**3. Results and Discussion**

## *3.1. Physical Properties*

The measured physical properties are shown in Figure 5. Figure 5a shows that the viscosities of PEG solutions are constant against shear rates, which means the PEG solutions are Newtonian. Figure 5b depicts the dependence of viscosity on the solute concentrations, and the inset shows the results of Na2SO4 for easy observation. The viscosities of PEG solutions depend greatly on the concentration of PEG, while the viscosities of Na2SO4 are very low compared to those of PEG solutions. Figure 5c shows the densities, which have a linear relationship with the concentration of the solution. In the horizontal Hele-Shaw cells, however, the density effect (gravitational effect) can be ignored when the gap between the cells is small enough.

**Figure 5.** (**a**) Viscosity of PEG solutions against shear rate, (**b**) viscosity of PEG and Na2SO4 solutions against concentration, (inset) viscosity of Na2SO4 solutions against concentrations, (**c**) density of PEG and Na2SO4 solutions against concentrations.

#### *3.2. Fluid Displacements*

We first investigated the pattern formation of three fluid systems during fluid displacement under different thermodynamic conditions and the same hydrodynamic conditions or different hydrodynamic conditions and similar thermodynamic conditions. System I is at 20 wt % PEG and 0 wt % Na2SO4, System II is at 20 wt % PEG and 20 wt % Na2SO4, and System III is at 40 wt % PEG and 20 wt % Na2SO4. System I lies in Region I, whereas Systems II and III lie in Region II in Figure 3. Figure 6a shows the time evolution of the pattern formation of System I. A circular pattern expands as time proceeds because System I is absent from the thermodynamic instability and hydrodynamic interfacial instability. Similarly, the immiscible system, which is thermodynamically and hydrodynamically stable, approaches a perfectly circular pattern because of the high interfacial tension. The displacement patterns in the immiscible system have already been reported in [10].

Figure 6b shows the time evolution of the pattern of System II. At the beginning of the formation, the color of the outer region of the displacing fluid changes from indigo blue to light blue at time *t* = 30 s. The color change means that water and Na2SO4 molecules diffuse from the interface into the displacing fluid to reduce the PEG concentration inside the diffusion region. At *t* = 50 s, many indigo blue domains newly form in the outermost part of the displacing fluid, and the light blue region is left inside the outer indigo domains to produce a light blue circular ring. The formation of the outer indigo blue domains and the light blue circular ring indicates the spontaneous formation of PEG-rich and Na2SO4-rich phases, respectively. The interface becomes distorted, and the annular-like pattern expands with distortion as time proceeds. System I has a similar viscous ratio to System II because the viscous ratio depends on the PEG concentration, as shown in Figure 5b. Thus, thermodynamic instability, i.e., phase separation, is estimated to contribute to the spontaneous formation of the outer indigo domains and to the distorted interfacial pattern.

**Figure 6.** Time evolution of the pattern formation at (**a**) System I, where 20 wt % PEG solution displaces 0 wt % Na2SO4 solution; (**b**) System II, where 20 wt % PEG solution displaces 20 wt % Na2SO4 solution; and (**c**) System III, where 35 wt % PEG solution displaces 20 wt % Na2SO4 solution.

The pattern formation of System III is shown in Figure 6c. The outer region of the displacing fluid becomes distorted without a color change, and the distorted interface grows continuously. System III is comparable in the extent of thermodynamic instability to System II because both fluid systems lie in Region II at the same Na2SO4 concentration, but they are different in terms of their viscous ratio. A high viscous ratio restricts the diffusion region to the periphery of the displacing fluid. Thus, phase separation occurs at the outmost interface of the displacing fluid, inducing the interfacial distortion. This fingering-like pattern is already reported by Suzuki et al. [10]. The distorted interfacial pattern is created by the Korteweg force exerted in a direction toward the higher region of PEG concentration [10]. Thus, the pattern of System II is a phase separation-dominated displacement in comparison with System III, where the process of viscous dissipation weakens the phase separation effect.

Figure 7 shows the results of the hydrodynamically stable displacement using various composition combinations. A circular, stable pattern forms as PEG concentration increases and Na2SO4 concentration decreases, i.e., the composition goes to the upper left in Figure 7. In contrast, the finger-like pattern forms as PEG concentration decreases and Na2SO4 concentration increases, i.e., the composition moves to the lower right in Figure 7. For PEG concentrations higher than 30 wt %, the light blue circular ring disappears, and the outer indigo blue interface becomes distorted. As the Na2SO4 concentration increases, the interface becomes sharp because the interfacial tension increases with the increase in Na2SO4 concentration [31].

It is reported that the Korteweg force becomes stronger as the concentration of Na2SO4 increases [10,31]. As shown in Figure 5b, the viscosity decreases as the concentration of PEG solution decreases, which means that the solutions with less viscosity easily move, and phase separation is thought to easily occur.

**Figure 7.** The results of the hydrodynamically stable displacement. The time shown in the right bottom corner is the time when the longest radius reached 42 mm.

Figure 8 shows the fluid displacement patterns with the phase diagram shown in Figure 3. The curve in the figure is from [36] and indicates the boundary between fully miscible and partially miscible zones. The patterns are categorized into three types: the circular pattern (•), the finger-like pattern (ɢ), and the annular-like pattern () in Figure 8. If we pay close attention to the boundary between the finger-like pattern (ɢ) and the annular-like pattern (), both PEG and Na2SO4 concentrations are related to the pattern formation, which means that the morphologies are affected by the complexity of the hydrodynamic effect, such as the viscosity and thermodynamic effect such as phase separation. On the other hand, the patterns are circular (•) when the Na2SO4 concentration is less than 5 wt % in Figure 8, regardless of the PEG and Na2SO4 concentrations, because the displacement patterns are determined only by the hydrodynamic effect of viscosity.

The extent of the phase separation can be expressed by the progression of the phase separation. The progression of the phase separation can be considered from the growth rate of the interfacial tension (IFT) because the formation of an interface due to phase separation induces the increase in interfacial free energy. Figure 9a shows the time evolution of IFT between 20 wt % Na2SO4 solution and PEG solutions with several concentrations. IFT, at all concentration ranges, increases with time. The steady value of the IFT increases with the decrease in PEG concentration. The IFT of the fully miscible systems decreases with time because the width of the interface becomes wider due to molecular diffusion [10]. In contrast, the IFT of partially miscible systems increases with time because the interface becomes sharp due to phase separation. Figure 9b depicts the rate constant, *k*, defined as <sup>γ</sup> = (γ<sup>0</sup> <sup>−</sup> <sup>γ</sup>∞)*e*−*kt* + <sup>γ</sup>∞, where <sup>γ</sup><sup>0</sup> and <sup>γ</sup><sup>∞</sup> are initial and steady values of IFT in Figure 9a, respectively. The relaxation process of IFT corresponds to that of phase separation. Therefore, the rate constant, *k*, is thought to represent the progress of phase separation. The rate is higher with the decrease in PEG concentration because the formation rate of the interface decreases with the increase in the viscosity of the displacing fluid. Therefore, the progress of phase separation is important for forming patterns, for example, the annular-like pattern (Figure 6b) for high rates and the finger-like patterns (Figure 6c) for low rates. Moreover, the progression is affected by the mobility-like viscosity contrast, and the

viscosity is affected by the concentrations of the components (here, PEG). Thus, we compared the patterns using a dimensionless number considering those effects, such as modified capillary number, Ca , including the viscosity, interfacial tension, and flow rate.

**Figure 8.** Phase diagram of the displacement patterns.

**Figure 9.** (**a**) Time evolution of the interfacial tension between PEG solutions and 20 wt % Na2SO4 solution; (**b**) the phase separation rate constant, *k*, of (**a**).

In order to explore the factors on which the fluid displacement patterns depend, we evaluated the effects of physicochemical and hydrodynamic properties on the patterns using two dimensionless numbers: the well-known capillary number, Ca , and the newly defined body force number, Bf. The modified capillary number, Ca , is defined as Ca = <sup>μ</sup>*<sup>q</sup> <sup>b</sup>*2<sup>γ</sup> , where <sup>μ</sup> (Pa·s) is viscosity, *<sup>q</sup>* (m3/s) is the flow rate, *b* (m) is the gap between the cells, and γ (N/m) is an interfacial tension between displacing and displaced liquids [37–41]. It is noted that Ca at fully miscible systems cannot be defined because IFT in the fully miscible systems is almost zero. A dimensionless number, Bf, represents the relative effect of the body force driven by thermodynamic instability versus the pressure gradient related to Darcy's law, which was introduced in [10]:

$$\mathbf{B}\_{\mathbf{f}} = \frac{\frac{\Delta \mathbf{y}}{2\pi r\_m b}}{\frac{\mu V}{\kappa}} = \frac{\Delta \mathbf{y} b^2}{\mu q},\tag{3}$$

where Δγ = γ<sup>∗</sup> − γ0, γ<sup>0</sup> is the initial value of the measured interfacial tension, γ<sup>∗</sup> is interfacial tension at a time when the longest radius, *rm*, reaches 42 mm, κ (= *b*2/12) is the permeability of the Hele-Shaw cell, μ is the viscosity of the displacing fluid, *V* (= *q*/2π*rmb*) is linear velocity, and *q* is the injected flow rate. We neglected the numerical coefficient on the right-hand side for simplicity. We introduced a deformation index as a quantitative evaluation index for the extent of interfacial deformation. The index, ρdi, is defined as the white area involving areas of Na2SO4-rich regions generated by phase separation and created by deformation, divided by the area of a circle with maximum radius of 42 mm as shown in Figure 10a. Here, we measured the patterns of the immiscible system, the conditions of which are described in [10], to better understand the mechanism of the pattern formation of all cases. The ρdi of the immiscible system is almost zero because the patterns are perfectly circular, as mentioned in the Introduction.

**Figure 10.** (**a**) The definition of ρdi. (**b**) The relationship between the modified capillary number, Ca , and the ρdi for the patterns. (**c**) The pattern evaluation with Bf. The fitted curves are for a better visualization of the effects of Ca and Bf.

Figure 10b shows the relationship between the deformation index and Ca in the partially miscible and the immiscible systems. The value of ρdi decreases with increasing Ca , but the immiscible system indicates that the value of ρdi remains at zero regardless of Ca because the displacement of the immiscible system is hydrodynamically and thermodynamically stable. Thus, the well-known Ca is not appropriate for a comprehensive analysis of the fluid displacement using various types of fluid systems. Therefore, we investigated the relationship between ρdi and the newly defined Bf. For the immiscible system, IFT cannot change with time because the system is under thermodynamic equilibrium, and the value of Bf is zero. Figure 10c shows the relationship between ρdi and Bf. The value of ρdi increases with increasing Bf, and the patterns change from circular to finger-like to

annular-like patterns. The data of the patterns seem to collapse well onto a single curve, even with the immiscible patterns with the variation in Bf, which is not with the case for the effects on Ca . This relation indicates that, as the thermodynamic instability is dominant over the viscous dissipation, the degree of deformation becomes larger. The increase in IFT due to phase separation enhances the interface deformation because the increase in IFT corresponds to the increase in Korteweg force, which is a driving force for spontaneous convection. The pattern formation is attributed to the combination of the hydrodynamic effect (viscosity), molecular diffusion, and phase separation (thermodynamic effect). Because Bf includes these effects, the ρdi can be expressed as a function of Bf on a single curve, regardless of the miscibility.
