Finite Element Simulation and Experimental Validation of Nickel Coating Thickness Distribution in Electroplated FCCZ Lattice Structures

Abstract

Metal electrodeposition on additively manufactured lattice structures enables the creation of functionally graded hybrid components with enhanced mechanical properties. However, predicting coating thickness distribution remains challenging due to complex current density fields in intricate geometries. This study develops and validates a finite element electrochemical simulation model for predicting coating thickness distribution in lattice structures using COMSOL Multiphysics 6.1. The model incorporates Butler–Volmer electrode kinetics, mass transport limitations, and the Laplace equation for current distribution. Experimental validation was performed using FCCZ lattice structures electrochemically coated with nickel for 24 h at 200 A/m². CT scanning analysis revealed mean absolute errors of 5.25% between simulation and experiment after model calibration. The validated model successfully captures the exponential coating gradient from exposed edges to internal regions and provides a robust predictive tool for coating thickness distribution, which is essential for the effective design and optimization of electrochemically metallized lattice structures.

1. Introduction

Additive manufacturing (AM) enables the fabrication of complex components through layer-by-layer material deposition, achieving unprecedented design freedom that is difficult to realize cost-effectively with conventional manufacturing processes [1]. This is particularly significant for future lightweight applications, as hollow structures or lattice structures can be manufactured directly [2,3]. Through additive manufacturing processes, such structures can be integrated directly into components and produced in one piece, enabling the exploitation of topological advantages for enhanced mechanical performance [4].

Additively manufactured metallic lattice structures exhibit excellent mechanical properties relative to their mass, making them a focus of current research [2,5]. The most commonly used manufacturing process is powder bed fusion, where metal powder is selectively melted by a laser to build components layer by layer [3,6]. An alternative approach is the hybrid manufacturing of lattice structures consisting of a polymer core structure with a thick metallic coating [7,8]. In this process, the polymer structure is first fabricated using stereolithography and subsequently electrochemically metallized through electroless plating followed by electrodeposition [9,10]. Through extended coating times, it is possible to achieve high metal content in the component, with recent studies demonstrating metal mass fractions exceeding 96% [11]. Recent investigations have explored electrochemical metallization of both regular and stochastic polymer lattice geometries, demonstrating that lattice design parameters significantly influence the resulting mechanical properties of bionic cell structures [12]. Current research results show that even thin metallic coatings lead to significant improvements in mechanical component properties, with reported increases in specific strength of up to 500% and specific stiffness improvements of 120% [10].

The mechanical enhancement mechanisms of metal-coated lattice structures have been extensively investigated. Song et al. [8] demonstrated that nickel-coated polymer meso-lattice composites exhibit 68% higher modulus and 35% higher strength compared to polymer-only structures. Zhao et al. [13] characterized hollow octet nickel lattices, revealing that strength, modulus, and energy absorption properties scale exponentially with density. Liu et al. [14] developed machine learning approaches for predicting equivalent elastic properties of metal-coated lattices, while Soleimanian et al. [15] investigated the optimal coating film thickness for maximizing specific modulus through multiscale evaluation methods.

A central characteristic of additively manufactured metal-hybrid structures produced through electrodeposition is the shape gradation that occurs within the component. Due to the electrochemical process, coating thickness is distributed nonuniformly across all lattice struts, with outer component surfaces being more heavily coated than inner ones [11]. Understanding and predicting this coating thickness distribution is essential for optimizing mechanical properties and enabling the design of functionally graded structures. The coating distribution follows complex electrochemical principles governed by current density distribution, mass transport phenomena, and electrode kinetics.

The complex three-dimensional geometry of lattice structures poses significant challenges for electrochemical modeling. Current density distribution is governed by the component geometry, electrolyte conductivity, and electrode kinetics, resulting in nonuniform coating deposition. The electrochemical process is fundamentally described by the Butler–Volmer equation, which relates current density to overpotential through electrode kinetics parameters. For complex geometries, accurate prediction requires finite element simulation that captures the electrochemical field distribution within the intricate lattice geometry, typically employing Laplace’s equation for potential distribution in the electrolyte. Recent investigations have demonstrated that geometric modifications, such as sheet covers, can significantly influence filling behavior and current distribution during electrochemical joining of additively manufactured components [16]. Recent modeling work has demonstrated that accurate prediction of nickel coating thickness on geometrically complex substrates requires resolving the coupled effects of local curvature, current redistribution, and deposition kinetics [17]. Advanced finite element studies further highlight that electrodeposition on three-dimensional components exhibits highly nonuniform current density fields, which must be captured to reliably predict the resulting nickel layer geometry [18]. Recent developments in next-generation electroforming simulations underline the need for physically consistent models that incorporate realistic geometries and process constraints to achieve predictive coating control in complex applications [19]. However, validation of such models against experimental data remains limited, particularly for extended coating times where substantial thickness build-up occurs and geometry evolution effects become significant.

This work develops a comprehensive electrochemical simulation framework for predicting coating thickness distribution in additively manufactured lattice structures. The finite element model, implemented in COMSOL Multiphysics, incorporates Butler–Volmer electrode kinetics, mass transport limitations through limiting current density calculations, and mesh convergence analysis to calculate local coating thickness using Faraday’s law.

Experimental validation employs stereolithographically manufactured FCCZ lattice structures with electrochemical nickel coating. The validation methodology combines cross-sectional microscopy and high-resolution computed tomography scanning to assess coating thickness distribution throughout the complex three-dimensional geometry, enabling systematic comparison between simulation predictions and measured coating distribution.

This study establishes a validated simulation framework for predicting coating thickness gradients in lattice structures, providing the foundation for design optimization of functionally graded hybrid components and addressing the critical need for predictive modeling tools in hybrid additive manufacturing.

2. Materials and Methods

2.1. Simulation Model Setup

The electrodeposition simulation was performed using COMSOL Multiphysics with the Electrodeposition Module. A 3D model was chosen as the electrodeposition process is influenced by layer growth in all three spatial directions and current flow lines from all directions in the electrolyte.

2.1.1. Geometry Generation and Lattice Selection

An FCCZ (face-centered cubic with Z-struts) lattice structure was modeled directly in COMSOL Multiphysics. Individual unit cells measuring 5 × 5 × 5 mm³ with strut diameters of 0.2 mm were arranged in a 6 × 6 × 4 configuration, resulting in a final structure size of 30 × 30 × 20 mm³. Support struts with 0.7 mm diameter were added to the top and bottom surfaces for manufacturing stability during stereolithography.

Due to symmetry properties, the simulation domain was reduced to 1/8 of the original size to decrease computational time. The electrolyte volume was modeled by subtracting the lattice geometry from the surrounding volume using Boolean operations. Figure 1 shows the final modeled FCCZ lattice geometry used for the electrochemical simulation.

2.1.2. Mesh Generation and Convergence Analysis

The finite element mesh was generated using tetrahedral elements in COMSOL Multiphysics, chosen for their ability to handle complex geometries through triangular faces and suitability for automated meshing algorithms. Due to the small strut diameter (0.2 mm) and expected high information gradient along the lattice structure, fine meshing was required near the component, while the electrolyte volume could be meshed more coarsely to reduce computational time.

A systematic mesh convergence analysis was performed to determine optimal mesh parameters. The convergence criterion was based on local current density values at the outermost lattice corner, where the highest current density was expected. The analysis confirmed that the local current density values converged at the chosen mesh parameters, ensuring mesh independence for the simulation. The optimized parameters are summarized in

Table 1. Optimized mesh parameters for electrochemical simulation.

Parameter	Value	Unit
Maximum element size ($e_{max}$)	0.03	m
Minimum element size ($e_{min}$)	$1\times {10}^{-5}$	m
Growth rate	1.5	-
Curvature resolution factor	0.5	-
Narrow region resolution	0.6	-

.

The maximum element size of 0.03 m applies only to electrolyte regions far from the cathode surface, where the potential field exhibits low spatial gradients. The adaptive meshing algorithm ensures fine resolution (minimum element size of $1\times {10}^{-5}$ m) at the lattice structure. The mesh convergence analysis confirmed that the chosen far-field mesh size does not influence the local current density distribution at the cathode. The dimensionless mesh parameters control element quality and adaptive refinement: growth rate limits size variation between adjacent elements, curvature resolution factor determines mesh density along curved surfaces, and narrow region resolution controls refinement in confined spaces between struts.

Figure 2 shows the generated finite element mesh for the FCCZ lattice structure in COMSOL Multiphysics, demonstrating the adaptive mesh refinement with fine discretization near the lattice surfaces and coarser elements in the surrounding electrolyte volume.

2.1.3. Current Density Distribution Models

The electrochemical modeling is based on the Nernst–Planck equation [20,21], which describes ion transport in the electrolyte through three mechanisms:

$${\mathbf{N}}_i=-D_i\nabla c_{i,L}-z_i{u}_{i}F{c}_{i,L}\nabla {\Phi }_l+c_{i,L}\mathbf{u}$$

(1)

where ${\mathbf{N}}_i$ is the material flux vector of charged particles, consisting of diffusion (first term), migration (second term), and convection (third term). The individual coefficients are defined as follows:

• $D_i$ is the diffusion coefficient of ion species i (m²/s).
• $c_{i,L}$ is the concentration of ion species i in the electrolyte (mol/m³).
• $z_i$ is the charge number of ion species i (dimensionless).
• $u_i$ is the mobility of ion species i (m²/(V·s)).
• F is Faraday’s constant (96,485 C/mol).
• ${\Phi }_l$ is the electric potential in the electrolyte (V).
• $\mathbf{u}$ is the fluid velocity vector (m/s).

The current density vector in the electrolyte is calculated according to Faraday’s law [21]:

$${\mathbf{i}}_l=F\sum _i{z}_i{\mathbf{N}}_i$$

(2)

The electrochemical model incorporates several simplifications to the general Nernst–Planck equation. The convection term is set to zero, which is valid for stagnant electrolyte conditions without forced fluid flow or significant natural convection. The diffusion term is neglected, assuming uniform ion concentration throughout the electrolyte, which is justified when the bulk electrolyte concentration is much larger than the concentration changes near the electrode surfaces and when mass transport limitations are primarily accounted for through the limiting current density. Additionally, electroneutrality is assumed throughout the electrolyte (${\sum }_i{z}_i{c}_i=0$), which is valid except in thin electrical double layers near electrode surfaces. Under these assumptions, the problem reduces to Laplace’s equation:

$${\nabla }^2{\Phi }_l=0$$

(3)

where ${\Phi }_l$ is the electric potential in the electrolyte. To provide realistic current distribution predictions, electrode kinetics are incorporated through the Butler–Volmer equation [20,21], and the limiting current density concept is introduced to restrict metal deposition to physically achievable values. The modeling approach follows established electrodeposition simulation techniques [22]:

$$i_{\lim }={\displaystyle \frac{z_{i}F{D}_i{c}_{i,L}}{{\delta }_N}}$$

(4)

where $D_i$ is the diffusion coefficient of the metal ion, $c_{i,L}$ is the metal ion concentration in the electrolyte, and ${\delta }_N$ is the thickness of the Nernst diffusion layer [20,23].

For the nickel electroplating system investigated, the limiting current density was calculated using experimental parameters derived from the electrochemical literature. The nickel ion concentration was determined from salt concentration measurements in the electrolyte (22 g/L) and the molar mass of nickel. The diffusion coefficient of Ni²⁺ ions (1.01 × 10⁻⁹ m²/s) was taken from ionic diffusion measurements [24], while the Nernst diffusion layer thickness (0.2 × 10⁻³ m) was obtained from the electroplating literature [20]. The number of electrons transferred for nickel deposition (z = 2) completes the parameter set required for the calculation.

The limiting current density was calculated as

$$i_{\lim }={\displaystyle \frac{z_{\mathrm{Ni}}F{D}_{\mathrm{Ni}}{c}_0}{{\delta }_N}}={\displaystyle \frac{z_{\mathrm{Ni}}F{D}_{\mathrm{Ni}}\frac{Sa{l}_l}{M_{\mathrm{Ni}}}}{{\delta }_N}}=360.6{\mathrm{A}/\mathrm{m}}^2$$

(5)

Electrode kinetics are described by the full Butler–Volmer equation:

$$i_{\mathrm{loc}}=i_0\left[exp\left({\displaystyle \frac{{\alpha }_{a}F\eta }{RT}}\right)-exp\left(-{\displaystyle \frac{{\alpha }_{c}F\eta }{RT}}\right)\right]$$

(6)

where $i_0$ is the exchange current density, ${\alpha }_a$ and ${\alpha }_c$ are the anodic and cathodic charge transfer coefficients, $\eta$ is the overpotential, R is the gas constant, and T is the absolute temperature.

The overpotential $\eta$ is defined as the difference between the electrode potential and the equilibrium potential:

$$\eta ={\Phi }_s-{\Phi }_l-E_{eq}$$

(7)

where ${\Phi }_s$ is the potential in the solid electrode, ${\Phi }_l$ is the potential in the electrolyte at the electrode surface, and $E_{eq}$ is the equilibrium potential for the nickel deposition reaction (Ni₂₊ + 2e₋ → Ni). The equilibrium potential and the exchange current density for nickel deposition were taken from the electrochemical literature [25], with the latter initially set to 0.1 A/m². The charge transfer coefficients were adopted from nickel electrochemistry studies [26].

2.1.4. Boundary Conditions

The electrochemical simulation employed specific boundary conditions to represent the experimental setup, as summarized in

Table 2. Boundary conditions for electrochemical simulation.

Surface Type	Boundary Condition
Anode surfaces	Constant total current: 0.12 A (galvanostatic control)
Cathode surfaces (lattice)	Butler–Volmer equation with variable local current density
Electrolyte boundaries	Insulating: $\partial {\Phi }_l/\partial n=0$
Reference potential	Cathode potential = 0 V (reference)

. A constant total current of 0.12 A was applied at the anode surfaces, corresponding to 1/8 of the experimental current (0.96 A) to account for the symmetry-reduced geometry. This maintains the experimental current density of 200 A/m² while scaling appropriately for the reduced cathode surface area. The Butler–Volmer equation was applied at all cathode surfaces to allow realistic current distribution based on local overpotential and electrode kinetics. Electrolyte boundaries are treated as insulating surfaces with zero normal current flux, while the cathode potential is set as the electrical reference point (0 V). The anode potential adjusts automatically to maintain the specified total current under galvanostatic control conditions.

2.1.5. Model Simplifications and Assumptions

The simulation model employs several simplifications to balance computational efficiency with physical accuracy. The model incorporates three key physical phenomena governing the coating deposition process. Potential distribution in the electrolyte is calculated using Laplace’s equation, accounting for ohmic losses due to electrolyte resistance. Electrode reaction kinetics are described by the Butler–Volmer equation, which relates local current density to overpotential based on exchange current density and charge transfer coefficients.

Mass transport limitations are incorporated through a limiting current density constraint that prevents unrealistically high current densities at geometrical singularities such as corners and edges. This constraint becomes particularly important for extended electrodeposition processes, where diffusion of ionic species to the electrode surface limits the maximum achievable current density. The model assumes steady-state conditions with uniform electrolyte composition and temperature. In this work, the electrodeposition process is treated as quasi-steady. The stationary current density distribution obtained for the initial geometry is used to compute the local deposition rate, while a fully coupled transient simulation with continuous geometry updates is not performed due to its prohibitive computational cost for complex three-dimensional lattice structures with large relative coating thicknesses. This modeling choice may introduce systematic deviations, because the growing coating alters the local curvature and available surface area, which would influence the current density distribution in a fully transient formulation. In particular, the quasi-steady approach does not capture the local redistribution of current that may occur as regions become more exposed or increasingly shielded over time. The calibration procedure described in Section 2.4.1 compensates for the global scaling error introduced by the nonlinear geometry growth, but it does not correct local variations in the material distribution. These geometric limitations are therefore explicitly acknowledged as part of the model’s assumptions.

2.1.6. Coating Thickness Calculation

Local coating thickness was calculated using Faraday’s law [20]:

$$h_{\mathrm{loc}}={\displaystyle \frac{i_{\mathrm{loc}}·t·M}{\rho ·n·F}}$$

(8)

where h is the coating thickness, $i_{\mathrm{loc}}$ is the local current density, t is the coating time, M is the molar mass of nickel, $\rho$ is the density of nickel [27], n is the number of electrons transferred, and F is Faraday’s constant.

2.2. Experimental Validation

2.2.1. Sample Preparation

The sample preparation process follows a systematic three-stage approach to enable electrodeposition of the insulating polymer lattice structures. Figure 3 illustrates the key stages of this process.

FCCZ lattice structures were manufactured using stereolithography (FormLabs Form 3) with standard clear resin. The printed structures underwent post-processing including washing in isopropanol and UV curing according to manufacturer specifications.

2.2.2. Surface Preparation and Conductivity Treatment

Prior to electrodeposition, the polymer surfaces were made conductive through spray application of conductive copper lacquer (Tifoo copper conductive coating, MARAWE GmbH & Co. KG, Regensburg, Germany). After 24 h of drying, specimens underwent pre-electroplating in a copper bath for 15 min at 75 A/m² current density to establish a uniform conductive base layer [11]. This two-step process ensures reliable electrical contact across the complex lattice geometry while maintaining coating uniformity.

2.2.3. Electrodeposition Process

Nickel electroplating was performed using a nickel sulfamate bath. The electrolyte composition is specified in

Table 3. Electrolyte composition.

Component	Percentage
Nickel sulfamate	15–25%
Boric acid	<5%
Nickel chloride	0.3–1%

, and the coating process parameters are detailed in

Table 4. Electrodeposition process parameters.

Parameter	Value	Unit
Current density	200	A/m2
Coating duration	24	hours
Bath temperature	50	°C
Applied current	0.96	A
Electrolyte conductivity	36	mS/cm
Salt concentration	22	g/L

.

2.3. Characterization Methods

2.3.1. Computed Tomography (CT) Analysis

Selected specimens were analyzed using CT scanning with a Phoenix V|tome|x S240 system (General Electric, now Waygate Technologies, Hürth, Germany) at the Institute for Materials Science (IfW), TU Darmstadt. The scanning parameters were optimized for nickel coating analysis and are specified in

Table 5. CT scanning parameters.

Parameter	Value	Unit
Voxel resolution	48.08	$\mathsf{\mu }$m
Tube voltage	130	kV
Tube current	220	$\mathsf{\mu }$A
Rotation angle	360	degrees

.

The CT scanning creates a digital representation where the nickel coating absorbs X-rays while the polymer substrate allows transmission, enabling clear material differentiation. Raw data processing involved conversion from .vol format to .raw format for import into ParaView software. Volume dimensions were extracted from accompanying .pcr files, and appropriate data scalar types were specified for accurate reconstruction.

For quantitative analysis, virtual cross-sections were generated at specific z-positions corresponding to experimental cutting planes, enabling direct comparison between CT measurements and cross-sectional microscopy results. This nondestructive analysis provides comprehensive coating thickness data across the entire lattice structure.

2.3.2. Cross-Sectional Analysis

Cross-sectional analysis was performed complementary to CT scanning to provide enhanced validation of coating thickness measurements and detect potential coating defects. While CT scanning offers comprehensive three-dimensional analysis, cross-sectional microscopy provides superior resolution for detecting thin coating layers in the interior regions of the lattice, where CT resolution (48.08 $\mathsf{\mu }$m voxel size) may not accurately capture very thin coatings. Additionally, cross-sectional analysis enables detection of coating formation errors, inclusions, porosity, and other microstructural features that are critical for coating quality assessment.

Sample preparation involved embedding lattice specimens in epoxy resin, which cured for 24 h. To facilitate easy removal from the embedding container, the container was pregreased with Vaseline.

During cross-section preparation, coarse sandpaper was initially used to grind down the resin to the actual specimen material. Subsequently, progressively finer sandpaper was employed to reach the target height of z = 9 mm from the lattice structure center. Special attention was paid to maintaining uniform grinding progress to avoid tilting of the specimen.

The z-position was selected to be just below the FCCZ node points at the upper (or lower) end of the unit cell. This position offers the advantage of containing the most individual struts, while the diagonal struts of the unit cell face can still be clearly assigned to their z-struts, which is relevant for subsequent analysis.

Cross-sectional images were acquired using a ZEISS Smartzoom 5 microscope with a PlanApoD 1.6x/0.1 FWD 36 mm objective (ZEISS, Oberkochen, Germany).

2.4. Comparison Methodology and Calibration

Simulation results were compared with experimental data using a systematic approach that examined metal coating cross-sectional areas at individual strut nodes in identical cutting planes. The quantitative comparison focused on the absolute average error between predicted and measured coating thickness distributions.

2.4.1. Surface Area Calibration Method

The electrodeposition of lattice struts with diameters on the order of $d=0.2\mathrm{mm}$ and coating thicknesses between $h=0.1$–$0.9\mathrm{mm}$ introduces a substantial geometric nonlinearity in the growth of the coated cross-sections. For such geometries, the commonly used first–order approximation that relates deposited volume to the product of local surface area and coating thickness,

$$\Delta V\approx hA,$$

(9)

does not hold. This approximation is only valid in the limit $h\ll R$, where R denotes the local radius of curvature. In the present case, however, the relative coating thicknesses reach $h/R\approx 1$–9, rendering the offset operation highly nonlinear and dominated by curvature effects.

Geometrically, an offset surface generated by shifting the mesh nodes along their surface normals does not conserve the correct volume. For curved geometries, the exact deposited volume is described by the second-order Steiner formula,

$$V\left(h\right)=V_0+h{A}_0+h^2\int H\mathrm{d}A+\mathcal{O}\left(h^3\right),$$

(10)

where $A_0$ is the original surface area and H is the mean curvature. The neglected higher-order terms $\mathcal{O}\left(h^2\right)$ and $\mathcal{O}\left(h^3\right)$ become significant for thick coatings. In lattice structures, two competing effects emerge: (1) the simplified volume calculation $\Delta V\approx hA$ neglects local curvature contributions, leading to volume underprediction, while (2) self-intersections at lattice node points where coating growth from multiple struts overlaps result in artificially inflated computed volumes due to double-counting of overlapping regions.

These competing geometric effects necessitate a calibration procedure that balances curvature-induced underprediction against intersection-induced overprediction to achieve accurate volume calculations in the deformed mesh geometry.

To compensate for this intrinsic geometric bias, an iterative surface-area calibration procedure was implemented. The calibration algorithm determines a global scaling factor $S_{\mathrm{area}}$ through the following iterative procedure:

$$S_{\mathrm{area}}^{(k+1)}=S_{\mathrm{area}}^{\left(k\right)}·{\displaystyle \frac{A_{\mathrm{measured}}}{A_{\mathrm{simulated}}^{\left(k\right)}}}$$

(11)

where k denotes the iteration number, $A_{\mathrm{measured}}$ is the experimentally determined coating area in the reference cross-sectional plane (z = 9 mm), and $A_{\mathrm{simulated}}^{\left(k\right)}$ is the simulated coating area in iteration k.

Starting with $S_{\mathrm{area}}^{\left(0\right)}=1.0$, the algorithm applies the scaled local coating thickness to deform the mesh, calculates the resulting cross-sectional area $A_{\mathrm{simulated}}^{\left(k\right)}$ in the reference plane, and updates the scaling factor according to the above equation. The iterations continue until the convergence criterion $|A_{\mathrm{simulated}}^{\left(k\right)}-A_{\mathrm{measured}}|/A_{\mathrm{measured}}<0.1\%$ is satisfied.

The final converged value $S_{\mathrm{area}}={lim}_{k\to \infty }{S}_{\mathrm{area}}^{\left(k\right)}$ represents the global surface area calibration factor that accounts for curvature-dependent nonlinear growth behavior. This factor is then applied uniformly to scale all local coating thickness predictions:

$$h_{\mathrm{calibrated}}=S_{\mathrm{area}}·h_{\mathrm{Faraday}}$$

(12)

where $h_{\mathrm{Faraday}}$ is the local coating thickness calculated from Faraday’s law based on the local current density, and $h_{\mathrm{calibrated}}$ is the corrected coating thickness applied during mesh deformation.

2.4.2. Electrochemical Parameter Studies

As exchange current density and limiting current density values could not be experimentally determined for the specific nickel sulfamate bath composition, a parametric investigation was conducted. The parameter study systematically varied these critical electrochemical parameters within physically reasonable ranges to optimize simulation accuracy.

The exchange current density $i_0$ was varied between 0.005 and 50 A/m² while the limiting current density $i_{\lim }$ was investigated over the range 250 to 650 A/m². Model accuracy was evaluated using mean absolute error (MAE) between simulation and experimental coating thickness measurements to identify optimal parameter combinations.

3. Results

This section presents the electrochemical simulation results and experimental validation, structured to clearly distinguish between current density distribution predictions and coating thickness results. The analysis follows the workflow from electrochemical modeling through geometric conversion to final coating thickness validation.

3.1. Current Density Distribution and Geometry Conversion

The electrochemical simulation was performed using COMSOL Multiphysics with the Butler–Volmer kinetic model described in Section 2.1. Figure 4 shows the current density distribution on the FCCZ lattice structure surface, demonstrating the expected gradient from high current density at exposed edges and corners to lower values in protected internal regions.

The electrochemical simulation reveals current density values ranging from 20 A/m² in the most protected interior regions to 220 A/m² at exposed corners and edges. This distribution reflects the fundamental physics of electrodeposition, where geometric accessibility determines the local electrochemical activity.

The local current density values were converted to coating thickness using Faraday’s law as described in Section 2.1.6. This conversion reveals significant coating thickness variations across the lattice structure, with maximum values of 600–650 $\mathsf{\mu }$m occurring at exposed corners and edges, while internal strut nodes exhibit coating thicknesses in the range of 60–70 $\mathsf{\mu }$m.

When the calculated coating thickness is applied through normal displacement of surface mesh nodes, geometric artifacts emerge due to the substantial thickness variations; see Section 2.4.1. Figure 5 illustrates this challenge, showing the deformed mesh geometry with surface self-intersections at node points where large coating thickness values cause geometric overlap.

These geometric intersections arise from the nonlinear nature of coating growth on curved surfaces with coating thicknesses comparable to the substrate radius. The intersections necessitate a systematic calibration approach to maintain volume consistency.

3.2. Experimental Validation and Calibration

As detailed in Section 2.4.1, the substantial geometric nonlinearity introduced by coating thicknesses comparable to strut radii requires a systematic calibration approach. The surface area calibration factor $S_{\mathrm{area}}$ was determined by comparing simulated coating areas with experimentally measured coating areas in reference cross-sectional planes. High-resolution CT scanning and cross-sectional microscopy were employed to obtain reference coating thickness data for calibration, providing comprehensive coating area measurements across the lattice structure.

Figure 6 presents the systematic analysis methodology for coating thickness distribution measurements and the resulting exponential gradient validation. The results demonstrate an exponential decay in coating thickness with distance from the structure boundary.

3.2.1. CT Scan Validation

High-resolution CT scanning revealed the actual coating thickness distribution on fabricated samples. Figure 7 shows the three-dimensional visualization of the scanned coated lattice structure with cross-sectional planes used for analysis.

The CT data processing sequence included raw data conversion to 3D volume with dimensions of 900 × 900 × 1000 voxels and voxel size 48.08 $\mathsf{\mu }$m, identification of metal coating through X-ray absorption contrast, cross-sectional analysis at z = 9 mm plane, and quantification of metal cross-sectional area per strut node.

To accurately determine coating areas, the cross-sectional plane was systematically subdivided into rectangular regions centered on individual strut nodes. As shown in Figure 6a, each rectangular analysis region encompasses a z-strut with its surrounding diagonal struts, progressing systematically from the structure boundary (#1) toward the interior (#4). This approach ensures accurate area quantification while avoiding measurement artifacts from oblique strut intersections.

The coating area within each rectangle is calculated using the negative voxel count multiplied by the square of the voxel size according to

$$A=N_{\mathrm{neg},\mathrm{Voxel}}·L_{\mathrm{Voxel}}^2$$

(13)

where A is the coating area, $N_{\mathrm{neg},\mathrm{Voxel}}$ is the number of negative voxels, and $L_{\mathrm{Voxel}}=0.04808$ mm is the voxel size. This exploits nickel’s X-ray absorption properties to distinguish coating from substrate material.

3.2.2. Cross-Sectional Microscopy Analysis

Figure 8 shows a representative cross-sectional image of a coated specimen, where the nickel coating appears bright white while the dark regions within the struts represent the original polymer substrate. The cross-sectional analysis reveals the characteristic coating thickness variation across different strut positions, with enhanced nickel accumulation observed at lattice node intersections due to the locally increased current density at these geometric features.

The microscopic analysis confirms the predicted coating thickness gradients. Individual strut examination reveals uniform coating adhesion without significant porosity or delamination, indicating successful surface preparation and stable electrochemical conditions throughout the deposition process.

3.2.3. Quantitative CT Scan Validation

For quantitative validation, coating areas in cross-sectional planes were compared between simulation and CT measurements. The exponential coating gradient was verified by analyzing coating areas from the structure boundary toward the interior. Rectangular analysis regions were positioned around strut nodes along diagonal paths from exterior corners to the center. The measured coating areas were fitted to an exponential function of the form $A\left(d\right)=a·e^{-b·d}+c$, where d is the distance from the boundary. This exponential decay follows theoretical predictions from the Butler–Volmer equation, confirming that local current density decreases exponentially with increasing overpotential toward the structure interior.

The iterative calibration procedure converged after six iterations to a final surface area calibration factor $S_{\mathrm{area}}=1.169$. The initial uncalibrated simulation (iteration 0) predicted a coating area of 123.92 mm² compared to the experimentally measured area of 140.01 mm² in the reference plane at z = 9 mm. This initial ratio $A_{\mathrm{measured}}/A_{\mathrm{simulated}}^{\left(0\right)}=1.130$ indicated a 13.0% underprediction by the uncalibrated model. Through the iterative procedure, the final surface area calibration factor converged to $S_{\mathrm{area}}=1.169$, which accounts for the systematic curvature effects in the electrodeposition process.

Figure 9 shows the quantitative comparison between simulated and measured coating distribution after calibration. The experimental coating follows the predicted exponential gradient, with correlation coefficient R² = 0.92 for the exponential fit function, confirming the theoretical framework.

3.2.4. Quantitative Validation and Error Analysis

For quantitative error analysis, the mean absolute error (MAE) and root mean square error (RMSE) were calculated using the following:

$$\mathrm{MAE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n|{\mathrm{relError}}_i|$$

(14)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{\mathrm{relError}}_i^2}$$

(15)

where n is the number of analyzed strut nodes, and ${\mathrm{relError}}_i$ is the relative error between simulated and measured coating areas at node i. Quantitative comparison between simulation and experimental results yielded mean absolute errors of 5.46% and root mean square errors of 7.13%. The RMSE is larger than the MAE due to its higher sensitivity to outliers. The error distribution shows no systematic pattern. Detailed accuracy metrics including maximum and minimum relative errors are summarized in

Table 6. Simulation accuracy metrics.

Error Metric	Value
Maximum relative error	+21.43%
Minimum relative error	−15.25%
Mean absolute error (MAE)	5.46%
Root mean square error (RMSE)	7.13%

.

3.3. Model Optimization Through Parametric Studies

3.3.1. Electrochemical Parameter Calibration

To address the observed deviations and improve simulation accuracy, a systematic parametric study was conducted to optimize the exchange current density ($i_0$) and limiting current density ($i_{lim}$). As these critical electrochemical parameters could not be experimentally determined for the specific nickel sulfamate bath composition, the calibration was performed through comparison with experimental coating thickness measurements.

The calibration process was carried out in two phases. The first parametric study involved 36 simulations covering parameter ranges of 0.005–50 A/m² for the exchange current density ($i_0$) and 250–550 A/m² for the limiting current density ($i_{lim}$). As the lowest error occurred at the upper boundary of the investigated $i_{lim}$ range, a second study was conducted with refined intervals around the optimal region, extending the limiting current density range to 525–650 A/m² while maintaining the same $i_0$ values.

To provide a quantitative assessment of parameter sensitivity, Figure 10 summarizes the results of both studies by visualizing the mean absolute error (MAE) as a function of $i_0$ (logarithmic scale) and $i_{lim}$. The heatmap clearly demonstrates that the model accuracy is strongly governed by the limiting current density, whereas variations in $i_0$ lead to only minor changes in MAE over several orders of magnitude. The region of lowest error is found near $i_0=1\mathrm{A}/{\mathrm{m}}^2$ and $i_{lim}=600\mathrm{A}/{\mathrm{m}}^2$, which is, therefore, selected for all subsequent simulations. The final calibrated model parameters are presented in

Table 7. Calibrated model parameters and performance.

Parameter	Optimized Value
Exchange current density $i_0$	1.0 A/m2
Limiting current density $i_{lim}$	600 A/m2
Mean absolute error (MAE)	5.25%
Root mean square error (RMSE)	6.60%
Maximum relative error	+19.01%
Minimum relative error	−14.91%

.

The calibrated model showed improved accuracy. The scaling factor between simulated and experimental total coating area improved from 1.169 to 1.168, indicating better overall agreement.

3.3.2. Demonstrator Validation

To verify the transferability of the simulation model to different geometries, a demonstrator component was designed and manufactured. The demonstrator consisted of a table structure with a lattice top plate composed of multiple FCCZ unit cells arranged in a planar configuration.

Figure 11 shows the modeled geometry in COMSOL and the corresponding simulation results. The coating thickness distribution exhibits the expected gradient pattern, with corner regions showing thicknesses of 0.4–0.7 mm while interior regions achieve approximately 0.1 mm coating thickness.

The demonstrator was manufactured and coated using identical procedures to the validation specimens.

Table 8. Demonstrator coating process parameters.

Parameter	Value	Unit
Applied current	0.6	A
Coating duration	24	hours
Mass before coating	0.67	g
Mass after coating	15.06	g

summarizes the key process parameters.

Validation was performed for two different cross-sectional planes (z = 9 mm and z = 6 mm) to assess the model’s ability to predict coating gradients in the vertical direction. Using the same calibration scaling factor determined from the initial validation specimens, the demonstrator achieved MAE = 6.50% (upper plane) and MAE = 5.64% (lower plane).

Figure 12 presents the quantitative comparison between simulation and CT measurements at the z = 9 mm cross-section. The percentage deviation analysis confirms the exponential gradient pattern predicted by the model, with systematic accuracy maintained across different cross-sectional planes.

These results confirm the model’s transferability to alternative geometric configurations while maintaining prediction accuracy consistent with the original validation studies.

4. Discussion

This study successfully developed and validated an electrochemical simulation model for predicting coating thickness distribution in electroplated lattice structures. The results demonstrate good agreement between simulation predictions and experimental measurements, with mean absolute errors of 5.25% after model calibration.

4.1. Simulation Accuracy and Limitations

The simulation model captures the fundamental characteristics of the electrodeposition process and reproduces the experimentally observed coating gradients with good accuracy. Nevertheless, several limitations arise from the modeling assumptions. First, surface roughness effects are not considered, as the simulation assumes perfectly smooth lattice struts. Real stereolithography surfaces exhibit submillimeter roughness features that can locally enhance current density and thereby increase deposition rates in a manner not captured by the model. Additionally, simplified assumptions regarding electrolyte conductivity, concentration homogeneity, and mass transport may contribute to deviations between simulation and experiment, particularly in regions with restricted electrolyte access.

A further limitation stems from the quasi-steady treatment of the electrochemical problem. In the present model, the current density distribution is computed only once for the initial geometry, although the coating growth alters the local curvature, surface accessibility, and effective electrode area over time. These geometric changes would lead to a gradual redistribution of current in a fully coupled transient formulation, including shielding effects in interior regions and prolonged exposure of outer regions. The quasi-steady approach does not capture these time-dependent electrochemical effects. The calibration method described in Section 2.4.1 compensates for the global scaling error introduced by the nonlinear geometric growth, but it does not correct local deviations in the material distribution that may arise from omitted transient current redistribution.

An additional aspect worth noting is that the two dominant sources of modeling error—the geometric approximation associated with offset-based coating growth (including curvature-dependent deviations and local self-intersections) and the neglect of time-dependent current redistribution—act in opposite directions. While the geometric approximation tends to underestimate deposited volume in regions of high curvature, the missing transient coupling would, in principle, lead to an overestimation of deposition in areas that remain highly exposed as the coating grows. It is, therefore, plausible that these effects partially compensate each other, which may contribute to the comparatively small global deviation observed in the validation. A quantitative investigation of this interaction, for example, through simplified transient test cases or stepwise geometry-updated simulations, represents a promising direction for future work.

4.2. Experimental Validation Challenges

The CT-based validation approach provided valuable quantitative data but also revealed measurement limitations. At low coating thicknesses (<50 $\mathsf{\mu }$m), the voxel resolution of 48.08 $\mathsf{\mu }$m becomes limiting, leading to quantization errors in thickness measurements. This explains the increasing scatter in experimental data toward the structure interior.

Additional measurement challenges include CT scanning artifacts and process-induced coating irregularities. Figure 13 illustrates typical issues encountered during validation: CT scan artifacts can create apparent voids or thickness variations (left), while electroplating process variations lead to dendrite formation at individual struts (right). These phenomena contribute to local measurement uncertainties but do not affect the overall coating gradient validation.

Cross-sectional microscopy confirmed the CT findings but was limited to 2D analysis. The automated image analysis successfully identified coating boundaries but struggled with merged coatings where adjacent struts grew together, a phenomenon correctly predicted by the simulation.

4.3. Model Calibration and Parameter Sensitivity

The parametric study revealed that the limiting current density has a stronger influence on coating uniformity than the exchange current density. The optimized limiting current density of 600 A/m² is 66% higher than the theoretical value calculated from the data in the literature (360.6 A/m²), suggesting that mass transport limitations are more severe in the complex lattice geometry than in the idealized conditions used for parameter determination.

The exchange current density showed minimal impact on overall accuracy, with values between 0.1 and 5 A/m² yielding similar results.

4.4. Comparison of Modeling Approaches

This study compared two electrochemical modeling approaches to evaluate their suitability for lattice structure coating prediction. Figure 14 demonstrates the current density distribution predicted by the simplified ohmic resistance model, which shows significant limitations when applied to complex geometries.

The simplified ohmic resistance model systematically overpredicts coating thickness at exposed corners and edges while underpredicting interior coating. This behavior results from the model’s neglect of electrochemical reaction kinetics, which become increasingly important as current density variations increase. The extreme current concentration at geometric singularities shown in Figure 14 leads to unrealistic coating predictions that fail to match experimental observations.

In contrast, the Butler–Volmer kinetic model (Figure 4) successfully captured the experimentally observed coating distribution patterns. The inclusion of electrode reaction kinetics through exchange current density and limiting current density parameters enabled realistic prediction of coating gradients across the complex lattice geometry. The model’s ability to account for activation and concentration overpotentials proved essential for accurate simulation of the electrodeposition process.

This comparison confirms that simplified resistance models are inadequate for complex three-dimensional geometries where significant current density variations occur. The incorporation of electrochemical kinetics, despite increased computational complexity, is necessary to achieve predictive accuracy suitable for engineering applications.

4.5. Model Performance and Validation

The achieved simulation accuracy demonstrates the effectiveness of the Butler–Volmer kinetic modeling approach for complex lattice geometries. The exponential coating gradient predicted by the model matches the experimental observations from both CT scanning and cross-sectional microscopy analysis. The iterative calibration procedure successfully reduced prediction errors.

The validation approach using dual measurement techniques (CT scanning for 3D quantitative analysis and microscopy for high-resolution 2D verification) provides robust experimental evidence for the simulation accuracy. The systematic analysis of edge effects and coating gradients confirms that the model captures the fundamental physics governing electrochemical deposition in complex geometries.

4.6. Future Research Directions

Several areas warrant further investigation to advance the simulation framework. Extension to alternative coating materials such as copper, silver, or zinc requires validation of material-specific electrochemical parameters and deposition kinetics under different electrolyte compositions. The transferability of the modeling approach should be established across diverse lattice topologies, including body-centered cubic, diamond, and gyroid structures, to understand the impact of different geometric complexities and surface-to-volume ratios. Multiphysics coupling should integrate thermal effects and bubble formation during electroplating to improve model accuracy for high-current-density processes. Dynamic modeling through time-dependent simulations would capture coating evolution and electrolyte aging effects over extended processing times. Integration of the validated coating thickness distributions with finite element mechanical analysis would enable prediction of resulting stiffness, strength, and failure modes of hybrid lattice structures.

The simulation framework provides a foundation for these extensions and demonstrates the feasibility of predictive electrochemical modeling for complex additive manufacturing applications.

5. Conclusions

This study successfully developed and validated a finite element electrochemical simulation model for predicting coating thickness distribution in electroplated FCCZ lattice structures. The 3D model incorporates Butler–Volmer electrode kinetics and current distribution under mass transport limitations, achieving a mean absolute error of 5.25% compared to experimental CT scanning data after iterative calibration.

The simulation accurately predicts the exponential coating thickness gradient from exposed edges to internal regions, with outer struts experiencing 7–8× higher coating thickness than internal struts. Parameter sensitivity analysis revealed that limiting current density has stronger influence on coating uniformity than exchange current density, indicating that mass transport limitations dominate over activation kinetics under the studied conditions.

The validated model enables predictive coating thickness distribution in metallized lattice structures, providing a foundation for iterative design optimization of the polymer substrate geometry to achieve targeted mechanical properties in functionally graded hybrid components. The methodology demonstrates the feasibility of predictive electrochemical modeling for complex additively manufactured structures and is transferable to lattice structures with comparable geometric complexity. This represents a significant advancement toward digital process control in electrochemical metallization of additively manufactured components, enabling the systematic design of lightweight structures with controlled coating thickness distribution.