Numerical Simulation of Particle Deposition on Superhydrophobic Surfaces with Randomly Distributed Roughness—A Coupled LBM-IMBM-DEM Method

Abstract

Dust pollution has emerged as a critical issue in a wide range of industrial applications, creating an urgent demand for effective strategies to mitigate particle deposition. Recent experimental studies have demonstrated that superhydrophobic coatings represent a promising class of self-cleaning materials, primarily attributed to their hierarchical rough structures and intrinsically low surface energy. Nevertheless, the underlying self-cleaning mechanisms of superhydrophobic surfaces have not yet been fully elucidated. This work examines particle deposition on superhydrophobic surfaces featuring stochastic roughness distributions through computational modeling. Surface topographies were generated using Fast Fourier Transform techniques. An integrated lattice Boltzmann–discrete element method (LBM–DEM) framework simulated particle transport in superhydrophobic-coated channels. Particle–fluid coupling was achieved via the immersed moving boundary approach, while particle–surface interactions employed a modified Johnson–Kendall–Roberts (JKR) adhesion model. Parametric studies quantified effects of particle size, interfacial energy, flow Reynolds number, and topographical statistics on deposition dynamics. Experimental validation demonstrates good agreement between numerical predictions and measurements. Smaller particles exhibit a lower tendency to deposit on superhydrophobic surfaces, whereas increasing surface energy significantly enhances particle deposition due to stronger adhesion forces and the suppression of particle resuspension. In addition, higher Reynolds numbers effectively reduce particle deposition. The revealed self-cleaning mechanisms provide theoretical guidance for the design of high-performance self-cleaning coatings, and the identified effects of particle and surface parameters offer practical insights for anti-pollution engineering applications.

1. Introduction

Particle transport and deposition represent ubiquitous phenomena across natural and industrial systems. The dust particles generated from daily life and engineering fields can lead to serious problems. Dust accumulation on the surfaces of industrial equipment can lead to a reduction in efficiency and increased operational downtime [1,2,3]. Similarly, in heat exchangers, accumulated particles increase thermal resistance, resulting in a considerable decrease in heat transfer efficiency, as reviewed in recent studies of particulate and crystallization fouling mechanisms that highlight how heterogeneous deposit layers impede thermal performance and elevate energy consumption in heat exchangers [4]. Experimental investigations under real operating conditions have demonstrated that natural dust deposition on PV systems significantly reduces electrical output, with power and current decreases observed after prolonged exposure without cleaning [5]. Hosseini et al. examined PV soiling impact on photovoltaic performance in field conditions, reporting measurable declines in module efficiency correlated with dust deposition density [6]. Heat exchanger fouling due to airborne particulates has also been studied numerically and experimentally, showing that adhesive dust particle deposition on finned-tube surfaces significantly degrades heat transfer, particularly at higher Reynolds numbers and for particles with stronger surface adhesion properties [7]. These findings underscore the critical importance of understanding particle size, surface characteristics, and fluid dynamics when assessing fouling impacts in thermal systems. Consequently, particle deposition has emerged as a critical issue that urgently requires effective management strategies.

Particle deposition characteristics have been extensively investigated through both experimental measurements and numerical simulations, and recent studies have continued to deepen understanding of the mechanisms governing deposition behavior. Experimental work has shown that flow conditions, particle properties, and geometric features strongly influence particle deposition in ventilation ducts and related flows. For example, Sun et al. demonstrated through experiments in air conditioning ventilation ducts that particle size and airflow velocity significantly affect local deposition hotspots and overall deposition efficiency [8]. In addition, experimental studies of particle deposition in industrial ventilation bends have highlighted the role of Reynolds number and curve geometry on deposition trends, with efficiency and Stokes number exhibiting an exponential relationship under varying conditions [9].

Numerical simulations have played a fundamental role in complementing experiments by providing detailed insights into flow–particle interactions and deposition mechanisms that are difficult or costly to measure directly. CFD studies have shown that inlet velocity is negatively correlated with deposition rate, whereas particle size tends to enhance deposition when other conditions are held constant [10]. For complex geometries, such as corrugated pipes or rough-walled channels, turbulence enhancement near wall surfaces can increase turbulent kinetic energy and promote particle deposition for small particles (<30 µm), indicating a significant interaction between surface features and particle inertia [11,12]. Moreover, Eulerian–Lagrangian simulations in curved pipes have revealed that inserting obstacles such as ribs can reduce deposition locally by modifying the flow field and secondary circulations, thus offering potential control strategies for engineering applications [13].

Beyond engineering ducts, recent investigations have extended to aerosols in biological and heat transfer contexts. Particle transport and deposition in wall-sheared thermal turbulence highlight complex deposit patterns across different Stokes numbers, revealing nonlinear deposition stages that depend on both buoyancy and shear effects [14]. Direct numerical simulations exploring charged inertial particles further show that electrostatic forces can significantly enhance near-wall accumulation and modify deposition dynamics relative to neutral particles [15]. These studies underscore the increasingly sophisticated modeling approaches available, such as large eddy simulation, discrete phase models, and direct numerical simulation, which together improve predictive capability while reducing dependence on costly experiments. Collectively, this body of work suggests that particle size, flow regime, surface geometry, and additional forces (e.g., electrostatic or thermophoretic) jointly determine particle deposition behavior in engineered and natural turbulent flows (e.g., high-fidelity DNS of particle transport and deposition in wall-sheared flows [16] and charged particle deposition influenced by electrostatic interactions in turbulent boundary layers [17]).

Recent research has increasingly focused on particle deposition over textured surfaces via experimental and computational methods. Lu and colleagues investigated deposition within ribbed-wall ducts using Reynolds stress turbulence modeling integrated with discrete particle tracking, demonstrating strong geometric sensitivity of deposition rates [18,19,20]. Hong et al. employed three-way coupling simulations for fine particle deposition on rib-textured substrates, finding that elevated flow velocities diminish normalized deposition rates [21]. Li et al. experimentally studied particle impacts on rough surfaces and demonstrated that increased surface roughness leads to a reduction in the normal restitution coefficient, accompanied by higher dissipation of adhesion energy and frictional losses [22]. Hemmati et al. investigated particle deposition in channels with roughness elements of varying shapes and heights and found that the spacing between roughness elements has a pronounced effect on the deposition behavior of small particles [23].

Overall, particle deposition characteristics have been extensively explored using both experimental and numerical approaches, and the influence of surface morphology on deposition behavior has been well recognized. In previous years, the lattice Boltzmann method (LBM), as an efficient second-order flow solver, was rapidly developed to investigate the multi-phase flow due to its explicit solution, easier parallelism implementation, and flexibility in boundary setting. Numerous studies have explored the interaction between fluids and solids by developing various models [24,25,26]. Xiong et al. employed a combined lattice Boltzmann method–discrete element method–discrete particle simulation approach to investigate particle–fluid flow dynamics, showing that this novel model effectively captures the particle–fluid interaction [27]. In comparison, traditional CFD methods, such as RANS or LES, have been integrated with LBM to study particle deposition processes [28]. These results demonstrate the lattice Boltzmann method’s efficacy for modeling particle transport and deposition across diverse surface morphologies.

Due to the severe particle pollution in industrial applications and human life, an efficient way for reducing the particle deposition needs to be urgently proposed. Recently, effective self-cleaning coatings have been developed to mitigate particle deposition, with superhydrophobic coatings being the most efficient. These coatings, due to their low surface energy, significantly reduce the adhesion between particles and the surface, thereby decreasing deposition. This finding has been validated by several experimental studies. Zhang et al. demonstrated that superhydrophobic coatings can substantially reduce particle deposition on glass surfaces at various tilt angles [29]. Similarly, Pan et al. conducted experiments comparing particle deposition behavior on different glass samples coated with various materials, revealing that the superhydrophobic surface exhibited the lowest dust deposition density [30]. Additionally, the superhydrophobic coatings consisting of different chemical components were studied as well. The coatings with fluoroalkylsilane show higher self-cleaning performance than those with other materials [31]. Consequently, experimental research on superhydrophobic coating self-cleaning has proliferated. Nevertheless, the underlying mechanisms remain incompletely understood. A critical limitation is the absence of robust contact models for particle–rough surface interactions. Classical theories—including Hertz, Johnson–Kendall–Roberts (JKR), Derjaguin–Muller–Toporov (DMT), Muller–Yushchenko–Derjaguin (MYD), and Maugis–Daniel (MD) models—predominantly address smooth surface contacts, offering limited applicability to textured substrates [32,33,34,35,36].

Based on the above review, although particle deposition and superhydrophobic surfaces have been extensively studied, the combined effects of randomly distributed roughness, particle–rough surface contact, and deposition dynamics remain insufficiently understood. To address these gaps, the main contributions of the present study are summarized as follows. Firstly, a randomly distributed superhydrophobic rough surface is reconstructed using an FFT-based spectral method with prescribed statistical parameters, including skewness, kurtosis, and standard deviation, enabling a more realistic description of practical surface morphologies compared with idealized smooth or regularly structured surfaces. Secondly, a particle–rough surface contact treatment is developed by extending the conventional JKR framework through surface discretization and DEM coupling, allowing detailed resolution of particle rebound, rolling, resuspension, and deposition processes on randomly rough superhydrophobic surfaces. Thirdly, the effects of surface statistical parameters on particle deposition are systematically quantified and linked to deposition morphology and deposition number, providing new physical insights into the role of random surface topography in controlling self-cleaning performance. These advances collectively extend the applicability of coupled LBM–IMBM–DEM approaches to realistic superhydrophobic surfaces and offer guidance for the design of anti-fouling and dust-mitigation coatings.

2. Numerical Methodology

The main modeling assumptions adopted in this study are summarized and justified as follows. The particles are modeled as rigid circular solids, and particle deformation is neglected because the impact velocities and elastic moduli considered in the present simulations are sufficiently small that deformation effects are minimal. The airflow is assumed to be incompressible and isothermal, which is justified by the low Mach numbers and the absence of significant thermal gradients in the microchannel configuration studied. The randomly distributed superhydrophobic rough surface is assumed to remain geometrically unchanged during the deposition process, as the particle concentration is low and deposited particles do not significantly modify the underlying surface morphology. Electrostatic forces are neglected, and gravitational acceleration is explicitly included in the particle momentum equation; however, its influence is secondary compared to aerodynamic drag, adhesion, and collision forces under the present conditions. These assumptions allow the essential physical mechanisms governing particle transport, collision, resuspension, and deposition on superhydrophobic rough surfaces to be captured with reasonable computational efficiency and accuracy.

The coupled LBM–IMBM–DEM framework employed in this study was implemented using an in-house numerical solver developed in C++. The gas phase flow was resolved using the lattice Boltzmann method with a standard D2Q9 lattice model, which is suitable for incompressible, low-Mach-number flows. Particle motion was simulated using the discrete element method (DEM), while the immersed boundary method (IMBM) was used to enforce fluid–solid coupling between the gas flow, particles, and rough surface structures.

At the inlet of the computational domain, a prescribed velocity boundary condition was applied to generate a fully developed laminar flow, while a convective (fully developed) boundary condition was imposed at the outlet. No-slip boundary conditions were enforced on the channel walls and on the rough superhydrophobic surface by the immersed boundary treatment. Periodic boundary conditions were applied in the spanwise direction to reduce finite-size effects. The gas was assumed to be incompressible and isothermal, consistent with the experimental conditions.

Particle–particle and particle–surface interactions were resolved using a JKR-based contact model implemented within the DEM framework. The time integration of particle motion was performed using an explicit time-stepping scheme, with the DEM time step chosen to be sufficiently small to resolve collision dynamics and ensure numerical stability. The coupling between the LBM and DEM solvers was realized through a momentum-exchange scheme, allowing two-way interaction between the fluid and particle phases.

2.1. Lattice Boltzmann Model for Air Flow Simulation

Airflow dynamics were simulated using the lattice Boltzmann method. Following temporal and spatial discretization of the Boltzmann equation, the widely adopted single-relaxation-time BGK formulation is expressed as:

$$f_i(x+e_i{\delta }_t,t+{\delta }_t)-f_i(x,t)=-{\displaystyle \frac{1}{\tau }}[f_i(x,t)-f_i^{eq}(x,t)]$$

(1)

where $f_i(x,t)$ denotes the particle distribution function and $f_i^{eq}(x,t)$ represents its equilibrium counterpart. The relaxation parameter τ governs collision timescales and relates to fluid viscosity through:

$$v={c_s}^2(\tau -0.5)$$

(2)

where c_s is the speed of sound and v is the kinematic of the fluid in the lattice system.

At present, the D2Q9 model is widely applied for LBM simulation. Balancing computational cost and accuracy, the D2Q9 lattice scheme was adopted. The equilibrium distribution function takes the form:

$${f_i}^{eq}(x,t)=\rho w_i[1+{\displaystyle \frac{e_i{u}^{eq}}{{c_s}^2}}+{\displaystyle \frac{{(e_i{u}^{eq})}^2}{2{c_s}^4}}-{\displaystyle \frac{{(u^{eq})}^2}{2{c_s}^2}}]$$

(3)

where w_i denotes the weighting factor associated with each discrete velocity direction. For the D2Q9 model, the weighting coefficients are given as w₀ = 4/9, w_1–4 = 1/9, and w_5–8 = 1/36. The discrete velocity set e_i is defined as:

$$e_i=\left\{\begin{array}{l}(0,0) i=0\\ c\cdot (\cos [(i-1)\cdot {\displaystyle \frac{\pi }{2}}],\sin [(i-1)\cdot {\displaystyle \frac{\pi }{2}}]) i=1\text{–}4\\ \sqrt{2}\cdot c\cdot (\cos [(2i-1)\cdot {\displaystyle \frac{\pi }{4}}],\sin [(2i-1)\cdot {\displaystyle \frac{\pi }{4}}]) i=5\text{–}8\end{array}\right.$$

(4)

Macroscopic fluid density and velocity are computed from distribution functions via:

$$\rho ={\displaystyle \sum _i{f}_i}$$

(5)

$$\rho u={\displaystyle \sum _i{e}_i{f}_i}$$

(6)

2.2. Immersed Boundary Method for Particle–Fluid Coupling

Air flow can obviously be affected by solid particles. In this paper, the interactions between the particle and gas flow were implemented by the immersed boundary condition. Computational cells may contain solid regions [37]. Accordingly, the lattice Boltzmann equation incorporates an additional source term:

$$f_i(x+e_i{\delta }_t,t+{\delta }_t)-f_i(x,t)=-{\displaystyle \frac{1}{\tau }}[f_i(x,t)-f_i^{eq}(x,t)]+\beta \cdot {\mathit{\Omega }}_i$$

(7)

$$\beta ={\displaystyle \frac{\epsilon (\tau -0.5)}{(1-\epsilon )+(\tau -0.5)}}$$

(8)

where ${\mathit{\Omega }}_i$ denotes the collision source term accounting for fluid-solid momentum transfer, expressed as:

$${\mathit{\Omega }}_i=\left[f_i^{\prime }\left(x,t\right)-f_i^{\prime eq}\left(\rho ,u\right)\right]-\left[f_i\left(x,t\right)-f_i^{eq}\left(\rho ,{\nu }_p\left(X\right)\right)\right]$$

(9)

Here, $f_i^{\prime }(x,t)$ represents the reversed-direction distribution function. v_p(X) specifies the particle velocity at cell position X, determined by:

$$v_p(X)=w_p\cdot (X+0.5\cdot e_i\cdot {\delta }_t-X_c)+v_p$$

(10)

where v_p(X) is the particle translational velocity, while the w_p is the angular velocity. X_c is the position of the particle’s center of mass.

Hydrodynamic forces and torques result from summing momentum exchange across all lattice directions within particle-occupied cells:

$$F_d={\displaystyle \frac{{{\delta }_x}^2}{{\delta }_t}}{\displaystyle \sum \beta ({\displaystyle \sum _i{\mathit{\Omega }}_i{e}_i)}}$$

(11)

$$M_F={\displaystyle \frac{{{\delta }_x}^2}{{\delta }_t}}{\displaystyle \sum [(X-X_c)\cdot \beta ({\displaystyle \sum _i{\mathit{\Omega }}_i{e}_i)]}}$$

(12)

2.3. DEM-Based Particle Dynamics Analysis

In this study, particles were modeled as circular solids. The discrete element method captured inter-particle and particle–wall collision dynamics. Newton’s second law governed particle motion according to:

$${\displaystyle \frac{d{X}_p}{dt}}=v_p$$

(13)

$$m_p{\displaystyle \frac{d{v}_p}{dt}}=F_d+mg+F_c+F_b$$

(14)

$$I_p{\displaystyle \frac{d{w}_p}{dt}}=M_F+M_A$$

(15)

Here, X_p and v_p are the location and velocity of the particle center, respectively. F_d denotes the hydrodynamic force acting on particles, computed from preceding equations. F_b represents Brownian diffusion forces arising from molecular collisions, and F_c captures contact interactions during particle–particle and particle–wall collisions. I_p signifies the particle moment of inertia, while M_F and M_A correspond to torques induced by fluid flow and collision events, respectively. In the following parts, the specific value calculations of different parameters are introduced.

2.3.1. Contact Force Calculation

In this study, the elastic modulus of particles is not large, and the surface energy is considered during particle collision. Consequently, the JKR framework computed contact forces arising from van der Waals adhesion and elastic deformation. Collision dynamics involve both perpendicular and tangential force components. The total contact force F_c and associated torque M_A are formulated as:

$$F_c=F_{cn}+F_{cs}$$

(16)

$$M_A=r{F}_{cs}+M_r$$

(17)

where F_cn denotes the perpendicular component, while F_cs represents the tangential component of the contact interaction. The rolling resistance torque is designated as M_r. The perpendicular component F_cn comprises two contributions: F_ne, which accounts for repulsive forces arising from particle deformation, and F_nd, which captures energy dissipation effects. The subsequent sections detail the formulation of contact interactions within the JKR framework.

The perpendicular elastic component F_ne incorporates van der Waals adhesive interactions and is formulated as follows:

$${\displaystyle \frac{F_{ne}}{{F_c}_r}}=4{({\displaystyle \frac{a}{a_0}})}^3-4({\displaystyle \frac{a}{a_0}}{)}^{3/2}$$

(18)

In the above equation, F_cr is the critical force and a₀ denotes the equilibrium contact radius at mechanical equilibrium, which represents the adhesive attraction in the contact areas, balanced with the elastic repulsion. a represents the actual contact radius determined by normal deformation. In the present implementation, the JKR relations are applied locally at each particle–asperity contact point (i.e., between the particle and the contacted solid node representing the rough surface) so that the adhesive and elastic response is determined by the local contact state rather than an ideal smooth wall.

F_cr, a₀, and a are calculated by

$$F_{cr}=3\pi \gamma R$$

(19)

$$a_0=({\displaystyle \frac{9\pi \gamma {(R)}^2}{E}}{)}^{1/3}$$

(20)

$${\displaystyle \frac{{\delta }_n}{{\delta }_c}}=6^{1/3}[2{({\displaystyle \frac{a}{a_0}})}^2-{\displaystyle \frac{4}{3}}({\displaystyle \frac{a}{a_0}}{)}^{1/2}]$$

(21)

Here, γ denotes the effective interfacial energy used in the adhesion term, R is the effective radius, and E is the effective Young’s modulus. In this study, R and E are computed from the particle and surface material properties using Equations (22) and (23), where (E1, γ1) and (E2, γ2) correspond to the particle and the substrate (or the contacted asperity node), respectively. These material parameters are specified from measured values or literature-reported properties (see

Table 1. Physical and computational parameters for experiments and simulations.

Particle		Fluid Phase		Surface
Density	2500 kg/m3	Viscosity	1.8 × 10−5 Pa·s	Surface energy	0.068 J/m2
Diameter	250 × 10−6 m	Density	1.29 kg/m3
Young’s modulus	100 MPa	Cavity Height	1.0 × 10−2 m
Released height	4.612 × 10−3 m	Cavity Width	1.0 × 10−2 m

for the validation case), and no additional fitting parameter is introduced in the JKR formulation.

The values of the parameters can be calculated by

$${\displaystyle \frac{1}{E}}={\displaystyle \frac{1-{p_1}^2}{E_1}}+{\displaystyle \frac{1-{p_2}^2}{E_2}}$$

(22)

$${\displaystyle \frac{1}{R}}={\displaystyle \frac{1}{R_1}}+{\displaystyle \frac{1}{R_2}}$$

(23)

$$\gamma ={\gamma }_1\cdot {\gamma }_2$$

(24)

$${\delta }_n=R_1+R_2-\left|X_1-X_2\right|$$

(25)

For particle–rough surface collisions, the overlap δ_n in Equation (25) is evaluated between the particle center and the contacted roughness node, enabling the JKR-based force to respond to the local asperity geometry.

$${\delta }_c={\displaystyle \frac{a_0^2}{2{(6)}^{1/3}R}}$$

(26)

where δ_c is the critical normal deformation in the JKR theory and is used to normalize the instantaneous overlap δn in Equation (21). For particle–surface collisions, the substrate is treated as a rigid wall (or an asperity node representing the rough surface); thus, R₂→∞ and the effective radius in Equation (23) reduce to the particle radius, i.e., R = R₁. For particle–particle collisions, the effective radius R and equivalent modulus E are computed using Equations (22) and (23) based on the material properties of both particles. p is Poisson’s ratio. The subscripts 1 and 2 represent the different surfaces. When the particle collides with the surface, the effective radius is the particle radius. Thus, the normal elastic force F_ne can be acquired by the previous equations.

When the particle collides with other particles or surfaces, the normal dissipation force F_nd cannot be negligible. It is defined as

$${F_n}_d=-{\eta }_n{v}_{pr}\cdot n$$

(27)

where v_pr·n denotes the normal component of the relative velocity at the contact point and n is the unit normal vector. The damping coefficient η_n is computed as

$${\eta }_n=\alpha {(m_p{k}_N)}^{1/2}$$

(28)

where m_p is the effective (reduced) mass of the contacting pair and k_N is the normal contact stiffness given by

$$k_N={\displaystyle \frac{4}{3}}Ea$$

(29)

The dimensionless parameter α controls the dissipation level and is calibrated by prescribing a target normal coefficient of restitution Ea (particle–particle and particle–surface), following the standard DEM practice reported in Refs. [37,38,39,40]. In the revised manuscript, we explicitly report the adopted Ea values and keep them constant across parametric studies to avoid introducing case-dependent fitting.

In this paper, the tangential force was calculated by

$$\Delta F_{cs}=8G\ast r_a{\theta }_k\Delta t+{(-1)}^k\mu \Delta F_{cn}(1-{\theta }_k)$$

(30)

$${\displaystyle \frac{1}{G^{\ast }}}={\displaystyle \frac{1-{{\nu }_1}^2}{G_1}}+{\displaystyle \frac{1-{{\nu }_2}^2}{G_2}}$$

(31)

where G* is the equivalent shear modulus, μ is the friction coefficient, and Δt is the DEM time step. The information of the adopted model can be referred to the literature [38,39,40,41]. The limiting tangential friction force is determined by:

$$F_{fcr}=\mu \left|F_{ne}+2F_{cr}\right|$$

2.3.2. Brownian Force Calculation

The transport and deposition characteristics of the microparticle are greatly affected by the Brownian force that is caused by the Brownian diffusion mechanisms [42]. In this paper, the Brownian force is considered, and it can be calculated by

$$F_b=\epsilon \sqrt{{\displaystyle \frac{216v\rho k_{B}T}{\pi {{\rho }_p}^2{d_p}^{5}dt}}}$$

(32)

where ε is the random vector with Gaussian random numbers. k_B is the Boltzmann constant, while T is the temperature. d_p is the particle diameter, while ρ_p is the density of the particle. d_t is the particle interval. Therefore, the Brownian force can be computed by the above equation. Hydrodynamic forces were computed using the immersed boundary approach. Combining all force contributions enabled the prediction of particle dynamics.

2.4. Surface Roughness Reconstruction

Superhydrophobic surfaces exhibit two defining characteristics: minimal interfacial energy and textured topography. Thus, the reconstruction of a rough surface is necessary in a numerical study. For a surface with regular roughness, the structure parameters, such as the height and space of the surface, can be artificially set to obtain the regular surface structures. However, for the randomly distributed rough structures, it is difficult to clearly determine the surface structures. Thus, in this study, the Fast Fourier Transform (FFT) method was used for the reconstruction of a randomly distributed surface [43,44,45]. The kurtosis, skewness, and standard deviation are used to ascertain the surface roughness. Skewness quantifies the asymmetry in surface topography. A skewness of zero indicates a symmetric, Gaussian height distribution. Negative skewness corresponds to surfaces dominated by deep valleys with flattened peaks, whereas positive skewness characterizes surfaces with pronounced peaks and shallow valleys. Greater absolute skewness values reflect increased topographical asymmetry. Kurtosis describes the sharpness of the height distribution. A Gaussian surface exhibits a kurtosis of 3.0. Values below 3.0 indicate a flatter, more uniform height distribution compared to the Gaussian case, while values exceeding 3.0 signify a more peaked distribution with extreme heights. Elevated kurtosis indicates a propensity for pronounced peaks and deep troughs in surface topography. Standard deviation quantifies the dispersion of height distributions relative to the mean elevation. Figure 1 depicts morphological variations resulting from different skewness, kurtosis, and standard deviation combinations.

The reconstruction methodology is detailed below. The randomly distributed rough surface can be characterized using an exponential autocorrelation function (R_z) and height distribution. The reconstructed domain is represented by an N × M grid with corresponding height arrays z (i, j). These parameters and the exponential autocorrelation function (R_z) are formulated as follows:

$$\overline{z}={\displaystyle \frac{1}{MN}}{\displaystyle \sum _{j=0}^{M-1}{\displaystyle \sum _{i=0}^{N-1}{z}_{ij}}}$$

(33)

$${R_q}^2={\displaystyle \frac{1}{MN}}{\displaystyle \sum _{j=0}^{M-1}{\displaystyle \sum _{i=0}^{N-1}{z_{ij}}^2}}$$

(34)

$$Sk={\displaystyle \frac{1}{MN{R_q}^3}}{\displaystyle \sum _{j=0}^{M-1}{\displaystyle \sum _{i=0}^{N-1}{z_{ij}}^3}}$$

(35)

$$K={\displaystyle \frac{1}{MN{R_q}^4}}{\displaystyle \sum _{j=0}^{M-1}{\displaystyle \sum _{i=0}^{N-1}{z_{ij}}^4}}$$

(36)

$$R_z(k,l)={R_q}^2\exp [-2.3\sqrt{{({\displaystyle \frac{k}{{\mathsf{\beta }}_x}})}^2+({\displaystyle \frac{l}{{\mathsf{\beta }}_y}}{)}^2}]$$

(37)

Hu and Tonder proposed the 2D digital filter technique to generate the randomly distributed rough surfaces with different skewness and kurtosis for a known R_q and β [46]. The input sequence η(I, j) is filtered by h(k,l) to obtain the height sequence z(I,j) through the transformation system described as:

$$z(i,j)={\displaystyle \sum _{k=1}^N{\displaystyle \sum _{l=1}^{M}h(k,l)\eta (i+k-}}⌈{\displaystyle \frac{i+k}{N}}-1⌉N,j+l-⌈{\displaystyle \frac{i+k}{N}}-1⌉M)$$

(38)

The computational procedure involves the following steps.

First, the Fourier transform method is applied to Equation (38), yielding Equation (39):

$$Z(I,J)=H(I,J)A(I,J)$$

(39)

$$H(I,J)={\displaystyle \sum _{k=1}^N{\displaystyle \sum _{l=1}^{M}h(k,l)e^{-jk{\omega }_x}}}{e}^{-jk{\omega }_y}$$

(40)

Next, the auto-correlation function (R_z) is used to calculate the probability density function of the surface roughness, as shown in Equations (41) and (42):

$$S_z(I,J)={\displaystyle \sum _{k=1}^N{\displaystyle \sum _{l=1}^M{R}_z(k,l)e^{-jk{\omega }_x}}}{e}^{-jk{\omega }_y}$$

(41)

$$S_z(I,J)={\left|H(I,J)\right|}^2{S}_{\eta }(I,J)$$

(42)

Finally, the filter function h(k,l) is calculated using Equation (43):

$$h(k,l)={\displaystyle \frac{1}{MN}}{\displaystyle \sum _{I=1}^N{\displaystyle \sum _{J=1}^{M}H(I,J)e^{jk{\omega }_x}{e}^{jl{\omega }_y}}}$$

(43)

Thus, the sequence η(i,j) must be transformed into sequence η′, which is a non-Gaussian input sequence [47,48]. The specific transformation can be seen in the literature [48].

The randomly distributed rough surfaces are generated using a two-step stochastic reconstruction procedure based on established methods reported in the literature [43,44,45]. In the first step, a Gaussian random surface is constructed using an FFT-based spectral method. A target power spectral density or autocorrelation function is prescribed in the frequency domain, and an inverse Fourier transformation is applied to obtain a spatial height field with zero mean and a Gaussian height distribution. The standard deviation of surface height, σ, is then controlled by scaling the amplitude of the generated height field, allowing direct adjustment of the overall roughness intensity.

In the second step, to generate non-Gaussian surface morphologies, the Gaussian height field is transformed using a Johnson transformation [46,47,48]. This transformation modifies the probability density function of surface heights while preserving the prescribed standard deviation. By appropriately selecting the transformation parameters, the skewness and kurtosis of the surface height distribution can be independently controlled. In this manner, surface skewness governs the asymmetry of asperity height distribution, while kurtosis determines the sharpness and extremeness of surface peaks and valleys. This approach enables systematic generation of random rough surfaces with identical standard deviation but different skewness and kurtosis, as illustrated in Figure 1.

Superhydrophobic surfaces are characterized not only by low surface energy but also by complex surface topographies. In the present study, randomly distributed rough surfaces are reconstructed using an FFT-based spectral method, in which the statistical properties of the surface are prescribed through the skewness, kurtosis, and standard deviation of the height distribution. These parameters uniquely define the global morphological characteristics of the reconstructed surface, while the detailed local features are determined by the stochastic nature of the reconstruction process.

Because the surface generation procedure is inherently stochastic, different realizations with identical statistical parameters may exhibit variations in local asperity distribution. In this work, for each prescribed set of surface statistical parameters, representative rough surface realizations are generated and used consistently in the corresponding particle deposition simulations. Although the local deposition morphology may vary depending on the specific random realization, the primary objective of this study is to investigate the influence of surface statistical parameters on particle deposition behavior. As demonstrated by the systematic parametric comparisons presented in the Results and Discussion section (Section 5), the overall deposition trends are governed by the prescribed surface statistics rather than by individual random surface features. Therefore, the conclusions drawn in this study reflect statistically meaningful trends associated with surface roughness characteristics rather than artifacts arising from a particular random surface realization.

It should be noted that, in the present LBM–IMBM–DEM framework, particle–surface contact is represented through interactions between particles and discretized surface nodes. The effective contact geometry is, therefore, influenced by the lattice resolution used to describe the rough surface. To ensure adequate resolution of contact interactions, the lattice spacing is selected to be sufficiently smaller than both the particle diameter and the characteristic roughness length scales. Within this resolution regime, the JKR-based adhesion model provides a physically consistent description of adhesive interactions.

2.5. The Contact Model

Recently, the JKR model has been extensively applied to compute contact forces in particle–particle and particle–wall collisions. Nevertheless, existing models are limited to smooth surface interactions. A computational framework for particle–rough surface impacts remains undeveloped. This study presents a novel contact model extending the JKR approach to address particle–rough surface collisions. The simulation assumes that deposited particles do not alter the rough surface morphology. The lattice Boltzmann method simulates airflow and particle–fluid interactions within a discretized computational domain. Grid nodes are classified as either solid roughness elements or fluid regions. Collision detection utilizes the spacing between particle centers and solid nodes representing surface asperities. Contact is identified when this spacing falls below the particle radius. Then, the JKR model introduced in previous parts can be employed to calculate the contact force. Here, we used the solid node to replace the solid surface; thus, the particle size must be larger than the grid length.

3. Numerical Model Validation

As summarized in Table 1, particles with a density of 2500 kg·m⁻³, a diameter of 250 μm, and a Young’s modulus of 100 MPa were released individually from a fixed height of 4.612 × 10⁻³ m above the bottom surface into a microchannel with a height and width of 1.0 × 10⁻² m. This release configuration ensures well-defined and repeatable initial conditions for particle–surface interactions. The carrier gas properties correspond to ambient air conditions, with a density of 1.29 kg·m⁻³ and a dynamic viscosity of 1.8 × 10⁻⁵ Pa·s.

Particle motion within the microchannel was recorded using a high-speed camera (PCO.dimax S1, PCO AG, Kelheim, Germany), which provides sufficient temporal and spatial resolution to capture particle impact, rebound, and deposition processes. The spatial resolution was calibrated using a reference target placed in the imaging plane. Particle centroid positions were extracted from the recorded image sequences using frame-by-frame image processing techniques, and particle velocities were subsequently obtained from the measured trajectories using a central finite difference scheme. This experimental procedure enables quantitative comparison between measured particle trajectories and numerical predictions.

To quantitatively assess model accuracy, the root-mean-square error (RMSE) between the measured and simulated particle height and velocity time histories shown in Figure 2 were calculated. The uncertainty in particle position was estimated from the imaging spatial resolution and centroid detection accuracy, while the time uncertainty was determined from the camera frame interval. These uncertainties were propagated to the velocity measurements using standard error propagation for finite differences.

The root-mean-square error (RMSE) was used to quantify the deviation between numerical predictions and experimental measurements and is defined as

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\left(X_i^{\mathrm{sim}}-X_i^{\exp }\right)}^2},$$

(44)

where X represents either particle height or velocity and N is the number of sampled data points.

Figure 2 compares experimentally measured and numerically predicted particle height and velocity as functions of time. The quantitative agreement is assessed using estimated error metrics based on digitized data, yielding RMSE ≈ 0.06 mm and mean relative error ≈ 3.8% for particle height, and RMSE ≈ 0.012 m/s and mean relative error ≈ 5.1% for particle velocity. Based on digitized experimental data from Figure 2, the RMSE and relative error remain below 6% for both particle height and velocity, confirming the satisfactory predictive accuracy of the proposed numerical framework.

It should be noted that the experimental validation presented in this study focuses on single-particle impact under quiescent conditions. This validation strategy is adopted to isolate and verify the fundamental particle–surface interaction mechanisms, including collision, rebound, and adhesion, which are essential components of the numerical model. While multi-particle deposition, agglomeration, and resuspension under flow conditions are central to the numerical investigations presented in Section 5.2, Section 5.3, Section 5.4 and Section 5.5, direct experimental validation of these complex coupled processes remains challenging due to limitations in experimental resolution and controllability, particularly for rough superhydrophobic surfaces. Therefore, the present validation is intended to ensure the physical consistency of the particle–surface interaction model, whereas the subsequent numerical simulations are employed to examine relative deposition trends under varying conditions.

4. Computational Case Description

In this paper, we aimed to numerically investigate the particle deposition on a superhydrophobic surface with a rough surface. Here, a microchannel system that mimics the real cases was built. A low-energy rough surface was positioned at the channel bottom to simulate superhydrophobic material characteristics. This configuration can efficiently display the effects of a superhydrophobic surface on particle deposition when the particles are transported inside the channel. The specific description of the computational case is introduced in the following parts. Particles enter the microchannel entrained in gas flow at constant velocity, with initial particle velocities matching the airflow and vertically randomized positions. Particle tracking ceases once they exit the computational domain. The inlet airflow exhibits a uniform velocity profile with constant horizontal velocity and zero vertical component. Boundary conditions are specified as follows: velocity inlet (left), fully developed outflow (right), and no-slip walls (top and bottom) as the Knudsen number is sufficiently small and continuum flow assumptions remain valid. For the lattice cells that are occupied by the rough surface, the velocity was set as 0, and the structures were set as a no-slip boundary condition as well.

5. Results and Discussion

5.1. Particle Transport and Deposition in Microchannels

Previous investigations have employed different simplifications based on size ratios. When particle diameters significantly exceed surface asperity heights, the substrate is approximated as smooth, neglecting roughness effects on both deposition behavior and flow patterns. Conversely, when particles are substantially smaller than roughness features, they are modeled as point masses under one-way coupling, omitting rebound, deposition, and resuspension dynamics. For superhydrophobic surfaces with textured topography, rough elements and particles share comparable length scales. Consequently, both particle and surface topography significantly alter flow distribution. Surface roughness modifies hydrodynamic forces acting on particles by disturbing the surrounding velocity field, as demonstrated in Figure 3. For the particle, its deposition behavior would obviously be affected by the roughness when it collides with the surface. The roughness structures may cause the particles to rebound off or deposit on the surface. When the particle rebounds off the surface, it may collide with other particles. This phenomenon is rarely seen in diluted gas–solid flow. However, in this study, the real particle transport and deposition behaviors inside the microchannel were predicted; thus, the particle–particle collision cannot be neglected. Moreover, we found that the deposit particles may rebound after contacting other particles. The reduced surface energy of superhydrophobic substrates results in minimal particle–surface adhesion. Consequently, deposited particles can resuspend and exit the channel, as illustrated in Figure 4. This resuspension phenomenon is critical for understanding self-cleaning mechanisms and demonstrates the superior dust-repellent performance of superhydrophobic coatings. Additionally, particles may roll along the surface before escaping the channel, facilitated by weakened adhesive interactions that inhibit particle attachment.

5.2. Influence of Particle Size on Deposition Patterns

Particle size significantly influences transport and deposition dynamics [49,50,51]. To investigate this effect, we analyzed deposition patterns on rough microchannel substrates using three particle radii, 8 μm, 10 μm, and 12 μm, selected based on measurements from South China University of Technology showing particle sizes ranging from 3 to 13 μm [52]. Figure 5a–c present the temporal evolution of deposition morphology for different particle sizes on superhydrophobic surfaces. At equivalent time intervals, larger particles occupy lower vertical positions due to enhanced settling velocities. This occurs because drag forces exert proportionally greater influence on smaller particles, causing larger particles to experience greater acceleration and descend more rapidly. Final deposition patterns reveal distinct morphological differences: small particles distribute dispersedly across the surface, while large particles exhibit agglomeration. Two mechanisms explain this behavior. First, larger particles experience a higher collision probability with neighboring particles. Since particle–particle adhesion exceeds particle–surface adhesion due to relative surface energies, colliding large particles tend to coalesce. Small particles rarely encounter each other, preventing cluster formation. Second, drag forces dominate small particle motion, extending their horizontal displacement before surface contact and further reducing agglomeration likelihood.

In the present simulations, the total number of injected particles is 300 for all cases. Based on the deposition numbers shown in Figure 6, the corresponding deposition rates are approximately 5.7%, 9.3%, and 10.7% for particle radii of 8 μm (red), 10 μm (green), and 12 μm (blue), respectively. These results indicate that larger particles exhibit a higher deposition probability due to enhanced inertia and increased particle–surface collision frequency. The deposition rate of the largest particles is approximately twice that of the smallest particles. This is because the small particles are suspended in the channel and then leave the channel before colliding with the bottom surface. Additionally, the lower particle agglomeration probability of small particles causes a lower particle deposition probability. Therefore, the smaller particles are more likely to leave the channel coated by superhydrophobic materials with the air flow. For larger particles, their larger diameter and higher surface energy lead to a higher accumulation probability, which can make them easier to agglomerate and deposit on the surface. It should be noted that the present simulations do not explicitly distinguish between the deposition of individual particles and particle agglomerates, nor do they quantify clustering statistics. Therefore, the influence of inter-particle interactions is discussed in a qualitative sense to interpret the observed deposition trends rather than as a fully resolved agglomeration mechanism.

5.3. Influence of Surface Energy on Deposition Behavior

Surface energy governs particle–surface adhesion strength, thereby controlling deposition dynamics at the microchannel substrate. Currently, surfaces spanning the wettability spectrum—from superhydrophilic to superhydrophobic—find widespread industrial application [53]. The surface energy of the superhydrophobic surface is about 0.003 J/m², while that of the hydrophilic surface is about 0.5 J/m². When two surfaces contact, the effective surface energy γ for adhesion calculations is determined by:

$$\gamma =\sqrt{{\gamma }_1\cdot {\gamma }_2}$$

(45)

where γ₁ and γ₂ represent two different surfaces. Based on realistic particle properties [54,55], we examined particle–surface energies of 0.001, 0.01, and 0.1 J/m², while maintaining inter-particle energy at 0.01 J/m². Figure 7 illustrates deposition morphologies across varying surface energies. Pronounced particle clustering occurs on high-energy surfaces due to enhanced adhesive forces that prevent low-velocity particles from rebounding. Additionally, when incoming particles strike deposited particles, insufficient elastic restitution inhibits resuspension of the deposited layer.

Figure 8 quantitatively illustrates the effect of particle–surface energy on deposition behavior. As the particle–surface energy increases from 0.001 to 0.01 and 0.1 J·m⁻², the deposition count increases nonlinearly from 29 to 64 and 73 particles, corresponding to deposition rates of approximately 9.7%, 21.3%, and 24.3%, respectively. First, elevated adhesion on high-energy surfaces suppresses particles’ rebound upon initial contact. Second, subsequent particle collisions with deposited material dissipate kinetic energy without dislodging adhered particles, increasing deposition probability for incoming particles. Notably, the deposition rate increment decreases at higher surface energies. This saturation occurs because particle surface energy becomes progressively lower than substrate energy, causing direct adhesion that eliminates resuspension. Consequently, high-energy surfaces promote deposition within microchannels. Comparing superhydrophobic and superhydrophilic energy ranges, superhydrophobic surfaces demonstrate superior self-cleaning performance.

The nonlinear variation in particle deposition with increasing surface energy can be attributed to a transition in the dominant particle–surface interaction mechanism. At relatively low surface energies (γ < 0.01 J·m⁻²), adhesive forces are weak compared to particle inertia and elastic restitution during impact. Under these conditions, particles frequently rebound or roll along the surface after collision, and deposition is governed by the competition between aerodynamic drag, collision dynamics, and intermittent adhesion. As the surface energy increases toward approximately 0.01 J·m⁻², the adhesive force predicted by the JKR model increases rapidly, leading to a pronounced reduction in particle rebound probability and resuspension events. Consequently, the deposition number increases sharply within this regime.

When the surface energy exceeds 0.01 J·m⁻², adhesion becomes the dominant interaction mechanism during particle–surface contact. Most impacting particles adhere to the surface upon first contact, and subsequent collisions with deposited particles no longer provide sufficient kinetic energy to overcome the adhesive pull-off force. Under these conditions, particle deposition approaches a saturation regime in which further increases in surface energy produce only marginal changes in deposition behavior. This adhesion-dominated regime explains the observed change in trend and the reduced sensitivity of deposition rate to surface energy beyond the critical threshold.

5.4. Effects of Structures

This study examined particle transport and deposition in microchannels using a coupled LBM-IMBM-DEM framework. For a superhydrophobic surface, the effects of the bottom surface structures need to be specifically discussed. In previous parts, we introduced the essential parameters for reconstructing the randomly distributed structures.

The analysis begins with skewness effects. Since skewness quantifies topographical asymmetry, three values were selected: −1.21, 0, and 3.25. Figure 9 presents deposition morphologies across these conditions. Positive skewness produces peak-dominated topographies with upward-oriented asperities, confirming skewness effects on surface geometry. Deposition patterns reveal that low absolute skewness values reduce particle clustering. Symmetric peak–valley distributions promote dispersed particle configurations. Conversely, negative skewness surfaces—characterized by valley-dominated topographies—exhibit pronounced particle accumulation within depressions. This clustering behavior undermines anti-fouling performance. For the surface with negative skewness, the existence of the peak may prevent the particle from rolling on the surface as well. Consequently, superhydrophobic surfaces with reduced skewness suppress particle clustering, thereby enhancing self-cleaning capability.

Following the skewness analysis, we examine kurtosis effects on deposition behavior. Kurtosis characterizes peak sharpness relative to Gaussian-distributed topographies. In this paper, three different kinds of kurtosis, which are 3.12, 5.05, and 7.18, are employed for investigation. Comparing the structure distribution displayed in Figure 10, we found that the higher kurtosis can significantly increase the overall roughness element of the rough surface. This is consistent with the effect of the kurtosis on the roughness distribution. Figure 11a reveals that low-kurtosis surfaces exhibit predominantly flat topography with sporadic elevated asperities. These protruding features obstruct particle motion during impact, inhibiting channel escape. Figure 11c illustrates deposition on high-kurtosis surfaces. Elevated kurtosis produces exaggerated topography with sharper peaks and deeper valleys, which trap particles and increase accumulation. Conversely, moderate-kurtosis surfaces (Figure 11b) feature gentler asperities insufficient to retain particles, allowing them to roll across and exit the channel. Thus, as shown in Figure 12, deposition exhibits a non-monotonic relationship with kurtosis: initially increasing, then decreasing. The results suggest that variations in skewness and kurtosis influence particle deposition behavior to some extent; however, the observed differences in deposition counts remain relatively small. Therefore, the present results are more suitable for illustrating relative sensitivity to roughness parameters rather than identifying an optimal roughness configuration. To assess standard deviation effects, three rough surfaces plus one smooth surface (σ = 0) were examined. Figure 13 demonstrates that at constant skewness and kurtosis, deposition increases with standard deviation. Higher σ generates more numerous and spatially irregular roughness elements with amplified peak–valley features that obstruct particle motion and promote retention (Figure 14). However, smooth surfaces (σ = 0) also exhibit poor self-cleaning due to elevated tangential shear and friction during collisions. This study indicates that minimizing standard deviation, enhancing topographical symmetry, and optimizing kurtosis collectively reduce particle deposition on superhydrophobic surfaces.

It should be noted that, due to the stochastic nature of surface reconstruction and the absence of a formal uncertainty analysis, the present results do not support statistically significant ranking or optimization of roughness parameters. The conclusions are, therefore, limited to comparative and trend-based observations.

5.5. Effects of the Reynolds Number

Airflow conditions directly govern particle deposition, particularly for larger particles where inertial effects dominate transport. This section examines Reynolds number effects on deposition morphology by varying inlet velocities. Figure 15 reveals that at low Reynolds numbers, particles undergo limited lateral displacement before surface contact. Rebounding particles frequently collide with incoming particles, forming clusters. These collisions dissipate kinetic energy, reducing velocities and facilitating deposition. Consequently, agglomeration intensifies under low-Re conditions. Figure 15 quantifies this trend: deposition counts decrease substantially at higher inlet velocities. Several mechanisms explain this behavior. First, elevated Reynolds numbers enable high-altitude particles to exit the channel without contacting surfaces. Second, higher inlet velocities increase particle inertia, resisting deposition. Third, reduced surface impact frequency at higher Re minimizes post-rebound inter-particle collisions, preserving kinetic energy and preventing accumulation. Therefore, increasing airflow velocity significantly mitigates particle deposition.

6. Conclusions

In this study, a coupled LBM–IMBM–DEM framework was developed to investigate particle deposition on superhydrophobic surfaces with randomly distributed roughness. The numerical model accounts for particle–fluid interaction, particle–surface adhesion, and inter-particle interactions, enabling a systematic, trend-based analysis of deposition behavior under varying particle size, surface energy, and surface roughness statistics. Based on the numerical results and experimental validation, the following conclusions can be drawn:

• Particle deposition on superhydrophobic surfaces is strongly influenced by surface energy. Increasing particle–surface energy significantly enhances particle retention, with deposition exhibiting a nonlinear increase as surface energy rises, indicating its dominant role in governing deposition trends.
• Surface roughness characteristics play a critical role in deposition behavior. An increase in roughness standard deviation promotes particle deposition by modifying local contact conditions and near-wall flow structures, while the stochastic nature of rough surfaces necessitates ensemble-averaged analysis to ensure statistical robustness.
• Particle size affects deposition primarily through changes in particle inertia, near-wall residence time, and collective particle behavior. Larger particles exhibit higher deposition probability due to reduced mobility near the surface, whereas inter-particle interactions contribute qualitatively to collective deposition trends rather than fully resolved agglomeration mechanisms.
• The single-particle impact validation confirms the physical consistency of the particle–surface interaction model. The subsequent multi-particle deposition simulations are intended to reveal comparative and trend-based behaviors rather than provide fully predictive quantitative deposition rates under all flow conditions.

7. Discussion

The numerical results obtained in this study are generally consistent with previously reported experimental and numerical investigations on particle deposition over superhydrophobic and rough surfaces. Numerous studies have demonstrated that superhydrophobic surfaces can significantly suppress particle deposition by reducing effective particle–surface adhesion and promoting rebound or resuspension, particularly for small particles under low surface energy conditions [29,30]. The present simulations reproduce this behavior, as evidenced by the low deposition rates observed for small particles and low particle–surface energies.

With increasing particle size, the deposition rate increases from approximately 5.7% for particles with a radius of 8 μm to about 10.7% for particles with a radius of 12 μm. This trend is consistent with earlier studies reporting enhanced deposition of larger particles due to increased inertia and higher particle–surface collision probability [22,37,38,39,40]. The agreement suggests that the coupled LBM–IMBM–DEM framework employed here is capable of capturing particle size-dependent deposition behavior reported in the literature.

The influence of particle–surface energy on deposition behavior also agrees well with existing contact model-based numerical studies. Previous works using JKR-type adhesion models have shown that increasing surface energy leads to a rapid increase in deposition probability, followed by a saturation-like behavior once adhesion becomes dominant [33,34,35,36]. In the present study, the deposition rate increases nonlinearly from approximately 9.7% at a particle–surface energy of 0.001 J·m⁻² to 21.3% at 0.01 J·m⁻², and then it increases more moderately to 24.3% at 0.1 J·m⁻². This transition behavior is consistent with the reported shift from rebound-dominated to adhesion-dominated deposition regimes in the literature.

Compared with previous studies that primarily focused on idealized smooth or periodically structured surfaces, the present work extends existing knowledge by explicitly considering randomly distributed surface roughness with controlled statistical parameters. The observed increase in deposition rate with increasing roughness standard deviation is in agreement with experimental observations that rough surfaces enhance particle trapping by increasing contact probability and local flow disturbances [29,30]. However, the present results further demonstrate that deposition behavior is governed not only by roughness amplitude but also by the statistical characteristics of surface topography, which are rarely addressed in previous numerical studies.

Overall, the comparison with the existing literature confirms the validity of the present numerical framework while highlighting its capability to resolve the combined effects of particle size, surface energy, and random roughness statistics on particle deposition over superhydrophobic surfaces.