A unified relationship for evaporation kinetics at low Mach numbers

We experimentally realized and elucidated kinetically limited evaporation where the molecular gas dynamics close to the liquid–vapour interface dominates the overall transport. This process fundamentally dictates the performance of various evaporative systems and has received significant theoretical interest. However, experimental studies have been limited due to the difficulty of isolating the interfacial thermal resistance. Here, we overcome this challenge using an ultrathin nanoporous membrane in a pure vapour ambient. We demonstrate a fundamental relationship between the evaporation flux and driving potential in a dimensionless form, which unifies kinetically limited evaporation under different working conditions. We model the nonequilibrium gas kinetics and show good agreement between experiments and theory. Our work provides a general figure of merit for evaporative heat transfer as well as design guidelines for achieving efficient evaporation in applications such as water purification, steam generation, and thermal management.


Supplementary Figures
Supplementary Figure 1

Supplementary Note 1: Vapour Transport Analysis
The gas kinetics in the Knudsen layer is governed by the Boltzmann Transport Equation (BTE).
where is the mass-based distribution function, t is time, and u is the molecular velocity. The first term on the right hand side represents the streaming of molecules and the second term is the change of distribution function due to collisions between molecules. These two terms both contribute to the time evolution of (the left hand side). Choosing the reference temperature to be the interface temperature T0, the reference sonic speed is then given by s 0 u RT (2) where is the ratio of specific heats and R is the specific gas constant. The reference pressure is set as P0, the saturation pressure at T0 and the reference density is set as 0, the saturation vapour density at T0.
We can nondimensionalize the distribution function as   3   0 , , , , where u u/us, x = x/l0 with l0 being the characteristic length, and = with being the characteristic relaxation time for molecular collisions. Accordingly, in the dimensionless form, the BTE can be rewritten as: where us /l0 is proportional to the Knudsen number.
The vapour distribution function at the interface 0 can be written as 1, 2, 3 : The vapour flow coming out of the Knudsen layer follows a drifted Maxwell-Boltzmann distribution, where we assume local thermodynamic equilibrium: where TK is the temperature at the boundary of the Knudsen layer. After nondimensionalization, we can rewrite Supplementary Eq. (9) as Assuming ideal gas behavior, we have K K K 0 0 0 P T P T (11) and where MK is the Mach number at the Knudsen boundary and MK = uK/( From the Eqs. (7), (8), and (10)-(12), we observe that the boundary conditions of Supplementary Eq. (4) are uniquely determined by PK/P0, TK/T0, and MK. Previous studies 1, 2, 3, 4 also recognized that these are the three key parameters that characterize the Knudsen layer problem. More specifically, it was demonstrated that for evaporation problems, one of the three parameters uniquely determines the other two. In other (3), (4), and (5) in the main text shows that is a function of PK/P0, TK/T0, and MK and thus can also be P/P0. Previous studies showed that P scales linearly with MK. 2,4,5 Meanwhile, for the gas expansion region, using the isentropic approximation, we can write Pv is the pressure measured >0.1 m away from the interface in the chamber whereas the expected vapour mean free path varied between 0.29 1.

Supplementary Note 2: Fabrication Process
The ultrathin nanoporous membrane was microfabricated starting from a double side polished silicon wafer with both sides coated with silicon nitride ( 300 nm thick) using low pressure chemical vapour deposition (Supplementary Figure 1a). A nanopore array was patterned in the front silicon nitride layer using Supplementary Figure 2 shows the cross-section image of the ultrathin nanoporous membrane. Also, since the contact angle of water on gold is below 90° 6 , liquid can still wick into the pore from the bottom and wet along the pore wall, and the meniscus is pinned at the top corner of the pore given some slight gold deposition on the pore wall.

Supplementary Note 3: Device Design Rationale
We identified several experimental challenges for characterizing the interfacial transport which include: (1) measuring the interface temperature accurately and non-invasively; (2) decoupling the interfacial thermal resistance from the thermofluidic resistance in the liquid phase and the diffusion resistance in the vapour phase; and (3) mitigating the risk of blockage of evaporating surface due non-evaporative contaminants.
The temperature variation along the interface 0 can be estimated as where is the interfacial heat flux, is the thermal conductivity of the working fluid, and L is the characteristic length for thermal conduction in the liquid (estimated as the pore radius in this case). For a reference condition where = 100 W/cm 2 , = 0.6 W/m-K (for water), and L = 70 nm, 0 = 0.12 K. This indicates that the pore diameter is small enough for the thermal resistance in the liquid to be negligible and the temperature of the gold layer is approximately the same as the interface, which allows us to measure the interface temperature accurately and non-invasively.
The pressure drop along the pore can be estimated using the Hagen-Poiseuille equation: l m vis 2 p l lv 32 t q P d h (15) where l is the liquid viscosity, tm is the membrane thickness, l hlv is the enthalpy difference between the two phases. Setting = 100 W/cm 2 , tm = 200 nm, dp = 140 nm, and using properties of water at the room temperature, we have Pvis < 100 Pa. This is much smaller than the characteristic capillary pressure lv/dp where lv is the surface tension. During operation, the interface is pinned at the top corner of the pore, creating a self-regulating system, i.e., the apparent contact angle on the pore wall (or the curvature of the interface) adjusts to the actual interfacial pressure difference. The very small pressure difference across the interface implies that the interface is almost flat during evaporation.
The ultrathin membrane also mitigates the clogging risk which nanoporous configurations are often prone to. The mass fraction of non-evaporative contaminants in the nanopore c(z) is governed by the 1-D steady state convection-diffusion equation: (16) where

Supplementary Note 4: Heat Loss Characterization
To characterize the heat loss of the system, we fabricated a control sample which had the same structure as the designed device except that the active part was impermeable, i.e., had no pores. With liquid supplied at 1 mL/min (same as in the evaporation experiment), we measured the heating power Qloss as a function of where is the mass of the membrane, cp is its specific heat, is the interface area, and U is the overall heat transfer coefficient. In our setup, < 10 -8 kg, cp < 2 J/g-K, and UA C, such that < 0.01 s. The heating power in the experiment was limited by the onset of nucleation beneath the membrane. Due to the low hydraulic resistance across the thin membrane separating the two phases, the liquid and vapour were at similar pressures during operation. Although most heat was dissipated through evaporation, the membrane temperature was still elevated as we increased the heat fluxes, resulting in superheated liquid adjacent to the membrane. As we approached larger superheats with higher heating powers, cavitation became more likely to occur in the metastable liquid, which would cause the mechanical failure of the membrane. The experimental data that we reported in this study corresponded to the largest superheats we achieved under each working condition without observing the onset of bubble nucleation.

Supplementary Note 7: Discussion on Evaporation and Condensation Coefficients
The dashed line in Figure 5 in the main text represents the least-square model fit with equal evaporation and condensation coefficients where e = c = = 0.31±0.03. Molecular dynamics simulations of water generally yielded the same order of magnitude. Although various values have been reported 6,7,8 , which may be due to different intermolecular potential models used, the simulation results generally suggest no significant variation of e and c for the temperature range that we considered. Previous experimental studies, on the other hand, reported e and c across three different orders of magnitude (0.002-1) 9 . Restricting the comparison to the studies with dynamically renewing interfaces 10,11,12,13,14 , the results become more similar to the current work (0.1-1), indicating contamination could be a severe challenge in many previous studies.
There are several transient evaporation studies with liquid water exposed to vacuum, which simplifies the vapour transport into the free molecular flow. Hickman 8  obtained similar e and c compared to many previous studies, it represents quite different interfacial heat