Radiative heat transfer to enhance the performance of liquid metal-based phase change heat sink under natural convection condition

Many studies have shown that the operating temperature of electronic devices needs to be controlled in order to extend their lifetime and achieve reliable performance.^1–3 Natural convection cooling is widely used as a reliable passive cooling technique to cool electronic devices with low power consumption because of the advantages of low maintenance cost and no noise.⁴ It is typically used in applications where vibration and noise need to be avoided and where equipment such as fans are sensitive. In addition, some electronic devices that operate outdoors, such as LED streetlights and communication base stations, are generally not suitable for active cooling means and are usually cooled by natural convection. However, the cooling capacity of natural cooling is very limited. Conventional low-power devices can still cope with it, while high power density devices and large thermal shocks are often unable to meet their cooling needs.⁵ 

Phase change cooling, as a passive thermal management technique, does not require additional drive components and extra power consumption.⁶ During the phase change process, the phase change material (PCM) can absorb or release a large amount of latent heat while maintaining the temperature constant.⁷ Currently, PCM-based heat sinks have been widely used in the field of electronic heat dissipation, especially for some electronic devices with intermittent operation, where phase change heat sinks can play a unique advantage.⁸ When the device is working, the PCM absorbs heat and melts; when the device stops working, the PCM releases heat and returns to the solid state. In order to return the PCM to its initial state as quickly as possible, phase change heat sinks are usually equipped with fins on the outer surface to increase the convective heat transfer area. Two typical passive thermal management technologies, natural convective heat dissipation and phase change cooling, are often used in combination in many applications for high reliability and maintenance-free operation. In addition, the phase change heat sink can effectively deal with thermal shocks with high heat flow density in a short period of time.^9–11 

Liquid metals, as a kind of inorganic metallic phase change materials, have received much attention due to a series of advantages such as high thermal conductivity, high specific volume phase change enthalpy, small rate of volume change during phase change, and stable physical properties.^12–14 A series of works conducted by Ge and Liu et al. have shown that low-melting-point metals represented by gallium have great potentials for the passive thermal management of mobile storage devices and smartphones.^15–17 Through numerical simulations, Yang et al.¹⁸ concluded that a gallium-based PCM heat sink can cope with a large thermal shock of 100 W/cm² at an ambient temperature of 25 °C, while the maximum temperature of the device only rises to 46 °C. In the subsequent work, they designed experiments to compare the thermal performance of E-BiInSn (Bi_32.5In₅₁Sn_16.5) and octadecanol with cross fins installed internally. The results show that the liquid metal outperforms octadecanol under all heating conditions. The E-BiInSn-based heat sink maintains the desired temperature below 80 °C for twice as long as compared to the octadecanol-based heat sink.¹⁹ Li et al. compared more than 200 phase change materials with phase change temperatures ranging from 0 to 100 °C and found that liquid metal has the highest figure of merit, which is 1–2 orders of magnitude higher than that of other phase change materials and pointed out that liquid metal is a promising material for thermal control of phase change.²⁰ In addition, the liquid metal is very suitable for space-compact applications due to its high volumetric latent heat. Wang et al.²¹ prepared phase-change microcapsules consisting of a liquid metal core and a polymethyl methacrylate (PMMA) shell, which were then mixed with a thermally conductive gel for the thermal management of LED chips. The experimental results showed that the composite containing liquid metal microcapsules can control the chip temperature at 71.1 °C, which is 11.2 °C lower than using a single thermally conductive gel. Yao et al. prepared octadecane/liquid metal composite phase change materials with expandable graphite as a carrier, whose properties can be adjusted over a wide range.²² Octadecane and gallium were stored in the microstructure of expandable graphite, which solved the problem of difficult direct mixing of organic PCM with the liquid metal. The latent heat of the prepared phase change composites ranged between 211 and 70.19 J/g, and the corresponding thermal conductivity ranged between 2.639 and 39.158 W/(m K).

Natural convection in air is usually accompanied by radiative heat transfer. For high emissivity surfaces, the two modes of heat transfer are of comparable magnitude so that the radiative heat exchange of such surfaces cannot be neglected. In this case, the total heat transfer coefficient is $h t= h c+ h r$⁠, where $h c$ is the convective heat transfer coefficient and $h r$ is the equivalent radiative heat transfer coefficient. Castro et al.²³ conducted an experimental and numerical study on natural convection heat transfer and radiative heat exchange for vertical and inclined plates. The results showed that when the emissivity increased from 0.05 to 0.85, the temperature of the heat source decreased from 394 to 365 K. The radiative heat exchange exceeds 35% under a heating power of 70 W, which affirms the importance of radiative heat transfer in natural convection problems. Hernández-Castillo et al.²⁴ performed a numerical study on the coupled heat transfer problem of natural convection and radiation in a large rectangular parallel-piped cavity. The results showed that when the wall temperature exceeds 80 °C, if the wall emissivity ɛ > 0.5, the radiative heat transfer will dominate. At this temperature, radiative heat exchange accounts for 63% of the total heat transfer. Therefore, when calculating heat transfer in large cavities, the radiation effect must be considered. Benyahia et al.²⁵ studied the combined heat transfer process of natural convection, surface radiation, and heat conduction in a 3D large cavity. The results indicated that the total heat flux along the vertical heated wall depends strongly on the combined effect of natural convection and surface radiation and significantly increases with the increase of the inner surface emissivity.

Currently, a single thermal management strategy struggles to meet the increasingly complex heat dissipation demands of electronic devices. Natural convection cooling, while simple and reliable, cannot handle high heat flux or short-term large heat shocks. Phase-change cooling can utilize latent heat to absorb large amounts of heat, but it faces the problem of thermal failure due to complete melting of the phase change materials during prolonged operation. Although radiative heat transfer is significant for high emissivity surfaces, it is highly temperature dependent, resulting in relatively low radiative heat exchange levels at lower temperatures. To address these challenges, a radiation-enhanced liquid metal phase-change heat sink is proposed in this study. By preparing high emissivity coating materials and applying them to the fin surfaces of the phase-change heat sink, it achieves improved heat dissipation performance under natural convection conditions.

Figure 1(a) is an exploded view of the liquid metal phase change heat sink showing its main components and the corresponding structural dimensions. Figure 1(b) is a schematic cross section of the liquid metal phase change heat sink with more dimensional details labeled on it.

The phase-change heat sink consists of seven parts: external fins, cover plate, internal fins, cavity, copper block, heating rods, and phenolic resin. Three heating rods are placed within the copper block as a heat source, and the copper is surrounded by phenolic resin as thermal insulation to minimize heat leakage. The cavity containing the liquid metal is in close contact with the surface of the heat source and the phenolic resin, and a layer of thermally conductive silicone grease is coated in the middle of the cavity and the heat source. The cover plate is divided into two layers, each connecting 13 external fins spaced at d₁ intervals and six internal fins spaced at d₂ intervals. The internal fins are inserted into the liquid metal to enhance heat conduction during phase change, accelerating the melting of the liquid metal and improving thermal control performance. The external fins are exposed to the ambient environment, relying on natural convection and radiation for heat exchange with the surroundings. Both the cover plate and external fins are coated with the radiative coating material to enhance radiative heat transfer. The liquid metal used in this experiment is E-BiInSn (Bi_31.6In_48.8Sn_19.6), which is a low-melting-point alloy characterized by high thermal conductivity, large volumetric latent heat of phase change, and non-toxicity.

The specific dimensions in Fig. 1(b) are shown in Table I.

Highly oriented pyrolytic graphite (HOPG) powder (99.9%, 800 mesh) and hexagonal boron nitride (h-BN) powder (99.9%, 325 mesh) were purchased from Aladdin (Shanghai, China). Aluminum flake (99.9%, average particle size: 22 μm) was purchased from Advanced Institute (Shenzhen) Technology Co., Ltd. FSi resin (RF-901, solid content: 50 wt. %) was purchased from Xinmuming International Trade Co., Ltd (Shanghai, China). Isocyanate trimer (N3390) was purchased from the Bayer China Co., Ltd.

In this study, hybrid nanosheets of h-BN and graphene (4:1, w/w) were selected as the additives for the preparation of high emissivity coatings. The above hybrid nanosheets were prepared by the 2ball milling method. In a typical experiment, the ball milling of HOPG and h-BN hybrid filler was carried out in a planetary ball-mill machine under nitrogen atmosphere. First, 10 g of HOPG and h-BN hybrid powder (4:1, w/w) was placed into a stainless steel container containing stainless steel balls of 5 mm in diameter. The container was filled with nitrogen, then sealed and fixed in the planetary ball-mill machine, and agitated with 500 rpm for 48 h.

To avoid damage to the graphene substrate area, the spinning speed of the planetary ball mill was maintained at 500 rpm to ensure that the shear force was dominated. Subsequently, the internal pressure was kept at around 5 bar and slowly released through the gas outlet. The resulting product was then subjected to Soxhlet extraction with 1M HCl aqueous solution to completely remove metallic impurities, if any. The final dark black powder product was washed repeatedly with water and then freeze-dried.

The high emissivity coating was prepared using a simple spray method. First, the prepared h-BN/graphene hybrid nanosheets were dispersed in FSi resin and then the mixture was stirred with a planetary centrifugal stirrer at 2000 rpm for 20 min. The spiral flow generated by rotation applied strong shear forces to the h-BN/graphene hybrid nanosheets, resulting in a uniform solution. The curing agent N3390 was gradually added to the above mixture and vigorously stirred for 10 min. The mass ratio of FSi resin and curing agent was 10:1. The coatings with different emissivity were prepared by adding different amounts of flake aluminum powder. Second, acetone diluent was added to the above solution to adjust the viscosity. The metal substrate was cleaned with ethanol and dried with hot-blast air, then the prepared solution was sprayed onto the substrate with compressed air at 0.5 MPa. The fins were coated before assembling and only the external fins were coated. In order to ensure the uniformity of the coating of different batches, the surface coating thickness of the parallel aluminum samples placed was tested to ensure the uniformity of the coating thickness under different processes. Finally, the coated metal substrate was cured at 25 °C for 48 h. The preparation and spraying method are shown in Fig. 2.

Using a Fourier Transform Infrared (FT-IR) spectrometer (Bruker VERTEX 80v, USA), the reflection intensity was recorded to indirectly measure the spectral emissivity (2.5–25 μm) of both coated and uncoated metal substrates. The measured spectral emissivity results are shown in Fig. 3, with the percentage of aluminum powder added. As can be seen from Fig. 3, the emissivity of the uncoated aluminum alloy substrate is very low, fluctuating around 0.1, with a maximum of 0.2. In contrast, the emissivity of the coated substrate can reach a maximum of 0.97, with an overall average spectral emissivity of 0.9298. The addition of aluminum powder can adjust the emissivity of the coating, and within a certain range, the higher the percentage of aluminum powder added, the lower the emissivity of the coating. By controlling the addition ratio of aluminum powder, coatings with emissivity adjustable between 0.53 and 0.9298 can be prepared.

The entire experimental module is shown in Fig. 4. Three K-type thermocouples, k₁, k₂, and k₃, are placed between the heat source and the heat sink, and their average value T_ave is used as the surface temperature of the heat source. Temperature data are collected using an Agilent 34972A with a sampling frequency of 0.2 Hz. The collected data are then transferred to a computer for storage and post-processing. A high-power variable DC power supply (eTM-12020C) is used to control the heating power of the heat source. This power supply can output up to 120 V, 20 A of DC power. To reduce contact thermal resistance, screws and nuts are used to fasten the cover plate to the cavity, and thermally conductive silicone grease is applied to the contact surface between the cavity and the heat source. Figure 4(b) illustrates the arrangement of the three thermocouples and their measured temperatures and the average temperature. Temperature data are obtained by heating with 120 W power for 90 s and then cooling naturally for 180 s. It can be seen that the temperature difference between the thermocouples is small, and it is reasonable to use the average temperature of the thermocouples as the average temperature of the surface of the heat source.

All experimental equipment was conducted in an environment with a temperature of 27 ± 2 °C. This study includes two sets of experiments: experiments of continuous heating and subsequent natural cooling and experiments of high-power cyclic heating. In the continuous heating and natural cooling experiments, three different heating powers of 120/180/240 W were set to compare the thermal response curves of liquid metal phase-change heat sinks with a radiative coating with an emissivity of 0.9298 and without radiative coating at different heating powers. Heating started at 0 s and stopped when the average temperature of the thermocouples exceeded 100 °C, then the power was cut off, and data acquisition stopped when the device naturally cooled to near room temperature. In the high-power cyclic heating experiments, three different heating powers of 120/180/240 W were also set up, and the experiments were conducted using liquid metal phase-change heat sinks with radiative coatings with emissivity values of 0.9298 and 0.6, as well as without radiative coating under each heating condition. Each working cycle lasted 270 s, with continuous heating for 90 s at the beginning of each cycle followed by natural cooling for 180 s. Data acquisition was stopped after 10 working cycles.

Figure 5(a) shows the curves of the surface temperature T_ave of liquid metal phase-change heat sinks with/without radiative coatings at different heating powers. Continuous heating starts from 27 °C and ends at 100 °C. It can be seen that at the beginning of heating, the surface temperature of the heat source rises sharply, which is mainly due to the heat absorbed by the heat sink and the sensible heat of the phase-change material, with a small amount of heat dissipated by natural convection. When the temperature reaches a certain value, the temperature at the phase-change material reaches its melting point, at this time, the phase-change material begins to melt, and the heat source surface temperature rise slows down. During the melting process, the phase-change material absorbs a large amount of heat in the form of latent heat. However, due to the lack of close contact during the melting process,²⁶ the phase-change material forms a continuously thickening liquid phase region, which hinders the heat transfer from the surface of the heat source to the solid region. As the phase-change material approaches complete melting, the surface temperature of the heat source begins to rise rapidly again. The second segment of temperature rise curve demonstrates the excellent thermal control performance of the liquid metal phase-change material. Comparison of the temperature rise curves for the coated/uncoated heat sinks at various heating powers shows that during the initial heating stage, the overall temperature of the liquid metal phase-change heat sink is relatively low, so the effect of the radiative coating on the thermal management performance is minimal. However, in the later stages of heating, especially when the phase-change material is close to complete melting, the radiative coating plays an important role in suppressing the temperature rise of the heat source surface. For example, with a heating power of 120 W, the total heating time until the surface temperature of the heat source reaches 100 °C is 459 s without coating. However, when part of the surface of the liquid metal phase-change heat sink is coated with a radiative coating of ɛ = 0.9298, the total heating time increases to 498 s, which is 8.5% longer than 459 s. At heating powers of 180 and 240 W, the introduction of the radiative coating prolongs the time for the surface temperature to reach 100 °C by 8.5% and 9.4%, respectively.

Figure 5(b) shows the temperature variation curves over time as the surface temperature of the heat source is first raised to 100 °C and then naturally cooled to room temperature under different conditions. Due to the absence of active cooling measures, the decrease in temperature during the natural cooling process and the heat released from the phase-change material as it re-solidifies into the solid state must be dissipated entirely through natural convection and radiation. At this point, radiation from high emissivity surfaces cannot be ignored. Time t from thermal shock to complete solidification of the phase change material is critical. If this time is too long, the phase-change material may not fully solidify before the next working cycle, resulting in a decrease in heat absorption during the working cycle and eventually leading to thermal runaway. As can be seen from Fig. 5(b), the performance recovery time t for the three heating conditions of 120/180/240 W without coating is 2816/2653/2249 s, while with coating, the values of t are 2397/2265/1970 s, respectively. It is evident that the coating reduces the performance recovery time by 419/388/279 s, corresponding to percentages of 14.9%/14.6%/12.4%, respectively, which represents a significant improvement for the phase-change heat sink.

The experimental results presented above not only demonstrate the excellent thermal management performance of liquid metal phase change materials but also prove the significant role played by high emissivity coatings during both heating and natural cooling processes. In fact, phase change heat sinks are commonly used in intermittent or cyclic working conditions to cope with thermal shocks or extend the operating time of electronic devices. The experiments of high-power cycle heating in this section are closer to real scenarios, and their results can better illustrate the optimization effect of radiative coatings on liquid metal phase-change heat sinks in practical applications. Each working cycle consists of a heating time of 90 s followed by a natural cooling time of 180 s.

Figures 6(a)–6(c), respectively, show the thermal response curves of phase change heat sinks with different radiative coatings under heating powers of 120, 180, and 240 W. Taking the example of the thermal response curve under a heating power of 120 W, in the first four working cycles, the influence of the radiative coating on the surface temperature of the heat source is minimal due to the incomplete melting of the phase-change material and the relatively low overall temperature of the phase-change heat sink. However, in the subsequent working cycles, as the phase-change material completely melts, the heat generated by the heat source can only be dissipated through natural convection and radiation, leading to an increased impact of the radiative coating on the surface temperature of the heat source, which gradually increases with temperature.

In the tenth working cycle, the radiative coating with ɛ = 0.9298 reduces the peak temperature by 8.57 °C (6.74%) and the temperature at the end of the working cycle by 7.96 °C (7.35%). Similarly, at the heating power of 180 W, the radiative coating with ɛ = 0.9298 reduces the peak temperature in the tenth working cycle by 10.59 °C (6.07%) and the temperature at the end of the working cycle by 10.49 °C (7.15%). At the heating power of 240 W, the radiative coating with ɛ = 0.9298 reduces the peak temperature in the tenth working cycle by 16.56 °C (7.57%) and the temperature at the end of the working cycle by 13.95 °C (7.75%). Higher heating power means the phase-change material melts completely earlier, and the temperature of phase change heat sink increases faster, allowing the beneficial effect of the radiative coating material to be seen earlier. However, it is difficult to determine the exact melting volume fraction of the phase-change material and the time of complete melting during the experiment. Therefore, a combination of numerical simulation and experiments is required to analyze the contribution of phase change and radiation to the surface temperature control of the heat source.

The simulation is based on the following assumptions: (1) air is treated as an incompressible fluid, and the Boussinesq approximation is used to consider the effect of natural convection and simplify the equations. With the height of the external fins as the characteristic dimension, Previous experimental results have shown that the Rayleigh number in the natural convection area of the air is lower than 10⁸, so the natural convection is laminar²⁷; (2) volume changes and natural convection during liquid metal phase change are neglected²⁸; (3) all surfaces involved in radiation are assumed to be diffuse gray bodies, with no consideration of scattering and refraction, and the fluid medium does not participate in radiation; (4) the effect of thickness and microstructure of the radiative coating on flow and heat transfer is disregarded, and the role of the radiative coating is reflected by modifying the surface emissivity of the aluminum alloy coated with the radiative coating. The surface emissivity of the surface without a radiative coating is assumed to be 0.15; and (5) the surface temperature of the insulating material, phenolic plastic, is lower, and its surface thermal radiation and natural convection are neglected.

The phase-change region adopts the enthalpy-porosity method to solve the phase-change process,^29–31 and radiation heat transfer is solved using the discrete ordinate (DO) method.³² Based on the above assumptions, the governing equations are as follows, expressed in the form of a Cartesian tensor:

• Liquid phase region:

• Continuity equation:

  $∂ ρ ∂ t + ∂ ( ρ u i ) ∂ x i = 0,$

  (1)

  where ρ is the density of the material, u is the velocity, and the subscript i (i = x, y, z) denotes the component of this vector in the i direction.
• Momentum equation:

  $∂ ( ρ u i ) ∂ t + ∂ ( ρ u i u j ) ∂ x = − ∂ p ∂ x i + μ ∂ 2 u i ∂ x i 2 − ρ g i β ( T − T r e f ) + A u i,$

  (2)

  where μ, β, and T_ref are the kinetic viscosity, the coefficient of volume expansion, and the reference temperature, respectively. A_ui is the source term of the kinetic loss of the phase change material in each direction in the enthalpy method, which is introduced to correct for the velocity in the solid region,²⁹ and in the air domain, this source term of kinetic loss is 0. The source term A is defined as follows:

  $A = − C ( 1 − γ ) 2 ε + γ 2,$

  (3)

  where γ is the local liquid phase fraction and C is a large number used to suppress momentum (usually taken to be 10⁵ and ɛ is a very small amount to prevent the denominator from being zero (usually taken to be 0.001).
• Energy equation:

  $∂ ( ρ H ) ∂ t + ∂ ( ρ u i H ) ∂ x i = ∂ ∂ x i ( k c p ∂ H ∂ x i ) + 1 c p ( S V + S r a d ),$

  (4)

  where H is the total enthalpy of the PCM and the two components included, latent heat H_f and sensible heat h of the PCM, namely,

  $H = h + γ L,$

  (5)

  $h = h r e f + ∫ r e f T c p d T,$

  (6)

  $γ = { 0 , T < T m , T − T m T l − T m , T m < T < T l , 1 , T l < T ,$

  (7)

  where h_ref is the enthalpy of the substance at the reference temperature, T_m is the melting temperature, and T_l is the temperature at the end of the melting process. In the region without phase change materials, the latent heat L is equal to 0. Equation (4) also contains two heat sources, S_V and S_rad. S_V is a volumetric heat source, applied only at the heating rod

  $S V = Q V,$

  (8)

  where V is the volume of the heating rod and S_rad is the source term through which the radiative heat transfer is introduced

  $S r a d = − ∇ ⋅ q rad,$

  (9)

  where q_rad is the radiative heat flow density, which can be derived from the radiative transfer equation (RTE)

  $d I η d s = κ η I b η − β η I η + σ s η 4 π ∫ 4 π ⁡ I η ( s ^ i ) ϕ η ( s ^ i , s ^ ) d Ω i,$

  (10)

  where the subscript η denotes the unit wavelength in the spectrum, I is the intensity, $κ$ is the coefficient of absorption, and $σ s$ is the scattering coefficient. $ϕ η( s ^ i , s ^)$ is the scattering phase function and describes the probability that a ray from one direction, $s ^ i$⁠, will be scattered into a certain other direction, $s ^$⁠. $I η( s ^ i)$ is the intensity of radiation in the direction of $s ^ i$⁠. $Ω i$ is the solid angle and $β η$ is the coefficient of extinction,

  $β η = κ η + σ s η.$

  (11)
• Integrating Eq. (10) yields

  $∇ ⋅ q r a d , η = ∇ ⋅ ∫ 4 π ⁡ I η ( s ^ ) d Ω = 4 π κ η I b η − β η ∫ 4 π ⁡ I η ( s ^ ) d Ω + σ s η ∫ 4 π ⁡ I η ( s ^ i ) d Ω i .$

  (12)

The continuity, momentum, and energy equations are solved by ansys fluent software based on the finite volume method. The pressure solver based separation algorithm PISO is used to solve the pressure and velocity equations. The PISO algorithm although requires the pressure correction equations twice in a single iteration round, the pressure field obtained in each iteration step is more accurate and reduces the number of iteration steps required for the solution. The pressure field is discretized using a volumetric force-weighted scheme to correct for the normal pressure gradient at the wall in the case of natural convection. The momentum and energy equations are discretized using a second-order windward difference scheme with higher accuracy, which can effectively reduce the numerical diffusion error. The computation is considered to have reached convergence when the residuals are less than 10⁻⁴ for the mass conservation equation and less than 10⁻⁶ for the momentum, radiation, and energy equations.

The overall computational region is shown in Fig. 7, with the overall height of the heat sink, H, as the characteristic dimension, and the far-field boundaries of the naturally convective air domain are determined by three dimensionless numbers, l_x/H, l_y/H, and l_z/H. The external boundaries of the air domain are all set as pressure outlets, the temperature of reversed flow and the external blackbody radiation are set as ambient temperatures, and the external emissivities are all considered as 1. The three heating rod regions are given the volumetric heat source S_V, and the rest of the regions S_V is equal to 0. The solid–liquid interface is a no-slip surface, and the emissivities of the heat sink surfaces are set to 0.15, except for the black portion that has a radiative coating applied. A contact thermal resistance of size 2.5 × 10⁻⁴ was applied between the two because they were adhered only by gravity.³³ The surfaces of the bottom insulation are all adiabatic, i.e., the surface heat flow density is 0. The physical parameters of the materials used in the numerical simulations are shown in Table II.

To ensure the accuracy of the calculations and reduce computational resources consumption, domain independence analysis, grid independence analysis, and time step independence analysis were conducted before performing numerical simulations. The measurement standard for all independence analyses was the surface temperature of the heat source after two working cycles. The results of the independence verification are shown in Tables III–V. Based on the independence verification results, a computational domain with $l x/H= l y/H=8/3$⁠, $l z/H=2$⁠, a grid of n = 1 446 230 and a time step t = 1 s were ultimately selected. It is worth mentioning that although the calculation results with a time step of 0.1 s are more accurate, the CPU computation time corresponding to 0.1 s is too long due to the total calculation time being 2700 s. Therefore, a time step of 1 s was chosen in the end.

The thermal response curves of phase-change heat sinks with different emissivity radiative coatings under a heating power of 240 W are used to compare simulation and experimental results. From Fig. 8, it can be seen that there is good agreement between the experimental and simulation results, with maximum errors of 2.2% and 2.5% for simulations and experiments, respectively, when the surface emissivity ɛ is equal to 0.15/0.9298.

To more intuitively illustrate the specific roles of phase change and radiation during a certain period of time, an additional set of numerical simulation results with no coating (i.e., surface emissivity of 0.15), without phase change latent heat but with other thermal properties the same (shown as blue dashed lines in Fig. 9), was added for all three heating conditions. These numerical simulation results were then compared with the corresponding experimental thermal response curves of two sets of phase-change heat sinks without coating and with an emissivity of 0.9298, respectively. Additionally, the numerical simulation obtained the variation curve of the liquid phase volume fraction of the phase change material over time, as shown by the black and red dashed lines in Fig. 9. It is worth noting that the dashed lines represent numerical simulation results, while solid lines represent experimental results. The right side of each figure shows an enlarged view of the purple dashed box in the left side of the figure.

Taking Fig. 9(a) as an example, it can be seen that there is good consistency among the three temperature response curves when the phase-change material is not melted. As heating progresses, the temperature of the phase-change material continues to rise until the melting point, at which point the phase-change material begins to melt. When a significant amount of phase change occurs, such as at the start of the third working cycle, the temperature of the heat source surface under the condition of no latent heat of phase change begins to rise sharply, far exceeding the temperature of the heat source surface without coating but with phase-change material. A temperature difference as high as 8.56 °C was observed at the end of the third working cycle. In the fourth working cycle, the phase-change material continues to melt, and the temperature difference continues to increase, reaching 18.24 °C at the end of the fourth working cycle. This clearly demonstrates the excellent thermal control performance of the liquid metal phase-change material.

From the zoomed-in graph, it can be clearly seen that the introduction of high emissivity radiative coatings can delay the melting process of the phase-change material to some extent during heating, while increasing the solidification rate of the phase-change material during natural cooling. This is consistent with the experimental results of continuous heating and natural cooling in Sec. III B. At the end of the third working cycle, the liquid phase volume fraction of the phase-change material corresponding to an emissivity of 0.9298 for the coating is 55.84%, 13.96% lower than the liquid phase volume fraction of the phase-change material without coating, which is 69.80%. This indicates that the phase-change heat sink with an emissivity of 0.9298 for the coating has more latent heat to withstand the next heat shock when the next working cycle arrives. During the heating process of the fourth working cycle, the heating rate of the heat source surface corresponding to an emissivity of 0.9298 for the coating is lower, and its peak temperature is 2.59 °C lower than that without coating. After the fourth working cycle, the effect of the radiative coating begins to increase. At this point, the temperature difference on the heat source surface with phase change material but different emissivity of the coating starts to increase, and the difference in temperature between the heat source surface with the same emissivity coating gradually decreases. By the tenth working cycle, the temperatures of the two are very close.

In this study, a liquid metal phase-change heat sink with both external and internal fins was designed, where the internal fins enhance the heat conduction of the liquid metal phase-change material, and the external fins were coated with a high emissivity coating for heat exchange with the surroundings through natural convection and enhanced radiation. A combination of experiments and numerical simulation was performed to investigate the effect of the radiative coating on the performance of the liquid metal phase change heat sink under natural convection conditions. The high emissivity of the radiative coating and the high volumetric latent heat of the liquid metal phase change material enable the heat sink to provide good thermal control performance in both the high and low temperature regions, lowering the maximum temperature of the heat source and prolonging the operation time of the equipment. The phase change heat sink is suitable for electronic equipment that operates at low power for a long period of time but at high power for a short period of time, such as communication base stations. This is of great significance for thermal management of electronic devices that cannot use active cooling outdoors and can only rely on natural cooling. It is worth noting that much radiation emitted from the channels will be absorbed by the fin itself, and if the mounts of fins increase, the effects of radiative heat transfer will be reduced for this fin arrangement discussed before, but there must be an interesting design space to be explored where the trade-off is between highly radiating fins and highly naturally convection fins. The next research work could be to change the shape, number, and arrangement of the fins to make the radiative and convective heat transfer optimal.

The main conclusions of this study are as follows:

1. The maximum emissivity of the coating prepared using graphite powder and hexagonal boron nitride can reach 0.9298, and the emissivity can be adjusted by doping aluminum powder.

2. In the experiments of continuous heating and natural cooling, the introduction of the radiative coating with an emissivity of 0.9298 can extend the time for the surface temperature of the heat source to reach 100 °C by up to 9.4% and can shorten the recovery time of the phase-change heat sink by up to 14.9%.

3. The results of high-power cyclic thermal shock test show that under the heating power of 240 W, the peak temperature in the tenth working cycle is reduced by 16.56 °C (7.57%), and the temperature at the end of the working cycle is reduced by 13.95 °C (7.75%), which effectively lowers the temperature of the heat source.

4. The numerical simulation results further confirm that the introduction of radiative coating with high emissivity can delay the melting process of the phase-change material to some extent during heating and, at the same time, increase the solidification rate of the phase-change material during natural cooling.

5. By adding a set of simulations without coating but with the same thermal properties except latent heat of phase change, the specific effects of phase change and radiation on the overall thermal management performance are intuitively obtained. The results indicate that liquid metal phase-change heat sinks with high emissivity radiative coatings provide good thermal control performance in both low and high temperature regions.

This work was supported by the National Key R&D Program of China under Grant/Award No.: 2022YFE0201200.

The authors have no conflicts to disclose.

Wei Li: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal); Writing – original draft (lead). Yuqing Li: Data curation (equal); Formal analysis (equal); Investigation (equal). Yuchen Yao: Data curation (equal); Formal analysis (equal). Yue Ren: Data curation (equal); Writing – original draft (supporting). Wendi Bao: Visualization (equal). Yong Li: Conceptualization (equal); Funding acquisition (equal); Supervision (equal); Writing – review & editing (equal). Jing Liu: Conceptualization (equal); Methodology (equal); Resources (equal). Zhongshan Deng: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Resources (equal); Software (equal); Supervision (equal); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.