**2. Experiment**

#### *2.1. Solutions*

We used an aqueous two-phase system (ATPS), the same as the one used in [10,31], where the more viscous fluid is PEG solution and the less viscous fluid is Na2SO4 solution. Figure 3 shows the phase diagram of the ATPS [36]. The partially miscible zone is the green Region II, where the system can be separated into two phases because of thermodynamic instability. Region II is called the spinodal region, where concentration fluctuations spontaneously grow due to thermodynamic instability as the system reduces the total free energy, but the interfacial free energy increases because the interfaces of domains gradually grow. After the domain growth, the domains coalesce with each other, and phase separation is eventually completed. The progression of phase separation becomes faster with the increase in the concentration of Na2SO4, according to the previous paper [10,31], as this leads the system far from equilibrium. Region I is a one-phase region where the system is fully miscible because of the thermodynamically stable region. Here, we consider that the progression of phase separation represents the extent of phase separation. The details are described in [10,31]. The solutions used here are shown in Table 1; seven more viscous solutions and 11 less viscous solutions were used for the fully and partially miscible systems and a combination of Phases L and H was used for the immiscible system. Since Phases L and H are the solution after the phase separation, they are thermodynamically stable. The composition of Phases L and H is 36.5 wt % PEG solution with 3.2 wt % Na2SO4 and 1.4 wt % PEG solution with 16.0 wt % Na2SO4 [36]. To visualize the displacement process, the more viscous solutions were dyed blue by 0.1 wt % indigo carmine, which cannot dissolve into Na2SO4 solution because of the salting-out effect.

**Figure 3.** Phase diagram of polyethylene glycol (PEG)–Na2SO4–water system.

The composition of the system is important to decide whether the solution system is in Region I or II. Here, we assumed that the concentration at the interface, where the displacing and displaced fluids are brought into contact with each other, can be estimated to be half of the initial concentrations. For example, as the displacing and displaced fluids are 40 wt % PEG solution and 10 wt % Na2SO4 solution, respectively, the concentration at the interface was considered to be 20 wt % PEG–5 wt % Na2SO4–75 wt % water. This assumption is very simple, but we think this is appropriate. In previous studies [10,31], the same assumption was taken into account, and it can be explained by the assumption

that the result of the fluid displacement, calculation of free energy, and the result of the interfacial tension measurement all have good agreement.


**Table 1.** The solution system used.
