1. Introduction
When a less viscous fluid displaces a more viscous one in porous media, the interface between the two fluids forms a finger-like pattern due to a viscous fingering (VF) or Saffman–Taylor instability (Saffman & Taylor Reference Saffman and Taylor1958; Homsy Reference Homsy1987). Such VF has received a lot of attention because of its importance in various fields such as secondary and tertiary oil recovery (Tchelepi & Orr Reference Tchelepi and Orr1994), chromatographic separation (Broyles et al. Reference Broyles, Shalliker, Cherrak and Guiochon1998; Haudin et al. Reference Haudin, Callewaert, De Malsche and De Wit2016), and even medical applications (Bhaskar et al. Reference Bhaskar, Garik, Turner, Bradley, Bansil, Stanley and LaMont1992).
Chemical reactions have been shown to affect VF when they modify in situ the interfacial tension for immiscible fluids (Fernandez & Homsy Reference Fernandez and Homsy2003) or viscosity in miscible interfaces (De Wit & Homsy Reference De Wit and Homsy1999; Nagatsu et al. Reference Nagatsu, Matsuda, Kato and Tada2007, Reference Nagatsu, Kondo, Kato and Tada2009; Hejazi et al. Reference Hejazi, Trevelyan, Azaiez and De Wit2010a ; Bunton et al. Reference Bunton, Tullier, Meiburg and Pojman2017; De Wit Reference De Wit2020; Escala & Muñuzuri Reference Escala and Muñuzuri2021; Tafur et al. Reference Tafur, Escala, Soto and Muñuzuri2021). As VF is usually a negative factor in terms of displacement efficiency, attention has also been paid to strategies for VF suppression via variable fluid injection rate (Dias et al. Reference Dias, Alvarez-Lacalle, Carvalho and Miranda2012), reactors with an elastic membrane (Pihler-Puzovic et al. Reference Pihler-Puzovic, Illien, Heil and Juel2012), or gradients of permeability (Al-Housseiny, Tsai & Stone Reference Al-Housseiny, Tsai and Stone2012), for instance.
Classically, in the reverse displacement, i.e. when the more viscous fluid displaces the less viscous one, the interface is stable with regard to VF, and remains planar. However, in some peculiar reverse cases, fingering can nevertheless develop. For immiscible systems, fingering of the interface has been observed in the reverse situation, driven by Marangoni effects (Chan & Liang Reference Chan and Liang1997; Krechetnikov & Homsy Reference Krechetnikov and Homsy2004) or accumulation of suspensions (Tang et al. Reference Tang, Grivas, Homentcovschi, Geer and Singler2000), for instance. In the case of partially miscible interfaces, destabilisation of reverse displacements can be triggered by spinodal decomposition (Suzuki et al. Reference Suzuki, Nagatsu, Mishra and Ban2019).
For miscible systems, Podgorski et al. (Reference Podgorski, Sostarecz, Zorman and Belmonte2007) demonstrated experimentally VF induced by a chemical reaction in the case of an aqueous reactant solution displacing another reactant solution of the same viscosity in a Hele-Shaw cell. The chemical reaction taking place in the contact zone between the two solutions produces a micellar material with viscoelastic properties, and fingering is then observed. Similar fingering between aqueous solutions of the same viscosity due to a precipitation reaction was reported by Nagatsu et al. (Reference Nagatsu, Ishii, Tada and De Wit2014). In this case, the local change in permeability is the source of the fingering. Possibility of destabilisation by a chemical reaction of a reverse miscible displacement has been suggested theoretically to occur if non-monotonic viscosity profiles develop because of a reaction inducing locally either a maximum or a minimum in viscosity (Hejazi et al. Reference Hejazi, Trevelyan, Azaiez and De Wit2010a ). Nonlinear simulations show that fingering develops then at the back of the maximum where the less viscous injected solution crashes into the zone of maximum viscosity, while forward fingering is rather obtained ahead of a minimum (Hejazi & Azaiez Reference Hejazi and Azaiez2010b ,Reference Hejazi and Azaiez c ; Nagatsu & De Wit Reference Nagatsu and De Wit2011; Sharma et al. Reference Sharma, Pramanik, Chen and Mishra2019). Experimental demonstration of both types of reaction-induced reverse fingering has been subsequently obtained using viscous polymer solutions displacing less viscous aqueous solutions of reactants (Riolfo et al. Reference Riolfo, Nagatsu, Iwata, Maes, Trevelyan and De Wit2012; Escala et al. Reference Escala, De Wit, Carballido-Landeira and Muñuzuri2019; Escala & Muñuzuri Reference Escala and Muñuzuri2021). Recent works have shown that reverse fingering can also be obtained in nanocatalytic reactive flows where the presence of nanocatalysts can affect the viscosity distribution (Dastvareh, Azaiez & Tsai Reference Dastvareh, Azaiez and Tsai2019). In absence of reactions, reverse miscible fingering can further be obtained by non-ideal mixing effects of separate pure fluids triggering a local maximum in viscosity, as observed experimentally in displacements in micro-pillar arrays (Haudin et al. Reference Haudin, Callewaert, De Malsche and De Wit2016).
Theoretically, double-diffusion has also been shown to be able to trigger VF in the reverse situation if the system involves two scalars that both affect viscosity independently, and diffuse at different rates (Pritchard Reference Pritchard2009; Mishra et al. Reference Mishra, Trevelyan, Almarcha and De Wit2010). Such double-diffusion effects can occur typically due to differences in diffusivity of heat and mass in porous media, where they can then trigger also different convective speeds (Azaiez & Sajjadi Reference Azaiez and Sajjadi2012). More generally, the presence of two solutes affecting viscosity independently and having different diffusion coefficients is predicted to destabilise reverse displacements and yield new types of fingered patterns (Mishra et al. Reference Mishra, Trevelyan, Almarcha and De Wit2010). Such pure double-diffusion-induced VF has, however, not been evidenced yet experimentally.
In this context, we evidence here experimentally a novel mechanism of VF destabilisation of a reverse miscible displacement in the case of a dyed, viscous polymer solution displacing water in the absence of any chemical reaction. Contrary to what is usually hypothesised, the dye is here actively decreasing the viscosity of the polymer solution, even if present in a minute quantity. This viscosity reduction coupled to differential diffusion of the polymer and of the dye induces a non-monotonic viscosity profile with a maximum at the origin of the fingering pattern.
We first present the experimental observation of reverse VF, and then proceed with viscosity measurements and theoretical reconstruction of viscosity profiles to explain the origin of the VF of the reverse displacement. Our results not only introduce a novel way to destabilise a reverse displacement of a less viscous solution by a more viscous one, they also present the first experimental evidence of an active role of a dye and of differential diffusion effects in VF dynamics.
2. Dye-induced VF
Experiments are conducted in a Hele-Shaw cell consisting of two parallel horizontal plates separated by a small gap
$b = 0.3$
mm in which radial displacements of one fluid by another miscible one of different viscosity are studied (Nagatsu et al. Reference Nagatsu, Matsuda, Kato and Tada2007). The less viscous fluid is here pure water, while the more viscous fluid is a solution of PAA (poly(acrylic acid) polymer from Aldrich, molecular weight (MW) 1.25 million) in concentration 0.5 wt %, mixed with Trypan Blue (TB) in concentration 0.1 wt %, a dye used for the visualisation of the pattern. For radial displacements in miscible systems, the Péclet number
$Pe$
is defined as
$Pe = q/2\unicode{x03C0} bD$
(Petitjeans et al. Reference Petitjeans, Chen, Meiburg and Maxworthy1999; Nagatsu et al. Reference Nagatsu, Matsuda, Kato and Tada2007), where
$q$
is the injection rate of the displacing fluid, and
$D$
is the diffusion coefficient, which we estimate to be equal to
$1 \times 10^{-9}$
m
$^2$
s–1, a representative value of
$D$
for chemical species in water. Here, the flow rate is constant and fixed at
$q = 3.0 \times 10^{-9}$
m
$^3$
s–1, which gives
$Pe = 1.6 \times 10^3$
. The shear rate at the finger tip can be estimated as
$\dot {\gamma }_f = q/\unicode{x03C0} R b^2$
(Vlad & Maher Reference Vlad and Maher2000; Nagatsu et al. Reference Nagatsu, Matsuda, Kato and Tada2007), where
$R$
is the radius of a stable displacement circle. In the present reactor,
$R$
is varied in the range
$2\, {\rm mm} \leq R \leq58$
mm, which leads to
$0.2\,\textrm{s}^{-1} \leq \dot {\gamma }_f \leq 5\,\textrm{s}^{-1}$
.
Fingering pattern formed when a solution of 0.5 wt % PAA (MW = 1.25 million) and 0.1 wt % TB displaces water for Pe =
$1.6\times 10^3$
and
$b = 0.3$
mm at (a) t = 60 s, R = 15 mm, (b) t = 120 s, R = 20 mm, (c) t = 180 s, R = 26 mm, (d) t = 240 s, R = 33 mm, (e) t = 300 s, R = 35 mm, ( f) t = 360 s, R = 40 mm, (g) t = 450 s, R = 44 mm, and (h) t = 510 s, R = 48 mm.
Figure 1 shows the temporal evolution of the displacement pattern when the more viscous dyed aqueous solution of PAA displaces water radially. Such a displacement of a less viscous fluid by a more viscous one is classically expected to be stable and hence feature a radially growing undeformed circular displacement. Yet a fingered deformation of the miscible interface, the amplitude of which grows in time, is clearly observed.
We emphasise that there is no chemical reaction taking place here, and that the only involved solutes are PAA and TB. This unexpected reverse VF is not a pure double-diffusion mechanism as studied before theoretically Mishra et al. (Reference Mishra, Trevelyan, Almarcha and De Wit2010) because the dye in itself does not change the viscosity of the water solvent when present alone. It is also different from reverse VF induced by non-ideal effects due to mixing of one fluid with another (Haudin et al. Reference Haudin, Callewaert, De Malsche and De Wit2016), as both the polymer and the dye are here initially present in the same solution, which is injected into water.
To clarify the origin of the instability, we have investigated the dependence of the solution’s viscosity on the concentration of PAA and TB, using a rheometer (AR-G2, TA Instruments). First, the polymer solution has a shear-thinning viscosity behaviour, i.e. its viscosity decreases when the shear rate increases. Figure 2 shows the viscosity as a function of the TB concentration for three different concentrations of PAA at a fixed shear rate of 1
$\textrm{s}^{-1}$
, a value chosen because it corresponds to the shear rate at approximately
$R=10{-}15$
mm where the instability starts to develop (see figure 1
a). The solution’s viscosity increases with the concentration of the polymer, which is quite classical. An unexpected trend is, however, that the dye is not passive here. On the contrary, the viscosity of each polymer solution decreases when the amount of TB increases, even if this amount is very small (less than 1 %). The dye plays here clearly an active role, and the viscosity of the solution depends on the simultaneous presence of both the polymer and the dye. The active role of TB is due to the fact that the addition of metal ions to an aqueous solution of anionic polymers reduces the solution viscosity (Escala et al. Reference Escala, Muñuzuri, De Wit and Carballido-Landeira2017; Escala et al. Reference Escala, De Wit, Carballido-Landeira and Muñuzuri2019; Tam & Tiu Reference Tam and Tiu1990). In the aqueous solutions used here, positively charged Na
$^+$
sodium ions are released by TB, while PAA behaves as an anionic polymer as many of the side chains of PAA lose their protons in water to become negatively charged. When the concentration of metal ions is sufficiently high, the electrostatic field established by the anionic polymer is screened out, and the apparent size of the polymer in solution becomes smaller, which leads to a decrease in the solution viscosity. Note that in absence of PAA, the viscosity of the dyed water is simply the viscosity of pure water, unaffected by the presence of the dye.
Viscosity of the solution of PAA (MW = 1.25 million) and TB in various concentrations at
$\dot{\gamma}=1\ \text{s}^{-1}$
.
3. Viscosity profiles
Previous works on reactive systems have shown that the destabilisation of the displacement of a less viscous fluid by a more viscous one can occur when a non-monotonic viscosity profile builds up thanks to the reaction changing the viscosity in situ (Hejazi et al. Reference Hejazi, Trevelyan, Azaiez and De Wit2010a ; Riolfo et al. Reference Riolfo, Nagatsu, Iwata, Maes, Trevelyan and De Wit2012). Here, there is no reaction, but the viscosity of the polymer solution decreases in the presence of TB. We will seek to understand how the spatial distribution of the viscosity varies in our system when PAA and TB are initially dissolved in the same solution but diffuse at different rates. As it is difficult to measure quantitatively the viscosity field in situ during the displacement (Bunton et al. Reference Bunton, Dice, Pojman, De Wit and Brau2016 Reference Bunton, Dice, Pojman, De Wit and Brau a ,Reference Bunton, Marin, Stewart, Meiburg and De Wit b ), we resort to reconstructing one-dimensional (1-D) viscosity profiles on the basis of 1-D analytical concentration profiles (Riolfo et al. Reference Riolfo, Nagatsu, Iwata, Maes, Trevelyan and De Wit2012).
Considering that in absence of fingering we have transverse symmetry, and assuming a constant velocity, the 1-D concentration profiles
$c_{\textit{PAA}}(x,t)$
and
$c_{\textit{TB}}(x,t)$
for PAA and TB, respectively, can be computed as solutions of Fick’s law of diffusion:
\begin{align} \frac {\partial c_{\textit{PAA}}}{\partial t} &= D_{\textit{PAA}} \frac {\partial ^2 c_{\textit{PAA}}}{\partial x^2 }, \end{align}
\begin{align} \frac {\partial c_{\textit{TB}}}{\partial t} &= D_{\textit{TB}} \frac {\partial ^2 c_{\textit{TB}}}{\partial x^2 }. \end{align}
We consider a solution of PAA at initial concentration
$c_{{\textit{PAA}}}^0=0.5\, \%$
containing TB at an initial concentration
$c_{\textit{TB}}^0=0.1\, \%$
so that the base state solutions for
$\textit{PAA}$
and
$\textit{TB}$
are
\begin{align} c_{\textit{PAA}}={c_{\textit{PAA}}^0 \over 2}\, \mbox{erfc}(\eta ) \quad \mbox{and} \quad c_{\textit{TB}}={c_{\textit{TB}}^0 \over 2}\, \mbox{erfc}(\eta /\sqrt {\delta }), \end{align}
where
$\eta =x/\sqrt {4D_{\textit{PAA}}t}$
and
$\delta =D_{\textit{TB}}/D_{\textit{PAA}}$
. These profiles are plotted in figures 3(a) and 3(b) for different and equal diffusion coefficients, respectively. Thanks to these concentration profiles, the spatial variation of the base state viscosity profile can be reconstructed. In practice, we select a dozen locations in space where the concentration of each species is known. Then we prepare in separate beakers a mixture of PAA and TP in the relevant proportions, and measure the viscosity of the solution by a rheometer as a function of the shear rate. The viscosities at the shear rate
$1\ \text{s}^{-1}$
are then used to reconstruct the 1-D spatial profile of viscosity in figure 3.
Concentration profiles of PAA and TB, and reconstructed viscosity profiles computed for 0.5 wt % PAA (MW = 1.25 million) and 0.1 wt % TB, as studied in figure 1 when (a)
$\delta=10$
, (b)
$\delta= 1$
.
Figure 3(a) focuses on the case where
$\delta$
is larger than 1. This assumption is reasonable as the polymer is a macromolecule that diffuses slower than the dye molecule, which has a much lower molecular weight. Here, we assume that the value of the diffusivities differ by one order of magnitude, i.e.
$\delta = 10$
. Figure 3(b) shows the profiles for equal diffusivities of PAA and TB (
$\delta = 1$
). We see that when TB diffuses faster than PAA, a non-monotonic spatial variation of the viscosity is obtained, i.e. the viscosity presents a maximum along the displacement direction, while the viscosity varies monotonically in space in the equal diffusion case. In figure 3(a), the local maximum develops around
$\eta = -1$
where
$c_{\textit{PAA}}$
is still very close to its initial value
$c_{\textit{PAA}}^0=0.5\, \%$
, while the concentration of TB has dropped to approximately 80 % of its bulk value
$c_{\textit{TB}}^0=0.1\, \%$
because TB diffuses faster than PAA. In comparison, around
$\eta =-3$
, the decrease in viscosity is larger because the concentration of TB is still larger as well. Much further downstream, i.e. for
$\eta \gt 0$
, the concentration of polymer has dropped so that the viscosity becomes very low again. This leads to the presence of a local maximum in viscosity at intermediate regions of the profile. In figure 3(b) with
$\delta = 1$
, the viscosity profile is monotonically decreasing because the ratio of the concentrations of PAA and TB is the same everywhere since the diffusivity of PAA and TB is the same. We note that for this value of shear rate (1 s
$^{-1}$
), elastic effects of the 0.5 wt
$\, \%$
PAA solutions can be neglected, as confirmed by measurements of the first normal stress difference (Nagatsu et al. Reference Nagatsu, Matsuda, Kato and Tada2007).
A careful analysis thus shows that the non-monotonic profile is possible thanks to the combination of three ingredients: (i) PAA increases the viscosity; (ii) TB decreases the viscosity of the PAA solution; and (iii) TB diffuses faster than PAA.
4. Parametric study
To further clarify the origin of the VF instability shown in figure 1, we have conducted a parametric study varying the concentrations of PAA and TB, keeping the injection rate constant. Figure 4(a) shows the displacement pattern for the same TB concentration as in figure 1 but a larger PAA concentration, while figure 4(b) features the displacement for the same PAA concentration as in figure 1 but a lower amount of dye. Note that the blue colour is here lighter because the dye concentration is smaller. In both cases, VF hardly occurs.
To understand why VF disappears for the experimental values of parameters considered in figure 4, we have reconstructed the viscosity profiles for those conditions using the methodology described above, assuming
$\delta =10$
. Figures 5(a) and 5(b) show the viscosity profiles corresponding to the values of parameters of figures 4(a) and 4(b), respectively. In both cases, the viscosity profiles still exhibit a non-monotonic behaviour, but the amplitude of the maximum is much smaller that in the unstable case of figure 3. To appreciate this, let us compare the viscosity of the displacing fluid denoted
$\mu _{bulk}$
to the viscosity of the local maximum
$\mu _{max}$
. Note that here the displaced less viscous fluid is always water, the viscosity of which is minimal. In the stable cases of figures 4 and 5, the ratio
$\mu _{max}/\mu _{bulk}$
is roughly equal to 1.1–1.3, while it is closer to 2.4 in the unstable dynamics corresponding to figures 1 and 3(a). These results suggest that the instability is indeed triggered by the extremum in viscosity but that its amplitude should exceed a given threshold to induce the destabilisation. Note that figure 1 shows that the effect of the dye on the viscosity of PAA is strongest for
$c_{\textit{PAA}}= 0.5$
wt
$\%$
and
$c_{\textit{TB}}$
in the range 0.01–0.1 wt
$\%$
, which are the values of parameters for which we get the highest
$\mu _{max}/\mu _{bulk}$
shown in figure 3(a). The less strong change of the viscosity of PAA on the dye for
$c_{\textit{PAA}}= 0.5$
wt
$\%$
in the regimes where
$c_{\textit{TB}} \lt 0.01$
wt
$\%$
or
$c_{\textit{TB}} \gt 0.1$
wt
$\%$
results in the lower
$\mu _{max}/\mu _{bulk}$
shown in figure 5(b). The amplitude of this
$\mu _{max}/\mu _{bulk}$
ratio is thus a subtle function of both
$c_{\textit{PAA}}$
and
$c_{\textit{TB}}$
in addition to the value of
$ \delta$
. Note that for shear rate 1 s
$^{-1}$
, measurements of the first normal stress difference as in Nagatsu et al. (Reference Nagatsu, Matsuda, Kato and Tada2007) show that elastic effects of the 2 wt
$\%$
PAA solution cannot be neglected. However, as elastic effects of the 0.5 wt
$\%$
PAA solution can be neglected, such elastic effects should not be responsible for the disappearance of the fingering.
Displacement patterns at
$t = 300$
s observed when a solution of PAA (MW = 1.25 million) and TB in various concentrations displaces water for
$Pe = 1.6 \times 10^3$
and
$b = 0.3$
mm: (a)
$C_{\textit{PAA}}= 2$
wt % and
$C_{\textit{TB}}= 0.1$
wt %; (b)
$C_{\textit{PAA}}= 0.5$
wt % and
$C_{\textit{TB}}= 0.01$
wt %.
The fact that the viscosity of the injected solution depends on two chemical species is a key prerequisite here to obtain a local extremum in its spatial profile. If the viscosity of the PAA solution becomes independent of the presence of TB, then the viscosity can only monotonically decrease along the displacement. To show this, we have repeated the experiments with a solution of PAA with a much smaller molecular weight (0.25 million, PAA from Wako, Japan) and larger concentrations of 2.0 and 4.0 wt %. As shown in figure 6, in that case the viscosity of the polymer does not vary when the TB dye is added. In that case, only monotonic decreasing viscosity profiles can build up, and no VF instability is observed, as seen in figure 7. This confirms that the destabilisation of the reverse displacement shown in figure 1 is obtained for some PAA of sufficient molecular weight for which the dye affects its viscosity.
For future possible modelling, we note that our system is different from the pure double-diffusive case of Mishra et al. (Reference Mishra, Trevelyan, Almarcha and De Wit2010) where the viscosity is modelled as
$\mu =\exp\,[R_s S + R_f F]$
, with
$S$
and
$F$
the concentrations of the slow and fast species, respectively, and
$R_s$
,
$R_f$
the constant log-mobility ratios. In their case, each species influences the viscosity of the solution independently of the other one. In the PAA/TB system presented here, the state equation of viscosity as a function of the concentration of
$A$
= PAA and
$B$
= TB is more complex and probably of the type
$\mu =\exp\,[R(B)\, A]$
such that TB alone does not change the viscosity of water, but the log-mobility ratio
$R$
in the viscosity of the PAA solution varies with the concentration
$B$
of TB.
Viscosities of the 2.0 and 4.0 wt % solutions of PAA (MW = 0.25 million) as functions of the concentration of TB, measured at
$\dot {\gamma } = 1\ \text{s}^{-1}$
.
Displacement patterns observed at
$t=300$
s for
$Pe = 1.6 \times 10^{3}$
and
$b = 0.3$
mm when water is displaced by a 4.0 wt % solution of PAA (MW = 0.25 million) containing TB in concentration (a) 0.01 wt %, (b) 0.1 wt %, (c) 1 wt %.
5. Conclusions
In conclusion, we have experimentally demonstrated viscous fingering (VF) of a reverse displacement of water by a more viscous solution of a poly(acrylic acid) (PAA) polymer dyed by Trypan Blue (TB) in a Hele-Shaw cell. Classically such a displacement is supposed to induce a circular displacement because the more viscous fluid is injected into the less viscous one, a hydrodynamically stable case. Yet we have shown that for PAA of sufficiently large molecular weight, a clear fingered pattern is obtained. The instability here is not due to a reaction as the displaced solution is pure water. A pure double-diffusion VF involving two solutes changing independently the viscosity as analysed by Mishra et al. (Reference Mishra, Trevelyan, Almarcha and De Wit2010) is also not at play as the TB dye alone does not change the viscosity of water. Non-ideal mixing effects between two solutes initially separated, as studied by Haudin et al. (Reference Haudin, Callewaert, De Malsche and De Wit2016), are also not relevant as the dye and the polymer are initially present in the same displacing fluid.
To clarify the origin of the instability, we have shown that the polymer solution’s viscosity increases with the concentration of PAA, whereas it decreases with that of TB. The viscosity profile has been reconstructed based on the concentration profiles of PAA and TB obtained from one-dimensional diffusion equations. If the polymer and the dye are assumed to diffuse at the same rate, then the viscosity profile remains monotonically decreasing, and no instability can develop. However, if the dye is assumed to diffuse faster than the long chain polymer, then the viscosity profile is non-monotonic with a local maximum. If the ratio of the local maximum viscosity to the viscosity of the displacing solution is sufficiently large, then a reverse VF instability is observed. However, when the ratio is small, the VF instability hardly occurs.
These results show that reverse VF is here obtained as a combination of a change in the viscosity of a polymer solution due to the presence of a dye, and double-diffusive effects. While the potential active role of dyes has already been evidenced in buoyancy-driven flows (Kuster et al. Reference Kuster, Riolfo, Zalts, El Hasi, Almarcha, Trevelyan, De Wit and D’Onofrio2011), we have evidenced here for the first time its active role in triggering a viscous instability, inviting us therefore to use dyes with caution in future experiments on viscous effects. Our results are also the first experimental evidence of a VF instability triggered by differential diffusion effects, which paves the way to analyse new fingering dynamics using solutions containing more than one solute.
Acknowledgements.
We thank S. Iwata and L. Riolfo for conducting preliminary experiments, and Y. Tada, P.M.J. Trevelyan and D. Escala for fruitful discussions.
Funding.
Y.N. acknowledges the financial support of KAKENHI from the Japan Society for the Promotion of Science (JSPS) (grant no. 22686020). A.D. acknowledges the financial support of the FRS-FNRS PHYLLO project and of PRODEX. We both gratefully acknowledge the Institute of Global Innovation research of TUAT for supporting financially our bilateral collaboration.
Declaration of interests.
The authors report no conflict of interest.
