Thermomechanical Modeling and Stress Analysis of Copper Inverse Opal (CIO) Structure for Capillary-Fed Boiling

Recently, a microporous copper inverse opal (CIO) wick material has been utilized in the thermal management of microelectronics as it significantly enhances heat transfer performance with capillary-fed boiling, providing a high critical heat flux of up to ~1100 W/cm2. In this study, we numerically investigate the thermomechanical reliability of the CIO structures with a finite-element method. The mechanical stress of ordered and graded CIO structures is simulated under the conditions of external shear load and boiling-induced thermal expansion. Different geometric parameters, including pore diameter and the neck-to-pore ratio of ordered CIO, and overlap distance and pore gradient of graded CIO, have been parametrically studied. Based on the effective mechanical properties extracted with the representative volume element (RVE) method, the mechanical response of the porous structure can be further understood. The von Mises yield criterion is applied to determine the fracture of the copper (Cu) structure. The reliability of ordered CIOs is assessed by evaluating the maximum von Mises stress and shear stress under the shear load, showing lower stresses with larger pore diameters and smaller pore-to-neck ratios. In contrast, a decrease in thermal stresses is found to appear in ordered CIOs with small pores and large neck-to-pore ratios under boiling conditions. The thermomechanical reliability of graded CIOs is analyzed with different pore gradients and overlap distances. The stress distribution of graded CIO structures in the shear test indicates that a well-coupled CIO interface could improve interfacial reliability and transfers the maximum stress point to pores near the silicon (Si)–Cu interface. The interfaces between the layers of CIOs are found to be more vulnerable under boiling conditions. The graded CIO with a larger gradient tends to be more fragile due to the stress concentration at the CIO interfaces induced by the increased differences in pore diameter.

Abstract-Recently, a microporous copper inverse opal (CIO) wick material has been utilized in the thermal management of microelectronics as it significantly enhances heat transfer performance with capillary-fed boiling, providing a high critical heat flux of up to ∼1100 W/cm 2 .In this study, we numerically investigate the thermomechanical reliability of the CIO structures with a finite-element method.The mechanical stress of ordered and graded CIO structures is simulated under the conditions of external shear load and boiling-induced thermal expansion.Different geometric parameters, including pore diameter and the neck-to-pore ratio of ordered CIO, and overlap distance and pore gradient of graded CIO, have been parametrically studied.Based on the effective mechanical properties extracted with the representative volume element (RVE) method, the mechanical response of the porous structure can be further understood.The von Mises yield criterion is applied to determine the fracture of the copper (Cu) structure.The reliability of ordered CIOs is assessed by evaluating the maximum von Mises stress and shear stress under the shear load, showing lower stresses with larger pore diameters and smaller pore-to-neck ratios.In contrast, a decrease in thermal stresses is found to appear in ordered CIOs with small pores and large neck-to-pore ratios under boiling conditions.The thermomechanical reliability of graded CIOs is analyzed with different pore gradients and overlap distances.The stress distribution of graded CIO structures in the shear test indicates that a well-coupled CIO interface could improve interfacial reliability and transfers the maximum stress point to pores near the silicon (Si)-Cu interface.The interfaces between the layers of CIOs are found to be more vulnerable under boiling conditions.The graded CIO with a larger gradient tends to be more fragile due to the stress concentration at the CIO interfaces induced by the increased differences in pore diameter.
Index Terms-Porous media, surface enhancement, thermal management, thermomechanical reliability, two-phase cooling.

I. INTRODUCTION
W ITH the continuous scaling down of electronic devices, the pursuit of high-performance, compactdesign microelectronics is causing an increment in power density and a decrease in the space available for cooling, resulting in severe thermal management challenges [1].Due to the lower pumping power and higher heat transfer efficiency compared to air cooling, liquid cooling has shown great potential in addressing high heat fluxes and has been widely applied to microelectronic systems [2].Two-phase liquid cooling can utilize the latent heat of the liquid-vapor phase transition and therefore can achieve a higher heat transfer coefficient (HTC) than single-phase liquid cooling [3], showing great potential in the thermal management of high-power density devices.To further improve the performance of two-phase heat transfer, surface enhancement methods, such as porous coating [4], [5], [6], [7] and micro-pin-fin structures [8], [9], [10], have been applied to two-phase flow boiling schemes.The surface microstructures can provide a substantial amount of nucleation sites for boiling and serve as surface extensions for convective heat transfer, leading to higher HTCs and critical heat fluxes (CHFs) than flat surfaces [11].
Recently, a microporous structure named copper inverse opal (CIO) wick material has been introduced as a surface enhancement approach for boiling heat transfer [5], [12], [13], [14], [15].CIO is a copper (Cu) structure grown around a matrix of regular sacrificial polystyrene microspheres by electrodeposition.The microspheres with a diameter of down to <1 µm are self-assembled onto a metal surface and form a regular face-centered cubic (FCC) lattice structure.A microporous structure with high porosity, high permeability, and high surface-to-volume ratio is generated after the dissolution of polystyrene [12].The microporous structure of CIO provides high-density nucleation sites and efficiently facilitates capillary-fed cooling, making it a promising wick material in two-phase cooling [16].However, the small feature size, complex structure, and high porosity of CIO make it fragile and are causing reliability concerns, especially in the boiling scenario with high temperatures and significant pressure oscillations [17].In addition to the ordered CIO structure with constant pore diameters, a graded CIO design has also been proposed, which appears to have better performance due to the vertical gradient of pore diameter [14].The graded CIOs facilitate the liquid delivery with small pores at the bottom while enhance the vapor extraction with large pores at the top of the porous layer, leading to larger CHF and capillary wicking distance.The ratio of graded pore diameter could be optimized to balance the vapor extraction and liquid capillary wicking to improve the heat transfer performance for different application scenarios.Despite the great cooling performance of graded CIO, the mechanical reliability of such structures needs closer scrutiny as the intersection between pores with different diameters may lead to stress concentration in the irregular and frangible structures.
The CIO structure can experience severe degradation in practical applications due to external loads [18], boiling conditions, or Cu oxidation [19], [20].A recent study by Pham et al. [19] showed that the CIO structure of small pore sizes in the order of 1 µm degraded after being used in boiling heat transfer, leading to a collapse of the regular structure and destruction of well-organized interconnects between pores.It was also found that Cu oxidation can lead to a reduction of effective pore diameter, which can affect the permeability of the structure.The reliability of CIOs was also numerically and experimentally examined under thermal cycling and lap shear tests by Singhal et al. [18].Stress concentration was found to appear at the necks of the CIO structure under both the shear test and thermal shock.In the experiment, the failure of the CIO structure showed up after around 150 thermal cycles, raising questions about the reliability of CIO in heat transfer scenarios.To address the reliability concerns of CIOs, studies have been performed to characterize and understand the mechanical characteristics of CIOs.Won et al. [13] utilized the resonator technique and finite-element method (FEM) simulation to determine the effective mechanical properties of the CIO structure.The measured Young's modulus is remarkably lower than bulk Cu, signifying the easily deformable nature of CIOs.
While some research has been conducted to study the mechanical reliability of CIO structures, the mechanical response of the porous structure has not been examined under boiling conditions, which are the major application scenarios for CIOs.In addition, although the graded CIO has shown great potential as a surface enhancement method for boiling, it also requires a thorough investigation from the aspect of thermomechanical reliability.A detailed analysis based on the mechanical properties and the heat transfer characteristics of the porous structure is critically needed.
In this study, we perform FEM simulations to study the mechanical response of two-phase cooling systems with ordered and graded CIO wick coatings.With the representative volume element (RVE) method [21], we first evaluate the effective mechanical properties of CIO structures.Different boundary conditions from experiments are then applied to assess the stress distribution in the ordered CIO structures under both the shear tests and boiling conditions.Combined with the effective properties of CIOs, the stress profile is analyzed to distinguish stress concentration locations and understand the effects of geometric parameters on mechan- ical characteristics.In addition, graded CIO structures with different pore gradients are studied to find crack locations and predict the impact of overlap distances and pore gradients on the level of stress.

A. Numerical Model
Thermomechanical analysis is performed with FEM simulation on a COMSOL Multiphysics 6.0 platform to investigate the mechanical reliability of CIOs.In capillary-fed boiling, a thin film of porous CIO structure is fabricated on the backside of the silicon (Si) chip [16].The liquid supplied from the side of the porous structure is heated up to the two-phase regime by the high heat flux dissipated by the chip, as depicted in Fig. 1(a) and (b).In order to characterize the large-scale porous layer with limited computational resources, a simplified model is built up for the surface porous structure, as shown in Fig. 1(c) and (d).The geometry of CIOs is created by inverting overlapped FCC pores, which is demonstrated to dominate the structure of CIOs in the experiments [22].The CIO model with FCC pores has been proven effective in describing the heat conduction [22], fluid transport [23], and mechanical properties [13] of the CIO structures.The simplified model only contains one single column of CIO unit cells and the Si substrate underneath it.The single-column model has been utilized by Singhal et al. [18] on investigating the thermomechanical characteristics of ordered CIOs and shows a good capability of capturing the thermomechanical response of ordered CIOs in experiments.As shown in Fig. 1(e), the geometry of the CIO unit cell can be described by a few geometric parameters, including the pore diameter (d p ), the neck diameter (d n ), and the unit cell length (a).Due to the face-centered cubic lattice structure of pores, the degree of freedom for the geometric parameters is reduced.Here, we defined the unit cell geometry with the pore diameter and pore-to-neck ratio (r = d p /d n ) to be consistent with other works [16].Apart from the ordered CIO structure, the graded CIOs are also investigated with a unit cell model shown in Fig. 2. To be consistent with experiments and maintain a Fig. 2. Geometries of ordered and graded CIOs.The impact of pore diameter on the structure of ordered CIO is illustrated.For graded CIOs, the overlap distance is defined as the distance between adjacent pores with different diameters and is fixed as half of the smaller diameter in this study.
fair comparison between different geometric parameters, the height of the ordered and graded CIO structure is designed as 30 µm in all the cases.Given that the typical Si chip substrate thickness is 500-700 µm, we assume that the thickness of the Si layer in our FEM model is kept at 5 µm with mechanically fixed boundary condition for the bottom surface to simplify our model.The overlap distance of graded CIOs, which is defined as the vertical overlap distance between nearest pores with different diameters, is fixed at half of the diameter of the smaller pores unless otherwise stated.
High-density tetrahedral meshes are generated in the solid domain to capture the thermal and mechanical characteristics of the porous structure.Local refinement of meshes is applied to the porous region and the interfacial region to accommodate the complicated structure of CIO.A mesh sensitivity study has been performed to balance the computation time and accuracy.The maximum von Mises stress is set as the value of interest here, whose variation with the number of mesh elements is studied as depicted in the Appendix.It can be found that compared to the finest mesh that we can achieve, the derivation of stress is lower than 1% for all other mesh sizes here.A maximum mesh size of d p /10 is selected based on the mesh convergence study.

B. Mechanical and Thermal Boundary Conditions
Mechanical and thermal boundary conditions are specified for different cases to simulate the mechanical response of the CIO structure under the conditions of shear load and capillary-fed boiling.In the shear test, the mechanical stresses result from the shear force applied on the top surface of the porous structure.Referring to the study of Singhal et al. [18], we applied the same mechanical boundary conditions to our CIO model as a reference case for our model validation.As shown in Fig. 3(a), the bottom surface of the Si substrate is fixed while the side surfaces parallel to the shear force are assumed as roller boundaries.The magnitude of the shear force F s is determined based on the horizontal dimensions of the CIO unit cell where τ is the average external shear stress applied on the top surface of the sample and a is the length of the model in the xand y-directions, which equals the unit cell length of CIO.The definition of both mechanical and thermal boundary conditions is required for evaluating the thermal stresses induced by capillary-fed boiling.In the thermomechanical simulation, the temperature distribution of the CIO structure is first calculated with the thermal boundary conditions.The thermal expansion induced by the temperature rise of the structure is then evaluated based on the temperature profile and coefficient of thermal expansion (CTE), enabling the assessment of thermal stress with the mechanical model and boundary conditions.
The effects of superheat temperature are considered by changing the heat transfer boundary conditions, namely, the HTC of the porous structure and the heat flux dissipated from the heating area.As the heat flux increases, the HTC varies, leading to the change of superheat temperatures and corresponding to different points on the boiling curve.We focused on the worst-case scenario corresponding to the CHF points, where the maximum heat flux leads to the highest superheat and largest thermal strain in the structure.Nonetheless, based on the simulation results at the CHF points, we found that the superheat temperature, although impacts the magnitude of thermal strain, is not the dominant influencing factor for the thermal strain.The thermal strain can be expressed as where T w is the solid surface temperature, T sat is the saturation temperature of water, and T 0 is the stress-free reference temperature, which is assumed as the room temperature here.Due to the high HTC of the porous structure, the magnitude of superheat stays below 4 • C for most cases, which is more than one magnitude lower than the temperature rise of the structure with respect to room temperature.Therefore, the superheat does not contribute to a major part of the thermal strain and thus is not a major influencing factor for the thermomechanical analysis here.
In order to evaluate the extreme thermomechanical stress under the worst-case scenario, the thermal boundary conditions are set based on the theoretical CHF condition, which is the maximum heat flux dissipation for the given CIO structure.We assume the effective wicking distance to be 350 µm, which is defined as the longest in-plane distance for the liquid to  travel from the edge.The theoretical model of CHF is limited by both the capillary pressure balance with the capillary pressure generated by the porous CIO structure and the boiling limit, which dictates that the vapor escaping pressure drop in the out-of-plane direction not exceeding the pressure elevation due to superheat temperature using the Clausius-Clapeyron equation.The superheat temperature ( T ) is assumed to be 12 K, as determined in prior studies that superheats in capillary-fed boiling of CIO do not vary significantly with CIO pore size and wicking distance.As shown in Table I and Fig. 4, once the CHF is determined, the effective HTC is calculated as As depicted in Fig. 3(b) and (c), different mechanical boundary conditions are applied to the model's outer boundaries to simulate the strain and stress distribution under boiling conditions.The bottom surface of the unit cell model is assumed to be fixed, and the boundaries of side surfaces are determined by the locations of the unit cell in the CIO structure.As depicted in Fig. 3(b), four, three, or two side surfaces are set as symmetric boundaries (roller boundary condition) to characterize the conditions that the model cell is located at the center, the edge, and the corner of the CIO layer.By varying the location of the CIO model, the mechanical boundary conditions help to identify the most fragile locations inside the CIO structure and allow us to distinguish the origin of cracks inside the system by comparing the stress level between different locations.The zero-stress temperature is set T 0 = 20 • C for the porous model as the CIOs are fabricated with room-temperature electroplating processes.

A. Representative Volume Element
The effective mechanical properties of the CIO structure are evaluated with the RVE method to understand the mechanical response of CIO structures with different geometric parameters.In the RVE approach, a small strain is applied to the outer boundaries of the CIO unit cell shown in Fig. 1(e), and the volumetric average of stress and strain in the CIO structure is calculated to assess the effective properties where ε i j and σ i j are the strain and stress in j-direction with regard to the external load applied on the surface with a normal vector in i-direction, and the coordinates i and j can be x, y, or z.Due to the isotropy of the CIO structure, the isotropic effective Young's modulus and Poisson's ratio can be extracted from the compliance form of Hooke's law as ( 6), shown at the bottom of the next page, [21].
With a constant pore-to-neck ratio, the variation of the pore diameter only changes the scale of the unit cell and has no impact on the structure of CIO.Therefore, the effective properties of CIO structure are only affected by the neckto-pore ratio, whose impact is quantitatively studied here.In this study, the neck-to-pore ratio of CIOs is defined in the range of r = 0.1-0.4 to align with experimental data from the literature [16].The effective Young's modulus and Passion's ratio are evaluated at different pore-to-neck ratios and plotted in Fig. 5.As the neck-to-pore ratio increases, a larger interconnect between pores increases the porosity of the porous structure, resulting in a less rigid body and a lower effective Young's modulus.In contrast, as shown in Fig. 5(b), the effective Poisson's ratio slightly increases with the neckto-pore ratio, showing properties closer to incompressible materials.
The temperature dependence of mechanical properties is not considered in this study due to the lack of temperature-dependent properties for CIO structure and the relatively narrow temperature range here (T = 20 • C to ∼100 • C).The variation of mechanical properties can be approximated with the data of bulk Cu [24], [25] for the temperature range used in this study and is found to be lower than 5% for Young's modulus, Poisson's ratio, and CTE.Despite the lack of temperature-dependent property data here, the conclusions obtained in this work can still predict the changing trends of mechanical reliability with CIO parameters and provide design guidelines for the experiments.
The effective properties calculated by the RVE method are compared with an analytical model by Phani and Sanyal [26] to validate our simulation.The analytical model is believed to well capture the characteristics of the CIO structure as the model was targeted at describing porous media with spherical pores, and it has been demonstrated that this analytical model showed a fairly good agreement with experimental data in a wide range of porosity.As shown in Fig. 5, the simulation Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.results are of great consistency with the analytical model, providing a solid validation for our calculation.

B. Shear Test With Effective Properties
Another commonly used approach has also been applied here to validate the effective properties obtained with the RVE method, where we compared the deformation of the CIO structure under the shear test in simulations with the real CIO structures and with a block using effective properties of CIOs, as shown in Fig. 6(a).Although the local stress distribution is significantly affected by the CIO structure and cannot be directly compared to the stress profile of the block, the effective properties of CIOs dominate the deformation of the CIO structure, creating an analogy between the x-direction displacements of the two models.As depicted in Fig. 6(b), the model using effective properties shows a good agreement with the complete CIO model in terms of the displacement in the x-direction, supporting the validity of the RVE method.Considering that Young's modulus of Si is around 160 GPa [27], which is two orders of magnitude larger than that of CIO structures here, the deformation of the Si substrate should be minimal compared to the porous structure.With the assumption that the Si substrate is regarded as a rigid body, the porous part of the model can be considered as a guided-end beam with a concentrated load, allowing an analytical estimation of the x-displacement with the effective properties [28] x where z is the distance from the Si-Cu interface and l is the height of the porous part in the model.As shown in Fig. 6(b), the displacement of the porous structure is evaluated with (7) and the effective properties from the RVE method.The model shows good accuracy for predicting displacement, especially for low-porosity structures as the beam theory assumes a small Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.beam deflection, which is more common for low-porosity structures.

IV. RESULTS AND DISCUSSION
A. Reliability of Ordered CIOs 1) Shear Test and Reliability Analysis of Ordered CIOs: The FEM simulations of shear tests are first performed to investigate the reliability of the CIO structure under external shear load.As shown in Fig. 7, we applied a shear load F y to the top surface of the CIO model and simulated the distribution of von Mises stress in the model.To distinguish the most fragile locations inside the CIO structure, the von Mises stress is evaluated among the entire porous structure.It is found that the maximum von Mises stress is located near the top surface where the shear load is applied.The CIO structure was constrained by the shear load applied at the top surface, which may contribute to a large internal stress there.The enlarged view of the stress contour in Fig. 7 indicates a significant stress concentration at the necks, which serves as interconnects between pores of the CIO structure, showing good consistency with the conclusions of Singhal et al. [18].Apart from the cracks in the CIO structure, another common issue for such metal-on-Si structures is the interfacial crack at the metal-Si interface resulting from the shear stress [29].The shear stress profile at the Cu-Si interface is analyzed here to understand the origin of the interfacial cracks.As shown in Fig. 7, the profile of interfacial stress shows a clear distribution due to the CIO structure, and large shear stress appears on the edges perpendicular to the shear load, which may be attributed to the porous structure in the first layer of the CIOs and the local reduction of Cu thickness due to the existence of the pores.
Parametric studies have been performed to study the effects of pore diameter and neck-to-pore ratio on the maximum stress levels.In this study, the reliability of CIO structures with a pore diameter ranging from 2 to 12 µm and a neck-to-pore ratio ranging from 0.2 to 0.4 has been evaluated.To ensure a relatively fair comparison, the height of the CIO structures is set as 30 µm in all the cases.The shear load defined by ( 1) can be applied to maintain a constant external shear stress for different CIO structures.The maximum von Mises stress of the porous Cu structure and the maximum shear stress on the Cu-Si interface are calculated to assess the probability of inter-CIO and interfacial cracks for different geometries.As shown in Fig. 8, the dependence of maximum stresses on both the neck-to-pore ratio and the pore diameter is shown in a 2-D contour, and the color filling the grids indicates the level of stress.It can be noted that the magnitude of both stresses shows a significant increase with the decreasing pore diameter and the increasing neck-to-pore ratio.
The trends of stress levels with geometric parameters can be understood with the effective mechanical properties obtained with the RVE methods and the abovementioned beam theory.According to ( 8), as we keep a similar external shear stress and model height among all the cases, the maximum deformation of the CIO structure is only affected by the effective Young's modulus and the unit cell length of the CIO structure.As the pore diameter increases, the CIO unit cell scales up, resulting in a large unit cell length.The deformation of the CIO structure, which is inversely proportional to unit cell length squared, will be, therefore, reduced, leading to a smaller deformation and lower stress levels.In contrast, the increase in neck-to-pore ratio will significantly reduce the effective Young's modulus of the CIO structure, causing a more predominant deformation and larger stresses.
2) Reliability Analysis of Ordered CIOs in Boiling: The thermomechanical reliability of the CIO coating has been numerically investigated in the capillary-fed boiling scenario.As shown in Fig. 9, the temperature distribution of the CIO model is calculated with the thermal boundary conditions.The calculation of HTCs and CHFs [14] allows a good estimation of the thermal profile.Due to the high convective HTC of capillary-fed boiling and the good thermal conduction in the Cu structure, the heat flux is efficiently dissipated to the coolant at the boiling point of 100 • C, maintaining a low-temperature difference among the entire CIO structures.Combined with the temperature profile, the mechanical boundary conditions are applied to investigate the maximum von Mises stresses of CIO models at different locations of the porous layer.
For all cases with boiling conditions, the maximum stress consistently appears at the neck of the CIO structure located near the Si-Cu interface, as depicted in Fig. 9, which may be Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.attributed to the mismatch of thermal expansion between Cu and Si.It has been reported that the CTE of Cu can be five times larger than that of Si, which can lead to severe interfacial cracks for Si-Cu structures, such as through silicon via (TSV) [30].For CIO structures on Si chips, despite the low effective Young's modulus of CIOs, the mismatch of thermal expansion may lead to significant interfacial stress at the Si-Cu interface, and the thermal strains caused by the mismatch can cause damage to the fragile CIO structure.Therefore, an interfacial layer is needed to mitigate the mechanical stress between the Si and Cu.
Similar to the shear test, the effects of two geometric parameters, including neck-to-pore ratio and pore diameter, are investigated under the boiling conditions.The maximum von Mises stresses are evaluated for different geometries and different locations of the CIO model.The variation of maximum von Mises stresses with the geometric parameters is depicted in Fig. 10.According to the von Mises yield criterion, a von Mises stress larger than the yield limit can lead to the plastic deformation of Cu and thus cause cracks in the Cu porous structure.The yield stress of Cu is reported to be size-dependent and can be as high as 0.53 GPa for microscale structures, which is around three times as large as that of bulk Cu [31].However, we found that the von Mises stress of CIO can achieve the magnitude of 1 GPa, which still exceeds the yielding limit of microscale Cu.
A notable difference is discovered between the three locations of the CIOs.The maximum stress level shows an obvious difference between the three locations.With four fully confined side boundaries, the CIOs at the center of the porous layer have a maximum stress that is nearly two times larger than those at the corners, whose side boundaries are only restricted in two directions.The CIOs at the edge are horizontally confined by three boundaries and have a stress level fall between the two other locations.A strong correlation between the stress level and the horizontal confinement is observed here, indicating that the enhanced restrictions of horizontal displacement can lead to a larger stress level in CIO structures.
Furthermore, the effects of geometric parameters are found to be inconsistent for cases with different CIO locations.For CIOs located at the center of the porous layer in Fig. 10(a), the maximum stress notably decreases with the increasing neckto-pore ratio while showing negligible dependence on the pore diameter.However, in the cases of edge and corner CIOs, the maximum stress is significantly related to both the neck-topore ratio and the pore diameter.The discrepancies between the geometric effects can also be attributed to the confinement effects.For the CIOs located at the center of the porous layer, all the horizontal displacement will be restricted by the nearby CIO cells.Therefore, all the horizontal thermal expansion will be converted to the z-direction displacement, resulting in the deformation of the porous structure and thermal stress inside the CIOs.In this case, the effect of CTE mismatch between Si and Cu is minimized as the horizontal deformation on the Si-Cu interface is limited.Given that the stress here is merely caused by the thermal expansion of Cu, the stress level is mainly affected by the porous structure and the effective mechanical properties of the CIOs, which are determined by the neck-to-pore ratio rather than the pore diameter.As the neck-to-pore ratio increases, the CIO structure will become more flexible and can thus relieve stress concentration and reduce stress levels.
In contrast, in the case that the CIOs are located at the edge in Fig. 10(b) and the corner in Fig. 10(c), expansion is allowed in at least one horizontal direction as some of the side boundaries are set as free surfaces.Although the effects of the neck-to-pore ratio are similar to the centered CIOs, the mismatch between Si and Cu will dominate the stress distribution of the bottom CIO structure, and the stress level will increase with the pore diameter for those cases.As the pore diameter increases, the unit cell size of CIOs increases, leading to Cu structures with a larger thickness close to the Si substrate.Due to the thicker Cu at the Cu-Si interface, the mismatch between Cu and Si can result in a more significant effect on the stress and increase the maximum stress level of the porous structure.
B. Reliability of Graded CIOs 1) Shear Test and Reliability Analysis of Graded CIOs: Despite the great potential in enhancing heat transfer, graded Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.CIOs may need a close examination of their mechanical reliability.FEM simulations have been performed to assess the maximum stress level and stress distribution in the graded CIO structures under both the shear test and the capillary-fed boiling conditions.The stress distribution of graded CIO models with different gradients is first examined with the shear test, which are composed of CIOs with pore diameters of 2-4-8, 2-5-10, and 2-6-12 µm, respectively.The neck-topore ratio of CIOs is kept at r = 0.4 for all the CIOs.
Mechanical boundary conditions similar to those of ordered CIOs are applied to study the reliability of graded CIOs under the shear load.The distribution of von Mises stress is first calculated for different structures, as shown in Fig. 11, to determine the location of the maximum von Mises stress.It is worth noting that for the graded CIOs where the ratio between adjacent pore diameters is an integer, such as the 2-4-8-and 2-6-12-µm cases in Fig. 11(a) and (b), the maximum von Mises stress appears at the smallest CIO pores near the Cu-Si interface.In contrast to ordered CIO cases, where the maximum stress is located near the top surface, the stresses in these graded CIOs are concentrated near the bottom of the structure, where the CIOs have the smallest pore diameter.Based on our previous analysis of ordered CIOs, the stress level of CIO structures increases with the decrease of pore diameter in the shear test, indicating that the CIOs with smaller pore diameters tend to be more fragile under external shear load.In the graded cases, with an external shear load, the smallest CIO cells also experience a more significant deformation compared to larger cells, leading to a localized increase in stress level.However, if the adjacent pores fail to form a good coupling, i.e., the pore size is not scaled with integer multiples, the maximum stress will be located at the poorly-coupled interface between two CIOs, as depicted in Fig. 11(c).The maximum von Mises stress on the interface between 2-and 5-µm CIOs can be attributed to the fragile structures generated by the poor coupling of pore diameters.The enlarged view in Fig. 11(c) shows the point with maximum von Mises stress on the 2-5-µm CIO interface, where the stress is concentrated on a thin beam-like structure led by the intersection between poorly-coupled pores.The magnitude of the maximum von Mises stress of Cu is evaluated for different gradients to illustrate the strength of CIO interfaces.As shown in Fig. 12, the graded CIO with poorly-coupled interface (2-5-10 µm) leads to a maximum von Mises stress that is four times as large as the graded CIOs with well-coupled interfaces.
Apart from pore diameter and neck-to-pore ratio, whose effects have been thoroughly discussed in the ordered CIO part, another important geometric parameter for graded CIOs is the overlap distance (h) between different-sized CIOs.The overlap distance h is defined based on the vertical overlap between adjacent CIOs with different diameters and is normalized by the diameter of the smaller pores here.The maximum von Mises stress levels are evaluated for different h.As depicted in Fig. 12, the maximum von Mises stress slightly decreases with the overlap distance if adjacent CIO diameters scale with integer multiples.The dependence of stress on the overlap distance arises from the change in the overall height of the porous structure.As the overlap between CIOs increases, the decreasing overall height of the porous structure leads to a reduction in the moment applied to the structure and lowers the deformation of the CIO structures, resulting in a lower stress level.Nonetheless, the change of overlap length only slightly affects the overall height and has no impact on the structure of CIOs at the bottom, where the maximum stress appears and therefore can only cause a minor influence on the maximum stress levels for the well-coupled graded CIOs.On the other hand, if the adjacent CIOs are poorly coupled, the interface property will dominate the maximum von Mises stress.As the overlap distance increases, the growing overlap between pores will lead to the thinner structure at the interface, resulting in a significant increase in maximum stress level.2) Reliability Analysis of Graded CIOs in Boiling: The reliability of graded CIOs is also assessed in capillary-fed boiling scenarios.The temperature and stress distribution are plotted for graded CIOs with different pore sizes.The graded CIOs located near the center of the porous layer are first focused here as the center cells are the most common cases in the porous layer and have been demonstrated to be the most fragile parts of the system in the above analysis.The temperature of the porous layer is first evaluated with the CHF and corresponding HTCs of the graded CIO structures.As shown in Fig. 13(a) and (b), due to the high HTC of the capillary-fed boiling, the structure shows a uniform temperature distribution with a maximum temperature rise lower than 2.5 • C for both cases.The distribution of von Mises stress is depicted as a 3-D contour and compared at different gradients.Although stress concentrations are still found at the neck of the pores like ordered CIOs, the maximum stress of the graded structure appears at the interfacial regions between CIO unit cells with different pore diameters, raising concerns about cracks between CIOs with different-sized pores in the graded structure.
To further examine the stress distribution and understand the origin of cracks in the graded CIO structure, the maximum stress is evaluated on various xy planes with different z-coordinates.Multiple horizontal section planes are created in the model, and the local maximum von Mises stresses are calculated for different section planes to show the variation of stress levels along the z-coordinate, as depicted in Fig. 13(c) and (d).It can be observed that the stress level keeps almost constant inside the CIOs with the same pore diameter (intra-CIO) but shows a significant increase in the interfacial regions, including the region near Si-Cu interface and those between CIOs with different pore diameters (inter-CIO).For ordered CIOs, we have concluded that the maximum stress level mainly depends on the neck-to-pore ratio while the effects of pore diameter are insignificant, which can explain the consistent intra-CIO stress level for different pore diameters inside the graded CIO structures here.The intersection of CIO unit cells with different diameters can lead to complicated and weak structures and can therefore contribute to the increase of stress levels and cause cracks in the interfacial regions.
Moreover, the magnitude of the maximum von Mises stress is evaluated and compared for different gradients and locations of graded CIOs.As shown in Fig. 14, the graded CIOs at the center exhibit predominantly larger stress levels than those at the edges or the corners as we concluded from ordered CIOs and expected for graded CIOs.The graded CIO structure with a larger gradient (2-6-12 µm) shows a more significant stress concentration at the intra-CIO interfaces, which is attributed to the larger difference of pore diameter there.

V. CONCLUSION
In this study, FEM simulations are performed to investigate the mechanical reliability of the microporous CIO under external load and capillary-fed boiling conditions.The effective mechanical properties of CIOs are calculated with the RVE method.The mechanical responses of CIO cells located at different parts of the porous layer are understood by varying mechanical boundary conditions.The origin of cracks in the CIO structures is distinguished by evaluating the stress profile of the porous structure.The effects of geometric parameters, including the pore diameter and the neck-to-pore ratio for the ordered CIO, and the overlap distance and gradient of pore diameters for graded CIOs, are analyzed.The main conclusions are as follows.
1) The effective Young's modulus and Poisson's ratio are evaluated for CIO structure with different neck-to-pore ratios.As the neck-to-pore ratio increases, the effective Young's modulus significantly decreases while the Poisson's ratio shows a notable increase.Analytical models and FEM simulations are applied to demonstrate the validity of the calculated properties.2) Under external shear load, the stress of the ordered CIOs appears to be concentrated at the necks near the top of the porous layer.Both the maximum von Mises stress and interfacial shear stress tend to decrease at a larger pore diameter and a lower neck-to-pore ratio.A CIO structure with a larger pore diameter and lower neckto-pore ratio will be preferred for application scenarios where external shear loads are major concerns.3) In the capillary-fed boiling scenario, an increase in the von Mises stress of ordered CIOs appears at the necks near the Si-Cu interface due to the mismatch of CTE.The CIOs show the largest magnitude of stress at the center of the porous layer rather than the edges or corners.The maximum stress is found to be larger than the von Mises yield criterion of Cu in some cases.In contrast to external load, CIOs under boiling conditions are believed to be more reliable with smaller pore diameters and larger neck-to-pore ratios.A design with a small pore diameter and large neck-to-pore ratio can be beneficial for CIO-assisted boiling heat transfer from the thermomechanical aspect.Moreover, to reduce the stress level between the Si-Cu interface, a dedicated interfacial stress buffer material layer should be carefully selected and investigated.4) The interfacial coupling of CIOs scaled with noninteger multiples can lead to a stress concentration on the interfaces of graded CIOs due to fragile thin features formed by the intersection of different-sized pores, reducing the reliability of graded CIOs in the shear test.For CIOs with well-coupled interfaces, the graded CIOs show maximum stress at the bottom of the porous layer in the shear test, where the pore diameter is the smallest.5) In capillary-fed boiling, the interfaces between CIO unit cells with different diameters are distinguished to be the most fragile parts of the graded CIOs.A larger variation of pore diameters at the interface can lead to a more significant increase in the stress level.Therefore, smooth transition with small pore diameter variation at the interface can mitigate the stress for graded CIOs.

Fig. 1 .
Fig. 1.Structures of CIO and model setup: (a) capillary-fed boiling with CIO; (b) boiling heat transfer inside the wick structure; (c) large area CIO structures on a Si substrate; (d) schematic of a simplified CIO model; and (e) enlarged view of the CIO internal structure with porous.

Fig. 3 .
Fig. 3. Boundary conditions for the mechanical model.(a) Mechanical simulation under shear load.(b) Boundary conditions for side surfaces are determined by the location of the simplified model in the porous layer.(c) Simulating stresses under boiling conditions with free/roller boundary conditions on four side surfaces.

Fig. 4 .
Fig. 4. Theoretical calculation of CHF for CIO structures regarding varied wicking distances.CIO structures are assumed a total thickness of 30 µm, and the superheat is assumed to be 12 K.Water is assumed to be the working fluid.

Fig. 5 .
Fig. 5. Effective properties of CIOs by RVE.(a) Young's modulus.(b) Poisson's ratio.The effective properties are validated by comparison with a theoretical model proposed by Phani and Sanyal [26].

Fig. 6 .
Fig. 6.Horizontal displacement in shear test.(a) Simulation setup for FEM with effective properties and complete CIO structures.(b) Comparison of x-displacement for CIOs with different neck-to-pore ratios at d p = 2 µm.

Fig. 7 .
Fig. 7. von Mises stress of the Cu porous stress and the distribution of shear stress on the Cu-Si interface.

Fig. 8 .
Fig. 8. Effects of geometric parameters of CIOs on stress.(a) Maximum von Mises stress.(b) Shear stress on Cu-Si interface.

Fig. 9 .
Fig. 9. Temperature and stress distribution on an ordered CIO structure at d p = 4 µm and r = 0.3.

Fig. 10 .
Fig. 10.Effects of geometric parameters on the maximum von Mises stress at different locations.(a) Center.(b) Edge.(c) Corner.

Fig. 11 .
Fig. 11.Profile of von Mises stress on graded CIO structures with different gradients under the external shear test.The maximum stress appears near the Cu-Si interface if the adjacent pore size scales by integer multiples, including graded CIOs with (a) 2-4-8 and (b) 2-6-12 µm, otherwise the CIO interface becomes the most fragile, such as the 2-5-µm interface in the (c) 2-5-10-µm graded CIO.

Fig. 12 .
Fig. 12. Variation of the maximum von Mises stress of the porous structure with the overlap distance.Cases with an overlap distance of h/R > 0.3 are excluded for the 2-5-10-µm graded CIO structure as it leads to extremely thin feature, causing low-quality meshing and unreliable results.

Fig. 13 .
Fig. 13.Temperature and von Mises stress distribution of the graded CIO structures with pore diameters of (a) 2-4-8 and (b) 2-6-12 µm in the capillary-fed boiling scenario.(c) and (d) Maximum von Mises stress is plotted for section planes with different z-coordinates.

Fig. 14 .
Fig. 14.Maximum von Mises stress for graded CIOs located in different parts of the porous layer.

Fig. 15 .
Fig. 15.Maximum von Mises stresses calculated with different numbers of mesh elements.A mesh element number of 1.33 × 10 6 is chosen, which falls into the 1% deviation range compared to the finest mesh we can achieve.

6 )
Two-phase capillary-fed boiling is a comprehensive phenomenon, involving complex heat transfer and fluid dynamics.The thermomechanical properties of CIOs can be affected by bubble dynamics, vapor pressure, fluid-structure interaction, and other localized effects.A more in-depth investigation on the thermomechanical characteristics of the CIO capillary-fed boiling process requires further experimental visualization of the capillary-fed boiling phenomena.APPENDIX See Fig. 15.
Thermomechanical Modeling and Stress Analysis of Copper Inverse Opal (CIO) Structure for Capillary-Fed Boiling Shuhang Lyu , Graduate Student Member, IEEE, Qianying Wu , Graduate Student Member, IEEE, Zheng Gong, Graduate Student Member, IEEE, Keyu Wang , Graduate Student Member, IEEE, Kenneth E. Goodson, Fellow, IEEE, and Tiwei Wei , Senior Member, IEEE

TABLE I HTC
AND CHF FOR DIFFERENT CIO STRUCTURES, ASSUMING AN EFFECTIVE WICKING DISTANCE OF 350 µM