Computational analysis of copper electrodeposition into a porous preform

Electroplating of metals into a porous preform with conductive walls is relevant in the fabrication of structural composites, fuel cells and batteries, and microelectronics. Electrodeposition process parameters, such as direct current or pulsed current, electric potential, and electrolyte concentration, as well as preform geometry, have important implications in the process outcomes including the filling process and the percentage of the infiltrated volume. Although electroplating into a vertical interconnect access (with nonconductive walls) for microelectronic applications has been extensively studied, the "flow-through" electroplating into a channel geometry with conducive walls has not been previously investigated. Here, copper infiltration into a such channel has been investigated using computational analysis for the first time. The effects of the inlet flow velocity, potential, electrolyte concentration, and microchannel geometry are systematically studied to quantify their influence on the electrodeposition rate, uniformity of the deposition front, and the infiltrated area within the channel. Computational results revealed that the unfilled area can be reduced to lower than 1% with a low applied potential, a high electrolyte concentration, and no inflow velocity. The results can be used to guide experiments involving electroplating metals into porous preforms toward reliable and reproducible manufacturing processes.


INTRODUCTION
Electroplating of metals into a porous scaffold (preform) has major applications in the fabrication of bioinspired structural composites, [1][2][3] in energy applications such as fuel cells and batteries, 4 filling the so-called vertical interconnect access (via) in microelectronics, and the production of multilayer printed wiring boards and advanced printed circuit boards (PCBs) with highdensity interconnects, 5,6 junction formation between thin metal film electrodes in a microfluidic channel, 7,8 among others.Depending on specific applications, different outcomes in terms of infiltration are desired.For example, if the process is used for composite fabrication, or filling interconnect vias, complete filling (infiltration) is desired.For applications in a fuel cell and battery, porosity is required for device function.Hence, an incomplete filling (with preferably open pores) is desired.In each case, however, it is desired that the channels remain open for the electroplating process to continue and for the electrolyte to replenish.Investigation of the through-hole plating including effects of organic acids, 9 through-hole copper electrodeposition by revision of plating cell configuration, 10 the effects of supporting electrolytes on copper electroplating for filling through-hole, 11 and the effect of channel geometries (cylindrical, V-, and X-shaped through holes) have been reported. 12][15] Computational analysis is a cost-effective way to investigate the effects of various process parameters of electroplating into a porous scaffold, such as current density distribution, thickness, shape, and uniformity of the deposited metal layer. 16,17This knowledge can be applied for process control to optimize the often time-consuming and costly experimentation.Zhang et al. proposed a physicochemical model for electroplating copper in through-silicon vias to investigate the filling mechanism, which includes the diffusion, adsorption, desorption, and incorporation of the additives and ions. 18Numerical analysis of electrodeposition in micro-vias has been reported. 19,20In this case, the micro-via has a metallic base and insulating side-walls with one side closed.In the current study, we investigate the "flowthrough" electrodeposition into a channel in which both walls of For the time evolution plot, the electrolyte concentration, the applied potential, and the inlet velocity were set to 0.5M, 0.2 V, and 0, respectively.x = 0 corresponds to the left corner of the bottom cathode.
the channel are conductive [see Fig. 1(a)].This configuration can be used for the fabrication of metal-ceramic composites 1 as well as for energy applications.For example, this case has been recently used for the fabrication of anodes in fuel cells, and the conductive walls are obtained by electroless deposition of metal on a ceramic scaffold. 4

THE METHODOLOGY
A schematic of the two-dimensional rectangular microchannel geometry used in this study is presented in Fig. 1(a).The width of the cathode (shown in blue color) is w and the distance between two cathodes is 2r.Two counter electrodes (anode), shown in orange color, are placed at a distance L from the two ends of the working electrodes (cathode).A constant voltage V is applied between the cathode and the anode.Far the upstream, the reactant concentration is assumed to be equal to its bulk value.Upstream was defined as a distance of 10 μm from the beginning of the channel.The initial concentration of the electrolyte was assumed to be 0.5M (500 mol/m 3 ).A steady-state and fully developed laminar flow is assumed to occur as the electrolyte moves from left to right.The velocity distribution is considered as where vx is the axial velocity, vmax is the maximum axial velocity, y is the radial coordinate from the center of the microchannel, and r is the radius of the microchannel.The electroplating process consists of two main steps: (i) the mass transport of copper ions through the electrolyte to the cathode surface and (ii) the reduction of copper ions at the cathode surface.Metal growth by electrodeposition process inside the microchannel is a multi-physics process in which the electrodeposition and fluid dynamics are simultaneously involved.The ionic flux within the electrolyte is governed by the diffusion, migration, and convection processes.We describe the electrochemical dynamics in the electrolyte by the Nernst-Planck equation, in which Ni is the transport vector [mol/(m 2 s)], In Eq. ( 2), Di denotes the diffusivity of the ionic species; ci, zi, and ui represent the concentration of ions, electronic charge of the ionic species, and the mobility of the charged species, respectively; F is the Faraday constant; φ l is the electrolyte potential; and u is the fluid velocity.The metal ions move from the bulk electrolyte toward the microchannel through three mechanisms: diffusion, migration, and convection, which overall control the deposition rate.These three fluxes are identified by the first, second, and third terms in Eq. ( 2), respectively.
We describe the electrode reaction kinetics by the Butler-Volmer expression, which describes the nonlinear relationship between the current density through the electrode and the overpotential, where ict is the local (electrode) current density (A/m 2 ), io is the exchange current density (the current in the absence of net electrolysis and at zero overpotential), αa and αc are the anodic and cathodic charge transfer coefficients, respectively, η is the activation overpotential (deviation from the equilibrium potential), and CR and C 0 are the dimensionless concentration of the reduced species and oxidized species, respectively.The first and second terms in Eq. ( 3) represent the anodic and cathodic components of the local current density, considering that both cathodic and anodic reactions occur on the same electrode.
The model considers ion transport in the electrolyte, the electron conduction in the electrodes, conservation of current and charge, and concentration overpotential.All the involved equations describing current trasnport and conservation, charge balances, electrochemical kinetics, and chemical species movement are coupled.
In the model, the thickness of the deposited metal layer on the cathode surface as a function of time is calculated using the Faraday's law.The time evolution of the electrodeposition front from one side of the cathode surface is shown in Fig. 1(c).For the time evolution plot, the electrolyte concentration, applied potential, and inlet velocity were 0.5M, 0.2 V, and 0, respectively.Because of geometrical changes as the growth front of each side of the microchannel moves toward each other with the deposition progression [Fig.1(c)], we used a moving mesh scheme in the simulations.The moving computational domain during electrodeposition was modeled using the arbitrary Lagrangian-Eulerian (ALE) method. 21The deposition rate or normal velocity of the boundary movement, v dep , is directly proportional to ict and can be expressed as where Mcu is the molar mass, ρ Cu is the density of the copper, ict is the local current density, zcu is the number of electrons of each ion, and F is the Faradays constant.
The inlet flow velocity generates a convective flow within the electrolyte domain in the channel.The Navier-Stokes equation governs the fluid flow, where u is the fluid velocity, p is the fluid pressure, ρ is the fluid density, μ is the dynamic viscosity of the fluid, and F is the external force.The Navier-Stokes equation is solved simultaneously with the continuity equation, We conducted the numerical simulations in the COMSOL Multi-physics commercial package using a 2D finite element model.Two-dimensional triangular elements were used to construct the geometry.A total number of 14 510 elements were used in the simulation, after a mesh sensitivity analysis to ensure that the results are independent of the mesh size.In this work, the electrolyte is copper sulfate.Copper reduction Cu 2+ + 2e − →Cu was considered as a single cathodic reaction.A no-slip condition was applied to the microchannel's internal wall.The details of the model for different physics are shown in Figs.1(a) and 1(b).The initial values and boundary conditions in the model were assigned based on the conventional experimental conditions.Conditions included assuming atmospheric pressure and ambient temperature for all domains.The initial concentration of the electrolyte was set to be 500 mol/m 3 .The diffusivity of the Cu 2+ was set to be 5 × 10 −10 m 2 /s. 22The following constants and variables in Butler-Volmer equation were used: ; and CR = 1, [23][24][25] where C cu 2+ and C cu 2+ ,ref are the concentration of Cu 2+ ions on the electrode surface and within the bulk electrolyte, respectively.The viscosity and density of the solution were assumed to be 0.001 009 3 Pa s and 1000 Kg/m 3 , respectively.The values of constants and variables used in this study are ambient temperature (20 ○ C) and atmospheric pressure.The diffusion coefficient and electrode kinetic parameters were treated to be constant, and the anode was assumed to be unpolarized with no change in shape during the process.The electroneutrality (∑ n i zici = 0) condition was considered in the model including copper and sulfate ions.It was assumed that the metal deposition process occurs at a steady-state and is governed by the complete Butler-Volmer equation.The simulated process was time-dependent because the boundaries of the cathode moved toward the center of the microchannel while the Cu 2+ ions were reduced and deposited at the growth front.The model predicts the metal deposition rate based on the Faraday's law to obtain the evolution of the deposition surface with time during the process, from which the microchannel fill time and the percentage of the unfilled area are determined (supplementary material).A through-microchannel having an aspect ratio of 3 was chosen for this investigation, with a channel radius of 1 μm and a length of 6 μm.

RESULTS AND DISCUSSION
A computational model was validated by comparing the predicted deposition rates from the model to the theoretical calculation based on the Faraday law (Fig. S1).In addition, the velocity of the electrolyte obtained from the computational analysis was also validated by the theoretical calculation (Fig. S2).The validation of the computational model is explained in detail in the supplementary material (validation of the computational model section).Based on the validated computational model, parametric studies were performed and explained in the rest of the article to investigate the influence of the inlet velocity of the electrolyte, applied potential, the concentration of the electrolyte, the aspect ratio (AR) of the microchannel on the deposition rate, the final shape of the growth front, percentage of porosity, concentration, and ionic flux distribution in the electrolyte.

Effect of inlet velocity on microchannel filling process
In these simulations, the length and height of the chosen microchannel are 6 μm, and 2 μm, respectively.7][28] Figure 2 shows the current density map for different values of inlet velocity.
For each inlet velocity, we calculated the profile of the deposited metal on the cathode surface at the time when the growth front reached the symmetry line of the microchannel [Fig.3(a)].It should be noted that the current model in COMSOL cannot fully capture the exact phenomenon that happens when the symmetry line has reached; rather, the model is only valid when the interfaces are relatively close to each other.We observed that except when the inlet velocity is 0 μm/s, the profile of the deposited metal is non-uniform, and this non-uniformity increases with an increase in the inlet velocity.The results showed that the case with an inlet velocity of 0 μm/s reaches the symmetry line at 58 s, 3 μm/s reaches at 57 s, 30 μm/s reaches at 46 s, and 300 μm/s reaches at 42 s.
Because of the non-uniform deposition, we calculated the average deposition rate (at each point along the normal to the cathode surface) by dividing the total thickness of the deposited metal by the corresponding simulation time [Fig.3(b)].Similarly, we calculated the electrolyte concentration for the different inlet velocities [Fig.3(c)].The concentration was normalized by the bulk concentration away from the channel inlet.The initial concentration was 0.5M, and it was set 10 μm away from the channel and was symmetric on both sides.From the results in Figs.3(a)-3(c), we conclude that the inlet velocity plays a significant role in the metal deposition profile.Under the convective flow, metal ions are transferred downstream of the channel, increasing the ionic concentration and hence the deposition rate at the channel exit.This non-uniform deposition causes the outlet of the channel to close since the metal growth fronts from both cathodes meet in the middle of the channel.
To further elaborate on the effects of the inlet velocity on the electrodeposition process, we compared the concentration and different fluxes within the electrolyte for two cases of zero and 30 μm/s inlet velocities (Fig. 4).The results show that the concentration distribution in the electrolyte is uniform at zero inlet flow velocity.On the other hand, when the inlet flow velocity is 30 μm, the electrolyte concentration is higher downstream than upstream [Fig.4(a)].
The total flux can be divided into migration flux, convection flux, and diffusion flux.The migration flux is nearly identical in both cases [Fig.4(b)].This is because the migration flux is essentially an electrostatic effect that arises from the applied voltage between the electrodes and depends on the strength of the electric field.The electric field strength is the same for both cases as the applied potential is constant.The inlet flows velocity results in a convective flow of the electrolyte toward the cathode surface.At zero inlet flow velocity, the convective flux is zero, and as the inlet flow velocity increases, the convective flow increases [Fig.4(c)].Notably, the convective flux is higher at the back corner of the cathode.
We note that the Reynolds number (Re = ρuL/μ, where ρ is the density of the fluid, u is the flow speed, L is the characteristic length, and μ is the dynamic viscosity of the fluid) for an inlet velocity of 30 μm/s is ∼2.87 × 10 −5 .The characteristic length of a rectangular channel, L = 2ab a+b , has been used to calculate the characteristic length of the considered geometry.As it is a 2D model, unit length (1 μm) is considered the width of the channel.Therefore, no turbulence flow, vortex formation, and back or eddy fluid flow is expected at such a low Re number.When the inlet flow velocity is zero, the diffusion flux is directed toward the cathode from both the inlet and outlet.However, when the inlet flow velocity is non-zero, the diffusion flux moves right to left [Fig.4(d)].This is because the ionic concentration is higher at the right end of the channel under the convective flux.For the zero-inlet flow velocity, the total flux [Fig.4(e)] is directed toward the cathode, similar to the migration flux and diffusion flux.For the non-zero inlet flow velocity, the total flux shows a more complex pattern.This is because the migration flux is directed toward the cathode, the convective flux is directed to the right, and the diffusion flux is directed to the left in this case.The total flux is higher downstream of the channel than upstream.The maximum total flux is ∼14 mM/m 2 s when the inlet velocity is zero, whereas it is ∼50 mM/m 2 s when the inlet flow velocity is 30 μm/s.
From the above discussion, we concluded that the migration flux density is the same in both cases, and the magnitude and direction of diffusion flux and convective flux depend on the inlet flow velocity.Thus, the flow-through electrodeposition into a microchannel is a convection-driven process when there is an inlet flow velocity.With further investigation, this phenomenon may be used to fill the microchannel to the extent possible, although electrolyte additives may be required in experimental conditions.The inlet flow velocity helps to fill the downstream end first, and metal is slowly deposited into the rest of the area of the microchannel.One important dimensionless variable for the flow through electrodeposition is the Peclet number, Pe.The Peclet number is a measure of the relative importance of the convective flux to the diffusive flux and is defined by the following equation, Pe = (vmax r)/D, where vmax is the maximum flow velocity, r is the radius of the microchannel hole, and D is the diffusion coefficient.A large number indicates a convectively dominated distribution, whereas a small number indicates a diffuse flow.By varying the electrolyte flow velocity, the process can be transferred to a convection driven process.The Pe number was calculated for inlet velocities of 3, 30, and 300 μm/s to be 0.006, 0.06, and 6, respectively.

Effects of the applied potential on the microchannel filling process
Generally, the plot of current density vs potential has four parts: linear, exponential, mixed control, and the limiting current density. 29The limiting or the maximum current density is given by iL = nFD δ c b , where D is the diffusion coefficient of the metal ions M z+ , C b is the bulk concentration of M z+ ions in the electrolyte, δ is the diffusion layer thickness, n is the number of electrons involved in the reaction, and F is the Faraday constant.Once the current density reaches the limiting current density, it becomes independent of the applied potential.Applying a voltage lower than the limiting value is a typical electrodeposition condition.In this situation, the applied voltage is one of the main parameters controlling the electrodeposition process.
To evaluate the effects of the applied potential on the electrodeposition in the channel, we computed the deposition rate, concentration distribution, current density distribution, and the percentage of the unfilled area and ionic flux distribution for three different applied potentials of 0.2, 0.3, and 0.4 V.The range of the applied potential was set based on the literature. 30,31The average current density range predicted on the cathode surface is ∼46-654 mA/cm 2 for the applied potential range of 0.2-0.4V.
Figure 5(a) depicts the average deposition rate along the cathode for different applied potentials.For a constant ionic resistance of the electrolyte, the current density increases with an increase in applied potential, and the deposition rate depends on the current density. 29The current density also varies with time as the microchannel fills gradually, which results in each deposition layer having a different plating rate.This current density and deposition rate variations can change the mechanical properties (e.g., elastic modulus, stress level, and hardness) of the deposited Cu.It has been observed that by changing the current density and eventually deposition rate, the average grain size and mechanical strength of the deposited Cu can be controlled. 32e observed that for an applied potential of 0.2 V, the deposition rate is uniform on the cathode surface.The deposition rate becomes nonuniform for 0.4 V with higher deposition rates at the front and back corners of the cathode [Fig.5(a)].The average deposition rate significantly increases from ∼15 to >200 nm/s by increasing the potential from 0.2 V to 0.4 V.As a result of the nonuniform deposition with a higher deposition rate at the two ends of the cathode for 0.4 V potential, the two ends of the growing front reach the symmetry line in the middle of the channel [Fig.5 potential results in a much smaller ∼1% unfilled region, whereas this value increases to 13% for a 0.4 V potential.
To further analyze the effects of the applied potential on the electrodeposition process, we computed the migration flux, diffusion flux, and total flux for these two applied potentials (Fig. 6).We note that the contribution of the convective flux to the total flux is zero since there is no convective flow.The results show that the magnitude of the migration flux [Fig.6(a)] significantly increases with the applied potential.Specifically, the maximum absolute value of the migration flux rate is ∼100 mM/m 2 s for the applied potential of 0.4 V, whereas it is ∼8 mM/m 2 s for the applied potential of 0.2 V.In the total flux map [Fig.6(c)], we observe the high concentration of flux in the back corner and front corner of the cathodes, while the maximum value of the flux increases more than 12 times by increasing the voltage from 0.2 V to 0.4 V.This observation correlates well with the observed high average deposition rate at these two corners, which results in the creation of an unfilled region.The electrolyte concentration plot (Fig. S3A) and the current density plot (Fig. S3B) further support this point.The nonuniform ionic flux density causes a higher deposition rate at the corners of the cathodes.As the deposition rate is high at a higher applied potential, metal ions are consumed at a faster rate and the concentration gradient along the cathode surface increases.The higher ionic concentration gradient in the electrolyte increases the diffusion flux rate [Fig.6(b)].Thus, both the migration flux and diffusion flux increase with an increase in the applied potential.
A local minimum is observed at the middle of the microchannel in the concentration profile (Fig. S3A) and the current density profile (Fig. S3B), which is more pronounced when the applied potential is high.Metal ions are consumed at the two ends of the channel before reaching the middle, and this consumption rate increases with an increase in applied potential, resulting in a nonuniform concentration profile.From this analysis, we concluded that a lower applied potential is essential to completely fill the microchannel with no to a low unfilled region.On the other hand, if a cavity is desired for certain applications, a higher applied potential should be used.

Effects of the electrolyte concentration
The electrolyte concentration is another parameter that affects the electrodeposition process.We investigated this effect for three different electrolyte concentrations, 0.1, 0.3, and 0.5M, by keeping the other parameters constant.electrolyte, which depends on the type and concentration of ionic species, is the mechanism through which current can be transferred across an electrolyte.An increase in the ionic strength causes a significant increase in the current transfer, resulting in a higher deposition rate.However, nonuniformity in the deposition rate increases with the decrease of concentration.The computational analysis shows that an increase in the conductivity of the solution would cause greater uniformity.This agrees with the theory that an increase in conductivity results in decreased ohmic losses in the electrolyte, allowing for higher efficiency in the current transfer.Thus, the decreased electrical resistance would cause the electrons to pass through the path into the microchannel more easily.
Figure 7(b) shows the variations in the shape of the growth front at the symmetry line.The calculated unfilled area for these three concentration values is shown in Fig. 7(c).The unfilled area is larger for the concentration of 0.1M than that for the other two concentrations.The increase in concentration from 0.1M to 0.5M resulted in a decrease in the unfilled area from ∼4.3% to ∼1%.
The electrolyte concentration along the cathode surface is shown in Fig. S4A for different electrolyte concentrations.The computational analysis predicts that a low concentration of electrolyte triggers the nonuniformity of the ionic concentration along the cathode.As the applied potential is the same, the ionic consumption rate is the same if there are enough ions near the cathode surface.When the initial concentration is low, if ions from other regions cannot easily refill the consumed region, the deposition rate will be slow in that region.On the other hand, when the initial electrolyte concentration is high, it would be much easier to maintain the required ionic density near the cathode surface and a uniform deposition rate regardless of ionic flow from other regions.A concentration that maintains the limiting current density everywhere on the cathode surface needs to be considered experimentally.Figure S4B shows the current density variations along the cathode surface.A variation in the current density along the cathode supports the deposition rate variation.The low concentration causes a nonuniform current density distribution along the cathode surface.
The total flux distribution plots [Fig.7(d)] show that the ionic flux density is higher near the two ends of the microchannel for two different initial concentrations of the electrolyte.The maximum total flux density is ∼12 mM/m 2 s when the initial concentration is 0.5M, whereas the maximum total flux density is ∼6 mM/m 2 s when the initial concentration is 0.1M.As there is no forced convection, the convective flux density is zero for all the cases.The total flux density is the summation of migration and diffusion fluxes in the electrolyte.Both the migration and diffusion fluxes increase with an increase in initial electrolyte concentration.Due to this change in migration and diffusion flux, the total flux increases.

Effects of channel geometry
Another distinctive aspect of electrodeposition in the microchannels is the aspect ratio (AR) of the microchannel.Based on the application requirements, the AR of the microchannel can vary.For example, the length of channels in a ceramic preform is not the same in the metal matrix composite manufacturing process. 1 In this section, the effect of the channel's AR on the percentage of unfilled area is elaborated.Different aspect ratios ranging from 3 to 30 were considered.Figure 8(a) shows the average deposition along the cathode for different AR values.It is predicted that nonuniformity in the deposition rate increases with an increase in AR of the microchannel.Figure 8(b) shows the variation in the shape of the growth front at the symmetry line.Computational analysis shows that an increase in AR while keeping the channel height unchanged would cause a greater nonuniformity in the concentration distribution [Fig.8(c)].When the aspect ratio is 3, the concentration is close to the bulk concentration everywhere in the channel.On the other hand, when the aspect ratio is 30, the concentration ratio is higher at the two ends of the channel and lower in the middle of the channel.This agrees with the hypothesis that an increase in aspect ratio results in a longer travel path for ions to reach the middle of the channel.A longer path leads to a nonuniform ionic distribution along the cathode surface, which eventually results in a nonuniform deposition rate.Figure 8(d) shows that the current density distribution along the cathode surface becomes nonuniform with an increase in AR.Variations in the current density along the cathode support the deposition rate and concentration variation.Nonuniform current density results in thicker deposits along the edges of the microchannel opening than that in the central part away from the edges.The results show an increasing trend in the percentage of unfilled areas with an increase in AR [Fig.8(e)].
Total flux distribution plots show that the ionic flux density is higher near the two ends of the microchannel for two different aspect ratios [Fig.8(f)].The maximum total flux density is ∼12 mM/m 2 s when the aspect ratio is 3, whereas the maximum total flux density is ∼70 mM/m 2 s when the aspect ratio is 30.Because there is no forced convection, the convective flux density is zero for all the cases.The electric field strength depends on the surface area of the anode and the cathode.Because the surface of the cathode increases with an increase in aspect ratio, the migration flux rate increases with an increase in AR.On the other hand, the diffusion flux rate will also increase with an increase in AR because the low concentration area increases, generating more driving force to flow more ions into the microchannel.The diffusion flux also links to the migration flux change.Due to this change in migration and diffusion fluxes, the total flux increases.The mass transport limitation of Cu ions lowers the deposition rate in the middle of the channel.The effect of dynamic aspect ratio and curvature of plating on the local electrodeposition rate needs to be considered in the discussion.Generally, the copper deposition at the edges and corners of through-holes (THs) electroplating is faster than that in the interior due to the uneven current density distribution and difficult mass transfer inside the THs.Pulse electrodeposition with a large t off time may allow more times for ions to transfer to the middle of the channel to achieve uniform electrodeposition.

CONCLUSIONS
In summary, we investigated the effects of inlet flow velocity, applied potential, electrolyte concentration, and the microchannel geometry on the deposition rate, uniformity of the growing front, and percentage of unfilled area in the flow-through microchannel copper filling process.Computational analysis shows that forced convection has a significant effect on the process.Specifically, the computational results predicted that the flux of ions near the back corner increased with an increase in inlet flow velocity.The analysis concludes that the other three variables selected for this study also have a substantial impact on the percentage of the unfilled area of the microchannel.The applied potential has a significant effect on the deposition rate and uniformity of the growing front in the flow-through microchannel filling.High concentration results in better conductivity of the electrolyte and has a large impact on the electron transfer across the electrolyte; thus, increasing this value leads to better outcomes.Our results revealed that the unfilled area can be reduced to lower than 1% with low applied potential, high electrolyte concentration, and no inflow velocity.Experimentally determining the optimum value of all the process parameters can be challenging since it depends on various factors, such as the type of electrolyte, electrolyte concentration, size of the microchannel, and applied potential.We believe that the computational analysis provides an insight into the various parameters involved in the process and can help guide future experimental design.In future studies, the effect of additives (inhibitors or accelerators) and supporting electrolytes on the current distribution and, eventually, deposition rate along the microchannel, can be considered.The change in the current waveform (i.e., pulse current) can generate a periodic relaxation time, which would effectively enhance the mass transfer of metal ions, contributing to a stable copper deposition.

SUPPLEMENTARY MATERIAL
See the supplementary material for figures showing the results of the model validation, electrolyte concentration, and current density for various process parameters and a table listing the parameters used in the model.

Figure S2
The computed electrolyte velocity vs. time at different points.For point 3, the theoretical velocity is shown with symbols overlaid on the computational results.

Table 1
Parameters used in the computational model.

Calculation of the unfilled area (porosity):
The unfilled area was calculated from the post-deposition shape of the microchannel.The metal deposits at the two sides of the

FIG. 1 .
FIG. 1.(a) The schematic diagram of the computational geometry and the boundary conditions for the fluid flow and mass transfer.(b) The boundary conditions for electrodeposition physics.(c) The time evolution of the electrodeposition front on the cathode.For the time evolution plot, the electrolyte concentration, the applied potential, and the inlet velocity were set to 0.5M, 0.2 V, and 0, respectively.x = 0 corresponds to the left corner of the bottom cathode.

FIG. 2 .
FIG. 2.Current density map for different values of inlet velocity.In all cases, the applied potential is 0.2 V, and the concentration is 0.5M.Results are shown at t = 20 s.

FIG. 3 .
FIG. 3.The effects of the inlet velocity on (a) the shape of the growth front, (b) the deposition rate, and (c) the electrolyte concentration ratio.Except for A, all the results are shown for t = 20 s. x/W represents the normalized length of the channel along the cathode surface.In all cases, the applied potential is 0.2 V, and the concentration is 0.5M.

FIG. 4 .
FIG. 4. The map of the (a) Cu ion concentration, (b) migration flux, (c) convection flux, (d) diffusion flux, and (e) total flux for the inlet velocity of 0 μm/s (left) and 30 μm/s (right).All the results are for t = 20 s.Applied potential is 0.2 V, and the concentration is 0.5M for both cases.

FIG. 5 .
FIG. 5.The effects of the applied potential on (a) the average deposition rate, (b) the shape of the growth front, and (c) the percentage of unfilled area vs the applied potential.In all cases, the inlet flow velocity is zero, and the concentration is 0.5M.Insects (b) and (c) show how the Cu fills the channel from both sides.

FIG. 6 .
FIG. 6.The distribution of (a) the migration flux, (b) the diffusion flux, and (c) the total flux during the microchannel filling process for the applied potential of 0.2 V (left) and 0.4 V (right).In both cases, the inlet flow velocity is zero, and the concentration is 0.5M.The scale bars in (b) and (c) are the same.

FIG. 7 .
FIG. 7. The effects of the electrolyte concentration on (a) the average deposition rate and (b) the shape of the growth front along the longitudinal direction of the cathode.(c) The percentage of the unfilled area vs the electrolyte concentration.(d) The distribution of the total flux for the electrolyte concentration of 0.1 and 0.5M.In this simulations, the applied potential was 0.2 V, and inlet flow velocity was zero.

FIG. 8 .
FIG. 8.The effects of the channel geometry (aspect ratio, AR) on (a) the average deposition rate, (b) the electrolyte concentration ratio, (c) the shape of the growth front, and (d) the normalized current density, along the longitudinal direction of the cathode.(e) The percentage of the unfilled area vs the AR.(f) The distribution of the total flux for the AR of 3 and 30.In all cases, the inlet flow velocity is zero, the applied potential is 0.2 V, and the concentration is 0.5M.

Figure
Figure S3 (A) The normalized electrolyte concentration, and (B) the normalized current density along the longitudinal direction of the cathode for various applied

Figure
Figure S4 (A) The electrolyte concentration ratio, and (B) the normalized current density along the cathode surface for different electrolyte concentrations.