# Contact Angle Effects on Pore and Corner Arc Menisci in Polygonal Capillary Tubes Studied with the Pseudopotential Multiphase Lattice Boltzmann Model

^{1}

^{2}

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^{*}

## Abstract

**:**

_{c}which is known as a key parameter for the existence of the two configurations. LBM succeeds in simulating the formation of a pore meniscus at θ > θ

_{c}or the occurrence of corner arc menisci at θ < θ

_{c}. The curvature of corner arc menisci is known to decrease with increasing saturation and decreasing contact angle as described by the Mayer and Stoewe-Princen (MS-P) theory. We obtain simulation results that are in good qualitative and quantitative agreement with the analytical solutions in terms of height of pore meniscus versus contact angle and curvature of corner arc menisci versus saturation degree. LBM is a suitable and promising tool for a better understanding of the complicated phenomena of multiphase flow in porous media.

## 1. Introduction

_{c}= π/n, in n-sided polygonal tubes based on the Rayleigh-Taylor interface instability. When the contact angle θ is between π/2 and the critical contact angle, i.e., θ

_{c}(= π/n) ≤ θ <π/2, the liquid wets the tube walls and the liquid meniscus spans the total tube, resulting in a configuration named the pore meniscus. In contrast, if the contact angle is smaller than the critical contact angle, i.e., θ < θ

_{c}(= π/n), in addition to the pore meniscus, the liquid also invades the edges or corners of the polygonal tube, forming corner arc menisci [1,2]. Corner arc menisci occur at each corner and move upward as a result of a capillary pressure gradient [3,4].

_{arc}is the radius of curvature of the corner arc meniscus. In this equation, full wetting conditions with contact angle θ = 0° are assumed. The Mayer and Stowe-Princen (MS-P) model [5,6,7,8] predicts the curvature radius of the arc meniscus as a function of the effective area and perimeter of the non-wetting phase (gas), resulting in a better estimation of the interfacial area as shown by [9] using experimental data. Ma et al. [10] investigated capillary flow in polygonal tubes during imbibition and drainage and suggested a relationship between liquid saturation and the curvature of arc menisci in corners based on the MS-P model. In recent work, Feng and Rothstein [11] studied the pore meniscus height as a function of the contact angle for polygonal capillary tubes for contact angles higher than the critical contact angle. Furthermore, they considered different geometries with either sharped or rounded corners, showing the effect of rounded corners and contact angle on meniscus height, and compared their simulation results using Surface Evolver, which is an open-source code for surface energy minimization.

## 2. Numerical Model

_{i}(

**x**,t) is the density distribution function and f

_{i}

^{eq}(

**x**,t) is the equilibrium distribution function in the ith lattice velocity direction, where

**×**denotes the position and t is the time. A relaxation time τ is introduced, which relates to the kinematic viscosity as v = c

_{s}

^{2}(τ − 0.5)Δt. The lattice sound speed c

_{s}is equal to c/$\sqrt{3}$, where the lattice speed c is equal to Δx/Δt, with Δx as the grid spacing and Δt as the time step. In this study, both grid spacing and time step are set equal to 1. The equilibrium distribution function for the D3Q19 lattice model is of the form:

_{i}are:

**e**

_{i}is given by:

**u**are calculated as:

_{i}is added into the right term of the equilibrium distribution function in Equation (2) and is defined as:

**u**is defined as:

**F**

_{total}equals the sum of the total forces. By averaging the moment force before and after a collision step, the real fluid velocity is calculated as:

**F**

_{m}between liquid particles is needed and this force causes phase separation [26]. The force is defined as:

**F**

_{a}between fluid and solid particles is obtained as follows [30]:

**x**) is obtained by choosing an equation of state (EOS) [35]. The EOS describes the relation between the density of the gas and liquid phases for a given pressure and temperature [35,36]. The choice of a suitable EOS is based on different criteria [35,37]. The first criterion is the choice of the maximum density ratio between liquid and gas phases. The second criterion is to avoid the appearance of spurious currents at the interface of different phases. Spurious currents are present in most multiphase models and higher density ratios promote larger spurious currents. The appearance of large spurious currents makes a numerical simulation unstable and leads to divergence. It is important in a LBM with a high density ratio to reduce the appearance of these spurious currents as much as possible. The third criterion relates to the choice of the temperature ratio T

_{min}/T

_{c}, where T

_{c}is the critical temperature. According to the Maxwell equal area construction rule, T < T

_{c}leads to the coexistence of two phases. At a lower temperature ratio, spurious currents appear and the simulation becomes less stable. The last criterion relates to the agreement between a mechanical stability solution and thermodynamic theory. Choosing a proper EOS model reduces the appearance of spurious currents and leads to a thermodynamically consistent behavior [35]. Recently, Yuan and Schaefer [35] investigated the incorporation of various EOS models in a single component multiphase LB model and, based on their study, we apply the Carnahan-Starling (C-S) EOS. The C-S EOS generates lower spurious currents and applies to wider temperature ratio ranges. The EOS is given below:

_{c})

^{2}/p

_{c}is chosen equal to 1 and the repulsion parameter b = 0.1873RT

_{c}/p

_{c}is chosen equal to 4, with T

_{c}= 0.094 and p

_{c}= 0.13044. The effective mass ψ is calculated by:

_{0}equals 1 and G equals –1 to obtain a positive value inside the square root of Equations (14) and (15).

## 3. Validation and Parametrization

#### 3.1. Dynamic Capillary Intrusion

_{L}is the dynamic viscosity of the liquid and × is the position of the interface. The surface tension σ in Equation (16) is determined from the Laplace law describing the pressure difference across the interface of a spherical droplet [41]. The dynamic viscosity is defined as the product of the kinematic viscosity ν and the liquid density, μ

_{L}= ν × ρ, with ν = 1/6 lattice units. Figure 2a illustrates the two-dimensional computational domain of 1600 × 80 lattices used for the capillary intrusion test. Periodic boundary conditions are imposed on all boundaries of the computational domain. The parallel plates of the capillary are positioned between lattices 400 to 1200 of the domain. The boundaries of the plates are treated as walls and are represented by thick black lines in Figure 2a. They have an equilibrium contact angle of 50°, equivalent to a solid-fluid interaction parameter w = −0.06. The density ratio equals ρ/ρ

_{c}= 9.4 at T/T

_{c}= 0.85. The time evolution of the interface position as obtained from the LBM shows a good agreement with Equation (16), as shown in Figure 2b. Based on this dynamic capillary intrusion test, we conclude that the Shan-Chen pseudopotential LB model is adequate to simulate capillary-driven flow.

#### 3.2. Contact Angle

_{c}= 0.85. After reaching steady state, the contact angle is measured using the method LB-ADSA in Image J [42].

## 4. Setup and Boundary Conditions

_{c}= π/n (45° for square, 60° for triangular tube) and only a pore meniscus is built. The second case is when the contact angle is smaller than the critical contact angle and both pore and corner arc menisci are formed.

_{c}= 9.4 at T/T

_{c}= 0.85. Different contact angle ranges are applied. For the square tube, the contact angle ranges from 42.6° to 136.5° as related to a solid-fluid interaction parameter w ranging from −0.08 to 0.06. For the triangular tube, the contact angle ranges from 59.8° to 125.6° as related to a solid-fluid interaction parameter w ranging from −0.05 to 0.05. As shown in Figure 4a,b, bounce-back boundary conditions are imposed on all sides, except for the top 100 lattices on the three or four vertical sides where periodic boundary conditions are imposed to simulate an open capillary tube.

_{cap}, defined at standard temperature and pressure to be [11,45]

## 5. Results and Discussion

#### 5.1. Pore Meniscus

_{c}, the liquid wets the tube walls and a pore meniscus is formed in the tube. Figure 5 shows, as an example, snapshots of pore menisci for square and triangular tubes with hydrophilic and hydrophobic surfaces after reaching steady state. For the square configuration, the meniscus is regular (Figure 5a–d), while for the triangular configuration (Figure 5e,f) the pore menisci show different heights at each corner, especially at a small contact angle (hydrophilic case). This observation is explained by the artificially introduced wall roughness for the triangular tube, as also observed by other authors such as Dos Santos et al. [28]. We found that corner 1 in Figure 4c, which has the highest roughness, shows the lowest height, while corners 2 and 3 show the same height.

**F**

_{a}between solid and fluid in Equation (12) increases and the pressure difference to maintain hydrostatic equilibrium increases, resulting in an increase of the height. At very high (low) contact angles, the height increases (decreases) even more, resulting in an S-shape curve.

_{eff}= cosθ/sinα and θ

_{eff}as the equivalent contact angle. Figure 6d shows the normalized height h/r versus cosine of the equivalent contact angle. We observe that the analytical solutions and LB results for square and triangular tubes collapse onto a single curve. This shows that the LB results for the triangular tube, although suffering from the artificial roughness introduced, agree well over the total hydrophobic and hydrophilic range with the results of the square tube, which does not suffer from an artificial roughness.

#### 5.2. Corner Arc Menisci

_{c}, the liquid invades the corners, forming corner arc menisci. We consider two contact angles of 22° and 32° (w = −0.12 and −0.10), both lower than the critical contact angle for the square and triangular tubes, in order to study the influence of the hydrophilic character of the surface in more detail. Figure 7a,b show snapshots of the pore and corner arc menisci as a function of time (iteration step) for the square and triangular tubes. Figure 7c shows diagonal profiles of the menisci over the height and cross-sections of the meniscus at one corner as a function of time for the square tube. For both tubes, at the early stage, the liquid invades the corners at a small thickness and reaches the top of the tube in a short time. With increasing time, the corner arc menisci thicken while their curvature decreases. At the same time, the pore menisci at the bottom evolve from a more flat shape to a concave shape. This process continues until equilibrium is reached. For the triangular tube, corner arc menisci develop only at two corners, while one corner does not show the presence of a corner arc meniscus, or it does so only at a late time. As mentioned before, this observation is attributed to the artificial roughness introduced by the zigzag surfaces, which is higher in corner 1 than in corners 2 and 3 (see Figure 4c), where the former corner is not invaded. The profiles in Figure 7c show that the thickness of the corner arc menisci is not constant over the height, since at the bottom its thickness is influenced by the pore meniscus, and at the top by the edge of the tube. We remark that the thickness of the corner arc meniscus at equilibrium depends on the initial liquid volume present in the tube. In the case of an infinite reservoir, the corner arc menisci of two adjacent corners join. The cross-sections show that the thickness and curvature for the more hydrophilic surface (θ = 22°) are higher compared to the less hydrophilic case (θ = 32°) at the same time step.

_{n}is given by [10]:

_{contact}is the side length of the corner arc meniscus wetting the side of the tube. The contact length L

_{contact}is determined from the LBM results at mid-height of the corner arc menisci. We note that the phase interface in LBM is not sharp but gradually decreases from liquid to gas density over three to five lattices. The position of a phase interface is evaluated at the average density between liquid and gas. Therefore, there is an uncertainty on the contact length L

_{contact}of around two lattices [26].

_{w}in the function of the curvature is given by [10]:

_{contact}. At a small contact length, an error of two lattices can have a non-negligible effect, as the length L

_{contact}appears in Equation (21) in the denominator. Based on these, we suspect that the contact length is slightly underestimated.

## 6. Conclusions

- When the contact angle is larger than the critical contact angle, θ ≥ θ
_{c}, only a pore meniscus develops and its height increases with the decreasing contact angle for both square and triangular tubes. The LB simulation results show good agreement with the analytical solution. At a very low contact angle in the triangular tube, the height is under-predicted due to the artificial roughness introduced. The LB heights normalized with the circumscribed radius for hydrophobic and hydrophilic surfaces as a function of the effective contact angle collapse into a single S-shaped curve for square and triangular tubes. - When the contact angle is smaller than the critical contact angle, θ < θ
_{c}, LB simulations predict that the liquid invades the corners, forming corner arc menisci. The relation between the degree of saturation and the curvature of the corner arc menisci follows the Mayer and Stoewe-Princen (MS-P) model. The study of the time-dependence of the degree of saturation shows corners filling faster at an early stage and corner arc menisci thickening at a later stage.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Schematic representation of the two liquid configurations in a square tube: (

**a**) pore meniscus when the contact angle is larger than the critical contact angle, θ ≥ θ

_{c}; and (

**b**) co-occurrence of pore and corner arc menisci when the contact angle is smaller than the critical contact angle, θ < θ

_{c}.

**Figure 2.**LBM validation of dynamic capillary intrusion test for T/T

_{c}= 0.85: (

**a**) computational domain; and (

**b**) comparison between the LB simulation results and analytical solution of the position of phase interface as a function of time (iteration step).

**Figure 3.**LBM results of contact angle test: equilibrium contact angles θ as function of solid-fluid interaction parameter w for T/T

_{c}= 0.85.

**Figure 4.**Schematic geometries of polygonal tubes: (

**a**) computational domains for pore meniscus case, (

**b**) computational domains for corner arc menisci case; and (

**c**) computational mesh details for the triangular tube at different corners.

**Figure 5.**Liquid configurations in square and triangular tubes for different contact angles after reaching steady state.

**Figure 6.**Height of the pore meniscus h as a function of cosine of contact angle θ. Comparison between simulation results and analytical solution: (

**a**) square tube; (

**b**) diagonal profiles for square tube, for different solid-fluid interaction value w; (

**c**) triangular tube. (

**d**) Normalized height of the pore meniscus as a function of cosine of effective contact angle θ

_{eff}for square and triangular tubes and comparison with analytical solutions.

**Figure 7.**Liquid configuration versus time (iteration count) for θ = 22°: (

**a**) square tube; (

**b**) triangular tube; and (

**c**) diagonal profiles and cross-sections of a corner arc menisci for square tube at different iteration steps for θ = 22° and 32°.

**Figure 8.**Log-log plot of degree of saturation S

_{w}versus time for contact angles θ = 22° and 32° for square and triangular tubes.

**Figure 9.**Log-log plot of curvature C

_{n}versus degree of saturation S

_{w}for square tube and comparison with analytical solution: (

**a**) for different contact angles θ = 22° and 32°; (

**b**) Grid sensitivity analysis for coarse, reference and fine mesh.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Son, S.; Chen, L.; Kang, Q.; Derome, D.; Carmeliet, J.
Contact Angle Effects on Pore and Corner Arc Menisci in Polygonal Capillary Tubes Studied with the Pseudopotential Multiphase Lattice Boltzmann Model. *Computation* **2016**, *4*, 12.
https://doi.org/10.3390/computation4010012

**AMA Style**

Son S, Chen L, Kang Q, Derome D, Carmeliet J.
Contact Angle Effects on Pore and Corner Arc Menisci in Polygonal Capillary Tubes Studied with the Pseudopotential Multiphase Lattice Boltzmann Model. *Computation*. 2016; 4(1):12.
https://doi.org/10.3390/computation4010012

**Chicago/Turabian Style**

Son, Soyoun, Li Chen, Qinjun Kang, Dominique Derome, and Jan Carmeliet.
2016. "Contact Angle Effects on Pore and Corner Arc Menisci in Polygonal Capillary Tubes Studied with the Pseudopotential Multiphase Lattice Boltzmann Model" *Computation* 4, no. 1: 12.
https://doi.org/10.3390/computation4010012