Biomimetic Copper Forest Structural Modiﬁcation Enhances the Capillary Flow Characteristics of the Copper Mesh Wick

: In a two-phase heat transfer device, achieving a high capillarity of the wick while reducing ﬂow resistance within a limited space becomes the key to improving the heat dissipation performance. As a commonly used wick structure, mesh is widely employed because of its high permeability. However, achieving the desired capillary performance often requires multiple layers to be superimposed to ensure an adequate capillary, resulting in an increased thickness of the wick. In this study, an ultra-thin biomimetic copper forest structural modiﬁcation of copper mesh was performed using an electrochemical deposition to solve the contradiction between the permeability and the capillary. The experiments were conducted on a copper mesh to investigate the effects of various conditions on their morphology and capillary performance. The results indicate that the capillary performance of the modiﬁed copper mesh improves with a longer deposition time. The capillary pressure drops can reach up to 1400 Pa when using ethanol as the working ﬂuid. Furthermore, the modiﬁed copper mesh demonstrates a capillary performance value ( ∆ P c · K ) of 8.44 × 10 − 8 N, which is 1.7 times higher than that of the unmodiﬁed samples. Notably, this enhanced performance is achieved with a thickness of only 142 µ m. The capillary limit can reach up to 45 W when the modiﬁed copper mesh is only 180 µ m. Microscopic ﬂow analysis reveals that the copper forest modiﬁed structure maintains the original high permeability of the copper mesh while providing a greater capillary force, thereby enhancing the overall ﬂow characteristics.


Introduction
In recent decades, there has been a rapid advancement in microelectronic technology and the telecommunications industry, leading to increasingly integrated, miniaturized, and high-performance electronic components. However, while the internal power consumption of these devices has significantly increased, their physical dimensions have been continuously shrinking [1]. Consequently, there is a challenge in efficiently dissipating the high heat flux generated by chips within confined spaces in a timely manner [2,3]. Elevated temperatures resulting from inadequate heat dissipation can lead to electronic component failures and pose serious safety hazards. To address this issue, ultra-thin two-phase heat transfer devices have gained widespread use in portable electronic devices due to their large heat dissipation area, high heat transfer efficiency, and lightweight characteristics.
The wick plays a crucial role in two-phase heat transfer devices, serving as both the driving force behind the circulation of the working fluid and the main source of resistance to fluid flow. In ultra-thin heat pipes and vapor chambers, as the thickness decreases, the vapor space and wick thickness also decrease significantly, especially when the thickness is less than 0.3 mm. Even slight variations in thickness can result in a sharp increase thin heat pipes [14][15][16]. Therefore, this paper combines the biomimetic copper forest structure with copper mesh to fabricate a composite ultra-thin wick with a high capillary performance. The copper forest structure is deposited on the surface of the copper mesh using electrodeposition, resulting in a copper forest modified copper mesh wick. The morphology and capillary performance of the modified wick are characterized and studied in this paper. Furthermore, to further investigate the mechanism of the copper forest structural modification on the capillary performance of the copper mesh, the capillary rising process of both the copper mesh wick and the copper forest modified copper mesh wick is analyzed through visualization.

The Preparation of Ultra-Thin Copper Mesh Wicks
The copper mesh wick (MW) was prepared using a high-temperature sintering method. The experimental process is illustrated in Figure 1. First, a single layer of copper mesh was cut into 170 mm × 10 mm pieces and cleaned with acetone, ethanol, and deionized water to remove the oil residues. The cleaned mesh was then smoothly placed on a 50 μm-thick copper foil (180 mm × 20 mm) made of pure copper (Cu 99%). Two graphite plates were used to apply pressure on the top and bottom of the assembly. The graphite plates, copper mesh, and copper foil were placed together in a high-temperature reduction atmosphere furnace for sintering. After the copper had melted and diffused under high-temperature conditions, the copper mesh became tightly bonded to the copper substrate.  Table 1 shows the information of three copper mesh wicks with different specifications. The morphology and structure of the copper mesh wick were characterized shown in Figure 2. The thickness of wick Δwick is measured by micrometer, and is the difference between the total thickness of the sample and the thickness of copper foil (shown in Equation (1)). The porosity is calculated by the Equation (2).
where m is the quality of the sample, ρ is the density of pure copper, Awick is the crosssectional area of the wick.   Table 1 shows the information of three copper mesh wicks with different specifications. The morphology and structure of the copper mesh wick were characterized shown in Figure 2. The thickness of wick ∆ wick is measured by micrometer, and is the difference between the total thickness of the sample and the thickness of copper foil (shown in Equation (1)). The porosity is calculated by the Equation (2).
where m is the quality of the sample, ρ is the density of pure copper, A wick is the crosssectional area of the wick.

The Preparation of Ultra-Thin Copper Mesh with Biomimetic Copper Forest Modification Wick
The biomimetic copper forest modified copper mesh wick (CFMW) was prepared using an electrochemical deposition method ( Figure 3). During the deposition process, the copper mesh, prepared as described earlier, functions as the cathode substrate. At the cathode, the simultaneous reactions of hydrogen production and copper electrodeposition take place. The growth and morphology of the deposited copper can be precisely regulated by controlling factors such as the solution concentration, reaction time, and current density. To create the biomimetic copper forest, the following conditions were employed. In an acidic electrolyte solution containing CuSO4 and H2SO4, a linearly increasing current was applied to deposit the CFMW samples with different thicknesses at various deposition times. The reaction conditions of all samples were optimized based on previous research [16]. The copper sulfate concentration was 0.6 M/L, and the sulfuric acid concentration was 0.75 M/L. An increasing current density was applied to deposit the copper forest structure in the cathode [16]. In this case, the hydrogen evolution reaction will be suppressed, and under the shielding effect of the electric field, a dendrite-shaped structure will be formed. After the deposition process, the wick surface was cleaned with deionized water, and then the wick was subjected to high-temperature sintering in a reducing gas furnace to enhance its mechanical strength.

The Flow Performance Test of Wicks
This study employed capillary rising experiment to test the flow performance, the setup is shown in Figure 4, included a cold light source, a high-speed camera (Phantom V211-8G-M), a lifting platform, and an iron frame. The sample was attached to the iron frame and fixed vertically to the ground. Ethanol was used as the working fluid, and when the lifting platform rose above the bottom of the wick, the working fluid quickly rising under the action of capillary force. The high-speed camera was used to track the rising height of the liquid level. The captured images were processed using the software provided by the high-speed camera (PCC 3.1) to measure the liquid rising height h and

The Preparation of Ultra-Thin Copper Mesh with Biomimetic Copper Forest Modification Wick
The biomimetic copper forest modified copper mesh wick (CFMW) was prepared using an electrochemical deposition method ( Figure 3). During the deposition process, the copper mesh, prepared as described earlier, functions as the cathode substrate. At the cathode, the simultaneous reactions of hydrogen production and copper electrodeposition take place. The growth and morphology of the deposited copper can be precisely regulated by controlling factors such as the solution concentration, reaction time, and current density. To create the biomimetic copper forest, the following conditions were employed. In an acidic electrolyte solution containing CuSO 4 and H 2 SO 4 , a linearly increasing current was applied to deposit the CFMW samples with different thicknesses at various deposition times. The reaction conditions of all samples were optimized based on previous research [16]. The copper sulfate concentration was 0.6 M/L, and the sulfuric acid concentration was 0.75 M/L. An increasing current density was applied to deposit the copper forest structure in the cathode [16]. In this case, the hydrogen evolution reaction will be suppressed, and under the shielding effect of the electric field, a dendrite-shaped structure will be formed. After the deposition process, the wick surface was cleaned with deionized water, and then the wick was subjected to high-temperature sintering in a reducing gas furnace to enhance its mechanical strength.

The Preparation of Ultra-Thin Copper Mesh with Biomimetic Copper Forest Modification Wick
The biomimetic copper forest modified copper mesh wick (CFMW) was prepared using an electrochemical deposition method ( Figure 3). During the deposition process, the copper mesh, prepared as described earlier, functions as the cathode substrate. At the cathode, the simultaneous reactions of hydrogen production and copper electrodeposition take place. The growth and morphology of the deposited copper can be precisely regulated by controlling factors such as the solution concentration, reaction time, and current density. To create the biomimetic copper forest, the following conditions were employed. In an acidic electrolyte solution containing CuSO4 and H2SO4, a linearly increasing current was applied to deposit the CFMW samples with different thicknesses at various deposition times. The reaction conditions of all samples were optimized based on previous research [16]. The copper sulfate concentration was 0.6 M/L, and the sulfuric acid concentration was 0.75 M/L. An increasing current density was applied to deposit the copper forest structure in the cathode [16]. In this case, the hydrogen evolution reaction will be suppressed, and under the shielding effect of the electric field, a dendrite-shaped structure will be formed. After the deposition process, the wick surface was cleaned with deionized water, and then the wick was subjected to high-temperature sintering in a reducing gas furnace to enhance its mechanical strength.

The Flow Performance Test of Wicks
This study employed capillary rising experiment to test the flow performance, the setup is shown in Figure 4, included a cold light source, a high-speed camera (Phantom V211-8G-M), a lifting platform, and an iron frame. The sample was attached to the iron frame and fixed vertically to the ground. Ethanol was used as the working fluid, and when the lifting platform rose above the bottom of the wick, the working fluid quickly rising under the action of capillary force. The high-speed camera was used to track the rising height of the liquid level. The captured images were processed using the software provided by the high-speed camera (PCC 3.1) to measure the liquid rising height h and

The Flow Performance Test of Wicks
This study employed capillary rising experiment to test the flow performance, the setup is shown in Figure 4, included a cold light source, a high-speed camera (Phantom V211-8G-M), a lifting platform, and an iron frame. The sample was attached to the iron frame and fixed vertically to the ground. Ethanol was used as the working fluid, and when the lifting platform rose above the bottom of the wick, the working fluid quickly rising under the action of capillary force. The high-speed camera was used to track the rising height of the liquid level. The captured images were processed using the software provided by the high-speed camera (PCC 3.1) to measure the liquid rising height h and time t. The liquid rising curve was plotted, and relevant parameters were calculated following the method below.  The fluid flow within the wick, without any additional driving force, occurs due to the presence of surface tension within the micro/nano pores. This phenomenon is similar to the capillary rise observed in capillary tubes. In other words, within the porous medium, numerous capillary tubes exist, and the fluid is driven by the surface tension to flow through these tubes. The momentum equation in a capillary tube can be represented as follows [17,18]: The equation corresponds from left to right to capillary pressure, inertial force, viscous pressure loss (Hagen-Poiseuille), and gravity (hydrostatic pressure). Σ is the surface tension of working fluid, θ is the contact angle, Reff is the effective capillary radius, ρ is the density of working fluid, h is the capillary rise height, t is the capillary rise time, μ is the viscosity of working fluid, g is the gravitational acceleration, and Ψ is the angle of inclination with the horizontal plane.
When a liquid flows through a capillary with a porous medium (excluding the evaporative action), according to Darcy's law, the viscous pressure loss can be expressed as: where vs is the volume averaged velocity (superficial velocity) and K is the permeability of the porous medium, v is the (interstitial) velocity of the liquid, and ε is the porosity of the porous medium. According to the Hagen-Poiseuille law: Combining Equations (4) and (5), Equation (6) can be obtained: The porosity is included as both laws are defined for the interstitial (Hagen-Poiseuille) and the superficial (Darcy) velocity. The Equation (3) can be written as: The fluid flow within the wick, without any additional driving force, occurs due to the presence of surface tension within the micro/nano pores. This phenomenon is similar to the capillary rise observed in capillary tubes. In other words, within the porous medium, numerous capillary tubes exist, and the fluid is driven by the surface tension to flow through these tubes. The momentum equation in a capillary tube can be represented as follows [17,18]: The equation corresponds from left to right to capillary pressure, inertial force, viscous pressure loss (Hagen-Poiseuille), and gravity (hydrostatic pressure). Σ is the surface tension of working fluid, θ is the contact angle, R eff is the effective capillary radius, ρ is the density of working fluid, h is the capillary rise height, t is the capillary rise time, µ is the viscosity of working fluid, g is the gravitational acceleration, and Ψ is the angle of inclination with the horizontal plane.
When a liquid flows through a capillary with a porous medium (excluding the evaporative action), according to Darcy's law, the viscous pressure loss can be expressed as: where v s is the volume averaged velocity (superficial velocity) and K is the permeability of the porous medium, v is the (interstitial) velocity of the liquid, and ε is the porosity of the porous medium. According to the Hagen-Poiseuille law: Combining Equations (4) and (5), Equation (6) can be obtained: The porosity is included as both laws are defined for the interstitial (Hagen-Poiseuille) and the superficial (Darcy) velocity. The Equation (3) can be written as: Energies 2023, 16, 5348 6 of 14 Because the inertial force is small and its action time is short, it can usually be ignored, so the neglected inertial force in the above is Equation (8), which can be written as Equation (10).
As shown in Equation (10), the rate of liquid ascent per unit time (dh/dt) is linearly related to the reciprocal of the liquid rising height (1/h). By considering the slope of the fitted curve, the capillary performance value ∆P c ·K can be calculated. This parameter integrates the capillary pressure ∆P c (representing the driving force of liquid flow in the wick) and the permeability K (representing the resistance to liquid flow) and effectively reflects the capillary performance of the wick.
At the end of the liquid rising process, the viscous force can also be neglected. At this time, from Equation (8), the capillary pressure drops are equal to the gravity, which is shown as Equation (11). We can obtain the ∆P c by measuring the maximum rising height h max . The permeability is obtained by the calculation of ∆P c ·K and ∆P c .

The Morphological Features of CFMW
The information about the CFMW samples is presented in Table 2. The data are drawn in Figure 5; as the electrodeposition time increases, the thickness of the wick grows. Similarly, reducing the wire diameter also leads to an increase in wick thickness. Although the porosity remains relatively stable, there is a slight improvement observed after modification. The left side of Figure 6 shows the typical copper mesh and the biomimetic copper forest structure. The copper mesh, as the most widely used wick structure, has advantages such as high permeability, strong stability, and ease of acquisition due to its large pore structure. However, single-layer copper mesh usually has a poor capillary performance, requiring multiple layers to increase the capillary force, which leads to an increased thickness. The biomimetic copper forest structure, with its vertical dendrites structure and rich secondary and tertiary nanostructures, forms abundant dendrites in the upper part and interconnected channels in the lower part, creating a connected "Ω" shaped channel. At the same time, the nanoscale pores formed between the secondary and tertiary branches provide a larger capillary force, and the "Ω" shaped channel structure effectively increases Energies 2023, 16, 5348 7 of 14 the permeability of the structure and reduces resistance during liquid flow [16]. However, when reducing the thickness of the biomimetic copper forest structure, it usually sacrifices permeability while maintaining a high capillary force. By combining the advantages of the copper mesh structure and the biomimetic copper forest structure, the high permeability of the copper mesh and the high capillary force of the biomimetic copper forest structure can be utilized, while significantly reducing the thickness of the wick to meet the requirements of ultra-thin heat dissipation devices. The left side of Figure 6 shows the typical copper mesh and the biomimetic copper forest structure. The copper mesh, as the most widely used wick structure, has advantages such as high permeability, strong stability, and ease of acquisition due to its large pore structure. However, single-layer copper mesh usually has a poor capillary performance, requiring multiple layers to increase the capillary force, which leads to an increased thickness. The biomimetic copper forest structure, with its vertical dendrites structure and rich secondary and tertiary nanostructures, forms abundant dendrites in the upper part and interconnected channels in the lower part, creating a connected "Ω" shaped channel. At the same time, the nanoscale pores formed between the secondary and tertiary branches provide a larger capillary force, and the "Ω" shaped channel structure effectively increases the permeability of the structure and reduces resistance during liquid flow [16]. However, when reducing the thickness of the biomimetic copper forest structure, it usually sacrifices permeability while maintaining a high capillary force. By combining the advantages of the copper mesh structure and the biomimetic copper forest structure, the high permeability of the copper mesh and the high capillary force of the biomimetic copper forest structure can be utilized, while significantly reducing the thickness of the wick to meet the requirements of ultra-thin heat dissipation devices. Figure 6 illustrates the morphological structure of the CFMWs. From top to bottom are the #1, #2, and #3 specifications of the copper mesh, and from left to right are the electrodepositions for 100 s, 120 s, and 150 s. The copper wires intersect to form micrometer-scale pores that serve as channels for liquid reflux. The copper forest structure deposited on the copper wires comprises nanoscale copper particles, and the interlocking voids created by the branching pattern provide the abundant capillary force. Due to the shielding effect, no deposited copper is observed at the intersections of the copper wires and beneath the copper foil substrate. Copper dendrites is only deposited on the copper wires. When viewed from left to right, for the same specification of copper mesh, the thickness of the deposited copper dendrites significantly increases with longer deposition times, forming elongated layers resembling dispersed hedgehogs. Shorter deposition times result in shorter dendrites. When viewed from top to bottom, for the same electric charge, the thinner the diameter of the copper wires in the copper mesh, the thicker the deposited copper dendrites and the longer the dendrites. In contrast, when the diameter of the copper wires is thicker, only small copper particles are deposited, resulting in a thinner layer on the copper wires.

The Capillary Performance of MWs and CFMWs
To determine the improvement in capillary performance of the CFMWs and further optimize the ratio between the MWs and the CFMWs, capillary rising tests were conducted on the samples made as described above. The capillary performance parameters and capillary rising curves when ethanol was used as the working fluid are shown in Figure 7.   Figure 6 illustrates the morphological structure of the CFMWs. From top to bottom are the #1, #2, and #3 specifications of the copper mesh, and from left to right are the electrodepositions for 100 s, 120 s, and 150 s. The copper wires intersect to form micrometerscale pores that serve as channels for liquid reflux. The copper forest structure deposited on the copper wires comprises nanoscale copper particles, and the interlocking voids created by the branching pattern provide the abundant capillary force. Due to the shielding effect, no deposited copper is observed at the intersections of the copper wires and beneath the copper foil substrate. Copper dendrites is only deposited on the copper wires. When viewed from left to right, for the same specification of copper mesh, the thickness of the deposited copper dendrites significantly increases with longer deposition times, forming elongated layers resembling dispersed hedgehogs. Shorter deposition times result in shorter dendrites. When viewed from top to bottom, for the same electric charge, the thinner the diameter of the copper wires in the copper mesh, the thicker the deposited copper dendrites and the longer the dendrites. In contrast, when the diameter of the copper wires is thicker, only small copper particles are deposited, resulting in a thinner layer on the copper wires.

The Capillary Performance of MWs and CFMWs
To determine the improvement in capillary performance of the CFMWs and further optimize the ratio between the MWs and the CFMWs, capillary rising tests were conducted on the samples made as described above. The capillary performance parameters and capillary rising curves when ethanol was used as the working fluid are shown in Figure 7.

The Capillary Performance of MWs and CFMWs
To determine the improvement in capillary performance of the CFMWs and further optimize the ratio between the MWs and the CFMWs, capillary rising tests were conducted on the samples made as described above. The capillary performance parameters and capillary rising curves when ethanol was used as the working fluid are shown in Figure 7. It can be observed that the CFMWs at different time intervals have different effects on the three types of MW. For #1 MW with a smaller wire diameter, the deposition of copper forest dendrites is more significant under the same applied charge, leading to an increase in the thickness of the deposition layer. Simultaneously, the capillary performance of the wick also increases. This indicates a good proportion between the pore size and the deposited dendrites. As the thickness of the deposition layer increases, the nano-scale dendrite structure can provide higher and more abundant capillary forces. However, due to the relatively smaller copper pore size, the volume of fluid flow channels decreases, resulting in a slight decrease in permeability. Nonetheless, within the It can be observed that the CFMWs at different time intervals have different effects on the three types of MW. For #1 MW with a smaller wire diameter, the deposition of copper forest dendrites is more significant under the same applied charge, leading to an increase in the thickness of the deposition layer. Simultaneously, the capillary performance of the wick also increases. This indicates a good proportion between the pore size and the deposited Energies 2023, 16, 5348 9 of 14 dendrites. As the thickness of the deposition layer increases, the nano-scale dendrite structure can provide higher and more abundant capillary forces. However, due to the relatively smaller copper pore size, the volume of fluid flow channels decreases, resulting in a slight decrease in permeability. Nonetheless, within the experimental range, the longer the deposition time, the better the capillary performance. At a deposition time of 150 s, the ∆P c ·K value of the CFMW reaches 8.44 × 10 −8 N. After copper forest modification, the capillary forces of the wicks significantly increased from the original 777 Pa to 1100-1400 Pa, while the permeability remained relatively unchanged. The capillary pressure drops of CFMWs are 1.45, 1.62, and 1.8 times than that of MWs. Therefore, it can be concluded that the appropriate copper forest modification can further enhance the capillary forces of the MW while maintaining its original permeability, thereby achieving an overall improvement in capillary performance.
For the #2 and #3 MWs with slightly larger pore sizes, when the deposition time is short (100 s, 120 s), only shorter "bush-like" dendrites is deposited, leading to a limited enhancement in capillary performance. The thickness of the deposition layer shows a linear increase with the increase in deposition time. When the deposition time reaches 150 s, lateral branches start to grow on the dendrites, resulting in a significant improvement in the capillary forces and a noticeable enhancement in overall capillary performance. The ∆P c ·K values for #2F3 and #3F3 reach 6.21 × 10 −8 N and 8.21 × 10 −8 N, respectively. Within the experimental range, samples #1F3, #2F3, and #3F3 exhibit the best capillary performance, reaching 8.44 × 10 −8 N, 6.41 × 10 −8 N, and 8.21 × 10 −8 N, respectively, which are 1.7, 1.2, and 1.6 times higher than the unmodified MWs (#1, #2, #3).

The Microscopic Flow Analysis for the Liquid in Wicks
To verify the improvement mechanism, visual analysis was conducted through a high-speed camera with a microscope lens to capture the images of the liquid rising process in both the MW and the CFMW. Figure 9a shows the liquid rising process of the MWs. When the liquid flows inside the wick, it is subjected to four forces: upward capillary force (dynamic force), downward gravity, viscous force, and inertia force (resistance). In general, the inertia force can be neglected due to its small magnitude. In the initial stage, the gravitational force acting on the working fluid is small, so under the influence of capillary force, the working fluid rises rapidly. As the rising time increases, the working fluid undergoes a transition from ascending in a continuous column (0-60 ms) to flowing from right to left along the copper mesh grid (80-160 ms). This change occurs because, with increasing height, the gravitational force acting on the working fluid becomes more significant. Simultaneously, the capillary force weakens due to the larger aperture of the copper mesh. As a result, the climbing force exerted on the working fluid

The Microscopic Flow Analysis for the Liquid in Wicks
To verify the improvement mechanism, visual analysis was conducted through a high-speed camera with a microscope lens to capture the images of the liquid rising process in both the MW and the CFMW. Figure 9a shows the liquid rising process of the MWs. When the liquid flows inside the wick, it is subjected to four forces: upward capillary force (dynamic force), downward gravity, viscous force, and inertia force (resistance). In general, the inertia force can be neglected due to its small magnitude. In the initial stage, the gravitational force acting on the working fluid is small, so under the influence of capillary force, the working fluid rises rapidly. As the rising time increases, the working fluid undergoes a transition from ascending in a continuous column (0-60 ms) to flowing Energies 2023, 16, 5348 10 of 14 from right to left along the copper mesh grid (80-160 ms). This change occurs because, with increasing height, the gravitational force acting on the working fluid becomes more significant. Simultaneously, the capillary force weakens due to the larger aperture of the copper mesh. As a result, the climbing force exerted on the working fluid becomes insufficient, requiring it to fill the entire grid before resuming its ascent. This leads to a significant decrease in the liquid rising rate. In the MW, as the working fluid climbs from the first row to the fourth row, its volumetric flow rate decreases from 171 µm 3 ·s −1 to 64 µm 3 ·s −1 .
the improvement in capillary performance of the modified copper mesh can be attributed mainly to the presence of dendrites. The large pores of the copper mesh provide flow channels for the flow of the liquid. By examining Figure 7a,b together, it is evident that there is a noticeable change in capillary performance before and after the modification. The modification significantly enhances the capillary force. However, when considering permeability, the change is relatively small, with a maximum of 25% (#1-#1F2). Nevertheless, the overall capillary performance is improved. Therefore, the utilization of the copper forest structure modification can significantly enhance the overall capillary performance. In summary, the copper forest modification can provide a richer capillary force, greater driving force, while maintaining the high permeability of the copper mesh, thereby enhancing its overall capillary performance.

The Capillary Limit of the Wicks
The capillary performance of the wick plays a critical role in determining its heat transfer capacity. During the operation of a heat pipe, in order for the heat pipe to operate, Equation (12) must be satisfied. The wick's capillary performance sets the boundaries for providing a sufficient driving force to facilitate the circulation of the working fluid. Consequently, it imposes limitations on the operation of ultra-thin heat pipes and directly affects their heat transfer capacity. This limitation is referred to as the capillary limit [30].
An expression for the maximum flow rate m  may readily be obtained in Equation (13), and the corresponding heat transport is given by Equation (14), when the angle of inclination angle of heat pipe Ψ is 0, it can be drawn as Equation (15) [30]. The operation of ultra-thin heat pipes is mainly constrained by the capillary limit, which indirectly reflects the heat transfer capacity of the ultra-thin heat pipes. The capillary limit of the wick is calculated according to Equation (15) [30].
,max  Figure 9b shows the liquid rising process of the CFMW. The working fluid climbs rapidly under the action of capillary force, ascending from bottom to top, without the phenomenon of flowing along the grid. This behavior can be attributed to the enhanced capillary force provided by the copper forest structure, which directs the working fluid to flow along the dendrites rather than the copper wires. As the height increases, the climbing rate does not decrease significantly, and the liquid level continues to rise uniformly and steadily from 100 ms to 160 ms. Even when the working fluid reaches higher positions and experiences increasing gravity, the substantial capillary force attracts the liquid to replenish through the lower copper forest dendrites. This replenishment process maintains the climbing rate of the working fluid.
From Figure 9, it is evident that the modification of the copper mesh structure results in the dendritic structure covering the surface of the copper wires, slightly reducing the space within the copper grid. However, upon closer inspection of the pink circle in Figure 9, it can be observed that during the liquid rising process, the liquid first appears between the gaps among the copper wires, and then it fills the holes within the MW. This observation suggests that the capillary force of the copper mesh primarily relies on the tiny gaps between the copper wires. Similarly, in the case of the CFMW, the liquid also initially appears in the dendritic parts of the structure. Once the dendritic parts are completely infiltrated, the larger pores within the MW are filled with the liquid. Hence, the improvement in capillary performance of the modified copper mesh can be attributed mainly to the presence of dendrites. The large pores of the copper mesh provide flow channels for the flow of the liquid.
By examining Figure 7a,b together, it is evident that there is a noticeable change in capillary performance before and after the modification. The modification significantly enhances the capillary force. However, when considering permeability, the change is relatively small, with a maximum of 25% (#1-#1F2). Nevertheless, the overall capillary performance is improved. Therefore, the utilization of the copper forest structure modification can significantly enhance the overall capillary performance. In summary, the copper forest modification can provide a richer capillary force, greater driving force, while maintaining the high permeability of the copper mesh, thereby enhancing its overall capillary performance.

The Capillary Limit of the Wicks
The capillary performance of the wick plays a critical role in determining its heat transfer capacity. During the operation of a heat pipe, in order for the heat pipe to operate, Equation (12) must be satisfied. The wick's capillary performance sets the boundaries for providing a sufficient driving force to facilitate the circulation of the working fluid. Consequently, it imposes limitations on the operation of ultra-thin heat pipes and directly affects their heat transfer capacity. This limitation is referred to as the capillary limit [30]. An expression for the maximum flow rate . m may readily be obtained in Equation (13), and the corresponding heat transport is given by Equation (14), when the angle of inclination angle of heat pipe Ψ is 0, it can be drawn as Equation (15) [30]. The operation of ultra-thin heat pipes is mainly constrained by the capillary limit, which indirectly reflects the heat transfer capacity of the ultra-thin heat pipes. The capillary limit of the wick is calculated according to Equation (15) [30]. .
Here, ρ l represents the density of water, σ is the surface tension of water, µ l is the viscosity of working fluid, L eff denotes the effective length, A wick is the cross-sectional area of the wick, R eff represents the effective capillary radius, and h fg is the latent heat of evaporation. Assuming A wick as the cross-sectional area of the liquid wick structure (∆ wick × 3 cm) instead of the cross-sectional area of the heat pipe, and taking L eff as 9 cm, the capillary limit for MWs and CFMWs is calculated as shown in Figure 10. The capillary limit of the CFMWs is 1.2-2.8 times that of the unmodified copper mesh wick. Figure 11 presents the capillary limits of different liquid wick structures. Compared to the composite parallel woven spiral mesh structure, the CFMW structure reduces the thickness by 68% while maintaining the same capillary limit (36 W). The CFMW can provide a significantly higher capillary limit at an ultra-thin thickness, which greatly reduces the thickness of the heat pipe and facilitates the preparation of ultra-thin heat pipes.
Here, ρl represents the density of water, σ is the surface tension of water, μl is the viscosity of working fluid, Leff denotes the effective length, Awick is the cross-sectional area of the wick, Reff represents the effective capillary radius, and hfg is the latent heat of evaporation. Assuming Awick as the cross-sectional area of the liquid wick structure (Δwick × 3 cm) instead of the cross-sectional area of the heat pipe, and taking Leff as 9 cm, the capillary limit for MWs and CFMWs is calculated as shown in Figure 10. The capillary limit of the CFMWs is 1.2-2.8 times that of the unmodified copper mesh wick. Figure 11 presents the capillary limits of different liquid wick structures. Compared to the composite parallel woven spiral mesh structure, the CFMW structure reduces the thickness by 68% while maintaining the same capillary limit (36 W). The CFMW can provide a significantly higher capillary limit at an ultra-thin thickness, which greatly reduces the thickness of the heat pipe and facilitates the preparation of ultra-thin heat pipes.   Figure 11. Comparison of the capillary limit Qmax,capillary of different wicks [18,19,[21][22][23]26,27,29].

Conclusions
This study employed an electrochemical deposition method to deposit a series of biomimetic copper forest modified copper mesh wicks on three different specifications of copper mesh (250 mesh, 200 mesh, 150 mesh) for deposition times of 100 s, 120 s, and 150 s, respectively. The morphology characterization and capillary performance testing were conducted on these wicks. For all three copper mesh structures, the capillary performance increased with an increasing deposition time. The ∆P c ·K values of the CFMWs for the three specifications reached 8.44 × 10 −8 N, 6.41 × 10 −8 N, and 8.21 × 10 −8 N, which were 1.7, 1.2, and 1.6 times higher than those of the unmodified MWs. Compared to other wick structures, the ∆P c ·K value of the CFMWs was 2.9 times higher than that of the 450 µm composite parallel woven spiral woven wick at a thickness of 142 µm. Compared to the composite parallel woven spiral woven structure, the CFMW structure reduces the thickness by 68% while maintaining the same capillary limit (40 W).
Analysis of its flow performance characteristics revealed that the deposition of nanoscale copper particles in the form of copper forest structures introduced dendrite and crisscrossing holes that enhanced the capillary action of the wick, and the big pores of the mesh provide the flow path for the liquid. Therefore, copper forest modification can provide a greater capillary force and working fluid replenishment while maintaining the high permeability characteristics of the copper mesh, thus improving its overall capillary performance.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.