Lifetime of micrometer-sized drops of oil pressed by buoyancy against a planar interface

Emulsion stability simulations are used to estimate the coalescence time of one drop of hexadecane pressed by buoyancy against a planar water/hexadecane interface. In the present simulations, the homophase is represented by a big drop of oil at least 500 times larger than the approaching drop $(1–10\mu \text{m})$. Both deformable and nondeformable drops are considered along with six different diffusion tensors. In each case, van der Waals, electrostatic, steric, and buoyancy forces are taken into account. The coalescence times are estimated as the average of 1000 random walks. It is found that the repulsive potential barrier has a significant influence in the results. The experimental data can only be reproduced assuming negligible repulsive barriers, as well as nondeformable drops that move with a combination of Stokes and Taylor tensors as they approach the interface.

Figure 1

Calculation of the diffusion tensor. (a) Spherical: inner region $\left(0<h\le r_i\right)$, outermost region $\left(h>r_i\right)$, $h=r_{ij}-r_i-r_j$. (b) Deformable: inner region $\left(0<h\le h_{\text{ini}}\right)$, intermediate region $\left(h_{\text{ini}}<h\le r_i\right)$, and outermost region $\left(h>r_i\right)$.

Figure 2

Model of the drop/interface system employed in the simulations. Here, $r$ is the radius of the small droplet, and $R$ the radius of the large drop resembling the interface. $d$ is the initial distance between the surfaces of the drops.

Figure 3

Steric potential between two 100 nm $\kappa$-casein micelles. Solid line: $V_{\text{st}}\left({\overline{\varphi }}_{\text{p}}=2.1\times {10}^{-2}\right)$; dashed line: $V_{\text{st}}\left({\overline{\varphi }}_{\text{p}}=1.0\times {10}^{-2}\right)$; dotted line: steric potential reported by Tuinier and Kruif (Fig. 2 in Refs. [50, 51]); dot-dashed line: $V_{\text{st}}\left({\overline{\varphi }}_{\text{p}}=4.7\times {10}^{-3}\right)$; dashed double-dots line: $V_{\text{st}}\left({\overline{\varphi }}_{\text{p}}=7.9\times {10}^{-4}\right)$.

Figure 4

Total potential of interaction between a large droplet of $500\mu \text{m}$ and a small droplet of micron-size. For this calculation, the volume fraction of protein around the drops was assumed to be equal to ${\overline{\varphi }}_{\text{p}}=2.1\times {10}^{-2}$. Solid line: $r=2\mu \text{m}$; dashed line: $r=4\mu \text{m}$; dotted line: $r=6\mu \text{m}$; dot-dashed line: $r=8\mu \text{m}$; dashed double-dots line: $r=10\mu \text{m}$.

Figure 5

Total potential of interaction between a large droplet of $500\mu \text{m}$ and a small droplet of micron-size. For this calculation, the volume fraction of protein around the drops was assumed to be equal to ${\overline{\varphi }}_{\text{p}}=1.0\times {10}^{-2}$. Solid line: $r=2\mu \text{m}$; dashed line: $r=4\mu \text{m}$; dotted line: $r=6\mu \text{m}$; dot-dashed line: $r=8\mu \text{m}$; dashed double-dots line: $r=10\mu \text{m}$.

Figure 6

Total potential of interaction between a large droplet of $500\mu \text{m}$, and a small droplet of micron-size. For this calculation, the volume fraction of protein around the drops was assumed to be equal to ${\overline{\varphi }}_{\text{p}}=4.7\times {10}^{-3}$. Solid line: $r=2\mu \text{m}$; dashed line: $r=4\mu \text{m}$; dotted line: $r=6\mu \text{m}$; dot-dashed line: $r=8\mu \text{m}$; dashed double-dots line: $r=10\mu \text{m}$.

Figure 7

Dependence of the average coalescence time as a function of the particle radius for an initial distance of: (a) $d=20\text{nm}$, (b) $d=100\text{nm}$, and (c) $d=750\text{nm}$. The total potential corresponds to the one depicted in Fig. 6 (Table ).

Figure 8

Dependence of the average coalescence time as a function of the particle radius for an initial distance of: (a) $d=20\text{nm}$, (b) $d=100\text{nm}$, and (c) $d=750\text{nm}$. The total potential corresponds to the one depicted in Fig. 9 (Table ).

Figure 9

Total potential of interaction between a large spherical droplet of $500\mu \text{m}$ and a small spherical droplet of micron-size. For this calculation, the volume fraction of protein around the drops was assumed to be equal to ${\overline{\varphi }}_{\text{p}}=7.9\times {10}^{-4}$. Solid line: $r=2\mu \text{m}$; dashed line: $r=4\mu \text{m}$; dotted line: $r=6\mu \text{m}$; dot-dashed line: $r=8\mu \text{m}$; dashed double-dots line: $r=10\mu \text{m}$.

Figure 10

Total potential of interaction between a large deformable droplet of $500\mu \text{m}$ and a small deformable droplet of micron-size. For this calculation, the volume fraction of protein around the drops was assumed to be equal to ${\overline{\varphi }}_{\text{p}}=7.9\times {10}^{-4}$. Solid line: $r=2\mu \text{m}$; dashed line: $r=4\mu \text{m}$; dotted line: $r=6\mu \text{m}$; dot-dashed line: $r=8\mu \text{m}$; dashed double-dots line: $r=10\mu \text{m}$.

Figure 11

Individual contributions to the total potential between a drop and a planar interface for a spherical droplet of $r=10\mu \text{m}$. The steric potential was calculated using a volume fraction of protein equal to ${\overline{\varphi }}_{\text{p}}=7.9\times {10}^{-4}$. Dashed line: electrostatic; dotted line: steric.

Figure 12

Individual contributions to the total potential between a drop and a planar interface for a deformable droplet of $r=10\mu \text{m}$. The steric potential was calculated using a volume fraction of protein equal to ${\overline{\varphi }}_{\text{p}}=7.9\times {10}^{-4}$. Dashed line: electrostatic; dotted line: steric; dot-dashed line: dilational; dashed double-dots line: bending.

Figure 13

Total interaction potential between two micron-sized drops with a radius of $10\mu \text{m}$. The parameters of the potentials are equal to the ones employed in the calculation of Fig. 9, 10 for the drop/interface model. $V_{\text{st}}\left({\overline{\varphi }}_{\text{p}}=7.91\times {10}^{-4}\right)$. Solid line: spherical drops; dashed line: deformable drops.

Figure 14

Average coalescence time vs droplet radius for spherical drops initially separated by a distance of $d=15\mu \text{m}$. Stars: experimental data (Ref. [6]); double-dashed dot line: tensor $A$; dashed double-dots line: tensor $B$; dotted line: tensor $C$; gray dashed line: tensor $D$; solid line: tensor $E$; dashed line: tensor $F$; dot-dashed line: tensor $G$; gray dot-dashed line: tensor $H$.

Figure 15

Fitting of Eq. (6) to the data of Dickinson et al. (Ref. [6]). Stars: experimental data; solid line: Stokes-Taylor equation

Figure 16

Average coalescence time vs droplet radius for deformable drops. Stars: experimental data (Ref. [6]); dashed line: tensor $J$ $\left(d=50\text{nm}\right)$; dot-dashed line: tensor $K$ $\left(d=50\text{nm}\right)$; dotted line: tensor $L$ $\left(d=50\text{nm}\right)$; dashed double-dots line: tensor $I\left(d=50\text{nm}\right)$; double-dashed dot line: tensor $I$ $\left(d=100\text{nm}\right)$.

Figure 17

Best simulation results for the coalescence time of a micron-sized drop pressed by buoyancy against a planar interface $\left(d=15\mu \text{m}\right)$. Stars: experimental data (Ref. [6]); solid line: Stokes-Taylor law (tensor $D$); dashed line: Stokes-Taylor law (tensor $G$). Error bars were approximated by the standard deviation of 1000 simulations