critical regime ( 𝛾 = 2𝜔0): 

        

𝑊(𝑡) = 24
𝑘𝐵𝑇

𝑀𝛾²
[1 − 𝑒 −𝛾𝑡/2 (1 +

𝛾𝑡

2
)]                                       (24) 

 
In contrary to the freely diffusing Brownian particle, MSD 
approaches a plateau value in the limit 𝑡 → ∞, independently 
of the dynamic regime, 

 

𝑊(𝑡) = 6
𝑘𝐵𝑇

𝑀𝜔0
2  ,                           𝑡 ≫ 2𝛾

−1 ≡ 𝑡 ∗∗            (25) 

 
Then, this corresponds to the plateau regime where 𝛼𝑝 = 0.  

 
Subdiffusive regime: Beyond the plateau regime, the random 
walker executes a slow motion, and the corresponding MSD 
obeys the law 

𝑊(𝑡) = 〈[𝒓(𝑡) − 𝒓(0)]²〉 = 2𝐷𝛼 𝑡
𝛼 ,    (0 < 𝛼 < 1)        (26) 

 
The normal diffusion with 𝑊(𝑡) ∼ 𝑡 (large-time) corresponds 
to a vanishing VACF, that is 𝑐𝑣𝑣 (𝑡) = 0 (large-time), whereas 
𝑐𝑣𝑣 (𝑡) ≠ 0 in the case of an anomalous diffusion. Specifically, 
𝑐𝑣𝑣 (𝑡) < 0 corresponds to the subdiffusion (since 𝛼 < 1), and 
𝑐𝑣𝑣 (𝑡) > 0 to the superdiffusion (since 𝛼 > 1).  
We note that for the determination of the diffusion exponent, 
𝛼, we shall use relation (26), expressed in log-log scale. This 
exponent is 1, for normal diffusion regime, 0, for the plateau 
regime, and between 0 and 1, for the subdiffusive one. 
To sum up, we say that we have now all necessary theoretical 
ingredients for the explanation of the results from our MD 
simulations. This is the aim of the next section. 

Results from MD simulation: 

Spirit of MD simulation: Within the framework of MD 
method for the description of the dynamic properties of 
Pickering emulsions, the equations of motion are solved in the 
canonical ensemble using the velocity Verlet algorithm (VA) 
[10] with the thermostat of Berendsen [11], in order to keep the 
temperature constant. Periodic boundary conditions are 
applied to remove the surface effects and simulate an infinite 
system. In the following, we will use dimensionless units, 

where the length unit is 𝜎, time in units of 𝜏 = 𝜎√(𝑀/𝜖), 𝑘𝐵 𝑇 is 

the energy unit, and 𝐿₀ = 𝑁 × (6𝑉/𝜋𝑁)1/3 is the box-size, where 
𝑉 is the volume of simulation box (in periodic conditions). MD 
simulations where carried out with 1728  particles and the 
dimensionless time-step for the velocity VA is chosen to be 
0.05. 
 
Surface charge effects: In this paragraph, we will look at 
the influence of the surface charge of the clothed oil-droplets 

on their dynamic properties. For the study, the size of the oil-
droplets, their reduced number density and the salt-

concentration are fixed to the values: 𝜎 = 20000 Å, 𝜌∗ = 0.0020 
and 𝐶𝑠 = 2.91 𝜇𝑀. Their valence 𝑍 is ranged from 1000 to 4500. 
Figure 2  shows the log-log plot of the reduced MSD upon 
dimensionless time, 𝑡/𝜏 , for various values of the oil-droplet 
surface charge. We first remark that, for times less than a 
(dimensionless) transition time 𝑡₁ ≃ 2.483, which is independent 
of the surface charge, the curves for different surface charges 
are superposed, and the random walker experiences a normal 
diffusion, that is 𝑊(𝑡) = 6𝐷𝑡 , The reduced normal diffusion 
coefficient is 𝐷 = 14.120 . This simulated normal regime is in 
perfect agreement with the theoretical predictions relatively to 
the normal diffusion regime discussed above, eq. (12). 
The existence of the normal diffusive regime can be understood 
as follows. In the intermediate times, the random walkers (oil-
droplets) do not feel to be correlated, whatever are the values 
of their surface charge. For larger times, however, the situation 
is quite different, and the dynamic of the oil-droplets depends 
heavily on the surface charge, and MSD behaves rather as 
𝑊(𝑡) ∼ 𝑡𝛼, with an average subdiffusion exponent 𝛼𝑐 = 0.418 (the 
subscript c is for charge). 

In agreement with the theory described above, we remark that 
the normal diffusion is followed by a transient plateau-like 
regime, which precedes the subdiffusion one. The existence of 
this plateau regime is conform with the theoretical eq. (25). This 
implies the formation of cages, where a given clothed oil-droplet 
is surrounded by their nearest neighbors. As shown in figure 2, 
the plateau is more and more pronounced, for strong surface 
charges. We also remark that the crossover time between the 
plateau-like regime and the subdiffusion one, 𝑡₂, depends on the 
value of the surface charge. This means that the moving oil-
droplet needs more time to break free of the cage of its nearest 
neighbors, as the surface charge increases. Physically, for 
higher surface charges, the available space for the oil-droplets 
becomes effectively reduced by the volume-excluding, because 
the repulsion becomes more important. This leads to a slowing 
down of the diffusion process of the oil-droplets, which stay 
localized for some time that increases with increasing surface 
charge. This behavior is reflected in VACFs displayed in figure 3, 
for the same values of the oil-droplet surface charge. VACFs are 
characterized by an underdamped (oscillatory) decay, more 
pronounced, for higher surface charges, in perfect agreement 
with the theoretical eq. (19). This underdamped feature can be 
related to the confinement of the particle in the cage formed by 
its nearest neighbors, and means that the diffusion process is 
not Markovian, but rather has memory [9]. For low-surface 
charges, however, we see that VACF is rather an overdamped 
function, which is conform with the theoretical expression (20). 

We also observe that, for any value of the surface charge, the 
tail or the long-time behavior of VACFs, which reflects the 
diffusional regime of the oil-droplets under consideration, 
approaches asymptotically zero, from negative values. 
Therefore, the oil-droplets display a subdiffusion. This negative 
region indicates that, on average, a displacement of the oil-
droplets toward its nearest neighbors is followed by a 
displacement back toward its initial position. In other words, it 
reflects the fact that the velocity of the oil-droplets is, on 
average, reversed by repulsion with the cage of nearest 
neighbors. 



El-Moudny et al – extensive study of cage effect and subdiffusion in Pickering emulsions using molecular dynamics simulations   

 

 

OAJ Materials and Devices, Vol 3 #1, 2404 (2018) – DOI: 10.23647/ca.md20182404 

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

 

 

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log(Time)

 Z=1000

 Z=1500

 Z=2000

 Z=2500

 Z=3000

 Z=3500

 Z=4000

 Z=4500

 
Figure 2 : MSD versus the dimensionless time, 𝒕/𝝉 , in 

log-log scale, for various values of surface charge 

0 2 4 6 8 10 12 14
-10

-5

0

5

10

15

20

 

 

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 F

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ti

o
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Time

 Z=1000

 Z=1500

 Z=2000

 Z=2500

 Z=3000

 Z=3500

 Z=4000

 Z=4500

 
Figure 3: VACF versus the dimensionless time, 𝒕/𝝉, for 

various values of surface charge 

 

Density effects: To study the influence of the density of the 
clothed oil-droplets on their dynamics, we keep fixed their 
surface charge and size, and the salt-concentration to the 

values: 𝑍 = 2000, 𝜎 = 20000 Å and 𝐶∗ = 2.91 𝜇𝑀, and vary this 
density.  
In figure 5, we depict the log-log plot of MSD against the 
dimensionless time, 𝑡/𝜏 , for various values of the oil-droplet 
density. We first remark that for 𝑡 < 𝑡₁′ , with 𝑡₁′ = 1.449 
(reduced transition time), MSDs are straight lines of the same 
slope, but decrease as the oil-droplet density is increased, in 
perfect agreement with the theoretical formula (13). Then, 
the random walker follows a normal Brownian diffusion, 
whose MSD is 𝑊(𝑡) = 6𝐷(𝜌)𝑡, for 𝑡 < 𝑡₁′. Here 𝐷 is the usual 
diffusion constant. 
Second, for 𝑡 > 𝑡₁′, we assist to a cage effect, where MSDs are 
very sensitive to the variation of the oil-droplet density, and 
exhibit a subdiffusive behavior, with an average subdiffusion 
exponent 𝛼𝑑 = 0.432 (the subscript d is for density). Before, 
MSDs exhibit a plateau regime that becomes more and more 
pronounced. The existence of such a regime agrees well with 
the theoretical formula (25). 

The subdiffusive behavior evoked above is clearly shown in 
figure 6 representing the variation of VACFs upon the reduced 
time, 𝑡/𝜏, for different values of the oil-droplet density. From 
these plots, we see that the negative region of VACFs 
becomes larger as the oil-droplet density increases. In 
particular, we observe that VACF is underdamped (oscillatory) 

for stronger densities, and overdamped for lower ones, in 
perfect agreement with theoretical formulae (19) and (20), 
respectively. In addition, the same plots indicate that VACFs tails 
approach asymptotically zero, from negative values. This means 
that the oil-droplets undergo a subdiffusion. 

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

 

 

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log(Time)

 

=0.0010

 

=0.0015

 

=0.0020

 

=0.0025

 

=0.0030

 

=0.0032

 
Figure 4: MSD versus the dimensionless time, 𝒕/𝝉 , in log-

log scale, for various values of the oil-droplet density 

0 2 4 6 8 10 12 14
-10

-5

0

5

10

15

20

 

 

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Time

 
*
= 0.0010

 
*
= 0.0015

 
*
= 0.0020

 
*
= 0.0025

 
*
= 0.0030

 
*
= 0.0032

 
Figure 5: VACF versus the dimensionless time, 𝒕/𝝉, for 

various values of the oil-droplet density 

  

Salt-concentration effects: Now, we fix the surface charge, 
size and reduced number density of the oil-droplets to the 

values: 𝑍 = 500 , 𝜎 = 5000 Å  and 𝜌∗ = 0.0020 , and vary 
progressively the salt-concentration, 𝐶𝑠. The aim is to quantify 
the effects of this concentration on the diffusion process 
executed by a given clothed oil-droplet. 

We represent in figure 8 the log-log plot of MSD upon the 
dimensionless time, 𝑡/𝜏  , for various values of the salt-
concentration. We first observe that, for 𝑡 < 𝑡₁′′, with 𝑡₁′′ = 1.439 
(reduced transition time), MSDs increase linearly with time, and 
have practically the same slope, whatever is the value of the 
salt-concentration. This means that the tracer experiences a 
normal Brownian diffusion, whose MSD is 𝑊(𝑡) = 6𝐷𝑡, for 𝑡 < 𝑡₁′′, 
with the (reduced) normal diffusion coefficient 𝐷 = 14.186, in 
perfect agreement with the theoretical equation (13). 

Second, for larger times ( 𝑡 > 𝑡₁′′ ), MSDs become very 
sensitive to the variation of the salt-concentration, and present 
a subdiffusive behavior, due to the cage effects, with an average 
subdiffusion exponent 𝛼𝑠 = 0.467  (the subscript s is for salt). 



El-Moudny et al – extensive study of cage effect and subdiffusion in Pickering emulsions using molecular dynamics simulations   

 

 

OAJ Materials and Devices, Vol 3 #1, 2404 (2018) – DOI: 10.23647/ca.md20182404 

Before this regime is attenuated, MSDs exhibit a plateau, 
which is more and more pronounced, rather for low salt-
concentrations. The existence of such a regime is in perfect 
agreement with the theoretical formula (25). 

The subdiffusive character of the random walker is also 
shown in figure 9 that represents VACFs versus the reduced 
time, 𝑡/𝜏 , for many values of the salt-concentration. These 
curves are underdamped (oscillatory), for low salt-
concentrations, and overdamped (non-oscillatory), for high 
salt-concentrations, before they reach a negative region of 
VACFs that becomes more and more pronounced, as the salt-
concentration is decreased. The existence of underdamped 
and overdamped behaviors is conform with the theoretical 
formulae (19) and (20). Thus, the cage effect is more 
important, decreases with increasing salt-concentration 
(mutual interactions between charged oil-droplets are 
drastically diminished).  Also, the curves show that VACFs 
tails approach asymptotically zero, from negative values. This 
indicates that the oil-droplets experience a subdiffusion. 

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0
 C

s
=0.470 

 C
s
=0.595 

 C
s
=0.735 

 C
s
=1.058 

 C
s
=1.440 

 C
s
=1.881 

 C
s
=2.381 

 

 

lo
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log (Time)
 

Figure 6: MSD versus the dimensionless time, 𝒕/𝝉 , in 

log-log scale, for various values of the salt concentration 

0 2 4 6 8 10 12 14
-10

-5

0

5

10

15

20

 

 

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Time

 C
s
=0.470 

 C
s
=0.595 

 C
s
=0.735 

 C
s
=1.058 

 C
s
=1.440 

 C
s
=1.881 

 C
s
=2.381 

 
Figure 7: VACF versus the dimensionless time, 𝒕/𝝉, for 

various values of the salt concentration 

 

Conclusion 

In this work, we focus on a dynamic problem that is related to 

the so-called Pickering emulsions. we are interested in how the 

stabilization process is accomplished in time. During this 

process, the clothed oil-droplets execute a diffusion motion we 

are interested in.  Contrary to the diffusion in homogeneous 

media, we find that, using MD simulations, the oil-droplets 

experience rather a slow motion, due to the complex structure 

of the Pickering emulsions. The origin of this anomalous 

diffusion, observed in many areas of science, is the existence of 

cages, made of nearest neighbors (traps) surrounding the 

random walker (target). To valid our MD simulation data, we 

proposed a memory diffusion theory that is based essentially on 

a generalized Langevin equation. We found that the results from 

MD simulations agree well with the predictions of this theory. 

 

REFERENCES 

1. B.P Binks, Curr Opin in Colloid & Interface Science, vol. 7, p 2 (2002) 
2. I. Sogami, Phys. Lett. A, vol 96, p 199 (1983) 
3. B.V.  Derjaguin, Trans. Faraday Soc., vol 36, p 203 (1940) 
4. B.V.R.  Tata, A.K, Sood, R. Kesavamoorthy, Pramana-J. Phys., vol 34, p 23 (1990) 
5. B.V.R. Tata, R. Kesavamoorthy, A.K. Sood, Mol. Phys., vol 61, p 943 (1987) 
6. A.K.  Arora, B.V.R.Tata, A.K. Sood, R.  Kesavamoorthy, Phys. Rev. Lett., vol 60, p 2438 (1988) 
7. G.M. Kepler, S. Fraden, Phys. Rev. Lett., vol 73 p. 356 (1994) 
8. B.V.R.  Tata,  A.K . Arora, Phys. Rev. Lett., vol 75, p 3200 (1995) 
9. R. Kubo, M.Toda, N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, Vol. 31, Springer Science 

& Business Media (2012) 

10. M.P.  Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford university Press (1989) 

11. H.J. Berendsen, J.P.M. Postma, W.F.  van Gunsteren, A.R.H.J DiNola, J.R Haa, J. Chem. Phys, vol 81, p 3684 (1984) 
 

 



El-Moudny et al – extensive study of cage effect and subdiffusion in Pickering emulsions using molecular dynamics simulations   

 

 

OAJ Materials and Devices, Vol 3 #1, 2404 (2018) – DOI: 10.23647/ca.md20182404 

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