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Hydrophobic copper nanowires for enhancing condensation heat transfer
Rongfu Wen12 Qian Li1 Jiafeng Wu13 Gensheng Wu14 Wei Wang15 Yunfei Chen6 Xuehu
Ma2 Dongliang Zhao1 and Ronggui Yang17
1Department of Mechanical Engineering University of Colorado Boulder CO 80309-0427
USA
2Liaoning Key Laboratory of Clean Utilization of Chemical Resources Institute of Chemical
Engineering Dalian University of Technology Dalian 116024 P R China
3Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education School of
Energy and Environment Southeast University Nanjing 210096 P R China
4 School of Mechanical and Electronic Engineering Nanjing Forestry University Nanjing 210037 P R China
5Advanced Li-ion Batteries Engineering Lab Ningbo Institute of Material Technology and
Engineering Chinese Academy of Sciences Ningbo 315201 P R China
6Jiangsu Key Laboratory for Design amp Manufacture of MicroNano Biomedical Instruments and
School of Mechanical Engineering Southeast University Nanjing 210096 P R China
7Materials Science and Engineering Program University of Colorado Boulder CO 80309-0596
USA
These authors contribute equally
Corresponding author RongguiYangColoradoEdu
1
Supplementary Information
S1 Surface Fabrication
S2 Experimental System for Condensation Heat Transfer
S3 Data Reduction
S4 Droplet Growth and Distribution at Different Surface Subcooling
S5 Theoretical Model for Droplet Jumping in Different Wetting States
Supplementary Videos
1 Droplet behaviors on the nanowired hydrophobic surface at Pv = 60kPa ΔT = 1 K
2 Droplet behaviors on the nanowired hydrophobic surface at Pv = 60kPa ΔT = 5 K
3 Droplet behaviors on the nanowired hydrophobic surface at Pv = 60kPa ΔT = 12 K
4 Droplet behaviors on the nanowired hydrophobic surface at Pv = 60kPa ΔT = 28 K
5 Droplet behaviors on the plain hydrophobic surface at Pv = 60kPa ΔT = 5 K
2
S1 Surface Fabrication
High purity copper (999 purity) is used to fabricate the condensing block (see
Supplementary Section 2) The condensing surface on the condensing block is polished by 2000
grit sandpaper cleaned in an ultrasonic bath with acetone for 10 minutes and then rinsed with
isopropyl alcohol ethanol and deionized water For comparison both plain hydrophobic surface
and nanowired hydrophobic surface has been fabricated as the condensing surface on the block
The plain hydrophobic surface is fabricated by dipping the condensing surface in a 25 mM
n-octadecanethiol (96 n-octadecyl mercaptan Acros Organics) in ethanol solution for 60
minutes at 70 A self-assembly monolayer (SAM) hydrophobic coating forms on the plain
copper surface
To fabricate nanowired hydrophobic surfaces copper nanowires are fabricated on the
condensing surface of the copper condensing block by a two-step porous anodic alumina (PAA)
template-assisted electro-deposition method as shown in Fig S1 The commercial PAA template
(nominal pore size 200 nm from GE Healthcare) formed by electrostatic oxidization is used [1]
Fig S2a shows the schematic of the PAA template used in this work The template has
hexagonal distributed holes with the diameter of ~220 nm and center-to-center spacing of ~380
nm on average (Fig S2b)
The first step of the electrodeposition process is to bond the PAA template onto the copper
condensing block For the bonding process the PAA template is first placed on the clean copper
condensing block The electrolyte solution composed of cupric pyrophosphate (Cu2P2O7xH2O
Sigma-Aldrich) potassium pyrophosphate (K4P2O7 Sigma-Aldrich) ammonium citrate tribasic
3
(C6H17N3O7 Sigma-Aldrich) and DI water (6252100 wt) is pipetted on top of the template
After the template is sufficiently wetted a filter pater wetted with electrolyte solution is placed
on the template which provide a conductive channel for the copper ions from the anode to
cathode (condensing block) Another copper plate is then placed on the filter paper as the counter
electrode Fig S1a shows the stacking structure consisting of the condensing block a PAA
template a filter paper wetted with electrolyte solution and another copper plane on the top of it
(Fig S1a) Uniform pressure is exerted to make sure that the PAA template adhere tightly on the
copper plate by a fastening device (Fig S1c) A constant voltage of -08 V (Electrochemical
Workstation CH Instrument CHI760C) is applied between the counter electrode (copper plane)
and the condensing block During this first-step bonding process copper nanowires are grown on
the copper substrate to serve as the screws that bonds PAA template on top of the copper sample
After 15 minutes of the bonding step the copper sample along with PAA template is released
from the stacking structure
4
Fig S1 Fabrication of nanowired surfaces a-b Schematic illustrating the two-step porous anodized
alumina (PAA)-templated electrodeposition process c Photographs of the fastening device for the first
electrodepostion step ie bonding
Fig S2 The structure of PAA a Schematic showing PAA template size feature b SEM image of the PAA
template
During the second electro-deposition step the condensing block along with PAA template
on the top is placed in a 3-electrode electroplating cell (Fig S1b) A constant voltage of -08 V
versus reference electrode (AgAgCl) is applied for the deposition of copper nanowires The
length of copper nanowires is controlled by the electro-deposition time Here the nanowires with
an average height (length) of 20 microm and 30 microm are fabricated after the electro-deposition in the
3-electrode electroplating cell for 150 and 240 minutes respectively By immersing the sample
in 2 molL NaOH solutions for 180 minutes the PAA template is released from the copper
nanowires grown on copper condensation block The sample is then washed with deionized
water to remove the residual solutions and dried in the vacuum chamber SAM hydrophobic
coating is then applied on copper nanowire surfaces by putting the sample into the 25 mM n-
octadecanethiol (96 n-octadecyl mercaptan Acros Organics) in ethanol solution which is
maintained at 70 for 60 minutes While the SAM hydrophobic coating thickens the diameter
5
of nanowires by ~3 nm due to the deposition of n-octadecyl mercaptan the general morphology
of nanowired surfaces remains unchanged [2]
Fig S3 shows the typical surface morphologies of both plain and nanowired hydrophobic
surfaces using a field-emission scanning electron microscope (FE-SEM JEOL JSM-7401F) For
the plain hydrophobic surface there is no obvious hierarchical and geometrical features except
some polished traces Nanowires on nanowired hydrophobic surfaces are hexagonal distributed
with a diameter (d) and center-to-center spacing (w) of ~220 nm and ~380 nm respectively
which is a replica of the PAA template Based on the nanowire distribution and geometry the
solid fraction is calculated as φ= π d2
2radic3 w2 to be 0282 The roughness factor r f=1+ 2 π dhradic3 w2 of the
nanowired hydrophobic surfaces are calculated to be 54 and 81 for nanowires with the length of
20 microm and 30 microm respectively Here d h and w are the diameter the length and the center-to-
center spacing of nanowires
The apparent contact angles of both plain and nanowired hydrophobic surfaces are
measured by the sessile droplet method with the values as 1146 plusmn 26deg (plain) 1392 plusmn 35deg (20-
μm long nanowires) and 1436 plusmn 13deg (30-μm long nanowires) respectively The images are
obtained using a high-speed camera (Photron FASTCAM SA4) with 5 μl droplets at atmospheric
pressure and room temperature
6
Fig S3 SEM images showing the surface morphology of (a) plain hydrophobic surface and (b)
nanowired hydrophobic surface (20 μm long nanowires) The insets show the apparent contact angle of
droplet on the plain and nanowired hydrophobic surfaces are 1146 plusmn 26deg and 1392 plusmn 35deg respectively
S2 Experimental System for Condensation Heat Transfer
Fig S4a shows the schematic of our custom-built setup for dropwise condensation heat
transfer measurements with in-situ visualization capability which consists of a steam generator
a condensation chamber a vacuum pump a chilled water bath and a data acquisition subsystem
The steam generator is a Ф76 mm times 900 mm vertical cylinder with an electric heater inside the
chamber The electric heater is designed to heat non-uniformly the larger heating power is
applied under the water level to boil the water and the upper part of the heater functions as a
super-heater The heating power is adjusted by a direct-current regulator (Agilent N5771A) The
steam temperature (Tse) at the outlet of the steam generator is always maintained to be 02~03 K
higher than the saturated water vapor temperature (Tv) measured in the condensing chamber The
superheated steam leaves the steam generator and enters the condensing chamber through a
connecting pipe (20 mm in diameter) To prevent condensation at the inside walls the
connecting pipe is heated by an adjustable heating tape (Omega HTWC 101-004) and covered
with fiberglass for the insulation
7
Condensation occurs in the condensing chamber which is a stainless steel cylinder with the
inner dimensions of Ф263 mm times 50 mm (Fig S4b) Flanges are used for sealing the end faces of
the condensing chamber A heating tape is wrapped around the exterior of the chamber walls to
prevent condensation at the inside walls where the exterior walls are insulated except a
transparent window for visualization The steam connection pipe is installed on the cover flange
and the condensing block is located below the inlet of steam At the bottom of the condensing
chamber a backflow pipe returns the condensate (water in this study) to the steam generator Fig
S4c shows the condensing block which is machined as two co-axial copper cylinders in different
diameters the smaller one (40 mm in diameter) is used as the condensing surface while the
larger one (90 mm in diameter) is designed to connect with the chiller (Thermo Electron
Corporation HAAKE Phoenix II) for dissipating the latent heat from the condensing surface A
flow meter (Proteus Industries 08004SN1) with an accuracy of plusmn 3 is integrated along the
cooling water inflow line to measure the flow rate G so that the heat transfer rate can be
calculated with the inlet (Ti) and outlet (To) temperatures measured for the cooling water loop
The condensing block is installed vertically at the center of condensing chamber Three holes (2
mm in diameter) are drilled in parallel on condensing block to measure the temperature
distribution in the condensing block during condensation (Fig S4c) All thermocouple holes
have the same distances of 35 mm to the condensing surface A transparent window (100 mm in
diameter) is installed about 50 mm in front of the condensing surface for the visualization using
high speed camera (Photron FASTCAM SA4)
T-type thermocouples and a pressure transducer (Omega PX409-050AI plusmn 008 accuracy)
are installed to monitor the temperature and pressure inside the chamber The thermocouple
bundles flow meter and pressure transducer are all connected to a data acquisition (Agilent
8
34970A) which is interfaced to a computer for data recording A vacuum pump (Edwards RV8)
is integrated into the vacuum line to pump down the experimental system to vacuum conditions
before water is filled into the steam generator
9
10
Fig S4 Experimental system a Schematic showing the custom-built setup for condensation heat
transfer with in-situ visualization capability b Photograph of the experimental setup (refrigeration
unit is not shown) c Photographs of the condensing block
S3 Data Reduction
Based on the energy balance the condensation heat flux q on the surface is calculated by
the temperature rise in the chilled water loop
q=cG (T ominusT i )
A (S1)
where c is the specific heat of cooling water (4200 Jmiddotkg-1K-1) A is the area of condensing surface
(1257 cm2) G is the mass flow rate of cooling water measured by the flow meter Ti and To are
the inlet and outlet temperatures of cooling water respectively Here the inlet temperature (Ti) of
cooling water is controlled by the chilled water tank with the fluctuation less than plusmn 01 K The
steady state of condensation is maintained for more than 40 minutes for each data point
The wall temperature (Tw) of condensing surface is calculated by assuming one-dimensional
heat conduction through the condensing block as
(S2)
where Tm is the average temperature measured by the three thermocouple in the condensing
block δ is the distance (35 mm) between the condensing surface and the measurement position
of thermocouples and k is the thermal conductivity of copper (398 Wm-1K-1 measured)
The surface subcooling ΔT is defined as the temperature difference between the steam
temperature (Tv) in condensing chamber and the wall temperature (Tw)
(S3)
Heat transfer coefficient h of condensation is defined as the ratio of heat flux to surface
subcooling which can be expressed as
11
(S4)
Table S1 summarizes the uncertainties of the measured quantities including the temperature
T the steam pressure Pv in condensing chamber the mass flow rate of cooling water G the
distance between condensing surface and the bottom of thermocouple holes δ the thermal
conductivity of condensing block k
Table S1 Parameters and uncertainties
Parameters Uncertainty (plusmn)
Temperature T (K) 015
Pressure P (kPa) 008
Flow rate G (kg) 03
Distance δ (mm) 025
Thermal conductivity k (Wm-1K-1) 1
Based on the error propagation the uncertainty of heat flux σ(q) is determined by
(S5)
The uncertainty of the mass flow rate (σ(G)) is plusmn 03 according the specifications of the
flow meter (Proteus Industries 08004SN1) The uncertainty of the inlet and outlet temperature of
cooling water (σ(Ti) and σ(To)) are both plusmn 015 K according the specifications of the
thermocouples (Omega T-type) The uncertainty of heat flux can then be obtained by the
measure quantities
The uncertainty of surface subcooling σ(ΔT) is determined by
(S6)
12
The uncertainty of the temperature of steam and condensing surface (σ(Tv) and σ(Tm)) are
both plusmn 015 K according the specifications of the thermocouples (Omega T-type) The
uncertainty of heat flux σ(q) is determined by equation (S5) The uncertainty of distance between
condensing surface and the bottom of thermocouple hole σ(δ) due to the mechanical processing
is plusmn 025 mm The uncertainty of the thermal conductivity of copper due to the change of
temperature in condensation experiments is plusmn 1
The uncertainty of heat transfer coefficient of condensation σ(h) is then determined by
(S7)
The uncertainty of heat flux σ(q) and surface subcooling σ(ΔT) are calculated by equation
(S5) and equation (S6) The derived quantities including heat flux q and surface subcooling ΔT
can be obtained by equation (S1) and equation (S3)
In the experiment all values of the temperature pressure and flow rate are collected by the
data acquisition simultaneously with the video capturing Error bars in Fig 2 indicate the
propagation of error associated with the temperatures pressure and flow rate measurements The
heat transfer error bars at small surface subcooling are largest due to the relatively low heat
fluxes measured corresponding to the small temperature difference between the outlet and outlet
of cooling water
S4 Droplet Growth and Distribution at Different Surface Subcooling
Representative droplet images at various times after the onset of condensation on both plain
(Fig S5a-c) and nanowired (Fig S5d-f) hydrophobic surfaces at different surface subcooling are
shown in Fig S5 Fig S5a-c show dropwise condensation on the plain hydrophobic surface
where discrete droplets form and grow to the sizes approaching the capillary length (~1 mm) 13
before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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Supplementary Information
S1 Surface Fabrication
S2 Experimental System for Condensation Heat Transfer
S3 Data Reduction
S4 Droplet Growth and Distribution at Different Surface Subcooling
S5 Theoretical Model for Droplet Jumping in Different Wetting States
Supplementary Videos
1 Droplet behaviors on the nanowired hydrophobic surface at Pv = 60kPa ΔT = 1 K
2 Droplet behaviors on the nanowired hydrophobic surface at Pv = 60kPa ΔT = 5 K
3 Droplet behaviors on the nanowired hydrophobic surface at Pv = 60kPa ΔT = 12 K
4 Droplet behaviors on the nanowired hydrophobic surface at Pv = 60kPa ΔT = 28 K
5 Droplet behaviors on the plain hydrophobic surface at Pv = 60kPa ΔT = 5 K
2
S1 Surface Fabrication
High purity copper (999 purity) is used to fabricate the condensing block (see
Supplementary Section 2) The condensing surface on the condensing block is polished by 2000
grit sandpaper cleaned in an ultrasonic bath with acetone for 10 minutes and then rinsed with
isopropyl alcohol ethanol and deionized water For comparison both plain hydrophobic surface
and nanowired hydrophobic surface has been fabricated as the condensing surface on the block
The plain hydrophobic surface is fabricated by dipping the condensing surface in a 25 mM
n-octadecanethiol (96 n-octadecyl mercaptan Acros Organics) in ethanol solution for 60
minutes at 70 A self-assembly monolayer (SAM) hydrophobic coating forms on the plain
copper surface
To fabricate nanowired hydrophobic surfaces copper nanowires are fabricated on the
condensing surface of the copper condensing block by a two-step porous anodic alumina (PAA)
template-assisted electro-deposition method as shown in Fig S1 The commercial PAA template
(nominal pore size 200 nm from GE Healthcare) formed by electrostatic oxidization is used [1]
Fig S2a shows the schematic of the PAA template used in this work The template has
hexagonal distributed holes with the diameter of ~220 nm and center-to-center spacing of ~380
nm on average (Fig S2b)
The first step of the electrodeposition process is to bond the PAA template onto the copper
condensing block For the bonding process the PAA template is first placed on the clean copper
condensing block The electrolyte solution composed of cupric pyrophosphate (Cu2P2O7xH2O
Sigma-Aldrich) potassium pyrophosphate (K4P2O7 Sigma-Aldrich) ammonium citrate tribasic
3
(C6H17N3O7 Sigma-Aldrich) and DI water (6252100 wt) is pipetted on top of the template
After the template is sufficiently wetted a filter pater wetted with electrolyte solution is placed
on the template which provide a conductive channel for the copper ions from the anode to
cathode (condensing block) Another copper plate is then placed on the filter paper as the counter
electrode Fig S1a shows the stacking structure consisting of the condensing block a PAA
template a filter paper wetted with electrolyte solution and another copper plane on the top of it
(Fig S1a) Uniform pressure is exerted to make sure that the PAA template adhere tightly on the
copper plate by a fastening device (Fig S1c) A constant voltage of -08 V (Electrochemical
Workstation CH Instrument CHI760C) is applied between the counter electrode (copper plane)
and the condensing block During this first-step bonding process copper nanowires are grown on
the copper substrate to serve as the screws that bonds PAA template on top of the copper sample
After 15 minutes of the bonding step the copper sample along with PAA template is released
from the stacking structure
4
Fig S1 Fabrication of nanowired surfaces a-b Schematic illustrating the two-step porous anodized
alumina (PAA)-templated electrodeposition process c Photographs of the fastening device for the first
electrodepostion step ie bonding
Fig S2 The structure of PAA a Schematic showing PAA template size feature b SEM image of the PAA
template
During the second electro-deposition step the condensing block along with PAA template
on the top is placed in a 3-electrode electroplating cell (Fig S1b) A constant voltage of -08 V
versus reference electrode (AgAgCl) is applied for the deposition of copper nanowires The
length of copper nanowires is controlled by the electro-deposition time Here the nanowires with
an average height (length) of 20 microm and 30 microm are fabricated after the electro-deposition in the
3-electrode electroplating cell for 150 and 240 minutes respectively By immersing the sample
in 2 molL NaOH solutions for 180 minutes the PAA template is released from the copper
nanowires grown on copper condensation block The sample is then washed with deionized
water to remove the residual solutions and dried in the vacuum chamber SAM hydrophobic
coating is then applied on copper nanowire surfaces by putting the sample into the 25 mM n-
octadecanethiol (96 n-octadecyl mercaptan Acros Organics) in ethanol solution which is
maintained at 70 for 60 minutes While the SAM hydrophobic coating thickens the diameter
5
of nanowires by ~3 nm due to the deposition of n-octadecyl mercaptan the general morphology
of nanowired surfaces remains unchanged [2]
Fig S3 shows the typical surface morphologies of both plain and nanowired hydrophobic
surfaces using a field-emission scanning electron microscope (FE-SEM JEOL JSM-7401F) For
the plain hydrophobic surface there is no obvious hierarchical and geometrical features except
some polished traces Nanowires on nanowired hydrophobic surfaces are hexagonal distributed
with a diameter (d) and center-to-center spacing (w) of ~220 nm and ~380 nm respectively
which is a replica of the PAA template Based on the nanowire distribution and geometry the
solid fraction is calculated as φ= π d2
2radic3 w2 to be 0282 The roughness factor r f=1+ 2 π dhradic3 w2 of the
nanowired hydrophobic surfaces are calculated to be 54 and 81 for nanowires with the length of
20 microm and 30 microm respectively Here d h and w are the diameter the length and the center-to-
center spacing of nanowires
The apparent contact angles of both plain and nanowired hydrophobic surfaces are
measured by the sessile droplet method with the values as 1146 plusmn 26deg (plain) 1392 plusmn 35deg (20-
μm long nanowires) and 1436 plusmn 13deg (30-μm long nanowires) respectively The images are
obtained using a high-speed camera (Photron FASTCAM SA4) with 5 μl droplets at atmospheric
pressure and room temperature
6
Fig S3 SEM images showing the surface morphology of (a) plain hydrophobic surface and (b)
nanowired hydrophobic surface (20 μm long nanowires) The insets show the apparent contact angle of
droplet on the plain and nanowired hydrophobic surfaces are 1146 plusmn 26deg and 1392 plusmn 35deg respectively
S2 Experimental System for Condensation Heat Transfer
Fig S4a shows the schematic of our custom-built setup for dropwise condensation heat
transfer measurements with in-situ visualization capability which consists of a steam generator
a condensation chamber a vacuum pump a chilled water bath and a data acquisition subsystem
The steam generator is a Ф76 mm times 900 mm vertical cylinder with an electric heater inside the
chamber The electric heater is designed to heat non-uniformly the larger heating power is
applied under the water level to boil the water and the upper part of the heater functions as a
super-heater The heating power is adjusted by a direct-current regulator (Agilent N5771A) The
steam temperature (Tse) at the outlet of the steam generator is always maintained to be 02~03 K
higher than the saturated water vapor temperature (Tv) measured in the condensing chamber The
superheated steam leaves the steam generator and enters the condensing chamber through a
connecting pipe (20 mm in diameter) To prevent condensation at the inside walls the
connecting pipe is heated by an adjustable heating tape (Omega HTWC 101-004) and covered
with fiberglass for the insulation
7
Condensation occurs in the condensing chamber which is a stainless steel cylinder with the
inner dimensions of Ф263 mm times 50 mm (Fig S4b) Flanges are used for sealing the end faces of
the condensing chamber A heating tape is wrapped around the exterior of the chamber walls to
prevent condensation at the inside walls where the exterior walls are insulated except a
transparent window for visualization The steam connection pipe is installed on the cover flange
and the condensing block is located below the inlet of steam At the bottom of the condensing
chamber a backflow pipe returns the condensate (water in this study) to the steam generator Fig
S4c shows the condensing block which is machined as two co-axial copper cylinders in different
diameters the smaller one (40 mm in diameter) is used as the condensing surface while the
larger one (90 mm in diameter) is designed to connect with the chiller (Thermo Electron
Corporation HAAKE Phoenix II) for dissipating the latent heat from the condensing surface A
flow meter (Proteus Industries 08004SN1) with an accuracy of plusmn 3 is integrated along the
cooling water inflow line to measure the flow rate G so that the heat transfer rate can be
calculated with the inlet (Ti) and outlet (To) temperatures measured for the cooling water loop
The condensing block is installed vertically at the center of condensing chamber Three holes (2
mm in diameter) are drilled in parallel on condensing block to measure the temperature
distribution in the condensing block during condensation (Fig S4c) All thermocouple holes
have the same distances of 35 mm to the condensing surface A transparent window (100 mm in
diameter) is installed about 50 mm in front of the condensing surface for the visualization using
high speed camera (Photron FASTCAM SA4)
T-type thermocouples and a pressure transducer (Omega PX409-050AI plusmn 008 accuracy)
are installed to monitor the temperature and pressure inside the chamber The thermocouple
bundles flow meter and pressure transducer are all connected to a data acquisition (Agilent
8
34970A) which is interfaced to a computer for data recording A vacuum pump (Edwards RV8)
is integrated into the vacuum line to pump down the experimental system to vacuum conditions
before water is filled into the steam generator
9
10
Fig S4 Experimental system a Schematic showing the custom-built setup for condensation heat
transfer with in-situ visualization capability b Photograph of the experimental setup (refrigeration
unit is not shown) c Photographs of the condensing block
S3 Data Reduction
Based on the energy balance the condensation heat flux q on the surface is calculated by
the temperature rise in the chilled water loop
q=cG (T ominusT i )
A (S1)
where c is the specific heat of cooling water (4200 Jmiddotkg-1K-1) A is the area of condensing surface
(1257 cm2) G is the mass flow rate of cooling water measured by the flow meter Ti and To are
the inlet and outlet temperatures of cooling water respectively Here the inlet temperature (Ti) of
cooling water is controlled by the chilled water tank with the fluctuation less than plusmn 01 K The
steady state of condensation is maintained for more than 40 minutes for each data point
The wall temperature (Tw) of condensing surface is calculated by assuming one-dimensional
heat conduction through the condensing block as
(S2)
where Tm is the average temperature measured by the three thermocouple in the condensing
block δ is the distance (35 mm) between the condensing surface and the measurement position
of thermocouples and k is the thermal conductivity of copper (398 Wm-1K-1 measured)
The surface subcooling ΔT is defined as the temperature difference between the steam
temperature (Tv) in condensing chamber and the wall temperature (Tw)
(S3)
Heat transfer coefficient h of condensation is defined as the ratio of heat flux to surface
subcooling which can be expressed as
11
(S4)
Table S1 summarizes the uncertainties of the measured quantities including the temperature
T the steam pressure Pv in condensing chamber the mass flow rate of cooling water G the
distance between condensing surface and the bottom of thermocouple holes δ the thermal
conductivity of condensing block k
Table S1 Parameters and uncertainties
Parameters Uncertainty (plusmn)
Temperature T (K) 015
Pressure P (kPa) 008
Flow rate G (kg) 03
Distance δ (mm) 025
Thermal conductivity k (Wm-1K-1) 1
Based on the error propagation the uncertainty of heat flux σ(q) is determined by
(S5)
The uncertainty of the mass flow rate (σ(G)) is plusmn 03 according the specifications of the
flow meter (Proteus Industries 08004SN1) The uncertainty of the inlet and outlet temperature of
cooling water (σ(Ti) and σ(To)) are both plusmn 015 K according the specifications of the
thermocouples (Omega T-type) The uncertainty of heat flux can then be obtained by the
measure quantities
The uncertainty of surface subcooling σ(ΔT) is determined by
(S6)
12
The uncertainty of the temperature of steam and condensing surface (σ(Tv) and σ(Tm)) are
both plusmn 015 K according the specifications of the thermocouples (Omega T-type) The
uncertainty of heat flux σ(q) is determined by equation (S5) The uncertainty of distance between
condensing surface and the bottom of thermocouple hole σ(δ) due to the mechanical processing
is plusmn 025 mm The uncertainty of the thermal conductivity of copper due to the change of
temperature in condensation experiments is plusmn 1
The uncertainty of heat transfer coefficient of condensation σ(h) is then determined by
(S7)
The uncertainty of heat flux σ(q) and surface subcooling σ(ΔT) are calculated by equation
(S5) and equation (S6) The derived quantities including heat flux q and surface subcooling ΔT
can be obtained by equation (S1) and equation (S3)
In the experiment all values of the temperature pressure and flow rate are collected by the
data acquisition simultaneously with the video capturing Error bars in Fig 2 indicate the
propagation of error associated with the temperatures pressure and flow rate measurements The
heat transfer error bars at small surface subcooling are largest due to the relatively low heat
fluxes measured corresponding to the small temperature difference between the outlet and outlet
of cooling water
S4 Droplet Growth and Distribution at Different Surface Subcooling
Representative droplet images at various times after the onset of condensation on both plain
(Fig S5a-c) and nanowired (Fig S5d-f) hydrophobic surfaces at different surface subcooling are
shown in Fig S5 Fig S5a-c show dropwise condensation on the plain hydrophobic surface
where discrete droplets form and grow to the sizes approaching the capillary length (~1 mm) 13
before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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S1 Surface Fabrication
High purity copper (999 purity) is used to fabricate the condensing block (see
Supplementary Section 2) The condensing surface on the condensing block is polished by 2000
grit sandpaper cleaned in an ultrasonic bath with acetone for 10 minutes and then rinsed with
isopropyl alcohol ethanol and deionized water For comparison both plain hydrophobic surface
and nanowired hydrophobic surface has been fabricated as the condensing surface on the block
The plain hydrophobic surface is fabricated by dipping the condensing surface in a 25 mM
n-octadecanethiol (96 n-octadecyl mercaptan Acros Organics) in ethanol solution for 60
minutes at 70 A self-assembly monolayer (SAM) hydrophobic coating forms on the plain
copper surface
To fabricate nanowired hydrophobic surfaces copper nanowires are fabricated on the
condensing surface of the copper condensing block by a two-step porous anodic alumina (PAA)
template-assisted electro-deposition method as shown in Fig S1 The commercial PAA template
(nominal pore size 200 nm from GE Healthcare) formed by electrostatic oxidization is used [1]
Fig S2a shows the schematic of the PAA template used in this work The template has
hexagonal distributed holes with the diameter of ~220 nm and center-to-center spacing of ~380
nm on average (Fig S2b)
The first step of the electrodeposition process is to bond the PAA template onto the copper
condensing block For the bonding process the PAA template is first placed on the clean copper
condensing block The electrolyte solution composed of cupric pyrophosphate (Cu2P2O7xH2O
Sigma-Aldrich) potassium pyrophosphate (K4P2O7 Sigma-Aldrich) ammonium citrate tribasic
3
(C6H17N3O7 Sigma-Aldrich) and DI water (6252100 wt) is pipetted on top of the template
After the template is sufficiently wetted a filter pater wetted with electrolyte solution is placed
on the template which provide a conductive channel for the copper ions from the anode to
cathode (condensing block) Another copper plate is then placed on the filter paper as the counter
electrode Fig S1a shows the stacking structure consisting of the condensing block a PAA
template a filter paper wetted with electrolyte solution and another copper plane on the top of it
(Fig S1a) Uniform pressure is exerted to make sure that the PAA template adhere tightly on the
copper plate by a fastening device (Fig S1c) A constant voltage of -08 V (Electrochemical
Workstation CH Instrument CHI760C) is applied between the counter electrode (copper plane)
and the condensing block During this first-step bonding process copper nanowires are grown on
the copper substrate to serve as the screws that bonds PAA template on top of the copper sample
After 15 minutes of the bonding step the copper sample along with PAA template is released
from the stacking structure
4
Fig S1 Fabrication of nanowired surfaces a-b Schematic illustrating the two-step porous anodized
alumina (PAA)-templated electrodeposition process c Photographs of the fastening device for the first
electrodepostion step ie bonding
Fig S2 The structure of PAA a Schematic showing PAA template size feature b SEM image of the PAA
template
During the second electro-deposition step the condensing block along with PAA template
on the top is placed in a 3-electrode electroplating cell (Fig S1b) A constant voltage of -08 V
versus reference electrode (AgAgCl) is applied for the deposition of copper nanowires The
length of copper nanowires is controlled by the electro-deposition time Here the nanowires with
an average height (length) of 20 microm and 30 microm are fabricated after the electro-deposition in the
3-electrode electroplating cell for 150 and 240 minutes respectively By immersing the sample
in 2 molL NaOH solutions for 180 minutes the PAA template is released from the copper
nanowires grown on copper condensation block The sample is then washed with deionized
water to remove the residual solutions and dried in the vacuum chamber SAM hydrophobic
coating is then applied on copper nanowire surfaces by putting the sample into the 25 mM n-
octadecanethiol (96 n-octadecyl mercaptan Acros Organics) in ethanol solution which is
maintained at 70 for 60 minutes While the SAM hydrophobic coating thickens the diameter
5
of nanowires by ~3 nm due to the deposition of n-octadecyl mercaptan the general morphology
of nanowired surfaces remains unchanged [2]
Fig S3 shows the typical surface morphologies of both plain and nanowired hydrophobic
surfaces using a field-emission scanning electron microscope (FE-SEM JEOL JSM-7401F) For
the plain hydrophobic surface there is no obvious hierarchical and geometrical features except
some polished traces Nanowires on nanowired hydrophobic surfaces are hexagonal distributed
with a diameter (d) and center-to-center spacing (w) of ~220 nm and ~380 nm respectively
which is a replica of the PAA template Based on the nanowire distribution and geometry the
solid fraction is calculated as φ= π d2
2radic3 w2 to be 0282 The roughness factor r f=1+ 2 π dhradic3 w2 of the
nanowired hydrophobic surfaces are calculated to be 54 and 81 for nanowires with the length of
20 microm and 30 microm respectively Here d h and w are the diameter the length and the center-to-
center spacing of nanowires
The apparent contact angles of both plain and nanowired hydrophobic surfaces are
measured by the sessile droplet method with the values as 1146 plusmn 26deg (plain) 1392 plusmn 35deg (20-
μm long nanowires) and 1436 plusmn 13deg (30-μm long nanowires) respectively The images are
obtained using a high-speed camera (Photron FASTCAM SA4) with 5 μl droplets at atmospheric
pressure and room temperature
6
Fig S3 SEM images showing the surface morphology of (a) plain hydrophobic surface and (b)
nanowired hydrophobic surface (20 μm long nanowires) The insets show the apparent contact angle of
droplet on the plain and nanowired hydrophobic surfaces are 1146 plusmn 26deg and 1392 plusmn 35deg respectively
S2 Experimental System for Condensation Heat Transfer
Fig S4a shows the schematic of our custom-built setup for dropwise condensation heat
transfer measurements with in-situ visualization capability which consists of a steam generator
a condensation chamber a vacuum pump a chilled water bath and a data acquisition subsystem
The steam generator is a Ф76 mm times 900 mm vertical cylinder with an electric heater inside the
chamber The electric heater is designed to heat non-uniformly the larger heating power is
applied under the water level to boil the water and the upper part of the heater functions as a
super-heater The heating power is adjusted by a direct-current regulator (Agilent N5771A) The
steam temperature (Tse) at the outlet of the steam generator is always maintained to be 02~03 K
higher than the saturated water vapor temperature (Tv) measured in the condensing chamber The
superheated steam leaves the steam generator and enters the condensing chamber through a
connecting pipe (20 mm in diameter) To prevent condensation at the inside walls the
connecting pipe is heated by an adjustable heating tape (Omega HTWC 101-004) and covered
with fiberglass for the insulation
7
Condensation occurs in the condensing chamber which is a stainless steel cylinder with the
inner dimensions of Ф263 mm times 50 mm (Fig S4b) Flanges are used for sealing the end faces of
the condensing chamber A heating tape is wrapped around the exterior of the chamber walls to
prevent condensation at the inside walls where the exterior walls are insulated except a
transparent window for visualization The steam connection pipe is installed on the cover flange
and the condensing block is located below the inlet of steam At the bottom of the condensing
chamber a backflow pipe returns the condensate (water in this study) to the steam generator Fig
S4c shows the condensing block which is machined as two co-axial copper cylinders in different
diameters the smaller one (40 mm in diameter) is used as the condensing surface while the
larger one (90 mm in diameter) is designed to connect with the chiller (Thermo Electron
Corporation HAAKE Phoenix II) for dissipating the latent heat from the condensing surface A
flow meter (Proteus Industries 08004SN1) with an accuracy of plusmn 3 is integrated along the
cooling water inflow line to measure the flow rate G so that the heat transfer rate can be
calculated with the inlet (Ti) and outlet (To) temperatures measured for the cooling water loop
The condensing block is installed vertically at the center of condensing chamber Three holes (2
mm in diameter) are drilled in parallel on condensing block to measure the temperature
distribution in the condensing block during condensation (Fig S4c) All thermocouple holes
have the same distances of 35 mm to the condensing surface A transparent window (100 mm in
diameter) is installed about 50 mm in front of the condensing surface for the visualization using
high speed camera (Photron FASTCAM SA4)
T-type thermocouples and a pressure transducer (Omega PX409-050AI plusmn 008 accuracy)
are installed to monitor the temperature and pressure inside the chamber The thermocouple
bundles flow meter and pressure transducer are all connected to a data acquisition (Agilent
8
34970A) which is interfaced to a computer for data recording A vacuum pump (Edwards RV8)
is integrated into the vacuum line to pump down the experimental system to vacuum conditions
before water is filled into the steam generator
9
10
Fig S4 Experimental system a Schematic showing the custom-built setup for condensation heat
transfer with in-situ visualization capability b Photograph of the experimental setup (refrigeration
unit is not shown) c Photographs of the condensing block
S3 Data Reduction
Based on the energy balance the condensation heat flux q on the surface is calculated by
the temperature rise in the chilled water loop
q=cG (T ominusT i )
A (S1)
where c is the specific heat of cooling water (4200 Jmiddotkg-1K-1) A is the area of condensing surface
(1257 cm2) G is the mass flow rate of cooling water measured by the flow meter Ti and To are
the inlet and outlet temperatures of cooling water respectively Here the inlet temperature (Ti) of
cooling water is controlled by the chilled water tank with the fluctuation less than plusmn 01 K The
steady state of condensation is maintained for more than 40 minutes for each data point
The wall temperature (Tw) of condensing surface is calculated by assuming one-dimensional
heat conduction through the condensing block as
(S2)
where Tm is the average temperature measured by the three thermocouple in the condensing
block δ is the distance (35 mm) between the condensing surface and the measurement position
of thermocouples and k is the thermal conductivity of copper (398 Wm-1K-1 measured)
The surface subcooling ΔT is defined as the temperature difference between the steam
temperature (Tv) in condensing chamber and the wall temperature (Tw)
(S3)
Heat transfer coefficient h of condensation is defined as the ratio of heat flux to surface
subcooling which can be expressed as
11
(S4)
Table S1 summarizes the uncertainties of the measured quantities including the temperature
T the steam pressure Pv in condensing chamber the mass flow rate of cooling water G the
distance between condensing surface and the bottom of thermocouple holes δ the thermal
conductivity of condensing block k
Table S1 Parameters and uncertainties
Parameters Uncertainty (plusmn)
Temperature T (K) 015
Pressure P (kPa) 008
Flow rate G (kg) 03
Distance δ (mm) 025
Thermal conductivity k (Wm-1K-1) 1
Based on the error propagation the uncertainty of heat flux σ(q) is determined by
(S5)
The uncertainty of the mass flow rate (σ(G)) is plusmn 03 according the specifications of the
flow meter (Proteus Industries 08004SN1) The uncertainty of the inlet and outlet temperature of
cooling water (σ(Ti) and σ(To)) are both plusmn 015 K according the specifications of the
thermocouples (Omega T-type) The uncertainty of heat flux can then be obtained by the
measure quantities
The uncertainty of surface subcooling σ(ΔT) is determined by
(S6)
12
The uncertainty of the temperature of steam and condensing surface (σ(Tv) and σ(Tm)) are
both plusmn 015 K according the specifications of the thermocouples (Omega T-type) The
uncertainty of heat flux σ(q) is determined by equation (S5) The uncertainty of distance between
condensing surface and the bottom of thermocouple hole σ(δ) due to the mechanical processing
is plusmn 025 mm The uncertainty of the thermal conductivity of copper due to the change of
temperature in condensation experiments is plusmn 1
The uncertainty of heat transfer coefficient of condensation σ(h) is then determined by
(S7)
The uncertainty of heat flux σ(q) and surface subcooling σ(ΔT) are calculated by equation
(S5) and equation (S6) The derived quantities including heat flux q and surface subcooling ΔT
can be obtained by equation (S1) and equation (S3)
In the experiment all values of the temperature pressure and flow rate are collected by the
data acquisition simultaneously with the video capturing Error bars in Fig 2 indicate the
propagation of error associated with the temperatures pressure and flow rate measurements The
heat transfer error bars at small surface subcooling are largest due to the relatively low heat
fluxes measured corresponding to the small temperature difference between the outlet and outlet
of cooling water
S4 Droplet Growth and Distribution at Different Surface Subcooling
Representative droplet images at various times after the onset of condensation on both plain
(Fig S5a-c) and nanowired (Fig S5d-f) hydrophobic surfaces at different surface subcooling are
shown in Fig S5 Fig S5a-c show dropwise condensation on the plain hydrophobic surface
where discrete droplets form and grow to the sizes approaching the capillary length (~1 mm) 13
before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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(C6H17N3O7 Sigma-Aldrich) and DI water (6252100 wt) is pipetted on top of the template
After the template is sufficiently wetted a filter pater wetted with electrolyte solution is placed
on the template which provide a conductive channel for the copper ions from the anode to
cathode (condensing block) Another copper plate is then placed on the filter paper as the counter
electrode Fig S1a shows the stacking structure consisting of the condensing block a PAA
template a filter paper wetted with electrolyte solution and another copper plane on the top of it
(Fig S1a) Uniform pressure is exerted to make sure that the PAA template adhere tightly on the
copper plate by a fastening device (Fig S1c) A constant voltage of -08 V (Electrochemical
Workstation CH Instrument CHI760C) is applied between the counter electrode (copper plane)
and the condensing block During this first-step bonding process copper nanowires are grown on
the copper substrate to serve as the screws that bonds PAA template on top of the copper sample
After 15 minutes of the bonding step the copper sample along with PAA template is released
from the stacking structure
4
Fig S1 Fabrication of nanowired surfaces a-b Schematic illustrating the two-step porous anodized
alumina (PAA)-templated electrodeposition process c Photographs of the fastening device for the first
electrodepostion step ie bonding
Fig S2 The structure of PAA a Schematic showing PAA template size feature b SEM image of the PAA
template
During the second electro-deposition step the condensing block along with PAA template
on the top is placed in a 3-electrode electroplating cell (Fig S1b) A constant voltage of -08 V
versus reference electrode (AgAgCl) is applied for the deposition of copper nanowires The
length of copper nanowires is controlled by the electro-deposition time Here the nanowires with
an average height (length) of 20 microm and 30 microm are fabricated after the electro-deposition in the
3-electrode electroplating cell for 150 and 240 minutes respectively By immersing the sample
in 2 molL NaOH solutions for 180 minutes the PAA template is released from the copper
nanowires grown on copper condensation block The sample is then washed with deionized
water to remove the residual solutions and dried in the vacuum chamber SAM hydrophobic
coating is then applied on copper nanowire surfaces by putting the sample into the 25 mM n-
octadecanethiol (96 n-octadecyl mercaptan Acros Organics) in ethanol solution which is
maintained at 70 for 60 minutes While the SAM hydrophobic coating thickens the diameter
5
of nanowires by ~3 nm due to the deposition of n-octadecyl mercaptan the general morphology
of nanowired surfaces remains unchanged [2]
Fig S3 shows the typical surface morphologies of both plain and nanowired hydrophobic
surfaces using a field-emission scanning electron microscope (FE-SEM JEOL JSM-7401F) For
the plain hydrophobic surface there is no obvious hierarchical and geometrical features except
some polished traces Nanowires on nanowired hydrophobic surfaces are hexagonal distributed
with a diameter (d) and center-to-center spacing (w) of ~220 nm and ~380 nm respectively
which is a replica of the PAA template Based on the nanowire distribution and geometry the
solid fraction is calculated as φ= π d2
2radic3 w2 to be 0282 The roughness factor r f=1+ 2 π dhradic3 w2 of the
nanowired hydrophobic surfaces are calculated to be 54 and 81 for nanowires with the length of
20 microm and 30 microm respectively Here d h and w are the diameter the length and the center-to-
center spacing of nanowires
The apparent contact angles of both plain and nanowired hydrophobic surfaces are
measured by the sessile droplet method with the values as 1146 plusmn 26deg (plain) 1392 plusmn 35deg (20-
μm long nanowires) and 1436 plusmn 13deg (30-μm long nanowires) respectively The images are
obtained using a high-speed camera (Photron FASTCAM SA4) with 5 μl droplets at atmospheric
pressure and room temperature
6
Fig S3 SEM images showing the surface morphology of (a) plain hydrophobic surface and (b)
nanowired hydrophobic surface (20 μm long nanowires) The insets show the apparent contact angle of
droplet on the plain and nanowired hydrophobic surfaces are 1146 plusmn 26deg and 1392 plusmn 35deg respectively
S2 Experimental System for Condensation Heat Transfer
Fig S4a shows the schematic of our custom-built setup for dropwise condensation heat
transfer measurements with in-situ visualization capability which consists of a steam generator
a condensation chamber a vacuum pump a chilled water bath and a data acquisition subsystem
The steam generator is a Ф76 mm times 900 mm vertical cylinder with an electric heater inside the
chamber The electric heater is designed to heat non-uniformly the larger heating power is
applied under the water level to boil the water and the upper part of the heater functions as a
super-heater The heating power is adjusted by a direct-current regulator (Agilent N5771A) The
steam temperature (Tse) at the outlet of the steam generator is always maintained to be 02~03 K
higher than the saturated water vapor temperature (Tv) measured in the condensing chamber The
superheated steam leaves the steam generator and enters the condensing chamber through a
connecting pipe (20 mm in diameter) To prevent condensation at the inside walls the
connecting pipe is heated by an adjustable heating tape (Omega HTWC 101-004) and covered
with fiberglass for the insulation
7
Condensation occurs in the condensing chamber which is a stainless steel cylinder with the
inner dimensions of Ф263 mm times 50 mm (Fig S4b) Flanges are used for sealing the end faces of
the condensing chamber A heating tape is wrapped around the exterior of the chamber walls to
prevent condensation at the inside walls where the exterior walls are insulated except a
transparent window for visualization The steam connection pipe is installed on the cover flange
and the condensing block is located below the inlet of steam At the bottom of the condensing
chamber a backflow pipe returns the condensate (water in this study) to the steam generator Fig
S4c shows the condensing block which is machined as two co-axial copper cylinders in different
diameters the smaller one (40 mm in diameter) is used as the condensing surface while the
larger one (90 mm in diameter) is designed to connect with the chiller (Thermo Electron
Corporation HAAKE Phoenix II) for dissipating the latent heat from the condensing surface A
flow meter (Proteus Industries 08004SN1) with an accuracy of plusmn 3 is integrated along the
cooling water inflow line to measure the flow rate G so that the heat transfer rate can be
calculated with the inlet (Ti) and outlet (To) temperatures measured for the cooling water loop
The condensing block is installed vertically at the center of condensing chamber Three holes (2
mm in diameter) are drilled in parallel on condensing block to measure the temperature
distribution in the condensing block during condensation (Fig S4c) All thermocouple holes
have the same distances of 35 mm to the condensing surface A transparent window (100 mm in
diameter) is installed about 50 mm in front of the condensing surface for the visualization using
high speed camera (Photron FASTCAM SA4)
T-type thermocouples and a pressure transducer (Omega PX409-050AI plusmn 008 accuracy)
are installed to monitor the temperature and pressure inside the chamber The thermocouple
bundles flow meter and pressure transducer are all connected to a data acquisition (Agilent
8
34970A) which is interfaced to a computer for data recording A vacuum pump (Edwards RV8)
is integrated into the vacuum line to pump down the experimental system to vacuum conditions
before water is filled into the steam generator
9
10
Fig S4 Experimental system a Schematic showing the custom-built setup for condensation heat
transfer with in-situ visualization capability b Photograph of the experimental setup (refrigeration
unit is not shown) c Photographs of the condensing block
S3 Data Reduction
Based on the energy balance the condensation heat flux q on the surface is calculated by
the temperature rise in the chilled water loop
q=cG (T ominusT i )
A (S1)
where c is the specific heat of cooling water (4200 Jmiddotkg-1K-1) A is the area of condensing surface
(1257 cm2) G is the mass flow rate of cooling water measured by the flow meter Ti and To are
the inlet and outlet temperatures of cooling water respectively Here the inlet temperature (Ti) of
cooling water is controlled by the chilled water tank with the fluctuation less than plusmn 01 K The
steady state of condensation is maintained for more than 40 minutes for each data point
The wall temperature (Tw) of condensing surface is calculated by assuming one-dimensional
heat conduction through the condensing block as
(S2)
where Tm is the average temperature measured by the three thermocouple in the condensing
block δ is the distance (35 mm) between the condensing surface and the measurement position
of thermocouples and k is the thermal conductivity of copper (398 Wm-1K-1 measured)
The surface subcooling ΔT is defined as the temperature difference between the steam
temperature (Tv) in condensing chamber and the wall temperature (Tw)
(S3)
Heat transfer coefficient h of condensation is defined as the ratio of heat flux to surface
subcooling which can be expressed as
11
(S4)
Table S1 summarizes the uncertainties of the measured quantities including the temperature
T the steam pressure Pv in condensing chamber the mass flow rate of cooling water G the
distance between condensing surface and the bottom of thermocouple holes δ the thermal
conductivity of condensing block k
Table S1 Parameters and uncertainties
Parameters Uncertainty (plusmn)
Temperature T (K) 015
Pressure P (kPa) 008
Flow rate G (kg) 03
Distance δ (mm) 025
Thermal conductivity k (Wm-1K-1) 1
Based on the error propagation the uncertainty of heat flux σ(q) is determined by
(S5)
The uncertainty of the mass flow rate (σ(G)) is plusmn 03 according the specifications of the
flow meter (Proteus Industries 08004SN1) The uncertainty of the inlet and outlet temperature of
cooling water (σ(Ti) and σ(To)) are both plusmn 015 K according the specifications of the
thermocouples (Omega T-type) The uncertainty of heat flux can then be obtained by the
measure quantities
The uncertainty of surface subcooling σ(ΔT) is determined by
(S6)
12
The uncertainty of the temperature of steam and condensing surface (σ(Tv) and σ(Tm)) are
both plusmn 015 K according the specifications of the thermocouples (Omega T-type) The
uncertainty of heat flux σ(q) is determined by equation (S5) The uncertainty of distance between
condensing surface and the bottom of thermocouple hole σ(δ) due to the mechanical processing
is plusmn 025 mm The uncertainty of the thermal conductivity of copper due to the change of
temperature in condensation experiments is plusmn 1
The uncertainty of heat transfer coefficient of condensation σ(h) is then determined by
(S7)
The uncertainty of heat flux σ(q) and surface subcooling σ(ΔT) are calculated by equation
(S5) and equation (S6) The derived quantities including heat flux q and surface subcooling ΔT
can be obtained by equation (S1) and equation (S3)
In the experiment all values of the temperature pressure and flow rate are collected by the
data acquisition simultaneously with the video capturing Error bars in Fig 2 indicate the
propagation of error associated with the temperatures pressure and flow rate measurements The
heat transfer error bars at small surface subcooling are largest due to the relatively low heat
fluxes measured corresponding to the small temperature difference between the outlet and outlet
of cooling water
S4 Droplet Growth and Distribution at Different Surface Subcooling
Representative droplet images at various times after the onset of condensation on both plain
(Fig S5a-c) and nanowired (Fig S5d-f) hydrophobic surfaces at different surface subcooling are
shown in Fig S5 Fig S5a-c show dropwise condensation on the plain hydrophobic surface
where discrete droplets form and grow to the sizes approaching the capillary length (~1 mm) 13
before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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Fig S1 Fabrication of nanowired surfaces a-b Schematic illustrating the two-step porous anodized
alumina (PAA)-templated electrodeposition process c Photographs of the fastening device for the first
electrodepostion step ie bonding
Fig S2 The structure of PAA a Schematic showing PAA template size feature b SEM image of the PAA
template
During the second electro-deposition step the condensing block along with PAA template
on the top is placed in a 3-electrode electroplating cell (Fig S1b) A constant voltage of -08 V
versus reference electrode (AgAgCl) is applied for the deposition of copper nanowires The
length of copper nanowires is controlled by the electro-deposition time Here the nanowires with
an average height (length) of 20 microm and 30 microm are fabricated after the electro-deposition in the
3-electrode electroplating cell for 150 and 240 minutes respectively By immersing the sample
in 2 molL NaOH solutions for 180 minutes the PAA template is released from the copper
nanowires grown on copper condensation block The sample is then washed with deionized
water to remove the residual solutions and dried in the vacuum chamber SAM hydrophobic
coating is then applied on copper nanowire surfaces by putting the sample into the 25 mM n-
octadecanethiol (96 n-octadecyl mercaptan Acros Organics) in ethanol solution which is
maintained at 70 for 60 minutes While the SAM hydrophobic coating thickens the diameter
5
of nanowires by ~3 nm due to the deposition of n-octadecyl mercaptan the general morphology
of nanowired surfaces remains unchanged [2]
Fig S3 shows the typical surface morphologies of both plain and nanowired hydrophobic
surfaces using a field-emission scanning electron microscope (FE-SEM JEOL JSM-7401F) For
the plain hydrophobic surface there is no obvious hierarchical and geometrical features except
some polished traces Nanowires on nanowired hydrophobic surfaces are hexagonal distributed
with a diameter (d) and center-to-center spacing (w) of ~220 nm and ~380 nm respectively
which is a replica of the PAA template Based on the nanowire distribution and geometry the
solid fraction is calculated as φ= π d2
2radic3 w2 to be 0282 The roughness factor r f=1+ 2 π dhradic3 w2 of the
nanowired hydrophobic surfaces are calculated to be 54 and 81 for nanowires with the length of
20 microm and 30 microm respectively Here d h and w are the diameter the length and the center-to-
center spacing of nanowires
The apparent contact angles of both plain and nanowired hydrophobic surfaces are
measured by the sessile droplet method with the values as 1146 plusmn 26deg (plain) 1392 plusmn 35deg (20-
μm long nanowires) and 1436 plusmn 13deg (30-μm long nanowires) respectively The images are
obtained using a high-speed camera (Photron FASTCAM SA4) with 5 μl droplets at atmospheric
pressure and room temperature
6
Fig S3 SEM images showing the surface morphology of (a) plain hydrophobic surface and (b)
nanowired hydrophobic surface (20 μm long nanowires) The insets show the apparent contact angle of
droplet on the plain and nanowired hydrophobic surfaces are 1146 plusmn 26deg and 1392 plusmn 35deg respectively
S2 Experimental System for Condensation Heat Transfer
Fig S4a shows the schematic of our custom-built setup for dropwise condensation heat
transfer measurements with in-situ visualization capability which consists of a steam generator
a condensation chamber a vacuum pump a chilled water bath and a data acquisition subsystem
The steam generator is a Ф76 mm times 900 mm vertical cylinder with an electric heater inside the
chamber The electric heater is designed to heat non-uniformly the larger heating power is
applied under the water level to boil the water and the upper part of the heater functions as a
super-heater The heating power is adjusted by a direct-current regulator (Agilent N5771A) The
steam temperature (Tse) at the outlet of the steam generator is always maintained to be 02~03 K
higher than the saturated water vapor temperature (Tv) measured in the condensing chamber The
superheated steam leaves the steam generator and enters the condensing chamber through a
connecting pipe (20 mm in diameter) To prevent condensation at the inside walls the
connecting pipe is heated by an adjustable heating tape (Omega HTWC 101-004) and covered
with fiberglass for the insulation
7
Condensation occurs in the condensing chamber which is a stainless steel cylinder with the
inner dimensions of Ф263 mm times 50 mm (Fig S4b) Flanges are used for sealing the end faces of
the condensing chamber A heating tape is wrapped around the exterior of the chamber walls to
prevent condensation at the inside walls where the exterior walls are insulated except a
transparent window for visualization The steam connection pipe is installed on the cover flange
and the condensing block is located below the inlet of steam At the bottom of the condensing
chamber a backflow pipe returns the condensate (water in this study) to the steam generator Fig
S4c shows the condensing block which is machined as two co-axial copper cylinders in different
diameters the smaller one (40 mm in diameter) is used as the condensing surface while the
larger one (90 mm in diameter) is designed to connect with the chiller (Thermo Electron
Corporation HAAKE Phoenix II) for dissipating the latent heat from the condensing surface A
flow meter (Proteus Industries 08004SN1) with an accuracy of plusmn 3 is integrated along the
cooling water inflow line to measure the flow rate G so that the heat transfer rate can be
calculated with the inlet (Ti) and outlet (To) temperatures measured for the cooling water loop
The condensing block is installed vertically at the center of condensing chamber Three holes (2
mm in diameter) are drilled in parallel on condensing block to measure the temperature
distribution in the condensing block during condensation (Fig S4c) All thermocouple holes
have the same distances of 35 mm to the condensing surface A transparent window (100 mm in
diameter) is installed about 50 mm in front of the condensing surface for the visualization using
high speed camera (Photron FASTCAM SA4)
T-type thermocouples and a pressure transducer (Omega PX409-050AI plusmn 008 accuracy)
are installed to monitor the temperature and pressure inside the chamber The thermocouple
bundles flow meter and pressure transducer are all connected to a data acquisition (Agilent
8
34970A) which is interfaced to a computer for data recording A vacuum pump (Edwards RV8)
is integrated into the vacuum line to pump down the experimental system to vacuum conditions
before water is filled into the steam generator
9
10
Fig S4 Experimental system a Schematic showing the custom-built setup for condensation heat
transfer with in-situ visualization capability b Photograph of the experimental setup (refrigeration
unit is not shown) c Photographs of the condensing block
S3 Data Reduction
Based on the energy balance the condensation heat flux q on the surface is calculated by
the temperature rise in the chilled water loop
q=cG (T ominusT i )
A (S1)
where c is the specific heat of cooling water (4200 Jmiddotkg-1K-1) A is the area of condensing surface
(1257 cm2) G is the mass flow rate of cooling water measured by the flow meter Ti and To are
the inlet and outlet temperatures of cooling water respectively Here the inlet temperature (Ti) of
cooling water is controlled by the chilled water tank with the fluctuation less than plusmn 01 K The
steady state of condensation is maintained for more than 40 minutes for each data point
The wall temperature (Tw) of condensing surface is calculated by assuming one-dimensional
heat conduction through the condensing block as
(S2)
where Tm is the average temperature measured by the three thermocouple in the condensing
block δ is the distance (35 mm) between the condensing surface and the measurement position
of thermocouples and k is the thermal conductivity of copper (398 Wm-1K-1 measured)
The surface subcooling ΔT is defined as the temperature difference between the steam
temperature (Tv) in condensing chamber and the wall temperature (Tw)
(S3)
Heat transfer coefficient h of condensation is defined as the ratio of heat flux to surface
subcooling which can be expressed as
11
(S4)
Table S1 summarizes the uncertainties of the measured quantities including the temperature
T the steam pressure Pv in condensing chamber the mass flow rate of cooling water G the
distance between condensing surface and the bottom of thermocouple holes δ the thermal
conductivity of condensing block k
Table S1 Parameters and uncertainties
Parameters Uncertainty (plusmn)
Temperature T (K) 015
Pressure P (kPa) 008
Flow rate G (kg) 03
Distance δ (mm) 025
Thermal conductivity k (Wm-1K-1) 1
Based on the error propagation the uncertainty of heat flux σ(q) is determined by
(S5)
The uncertainty of the mass flow rate (σ(G)) is plusmn 03 according the specifications of the
flow meter (Proteus Industries 08004SN1) The uncertainty of the inlet and outlet temperature of
cooling water (σ(Ti) and σ(To)) are both plusmn 015 K according the specifications of the
thermocouples (Omega T-type) The uncertainty of heat flux can then be obtained by the
measure quantities
The uncertainty of surface subcooling σ(ΔT) is determined by
(S6)
12
The uncertainty of the temperature of steam and condensing surface (σ(Tv) and σ(Tm)) are
both plusmn 015 K according the specifications of the thermocouples (Omega T-type) The
uncertainty of heat flux σ(q) is determined by equation (S5) The uncertainty of distance between
condensing surface and the bottom of thermocouple hole σ(δ) due to the mechanical processing
is plusmn 025 mm The uncertainty of the thermal conductivity of copper due to the change of
temperature in condensation experiments is plusmn 1
The uncertainty of heat transfer coefficient of condensation σ(h) is then determined by
(S7)
The uncertainty of heat flux σ(q) and surface subcooling σ(ΔT) are calculated by equation
(S5) and equation (S6) The derived quantities including heat flux q and surface subcooling ΔT
can be obtained by equation (S1) and equation (S3)
In the experiment all values of the temperature pressure and flow rate are collected by the
data acquisition simultaneously with the video capturing Error bars in Fig 2 indicate the
propagation of error associated with the temperatures pressure and flow rate measurements The
heat transfer error bars at small surface subcooling are largest due to the relatively low heat
fluxes measured corresponding to the small temperature difference between the outlet and outlet
of cooling water
S4 Droplet Growth and Distribution at Different Surface Subcooling
Representative droplet images at various times after the onset of condensation on both plain
(Fig S5a-c) and nanowired (Fig S5d-f) hydrophobic surfaces at different surface subcooling are
shown in Fig S5 Fig S5a-c show dropwise condensation on the plain hydrophobic surface
where discrete droplets form and grow to the sizes approaching the capillary length (~1 mm) 13
before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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of nanowires by ~3 nm due to the deposition of n-octadecyl mercaptan the general morphology
of nanowired surfaces remains unchanged [2]
Fig S3 shows the typical surface morphologies of both plain and nanowired hydrophobic
surfaces using a field-emission scanning electron microscope (FE-SEM JEOL JSM-7401F) For
the plain hydrophobic surface there is no obvious hierarchical and geometrical features except
some polished traces Nanowires on nanowired hydrophobic surfaces are hexagonal distributed
with a diameter (d) and center-to-center spacing (w) of ~220 nm and ~380 nm respectively
which is a replica of the PAA template Based on the nanowire distribution and geometry the
solid fraction is calculated as φ= π d2
2radic3 w2 to be 0282 The roughness factor r f=1+ 2 π dhradic3 w2 of the
nanowired hydrophobic surfaces are calculated to be 54 and 81 for nanowires with the length of
20 microm and 30 microm respectively Here d h and w are the diameter the length and the center-to-
center spacing of nanowires
The apparent contact angles of both plain and nanowired hydrophobic surfaces are
measured by the sessile droplet method with the values as 1146 plusmn 26deg (plain) 1392 plusmn 35deg (20-
μm long nanowires) and 1436 plusmn 13deg (30-μm long nanowires) respectively The images are
obtained using a high-speed camera (Photron FASTCAM SA4) with 5 μl droplets at atmospheric
pressure and room temperature
6
Fig S3 SEM images showing the surface morphology of (a) plain hydrophobic surface and (b)
nanowired hydrophobic surface (20 μm long nanowires) The insets show the apparent contact angle of
droplet on the plain and nanowired hydrophobic surfaces are 1146 plusmn 26deg and 1392 plusmn 35deg respectively
S2 Experimental System for Condensation Heat Transfer
Fig S4a shows the schematic of our custom-built setup for dropwise condensation heat
transfer measurements with in-situ visualization capability which consists of a steam generator
a condensation chamber a vacuum pump a chilled water bath and a data acquisition subsystem
The steam generator is a Ф76 mm times 900 mm vertical cylinder with an electric heater inside the
chamber The electric heater is designed to heat non-uniformly the larger heating power is
applied under the water level to boil the water and the upper part of the heater functions as a
super-heater The heating power is adjusted by a direct-current regulator (Agilent N5771A) The
steam temperature (Tse) at the outlet of the steam generator is always maintained to be 02~03 K
higher than the saturated water vapor temperature (Tv) measured in the condensing chamber The
superheated steam leaves the steam generator and enters the condensing chamber through a
connecting pipe (20 mm in diameter) To prevent condensation at the inside walls the
connecting pipe is heated by an adjustable heating tape (Omega HTWC 101-004) and covered
with fiberglass for the insulation
7
Condensation occurs in the condensing chamber which is a stainless steel cylinder with the
inner dimensions of Ф263 mm times 50 mm (Fig S4b) Flanges are used for sealing the end faces of
the condensing chamber A heating tape is wrapped around the exterior of the chamber walls to
prevent condensation at the inside walls where the exterior walls are insulated except a
transparent window for visualization The steam connection pipe is installed on the cover flange
and the condensing block is located below the inlet of steam At the bottom of the condensing
chamber a backflow pipe returns the condensate (water in this study) to the steam generator Fig
S4c shows the condensing block which is machined as two co-axial copper cylinders in different
diameters the smaller one (40 mm in diameter) is used as the condensing surface while the
larger one (90 mm in diameter) is designed to connect with the chiller (Thermo Electron
Corporation HAAKE Phoenix II) for dissipating the latent heat from the condensing surface A
flow meter (Proteus Industries 08004SN1) with an accuracy of plusmn 3 is integrated along the
cooling water inflow line to measure the flow rate G so that the heat transfer rate can be
calculated with the inlet (Ti) and outlet (To) temperatures measured for the cooling water loop
The condensing block is installed vertically at the center of condensing chamber Three holes (2
mm in diameter) are drilled in parallel on condensing block to measure the temperature
distribution in the condensing block during condensation (Fig S4c) All thermocouple holes
have the same distances of 35 mm to the condensing surface A transparent window (100 mm in
diameter) is installed about 50 mm in front of the condensing surface for the visualization using
high speed camera (Photron FASTCAM SA4)
T-type thermocouples and a pressure transducer (Omega PX409-050AI plusmn 008 accuracy)
are installed to monitor the temperature and pressure inside the chamber The thermocouple
bundles flow meter and pressure transducer are all connected to a data acquisition (Agilent
8
34970A) which is interfaced to a computer for data recording A vacuum pump (Edwards RV8)
is integrated into the vacuum line to pump down the experimental system to vacuum conditions
before water is filled into the steam generator
9
10
Fig S4 Experimental system a Schematic showing the custom-built setup for condensation heat
transfer with in-situ visualization capability b Photograph of the experimental setup (refrigeration
unit is not shown) c Photographs of the condensing block
S3 Data Reduction
Based on the energy balance the condensation heat flux q on the surface is calculated by
the temperature rise in the chilled water loop
q=cG (T ominusT i )
A (S1)
where c is the specific heat of cooling water (4200 Jmiddotkg-1K-1) A is the area of condensing surface
(1257 cm2) G is the mass flow rate of cooling water measured by the flow meter Ti and To are
the inlet and outlet temperatures of cooling water respectively Here the inlet temperature (Ti) of
cooling water is controlled by the chilled water tank with the fluctuation less than plusmn 01 K The
steady state of condensation is maintained for more than 40 minutes for each data point
The wall temperature (Tw) of condensing surface is calculated by assuming one-dimensional
heat conduction through the condensing block as
(S2)
where Tm is the average temperature measured by the three thermocouple in the condensing
block δ is the distance (35 mm) between the condensing surface and the measurement position
of thermocouples and k is the thermal conductivity of copper (398 Wm-1K-1 measured)
The surface subcooling ΔT is defined as the temperature difference between the steam
temperature (Tv) in condensing chamber and the wall temperature (Tw)
(S3)
Heat transfer coefficient h of condensation is defined as the ratio of heat flux to surface
subcooling which can be expressed as
11
(S4)
Table S1 summarizes the uncertainties of the measured quantities including the temperature
T the steam pressure Pv in condensing chamber the mass flow rate of cooling water G the
distance between condensing surface and the bottom of thermocouple holes δ the thermal
conductivity of condensing block k
Table S1 Parameters and uncertainties
Parameters Uncertainty (plusmn)
Temperature T (K) 015
Pressure P (kPa) 008
Flow rate G (kg) 03
Distance δ (mm) 025
Thermal conductivity k (Wm-1K-1) 1
Based on the error propagation the uncertainty of heat flux σ(q) is determined by
(S5)
The uncertainty of the mass flow rate (σ(G)) is plusmn 03 according the specifications of the
flow meter (Proteus Industries 08004SN1) The uncertainty of the inlet and outlet temperature of
cooling water (σ(Ti) and σ(To)) are both plusmn 015 K according the specifications of the
thermocouples (Omega T-type) The uncertainty of heat flux can then be obtained by the
measure quantities
The uncertainty of surface subcooling σ(ΔT) is determined by
(S6)
12
The uncertainty of the temperature of steam and condensing surface (σ(Tv) and σ(Tm)) are
both plusmn 015 K according the specifications of the thermocouples (Omega T-type) The
uncertainty of heat flux σ(q) is determined by equation (S5) The uncertainty of distance between
condensing surface and the bottom of thermocouple hole σ(δ) due to the mechanical processing
is plusmn 025 mm The uncertainty of the thermal conductivity of copper due to the change of
temperature in condensation experiments is plusmn 1
The uncertainty of heat transfer coefficient of condensation σ(h) is then determined by
(S7)
The uncertainty of heat flux σ(q) and surface subcooling σ(ΔT) are calculated by equation
(S5) and equation (S6) The derived quantities including heat flux q and surface subcooling ΔT
can be obtained by equation (S1) and equation (S3)
In the experiment all values of the temperature pressure and flow rate are collected by the
data acquisition simultaneously with the video capturing Error bars in Fig 2 indicate the
propagation of error associated with the temperatures pressure and flow rate measurements The
heat transfer error bars at small surface subcooling are largest due to the relatively low heat
fluxes measured corresponding to the small temperature difference between the outlet and outlet
of cooling water
S4 Droplet Growth and Distribution at Different Surface Subcooling
Representative droplet images at various times after the onset of condensation on both plain
(Fig S5a-c) and nanowired (Fig S5d-f) hydrophobic surfaces at different surface subcooling are
shown in Fig S5 Fig S5a-c show dropwise condensation on the plain hydrophobic surface
where discrete droplets form and grow to the sizes approaching the capillary length (~1 mm) 13
before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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Fig S3 SEM images showing the surface morphology of (a) plain hydrophobic surface and (b)
nanowired hydrophobic surface (20 μm long nanowires) The insets show the apparent contact angle of
droplet on the plain and nanowired hydrophobic surfaces are 1146 plusmn 26deg and 1392 plusmn 35deg respectively
S2 Experimental System for Condensation Heat Transfer
Fig S4a shows the schematic of our custom-built setup for dropwise condensation heat
transfer measurements with in-situ visualization capability which consists of a steam generator
a condensation chamber a vacuum pump a chilled water bath and a data acquisition subsystem
The steam generator is a Ф76 mm times 900 mm vertical cylinder with an electric heater inside the
chamber The electric heater is designed to heat non-uniformly the larger heating power is
applied under the water level to boil the water and the upper part of the heater functions as a
super-heater The heating power is adjusted by a direct-current regulator (Agilent N5771A) The
steam temperature (Tse) at the outlet of the steam generator is always maintained to be 02~03 K
higher than the saturated water vapor temperature (Tv) measured in the condensing chamber The
superheated steam leaves the steam generator and enters the condensing chamber through a
connecting pipe (20 mm in diameter) To prevent condensation at the inside walls the
connecting pipe is heated by an adjustable heating tape (Omega HTWC 101-004) and covered
with fiberglass for the insulation
7
Condensation occurs in the condensing chamber which is a stainless steel cylinder with the
inner dimensions of Ф263 mm times 50 mm (Fig S4b) Flanges are used for sealing the end faces of
the condensing chamber A heating tape is wrapped around the exterior of the chamber walls to
prevent condensation at the inside walls where the exterior walls are insulated except a
transparent window for visualization The steam connection pipe is installed on the cover flange
and the condensing block is located below the inlet of steam At the bottom of the condensing
chamber a backflow pipe returns the condensate (water in this study) to the steam generator Fig
S4c shows the condensing block which is machined as two co-axial copper cylinders in different
diameters the smaller one (40 mm in diameter) is used as the condensing surface while the
larger one (90 mm in diameter) is designed to connect with the chiller (Thermo Electron
Corporation HAAKE Phoenix II) for dissipating the latent heat from the condensing surface A
flow meter (Proteus Industries 08004SN1) with an accuracy of plusmn 3 is integrated along the
cooling water inflow line to measure the flow rate G so that the heat transfer rate can be
calculated with the inlet (Ti) and outlet (To) temperatures measured for the cooling water loop
The condensing block is installed vertically at the center of condensing chamber Three holes (2
mm in diameter) are drilled in parallel on condensing block to measure the temperature
distribution in the condensing block during condensation (Fig S4c) All thermocouple holes
have the same distances of 35 mm to the condensing surface A transparent window (100 mm in
diameter) is installed about 50 mm in front of the condensing surface for the visualization using
high speed camera (Photron FASTCAM SA4)
T-type thermocouples and a pressure transducer (Omega PX409-050AI plusmn 008 accuracy)
are installed to monitor the temperature and pressure inside the chamber The thermocouple
bundles flow meter and pressure transducer are all connected to a data acquisition (Agilent
8
34970A) which is interfaced to a computer for data recording A vacuum pump (Edwards RV8)
is integrated into the vacuum line to pump down the experimental system to vacuum conditions
before water is filled into the steam generator
9
10
Fig S4 Experimental system a Schematic showing the custom-built setup for condensation heat
transfer with in-situ visualization capability b Photograph of the experimental setup (refrigeration
unit is not shown) c Photographs of the condensing block
S3 Data Reduction
Based on the energy balance the condensation heat flux q on the surface is calculated by
the temperature rise in the chilled water loop
q=cG (T ominusT i )
A (S1)
where c is the specific heat of cooling water (4200 Jmiddotkg-1K-1) A is the area of condensing surface
(1257 cm2) G is the mass flow rate of cooling water measured by the flow meter Ti and To are
the inlet and outlet temperatures of cooling water respectively Here the inlet temperature (Ti) of
cooling water is controlled by the chilled water tank with the fluctuation less than plusmn 01 K The
steady state of condensation is maintained for more than 40 minutes for each data point
The wall temperature (Tw) of condensing surface is calculated by assuming one-dimensional
heat conduction through the condensing block as
(S2)
where Tm is the average temperature measured by the three thermocouple in the condensing
block δ is the distance (35 mm) between the condensing surface and the measurement position
of thermocouples and k is the thermal conductivity of copper (398 Wm-1K-1 measured)
The surface subcooling ΔT is defined as the temperature difference between the steam
temperature (Tv) in condensing chamber and the wall temperature (Tw)
(S3)
Heat transfer coefficient h of condensation is defined as the ratio of heat flux to surface
subcooling which can be expressed as
11
(S4)
Table S1 summarizes the uncertainties of the measured quantities including the temperature
T the steam pressure Pv in condensing chamber the mass flow rate of cooling water G the
distance between condensing surface and the bottom of thermocouple holes δ the thermal
conductivity of condensing block k
Table S1 Parameters and uncertainties
Parameters Uncertainty (plusmn)
Temperature T (K) 015
Pressure P (kPa) 008
Flow rate G (kg) 03
Distance δ (mm) 025
Thermal conductivity k (Wm-1K-1) 1
Based on the error propagation the uncertainty of heat flux σ(q) is determined by
(S5)
The uncertainty of the mass flow rate (σ(G)) is plusmn 03 according the specifications of the
flow meter (Proteus Industries 08004SN1) The uncertainty of the inlet and outlet temperature of
cooling water (σ(Ti) and σ(To)) are both plusmn 015 K according the specifications of the
thermocouples (Omega T-type) The uncertainty of heat flux can then be obtained by the
measure quantities
The uncertainty of surface subcooling σ(ΔT) is determined by
(S6)
12
The uncertainty of the temperature of steam and condensing surface (σ(Tv) and σ(Tm)) are
both plusmn 015 K according the specifications of the thermocouples (Omega T-type) The
uncertainty of heat flux σ(q) is determined by equation (S5) The uncertainty of distance between
condensing surface and the bottom of thermocouple hole σ(δ) due to the mechanical processing
is plusmn 025 mm The uncertainty of the thermal conductivity of copper due to the change of
temperature in condensation experiments is plusmn 1
The uncertainty of heat transfer coefficient of condensation σ(h) is then determined by
(S7)
The uncertainty of heat flux σ(q) and surface subcooling σ(ΔT) are calculated by equation
(S5) and equation (S6) The derived quantities including heat flux q and surface subcooling ΔT
can be obtained by equation (S1) and equation (S3)
In the experiment all values of the temperature pressure and flow rate are collected by the
data acquisition simultaneously with the video capturing Error bars in Fig 2 indicate the
propagation of error associated with the temperatures pressure and flow rate measurements The
heat transfer error bars at small surface subcooling are largest due to the relatively low heat
fluxes measured corresponding to the small temperature difference between the outlet and outlet
of cooling water
S4 Droplet Growth and Distribution at Different Surface Subcooling
Representative droplet images at various times after the onset of condensation on both plain
(Fig S5a-c) and nanowired (Fig S5d-f) hydrophobic surfaces at different surface subcooling are
shown in Fig S5 Fig S5a-c show dropwise condensation on the plain hydrophobic surface
where discrete droplets form and grow to the sizes approaching the capillary length (~1 mm) 13
before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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Condensation occurs in the condensing chamber which is a stainless steel cylinder with the
inner dimensions of Ф263 mm times 50 mm (Fig S4b) Flanges are used for sealing the end faces of
the condensing chamber A heating tape is wrapped around the exterior of the chamber walls to
prevent condensation at the inside walls where the exterior walls are insulated except a
transparent window for visualization The steam connection pipe is installed on the cover flange
and the condensing block is located below the inlet of steam At the bottom of the condensing
chamber a backflow pipe returns the condensate (water in this study) to the steam generator Fig
S4c shows the condensing block which is machined as two co-axial copper cylinders in different
diameters the smaller one (40 mm in diameter) is used as the condensing surface while the
larger one (90 mm in diameter) is designed to connect with the chiller (Thermo Electron
Corporation HAAKE Phoenix II) for dissipating the latent heat from the condensing surface A
flow meter (Proteus Industries 08004SN1) with an accuracy of plusmn 3 is integrated along the
cooling water inflow line to measure the flow rate G so that the heat transfer rate can be
calculated with the inlet (Ti) and outlet (To) temperatures measured for the cooling water loop
The condensing block is installed vertically at the center of condensing chamber Three holes (2
mm in diameter) are drilled in parallel on condensing block to measure the temperature
distribution in the condensing block during condensation (Fig S4c) All thermocouple holes
have the same distances of 35 mm to the condensing surface A transparent window (100 mm in
diameter) is installed about 50 mm in front of the condensing surface for the visualization using
high speed camera (Photron FASTCAM SA4)
T-type thermocouples and a pressure transducer (Omega PX409-050AI plusmn 008 accuracy)
are installed to monitor the temperature and pressure inside the chamber The thermocouple
bundles flow meter and pressure transducer are all connected to a data acquisition (Agilent
8
34970A) which is interfaced to a computer for data recording A vacuum pump (Edwards RV8)
is integrated into the vacuum line to pump down the experimental system to vacuum conditions
before water is filled into the steam generator
9
10
Fig S4 Experimental system a Schematic showing the custom-built setup for condensation heat
transfer with in-situ visualization capability b Photograph of the experimental setup (refrigeration
unit is not shown) c Photographs of the condensing block
S3 Data Reduction
Based on the energy balance the condensation heat flux q on the surface is calculated by
the temperature rise in the chilled water loop
q=cG (T ominusT i )
A (S1)
where c is the specific heat of cooling water (4200 Jmiddotkg-1K-1) A is the area of condensing surface
(1257 cm2) G is the mass flow rate of cooling water measured by the flow meter Ti and To are
the inlet and outlet temperatures of cooling water respectively Here the inlet temperature (Ti) of
cooling water is controlled by the chilled water tank with the fluctuation less than plusmn 01 K The
steady state of condensation is maintained for more than 40 minutes for each data point
The wall temperature (Tw) of condensing surface is calculated by assuming one-dimensional
heat conduction through the condensing block as
(S2)
where Tm is the average temperature measured by the three thermocouple in the condensing
block δ is the distance (35 mm) between the condensing surface and the measurement position
of thermocouples and k is the thermal conductivity of copper (398 Wm-1K-1 measured)
The surface subcooling ΔT is defined as the temperature difference between the steam
temperature (Tv) in condensing chamber and the wall temperature (Tw)
(S3)
Heat transfer coefficient h of condensation is defined as the ratio of heat flux to surface
subcooling which can be expressed as
11
(S4)
Table S1 summarizes the uncertainties of the measured quantities including the temperature
T the steam pressure Pv in condensing chamber the mass flow rate of cooling water G the
distance between condensing surface and the bottom of thermocouple holes δ the thermal
conductivity of condensing block k
Table S1 Parameters and uncertainties
Parameters Uncertainty (plusmn)
Temperature T (K) 015
Pressure P (kPa) 008
Flow rate G (kg) 03
Distance δ (mm) 025
Thermal conductivity k (Wm-1K-1) 1
Based on the error propagation the uncertainty of heat flux σ(q) is determined by
(S5)
The uncertainty of the mass flow rate (σ(G)) is plusmn 03 according the specifications of the
flow meter (Proteus Industries 08004SN1) The uncertainty of the inlet and outlet temperature of
cooling water (σ(Ti) and σ(To)) are both plusmn 015 K according the specifications of the
thermocouples (Omega T-type) The uncertainty of heat flux can then be obtained by the
measure quantities
The uncertainty of surface subcooling σ(ΔT) is determined by
(S6)
12
The uncertainty of the temperature of steam and condensing surface (σ(Tv) and σ(Tm)) are
both plusmn 015 K according the specifications of the thermocouples (Omega T-type) The
uncertainty of heat flux σ(q) is determined by equation (S5) The uncertainty of distance between
condensing surface and the bottom of thermocouple hole σ(δ) due to the mechanical processing
is plusmn 025 mm The uncertainty of the thermal conductivity of copper due to the change of
temperature in condensation experiments is plusmn 1
The uncertainty of heat transfer coefficient of condensation σ(h) is then determined by
(S7)
The uncertainty of heat flux σ(q) and surface subcooling σ(ΔT) are calculated by equation
(S5) and equation (S6) The derived quantities including heat flux q and surface subcooling ΔT
can be obtained by equation (S1) and equation (S3)
In the experiment all values of the temperature pressure and flow rate are collected by the
data acquisition simultaneously with the video capturing Error bars in Fig 2 indicate the
propagation of error associated with the temperatures pressure and flow rate measurements The
heat transfer error bars at small surface subcooling are largest due to the relatively low heat
fluxes measured corresponding to the small temperature difference between the outlet and outlet
of cooling water
S4 Droplet Growth and Distribution at Different Surface Subcooling
Representative droplet images at various times after the onset of condensation on both plain
(Fig S5a-c) and nanowired (Fig S5d-f) hydrophobic surfaces at different surface subcooling are
shown in Fig S5 Fig S5a-c show dropwise condensation on the plain hydrophobic surface
where discrete droplets form and grow to the sizes approaching the capillary length (~1 mm) 13
before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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34970A) which is interfaced to a computer for data recording A vacuum pump (Edwards RV8)
is integrated into the vacuum line to pump down the experimental system to vacuum conditions
before water is filled into the steam generator
9
10
Fig S4 Experimental system a Schematic showing the custom-built setup for condensation heat
transfer with in-situ visualization capability b Photograph of the experimental setup (refrigeration
unit is not shown) c Photographs of the condensing block
S3 Data Reduction
Based on the energy balance the condensation heat flux q on the surface is calculated by
the temperature rise in the chilled water loop
q=cG (T ominusT i )
A (S1)
where c is the specific heat of cooling water (4200 Jmiddotkg-1K-1) A is the area of condensing surface
(1257 cm2) G is the mass flow rate of cooling water measured by the flow meter Ti and To are
the inlet and outlet temperatures of cooling water respectively Here the inlet temperature (Ti) of
cooling water is controlled by the chilled water tank with the fluctuation less than plusmn 01 K The
steady state of condensation is maintained for more than 40 minutes for each data point
The wall temperature (Tw) of condensing surface is calculated by assuming one-dimensional
heat conduction through the condensing block as
(S2)
where Tm is the average temperature measured by the three thermocouple in the condensing
block δ is the distance (35 mm) between the condensing surface and the measurement position
of thermocouples and k is the thermal conductivity of copper (398 Wm-1K-1 measured)
The surface subcooling ΔT is defined as the temperature difference between the steam
temperature (Tv) in condensing chamber and the wall temperature (Tw)
(S3)
Heat transfer coefficient h of condensation is defined as the ratio of heat flux to surface
subcooling which can be expressed as
11
(S4)
Table S1 summarizes the uncertainties of the measured quantities including the temperature
T the steam pressure Pv in condensing chamber the mass flow rate of cooling water G the
distance between condensing surface and the bottom of thermocouple holes δ the thermal
conductivity of condensing block k
Table S1 Parameters and uncertainties
Parameters Uncertainty (plusmn)
Temperature T (K) 015
Pressure P (kPa) 008
Flow rate G (kg) 03
Distance δ (mm) 025
Thermal conductivity k (Wm-1K-1) 1
Based on the error propagation the uncertainty of heat flux σ(q) is determined by
(S5)
The uncertainty of the mass flow rate (σ(G)) is plusmn 03 according the specifications of the
flow meter (Proteus Industries 08004SN1) The uncertainty of the inlet and outlet temperature of
cooling water (σ(Ti) and σ(To)) are both plusmn 015 K according the specifications of the
thermocouples (Omega T-type) The uncertainty of heat flux can then be obtained by the
measure quantities
The uncertainty of surface subcooling σ(ΔT) is determined by
(S6)
12
The uncertainty of the temperature of steam and condensing surface (σ(Tv) and σ(Tm)) are
both plusmn 015 K according the specifications of the thermocouples (Omega T-type) The
uncertainty of heat flux σ(q) is determined by equation (S5) The uncertainty of distance between
condensing surface and the bottom of thermocouple hole σ(δ) due to the mechanical processing
is plusmn 025 mm The uncertainty of the thermal conductivity of copper due to the change of
temperature in condensation experiments is plusmn 1
The uncertainty of heat transfer coefficient of condensation σ(h) is then determined by
(S7)
The uncertainty of heat flux σ(q) and surface subcooling σ(ΔT) are calculated by equation
(S5) and equation (S6) The derived quantities including heat flux q and surface subcooling ΔT
can be obtained by equation (S1) and equation (S3)
In the experiment all values of the temperature pressure and flow rate are collected by the
data acquisition simultaneously with the video capturing Error bars in Fig 2 indicate the
propagation of error associated with the temperatures pressure and flow rate measurements The
heat transfer error bars at small surface subcooling are largest due to the relatively low heat
fluxes measured corresponding to the small temperature difference between the outlet and outlet
of cooling water
S4 Droplet Growth and Distribution at Different Surface Subcooling
Representative droplet images at various times after the onset of condensation on both plain
(Fig S5a-c) and nanowired (Fig S5d-f) hydrophobic surfaces at different surface subcooling are
shown in Fig S5 Fig S5a-c show dropwise condensation on the plain hydrophobic surface
where discrete droplets form and grow to the sizes approaching the capillary length (~1 mm) 13
before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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10
Fig S4 Experimental system a Schematic showing the custom-built setup for condensation heat
transfer with in-situ visualization capability b Photograph of the experimental setup (refrigeration
unit is not shown) c Photographs of the condensing block
S3 Data Reduction
Based on the energy balance the condensation heat flux q on the surface is calculated by
the temperature rise in the chilled water loop
q=cG (T ominusT i )
A (S1)
where c is the specific heat of cooling water (4200 Jmiddotkg-1K-1) A is the area of condensing surface
(1257 cm2) G is the mass flow rate of cooling water measured by the flow meter Ti and To are
the inlet and outlet temperatures of cooling water respectively Here the inlet temperature (Ti) of
cooling water is controlled by the chilled water tank with the fluctuation less than plusmn 01 K The
steady state of condensation is maintained for more than 40 minutes for each data point
The wall temperature (Tw) of condensing surface is calculated by assuming one-dimensional
heat conduction through the condensing block as
(S2)
where Tm is the average temperature measured by the three thermocouple in the condensing
block δ is the distance (35 mm) between the condensing surface and the measurement position
of thermocouples and k is the thermal conductivity of copper (398 Wm-1K-1 measured)
The surface subcooling ΔT is defined as the temperature difference between the steam
temperature (Tv) in condensing chamber and the wall temperature (Tw)
(S3)
Heat transfer coefficient h of condensation is defined as the ratio of heat flux to surface
subcooling which can be expressed as
11
(S4)
Table S1 summarizes the uncertainties of the measured quantities including the temperature
T the steam pressure Pv in condensing chamber the mass flow rate of cooling water G the
distance between condensing surface and the bottom of thermocouple holes δ the thermal
conductivity of condensing block k
Table S1 Parameters and uncertainties
Parameters Uncertainty (plusmn)
Temperature T (K) 015
Pressure P (kPa) 008
Flow rate G (kg) 03
Distance δ (mm) 025
Thermal conductivity k (Wm-1K-1) 1
Based on the error propagation the uncertainty of heat flux σ(q) is determined by
(S5)
The uncertainty of the mass flow rate (σ(G)) is plusmn 03 according the specifications of the
flow meter (Proteus Industries 08004SN1) The uncertainty of the inlet and outlet temperature of
cooling water (σ(Ti) and σ(To)) are both plusmn 015 K according the specifications of the
thermocouples (Omega T-type) The uncertainty of heat flux can then be obtained by the
measure quantities
The uncertainty of surface subcooling σ(ΔT) is determined by
(S6)
12
The uncertainty of the temperature of steam and condensing surface (σ(Tv) and σ(Tm)) are
both plusmn 015 K according the specifications of the thermocouples (Omega T-type) The
uncertainty of heat flux σ(q) is determined by equation (S5) The uncertainty of distance between
condensing surface and the bottom of thermocouple hole σ(δ) due to the mechanical processing
is plusmn 025 mm The uncertainty of the thermal conductivity of copper due to the change of
temperature in condensation experiments is plusmn 1
The uncertainty of heat transfer coefficient of condensation σ(h) is then determined by
(S7)
The uncertainty of heat flux σ(q) and surface subcooling σ(ΔT) are calculated by equation
(S5) and equation (S6) The derived quantities including heat flux q and surface subcooling ΔT
can be obtained by equation (S1) and equation (S3)
In the experiment all values of the temperature pressure and flow rate are collected by the
data acquisition simultaneously with the video capturing Error bars in Fig 2 indicate the
propagation of error associated with the temperatures pressure and flow rate measurements The
heat transfer error bars at small surface subcooling are largest due to the relatively low heat
fluxes measured corresponding to the small temperature difference between the outlet and outlet
of cooling water
S4 Droplet Growth and Distribution at Different Surface Subcooling
Representative droplet images at various times after the onset of condensation on both plain
(Fig S5a-c) and nanowired (Fig S5d-f) hydrophobic surfaces at different surface subcooling are
shown in Fig S5 Fig S5a-c show dropwise condensation on the plain hydrophobic surface
where discrete droplets form and grow to the sizes approaching the capillary length (~1 mm) 13
before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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S3 Data Reduction
Based on the energy balance the condensation heat flux q on the surface is calculated by
the temperature rise in the chilled water loop
q=cG (T ominusT i )
A (S1)
where c is the specific heat of cooling water (4200 Jmiddotkg-1K-1) A is the area of condensing surface
(1257 cm2) G is the mass flow rate of cooling water measured by the flow meter Ti and To are
the inlet and outlet temperatures of cooling water respectively Here the inlet temperature (Ti) of
cooling water is controlled by the chilled water tank with the fluctuation less than plusmn 01 K The
steady state of condensation is maintained for more than 40 minutes for each data point
The wall temperature (Tw) of condensing surface is calculated by assuming one-dimensional
heat conduction through the condensing block as
(S2)
where Tm is the average temperature measured by the three thermocouple in the condensing
block δ is the distance (35 mm) between the condensing surface and the measurement position
of thermocouples and k is the thermal conductivity of copper (398 Wm-1K-1 measured)
The surface subcooling ΔT is defined as the temperature difference between the steam
temperature (Tv) in condensing chamber and the wall temperature (Tw)
(S3)
Heat transfer coefficient h of condensation is defined as the ratio of heat flux to surface
subcooling which can be expressed as
11
(S4)
Table S1 summarizes the uncertainties of the measured quantities including the temperature
T the steam pressure Pv in condensing chamber the mass flow rate of cooling water G the
distance between condensing surface and the bottom of thermocouple holes δ the thermal
conductivity of condensing block k
Table S1 Parameters and uncertainties
Parameters Uncertainty (plusmn)
Temperature T (K) 015
Pressure P (kPa) 008
Flow rate G (kg) 03
Distance δ (mm) 025
Thermal conductivity k (Wm-1K-1) 1
Based on the error propagation the uncertainty of heat flux σ(q) is determined by
(S5)
The uncertainty of the mass flow rate (σ(G)) is plusmn 03 according the specifications of the
flow meter (Proteus Industries 08004SN1) The uncertainty of the inlet and outlet temperature of
cooling water (σ(Ti) and σ(To)) are both plusmn 015 K according the specifications of the
thermocouples (Omega T-type) The uncertainty of heat flux can then be obtained by the
measure quantities
The uncertainty of surface subcooling σ(ΔT) is determined by
(S6)
12
The uncertainty of the temperature of steam and condensing surface (σ(Tv) and σ(Tm)) are
both plusmn 015 K according the specifications of the thermocouples (Omega T-type) The
uncertainty of heat flux σ(q) is determined by equation (S5) The uncertainty of distance between
condensing surface and the bottom of thermocouple hole σ(δ) due to the mechanical processing
is plusmn 025 mm The uncertainty of the thermal conductivity of copper due to the change of
temperature in condensation experiments is plusmn 1
The uncertainty of heat transfer coefficient of condensation σ(h) is then determined by
(S7)
The uncertainty of heat flux σ(q) and surface subcooling σ(ΔT) are calculated by equation
(S5) and equation (S6) The derived quantities including heat flux q and surface subcooling ΔT
can be obtained by equation (S1) and equation (S3)
In the experiment all values of the temperature pressure and flow rate are collected by the
data acquisition simultaneously with the video capturing Error bars in Fig 2 indicate the
propagation of error associated with the temperatures pressure and flow rate measurements The
heat transfer error bars at small surface subcooling are largest due to the relatively low heat
fluxes measured corresponding to the small temperature difference between the outlet and outlet
of cooling water
S4 Droplet Growth and Distribution at Different Surface Subcooling
Representative droplet images at various times after the onset of condensation on both plain
(Fig S5a-c) and nanowired (Fig S5d-f) hydrophobic surfaces at different surface subcooling are
shown in Fig S5 Fig S5a-c show dropwise condensation on the plain hydrophobic surface
where discrete droplets form and grow to the sizes approaching the capillary length (~1 mm) 13
before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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(S4)
Table S1 summarizes the uncertainties of the measured quantities including the temperature
T the steam pressure Pv in condensing chamber the mass flow rate of cooling water G the
distance between condensing surface and the bottom of thermocouple holes δ the thermal
conductivity of condensing block k
Table S1 Parameters and uncertainties
Parameters Uncertainty (plusmn)
Temperature T (K) 015
Pressure P (kPa) 008
Flow rate G (kg) 03
Distance δ (mm) 025
Thermal conductivity k (Wm-1K-1) 1
Based on the error propagation the uncertainty of heat flux σ(q) is determined by
(S5)
The uncertainty of the mass flow rate (σ(G)) is plusmn 03 according the specifications of the
flow meter (Proteus Industries 08004SN1) The uncertainty of the inlet and outlet temperature of
cooling water (σ(Ti) and σ(To)) are both plusmn 015 K according the specifications of the
thermocouples (Omega T-type) The uncertainty of heat flux can then be obtained by the
measure quantities
The uncertainty of surface subcooling σ(ΔT) is determined by
(S6)
12
The uncertainty of the temperature of steam and condensing surface (σ(Tv) and σ(Tm)) are
both plusmn 015 K according the specifications of the thermocouples (Omega T-type) The
uncertainty of heat flux σ(q) is determined by equation (S5) The uncertainty of distance between
condensing surface and the bottom of thermocouple hole σ(δ) due to the mechanical processing
is plusmn 025 mm The uncertainty of the thermal conductivity of copper due to the change of
temperature in condensation experiments is plusmn 1
The uncertainty of heat transfer coefficient of condensation σ(h) is then determined by
(S7)
The uncertainty of heat flux σ(q) and surface subcooling σ(ΔT) are calculated by equation
(S5) and equation (S6) The derived quantities including heat flux q and surface subcooling ΔT
can be obtained by equation (S1) and equation (S3)
In the experiment all values of the temperature pressure and flow rate are collected by the
data acquisition simultaneously with the video capturing Error bars in Fig 2 indicate the
propagation of error associated with the temperatures pressure and flow rate measurements The
heat transfer error bars at small surface subcooling are largest due to the relatively low heat
fluxes measured corresponding to the small temperature difference between the outlet and outlet
of cooling water
S4 Droplet Growth and Distribution at Different Surface Subcooling
Representative droplet images at various times after the onset of condensation on both plain
(Fig S5a-c) and nanowired (Fig S5d-f) hydrophobic surfaces at different surface subcooling are
shown in Fig S5 Fig S5a-c show dropwise condensation on the plain hydrophobic surface
where discrete droplets form and grow to the sizes approaching the capillary length (~1 mm) 13
before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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The uncertainty of the temperature of steam and condensing surface (σ(Tv) and σ(Tm)) are
both plusmn 015 K according the specifications of the thermocouples (Omega T-type) The
uncertainty of heat flux σ(q) is determined by equation (S5) The uncertainty of distance between
condensing surface and the bottom of thermocouple hole σ(δ) due to the mechanical processing
is plusmn 025 mm The uncertainty of the thermal conductivity of copper due to the change of
temperature in condensation experiments is plusmn 1
The uncertainty of heat transfer coefficient of condensation σ(h) is then determined by
(S7)
The uncertainty of heat flux σ(q) and surface subcooling σ(ΔT) are calculated by equation
(S5) and equation (S6) The derived quantities including heat flux q and surface subcooling ΔT
can be obtained by equation (S1) and equation (S3)
In the experiment all values of the temperature pressure and flow rate are collected by the
data acquisition simultaneously with the video capturing Error bars in Fig 2 indicate the
propagation of error associated with the temperatures pressure and flow rate measurements The
heat transfer error bars at small surface subcooling are largest due to the relatively low heat
fluxes measured corresponding to the small temperature difference between the outlet and outlet
of cooling water
S4 Droplet Growth and Distribution at Different Surface Subcooling
Representative droplet images at various times after the onset of condensation on both plain
(Fig S5a-c) and nanowired (Fig S5d-f) hydrophobic surfaces at different surface subcooling are
shown in Fig S5 Fig S5a-c show dropwise condensation on the plain hydrophobic surface
where discrete droplets form and grow to the sizes approaching the capillary length (~1 mm) 13
before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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before being removed by gravity Despite that the growth rates of condensed droplets are
different at different surface subcooling the condensation mode is independent of the surface
subcooling In contrast the condensation on nanowired hydrophobic surfaces shows significant
dependence on the surface subcooling (Fig S5d-f) At low surface subcooling (ΔT = 3 K)
droplets on the nanowired hydrophobic surface are highly efficiently removed by the
coalescence-induced jumping mechanism with numerous microscale droplets populating on the
surface (Fig S5d) However as the surface subcooling is increased to 11 K the droplet
morphology transitions from highly mobile jumping droplets to the pinned wetting droplets (Fig
S5e) With the surface subcooling further increased to 28 K (Fig S5f) a large number of pinned
droplets stay on the surface until the gravity-induced removal leading to a flooding condensation
mode
14
Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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Fig S5 Sequential images of droplet growth and distribution a-c Time-lapse images of droplet growth
and distribution on the plain hydrophobic surface at the surface subcooling of 3 K 14 K and 25 K
respectively d-f Time-lapse images of droplet growth and distribution on the nanowired hydrophobic
surface (20 μm long) at the surface subcooling of 3 K 11 K and 28 K respectively Scale bar 800 μm
S5 Theoretical Model for Droplet Jumping in Different Wetting States
The above discussion clearly shows that the nucleation-induced wetting transition of
condensed droplets plays a crucial role in condensation heat transfer To gain further insight of
how the dynamic behavior of droplets is controlled by the wetting states we proposed a simple
model based on energy balance to quantify the jumping behaviors of condensed droplets in
15
different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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different wetting states As shown in Fig S6 when the mobile state droplets coalesce on
nanostructured hydrophobic surfaces at small surface subcooling (ΔT = 05 K) the merged
droplet can jump away from the surface in a process that is termed coalescence-induced droplet
jumping In general the coalesced jumping droplets caused by coalescence range from several
microns to hundreds of microns [3] Droplet jumping is a process governed by the conversion of
excess surface energy into kinetic energy when droplets coalesce
Fig S6 Coalescence-induced droplet jumping High-speed time-lapse images capturing coalescence-
induced droplet jumping Conditions Pv = 60 kPa ΔT = 05 K The red line shows trajectory of droplet
Several studies have attempted to explain the underlying physical mechanisms governing
coalescence-induced droplet jumping based on the balance of surface energy kinetic energy
and energies dissipated by viscous and surface-adhesion effects before and after coalescence [3-
7] Boreyko and Chen [4] were the first to report droplet jumping in condensation on
superhydrophobic surfaces They developed a simple capillary-inertial scaling to assess the
droplet jumping velocity by assuming that all of the surface energy of spherical droplets released
was converted into kinetic energy Wang et al [8] included the additional viscous dissipation
caused by the flow during coalescence on a superhydrophobic surface which showed that
droplet jumping can occur only for coalescence of droplets within a certain range of size
16
Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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Subsequently Lv et al [9] considered the additional surface adhesion-induced energy dissipation
caused by contact angle hysteresis in the analysis and showed that surface adhesion limits the
critical size range for jumping droplets Recently the expression for adhesion-induced energy
dissipation was modified for the coalescence of multiple droplets and the results showed that the
jumping velocity increases with the number of coalescing droplets [3] None of the previous
models includes the effect of droplet wetting states caused by surface subcooling on the droplet
dynamic behaviors Here we develop a model to investigate the coalescence of condensed
droplets with different wetting states
Accounting for viscous dissipation energy Evis and work of adhesion between liquid and
nanostructures ΔEw after releasing the droplet interfacial surface free energy ΔEs the kinetic
energy of the coalesced droplet Ek can be expressed as [4 10]
(S8)
To simplify the mathematical derivation it is assumed that the coalescence occurs between
two droplets with the same radius Here the surface free energy difference of droplet before and
after the coalescence is given by ΔEs = σlgΔAlv + σslΔAsl where A is the interfacial surface area
before and after coalescence σ is the interfacial tension and subscripts s l and v denote the
solid liquid and gas phase respectively
Considering the high aspect ratio nanowires it is assumed that the nucleation position
extends from the top to bottom of the separation with the increase of surface subcooling Here
we define the immersion factor ξ as the ratio of immersion depth hw to the height of nanowires h
as shown in Fig S7 to describe the different wetting states of droplets The immersion factor ξ is
the percentage of immersion height hw within nanostructures over the height of nanowires h ξ =
0 corresponds to a Cassie state while ξ = 1 refers to a Wenzel state
17
Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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Fig S7 Schematic illustrating droplet wetting state on nanowired hydrophobic copper surfaces
The surface free energy difference ΔEs can now be estimated as
(S9)
where r Rc σlv and θ are the radius of droplets before coalescence the radius of droplet after
coalescence surface tension and apparent contact angle of water Here f(ξ) is related to the
immersion depth of condensed droplets on nanowired hydrophobic surfaces and is obtained as
(S10)
where θw and θc are contact angle of droplets in Wenzel state and Cassie state θY is the
equilibrium contact angle on a nanowired hydrophobic surface rf and φs are the roughness factor
and the fraction of solid-liquid contact area
The viscous dissipation energy for each droplet is obtained as[8]
(S11)
where μ and ρl are the viscosity and density of condensed droplets
The work of adhesion between droplet and nanowires can be shown as [10]
(S12)
18
where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
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where Asl is obtatin by Asl = 2πr2sin2θ[ξrf + (1 - ξ)φs]
The kinetic energy of the coalescent droplet after merging tow identical droplets can then be
obtained as
(S13)
Then the jumping velocity can be found to be
(S14)
The jumping velocity obtained from equation (S14) can quantitatively describe the dynamic
behavior of condensed droplets in different wetting states on nanowired hydrophobic surfaces
When a kinetic energy (jumping velocity) is larger than zero droplets can jump from the
condensing surface after coalescence Otherwise droplet remains on the surface after
coalescence until growing to a size that is removed by gravity Fig S8a shows that the droplet
jumping velocity increases to a maximum and then decreases with the increase of droplet size for
various immersion depth A too small or too large droplet cannot jump because of viscous
dissipation and the work needed to acount for adhesion With the increase of immersion depth
from 0 nm to 600 nm the maximum of droplet jumping velocity decreases from 027 to 002 ms-1
and the smallest size of droplets that can jump increases from 10 μm to 230 μm More
importantly when the immersion depth is larger than 600 nm the released surface energy after
the coalescence of two droplets is not sufficient to overcome the dissipative forces resulting in
no jumping Fig S8b shows that the available jumping velocity of coalesced droplets is inversely
19
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
![Page 20: ars.els-cdn.com€¦ · Web view4 School of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, P. R. China 5Advanced Li-ion Batteries Engineering](https://reader030.fdocuments.in/reader030/viewer/2022040623/5d4db2ae88c993af7b8b7ab3/html5/thumbnails/20.jpg)
proportional to the degree of droplet immersion and the size range of available jumping droplets
increases with the increase of immersion factor Based on the assumption that the nucleation
position extends from the top to bottom of the separation with the increase of surface subcooling
a smaller immersion factor can be achieved at small surface subcooling which facilitates droplet
jumping Large surface subcooling could lead to the increase of immersion factor as well as the
decreased droplet jumping velocity which results the failure of of nanowired hydrophobic
surfaces
Fig S8 Dynamic behavior of condensed droplets in different wetting states a Droplet jumping velocity
as a function of droplet size with different immersion depth b Available velocity range and size range of
jumping droplets as a function of immersion factor for condensed droplets
20
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21
![Page 21: ars.els-cdn.com€¦ · Web view4 School of Mechanical and Electronic Engineering, Nanjing Forestry University, Nanjing 210037, P. R. China 5Advanced Li-ion Batteries Engineering](https://reader030.fdocuments.in/reader030/viewer/2022040623/5d4db2ae88c993af7b8b7ab3/html5/thumbnails/21.jpg)
References
[1] M Tian W Wang Y Wei R Yang J Power Sources 211 (2012) 46-51[2] MD Porter TB Bright DL Allara CED Chidsey J Am Chem Soc 109 (1987) 3559-3568[3] X Chen RS Patel JA Weibel SV Garimella Sci Rep 6 (2016) 18649[4] JB Boreyko CH Chen Phys Rev Lett 103 (2009) 184501[5] KM Wisdom JA Watson XP Qu FJ Liu GS Watson CH Chen Natl Acad Sci USA 110 (2013) 7992-7997[6] R Enright N Miljkovic J Sprittles K Nolan R Mitchell EN Wang ACS Nano 8 (2014) 10352-10362[7] BL Peng SF Wang Z Lan W Xu RF Wen XH Ma Appl Phys Lett 102 (2013) 151601[8] FC Wang FQ Yang YP Zhao Appl Phys Lett 98 (2011) 053112[9] C Lv P Hao Z Yao Y Song X Zhang F He Appl Phys Lett 103 (2013) 021601[10] GQ Li MH Alhosani SJ Yuan HR Liu A Al Ghaferi TJ Zhang Langmuir 30 (2014) 14498-14511
21