Effects of the Inner Droplet of Double Emulsions on the Film Drainage
during a Head-on Collision ― Supplementary Materials
Section 1: The boundary element method
The governing equations of the flow system as shown in Figure 1b in the paper are the Stokes
equations and the continuity equation
  σ= -P  μ2 u=0
(S1)
 u=0
(S2)
where σ is the stress tensor, P is the dynamic pressure combining the pressure and gravitational
terms, u is the velocity of the continuous phase (CP), and μ is the viscosity of CP. These equations
can be applied not only for the external continuous phase but also for all droplets of the multiple
emulsions. Nonetheless, physical parameters in these equations must be replaced by the
corresponding parameters of those droplets.
The velocity u and surface stress f=·n on Sij satisfy the boundary conditions (BCs)
ui,j  ui,jijmom ,
(S3)
f i,j  f i,jijmom  f i,j
  i,j    n  n   ijmom   i,j    g  x  n ,
(i=1,2, , n;
(S4)
j=1,2, , m i )
where g is the gravity acceleration, γ is the interfacial tension, and the subscripts i and j indicate
the j-th droplet of the i-th layer. The superscript “ijmom” indicates the mother droplet of the ij-th
droplet (when i=1, “ijmom” indicates CP). fi,jijmom is the surface stress of the mother droplet side
of the surface of the j-th droplet of the i-th layer; and fi,j is the surface stress of the internal-phase
side of the surface of the j-th droplet of the i-th layer. fi,jijmom and fi,j are different, which is mainly
due to the surface curvature. x is the position vector of each discrete point on the boundary. Eq.
(S4) describes the stress difference across the surface of the j-th droplet of the i-th layer. At walls
(S0) of the cross slot, the nonslip boundary condition gives
u0  0 .
(S5)
At the inlets and outlets (S0) of the cross slot, the undisturbed flows are specified as the parabolic
pressure-driven flows. The velocity profile is
u0  G
2
R0   R   ,
1     n
2   R0  


(S6)
where G is the shear rate at the wall of inlets or outlets; positive sign is for inlets and negative sign
is for outlets. The average velocity of continuous phase at inlets is U = R0G/3, which is equivalent
to capillary number Ca = μU/γ when μ and λ are fixed. The velocity at a point x0 on the surface Sij
and outer boundaries S0 can be described by the generalized boundary integral equation (GBIE).
LHS    [ S  f  T  u  n]dS
S0
m1
  [ S  f1,j  (1  1,j ) T  u  n]dS
j=1
n
S1,j
mi
  [ S  f i,j  ( ijmom  i,j ) T  u  n]dS
i=2 j=1
Si,j
,
(S7)
where LHS is given by
2 u( x0 )


2 (1  1,j ) u( x0 )
LHS  

2 ( ijmom  i,j ) u( x0 )

(i  2, 3,       n  1, n
j = 1,2, 3,       mi  1, mi )
x0  S0
x0  S1,j
x0  Si,j ,
(S8)
where n, m1 and mi can be any integer. S is the fundamental solution of Stokes equations and T is
the associated stress kernel. The interfaces of the cross-slot and droplets are discretized by the
spectral elements method.
In order to determine the evolution of the droplet shape, an explicit time-integration algorithm
is employed to solve the kinematic condition at the interface. The fourth-order Runge-Kutta
method are employed to lower the numerical error associated with the time integration.
The half width r0 of the cross-slot inlets which equals the diameter of the globule is used as the
length scale. When the volume flow rate Q=2r02/3, the shear rate is unit, which is selected as the
scale G0; thus, the time is scaled with the flow time scale G0-1; the scale of the flow rate is r0G0;
the scale of the volume flow rate will be r02G0. The viscosity of CP is the viscosity scale μ0, the
scale of the interface tension is μ0r0G0 and the scale of the pressure is μ0G0.
Since the physical parameters λij, γij and κij are all disrelated for different droplets, each droplet
can contain distinct components and have its own properties. However, for the common situation,
i.e., only water and one kind of oil involved, although the internal structure can still be of any
pattern, the parameters are simplified to only one λ, one γ and one κ. When effects of gravities and
buoyancies are neglected, density differences in Eq. S4 vanish. In the calculations of our paper,
these simplifications are applied.
Furthermore, although an amount of surfactants are added to keep the inner droplets stable in
the real preparation of multiple emulsions, there is no surfactant involved in the calculations in the
paper for simplification.
Section 2: The collision of simple droplets
Figure S1. The entire pressure fields and the flow fields near the contact region at different time
for the collision of simple droplets.
At time t=1.61, the droplet is in the early stage of the approach when the inner flows move
forward and turn outwards near the front interface due to the high pressure in the contact region.
Meanwhile, the fluid in the contact region is discharged out of the film due to the high pressure
around the film center. At time t=2.685, only the inner flows very close to the front interface move
forward and turn outwards. Out of this narrow region, a new circulation in which the flows move
inwards and then turn upwards near the centerline is formed. At this moment, the fluid in the film
is still squeezed out of the contact region. Then, at t=4.025, the inner circulation in MD is almost
fully developed. As the flows near the front and around the centerline begin to move inwards, a
critical state of the film is formed. In this state, the fluid near the film center will not be discharged,
but moves inwards. Thus, a dimple near the centerline begins to grow. After that (t=7.75 and 10),
under the driving of the fully-developed inner circulation, the fluid is pumped into the film
reversely although the pressure at the film center is still high, and the dimple enlarges gradually.
Section 3: The effect of the viscosity ratio λ
Figure S2. The effects of the viscosity ratio λ on the film drainage at χ=0.20 and Φ=16%, (a) on
hcent and (b) on hmin.
The effects of the viscosity ratio λ on the film drainage are shown in Fig. S2. It is obvious that the
shift of λ does not change the overall pattern of the curves for hcent and hmin. Thus, although the
variation of λ will vary the strength of the inner circulation, the pattern of the inner circulation will
not change, and all the film drainages for the collision of double emulsions at various viscosity
ratios have the same three stages (drainage, drainage halt and second drainage). Certainly, as the
droplet gets harder to deform at the smaller λ, the two globules of double emulsions could
approach each other closer, and the corresponding values of hcent and hmin are lower as shown in
Fig. S2.
Section 4: The effect of the size ratio Φ
Figure S3. The effects of the size ratio Φ of ID on the film drainage at λ=1.5 and χ=0.60, (a) on
hcent and (b) on hmin. The stars represent the separation points of hmin and hcent。
The effects of the size ratio Φ of ID on the film drainage are shown in Fig. S3. The curves of hcent
for various size ratio
Φ
of ID are shown in Fig. S3a. It could be seen that the curves have
approximately the same pattern (declining to the lowest point first, increasing slightly to the
highest point and then decreasing gradually) although the size ratio Φ changes from 9% to 25%.
Thus, it could be asserted that the film drainages for the collision of double emulsions with
various size ratios all have three same stages (drainage, drainage halt and second drainage).
Moreover, it is also obvious that the inner droplet with a larger size will have stronger effects on
the film drainage for the collision of double emulsions.