**1. Introduction**

The displacement of one fluid by another in porous media is ubiquitous in several processes such as chromatographic separation [1], transportation of digestive juice [2], CO2 sequestration [3], frontal polymerization [4], and secondary and tertiary oil recovery [5]. It is generally accepted that when the displacing fluid is more viscous, no interfacial instability occurs, while a finger-like pattern forms at the interface in the reverse situation (Figure 1). The latter case, where a less viscous fluid displaces a more viscous one in porous media, is called Saffman–Taylor instability [6] or viscous fingering (VF) [7]. Saffman–Taylor instability, based on Darcy's law [8,9], can be explained by Figure 2 and Equation (1):

$$
\mu\_A - \mu\_B = \left[\frac{\mu\_B(\mu\_1 - \mu\_2)}{\mu\_2 + (\mu\_1 - \mu\_2)\frac{\tilde{\mathbf{L}}}{\tilde{\mathbf{L}}}}\right] \frac{\delta \mathbf{z}}{L}'\tag{1}
$$

where *u* is flow velocity, μ is viscosity, δ*z* is an initial disturbance, *L* is a length scale of porous media, and indexes A and B represent location. *uA* and *uB* represent the velocity at the different locations in the interface between fluid 1 and fluid 2 (Figure 2a). We assume that a velocity fluctuation occurs at the interface, and the velocity at location B becomes faster than the remaining interface at the initial state. Therefore, the interface at location B protrudes from the initial flat interface by the displacement δ*z*. We have two different combinations of viscosity of the fluids. When displacing fluid 1 is more viscous than the other (μ<sup>1</sup> > μ2), the right-hand side of Equation (1) becomes positive, and then *uA* > *uB* according to Equation (1). In this case, the initial disturbance δ*z* becomes small. As a result, the disturbance is suppressed, and hydrodynamically stable displacement proceeds. However, when the displacing fluid is less viscous than the other (μ<sup>1</sup> < μ2), the right-hand side of Equation (1) becomes negative, and then *uA* < *uB*, and thus the initial disturbance δ*z* is enhanced. Therefore, the interface at location B protrudes further, and a finger-like pattern (VF) appears.

**Figure 1.** Fluid displacements for (**a**) a hydrodynamically stable condition of the interface, (**b**) a hydrodynamically unstable condition of the interface. The blue solution displaces the surrounding white solution. A more viscous solution is displacing a less viscous one in (**a**) while a less viscous solution is displacing a more viscous one in (**b**). The patterns like fingering in (**b**) are called "Saffman–Taylor instability" or "viscous fingering" (VF).

**Figure 2.** The explanation for Saffman–Taylor instability. The solution flows from left to right in the figures. (**a**) The setup situation and the results under (**b**) a hydrodynamically stable condition (μ<sup>1</sup> > μ2) and (**c**) a hydrodynamically unstable condition (μ<sup>1</sup> < μ2), i.e., Saffman–Taylor instability. *p* represents pressure at the position *z*.

The displacement fluid pair can be categorized into three types: fully miscible, partially miscible, and immiscible. The fully miscible system has infinite mutual solubility, like glycerol–water, the partially miscible system has finite mutual solubility, and the immiscible system has zero mutual solubility, like oil–water. The hydrodynamically stable displacement in fully miscible and immiscible systems makes the pattern circular for a radial geometry or flat for a rectangular geometry. Any fluctuation in the interface should always decay as fluid displacement proceeds. However, the hydrodynamically stable displacement in the partially miscible system shows not a circular but a finger-like pattern [10]. There are many studies on the displacement in the porous media or Hele-Shaw cells under hydrodynamically unstable and stable conditions of the interface with and without chemical reactions [7,11–14]. Although the unstable interfaces on the fully miscible and immiscible systems in hydrodynamically stable displacement have been reported by many papers, research on specialized conditions such as prewetting condition [15–17], including particles [18–21], viscosity-change reaction at the interface [22–25], precipitation reaction [26], and double diffusivity effect [27] is limited. Studies on the partially miscible system have recently received a lot of attention because of its application in enhanced oil recovery [5]. However, past work was unable to acquire an appropriate partially miscible system because changes in the composition and temperature lead to changes in multiple hydrodynamic properties of the fluid systems. These undesirable changes, which occur simultaneously, represent a significant obstacle in elucidating the exact properties of the miscibility that play a crucial role in fingering pattern formation. Therefore, hydrodynamically unstable displacement (Saffman–Taylor instability or VF) in the partially miscible systems has mostly been studied through numerical simulations [28–30]. One simulation [30] shows that the solubility in a partially miscible system greatly influences the degree of fingerings and also demonstrates that fluid dissolution or exsolution due to partial miscibility can hinder or enhance viscous fingering, respectively. However, partial miscibility is not mutual solubility but one-direction solubility, where one species dissolves into the other one. Very recently, Suzuki et al. have experimentally shown that a partially miscible system affects displacement such as a fingering pattern occurring in a hydrodynamically stable displacement [10] and a droplet formation in Saffman–Taylor instability or viscous fingering [31]. They have proven that the morphologies are driven by spontaneous convection induced by Korteweg force due to chemical potential gradient during spinodal decomposition-type phase separation. The force, first proposed by Korteweg in 1901 [32], is thermodynamically defined as the functional derivative of free energy [33] and is characterized as a body force. Korteweg force tends to minimize the free energy stored at the interface and induce spontaneous convection. The free energy is closely involved with an interfacial tension because the interfacial tension is defined as free energy per unit area. Therefore, measuring the interfacial tension is important for considering a partially miscible system or Korteweg force. We note that in Rayleigh–Taylor instability, where a more dense fluid is on a less dense fluid in gravitational field, the experimental studies and theoretical model for a partially miscible system have been reported [34,35].

Detailed experimental studies without any hydrodynamic interfacial instabilities are needed to thoroughly understand the influence of the partially miscible system. Here, we investigated the effect of different progressions of phase separation in a partially miscible system on the patterns in a hydrodynamically stable condition of the interface, like inverse Saffman–Taylor instability. An aqueous two-phase system was employed by following the previous reports [10,31], where the system consisted of polyethylene glycol (PEG; weight-average molecular weight *Mw* = 8000), Na2SO4, and water, allowing quantitative and qualitative control over the thermodynamic stability of the fluid system as well as the hydrodynamic interfacial stability. We changed the progression of the phase separation by changing the concentrations of PEG and Na2SO4 and investigated the effect on the fluid patterns to make a phase diagram of patterns. In addition, we measured the dynamic interfacial tension between the displacing and the displaced fluids to clarify the mechanism of pattern formation and to evaluate the patterns quantitatively.
