The Role of Network Topology on Electro‐Thermal and Optical Characteristics of Transparent Thermal Heaters Based on Embedded Random Metallic Mesh

Transparent thermal heaters based on metallic networks have gained considerable attention in the last few years as a result of their superior response time, low sheet resistance, and low cost of manufacturing. To increase the mechanical stability and reliability of the thermal heater, it is desirable to embed the metallic network in some form of matrix. Embedding the network, however, changes the nature of thermal conduction making both in‐plane and out‐of‐plane thermal conduction important for ensuring reliability and uniform thermal distribution. The performance of embedded thermal heaters is also significantly influenced by the geometry of the metallic network, both in terms of optical transparency and thermal performance. In this paper, a coupled electro‐thermal model and an electromagnetic model are developed to investigate the properties of an embedded metallic mesh in a polymer matrix. Infrared thermal imaging and ultraviolet‐visible spectrophotometry are used to quantify thermal transport and transparency in the system and to verify the performance of finite element models. A systematic study is then performed to assess the role of network topology both on in‐plane and out‐of‐plane thermal distribution and optical performance. According to numerical analysis, a structure‐property relationship is established which can provide desirable network configurations to optimize performance.


Introduction
Transparent thermal heaters (TTH) or transparent conductive films (TCF) play an important role in many modern devices, such as smart windows, defoggers, deicers and displays.They are also used in thermal management in electronics, sensors, and DOI: 10.1002/adts.202301221lenses of optical instruments in which the components need to exhibit uniform operating temperatures, fast heating time, and sufficient operating temperature at low voltage.Currently, doped metal oxide films, such as indium tin oxide (ITO) and fluorinedoped tin oxide (FTO), are the dominant materials in the market for transparent electrodes and transparent heating systems. [1,2]ommonly used as a thin film coating on glass or other substrates, these layers can generate heat when an electrical current is applied to them.
There are, however, several key drawbacks for doped metal oxide films including high cost, scarcity, high processing temperature, and brittleness which limit their applications. [3,4]][12][13][14] For transparent heaters to be suitable for everyday applications, they must exhibit excellent mechanical flexibility, durability, and operational stability when incorporated into diverse optoelectronic devices. [5]Carbon-based transparent heaters, employing carbon nanotubes (CNT) and graphene, have shown excellent heating performance with uniform heat distribution. [8]However, as their electrical sheet resistance is relatively high, the operating voltage and the power consumption are usually high. [8]For networks based on nanowires, the presence of junctions between wires results in high junction resistances, which affects the durability and operational stability of heaters due to enhanced joule heating at the junction [15] and local failure.
To tackle the limitations mentioned above, template-based techniques have recently been developed to produce a continuous interconnected Metallic Network (MN) with seamless junctions.Because of the absence of junction resistance, thermal heaters with a superior response time, good optical transparency, and low sheet resistance become available. [16]A common process for manufacturing such structures is first to use a colloidal dispersion that is spread onto a substrate and dried.A removable crack pattern is formed, with crack regions exposing the substrate sur- face underneath.The highly scalable crackle pattern will then act as a template for metal physical vapor deposition (PVD) processes, e.g.sputtering techniques.The geometric parameters of derived MNs, e.g.wire width and densities, depend entirely on those of the templates which in turn is closely related to the colloidal particle size, concentration of the dispersion, processing temperature and coating methods, etc.Therefore, the shape and "regularity" of crackle-formed polygons can be controlled within different fabrication methods of templates. [17]For example, TiO 2 gel-formed templates are reported to have uniform wire width but distribution in polygon sizes [14] ; meanwhile, acrylic resin-prepared templates contain non-uniform wire width and distribution in polygon sizes. [18]Cheuk et al. [10] suggested that the reliability of the network strongly depends on the regularity and percentage of areal coverage.By using CCD-based thermoreflectance microscopy they investigated the temperature variations of the network to locate the hotspots at which failures occurred and less hotspots were observed in MNs of high regularities/coverage.
It is also very desirable to embed metallic networks in some sort of matrix to overcome the issue with low adhesion to the substrate (as a result of increase in the contact surface area), im-prove mechanical stability, enhance surface smoothness, [19] and increase resistance to oxidation. [20]By embedding the MN, the nature of heat propagation will be completely changed as now in-plane and out-of-plane thermal conduction is affected differently.Embedding the network could also affect the performance and reliability of transparent heaters.
Experimental studies on embedded networks are challenging as a conventional infrared camera can only obtain the surface temperature distribution of the thermal heater during electric heating while localized current density in MNs is difficult to measure or predict, [5,21] especially when a nano-scaled structure is embedded into a polymer substrate.Therefore, a numerical finite element simulation is a useful tool to study metallic networks and their performances -allowing achievable measurements to be used to provide information on internal thermal distributions.In this study, a coupled electro-thermal model has been developed to explore the current and temperature distribution, and an electromagnetic (EM) wave model to estimate the transparency of the thermal heaters.The performance of the FE models was evaluated by comparisons between the FE numerical predictions (including temperature profiles, current density distributions, and transparency) and experimental results.After model validation, a systematic numerical study was performed to assess the effects of various geometric features, such as wire width, thickness, segment density, and topological regularity on electro-thermal and optical properties.

Experimental Methods and Model Validation
The experimental metallic network used was prepared using the techniques described in. [14]The process was first to form a crack pattern as a template via drying of a spread colloidal dispersion of TiO 2 nanoparticles (0.16 g mL −1 in a solution of acrylic resin, ethanol, and ethyl acetate (2:4:4), all from Sigma-Aldrich) on a polyethylene terephthalate, PET, substrate (0.17 mm in thickness).The random metallic network structure could then be obtained after the deposition of silver by E-beam evaporation.The evaporating current was first at 18 mA/8 kV@0.5 nm s −1 then rise to 24 mA/8kV@2 nm s −1 after reaching a thickness of 100 nm.By applying a potential between the two electrodes, Ag was deposited over the network and the reaction was stopped after obtaining the required thickness, e.g.700 nm.After template removal by immersion in acetone and 15 min of ultrasonic agitation (shown in Figure 1a), a transparent thermal heater was then formed by sandwiching the MN on PET between two layers of polyvinyl butyral, PVB, with a thickness of 1.0 mm (as illustrated in Figure 1b) by heat-pressing at 4 MPa at 120 °C for 15 min.
Thermal heaters with and without PVB lamination were examined in detail using infra-red (IR) imaging by using a FLIR A6750 IR camera, schematically shown in Figure 1c.Thermal images were obtained at a rate of 120 frames s −1 during Joule heating under static applied voltage.Transmittances of thermal heaters without PVB lamination (MNs on PET substrate only) were measured using a Cary 5000 UV/visible/near-IR spectrophotometer.
FE models were developed, based on experimentally produced samples.There were three main steps to build the model: first, digital MNs were constructed, using MATLAB and Auto-CAD, from optical images obtained using a Keyence microscope and corresponding thickness measurements obtained from map scans via a Bruker DektakXT.The measured thickness was 689 ± 72 nm; second, the digital 3D networks were imported into COMSOL Multiphysics and all the parameters were input according to the experiment settings (an example of a 3D reconstruction from its optical image, both shown in Figure 1d and finally, the boundary conditions were imposed before calculation.For the model validation, the electro-thermal results obtained from IR imaging experiments and transparency measurement on pure PET films and thermal heaters without lamination were compared with model results.Different wire widths and segment densities, defined as wire segment per unit area (WSPA) in this research, were used to ensure the robustness of the model.

Definition of Regularity and Geometric Features of Metallic Network
Alongside the experimentally derived digital 3D networks, random metallic networks with topological randomness were gen-erated by applying the Voronoi diagram in MATLAB.A Voronoi diagram, also known as a Voronoi tessellation, is a mathematical construct that partitions a plane into regions based on the proximity to a given set of points.It divides the space into various regions, one for each point.The region associated with point "x" consists of all the points in the space that were closer to that point than to any other point in that domain.The line segments of the diagram were all the points in the plane that were equidistant to the two nearest sites.Voronoi diagram is widely used to analyze the properties of many complex entities whose structures can be defined as a geometric arrangement of basic structural units as shown in Figure 2a.
Here, the geometry of the network is controlled by a defined number of seed points, n, in a given area, A, while its regularity of points distribution is controlled by : where s is the inhibition distance indicating the maximum distance between every two points and r is the distance between two points in a regular and planar arrangement which can be calculated as Equation 2: The regularity  changes in the range of (0,1).MNs of different wire widths, thickness, WSPA, and regularity were generated via AutoCAD with adjustable metal coverage before being imported into COMSOL Multiphysics as shown in Figure 2b.Details of the generated geometries using Voronoi tessellation can be found in Table 1.

Definition of Embedded TCF Structures
The 3D MN structures, constructed from Voronoi tessellation with a dimension of 1200 × 1200 × 500 μm as illustrated in Figure 3, were first imported into COMSOL for the electro-thermal study.In this model, a silver MN (with various thicknesses according to Table 1) was located on a PET substrate (170 μm in thickness) and both were sandwiched between two layers of PVB (500 μm in thickness).This is the same arrangement used for experiments.The material properties used for the three materials are listed in Table 2.

Definition of Boundary Conditions
The following assumptions were made about the electrical conduction and heat transfer in the network: 1) material properties of the metal were considered homogeneous and isotropic.
2) The heat transfer process of the model, from top and bottom surfaces, was simplified by integrating radiation losses into convection heat transfer. [30]3) The boundary condition for the lateral faces of the samples (whole laminates) are considered adiabatic.Based on the charge conservation, Equation 3 describes the charge transport in the metallic network, where V is the electric potential field and  is the electrical conductivity of the materials with a linearized relationship to the material temperature as shown in Equation 4,  0 is the resistivity of silver at 293.15 K and  is the resistivity temperature coefficient.
Based on the assumptions, Fourier's law defining the conductive heat flux, Qc, at internal boundaries is given by Equation 5: where k is thermal conductivity, A T is heat transfer area, dT is temperature difference and dx is the distance across which conduction takes place, Equation 6 describes the heat transfer in the metallic network and the substrate in which Qc is specific heat: As a coupled model, Q comes from Joule heating which can be expressed as: where J is the current density, E f is the electric field and ∇V represents the potential gradient.
For transparency in the TCFs, electromagnetic (EM) wave equation with proper boundary conditions was solved in the frequency domain in COMSOL.By assuming the time-harmonic wave and that radio-frequency (RF) quantities oscillate with e −it and all material properties are linear with respect to field Heat Capacity [J (kg K) −1 ] Silver͢ 10 490 [ 22] 1.59E-8 [ 22] 0.0038 [ 23] 429 [ 24] 233 [ 25] PET [26] 1400 1E14 -0.15 1300 PVB 1110 [ 27] 1E11 [ 28] -0 . 2 2 [ 28] 1850 [ 29]  strength, the EM wave equation governed by Maxwell's equations is given as: where the material properties are:  −1 r , the relative permeability;  r , the relative permittivity; and , the electrical conductivity.With the speed of light in vacuum, c 0 , the above equation is solved for the EM waves with frequency  throughout the modeling domain.The 3D model for the EM wave in the air and thermal heaters (MN+PET, 600 nm and 0.17 mm in thickness respectively) (isotropic and uniform medium) and the Cartesian coordinate system are shown in Figure 4.
The wave Equation 8 can be solved in a 3D model if the direction of the wave vector is chosen to be perpendicular in the xy plane and the medium is uniform in the z-direction.The yz boundaries (green line) of the structures are set to be the periodic boundary conditions which are consistent with the plane wave excitation denoted by the red line.The plane wave excitation is placed in the air region closer to the bottom.Perfectly Matched Layers (PMLs) are applied to simulate the infinite boundary conditions for EM waves.The excitation used here cannot be absorbed and has no influence on the reflected wave if it exists.The direction of the wave vector is perpendicular to the xy plane.The upper and lower domains are PMLs which can simulate the infinite boundary conditions for EM waves.The transmitted and reflected powers are calculated by defining integration planes at the AIR-PML boundaries.The relevant parameters regarding thickness of each layer are listed in Table 3.
The transparency, T, and reflection, R, of the thermal heaters are defined as: where P T , P R , and P S are the transmission, reflection, and wave excitation power, respectively.Mesh size in the air and PET do-mains also determines the spatial resolution of the field distributions and influences the calculation results.The calculation error can be controlled in ± 1% if the mesh size satisfies (Δx; Δy; Δz) < (0.1x; 0.1y; 0.1z), approximately.x, y and z are the EM wavelength in the x, y, and z directions.As we consider the optical transmittance of the thermal heaters, wavelengths ranging from 400 to 800 nm were investigated.

Model Validation
In this section, IR imaging is first used to validate the electrothermal FE models of thermal heaters.Before applying the bias voltage, the radiance emitted from the heater surface at room temperature was measured to adjust the emissivity in the settings.The average emissivity of the PET surface observed was 0.90 and that of the PVB is 0.94. [26,31]After emissivity calibration, electro-thermal response of MN deposited PET was measured.Thermal images of thermal heaters without PVB lamination were obtained from IR measurements and FE modeling during Joule heating under 0.69 V@1.0 s in Figure 5a.It can be seen from Figure 5a that silver wires are distinguishable as it has a lower IR emissivity (≈0.03) compared with the PET substrate, as expected.From the images, hotspots are found to occur at similar positions in the experimental and simulated results.Thermal maps at other applied voltage and its corresponding transient temperature profiles are shown in Figure S1a (Supporting Information).The transient temperature profiles plotted in Figure 5c and Figure S1b (Supporting Information) show good agreement between the FE model and the experimentally measured temperature, 10.75% and 6.72% mean absolute percentage error (MAPE) in mean and max., respectively.
The performances of the FE model for PVB laminated thermal heaters are shown in Figure 5b and Figure S2 (Supporting Information).The comparison is made between experimental and simulated electro-thermal response of PVB surfaces facing the IR camera.As PVB is transparent to the operational spectrum of the thermal imaging camera lens [32] the camera is able to capture the radiance and profile of the max.temperature values of MN (due to its transparency the PVB surface does not radiate heat when it is heated-up).In general, a good agreement exists between the experimental results and the FE predictions.However, some discrepancy for temperature distributions also exists especially in the region close to the contact point at the right electrode, as shown in Figure 5b.Besides some evident and comparable hotspots, the temperature distribution in the FE model results is much more uniform whilst that of experimental results is less heated.Closer inspection of the reconstructed metallic network in the FE model and comparison with microscopic images in Figure 5e reveals that some of the extra fine or dangled branches have not been fully reconstructed in the model and can be the reason for the existing discrepancies between the temperature data from the FE model and experimental results in those regions.
Figure 6 shows the comparison between the measured and the simulated transparency values of the pure PET and the thermal heaters without lamination having different WSPA of MNs.The results demonstrate a good agreement with maximum of 4.04% difference according to data listed in Table 4.

The Influence of Network Geometry on Electro-Thermal Response
In this section, the effects of key geometric features including wire width and WSPA on sheet resistance, current density, power density (W cm −2 ), and temperature distribution of transparent thermal heaters will be systematically investigated using numer-  ical simulation.For this purpose, variations in geometry were investigated against various geometrical features, each time fixing all parameters apart from one.The experimental process of creating crackle has an intrinsic stochastic nature.To replicate this randomness of the network in the numerical simulations, each kind of networks were generated four times and the initial random seed points for Voronoi tessellation were different, for a targeted set of geometrical features.The average of sheet resistances, fill factor, and effective cross-sectional area were then plotted with one-time standard deviation as error bars are shown in Figure 7.

Wire width
Increasing the average wire width from 2 to 10 μm causes a linear increase in the fill factor from 3% to 15% and correspondingly, leads to a decrease in R s from 1.81 to 0.33 Ω sq −1 (Figure 7a).The effect of geometrical features on the in-plane thermal distribution is demonstrated in Figures 8 and 9. Samples in Figures 8a-c share the same characteristics for thickness, 600 nm, regularity of  = 0.8, and WSPA 185 mm −2 ; all excited under 0.5 V.The main difference between them is the wire thickness.It can be clearly seen that under the same excitation, higher temperatures can be achieved for a network with a larger wire width which leads to lower R s thus higher current density.The location of the hot spots remains relatively similar for all three configurations (Figure 8) while the spatial variation of temperature, in terms of difference between min and max temperatures, increases with increasing the wire's width (Figure 9a).
Increasing the wire's width has a positive effect on lowering the current density as shown in Figure 9c.It is very important to reduce the possibility of high current density while designing and fabricating Ag networks, for instance, with current density in the order of 10 10 A m −2 or even lower.Excessive current density will cause breakdown of Ag wires at the nanoscale [33] and thus create an electrical discontinuity in the film.Of course, increasing the wire's width has a considerable negative effect on the transparency across all investigated wavelengths, as shown in Figure 10a.

Wire Segment Per Unit Area (WSPA)
For WSPA, a range of 185-2408 mm −2 are investigated in Figure 7c, which corresponds to polygon numbers of 100-1200 in 1 mm 2 .Increasing WSPA, generally leads to lower R s with a more significant reduction at the lower WSPAs (Figure 7c).A comparison is made between the in-plane spatial temperature distribution between two networks with the same wire width of 2 μm but different WSPA (of 185 and 2408 mm −2 ) in Figure 8a,d.It can be clearly seen that higher temperatures and more uniform thermal distribution can be achieved using higher WSPAs.The effect of WSPAs on temperature variation can be seen more clearly in Figure 9b where the minimum, maximum, and mean temperature values are plotted against time.Similar to increasing the wire's width, increasing the WSPA has a positive effect in lowering the maximum current density (Figure 9d) but with less negative effect on transparency (Figure 10b).

Fill factor and Network regularity
The effect of network regularity is investigated in Figure 7d.R s slightly decreases with the growth of regularity from 0.2 to 0.8 and then a more considerable reduction when the regularity reaches 1.The effect of increasing the regularity on lowering the sheet resistance is very similar to its effect on lowering the effective cross-sectional area in Figure 7d.This is mainly because there is a clear correlation between sheet resistance R S and effective cross-sectional area A eff as expressed by. [12] For example, samples of regularity = 1 have higher WSPA which leads to higher conductivity in the MNs and corresponding lower Rs, therefore, the trends of decreasing Rs over regularity growth.This could also be concluded according to Equations 11 and 12. where A eff can be expressed as: Here,  is the resistivity of the metal, w is the wire width, d is the film thickness, and N S is the number of wire segments per unit area.This linear relationship between R S and 1/A eff , as stated in Equation 11, can be clearly seen in Figure 10c.The data for sheet resistance, transparency, and power density is plotted against fill factor in Figure 10d and extended to fill factors as high as 30%.Both transparency and power density demonstrate a linear dependency on fill factor with transparency decreasing while the power density increases with increasing fill factor.It should also be mentioned that the scatter in the data for power density also increases with increasing fill factor.The sheet resistance on the other hand shows a nonlinear dependency on fill factor with the strongest sensitivity at the very low fill factors which of course is the region of interest for transparent heaters.It is worthwhile noting that transparency and fill factor during fabrication could be estimated according to 1/A eff which corresponds to the transparency curves.
In order to gain a deeper understanding of how regularity influences the electro-thermal response of MNs and heat dissipa-tions within the embedded system, temperature distribution will be discussed both in-plane and out-of-plane.Figure 11 shows the comparison of in-and out-of-plane temperature@1s between embedded thermal heaters with MN that has 0.2 and 0.8 in regularity and WSPA≈2408 mm −2 (systems with regularity values of 0.4 and 0.6 are presented in Figure S4, Supporting Information, and MN of WSPA≈185 mm −2 are shown in Figures S5 and  S6, Supporting Information).Here, the in-plane temperature distribution of MN with regularity value of 0.2 shows an obvious hotspot with a temperature difference (c.a. 13 °C) inside the network whilst MN with regularity value of 0.8 displays a more uniform temperature distribution with only 5 °C difference between min.and max.temperatures.
Isothermal surfaces are used to demonstrate heat dissipation through the thickness of PVB.The highly random geometry of the network leads to an uneven internal temperature along with the location of the hotspot in the network.The transient temperature responses of the two networks are shown in Figure 12a for in-plane and in Figure 12b for out-of-plane where the minimum, maximum and mean temperatures are plotted against time.The effect of regularity on variation in current and power densities are also plotted in Figure 12c,d, respectively.The values were calculated from the current density/power of the structures over   12e again for both in-plane and out of plane regions with fitted curves in which scale parameter (µ) indicates the average temperature and larger shape parameter (b) means higher homogeneity (listed in Table 5).
Low regularity in MN will also lead to the inhomogeneity of the current density and high current power density in the main horizontal branches and the inner corners of metallic wire junctions. [13]As seen in Figure 12c, there is a 27.87% reduction in maximum current density while the regularity of MNs increases from 0.2 to 0.8 (39.66% in MNs of WSPA≈185 mm −2 shown in Figure S7, Supporting Information).There is also a gradual increase in the power density with increasing regularity (Figure 12d).This results in a slightly higher average temperature for a more uniform network for both in-plane (Figure 12a) and out-of-plane (Figure 12b).This also can be more clearly seen in Figure 12e and Table 5.The difference becomes less significant between the regularity values of 0.6 and 0.8.Therefore, considering the breakdown mechanisms of the metallic network, uniform MN geometry could help prevent not only failures caused by thermal-assisted electronic migration, but also the breakdown of metallic wires resulting from local bulging and poor adhesion with the substrate, which may lead to inefficient heat transfer and melting of PVB at these places with high temperature.

Figure of Merit
The Figure of Merit (FoM), defined as a ratio of transmittance at 550 nm to sheet resistance, is plotted in Figure against the inverse of the effective cross-sectional area.This graph includes both data with constant WSPA and varying width and the ones with constant width and varying WSPA.It can be seen in Figure 13 that data for FOM forms a single curve with the FOM increases as the 1/A eff decreases.A very similar trend can also be observed for power density in Figure 13.

Conclusion
In this paper, a coupled FE model has been developed to simulate the transient electro-thermal response and electromagnetic response of the template-based silver networks and their TCF systems.In practical applications, transparent heaters with uniform temperature distribution, high transparency, high achievable temperatures, and low operating voltage are needed, therefore, the effects of geometric features, including wire width, wire segment per unit area, network regularity, and fill factor on the electric properties, temperature distribution and transparency of TCF systems were investigated using numerical approaches.IR thermal microscopy and UV spectrometer have been used to validate the models.A good agreement was observed between the FE numerical predictions and the experimental results.
Increasing wire width was observed to improve the network performance in terms of lowering the sheet resistance, increasing the maximum achievable temperature, and lowering the current density.However, all these will be at the expense of diminishing transparency.Increasing the wire segment per unit area, however, was found to be a more effective approach to achieve all the benefits mentioned above in addition to improving the temperature uniformity with much less negative impact on transparency.Increasing the wire thickness also can provide similar benefits in lowering the current density and consequently improving the reliability of the network and its resilience to damage.The regularity of the network was found to influence the temperature distribution in the conductive plane and heat propagation through its surrounding encapsulation.The non-uniform power density in the network with lower regularity led to higher current density (up to ≈40%) and thus hotspots, which potentially resulted in the breakdown of the network.Therefore, high regularity values could also improve resilience of the network to damage.

Figure 1 .
Figure 1.a) Optical microscope image of the network; b) Illustration of a laminated thermal heater sample; c) Experimental setup for IR thermal imaging consisting of a thermal heater connected to a power supply and placed under the IR imaging; d) A typical 3D optical profilometric image of a thermal heater consisting of Ag network in thickness c.a. 600 nm and 3D reconstruction in COMSOL.

Figure 4 .Table 3 .
Figure 4. Modelling of EM waves in the air and thermal heaters without PVB lamination.

Figure 5 .
Figure 5. Experimental and simulated electro-thermal response of a thermal heater a) without PVB lamination, 0.69 V@1.0 s and b) with PVB lamination, 3.15 V@5.0 s; transient mean/max.temperature profile c) without PVB lamination under 0.69 V and d) with PVB lamination under 3.15 V e) Silver network reconstructed in the FE model and right part from microscopy scanning.

Figure 6 .
Figure 6.Experimental and simulated transparency across different wavelengths for PET and TCFs with different WSPA values, in which pure refers to structures of no MNs.

Figure 10 .
Figure 10.Variations in transparency of thermal heaters wire thickness = 600 nm, a) wire width from 2 μm to 20 μm and b) WSPA from 185 mm −2 to 2408 mm −2 ; Variations in the sheet resistance, transparency, and power density of the metallic network and the substrate with c) Fill factor and d) 1/A eff .

Figure 12 .
Figure 12.Thermal heaters with different regularity (width = 2 μm; thickness = 600 nm; WSPA2408 mm −2 ) under 0.5 V: a) Temperature response, in-plane; b) Temperature response, out-of-plane; c) current and d) power density; e) Histograms of elements' temperature in FEA model from in-plane to out-of-plane.

Figure 13 .
Figure 13. Figure of merit, indicating relationship between power output and structure geometry in the TCFs.

Table 1 .
Different geometrical parameters of metallic networks used in this research.
Figure 3. Embedded thermal heater model for finite element simulation in COMSOL.

Table 2 .
List of materials properties used in the numerical simulations.

Table 4 .
Comparison in experimental and simulated transparency of thermal heaters without PVB lamination.

Table 5 .
Scale (µ) and shape (b) parameter of temperature distribution of in/out-of-plane elements in thermal heaters with MN of different regularities.