Numerical Simulation of Contact Angle on Gradient Sodium-Philic Nanostructured Surfaces Based on LBM

WANG Can¹², TANG Simiao¹²,*, ZHANG Luteng¹², MA Zaiyong¹², SUN Wan¹², ZHU Longxiang¹², LIAN Qiang¹², PAN Liangming¹²

¹Key Laboratory of Low-grade Energy Utilization Technologies and Systems, Ministry of Education, Chongqing University, Chongqing 400044, China
²Department of Nuclear Engineering and Technology, Chongqing University, Chongqing 400044, China

Abstract

Heat pipe cooled reactors have emerged as a research hotspot in nuclear energy due to their inherent safety and efficient heat transfer capabilities. As the core component of heat pipes, the wick's surface wettability directly influences capillary force and heat transfer efficiency. This study investigates the influence mechanism of gradient sodium-philic nanostructured surfaces on liquid sodium contact angles using the Lattice Boltzmann Method (LBM). By employing an improved Multi-Relaxation-Time (MRT) model and pseudo-potential interaction force model, we examine how geometric parameters—including micropillar height, spacing, and width—affect the contact angle of liquid sodium, achieving stable simulations of wetting and spreading processes on both flat and inclined micropillar-structured surfaces. The results provide a theoretical foundation for optimizing wick surface design and offer valuable insights for enhancing heat pipe thermal performance.

Keywords: Lattice Boltzmann Method; Wettability; Contact Angle; Gradient Nanostructure; Sodium Heat Pipe

Introduction

The development of advanced technologies such as deep-sea exploration and aerospace has created an urgent demand for highly reliable, long-life energy systems. Heat pipe cooled reactors, with their inherent safety, modular design, and efficient heat transfer characteristics, have become an ideal alternative to conventional energy systems [1]. As the core component of heat pipes, the wick drives the working fluid cycle through capillary force, and its performance directly affects reactor heat transfer efficiency and operational stability. Research indicates that the capillary performance of wicks is closely related to the wetting behavior of liquid sodium on their surfaces, and the regulation mechanism of surface micro/nanostructures on wettability is key to optimizing wick design [2]. However, the high density ratio of liquid sodium in high-temperature sodium heat pipes and the complex solid-liquid-gas interactions make it difficult to precisely characterize the dynamic wetting properties of gradient nanostructured surfaces using traditional experimental methods, necessitating the development of high-precision numerical simulation approaches.

Recent years have witnessed significant progress in surface wettability regulation research. Cansoy et al. [3] prepared square and cylindrical micropillar structures and found that contact angles are largely influenced by surface chemical composition, with the Cassie-Baxter equation being applicable only under specific geometric parameters. Hemeda et al. [4] established a mathematical model revealing the critical pressure mechanism for wetting state transitions on superhydrophobic surfaces. The Lattice Boltzmann Method has demonstrated unique value in multiphase flow interface dynamics simulation due to its mesoscopic scale advantages [5]. For instance, Hyväluoma et al. [6] conducted two-phase lattice Boltzmann simulations of flow on structured surfaces with attached bubbles, investigating the effects of geometry, pressure, and shear rate on slip. Lin et al. [7] used this method to analyze droplet spreading characteristics on gradient microstructured surfaces. Hu et al. [8] performed three-dimensional simulations of condensation on nanostructured superhydrophobic surfaces using LBM, examining how geometric dimensions of nanoarrays and local wettability heterogeneity affect nucleation sites and final wetting states of condensate droplets. Fang et al. [9] incorporated temperature fields to simulate droplet spreading and evaporation on surfaces with different wettabilities. However, current LBM-based simulation studies primarily focus on room-temperature working fluids, with relatively few investigations on high-temperature alkali metal sodium. Moreover, traditional single-relaxation-time LBM models are prone to numerical instability under gradient structures and large density ratio conditions, limiting accurate simulation of complex operating conditions [10-11].

Based on these considerations, this study improves the S-C pseudo-potential model and constructs a parametric model of gradient micropillar structures to systematically investigate the influence of micropillar spacing, width, and height on contact angles, revealing the dynamic mechanism of gradient structures in regulating wetting states. The research findings can provide theoretical support for nanostructure optimization of high-temperature sodium heat pipe wicks.

1.1 MRT-LBM Model

For multiphase flow problems, numerous models are available in the Lattice Boltzmann Method. This study employs an improved pseudo-potential model to simulate gas-liquid two-phase flow with large density ratios [12]. The single-relaxation-time model is replaced with a Multi-Relaxation-Time (MRT) model, which ensures computational stability by adjusting multiple relaxation times, making program modifications controllable [13]. The standard lattice Boltzmann equation with the collision matrix incorporated is:

$$
f_i(\mathbf{r} + \mathbf{c}_i\Delta t, t + \Delta t) - f_i(\mathbf{r}, t) = -\mathbf{M}^{-1}\mathbf{S}\mathbf{M}[f_i(\mathbf{r}, t) - f_i^{eq}(\mathbf{r}, t)]
$$

where $\mathbf{r}$ represents a lattice node in the computational domain; $\Delta t$ is the time step; $\mathbf{c}_i$ is the discrete velocity of fluid particles; $\mathbf{M}^{-1}\mathbf{S}\mathbf{M}$ constitutes the collision matrix operator, which can be expressed as $\Omega = \mathbf{M}^{-1}\mathbf{S}\mathbf{M}$; $\mathbf{M}$ is the orthogonal transformation matrix, and $\mathbf{M}^{-1}$ is its transpose, both related to the selection of the velocity discretization model; $\mathbf{S}$ is a diagonal matrix; $f_i$ is the velocity distribution function of fluid particles; and $f_i^{eq}$ is the equilibrium distribution function, expressed as:

$$
f_i^{eq} = w_i \rho \left[1 + \frac{\mathbf{c}_i \cdot \mathbf{u}}{c_s^2} + \frac{(\mathbf{c}_i \cdot \mathbf{u})^2}{2c_s^4} - \frac{\mathbf{u}^2}{2c_s^2}\right]
$$

where $\mathbf{u}$ is the macroscopic fluid velocity at the node, $c_s$ is the lattice sound speed (taken as $c_s = \sqrt{3}/3$ in the D2Q9 model), and each direction has a corresponding weight coefficient $w_i$. The weight coefficients for the D2Q9 model [14] can be expressed as:

$$
w_i = \begin{cases}
4/9 & i = 0 \
1/9 & i = 1,2,3,4 \
1/36 & i = 5,6,7,8
\end{cases}
$$

The collision process is difficult to implement in velocity space, so particle distribution functions must be mapped to moment space. To better satisfy thermodynamic consistency, this study adopts the new forcing term proposed by Li et al. [15] and introduces an additional term to conveniently adjust surface tension while ensuring program stability. The modified evolution equation takes the form:

$$
\mathbf{m}(\mathbf{r} + \mathbf{c}_i\Delta t, t + \Delta t) - \mathbf{m}(\mathbf{r}, t) = -\mathbf{S}[\mathbf{m}(\mathbf{r}, t) - \mathbf{m}^{eq}(\mathbf{r}, t)] + \delta_t \left(\mathbf{I} - \frac{\mathbf{S}}{2}\right)\mathbf{F}_L
$$

The equilibrium distribution function in moment space is:

$$
\mathbf{m}^{eq} = \rho \left(1, -2 + 3|\mathbf{u}|^2, 1 - 3|\mathbf{u}|^2, u_x, -u_x, u_y, -u_y, u_x^2 - u_y^2, u_x u_y\right)^T
$$

The diagonal matrix $\mathbf{S}$ contains multiple relaxation times and takes the form:

$$
\mathbf{S} = \text{diag}(s_\rho, s_e, s_\zeta, s_j, s_q, s_j, s_q, s_\nu, s_\nu)^{-1}
$$

The relaxation times are set as $s_\rho^{-1} = s_j^{-1} = 1.0$, $s_e^{-1} = s_\zeta^{-1} = 0.8$, and $s_q^{-1} = 1.1$. Here, $s_\nu$ represents the relaxation time related to kinematic viscosity, with $s_\nu = 3\nu + 0.5$.

The term $\mathbf{C}$ is introduced as:

$$
\mathbf{C} = \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 1.5(\tau_\nu^{-1} - 1) & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 1.5(\tau_\nu^{-1} - 1) & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & \xi_{xx} & \tau_\nu^{-1} - 1 \
0 & 0 & 0 & 0 & 0 & 0 & 0 & \tau_\nu^{-1} - 1 & \xi_{yy}
\end{pmatrix}
$$

The variables $Q_{xx}$, $Q_{yy}$, and $Q_{xy}$ are calculated as:

$$
Q_{xx} = \frac{1}{2}\sum_{i=0}^8 w_i c_{ix}^2 \psi^2, \quad Q_{yy} = \frac{1}{2}\sum_{i=0}^8 w_i c_{iy}^2 \psi^2, \quad Q_{xy} = \frac{1}{2}\sum_{i=0}^8 w_i c_{ix}c_{iy} \psi^2
$$

where parameter $\kappa$ is used to adjust surface tension.

The forcing term $\mathbf{F}_L$ is expressed as:

$$
\mathbf{F}_L = \left(0, 6\mathbf{u}\cdot\mathbf{F} + \frac{12\mathbf{F}^2}{\psi^2}, -6\mathbf{u}\cdot\mathbf{F} - \frac{12\mathbf{F}^2}{\psi^2}, F_x, -F_x, F_y, -F_y, 2(u_xF_x - u_yF_y), u_xF_y + u_yF_x\right)^T
$$

where $\sigma$ is a parameter for adjusting thermodynamic stability, taken as 0.12.

The force $\mathbf{F}$ can be calculated as:

$$
\mathbf{F} = \mathbf{F}{int} + \mathbf{F}_g} + \mathbf{F
$$

The fluid-fluid interaction force takes the form:

$$
\mathbf{F}{int}(\mathbf{r}) = -G_i\Delta t}\psi(\mathbf{r})\sum_{i=0}^8 w_i \psi(\mathbf{r} + \mathbf{c}_i\Delta t)\mathbf{c
$$

where $G_{int}$ is a constant and $w_i$ are the weight coefficients for interparticle forces, which in D2Q9 can be expressed as:

$$
w_i = \begin{cases}
1/3 & i = 0 \
1/12 & i = 1,2,3,4 \
1/12 & i = 5,6,7,8
\end{cases}
$$

$\psi$ represents the interaction potential between particles, primarily determined by density $\rho$ and pressure $p$:

$$
\psi(\rho) = \sqrt{\frac{2(p - c_0^2\rho)}{c_0^2 G_{int}}}
$$

where $c_0$ is a force coefficient, taken as $c_0 = 6$.

The fluid-solid interaction force is:

$$
\mathbf{F}{ads}(\mathbf{r}) = -G_i\Delta t}\psi(\mathbf{r})\sum_{i=0}^8 w_i \psi(\rho_w)s(\mathbf{r} + \mathbf{c}_i\Delta t)\mathbf{c
$$

where $s(\mathbf{r} + \mathbf{c}_i\Delta t)$ is an indicator function with value 1 for solid nodes and 0 for fluid nodes.

Gravity $\mathbf{F}_g$ is expressed as:

$$
\mathbf{F}_g = \rho \mathbf{g}
$$

where $\mathbf{g}$ is the gravitational acceleration vector.

Within the simulated temperature range, the Peng-Robinson (P-R) equation of state provides satisfactory results:

$$
p = \frac{\rho RT}{1 - b\rho} - \frac{a\alpha(T)\rho^2}{1 + 2b\rho - b^2\rho^2}
$$

where $a = 0.45724(R T_c)^2/p_c$, $b = 0.0778RT_c/p_c$, and $\omega$ is the acentric factor. For sodium, $\omega = -0.10546$. In the simulations, we take $a = 3/49$, $b = 2/21$, and $R = 1.0$.

Fluid density and macroscopic velocity are calculated as:

$$
\rho = \sum_i f_i, \quad \rho\mathbf{u} = \sum_i \mathbf{c}_i f_i
$$

1.2 Model Validation

The Lattice Boltzmann Method employs particle groups consistent with statistical mechanics to simulate actual fluid motion. For computational convenience, all physical parameters—including velocity, viscosity, length, surface tension, and pressure—are expressed in dimensionless lattice units. The lattice units used in numerical simulations are shown in Table 1 [TABLE:1]. Other lattice parameters are listed in Table 2 [TABLE:2].

1.2.1 Laplace Validation

The Laplace formula states that when surface tension remains constant, the pressure difference across a droplet interface is inversely proportional to its radius of curvature $R$ in stable equilibrium. This section initializes elliptical droplets of various shapes, setting the semi-major axis $a = 2R$ and semi-minor axis $b = R/2$ based on mass conservation. The final droplet radius ranges from 15 to 35 lattice units (lu) with an increment of 1 lu. The computational domain is $150 \times 150$ lu, with $G_{int} = -1$ and $\kappa = 0.5$. After the droplet becomes spherical and reaches steady state, the simulation results are fitted to a straight line, as shown in Figure 1 [FIGURE:1]. The slope satisfies the Laplace formula, showing a linear relationship between additional pressure and the reciprocal of droplet radius. The fitted relationship is $\Delta P = 5.04 \times 10^{-4}/R + 2.71 \times 10^{-7}$, yielding a surface tension of $\sigma = 5.04 \times 10^{-4}$ mu·ts$^{-2}$.

By adjusting the surface tension coefficient, the droplet's surface tension can be modified. Calculating the fluid surface tension under multiple $\kappa$ values yields the relationship between parameter $\kappa$ and surface tension shown in Figure 2 [FIGURE:2], which follows a linear relationship $\sigma = -0.0016\kappa + 0.00129$.

1.2.2 Smooth Wall Contact Angle Validation

The droplet radius is set to 30 lu with its center at $(x = n_x/2, y = r/2)$. The lower boundary is set as a smooth wall, while the other boundaries are treated as half-bounce-back boundaries. By varying $G_{ads}$ and recording the contact angle when the droplet reaches stable equilibrium, the data points are fitted. The final results, shown in Figure 3 [FIGURE:3], demonstrate a linear relationship between contact angle and fluid-solid interaction coefficient, consistent with previous studies: $\theta = 275.314 G_{ads} + 238.207$.

2 Results and Analysis

Based on the theoretical analysis in Section 2, the MRT-SC modified model is employed to simulate the wetting characteristics of liquid sodium on micro/nanostructures, with the entire simulation implemented in C++.

2.1 Spreading Process of Liquid Sodium on Micropillar-Structured Surfaces

This section simulates surfaces with micropillar structures, where $a$ is micropillar width, $b$ is micropillar spacing, and $h$ is micropillar height. The parameters are set as $a = b = 2$ lu and $h = 10$ lu, with a droplet radius of 35 lu. The spreading process is illustrated in Figure 4 [FIGURE:4]. Initially, the liquid sodium just contacts the micropillars. By 200 ts, droplet attraction due to liquid-solid interaction becomes observable. During 500–2000 ts, the bottom liquid sodium continuously spreads laterally and penetrates into micropillar gaps, while the upper droplet collapses downward due to internal liquid-phase forces, unable to maintain its original spherical shape. By 3500 ts, the liquid sodium has spread into an approximately hemispherical droplet in the Cassie-Baxter state. From 5500 ts until reaching stable equilibrium, vertical oscillations occur as surface tension and pressure difference balance, requiring an extended period to achieve stability.

Figure 5 [FIGURE:5] shows the dynamic spreading process of liquid sodium on inclined micropillar surfaces, with parameters $h = 6$ lu, $a = 3$ lu, and $b = 5$ lu. At $T = 250$ ts, the droplet bottom contacts the micropillar structure and begins penetrating the gaps. By $T = 500$ ts, an asymmetric wetting region forms at the droplet base, with greater left-side film extension. As time progresses, the droplet shifts overall toward the incline direction, with the upper portion accelerating collapse due to gravity, transforming the shape from an initially symmetric sphere to a left-leaning ellipse. By $T = 5000$ ts, the droplet bottom completely covers the micropillar gaps, entering the Wenzel state. During $T = 7500–15000$ ts, the droplet essentially stabilizes under the balance of surface tension and gravity, finally forming a left-biased hemispherical shape that exhibits contact angle asymmetry induced by gravity.

2.2 Effect of Micropillar Spacing on Liquid Sodium Contact Angle

Micropillar geometry is set with $h = 4$ lu and $h = 10$ lu, $a = 2$ lu, and spacing $b$ ranging from 2 to 10 lu. Figure 6 [FIGURE:6] shows the wetting and spreading states on micropillar structures with different gaps. Combined with Figure 7 [FIGURE:7], when $h = 4$ lu, micropillar spacing significantly affects the final contact angle. As spacing increases, the contact area where liquid sodium fully immerses between gaps enlarges, and solid-liquid interaction forces strengthen, creating greater resistance to lateral spreading and ultimately increasing the contact angle while reducing wettability. When $h = 10$ lu, the contact angle variation follows the same trend, but the static contact angle undergoes a transition process, with $b = 5$ lu serving as the inflection point. At spacings smaller than 5 lu, the contact angle remains in a Wenzel-Cassie intermediate state, while at spacings larger than 5 lu, it completely transitions to the Cassie state.

Further investigation on inclined surfaces with different left-right micropillar spacings is conducted using $h = 6$ lu, $a = 3$ lu, $b_1 = 4$ lu, and $b_2 = 8$ lu. Figure 8 [FIGURE:8] presents the wetting characteristics of liquid sodium on gradient micropillar inclined surfaces. Initially, the droplet center is positioned at the junction between left and right structures. The left side, with smaller micropillar spacing, experiences rapid liquid penetration and continuous film formation, while the right side, with larger spacing, exhibits higher penetration resistance and limited film extension. By $T = 500$ ts, the left-side film coverage is significantly larger than the right, causing the droplet to shift leftward to balance gravity. During $T = 1000–2500$ ts, the left side enters the Wenzel state with enhanced wettability, while the right side remains in a Cassie-Wenzel state due to excessive spacing, showing a pronounced convex gas-liquid interface. By $T = 5000–10000$ ts, gravity and surface tension jointly drive the droplet to migrate left-downward, establishing a wetting gradient.

2.3 Effect of Micropillar Width on Liquid Sodium Contact Angle

In this subsection, micropillar geometry is set with $h = 4$ lu and $h = 10$ lu, $b = 2$ lu, and width $a$ ranging from 2 to 10 lu. Figure 9 [FIGURE:9] illustrates the wetting and spreading states on micropillar structures with different widths, with the line plot shown in Figure 10 [FIGURE:10]. The results indicate that increasing micropillar width tends to increase the contact angle of liquid sodium on the structured surface. When width increases sufficiently, the contact angle stabilizes around 96.5°. As micropillar width increases, fewer micropillars are covered at the droplet base, reducing solid-liquid contact area and hindering lateral spreading. At very large widths, the micropillar structure surface can be approximated as a flat plane, where further width increases produce negligible changes in contact angle.

Additionally, a left-right width gradient is established with $h = 6$ lu, $b = 5$ lu, $a_1 = 2$ lu, and $a_2 = 5$ lu. Initially, liquid sodium spreads on both sides, but over time, the right side—with larger micropillar width—exhibits reduced solid-liquid contact area and slower spreading. By $T = 5000$ ts, the right side of the droplet ceases forward spreading while the left side maintains the Wenzel state. At final equilibrium, the droplet slides downward as a whole.

Figure 11 [FIGURE:11] shows the wetting and spreading process of liquid sodium on this two-level gradient inclined surface.

2.4 Effect of Micropillar Height on Liquid Sodium Contact Angle

Micropillar geometry is set with $a = b = 2$ lu and height ranging from 2 to 15 lu in increments of 1 lu. Figure 12 [FIGURE:12] shows the wetting and spreading states on micropillar structures of different heights. Figure 13 [FIGURE:13] presents the contact angles for all height variations. The line plot reveals that when micropillar height is between 2–6 lu, the contact angle gradually decreases with increasing height, as the concave gas-liquid interface more easily contacts the gap bottom, forming the Wenzel state. As micropillar height increases, the actual contact area between liquid sodium and the solid wall enlarges, reducing the contact angle. This height variation can render the structured surface sodium-philic. Once micropillar height reaches 6 lu, the contact angle gradually stabilizes around 80°, with the liquid sodium in a Cassie-Wenzel intermediate state, indicating that beyond a certain height, further increases have minimal effect on the contact angle.

3 Conclusions and Outlook

This study simulates liquid sodium contact angles on gradient sodium-philic nanostructured flat and inclined surfaces using an improved multi-relaxation-time pseudo-potential model based on the Lattice Boltzmann Method. Model accuracy is verified through Laplace's law and smooth wall contact angle validation. The results demonstrate that inclined surfaces cause droplets to become asymmetric, tending toward left-leaning elliptical shapes. The influence of micropillar geometric parameters on wetting characteristics is systematically investigated: increasing micropillar spacing enlarges the liquid sodium contact angle, potentially transforming the surface from sodium-philic to sodium-phobic; micropillar width exhibits similar trends to spacing changes, but when width becomes sufficiently large, the surface approximates a flat plane and the contact angle stabilizes; appropriately increasing micropillar height in the Wenzel state reduces the liquid sodium contact angle and improves wettability. This work provides theoretical support for optimizing wick surface design in heat pipes.

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