The experimental observation and modelling of film thinning and film retraction
during the interfacial coalescence of biodiesel and glycerol droplets
Electronic supplementary material
Method for measuring droplet dimensions
Supplementary Fig. 1 shows images of a biodiesel droplet and a glycerol droplet, annotated to
illustrate the method for measuring the droplets dimensions. The circles overlaid onto the
droplets demonstrate the method for estimating film area as that of a spherical cap with radius
of curvature rc. The height of the spherical cap was taken to be c, half the height of the
droplet. The width, 2b, of each droplet was estimated as shown in the images; an accurate
measurement of 2b was not possible because the meniscus obscured the edge of the droplet.
Supplementary Fig. 1: Photographs of (a) a rising biodiesel droplet and (b) a falling glycerol
droplet, both on a biodiesel/glycerol interface.
Method for determining the rim velocity during film retraction
Supplementary Fig. 2 shows two frames of a biodiesel droplet during film retraction, 0.004 s
apart. At the interface the droplets were elliptical, so calculating the distance the rim travelled
between frames was not straightforward. The calculation was simplified by overlaying a
circle of radius rc onto the droplet, where rc was the radius of curvature of the film, (see
Supplementary Fig. 2). The distance travelled by the rim was then the arc length L, which
was obtained by measuring the coordinates x1 and x2 and using Eq. (34):
𝐿 = 𝑟[arccos(𝑥1 /𝑟𝑐 ) − arccos⁡(𝑥2 /𝑟𝑐 )]
(34)
The velocity of the rim was thus L/0.004. Rim velocities were calculated for several
successive frames for each droplet.
Supplementary Fig. 2: Images of a biodiesel droplet at a biodiesel/glycerol interface after the
rupture of the film. The rim of the film can be seen as a dark ring. The time interval between
(a) and (b) was 0.004 s, when the rim travelled distance L.
This approach was suitable for biodiesel droplets, which tended to rupture near the top.
However, glycerol droplets often ruptured on the side, (see Supplementary Fig. 3). The rim
velocity was then determined from the distance travelled in the x direction. A plan view of
the droplet in the plane of z = 0, which intersects the expanding hole, is shown in
Supplementary Fig. 4. It is assumed that the rim followed a circular path as it retracted
around the droplet. From Supplementary Fig. 4, it can be seen that the arc length L travelled
by the rim between frames is given by:
𝐿 = 𝑟[arccos(𝑥2 /𝑟ℎ ) − arccos⁡(𝑥1 /𝑟ℎ )]
(35)
where rh is the droplet radius in the plane of z = 0, as shown in Supplementary Fig. 4. The x
positions of the rim were recorded for frames 0.04 s apart, so that the velocities were L/0.04.
Supplementary Fig. 3: Images of a glycerol droplet at a biodiesel/glycerol interface after the
rupture of the film. The rim of the film can be seen as a dark ring.
Supplementary Fig. 4: View from above the glycerol droplet in Supplementary Fig. 3 at the
plane of z = 0.
Supplementary data
Tables 1 to 6 provide experimental data used in the film thinning and film retraction analysis.
Table 1: Viscosities and densities of biodiesel and glycerol
Viscosity
(mPa.s)
Density
(kg/m3)
Biodiesel
6.5
880
Glycerol
53
1090
Table 2: Measured dimensions of biodiesel and glycerol droplets. The errors reflect the limits
of the measurement accuracy, which was due to the definition of the photographs or
obscuration by the meniscus. Standard deviations reflect the spread of the measurements.
Biodiesel droplet
Dimension
Standard
deviation
Mean
Number of droplets
Glycerol droplet
Standard
deviation
Mean
37
26
Droplet width, 2b (mm)
1.5 ± 0.1
0.15
2.4 ± 0.1
0.88
Droplet height, 2c (mm)
1.10 ± 0.06
0.08
1.42 ± 0.06
0.36
Radius of curvature, rc (mm)
0.92 ± 0.05
0.08
1.46 ± 0.07
0.60
Film area (mm2)
Initial droplet radius, ri (mm)
3.2 ± 0.4
1.36 ± 0.05
6.5 ± 0.6
0.08
Not
measured
Table 3: Experimental data of rim velocities measured for four different types of droplet.
Number of
droplets
analysed
Number of
velocity
measurements
Mean rim
velocity
(mm/s)
Maximum
velocity
(mm/s)
Minimum
velocity
(mm/s)
Biodiesel droplet in
crude glycerol
19
187
17
60
4
Crude glycerol
droplet in biodiesel
12
116
4
7
3
PDMS droplet in
aqueous glycerol
5
21
387
620
78
Aqueous glycerol
droplet in PDMS
5
14
255
329
208
Droplet type
Table 4: Dimensions of droplets in rim velocity study.
Droplet type
Number of
droplets
analysed
Droplet width (mm)
Droplet height (mm)
Mean
Standard
deviation
Mean
Standard
deviation
Biodiesel droplet in
crude glycerol
19
1.4 ± 0.1
0.1
1.12 ± 0.05
0.07
Crude glycerol
droplet in biodiesel
12
3.5 ± 0.1
0.5
1.95 ± 0.05
0.20
PDMS droplet in
aqueous glycerol
5
5.0 ± 0.1
0.03
4.09 ± 0.05
0.04
Aqueous glycerol
droplet in PDMS
5
3.5 ± 0.1
0.2
3.10 ± 0.05
0.18
Table 5: Parameters used to solve Eq. (14) and Eq. (18) for rim velocity. The rim radius, a,
could not be measured but an assumed value of 10 μm was used in Eq. (14).
σ
(mN/m)
μf
(mPa.s)
μc
(mPa.s)
ρc
(kg/m3)
a (μm)
Re
Biodiesel droplet in
crude glycerol
0.6a
53
6.5
880
10.0
0.0035
Crude glycerol
droplet in biodiesel
0.6a
6.5
53
1090
10.0
0.0054
PDMS droplet in
aqueous glycerol
28.7a
60b
4.8c
963
10.0
0.0078
10.0
0.5084
Droplet type
Aqueous glycerol
28.7a
4.8c
60b
1208
droplet in PDMS
a
Measured by the du Noüy ring method using a Krüss K12 Processor Tensiometer.
b
Obtained from Viscosity of aqueous glycerol solutions, Dow Chemical Company.
c
Measured by strain rate sweep using an ARES viscometer with Couette cell.
Table 6: Results of experimental measurements compared with predictions from Eq. (14) and
Eq. (18) for the rim velocities of four types of droplet.
Droplet type
Measured rim
velocity (mm/s)
Solution to Eq. (14)
(mm/s)
Solution to Eq. (18)
(mm/s)
Mean
Standard
deviation
Biodiesel droplet in
crude glycerol
17
10
58
11
Crude glycerol
droplet in biodiesel
4
0.8
13
92
PDMS droplet in
aqueous glycerol
387
138
862
517
Aqueous glycerol
droplet in PDMS
255
32
336
6458