Rémy Mensire

Jesse T Ault

Elise Lorenceau

Howard A Stone

Point-source imbibition into dry aqueous foams

Keywords: 47, 55, nb -Capillary flows 82, 70, Rr -Foams

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has the ability to absorb liquids similar to a sponge. In-20 deed, the interfacial energy of a wet foam is lower than the 21 interfacial energy of a dry foam since the bubbles are more 22 spherical than in a dry foam. Thus, when a dry aqueous 23 foam is put into contact with the same miscible liquid, imbibition is driven by the reduction of surface energy that occurs when going from a dry to a wetter foam.

Not surprisingly, imbibition phenomena for undeformable and deformable solid porous media have been studied extensively in the literature [START_REF] Lucas | [END_REF][5][START_REF] Gennes | Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls and Waves[END_REF][START_REF] Reyssat | [END_REF][8][9][10][11]. Also, imbibition of aqueous foams has been studied theoretically and experimentally for different configurations: onedimensional imbibition of the same foaming liquid at atmospheric pressure and pulsed imbibition at constant volumes and 2D foam drainage at constant flow rate [12][13][14][15][START_REF] Hutzler | Colloids and Surfaces A: Physicochem[END_REF]. With recent chemical formulations [17,18] making the liquid/air interfaces more rigid, foams absorb not only miscible liquids [12], but also immiscible liquids [19,20], such as organic oils, for the right oil-surfactant combination. This is of considerable interest in oil recovery and soil remediation processes. For those potential applications, oil is generally trapped into micropores at a defined pressure (generally lower than the atmospheric pressure).

We develop a mathematical model and compare with experimental results for the radial imbibition from a point source (at imposed pressure), which mimics liquid extraction from a micropore. Also, we develop a model to ex- plain why the imbibition of oil is possible in some cases despite the creation of new oil-water interfaces.

Experiments. -We use a foaming solution of viscosity η = 1.4 mPa.s and density ρ = 1.016 g.cm -3 , based on 0.66 wt.% of sodium lauryl-dioxyethylene sulfate (SLES;

Stepan Co.), 0.34 wt.% cocoamidopropyl betaine (CAPB;

Stepan Co. and Evonik), 0.04 wt.% myristic acid (MAc; Sigma-Aldrich), and 10 wt.% glycerol [17,18]. As we use different isomers for SLES (because they come from different sources), the air/water surface tension of the foaming solution varies from γ ow = 23.7 to 25.5 ± 1 mN/m (measured with the pendant drop method).

As imbibing liquids, we use olive oil and two glycerol solution mixtures, whose wt.% of glycerol are different; the properties (oil-water surface tension, dynamic viscosity, density) are summarised in Table 1. We also add a small quantity of fluorescent dyes (1:50) to the imbibing liquid (Yellow Black from Rohm and Haas, Fluorescein from Sigma-Aldrich and Tracerline) to enhance the contrast between the foam and the imbibing liquid. We checked that the addition of dyes does not change the value of γ ow .

We generate foams with a well-controlled average bubble radius R b = 1 -2 mm by injecting nitrogen or compressed air through a needle into the foaming solution.

The rigid interfaces owing to the type of surfactants used yield a stable, monodisperse foam (deviations from the mean bubble radius remain below 5 %). After generating the foam in a 20 cm-high rectangular column with a 4.5 cm-wide square base, we let it drain and extract at random times at the top of the column a foam sample for which the volume V and the weight m are known. The sample is turned upside down to invert the drainage process, leading to the homogenization of the liquid fraction.

The initial liquid fraction of the foam sample is deduced from i = m/(ρV ).

For the imbibing reservoir, we use a polycarbonate plate into which a 1 mm-wide hole is drilled. A small capillary tube of 1 mm diameter is inserted into the hole and slightly displaced upwards into the foam to ensure complete contact between the foam and the reservoir. This capillary tube is connected to a 20 cm-long and a 2 mm-diameter 87 PTFE tube (Fig. 1). This set of tubes is filled with the 88 imbibing liquids, which completely wet the tubes. The 89 bigger tube is attached to a motion controller that allows 90 the adjustment of the input liquid level at the same height 91 as the output liquid level while the liquid flows through 92 the foam. This system sets a constant atmospheric pres-93 sure p atm at the outlet of the capillary tube. We also use 94 another system without feedback for comparison, by using 95 a 5 cm-wide funnel instead of a motion controller. Indeed, 96 due to the width of the funnel, input pressure remains con-97 stant because the output level does not vary much when 98 the liquid is imbibed. Both systems give the same results 99 as shown below.

100

At time t = 0, the bottom of the foam touches the point 101 source of the liquid. We record (at 24 frames per second 102 for 3 to 4 minutes) foam imbibition by using fluorescent 103 imaging. Fig. 2 shows a typical image sequence of the 104 imbibition process. The same experiment is repeated for Miscible liquids. To explain experimental observations, we recall the foam drainage equation [13,21,22], which describes foam imbibition and drainage. We make the following assumptions:

-We consider a dry foam with a constant mean bubble radius R b and a low liquid fraction w < 5 x 10 -3 .

-The size of the point source is much smaller than the typical dimensions of the foam and the foam domain can be considered as an infinite space for imbibition.

-We assume isotropy of the foam. We use cylindrical coordinates (radial coordinate r and axial coordinate z).

Using Darcy's law, we relate the average liquid velocity u Darcy to the pressure gradient,

u Darcy = w u P B = k η w (-∇p + ρg) (1)
where u P B is the mean velocity of a liquid flowing into a Plateau border in a Poiseuille-like flow, k the foam permeability (m 2 ), w the liquid fraction, η w the dynamic viscosity of the invading miscible liquid, p the pressure in the Plateau borders and ∇ depends on r and z only.

To describe the pressure gradient, we resort to a microscopic analysis of the imbibition in the Plateau border.

When a dry aqueous foam is wetted by a miscible liquid, 142 the physical mechanism that enables the imbibition is the 143 reduction of the surface area of the air-water interfaces in-144 side the foam, i.e, the total energy of the system decreases 145 [23]. The liquid flow swells the Plateau borders, creating 146 more surface area. However, if we assume a constant vol-147 ume of gas in the bubbles, the surface area of the films 148 needs to decrease.

149

The reduction of interfacial energy is used by the foam to pump a volume dV of liquid at the osmotic pressure Π = p atm -p (Π is identical to the capillary pressure if we assume that the pressure in the bubbles is equal to atmospheric pressure). Thus, we have:

ΠdV = -γdS , dS < 0 , ( 2
)
where γ is the interfacial tension, dS = dS f + dS P B is the 150 variation of the air-water area, and S f and S P B are the 151 film and Plateau border area, respectively (Fig. 4a). For a dry aqueous and an ordered monodisperse foam, the radius of curvature of a Plateau border r P B is related to the bubble radius R b as 24], where δ b = 1.74 is a numerical factor deduced from the geometry of a Kelvin cell. As shown analytically in the limit of dry foam in [24], the osmotic pressure can be written as:

r P B = δ b R b 1/2 w [2,
Π = p atm -p ≈ p b -p ≈ γ aw r P B ≈ γ aw δ b R b 1/2 w (3)
where p b is the pressure in the bubbles and γ aw the airwater interfacial tension. For rigid interfaces, we consider a channel-dominated model for the foam permeability k [3, [25][26][27][28]:

k = δ a δ 2 b R 2 b 2 w 150 ( 4
)
with δ a = √ 3 -π 2 , a numerical factor deduced from the 153 cross-sectional area of a Plateau border A = δ a r 2 P B .

154 Differential equation for the cross-sectional area of a Plateau border A. By combining (1), ( 3) and (4) with the unit vector e z directed upwards, we have:

u P B = - γ aw δ a δ b R b 300η w 1/2 w ∇ w - δ a δ 2 b R 2 b ρg w 150η w e z (5)
We consider mass conservation in the Plateau border for the liquid phase, which is expressed as:

∂ w ∂t + ∇• ( w u P B ) = 0 (6)
From ( 5), ( 6) and w = A/(δ a δ 2 b R 2 b ), we deduce a nonlinear partial differential equation for the time-space evolution of the cross-sectional area of a Plateau border [13],

∂A ∂t = γ aw δ 1/2 a 300η w ∇• A 1/2 ∇A + ρg 150η w ∂A 2 ∂z (7)
For the initial condition, we assume that the foam is dry, w = 0. For the boundary conditions, we assume that the liquid fraction and thus the cross-sectional area of a Plateau border far from the point-source is close to 0. At the point-source, A = δ a R 2 b , which is the cross-sectional area for the maximum packing of undeformed bubbles. Thus, we have the initial and boundary conditions:

A(r > 0, z > 0, 0) = 0 , A(r, z → +∞, t) = 0 A(r → +∞, z, t) = 0 , A(0, 0, t) = δ a R 2 b ( 8
) and the no-flux condition at z = 0:

∂A ∂z = -2ρg δ 1/2 a γaw A 3/2 .

Solution to the partial differential equation (PDE).

We non-dimensionalize A, r, z and t as α

= A/(δ a R 2 b ), R = r/R b , Z = z/
∂α ∂τ = B ∂α 2 ∂Z + 1 2 1 R ∂ ∂R Rα 1/2 ∂α ∂R + ∂ ∂Z α 1/2 ∂α ∂Z ( 9
) with the initial and boundary conditions:

α(R > 0, Z > 0, 0) = 0 , α(R, Z → +∞, τ ) = 0 α(R → +∞, Z, τ ) = 0 , α(0, 0, τ ) = 1 (10) and the no-flux condition in Z = 0: ∂α ∂Z = -2Bα 3/2 .
To solve Eq. ( 9) with the conditions (10), we use a discretization by finite differences on a 2D spatial mesh and solve the PDE with Matlab. The numerical solution gives α(R, Z, τ ), as shown in Fig. 5a. The front position, as well as the entire front, is determined when α = 0 as α decreases from the point source to the outer boundary of the mesh, which gives the front profile for different times and Bond numbers (Fig. 5b-d). The simulations show that the front profile flattens out as B and τ increase.

Immiscible liquids.

For miscible liquids, the decrease of interfacial energy drives the imbibition into a dry aqueous foam. For immiscible liquids, such as organic oils, we sketch the liquid flow (in the dry limit) by an oil slug that penetrates the Plateau border, as shown in Fig. 4b; new air-water surfaces in the Plateau border are created and the surface area of the films decreases. However, new oil-water interfaces are also created, which are ener-178 getically costly. Therefore, we develop another model to 179 add the influence of these oil-water interfaces.

180

Rewriting the osmotic pressure from (2), by using the interfaces dS w and dS o for the variations of the air-water and oil-water interfaces, respectively, yields:

ΠdV = -(γ aw dS w + γ ow dS o ) with dS w < 0 and dS o > 0 (11)
Equation ( 11) can also be expressed as:

ΠdV = -γ aw dS w 1 + γ ow γ aw dS o dS w = -γ ef f dS w (12)
with γ ef f an effective interfacial tension:

γ ef f = γ aw 1 + γ ow γ aw dS o dS w (13)
For miscible liquids, γ ow = 0, so γ ef f = γ aw and the os-181 motic pressure is related to the curvature of the Plateau 182 borders by (3). However, if γ ow = 0, then γ ef f ac-183 counts for the oil-water interfacial tension and the cre-184 ation of oil-water interfaces. The osmotic pressure be-

185 comes Π = γ ef f /r P B ≈ γ ef f /(δ b R b 1/2
) where is the 186 combined liquid fraction for oil and water. Since dS w < 0, 187 then γ ef f < γ aw . With immiscible liquids, the osmotic 188 pressure is reduced compared to miscible liquids, i.e., the 189 driving force for imbibition is weaker. The value of γ ef f 190 can be estimated. In particular, the variation of the air-191 water interfaces dS w can be decomposed into two contri-192 butions, one from the films dS f , and the other from the 193 Plateau borders dS P B (Fig. 4b).

194

Pitois et al. [29] and Hilgenfeldt et al. [30] provide estimates of the film and Plateau border surface areas with p-4 regard to the liquid fraction in the foam,

S f ≈ 3.3 R b V f oam (1 -)(1 -1.52 1/2 ) 2 (14a) S P B ≈ V f oam 1.5R b π √ 3δ a 1/2 (14b)
where V f oam is a volume of foam. Differentiating (14a) and (14b) with respect to in the dry limit yields,

dS f ≈ - V f oam R b (3.3)(3.04) 2 1/2 d (15a) dS P B ≈ V f oam R b π 3 √ 3δ a 1/2 d (15b)
In the dry limit, we can assume as a first approximation that dS o ≈ dS P B and neglect the oil-water interface at the top of the oil slug (Fig. 4b). This assumption is in agreement with recent Surface Evolver simulations of the shape of an oil slug in a single Plateau border [27]. In our case, the ratio γ ow /γ aw ≈ 0.2 is low and the ratio between the equivalent spherical radius for an oil slug embedded within the Plateau border and the radius of curvature of the Plateau border is close to 1 due to the very low liquid fraction. Thus, by using (15a) and (15b), we have:

γ ef f ≈ γ aw + γ ow dS P B dS f + dS P B ≈ γ aw 1 -3 γ ow γ aw (16)
With this effective interfacial tension, the osmotic pressure is reduced by a factor that depends on the ratio between the oil-water and the air-water interfacial tensions. More precisely, the governing equation for A is the same as developed previously for aqueous liquids in [START_REF] Reyssat | [END_REF] with A, the cross-sectional area of the Plateau border for both oil and water phases, γ ef f , instead of γ aw , and η o the oil viscosity:

∂A ∂t = γ ef f δ 1/2 a 300η o ∇• A 1/2 ∇A + ρg 150η o ∂A 2 ∂z (17)
We only use η o because the viscous dissipation occurs predominantly in the oil phase as shown by Piroird et al. [19] in the limit of η o /η w 1. Eq. ( 17) is identical to [START_REF] Reyssat | [END_REF] and can be solved numerically, except with a smaller air-water interfacial tension. The smaller capillary pressure induced by the oil phase is in agreement with our experiments: for the same elapsed time, the vertical front position of the oil is smaller than the aqueous case, as shown in Fig. 3.

In Fig. 6, we rescale z In both miscible and immiscible cases, the numerical 213 solutions that include the capillary pressure gradient and 214 gravity are comparable to the experimental data. Gravity 215 should be considered, since the numerical solution with 216 B = 0 largely deviates from the experimental data, and 217 gravitational effects flatten out the imbibition profiles very 218 quickly as shown in Fig. 5. However, our model devi-219 ates from the data at short times. Indeed when the front 220 position is of the order of magnitude of one bubble size 221 (R * < 2), the Darcy model for the average velocity is not 222 adequate, as imbibition occurs in individual Plateau bor-223 ders. Also, = 0 around the point source.

224

Analytical solution with B = 0.

An analytical scaling for the front position in the no-gravity case (B = 0), which is encountered in microgravity conditions, can be found by assuming spherical symmetry with the dimensionless spherical distance R. We introduce the dimensionless self-similar variable ζ = R/τ 1/2 . Substituting ζ into (9), written with γ ef f and η o/w , we find that α is the solution of the ordinary differential equation: f α * = α yields the same equation as (18) except with ζ f = 1. Therefore, we can take ζ f = 1 without loss of generality. The wetting front r f is given by:

r f (t) R b = δ a γ ef f 150η o/w R b t 1/2 (19)

  liquid phases. When the liquid fraction (ratio between the 13 liquid volume and the total volume of the foam) is low, an 14 aqueous foam is a dense assembly of bubbles. The inter-15 sections between the bubbles consist of films, vertices and 16 liquid microchannels called Plateau borders, which have a 17 curvature creating a capillary underpressure in the liquid 18 phase. Due to this pressure difference, an aqueous foam

19

 19

Fig. 1 :

 1 Fig. 1: Experimental set-up. A 30 cm-long PTFE tube (2 mmdiameter) is filled with the imbibing liquid (mixtures of foaming solution-glycerol or oil) and terminated by a 1 cm-long capillary tube (1 mm diameter) that enables the connection with the foam. A motion controller allows maintainence of the pressure of the liquid phase input at atmospheric pressure patm.

Fig. 2 :

 2 Fig. 2: Snapshots of the imbibition front for a 30-second timescale.The imbibing liquid (glycerol-foaming solution) is fluorescent and the front is defined as the boundary between the brighter and darker fluorescent areas. The liquid-filled tube is displaced upwards by a millimeter in the first snapshot to ensure contact between the source and the foam.

105

  the different imbibing liquids of Table1. Thus, different 106 liquids, different viscosities η, bubble radii R b , air-water 107 (γ aw ) and oil-water (γ ow ) interfacial tensions, and initial 108 liquid fractions i are tested. We determine the imbibition 109 front by applying a threshold at the boundary between the 110 brighter and darker fluorescent areas. Using ImageJ soft-111 ware for image processing, we plot the maximum vertical position z f of the front with respect to time.Results and discussion. -Fig.3shows two data sets of for the evolution of the vertical front position z f (t).The experimental data for two different liquids typically have error bars of ±0.5 mm, owing to the determination of the front, especially in the case of miscible liquids, where light diffusion by the films can create a small front width.For both types of liquid, the position of the front moves faster at short times and slows down as time increases. For similar viscosities, oil imbibition is slower than aqueous imbibition.

Fig. 3 :

 3 Fig. 3: Evolution of the vertical front position z f with respect to time t for two data sets. The round dots correspond to a glycerolfoaming solution and the square dots to olive oil. The bubble radius R b , the initial liquid fraction i , and the viscosity η o/w of the imbibing oil or aqueous solution are indicated, respectively, in the legend.

152Fig. 4 :

 4 Fig. 4: a. Cross-section of a Plateau border for a dry aqueous foam. b. Cross-section of a Plateau border filled with oil for a dry aqueous foam. S P B and S f , respectively, denote the areas of the side surfaces of the Plateau borders and the surface of the films that are connected to the Plateau borders. So represents the area of the oil-water interfaces.

  R b and τ = (δ a /150)t/(η w R b /γ aw ), and introduce the Bond number B = ρgR 2 b /γ aw , transforming (7) into:

Fig. 5 :

 5 Fig. 5: a. Typical numerical solution of (9) for the dimensionless Plateau border area α for B = 0 and τ = 1000. b-d. Numerical solutions of the imbibition front for different times τ = 10, 50, 100 and 500 with Bond numbers B = 0, 1.6 and 5.

  f by R b and t by the capillary time (150/δ a )η o/w R b /γ ef f , which is the same scaling as in (9) written with γ ef f instead of γ aw , and the viscosity of oil or aqueous solutions η o/w . The log-log plot in the dimensionless variables displays a reasonable collapse between the experimental data and the numerical solution for all miscible and immiscible imbibing liquids for the range of non-zero Bond numbers between 0.5 and 5. The collapse occurs at shorter τ for immiscible liquids, due to the smaller scaling in time induced by γ ef f .

Fig. 6 :

 6 Fig. 6: Vertical front position z f with respect to time in dimensionless coordinates for immiscible oils (open dots) and miscible aqueous liquids (closed dots). The experimental curves are obtained for two R b = 1-2 mm, different i , η o/w and B. The numerical solutions of (9) for Bond numbers B = 0, 1.6 and 5, deduced from (9), are plotted by the dashed lines. The self-similar power law evolution in τ 1/2 in the no-gravity case (Eq. 19) is shown.

150η o/w γ ef f δaR b 1 /2 r f t 1 / 2

 112 condition is α(ζ f ) = 0, where ζ f = 225 is a constant which then yields the 226 spherical front radius r f (t). A local analysis at the front 227 provides a second boundary condition and uniquely deter-228 mines the solution. 229 Setting ζ 4

Table 1 :

 1 Characteristics of the imbibing liquids. γow (mN/m) is the oil-water interfacial tension, η (mPa.s) the dynamic viscosity, ρ (g/cm 3 ) the density, R b (mm) the average bubble radius of the foam and B the Bond number defined in(9).

	Imbibing	γ ow	η	ρ	R b B
	liquid
	CAPB+SLES	0	64	1.21	1	0.5
	MAc+80% gly
	CAPB+SLES	0	15	1.17	2	1.8
	MAc+65% gly
	CAPB+SLES	0	1.4 1.016	2	1.7
	MAc+10% gly
	Olive oil 1	6	61	0.88	1	1.5
	Olive oil 2	6.2 68.5 0.88	2	5
	Sunflower oil	4.5	55	0.91	2	3

Princeton University, and Paris-Est University.

result recalls the 1D diffusive imbibition in a Hele-Shaw 231 cell observed in [12,31]. However, the collapse between 232 the PDE solution with B = 0 and the self-similar solution 233 in t 1/2 occurs at long times for τ > 10 4 . The value of α at 234 the point source is equal to 1. Yet, the self-similar solution blows up and is not consistent around the point source 236 [START_REF] Christov | Proc. Natl. Acad. Sci[END_REF]. Also, given the values of the Bond numbers from 237 0.5 to 5, the diffusive solution (and thus the no-gravity 238 approximation) is not valid over our experimental range.

Conclusions.

The which are neglected in [START_REF] Xiao | [END_REF], since B 1 due to the micron 254 size of the glass beads in their porous matrix.

255

From our results, we identify two criteria that define