Tian
Yu
^{a},
Qicheng
Sun
^{a},
Chen
Zhao
^{bc},
Jiajia
Zhou
*^{bd} and
Masao
Doi
^{b}
^{a}State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China
^{b}Center of Soft Matter Physics and its Applications, Beihang University, Beijing 100191, China. E-mail: jjzhou@buaa.edu.cn
^{c}School of Physics, Beihang University, Beijing 100191, China
^{d}School of Chemistry, Key Laboratory of Bio-Inspired Smart Interfacial Science and Technology of Ministry of Education, Beihang University, Beijing 100191, China

Received
26th September 2020
, Accepted 3rd December 2020

First published on 7th December 2020

When a capillary channel with corners is wetted by a fluid, there are regions where the fluid fills the whole cross-section and regions where only the corners are filled by the fluid. The fluid fraction of the partially-filled region, s*, is an important quantity related to the capillary pressure. We calculate the value of s* for channels with a cross-section slightly deviated from a rectangle: the height is larger in the center than those on the two short sides. We find that a small change in the cross-section geometry leads to a huge change of s*. This result is consistent with experimental observations.

An important quantity governing the capillary phenomena in porous media is the capillary pressure p_{c},^{1} defined as the pressure difference across the interfaces separating two phases (normally one is air and the other one is fluid). If the shape of the meniscus is known, the capillary pressure is given by the Laplace pressure

(1) |

Two recent studies in rectangular channels motivated our study. Keita et al.^{11} studied the drying dynamics in a nearly-rectangular channel, the thickness of which is slightly larger in the middle than in the edges (the thickness is 115 μm in the middle and 95 μm at the edges; the width of the channel is 2 mm). During the evaporation, two thick fluid fingers were present in the partially-saturated region (left column in Fig. 1). This is in contrast to a later study by Seck et al.^{12} In this case, the rectangular channel was nearly perfect, and very thin fluid columns were observed (right column in Fig. 1). These two experiments clearly showed that a slight deviation from the rectangular geometry changes the value of s* enormously. The purpose of this paper is to explain these phenomena based on a free energy model.

Fig. 1 Top views of the nearly-rectangular channel (open to the left) at different drying times. The left column is from ref. 11, in which the height of the cross-section is large in the middle. The right column is from ref. 12, in which the cross-section is a nearly-perfect rectangle. (Reprinted from ref. 11 and 12 with permission from the Springer Nature.) |

Now we place some wetting fluid into the channel and seal both ends. The amount of fluid is chosen such that the fluid only partially fills the channel, as shown schematically in Fig. 2(a). In the thermodynamic equilibrium, some portion of the channel is fully filled by the fluid with the saturation s = 1 (the saturation s is defined as the area fraction occupied by the fluid in the x–y cross-section), and some portion is only partially filled with a saturation s* < 1. Our goal is to determine the value of s*, and how its value varies when other geometrical parameters, such as α and β, change.

The coexistence between the fully-filled region and partially-filled region can be seen more clearly in the cutting x–z plane shown in Fig. 2(b): The AB part corresponds to full saturation s = 1, while the CD part corresponds to the partial saturation s*. We shall refer the fully-saturated region “bulk” and the partially-saturated region “finger”. There is a transition region BC going from the bulk to the finger. The length of the transition region is finite and with a characteristic size of a. The effect of the transition zone can be neglected in the thermodynamic limit when the length of the bulk (AB) and the finger (CD) both become larger than the length of the transition zone (BC).^{13,14} The fluid has a surface tension γ and an equilibrium contact angle θ_{E}. In the cornered geometry, only when the condition

(2) |

(3) |

(4) |

S_{0} = 2acosαb + 2acosαasinα = 2a^{2}cosα(sinα + β), | (5) |

(6) |

Since there are solid surfaces exposed to the vapor, the range of is limited by < b/2. This leads to the following constraint on the saturation

0 < s < s_{c1}, | (7) |

(8) |

The free energy density f(s) is expressed as

(9) |

Using eqn (3) and (6), the free energy density can be written as a function of s

(10) |

(11) |

(12) |

(13) |

(14) |

The saturation s in this case has two limits. The lower bound s_{c1} is given by eqn (8) and the upper bound is constrained by < a,

s_{c1} ≤ s ≤ s_{c2}, | (15) |

(16) |

The free energy density f(s) is expressed as

(17) |

(18) |

(19) |

(20) |

s_{c3} < s ≤ 1, | (21) |

(22) |

The free energy density f(s) is expressed as

f(s) = −(4a + 2b)γ + 2πrγ, | (23) |

(24) |

The free energy density at full saturation (s = 1) is

(25) |

As a summary, the free energy densities for the three cases are listed below

Fig. 4 Example free energy density f(s) as a function of the saturation s for β = 0.095. (a) tanα = 0.002. (b) tanα = 0.1. The different colors of the curves correspond to the three cases [red (I), green (II), and blue (III)]. The coexistence conditions of (32) are shown with gray lines. (c) and (d) The function g(s) [eqn (33)] corresponding to (a) and (b). In (c), s* locates in case (I) and in (d), s* locates in case (II). |

F_{total} = Lϕf(s) + L(1 − ϕ)f(1). | (26) |

LϕS_{0}s + L(1 − ϕ)S_{0} = V_{l}. | (27) |

Introducing a Lagrangian multiplier p_{c}, we can minimize the following function with respect to the fraction ϕ and the saturation s

= F_{total} + p_{c}(LϕS_{0}s + L(1 − ϕ)S_{0} − V_{l}) |

= Lϕf(s) + L(1 − ϕ)f(1) + p_{c}(LϕS_{0}s + L(1 − ϕ)S_{0} − V_{l}) | (28) |

(29) |

(30) |

The second eqn (30) leads to the identification of the Lagrangian multiplier as the capillary pressure

(31) |

(32) |

Graphically, the coexistence condition (32) corresponds to drawing a straight line passing through the free energy curve at both points (s*,f(s*)) and (1,f(1)), and the line should also be tangential to the free energy curve at s = s*. Example lines are shown as the gray lines in Fig. 4(a) and (b).

Since the value of f(s) changes a lot from s = 0 to 1, it is difficult to see the tangential line in the f(s) plots. For the purpose of illustration, we define another function g(s)

g(s) = f(s) + (1 − s)f′(s*) − f(1). | (33) |

Since the free energy curves can be separated into three regions, the position of s* might be in each of those regions. However, for case (III), the free energy density (24) has the form with F_{3} > 0. The second derivative f′′(s) < 0, so s* cannot exist in the range of case (III). We will discuss two cases that either s* is located in case (I) or case (II).

(34) |

Substituting the above expression into eqn (32) and using f(1) from eqn (25), we obtain an equation for s*

(35) |

(36) |

0 < s* < s_{c1}. | (37) |

For a perfect rectangular channel (α = 0), we obtain

(38) |

For a square cross-section (β = 2), we obtain s* ≃ 0.06. This is consistent with ref. 17 and 18.

In Appendix B, we derive an analytic expression of s_{⋄}* for the diamond channel,

(39) |

S_{⋄0}s_{⋄}* − 2S_{△} = S_{0}s*, | (40) |

(41) |

(42) |

s_{c1} ≤ s* ≤ s_{c2}. | (43) |

When α is close to zero (not equal to zero), the trigonometric functions take simple form sinα ≃ α, tanα ≃ α, cosα ≃ 1. Using these simplifications, approximated expressions for the diamond and nearly-rectangle can be obtained

(44) |

(45) |

(46) |

Fig. 6 shows the value of s* as a function of tanα for different values of β. For the special case of the diamond channel, β = 0, s* is given by eqn (39). This function is plotted with the solid black line in Fig. 6. It is a monotonically decreasing function when tanα increases. Also plotted is the approximated solution eqn (44). The approximated solution has good agreement with the exact formulae at small tanα, but shows noticeable deviation at large tanα.

Fig. 6 The s* value as a function of tanα and β. The solid lines are exact solutions: black, β = 0, diamond channel, eqn (39); red, β = 0.02, nearly-rectangular channel, eqn (36) and (42); green, β = 0.06; blue, β = 0.095; violet, β = 0.20. The dashed lines are approximated solutions given in eqn (44) and (45). There is no approximated solution for β = 0.20. The inset shows the enlarged part near small tanα, where the transition from case (I) to case (II) takes place at tanα_{crit}. |

For nearly-rectangular channel, β > 0, representative curves are shown in solid lines in Fig. 6. The value of s* shows two different regions. When tanα is small, s* corresponds to the four-finger configuration of case (I), and the expression is given in eqn (36). In this case, the value of s* is also very small (see the inset for the absolute values). When tanα is large, s* is located in the range of case (II) and its expression is given by eqn (42). In this case, the value of s* is large and has a non-monotonic dependence on tanα. We also plot the approximated solution (45) in dashed lines. The agreement between the approximated and exact solutions is good when α is small or β is small.

The two regions of s* are connected at a critical α_{crit} value. When β increases, i.e. as the constriction becomes less elongated, the critical α moves to a larger value. This trend is also demonstrated in Fig. 7(a) with the red solid line.

When s* is located in the range of case (II), the value of s* is very sensitive to α. For example, for β = 0.02 shown as the red line in Fig. 6, s* increases from nearly zero to about 0.7 when tanα changes from 2 × 10^{−4} to 5 × 10^{−3} (less than 0.2° change). In this range, the value s* increases first with increasing tanα, reaching a maximum, then decreases when tanα increases further. The location of the maximum α_{max} increases with increasing β. This is shown as the blue solid line in Fig. 7(a). Also shown is the approximated value (α_{max})_{app} = β/3 in dashed line, which again has better agreement with the exact solution at small β. The maximum value of (s*)_{max} is a decreasing function of β, shown in Fig. 7(b). When tanα increases further after passing the maximum, the cross-section of the channel resembles a diamond with two tips removed (small S_{Δ} in Fig. 5). In this case, s* curves of different β values all approach the curve of the diamond channel.

In the experiment of Keita et al.,^{11} the aspect ratio is β = 0.095 and the deviation from the rectangle tanα = 0.01. From Fig. 6, we can obtain s* ≃ 0.2987 by eqn (42). This value of s* is located in case (II) and the menisci are composed of two fingers. Our result is consistent with the observation in ref. 11 (see the left column of Fig. 1). The experimental value of s* (at the finger/bulk interface) is about 0.38, which is close to our prediction.

In the experiment of Seck et al.,^{12} the aspect ratio is β = 0.2 and the cross-section is nearly perfect rectangle, so α is very small. In this case, Fig. 6 shows that a very small s* exists over a large range of tanα values, corresponding to case (I) with four fingers. The value of s* given by eqn (36) is s* ≃ 0.018, again this is consistent with the observation in ref. 12 (also see the right column of Fig. 1).

(1) When the cross-section is a nearly perfect rectangle, angle α is close to zero, the saturation s* is located in the four-finger region and with a very small value.

(2) When the angle α is large but still less than 1° (tan1° = 0.017), the value of s* is very sensitive with respect to even a tiny change of α. The meniscus takes the two-finger configuration. The value of s* can be as large as 0.8, which is significantly larger than that in the four-finger case.

In this work, we have focused on the equilibrium case and discussed about the coexistence between the bulk region and finger region. When the system has a slow dynamics or the capillary number is small, our results of the saturation s* should apply at the interface between the bulk and the finger. We indeed found good agreement with the experimental observations.^{11,12} For simplicity, we have only discussed the case that the equilibrium contact angle is zero. The extension to a finite contact angle is straightforward. For fully wetting, a precursor film of molecular thickness might exist on the surface. In this case the free energy functions take the same forms, thus our prediction of s* should remain valid. An important and related problem is how the change of s* influences the imbibition dynamics.^{18–20} We expect that our framework can provide theoretical support to the understanding of the wetting and drying in porous media, and in the design of micro/nanofluidic devices.

Mason and Morrow used the virtual work balance to derive a condition for the capillary pressure (eqn (4) in ref. 10). Written in our notation, the condition is

p_{c}(S_{0} − S) = γ(P_{S}cosθ_{E} + P_{L}). | (47) |

Using the definition of the capillary pressure (31), we write the LHS of eqn (47)

(48) |

f(s) = (P_{0} − P_{S})γ_{SL} + P_{S}γ_{SV} + P_{L}γ, | (49) |

f(1) = P_{0}γ_{SL}, | (50) |

f(s) − f(1) = P_{S}(γ_{SV} − γ_{SL}) + P_{L}γ = P_{S}γcosθ_{E} + P_{L}γ. | (51) |

Combining eqn (48) and (51), we obtain the same condition (32) for s*

(52) |

r = tanα. | (53) |

(54) |

S_{⋄0} = 2a_{0}^{2}sinαcosα. | (55) |

(56) |

(57) |

The free energy density is

f(s) = (−4 + (2π − 4α)tanα)γ | (58) |

(59) |

(60) |

S = S_{⋄0} − πr^{2}. | (61) |

(62) |

(63) |

The free energy density is

f(s) = −4a_{0}γ + 2πrγ, | (64) |

(65) |

For the complete saturation, the free energy density is

(66) |

(67) |

(68) |

(69) |

(70) |

The free energy functions in these two channels are related by

(71) |

S = S_{⋄} − 2S_{Δ}. | (72) |

(73) |

(74) |

From eqn (71) and (72), we see that the free energy and the area for two cases differ only by a constant term, thus

f(1) − f(s) = f_{⋄}(1) − f_{⋄}(s_{⋄}), | (75) |

S_{0}(1 − s) = S_{⋄0}(1 − s_{⋄}). | (76) |

(77) |

When the saturation in the diamond channel satisfies the coexistent condition (32),

(78) |

(79) |

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