# Coarse Grained Modeling of Multiphase Flows with Surfactants

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dissipative Particle Dynamics

#### 2.2. Velocity Distribution of Two Immiscible Liquids in Nanoslits

#### 2.3. Details of the Computational Model

_{17}H

_{36}) molecules and each molecule included three oil beads (O) to represent the same volume of fluid as each of the water beads. The first surfactant was SDS, where each molecule contained one head (H) and two tail (T) beads, while four head and two tail beads were used for modeling C12E8 as the second surfactant. The schematic configuration of each molecule in the DPD simulations is shown in Figure 2. For the simulations of oil–water with C12E8, six water molecules were lumped in one DPD bead with the purpose of keeping comparable volumes of beads in the computations for all molecules. In addition, for the surfactant C12E8, the angle potentials that were applied between beads needed to be considered. Based on the C12E8 molecular structure, the tail–tail–head angle was fixed as 180° and all other angles were kept at 130° in order to meet the need for interfacial stability when the surfactant was added [16].

_{c}= k

_{B}T = 1. Since non-equilibrium computations were conducted, the temperature in the simulations was kept constant by employing the particle velocity rescaling thermostat available in LAMMPS. Every 200 time steps, the velocity of the beads was rescaled so that the temperature was kept constant at T = 1 (in DPD units of temperature). Furthermore, the bead number density was selected as ρ = 5 beads per unit volume. Using these parameters, the time scale was determined as $\tau ={r}_{c}\sqrt{m/{k}_{B}T}$

**[25]**. Conversion of time and length scales from DPD units to physical units in the case of C12E8 simulations was conducted using the same approach as for the SDS, and are shown in Table 1 [16,26].

## 3. Results

#### 3.1. Single Phase Flow, No-Slip Boundary Conditions, and Determination of Fluid Viscosity

#### 3.2. Flow of Two Immiscible Fluids

#### 3.3. Dynamic Oil–Water System with Surfactants

#### 3.3.1. Velocity Profile of Oil–Water System with Surfactants

#### 3.3.2. Micelle Formation

#### 3.3.3. The Effect of Computational Box Size

## 4. Summary and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Flow of two immiscible fluids between a pair of plates under (

**a**) Poiseuille, and (

**b**) Couette flow conditions.

**Figure 2.**Schematic configuration of heptadecane, water, SDS, and C12E8 surfactant molecules as beads of the DPD simulations. Hydrogen, carbon, oxygen, and sulfur molecules are shown as white, gray, red, and yellow spheres, respectively. The circles represent the DPD beads that group atoms or molecules together.

**Figure 3.**Poiseuille flow velocity profile for water with different dissipative parameter values between the wall and the water beads.

**Figure 4.**(

**a**) Relation between the dissipative parameter and computed fluid viscosity, and (

**b**) Poiseuille flow velocity profile for oil flow when ${\gamma}_{oil-oil}=22.5$ and ${\gamma}_{oil-wall}=45.0$.

**Figure 5.**Velocity profile for two immiscible fluids with various percentages of oil (0.0%, 28.8%, 48.4%, 65.6%, and 100.0%) under (

**a**) Poiseuille and (

**b**) Couette flow conditions. The oil flows through the bottom side of the channel and the water through the top side of the channel.

**Figure 6.**Snapshots of (

**a**) SDS and (

**b**) C12E8 surfactant at the oil–water interface with 50% of oil and water between two solid walls under Poiseuille flow. The wall, water, and oil beads are shown as ochre, blue, and yellow, respectively. The surfactant tails are purple, while green beads represent the head beads of the surfactants.

**Figure 7.**Velocity profile of oil–SDS–water and oil–C12E8-water in (

**a**) Poiseuille and (

**b**) Couette flows. The oil flows in the bottom side of the channel (z/h < 0) and the water flows through the top side (z/h > 0). The results are for a computational box with a size of 20$\times 20\times 21{r}_{c}^{3}$.

**Figure 8.**$\mathrm{ln}\left({\mu}_{eff}/{\mu}_{oil}\right)$ as a function of mole fraction ${x}_{i}$ of (

**a**) SDS and (

**b**) C12E8 used to stabilize the oil–water interface in the case of the oil phase and $\mathrm{ln}\left({\mu}_{eff}/{\mu}_{water}\right)$ as a function of mole fraction ${x}_{i}$ of (

**c**) SDS and (

**d**) C12E8 in the case of the water phase.

**Figure 9.**Shear rate and fluid behavior of Poiseuille flow for (

**a**,

**e**) oil–SDS–water and (

**c**,

**f**) oil–C12E8–water, and Couette flow for (

**b**) oil–SDS–water and (

**d**) oil–C12E8–water (box size 20$\times 20\times 21{r}_{c}^{3}$). Oil flows through the bottom of the channel and water through the top side of the channel (

**a**–

**d**). The color code for the DPD beads in (

**e**,

**f**) follows the colors in Figure 6.

**Figure 10.**Fluid behavior of Poiseuille flow for (

**a**) oil–SDS–water (${C}_{SDS}=$ 50.3% CMC; ${g}_{x}$ = 0.09) and (

**b**) oil–C12E8–water (${C}_{C12E8}=$ 55.1% CMC; ${g}_{x}$ = 0.17). The color code for the DPD beads follows the colors in Figure 6. Water and oil beads have been hidden. Note the high local concentration of the surfactants in the circled areas. The contour plots of the density of SDS surfactant and C12E8 surfactant are presented in (

**c**,

**d**) to show the local increase in surfactant concentration along the interface.

**Figure 11.**Velocity profile of the SDS or C12E8 surfactant at the interface of oil–water flow with a computational box size of $20\times 20\times 51{r}_{c}^{3}$ under (

**a**) Poiseuille flow and (

**b**) Couette flow configurations.

**Figure 12.**Shear rate and fluid behavior of Poiseuille flow for (

**a**) oil–SDS–water and (

**c**) oil–C12E8–water, and Couette flow for (

**b**) oil–SDS–water and (

**d**) oil–C12E8–water (box size: $20\times 20\times 51{r}_{c}^{3}$). Oil flows through the left side of the channel and water through the right side of the channel. The inset is a density plot with the y-axis as the density of the oil–surfactant–water system.

Type of Surfactant | Density | Number of Water Molecules in One Bead | Mass Scale $({10}^{-25}\mathbf{kg})$ | Length Scale $({10}^{-10}\mathbf{m})$ | Temperature Scale (K) | Time Scale $({10}^{-12}\mathbf{s})$ |
---|---|---|---|---|---|---|

SDS | 5 | 5 | 1.5 | 9.09 | 298 | 5.48 |

C12E8 | 5 | 6 | 1.8 | 9.66 | 298 | 6.38 |

None | 5 | 5 | 1.5 | 9.09 | 298 | 5.48 |

W | O | Wall | |
---|---|---|---|

W | 15 | 90 | 15 |

O | 15 | 20 | |

Wall | 15 |

**Table 3.**Set of repulsion parameters used in the water–oil–SDS simulations. H and T represent the head and tail beads of the SDS surfactant, while W and O denote the water and oil, respectively.

H | T | W | O | Wall | |
---|---|---|---|---|---|

H | 20 | 42 | 10 | 50 | 35 |

T | 15 | 25 | 12 | 15.5 | |

W | 15 | 90 | 15 | ||

O | 15 | 20 | |||

Wall | 15 |

**Table 4.**Details of parameters used in the water–oil–C12E8 simulations. H and T represent the head and tail of the C12E8 surfactant, while W and O denote the water and oil, respectively.

H | T | W | O | Wall | |
---|---|---|---|---|---|

H | 15 | 25 | 14 | 25 | 35 |

T | 15 | 54 | 14.5 | 15.5 | |

W | 15 | 100 | 15 | ||

O | 15 | 20 | |||

Wall | 15 |

Percentage of oil | 0.0% | 28.8% | 45.4% | 65.6% | 100.0% |

$\mathrm{Normalized}\mathrm{error}{\overline{e}}_{{L}_{2}}$(Poiseullie flow) | 0.024 | 0.023 | 0.032 | 0.029 | 0.022 |

$\mathrm{Normalized}\mathrm{error}{\overline{e}}_{{L}_{2}}$(Couette flow) | 0.007 | 0.009 | 0.017 | 0.021 | 0.007 |

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**MDPI and ACS Style**

Nguyen, T.X.D.; Vu, T.V.; Razavi, S.; Papavassiliou, D.V.
Coarse Grained Modeling of Multiphase Flows with Surfactants. *Polymers* **2022**, *14*, 543.
https://doi.org/10.3390/polym14030543

**AMA Style**

Nguyen TXD, Vu TV, Razavi S, Papavassiliou DV.
Coarse Grained Modeling of Multiphase Flows with Surfactants. *Polymers*. 2022; 14(3):543.
https://doi.org/10.3390/polym14030543

**Chicago/Turabian Style**

Nguyen, Thao X. D., Tuan V. Vu, Sepideh Razavi, and Dimitrios V. Papavassiliou.
2022. "Coarse Grained Modeling of Multiphase Flows with Surfactants" *Polymers* 14, no. 3: 543.
https://doi.org/10.3390/polym14030543