Brushed lubricant-impregnated surfaces (BLIS) for long-lasting high condensation heat transfer

Recently, lubricant-impregnated surfaces (LIS) have emerged as a promising condenser surface by facilitating the removal of condensates from the surface. However, LIS has the critical limitation in that lubricant oil is depleted along with the removal of condensates. Such oil depletion is significantly aggravated under high condensation heat transfer. Here we propose a brushed LIS (BLIS) that can allow the application of LIS under high condensation heat transfer indefinitely by overcoming the previous oil depletion limit. In BLIS, a brush replenishes the depleted oil via physical contact with the rotational tube, while oil is continuously supplied to the brush by capillarity. In addition, BLIS helps enhance heat transfer performance with additional route to droplet removal by brush sweeping. By applying BLIS, we maintain the stable dropwise condensation mode for > 48 hours under high supersaturation levels along with up to 61% heat transfer enhancement compared to hydrophobic surfaces.


S-4
To demonstrate the changes in the droplet wetting morphology during oil depletion, we calculate the modified roughness factor. Figures S1A and S1B show the geometry of a single CuO nanostructure where the lubricant oil is depleted as much as depletion fraction ϕd. ϕd = 1 or 0 indicates that the oil within the nanostructure fills as much as the height of the structure or is completely removed. When we assume the shape of the CuO nanostructure as a very thin triangle, the total surface area of the CuO nanostructure ACuO can be expressed as, where h is the height of the CuO nanostructure (~1 μm), b is the bottom length of the CuO nanostructure and α is the shape angle of the CuO nanostructure. The surface area of the where bd is the bottom length of the exposed CuO nanostructure. When we assume that the surface roughness factor is proportional to the surface area of the CuO nanostructures, the roughness factor r can be expressed as, Then the modified roughness factor rd, which is indicate of the surface roughness factor S-5 when the impregnated lubricant is depleted as much as ϕd, can be expressed as, Then the switch of the wetting morphology between stage 2 and 3 can be estimated by the normalized dimensionless energy criteria E*, which is expressed as This criterion is based on the comparison between wetting energies for homogeneous wetting (cosθa W = rcosθa, where θa W is the advancing angle of a Wenzel state droplet.) and non-wetting (cosθa C = −1, where θa C is the advancing angle of a Cassie state droplet.).
When E* < 1, the droplet tries to minimize the contact with the surface by growing over nanostructures rather than spreading in between nanostructures. We introduce the change in the lubricant thickness into the dimensionless energy criteria by replacing r with the modified roughness factor rd = ϕd 2 r like the following,

Robustness of CuO nanostructures during brushing
To prevent damage to CuO nanostructures due caused by continuous brushing, we used a very soft brush and it was placed in light contact with the surface. In addition, the brush is coated with lubricant oil, which can minimize the abrasion between the brush and the S-10 surface. Figure S5 shows SEM images of CuO nanostructures before and after the brushing tests. Even after 48 hours, the CuO nanostructures were not damaged.

Supplementary Note 6 Calculation for condensation heat transfer coefficient
To calculate the condensation heat transfer coefficient hc, the overall heat transfer coefficient U which indicates the heat transfer performance between surrounding vapor and the cold water inside the test tube is first calculated as follow, Here, q" is the heat flux through the tube surface, ṁ is the mass flow rate of the cooling water, cp,l is the specific heat of cooling water, Tout is the tube outlet temperature, Tin is the tube inlet temperature, Ao is the tube outer surface area (= πdoL, where do is the tube outer diameter and L is the tube length.) and ΔTLMTD is the logarithmic mean temperature difference (LMTD) which is defined as, where Tv is the vapor saturation temperature which is measured by wet-bulb thermocouples inside the condensation chamber. The hc which represents the heat transfer performance from surrounding vapor to the tube surface is calculated by excluding the effects of the internal cooling water flow and conduction through the tube as follow, where Ai is the internal tube surface area (= πdiL, where di is the tube inner diameter), kCu is the Cu thermal conductivity and hi is heat transfer coefficient of the internal cooling water flow expressed as, where ΔT is the surface subcooling temperature (= Tv -Ts). The surface heat flux q" can be calculated by combining the above heat transfer rate through the single droplet with the S-16 droplet size distribution as follow, Here, n(R) and N(R) are the droplet size distribution for small droplets of Rmin < R < Re and large droplets of Re < R < ̂, respectively. The Re is the radius when droplets growing by direct vapor addition begins to coalesce and grow by coalescence, and it is given by Re =

Supplementary Note 13
The lifetime for LIS The lifetime for LIS defined as tstage1 seems very short ( Figure 2E). We understand the reason as the criteria for evaluating the lifetime for LIS. In the manuscript, we defined the lifetime for LIS as tstage1 when the sliding droplet size is maintained to be about 2Rmax = ~1.2 mm. If the lifetime for LIS is set for 2Rmax = 2 mm, the lasting time for LIS can be over twice as long. As such, the lifetime for LIS can be different according to the criteria for the lifetime.
Second, as we reported in the manuscript, the oil depletion rate is very sensitive to the condensation rate. The fast oil depletion of tstage1 < 3 min is only for very harsh condensation conditions where non-condensable gases (NCGs) are removed. If LIS was S-22 tested under NCGs, stage 1 can last for tens of minutes. If the criteria for 2Rmax = 2 mm is considered here, LIS can last more than one hour under NCGs.
Third, the lifetime for LIS highly depends on surface structures 8 , lubricant properties such as interfacial tension and viscosity 9 , and lubricant thickness. Krytox 1506 used in our work has relatively small kinematic viscosity of ~62 mm 2 /s. And the thickness of the applied oil was maintained thinly by removing excess oil on the surface so that the conduction resistance across the impregnated lubricant layer was minimized. Smaller and denser structures, higher viscosity and thicker oils can show a longer lifetime than our results.