Liquid-vapor phase-change cooling has a significant potential to facilitate the development of highly dense electronics by leveraging latent heat during the phase transition to remove heat from hotspots. A promising form of liquid–vapor phase-change cooling is coalescence-induced jumping droplet condensation, where droplet growth results in coalescence and gravity-independent jumping from the cold surface due to capillary-inertial energy conversion. Once the departed droplets reach the hotspot, heat is extracted via evaporation and through vapor return, subsequently spreading to the cold surface via condensation. Realizing the full potential of jumping droplet cooling requires a detailed understanding of the physics governing the process. Here, we examine the fundamental thermal and hydrodynamic limits of jumping droplet condensation. We demonstrate that jumping is mainly governed by the rate of droplet growth and fluid thermophysical properties. Timescale analysis demonstrates that the upper bound of water vapor jumping droplet condensation critical heat flux is $\u223c$ 20 kW/cm^{2}, significantly higher than that experimentally observed thus far due to surface structure limitations. Analysis of a wide range of available working fluids shows that liquid metals such as Li, Na, and Hg can obtain superior performance when compared to water.

Heat generation is the key obstacle to the electrification of mobile and stationary systems classically controlled or propelled by mechanical and pneumatic processes.^{1,2} As the surface area for cooling these electronic systems reduces due to packaging and integration constraints, traditional cooling pathways, such as conduction through a thermal interface material (TIM) and convection either to the air or liquid inside a cold plate, are unable to remove the required amount of heat while keeping the package within the safe temperature operating limits.^{2–6} In contrast, thermal management approaches such as phase-change cooling have found growing industrial application due to the potential for extracting heat as well as providing several unique system-level design benefits like buffering energy from temporally and spatially changing hotspots via the deployment of thermal energy storage.^{4–8} Liquid-vapor phase-change cooling has a significant potential to facilitate the development of highly dense electronics by leveraging the large latent heat ($hfg$) of working fluids during phase transition.^{8–21} As shown in Fig. 1(a), a promising form of liquid-vapor phase-change is jumping droplet condensation.^{9–21} Droplet jumping initiates with isolated droplet nucleation and growth during condensation on a cooled, suitably designed superhydrophobic surface. When neighboring droplets begin to coalesce, the merged droplet converts excess surface energy into kinetic energy, and ejects from the cold surface independent of gravity due to capillary-inertial energy conversion.^{10–21} Once cold droplets reach the hotspot, heat is extracted from the electronic components via evaporation and is subsequently returned and spread to the original cold surface via condensation.^{10–21}

While the upper bound of boiling heat flux has been identified as the sonic limit,^{22,23} the fundamental limits of jumping droplet condensation have not been explored. In this work, we demonstrate that a delicate combination of droplet growth, coalescence dynamics, and fluid thermophysical properties are key parameters governing jumping droplet heat transfer performance. A majority of recent studies^{10–20} have focused exclusively on water as the working fluid. Here, in addition to water, we analyze and identify a wide range of available working fluids, such as liquid metals, and demonstrate how these materials perform in directed cooling applications.

In order to elucidate the heat transfer physics governing jumping droplet condensation, we first consider that the maximum amount of heat extracted via evaporation from hot transistors is determined primarily by the number of cold droplets that strike the surface per unit time. As a vapor chamber is a closed system, the control volume can be drawn at the perimeter of the cold surface and the number of droplets leaving the surface per unit time can be examined without the loss of generality. Figure 1(a) depicts the four key steps and their associated timescales governing droplet jumping: individual droplet growth ($\tau g$), coalescence ($\tau c$), initial jumping ($\tau j$), and the time required for the ejected droplet to reach a distance of one radius from the cold surface, or release ($\tau r$). After initial nucleation, $\tau g$ captures the amount of time required for a droplet to grow large enough for coalescence with a neighboring droplet of equivalent size.^{24,25} Once the neighboring droplets have bridged and combined, $\tau c$, additional time is required for the resulting droplet to eject from the surface, $\tau j$.^{26,27} The release time portion of this process seeks to avoid the intersection of a droplet leaving the surface with a droplet growing rapidly underneath it due to the presence of residual liquid within structures remaining after departure and hence re-nucleation and growth at the same spot.^{13,16,20,21} The release time consideration accounts for the case of immediate recoalescence and infinitely fast droplet growth. This is an important modeling assumption required to bound the complexity of the jumping dynamics considered in this work. When combined, these four timescales comprise the complete jumping time scale, $\tau t=\tau g+\tau c+\tau j+\tau r$. While complex expressions are employed in the past literature to capture the behavior of each portion of this process,^{24–27} scaling analysis (Sec. S1, supplementary material) shows that $\tau t\u2248\tau g+3.2\tau c+\tau r$, which considers that fluid is a function of condensate thermophysical properties (liquid–vapor surface tension, $\gamma $, the liquid density, $\rho $, and the critical nucleation radius, $Rm$) and geometric properties at the solid-liquid interface (droplet radius, $R$, and droplet growth-rate, $dR/dt$). Hence, the complete jumping time scale can be expressed as

Scaling analysis^{24,25} shows that the growth rate of the droplet dominates ($\tau t\u223c\tau g)$ and is governed namely by the vapor-to-surface temperature difference, $\Delta T$, geometric aspects of the coating, and the average droplet coalescence radius (Sec. S1, supplementary material).

The rate of droplet heat transfer, $q\u0307$, in one time interval is based on the condensate mass flux, $m\u0307$, and the latent heat of the working fluid, $hfg$,

Figure 1(b) highlights the method used to translate the rate of droplet heat transfer ($q\u0307$) into the overall surface heat flux ($q\u2033$) by defining a large reference area composed of many unit cells where condensate droplets grow and coalesce with neighboring cells. Thus, the heat flux for jumping condensation can be shown to scale as (Sec. S2, supplementary material)

Given our understanding of the material-dependent timescales governing droplet jumping, we explored the fundamental heat transfer limits of droplet jumping condensation. To characterize the limits, we determined when timescales associated with jumping begin to approach one another. During droplet jumping, the growth of individual droplets prior to coalescence governs the growth time scale ($\tau g$). If the growth time scale approaches or is smaller than either the coalescence ($\tau c$) or jumping ($\tau j$) timescales (i.e., $\tau g<\tau c$ or $\tau g<\tau j$), the jumping process cannot be analyzed as an isolated process with discrete phases independent of one another. If $\tau g$ approaches $\tau c$, the growth of condensate liquid on the coalescing droplets during merging will interrupt the hydrodynamic processes governing jumping, and may impede departure. Similarly, if droplet re-nucleation and growth is so rapid as to not allow a coalesced droplet to depart prior to merging with the newly formed droplet beneath it ($\tau g<\tau j$), jumping will be impeded. Interestingly, the latter is a common observation in pool boiling studies, where $\tau g\u226a\tau j$, results in multiple vapor bubble coalescence events with vertically departing bubbles from the surface, and the formation of vapor columns.^{28}

The key parameter governing $\tau g$ is $\Delta T$. As $\Delta T$ increases, $\tau g$ reduces and approaches $\tau c$ or $\tau j$. Figure 2 highlights the number of complete cycles per second per unit cell as a function of $\Delta T$ and average droplet jumping radius ($R$) for the room temperature water working fluid. For example, approximately 10 droplets per unit cell will be ejected per second for an average droplet radius of $R$ = 100 nm at a temperature difference of $\Delta T$ = 10 K. A lower $\Delta T$ for the same $R$ yields fewer complete cycles, while a higher $\Delta T$ yields more droplets departing per second. Yet, as $R$ increases, the number of complete cycles rapidly decreases. For example, a 10X increase in $R$ requires a 10× increase in $\Delta T$ to maintain the same rate of droplets departing the condensing surface. Figure 2 also underscores how the maximum possible attainable heat flux during droplet jumping can be achieved by either increasing $\Delta T$ or decreasing $R$ in order to increase the number of droplet jumping events per second. The stepwise nature of this plot is a result of rounding down to ensure only whole droplets completing the entire jumping process were counted during a complete cycle. Interestingly, the $q\tau \u2033$ limits are quite high, exceeding ∼20 kW/cm^{2} for reasonable operating temperature ranges, departure radii, and assumptions for the dominant growth time scale (Sec. S1, supplementary material).

In order to verify the critical heat flux limitations from the vapor side perspective, we also considered vapor-side hydrodynamics on the jumping and departure process. As was previously reported,^{13,29} the presence of high heat flux at the condensing surface results in a rapid vapor flow normal to the surface and drag of droplets back to the wall. Scaling the maximum droplet jumping acceleration as: $a\u2009\u223c\u2009vj/\tau j$ (where $vj$ is the jumping departure velocity), and the Stokes drag force^{30} as: $FD\u2009\u223c\u2009\mu vR(vv+vj)$ (where $\mu v$ is the vapor viscosity and $vv$ is the vapor velocity), one obtains a closed form solution for the critical heat flux not to be exceeded in order to allow droplet departure away from the surface: $ma\u2009\u223c\u2009R3\rho vj/\tau c\u2009\u223c\u2009\mu vR(vv+vj)$. Relating the vapor velocity to the heat flux, we obtain: $vv\u2009\u223c\u2009q\u2033/(\rho vhfg)$, where the $v$ subscript corresponds to a material parameter for the vapor phase. Finally, equating the limits, a simplified closed form solution for the orientation-independent, hydrodynamic critical heat flux is (Sec. S3, supplementary material)

For room temperature water ($\mu v$ ≈ 0.01 mPa·s, $\gamma $ ≈ 72.8 mN/m, $hfg$ ≈ 2.5 MJ/kg, and $\rho v$ ≈ 0.022 kg/m^{3}), $qh\u2033\u2009\u223c$ 35 kW/cm^{2}. Assuming that the velocity of vapor cannot exceed the speed of sound ($vv<vsonic$ < 300 m/s), $qh\u2033$ reduces to $qs\u2033\u2248vsonic\rho vhfg\u2248$ 1.5 kW/cm^{2}, much lower than the aforementioned time scale-based limitations (Fig. 2).

The determination of the hydrodynamic and thermodynamic limits is important since it is well-established from experimental observations that $R$ during jumping droplet condensation increases over time as the superhydrophobic structured surface degrades and the condensation mode transitions to flooding.^{13,24,26,30,31} Progressive flooding characterizes surfaces where an increasing number of droplets become too large to jump and cover the cold surface, which prevents the nucleation of new droplets. Experimental observations highlight that flooding occurs mainly due to three mechanisms. The long-term robustness of the low surface energy functional conformal coating is limited and can degrade due to imperfections in the coatings.^{13–16,24–27,31,32} Furthermore, localized pinning due to imperfections on the surface promotes continued droplet growth without jumping, which ultimately leads to flooding.^{18–21,24,25} In addition to surface imperfections, progressive flooding can occur due to vapor-side hydrodynamics since droplets that jump can immediately return to the surface due to vapor shear.^{13,20,31,32} These flooding limitations explain the relatively low heat fluxes measured for electronic hotspot cooling (<10 W/cm^{2}) using jumping droplets with water.^{18,19} Namely, fundamental hydrodynamic and thermodynamic limits are yet to be reached, indicating that significant work is needed on the fabrication of the superhydrophobic surfaces to promote smaller $R$ via the rational engineering of well-defined nucleation sites to avoid flooding.

Switching of the working fluid is another avenue for increasing the maximum jumping droplet condensation heat flux (Fig. S3). Since the overall surface heat flux scales with the product of the latent heat and liquid density of the working fluid ($q\tau \u2033\u223c\u2009\rho hfg$), the performances of different elements were normalized against the baseline behavior of water. Moreover, a significant number of elements have melting temperatures greater than the useful range for many cooling applications, so only the elements with melting temperatures less than 873.15 K were preserved for further analysis. Figure S3 categorizes the elements based on their supply chain and cradle-to-grave sustainability.^{33} Ignoring sustainability of the working fluid selection, gallium (Ga) or a Ga alloy would be the best possible choice since it has an order of magnitude better performance than water for a similar working temperature range (see Sec. S4 of the supplementary material for the discussion of utilizing other working fluids). Yet, based on the current rate of global consumption, a serious supply chain risk exists for Ga in the next century.^{33} Interestingly, other materials with greater than 2× improvement in a reasonable working temperature range for electronics cooling such as indium (In), tin (Sn), and lithium (Li) are all either serious or potential supply chain risks. However, metals like mercury (Hg) and sodium (Na) offer a nearly 2× improvement in performance with either limited or no supply chain risk, respectively.

Melting temperature does not capture all of the necessary thermophysical property requirements of a working fluid to undergo droplet jumping. By further sorting the subset of fluids with melting temperatures less than 473.15 K and normalized material performance similar or greater than that of water based on vapor pressure,^{34} the capacity of the working fluid to condense with a reasonable saturation pressure (diffusion transport) and temperature can be identified. Figure 3 underscores that a vast majority of the superior working fluids highlighted in Fig. S3 (i.e., Ga, In, Li, and Na) require temperatures several hundred degrees higher than water to achieve similar vapor pressures and avoid pressure-based limitations.^{33} While these working temperatures are vastly greater than the safe operating temperature of modern-day electronics, they still can be operated at reasonable temperatures to serve as cooling mechanisms for other high heat flux, extreme environment systems.^{4,34,35} Even Hg requires temperatures of $\u2248$ 423 K to achieve a modest vapor pressure on the order of 1 kPa, which is reasonable for the operation junction temperature of wide bandgap power transistors like gallium-nitride (GaN) and silicon carbine (SiC), but is significantly hotter than the standard operating temperature of silicon transistors, <393 K. The use of liquid metals as working fluids presents a promising approach from the perspective of high temperature electronics. Furthermore, the high surface tension of liquid metals makes them ideal candidates for droplet jumping owing to their efficient capillary-to-inertial energy conversion and ability to be removed from un-structured smooth surfaces,^{36} and hence eliminate structure-based nucleation mediated flooding.

Given the limitation of liquid metals to high temperatures, we expanded our material survey to include other common working fluids for heat transfer applications. Acetone, ethanol, ammonia (NH_{3}), and methanol all exhibited inferior performance to water, but would have sufficient vapor pressure to be useful for the temperature ranges relevant for electronics cooling. Refrigerants and hydrocarbons all exhibited heat transfer performance at least 10× lower than water, and therefore are not promising to investigate further as jumping droplet cooling fluids. Furthermore, the ultra-low surface tension (<10 mN/m) of the majority of refrigerants makes them incapable of jumping.^{37} Interestingly, past studies have been able to demonstrate droplet jumping using ethanol and ethylene glycol.^{38} As a guideline, to ensure droplet jumping for common working fluids, we assumed a requirement of surface tension, $\gamma $ > 22 mN/m. Although demonstrated for deposited droplets, the development of omniphobic coatings capable of sustaining droplet jumping with fluids such as ethanol are assumed to be reasonable here from a theoretical standpoint. Re-examining this survey of potential working fluids with respect to $q\u2033h$ (Fig. S4) highlights that only Hg and Li performed better than water for vapor pressures greater than ∼100 kPa (i.e., > 523 K). Even though Na has a high $\rho $ and $hfg$, the $q\u2033crit$ of water dominated Na in contrast to the behavior exhibited in Fig. 3 since the $\rho v$ of Na was an order of magnitude smaller than water for a majority of the vapor pressure range. While the $\rho v$ of ammonia increases four orders of magnitude over this vapor pressure range, the performance of ammonia was limited by the large decrease in both $hfg$ and $\gamma $ with increasing vapor pressure. It is worth noting that many of these alternative working fluids would require engineering considerations for factors such as health risks, corrosion, and material compatibility with the vapor chamber casing. Overall, Fig. S4 highlights that water, from a hydrodynamic perspective, has very competitive performance as a working fluid for jumping droplet condensation for temperatures relevant in many thermal management applications such as electronics cooling.

To put the material survey in context with all three fundamental limitations for achieving the maximum heat flux [i.e., $qcrit\u2033$ = min($q\tau \u2033$, $qh\u2033,qs\u2033$)], we computed the time scale-based, hydrodynamic, and sonic limitations for the most promising fluids including ethanol, water, ammonia, Na, Hg, and Li (for individual material plots, see Sec. S6, supplementary material). Figure 4 underscores the importance of incorporating all three limits in the jumping droplet heat flux analysis since unique shapes exist for the $qcrit\u2033$ behavior of all the surveyed fluids. For example, our calculations demonstrate that ethanol offers an enhanced $q\u2033crit$ for pressures between ∼7 and ∼30 kPa for a similar temperature range while Hg, Na, and Li outperform water for pressures above ∼100 kPa for higher temperature ranges. The inset table of Fig. 4 summarizes the limitations, showing analogous results to water, mainly that room exists to increase the jumping droplet heat flux prior to reaching the fundamental limitations developed here.

In summary, we developed the fundamental performance bounds of jumping droplet condensation. Similar to the theoretical critical heat flux limit for pool boiling, the heat flux for jumping droplet condensation increases with the mass flux of droplets and the latent heat of the working fluid. Exploring the physics behind the jumping process highlights the process limitations due to the individual droplet growth rate as well as structured surface design for anti-flooding. While this model assumes a uniform droplet distribution, which acts as an idealized distribution for estimating the upper bound for jumping droplet heat flux, the trends highlighted from exploring the limits provide important physical insights and serve as idealized targets. On realistic surfaces, droplets grow and coalesce following various size distributions. Improvements in surface engineering are still needed to translate monodisperse droplet nucleation site density^{31} into idealized monodisperse distributions for coalescence and jumping events assumed in this analysis. Future work can extend our framework for the theoretical bounds of maximum heat flux to incorporate different droplet size distributions and obtain more conservative estimates given the additional constraints of specific structured-surface designs. Timescale analyses reveal that fundamental limitations have yet to be achieved, pointing to the need for surface-structure design optimization to minimize flooding seen in past experiments. In addition to controlling the size scale of nanostructures, surface engineering must also minimize surface defects so that localized droplet pinning is eliminated. Additionally, any impurities including non-condensable gases must be removed during manufacturing in order to avoid premature flooding on the surface and degradation of the nanostructured coating during operation.^{10–21,24–27,31,32} Examining the impact of thermophysical fluid properties on the limits of jumping droplet condensation heat flux reveals that water is a competitive working fluid for many thermal management applications, with only Li, Na, and Hg offering enhanced performance for cooling systems at higher temperatures, > 523 K, and ethanol as an alternative for temperatures < 373 K.

See the supplementary material for detailed derivation of equations and comprehensive examination of supporting expressions for the timescales and heat flux limitations.

The authors gratefully acknowledge funding for this work in part from the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE–1144245, the Power Optimization of Electro-Thermal Systems (POETS) National Science Foundation Engineering Research Center with Cooperative Agreement No. EEC-1449548, and the International Institute for Carbon Neutral Energy Research (No. WPI-I2CNER), sponsored by the Japanese Ministry of Education, Culture, Sports, Science and Technology.