# Growing Interface with Phase Separation and Spontaneous Convection during Hydrodynamically Stable Displacement

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## Abstract

**:**

_{2}sequestration, but also material processing, such as Lost Foam Casting. Even during hydrodynamically stable fluid displacement where a more viscous fluid displaces a less viscous fluid in porous media or in Hele-Shaw cells, the growing interface fluctuates slightly. This fluctuation is attributed to thermodynamic conditions, which can be categorized as the following systems: fully miscible, partially miscible, and immiscible. The dynamics of these three systems differ significantly. Here, we analyze interfacial fluctuations under the three systems using Family–Vicsek scaling and calculate the scaling indexes. We discovered that the roughness exponent, $\alpha $, and growth exponent, $\beta $, of the partially miscible case are larger than those of the immiscible and fully miscible cases due to the effects of the Korteweg convection as induced during phase separation. Moreover, it is confirmed that fluctuations in all systems with steady values of $\alpha $ and $\beta $ are represented as a single curve, which implies that accurate predictions for the growing interface with fluctuations in Hele-Shaw flows can be accomplished at any scale and time, regardless of the miscibility conditions.

## 1. Introduction

_{2}SO

_{4}, and water, for which spinodal decomposition phase separation occurred at the boundary of the displacing and displaced liquids. They performed hydrodynamically stable displacements in a radial Hele-Shaw cell and observed interfacial deformation only in the partially miscible system [13]. They proved that interfacial fluctuations are driven by spontaneous convection as induced by the Korteweg force [17,18] due to the chemical potential gradient during spinodal decomposition-type phase separation. The Korteweg force was first proposed by Korteweg in 1901 [19]. It is defined thermodynamically as the functional derivative of free energy [20], and is characterized as a body force. The Korteweg force tends to minimize the free energy stored at the interface and induces spontaneous convection. Suzuki et al. found that the direction of Korteweg convection differed from that of injection, which roughened the interface [13].

## 2. Materials and Methods

#### 2.1. Solutions

_{2}SO

_{4}solution system as fully miscible, a 36.5 wt% PEG–20 wt% Na

_{2}SO

_{4}solution system as partially miscible, and a phase L–phase H (explained below) as immiscible (Figure 1), where PEG represents polyethylene glycol 8000, whose average molecular weight is approximately 8000. Figure 1 presents the phase diagram [29] of PEG–Na

_{2}SO

_{4}–water at 25 °C. When the composition of the solution is in Region I, the solution approaches a single phase, which is known as a fully miscible system. The displacement of a fully miscible system is thermodynamically stable, but the molecular diffusion progresses at the interface and the fluids mix together. When the composition is in Region II, the solution separates into two phases. The red solid curve in the figure shows the immiscible composition. Thus, when the initial solution compositions are on the curve, they are immiscible. For example, when the solution is comprised of 10 wt% PEG and 13 wt% Na

_{2}SO

_{4}(closed black triangle in Figure 1), it separates into phases L (closed red circle) and H (open red circle). Phase L is comprised of 36.5 wt% PEG and 3.2 wt% Na

_{2}SO

_{4}, and phase H is comprised of 1.4 wt% PEG and 16.0 wt% Na

_{2}SO

_{4}. Phases L and H are in thermodynamic equilibrium with each other. and considered immiscible. The displacement between the two fluids with a composition within Region II becomes thermodynamically unstable, and the mixing of the two fluids causes phase separation to occur at the interface. Therefore, the composition of a system is important to determine the thermodynamic state of a solution. Using the phase diagram in Figure 1 allows for the easy controlling the thermodynamic state of the solution system. The physical properties used in the system, such as the density and viscosity, are listed in Table 1.

#### 2.2. Displacement

_{2}SO

_{4}solution for the fully miscible case; a 36.5 wt% PEG solution to displace a 20 wt% Na

_{2}SO

_{4}solution for the partially miscible case; and phase L to displace phase H for the immiscible case (shown in Table 1). A radial Hele-Shaw cell comprises two square transparent glass plates (140 mm × 140 mm × 10 mm) with a gap of 0.3 mm. The gap was achieved using four metal plates located in four corners of the cell. The top glass plate had a small hole (4 mm diameter) drilled into the center for fluid injection. We first filled the cell with the less viscous liquid, then injected the more viscous liquid into it, as shown in Figure 2. The displacement experiments were recorded from the bottom of the setup using a video camera. To visualize the displacement, the more viscous fluids (PEG solution and phase L) were dyed blue with 0.1 wt% indigo carmine, which could not dissolve into the Na

_{2}SO

_{4}solution due to the salting-out effect. All experiments were performed at room temperature (25 $\pm $ 1 °C) and atmospheric pressures at a constant injection flow rate of $q=7.07\times {10}^{-10}{\text{}\mathrm{m}}^{3}/\mathrm{s}.$ We conducted the fluid displacement under same conditions three times to confirm the reproductivity.

#### 2.3. Family–Vicsek Scaling

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Phase diagram of the polyethylene glycol (PEG)–Na

_{2}SO

_{4}–water system [15]. The different color symbols show the three systems having different three thermodynamic states: the red circles for the immiscible system, the blue circles for the fully miscible system, and the green circles for the partially miscible system. The closed circles mean more viscous solutions and open ones mean less viscous solutions. The black line is the tie-line, whose end points determine the equilibrium phase compositions. The red curve represents the binodal curve, indicating equilibrium compositions of the two immiscible phases after separation.

**Figure 2.**Schematic illustration of fluid displacement experiments [15].

**Figure 3.**Example of the time evolution of the interface at $t$ = 500–1900 s. The value $l$ is determined by $r$ and $x$ and has the unit of $\mathrm{rad}$.

**Figure 4.**Time evolution of the interface for the (

**a**) fully miscible, (

**b**) partially miscible, and (

**c**) immiscible systems. The result lines are shown every 200 s ranging from t = 500−2500 s.

**Figure 5.**Scaling of the width $w\left(l,t\right)$ at different times $t$ for the (

**a**) fully miscible, (

**b**) partially miscible, and (

**c**) immiscible systems. The various colors indicate different times, and the dashed lines serve as visual guides.

**Figure 6.**Time evolution of the overall width $W\left(t\right)$. The various colors indicate different miscibilities, and the dashed lines serve as visual guides. The slope is calculated at the region where the value of $\alpha $. is constant.

**Figure 7.**Collapse of data using Equation (1) for the (

**a**) fully miscible, (

**b**) partially miscible, and (

**c**) immiscible systems presenting Family−Vicsek scaling.

System | Displacing More Viscous Liquid (Density, Viscosity) | Displaced Less Viscous Liquid (Density, Viscosity) |
---|---|---|

Fully miscible | 36.5 wt% PEG solution (1.07 g/cm ^{3}, 112 mPa·s) | 0 wt% Na_{2}SO_{4} solution(0.997 g/cm ^{3}, 0.972 mPa·s) |

Partially miscible | 36.5 wt% PEG solution (1.07 g/cm ^{3}, 112 mPa·s) | 20 wt% Na_{2}SO_{4} solution(1.19 g/cm ^{3}, 2.08 mPa·s) |

Immiscible | Phase L (1.08 g/cm ^{3}, 125 mPa·s) | Phase H (1.16 g/cm ^{3}, 1.76 mPa·s) |

Fully Miscible | Partially Miscible | Immiscible | |
---|---|---|---|

$\alpha $ | 0.35 ± 0.02 | 0.86 ± 0.03 | 0.49 ± 0.02 |

$\beta $ | 0.44 ± 0.12 | 0.84 ± 0.17 | 0.69 ± 0.05 |

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**MDPI and ACS Style**

Ban, T.; Tanaka, R.; Suzuki, R.X.; Nagatsu, Y.
Growing Interface with Phase Separation and Spontaneous Convection during Hydrodynamically Stable Displacement. *Materials* **2021**, *14*, 6089.
https://doi.org/10.3390/ma14206089

**AMA Style**

Ban T, Tanaka R, Suzuki RX, Nagatsu Y.
Growing Interface with Phase Separation and Spontaneous Convection during Hydrodynamically Stable Displacement. *Materials*. 2021; 14(20):6089.
https://doi.org/10.3390/ma14206089

**Chicago/Turabian Style**

Ban, Takahiko, Ryohei Tanaka, Ryuta X. Suzuki, and Yuichiro Nagatsu.
2021. "Growing Interface with Phase Separation and Spontaneous Convection during Hydrodynamically Stable Displacement" *Materials* 14, no. 20: 6089.
https://doi.org/10.3390/ma14206089