# Highly Anistoropic Thermal Conductivity of Arsenene: An  ab initio Study

M. Zeraatia S. M. Vaez Allaeia,b I. Abdolhosseini Sarsaric M. Pourfathd,e D.
Donadiof,g,h,i a) Department of Physics, University of Tehran, Tehran
14395-547, Iran
b)School of Physics, Institute for Research in Fundamental Sciences (IPM),
Tehran 19395-5531, Iran
c)Department of Physics, Isfahan University of Technology, Isfahan,
84156-83111, Iran
d)School of Electrical and Computer Engineering, University of Tehran, Tehran
14395-515, Iran
e)Institute for Microelectronics, TU Wien, Gußhausstraße 27–29/E360, 1040
Vienna, Austria
f)Department of Chemistry, University of California Davis, One Shields Ave.
Davis, CA, 95616
g)Max Planck Institut für Polymerforschung, Ackermannweg 10, D-55128 Mainz,
Germany
h)Donostia International Physics Center, Paseo Manuel de Lardizabal, 4, 20018
Donostia-San Sebastian, Spain
i)IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao, Spain

###### Abstract

Elemental 2D materials exhibit intriguing heat transport and phononic
properties. Here we have investigated the lattice thermal conductivity of
newly proposed arsenene, the 2D honeycomb structure of arsenic, using ab
initio calculations. Solving the Boltzmann transport equation for phonons, we
predict a highly anisotropic thermal conductivity, of $30.4$ and $7.8$ W/mK
along the zigzag and armchair directions, respectively at room temperature.
Our calculations reveal that phonons with mean free paths between $20$ nm and
$1$ $\mu$m provide the main contribution to the large thermal conductivity in
the zig-zag direction, mean free paths of phonons contributing to heat
transport in the armchair directions range between $20$ and $100$ nm. The
obtained low and anisotropic thermal conductivity, and feasibility of
synthesis, in addition to other reports on high electron mobility, make
arsenene a promising material for a variety of applications, including thermal
management and thermoelectric devices.

The discovery of graphene as a stable atomically thin material has led to
extensive investigation of similar 2D systems. Its properties such as high
electron mobility 1, and very high thermal conductivity 2; 3; 4; 5 make
graphene very appealing for applications in electronics, for packaging and
thermal management 6; 7; 8; 9; 10; 11. The successful isolation of single-
layer graphene fostered the search for further ultra-thin 2D structures, such
as silicene, germanene, phosphorene, and transition metal dichalcogenides,
e.g. MoS2 and WS2 12; 13. These materials are now considered for various
practical usages due to their distinguished properties stemming from their low
dimensionality. Thermal transport in 2D materials has recently attracted the
attention of the scientific community, as anomalous heat conduction has been
predicted to occur in systems with reduced dimensionality 14. Phononic
properties and thermal conductivity vary significantly from one 2D system to
another 15; 16; 17; 18. For example, silicene has a buckled structure and a
lower thermal conductivity 19; 20 compared to graphene 21; 22; 12.

2D structures of arsenic and phosphorous have been recently investigated 23;
24; 25; 26; 27. Arsenic and phosphorus are in the 5th group of the periodic
table and both have different allotropes. Black phosphorus is a layered
allotrope of phosphorus similar to graphite, and the stability of its single
layer form, named phosphorene has been probed both theoretically and
experimentally 13; 28. Gray arsenic is one of the most stable allotropes of
arsenic with a buckled layered structure 27; 29. In addition, arsenic has an
orthorhombic phase (puckered) similar to black phosphorus 23; 25; 26, and its
monolayer is called arsenene (see Fig. 1). Experimental observations have
shown that gray arsenic undergoes a structural phase transition to the
orthorhombic precursor of arsenene at temperatures of about $T=370$ K 30. As a
monolayer arsenene has a direct band gap as opposed to the multilayer
allotrope, which exhibits an indirect band gap of the order of $1$ eV 26.
According to our calculations, monolayer of arsenene is stable also near zero
temperature, in agreement with previous reports 23; 25; 26. Here we
investigate heat conduction in arsenene and we elucidate the anisotropy of its
thermal conductivity. Since in semiconductors phonons are the predominant heat
carriers, we use first-principles anharmonic lattice dynamics calculations and
the Boltzmann transport equation to compute phonon dispersion relations and
thermal conductivity.

We employ density functional theory as implemented in the VASP code 31 with
projector-augmented-wave (PAW) 32 pseudopotentials and the Perdew-Burke-
Ernzerhof (PBE) generalized gradient functional 33 exchange-correlation
functional. Integration over Brillouin zone is performed using $15\times
15\times 1$ $\Gamma$-center Monkhorst-Pack mesh of k-points 34 and plane waves
up to an energy cutoff of $360$ eV are used as basis set to represent the
Kohn-Sham wave functions. The simulation cell is built with a vacuum layer of
$10$ Å, which is sufficient to avoid interactions between periodic images. The
structural relaxations were performed by using the conjugate-gradient (CG)
algorithm with a convergence criterion of $10^{-6}$ eV/Å for the maximum
residual force component per atom.

Figure 1: Top and side view of puckered arsenene.

As shown in Fig. 1, arsenene has a puckered structure arranged in a
rectangular unit cell with lattice vectors $a=3.686$ Å and $b=4.769$ Å. The
bond length between atoms in the same plane (upper panel or lower) is $2.510$
Å, while the bonding length between atoms in different planes is $2.496$ Å.
The distance between the two planes is denoted by $\Delta$ and is $2.402$ Å.
Our structural parameters are compared with previous studies in Table 1.

Table 1:  The lattice parameters of arsenene. $a$ and $b$ are the lattice
vectors, $\Delta$ is the puckering distance, $c_{1}$ is the bond length
between atoms that are in the same plane, $c_{2}$ is the bond length between
atoms that are in different planes. Mono-layer of arsenene
---
Method |  $a$ (Å) |  $b$ (Å) |  $\Delta$ (Å) |  $c_{1}$ (Å) |  $c_{2}$ (Å)
PBE |  $3.686$ | $4.769$ | $2.402$ | $2.510$ | $2.496$
PBE (Ref. 23) |  $3.677$ | $4.765$ | – | $2.521$ | $2.485$
revPBE-vdW (Ref. 26) |  $3.68$ | $4.80$ | – | $2.51$ | $2.49$

We have utilized the PHONOPY package 35 interfaced with VASP to compute inter-
atomic force constants (IFCs). After testing convergence of the phonon
frequencies as a function of the size of the supercell, a $8\times 8\times 1$
super-cell has been adopted, with a $3\times 3\times 1$ $\Gamma$-center
Monkhorst-Pack mesh of k-points, to build the dynamical matrix $\mathcal{D}$.
The frequencies of the vibrational modes are obtained by diagonalizing
$\mathcal{D}$, and the phonon dispersion curves are represented in Fig. 2. The
absence of phonon branches with negative phonon frequencies indicates the
stability of the system at zero temperature.

Figure 2: (a) Phonon dispersion curves along the high-symmetry directions of
the first Brillouin zone for puckered arsenene.

The phonon bands exhibit two acoustic modes, a longitudinal one (LA) and an
in-plane transverse one (TA), with linear dispersion ($\omega\propto q$) for
$q~{}\rightarrow 0$, and an out-of-plane flexure mode (ZA) with quadratic
dispersion in the long wavelength limit, which is a general feature of 2D
membranes 36 and have been characterized in graphene 37; 15, silicene 38; 16,
hBN 39, MoS2 40 and ultra thin silicon membranes 41; 42. Flexural modes have a
major role in contributing to the thermal conductivity of graphene, both as
carriers 15 and as scatterers 43. In addition, we observe a gap of about $2$
THz between the low-frequency and high-frequency optical branches.

Figure 3: The thermal conductivity of arsenene as a function of temperature
along the zigzag and armchair direction.

We have computed the thermal conductivity of arsenene by solving the phonon
Boltzmann transport equation (BTE) with an iterative self-consistent algorithm
44; 45, using the ShengBTE package 46. In our calculations we have considered
third-order interatomic force constants up to the forth shell of neighbors in
a $5\times 5\times 1$ super-cell with a $3\times 3\times 1$, $\Gamma$-centered
Monkhorst-Pack mesh of k-points 46; 47. Intrinsic three phonon scattering
processes have been considered in the calculation of phonon relaxation times:

$\frac{1}{\tau^{0}_{\lambda}}=\frac{1}{N}\left(\sum_{\lambda^{\prime}\lambda^{\prime\prime}}^{+}\Gamma^{+}_{\lambda\lambda^{\prime}\lambda^{\prime\prime}}+\frac{1}{2}\sum_{\lambda^{\prime}\lambda^{\prime\prime}}^{-}\Gamma^{-}_{\lambda\lambda^{\prime}\lambda^{\prime\prime}}+\sum_{\lambda^{\prime}}\Gamma^{\mathrm{ext}}_{\lambda\lambda^{\prime}}\right)\
,$ (1)

where $\Gamma^{+}_{\lambda\lambda^{\prime}\lambda^{\prime\prime}}$ and
$\Gamma^{-}_{\lambda\lambda^{\prime}\lambda^{\prime\prime}}$ represent the
scattering rates due to absorbing and emitting three-phonon processes,
respectively. The extrinsic scattering term
$\Gamma^{\mathrm{ext}}_{\lambda\lambda^{\prime}}$ is zero in our calculations,
since arsenic does not present isotopic disorder, and neither defect or
boundary scattering processes were considered. Knowing the distribution
function, the phononic thermal conductivity tensor ($\kappa$) is expressed as:

$\kappa_{\alpha\beta}=\frac{1}{V}\sum_{\lambda}\hbar\omega_{\lambda}\frac{\partial
f}{\partial T}v^{\alpha}_{\lambda}v^{\beta}_{\lambda}\tau_{\lambda}\ ,$ (2)

where $f$ is the phonon distribution function, $\omega_{\lambda}$ is the
phonon angular frequency, $v^{\alpha}_{\lambda}$ is the velocity component
along the $\alpha$-direction, and $\tau_{\lambda}$ is the phonon relaxation
time.

The two diagonal components of the thermal conductivity at temperatures
between $20$ and $500$ K are reported in Fig. 3. The thermal conductivity
along the zigzag and armchair direction at $T=300$ K is $30.4$ and $7.8$
Wm-1K-1, respectively. The nearly 3-fold ratio between the two components
indicates a high anisotropy. The thermal conductivity of aresenene in
comparison to graphene and hexagonal boron nitride is very low: for example
both experimental and theoretical studies indicate the intrinsic thermal
conductivity of graphene is larger than $3000$ W/mK 2; 3. Due to the larger
atomic weight of As, $\kappa$ of arsenene is also lower than that of
phosphorene, which is also strongly anisotropic, theoretically predicted as
$110$ and $36$ Wm-1K-1 in the zigzag and armchair directions, respectively 18.

Table 2: Sound velocity (group velocity) of acoustic modes of arsenene close
to the zone center ($\Gamma$-point). (*) Due to quadratic form of ZA mode near
$\Gamma$ point, the magnitude has been averaged over a small reigon interval.
Sound velocity km/s
---
| Arsenene | Phosphorene
| ZA (*) | TA | LA | LA (ref. 48)
Armchair | $0.72$ |  $2.41$ |  $2.44$ |  $4.17$
Zigzag | $0.34$ | $2.41$ | $4.77$ | $7.73$

We note indeed that the group velocity of the acoustic modes of arsenene along
both zigzag and armchair directions near the zone center are about twice as
smaller as those of phosphorene (see Table 2). Since according to Equation 2
group velocities contribute to $\kappa$ quadratically, we can argue that the
smaller $v_{\lambda}$ is responsible for the difference between the thermal
conductivities of arsenene and phosphorene.

Figure 4: Normalized contribution of each phonon branch to the total thermal
conductivity, as a function of temperature along the (a) zigzag and (b)
armchair directions.

The contribution of each phonon mode to the total thermal conductivity as a
function of temperature is shown in Fig. 4. The ZA mode provides the largest
contribution to thermal conductivity along both directions at low
temperatures. However, this contribution decreases as the temperature
increases and reaches $7.8$% (armchair direction) for temperatures above $300$
K. This is in contrast to the case of graphene where the ZA modes provide the
main contribution to heat conduction 15. Due to the special selection rules of
phonon scattering in graphene, the ZA mode has a relatively large relaxation
time compared to other phonon modes 15, while due to the puckered structure of
arsenene, similar to silicene and phosphorene 16; 18, the hexagonal symmetry
of graphene is broken, resulting in larger scattering rates and shorter
relaxation times of the ZA modes at high temperatures. The contribution of the
ZA modes remains however substantial and roughly equivalent to that of the TA
modes in the zigzag direction, while it drops below 10$\%$ in the armchair
direction.

At temperatures higher than $100$ K, the contribution of LA modes to the
thermal conductivity is the largest in both directions. The group velocity of
LA mode near the zone center is the largest along the zigzag direction, while
it is comparable to the TA mode along the armchair direction (see Table 2).
Furthermore, the relaxation times of the LA modes at low frequencies, which
play a significant role in the process of heat conduction, are larger than
that of other acoustic modes (see Fig. 5). All these effects combined together
result in a larger contribution of LA modes to the thermal conductivity at
high temperatures.

Figure 5: Total relaxation time of phonon modes as a function of frequency
$\nu$ along the (a) zigzag and (b) armchair direction.

Finally we computed the cumulative thermal conductivity as a function of
phonon mean free path, defined as

$\kappa_{l}(\Lambda\leqslant\Lambda_{\mathrm{max}})=\frac{\kappa_{l}}{1+\frac{\Lambda_{0}}{\Lambda_{max}}},$
(3)

where $\Lambda_{\mathrm{max}}$ is the maximum mean free path and $\Lambda_{0}$
is a fitting parameter that can be interpreted as the mean free path of heat-
carrying phonons of infinite system size. By fitting Eq.(3) to the calculated
values one can evaluate $\Lambda_{0}$, see Fig. 6. This quantity shows which
mean free paths contribute the most to the thermal conductivity. Phonon modes
with mean free paths larger than $59.5$ nm along the armchair direction and
$91.8$ nm along the zigzag direction have 50% contribution to the total
thermal conductivity. Hence we can argue that in a polycrystalline structure
with a grain size of the order of $50$ nm the thermal conductivity can be
significantly further reduced. However, given the broad range of mean free
paths that contribute, the most efficient approach to abate $\kappa$ would be
hierarchical nanostructuring 49.

Figure 6: Cumulative thermal conductivity of arsenene as a function of the
phonon mean free path at $T=300$ K along the zigzag and armchair directions.

In summary we have theoretically investigated the phonon properties and the
thermal conductivity of arsenene by ab initio anharmonic lattice dynamics. Our
calculations evidence the lowest thermal conductivity among the 2D elemental
materials found so far. The larger mass of arsenic atoms compared to
phosphorus leads to a 3 times smaller thermal conductivity of this material
compared to phosphorene. Similar to phosphorene, the thermal conductivity of
arsene is strongly anisotropic. Such anisotropy could be utilized in thermal
engineering application, to establish directional heat transport at the
nanoscale. Unlike graphene, the longitudinal acoustic phonon modes provide the
main contribution to the thermal conductivity of arsenene at temperatures
above $100$ K, which is mainly due to the puckered structure of this material.
From this observation, we expect that $\kappa$ of arsenene would be less
affected by the interaction with a substrate than in those materials for which
the main contribution comes from flexural modes. In addition, given the high
electron mobility and the reduced dimensionality, which should enhance the
Seebeck coefficient, its relatively low and strongly directional thermal
conductivity makes arsenene a good candidate from thermoelectric energy
conversion.

The work of S.M.V.A. was supported in part by the Research Council of the
University of Tehran.

## References

  * Novoselov et al. (2005) K. S. Novoselov, a. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and a. a. Firsov, Nature 438, 197 (2005), ISSN 1476-4687.
  * Xu et al. (2014) X. Xu, L. F. C. Pereira, Y. Wang, J. Wu, K. Zhang, X. Zhao, S. Bae, C. Tinh Bui, R. Xie, J. T. L. Thong, et al., Nat Commun 5 (2014).
  * Fugallo et al. (2014) G. Fugallo, A. Cepellotti, L. Paulatto, M. Lazzeri, N. Marzari, and F. Mauri, Nano letters (2014), ISSN 1530-6992.
  * Lindsay et al. (2014) L. Lindsay, W. Li, J. Carrete, N. Mingo, D. A. Broido, and T. L. Reinecke, Phys Rev B 89, 155426 (2014).
  * Cepellotti et al. (2015) A. Cepellotti, G. Fugallo, L. Paulatto, M. Lazzeri, F. Mauri, and N. Marzari, Nature Communications 6, 6400 (2015).
  * Wei et al. (2009) P. Wei, W. Bao, Y. Pu, C. N. Lau, and J. Shi, Phys. Rev. Lett. 102, 166808 (2009), ISSN 0031-9007.
  * Zuev et al. (2009) Y. M. Zuev, W. Chang, and P. Kim, Phys. Rev. Lett. 102, 096807 (2009), ISSN 0031-9007.
  * Wang et al. (2011) C.-R. Wang, W.-S. Lu, L. Hao, W.-L. Lee, T.-K. Lee, F. Lin, I.-C. Cheng, and J.-Z. Chen, Phys. Rev. Lett. 107, 186602 (2011), ISSN 0031-9007.
  * Rajabpour et al. (2011) A. Rajabpour, S. M. Vaez Allaei, and F. Kowsary, Appl. Phys. Lett. 99, 2011 (2011), ISSN 00036951.
  * Rajabpour and Vaez Allaei (2012) a. Rajabpour and S. M. Vaez Allaei, Appl. Phys. Lett. 101, 053115 (2012), ISSN 00036951\.
  * Yan et al. (2012) Z. Yan, G. Liu, J. M. Khan, and A. A. Balandin, Nature Communications 3, 827 (2012).
  * Geim and Grigorieva (2013) a. K. Geim and I. V. Grigorieva, Nature 499, 419 (2013), ISSN 1476-4687.
  * Guan et al. (2014) J. Guan, Z. Zhu, and D. Tománek, Phys. Rev. Lett. 113, 046804 (2014), ISSN 0031-9007.
  * Lepri et al. (2003) S. Lepri, R. Livi, and A. Politi, Phys Rep 377, 1 (2003).
  * Lindsay et al. (2010) L. Lindsay, D. a. Broido, and N. Mingo, Phys. Rev. B 82, 115427 (2010), ISSN 1098-0121.
  * Xie et al. (2014a) H. Xie, M. Hu, and H. Bao, Appl. Phys. Lett. 104, 131906 (2014a), ISSN 0003-6951.
  * Li et al. (2013) W. Li, J. Carrete, and N. Mingo, Appl. Phys. Lett. 103, 253103 (2013), ISSN 00036951\.
  * Jain and McGaughey (2015a) A. Jain and A. J. H. McGaughey, Scientific reports 5, 8501 (2015a), ISSN 2045-2322.
  * Xie et al. (2014b) H. Xie, M. Hu, and H. Bao, Appl. Phys. Lett. 104, 131906 (2014b), ISSN 0003-6951.
  * Yang et al. (2014) K. Yang, S. Cahangirov, A. Cantarero, A. Rubio, and R. D’Agosta, Phys. Rev. B 89, 125403 (2014), ISSN 1098-0121.
  * Chen et al. (2012) L. Chen, C.-C. Liu, B. Feng, X. He, P. Cheng, Z. Ding, S. Meng, Y. Yao, and K. Wu, Physical review letters 109, 56804 (2012).
  * Vogt et al. (2012) P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis, M. C. Asensio, A. Resta, B. Ealet, and G. Le Lay, Phys. Rev. Lett. 108, 155501 (2012), ISSN 0031-9007.
  * Kamal and Ezawa (2015) C. Kamal and M. Ezawa, Phys. Rev. B 91, 85423 (2015).
  * Zhang et al. (2015a) S. Zhang, Z. Yan, Y. Li, Z. Chen, and H. Zeng, Angewandte Chemie 127, 3155 (2015a), ISSN 00448249\.
  * Zhang et al. (2015b) Z. Y. Zhang, H. N. Cao, J. C. Zhang, Y. H. Wang, D. S. Xue, and M. S. Si (2015b), eprint 1501.06044.
  * Zhang et al. (2014) Z. Y. Zhang, J. Xie, D. Z. Yang, D. S. Xue, M. S. Si, and W. Ji, 1 (2014), eprint 1411.3165.
  * Zhu et al. (2014) Z. Zhu, J. Guan, and D. Tomanek, p. 5 (2014), eprint 1410.6371.
  * Liu et al. (2014) H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, ACS nano 8, 4033 (2014), ISSN 1936-086X.
  * Madelung (2004) O. Madelung, _Semiconductors: Data Handbook_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004), ISBN 978-3-642-62332-5.
  * Krebs et al. (1957) H. Krebs, W. Holz, and K. H. Worms, Zeitschrift für Naturforschung B 12 (1957), ISSN 1865-7117.
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
  * Blöchl (1994) P. E. Blöchl, Phys. Rev. B 50, 17953 (1994), ISSN 0163-1829.
  * Perdew et al. (1996) J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
  * Monkhorst and Pack (1976) H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976).
  * Togo et al. (2008) A. Togo, F. Oba, and I. Tanaka, Phys. Rev. B 78, 134106 (2008).
  * Zabel (2001) H. Zabel, Journal of Physics: Condensed Matter 13, 7679 (2001), ISSN 0953-8984.
  * Jiang et al. (2015) J.-W. Jiang, B.-S. Wang, J.-S. Wang, and H. S. Park, Journal of physics. Condensed matter : an Institute of Physics journal 27, 083001 (2015).
  * Cahangirov et al. (2009) S. Cahangirov, M. Topsakal, E. Aktürk, H. Şahin, and S. Ciraci, Phys. Rev. Lett. 102, 236804 (2009), ISSN 0031-9007.
  * Lindsay and Broido (2011) L. Lindsay and D. a. Broido, Phys. Rev. B 84, 155421 (2011), ISSN 1098-0121.
  * Cai et al. (2014) Y. Cai, J. Lan, G. Zhang, and Y.-W. Zhang, Phys. Rev. B 89, 035438 (2014), ISSN 1098-0121.
  * Neogi and Donadio (2015) S. Neogi and D. Donadio, The European Physical Journal B 88, 73 (2015).
  * Neogi et al. (2015) S. Neogi, J. S. Reparaz, L. F. C. Pereira, B. Graczykowski, M. Sledzinska, A. Shchepetov, M. Prunnila, J. Ahopelto, C. M. Sotomayor-Torres, and D. Donadio, ACS Nano 9, 3820 (2015).
  * Pereira and Donadio (2013) L. F. C. Pereira and D. Donadio, Phys. Rev. B 87, 125424 (2013), URL http://link.aps.org/doi/10.1103/PhysRevB.87.125424.
  * Peierls (1929) R. Peierls, Ann. Phys 3, 1055 (1929).
  * Omini and Sparavigna (1996) M. Omini and A. Sparavigna, Phys. Rev. B 53, 9064 (1996).
  * Li et al. (2014) W. Li, J. Carrete, N. Katcho, and N. Mingo, Computer Physics Communications 185, 1747 (2014), ISSN 00104655.
  * Esfarjani and Stokes (2008) K. Esfarjani and H. T. Stokes, Phys. Rev. B 77, 144112 (2008), ISSN 1098-0121, URL http://link.aps.org/doi/10.1103/PhysRevB.77.144112.
  * Jain and McGaughey (2015b) A. Jain and A. J. H. McGaughey, Scientific reports 5, 8501 (2015b), ISSN 2045-2322, URL http://www.ncbi.nlm.nih.gov/pubmed/25686917.
  * Biswas et al. (2012) K. Biswas, J. He, I. D. Blum, C.-I. Wu, T. P. Hogan, D. N. Seidman, V. P. Dravid, and M. G. Kanatzidis, Nature 489, 414 (2012).


# Atmospheric water vapor condensation on engineered interfaces: Busting the
myths

Tibin M. Thomas
Department of Mechanical Engineering
Indian Institute of Technology Madras
Chennai 600036, India
<EMAIL_ADDRESS>
&Pallab Sinha Mahapatra
Department of Mechanical Engineering
Indian Institute of Technology Madras
Chennai 600036, India
<EMAIL_ADDRESS>
Ranjan Ganguly
Department of Power Engineering
Jadavpur University
Kolkata 700106, India
<EMAIL_ADDRESS>
&Manish K. Tiwari
Nanoengineered Systems Laboratory;
Wellcome/EPSRC Centre for
Interventional and Surgical Sciences
UCL, UK
<EMAIL_ADDRESS>
###### Abstract

Condensing atmospheric water vapor on surfaces is a sustainable approach to
potentially address the potable water crisis. However, despite extensive
research, a key question remains: what is the physical mechanism governing the
condensation from humid air and how significantly does it differ from pure
steam condensation? The answer may help define an optimal combination of the
mode and mechanism of condensation as well as the surface wettability for best
possible water harvesting efficacy. Here we show that this lack of clarity is
due to the differences in heat transfer characteristics during condensation
from pure vapor and humid air environments. Specifically, during condensation
from humid air, the thermal resistance across the condensate is non-dominant
and the energy transfer is controlled by vapor diffusion and condensate
drainage. This leads to filmwise condensation on superhydrophilic surfaces,
offering the highest water collection efficiency. To demonstrate this, we
measured condensation rate on different sets of superhydrophilic and
superhydrophobic surfaces in a wide degree of subcooling (10 - 26
${}^{\circ}C$) and humidity-ratio differences (5 - 45 g/kg of dry air). The
resulting condensation rate is enhanced by 57 - 333 % on the superhydrophilic
surfaces as compared to the superhydrophobic ones. The findings of this study
challenges the nearly century-old scientific ambiguity about the mechanism of
vapor condensation from humid air. Our findings will lead to the design of
efficient atmospheric water harvesting systems.

_Keywords_ Atmospheric water harvesting $\cdot$ dropwise and filmwise
condensation $\cdot$ humid air $\cdot$ hierarchical surface $\cdot$
superhydrophobic and superhydrophilic surfaces

##  1 INTRODUCTION

The dwindling reserve of potable water on the planet as a result of
industrialization, climate change, and environmental pollution is a major
challenge in the coming decades. Recently, it has been estimated that half a
billion of the global population is facing severe water scarcity throughout
the year, while about four billion people are facing the same for at least one
month of the year [1]. Capturing water vapor from the humid air by
condensation is a sustainable water-production technology, particularly in
arid environments. [2, 3, 4]. Condensing water from the atmosphere is an
attractive means to address this potable water shortage. In the earth’s
atmosphere, the mass fraction of water vapor is always below 5 %, with more
than 95 % being non-condensable gases (NCG). Condensation of the vapor
requires a subcooled surface for both pure vapor and vapor in humid air. It
begins with the formation of liquid nuclei on the surface once its temperature
falls below the saturation temperature. The mechanism of vapor condensation in
the presence of NCG is distinct than pure vapor condensation due to the
formation of thermal and concentration boundary layers outside the liquid-
vapor interface, which are not observed during pure vapor condensation. Along
with the growth of liquid nuclei, the NCG molecules present in the humid air
are accumulated over the interface of the nuclei and form an NCG diffusion
layer [5, 6]. This layer acts as a barrier to water vapor diffusion from the
bulk humid air to the interface, and even a small quantity of NCG (mass
fraction of around 0.5 %) can reduce heat transfer rates by 50 % when compared
to a pure vapor environment [7, 8]. Furthermore, depending on the water vapour
concentration, the condensation heat transfer rate in humid air can be 2 to 3
orders of magnitude lower than that in pure vapour [9, 10]. Prior research
over the past century has shown that the condensation heat transfer rate of
the dropwise condensation (DWC) mode can be one order of magnitude higher than
the filmwise condensation (FWC) mode under pure vapor conditions [11, 12, 13].
However, it remains unclear which mode of condensation is better for water
harvesting from humid air, despite decades of research.

Ref |  | Substrate
---
type
| Surface
---
morphology
| Environmental
---
condition
| Subcooling
---
$\Delta T$ (${}^{\circ}C$)
| Difference in
---
humidity ratio
$\Delta\omega$ ($g/kg$)
| Percentage
---
change in
FWC mass flux
with DWC (%)
[14] |  Silicon substrate |  $TiO_{2}$ nanorods | T=25 ${}^{\circ}C$, RH=93.5 % | 20.87 | 14.06 | +109
[15] |  Aluminum substrate |  Hierarchical $Al_{2}O_{3}$ | T=20 ${}^{\circ}C$, RH=80 % | 14.15 | 7.24 | +50
[15] |  Aluminum substrate |  Hierarchical $Al_{2}O_{3}$ | T=35 ${}^{\circ}C$, RH=80 % | 8.6 | 11.84 | +37.5
[16] |  Copper tube |  $CuO$ nanostructures | T=40 ${}^{\circ}C$, RH=80 % | 32.5 | 33.69 | +4.5
[17] |  Copper foil |  $CuO$ nanostructures | T=26 ${}^{\circ}C$, RH=50 % | 12.8 | 6.13 | +26.4
[18] |  Steel tube |  Carbon nanotubes | T=40 ${}^{\circ}C$, RH=80 % | 31.9 | 33.48 | -29.4
[19] |  Steel tube |  Carbon nanotubes | T=30 ${}^{\circ}C$, RH=80 % | 14.2 | 12.85 | -34.4
[20] |  Aluminum substrate |  Hierarchical $Al_{2}O_{3}$ | T=24.5 ${}^{\circ}C$, RH=70 % | 13.7 | 8.11 | +21
[20] |  Aluminum substrate |  Hierarchical $Al_{2}O_{3}$ | T=24.5 ${}^{\circ}C$, RH=80 % | 15.8 | 10.08 | -1.2
[20] |  Aluminum substrate |  Hierarchical $Al_{2}O_{3}$ | T=24.5 ${}^{\circ}C$, RH=90 % | 17.7 | 12.07 | -27.8
[21] |  Aluminum substrate |  Hierarchical $Al_{2}O_{3}$ | T=14 ${}^{\circ}C$, RH=100 % | 9 | 4.57 | +78.5
[22] |  Silicon substrate |  Rough nanostructures | T=25 ${}^{\circ}C$, RH=92.5 % | 13.69 | 10.91 | -12.1
[23] |  Aluminum substrate |  Hierarchical $Al_{2}O_{3}$ | T=23.6 ${}^{\circ}C$, RH=50 % | 8 | 5.25 | +8.9

Table 1: Summary of previous studies reporting the overall mass flux during
condensation on sub-cooled superhydrophobic and superhydrophilic surfaces in
vertical orientation. The reported results are too dispersed and cannot be
correlated with operating conditions such as degree of subcooling or humidity
ratio difference. The influence of several experimental uncertainties, such as
dynamic changes in relative humidity within the condensation chamber,
perturbation or velocity of the condensing fluid, edge effects of the
condensing surface, surface aspect ratio, and so on, could be implicated for
the inconclusive trend. The compiled data reveals a lack of understanding
regarding the overall impact of wettability modification on water yield during
humid air condensation.

Over the last two decades, advances in surface chemistry and micro/nano
fabrication techniques have inspired researchers to custom-engineer the
wettability of condensing surface for better condensation effectiveness, which
is required for water-energy nexus applications [24, 25, 26, 27, 28, 29, 30,
31]. Dropwise condensation on a superhydrophobic surface has been proposed for
water harvesting in several publications due to the early departure or
enhanced mobility of condensing droplets, allowing greater access to the dry
condensing area for continuous nucleation. In addition, filmwise condensation
on superhydrophilic surfaces has also been proposed for water harvesting.
Table 1 summarises the research articles that reported the overall
condensation rate on the non-wetting superhydrophobic and wetting
superhydrophilic surfaces in the atmospheric humid air conditions. Clearly,
the majority of the works cited in Table 1 report that filmwise mode is more
effective than dropwise mode for humidity harvesting. However, these
observations did not receive comprehensive attention and, as a consequence,
has remained a mystery. This result is contradictory to the general perception
about the dropwise and filmwise modes of condensation under pure vapor
conditions, but a plausible reason has not yet been addressed. Furthermore,
because these investigations are limited to a narrow range of ambient
conditions, making a firm conclusion about the most effective mode of
condensation in humid air is difficult from Table 1.

To bridge this gap, we performed the experiments on two superhydrophobic and
two superhydrophilic surfaces under humid air conditions in a much more
comprehensive range of degrees of subcooling (10 - 26 ${}^{\circ}C$) and
humidity ratio difference (5 - 45 g/kg of dry air). Wherein, the humidity
ratio difference ($\Delta\omega$) is defined as the difference between the
specific humidity of the bulk humid air and the saturated vapor at the
substrate temperature conditions. We observed that, depending on the specific
combination of the experimental conditions investigated, filmwise mode on
superhydrophilic surfaces can yield 57-333 % greater condensate yield than the
dropwise mode on superhydrophobic surfaces. To substantiate these
observations, a detailed assessment has been performed using basic
condensation heat transfer theory for computing the thermal resistance offered
by coating layer, condensate, liquid-vapor interface, diffusion layer, etc.
[32, 33]. This shows that in the presence of NCGs (specifically above 95 %),
the thermal resistance across the condensate and coating layer becomes
negligible, and vapor diffusion becomes the dominant mechanism. We also
investigated the different rivulet structures of the condensate forming on
superhydrophilic surfaces due to the surface morphology, spreading dynamics,
roughness, etc., and provide first insights into how they influence the
condensation rate in a humid air environment. Surprisingly enough, the
superhydrophilic surface with vertical rivulet channels outperformed those
with branched rivulet channels by 10-20 %. The current study also sheds light
on the previously unknown benefits of the filmwise mode in humid air, which
have far-reaching implications for developing efficient engineered interfaces
for humidity harvesting.

## Results

### Surface morphology

To investigate the condensation performance on completely wetting and non-
wetting surfaces under humid air conditions, hierarchical surfaces of copper
and aluminum were fabricated. Figures 1(A and B) show the surface morphology
of the micro/nano textured copper and aluminum substrates, both of which
exhibit superhydrophilic nature. The hierarchically rough substrates were
functionalized with perfluorooctyl-triethoxy silane (PFOTES) to achieve
superhydrophobicity. The surface morphology of the copper and aluminum
superhydrophobic surfaces was similar to the corresponding superhydrophilic
surfaces since the self-assembled monolayer of PFOTES does not modify the
hierarchical structures [34]. The copper substrate consists of micro-flower
shaped cupric oxide structures ranging in size from 2 - 4 $\mu m$, that are
orderly arranged on the surface to some extent (Fig. 1A) [35, 36]. The orders
of the structures can be visualised in the profilometer micrograph, as shown
in Fig. 1C. The spacing between the micro-flowers was in the range of 0.5 - 15
$\mu m$. Conversely, the surface features of the aluminum sample were random
and consisted of micro-bumps of different sizes, as shown in Figs. 1(B and D).
The height and width of these micro-bumps were in the range of 5 - 30 $\mu m$
and 5 - 60 $\mu m$, respectively. Flower-shaped features of size 0.5 - 1 $\mu
m$ are found to surround the micro-bumps (Fig. 1B). The micro-flower
structures on the copper substrate consist of thin sheets of nanostructures,
and the aluminum substrate consists of blade-shaped nanostructures.

Figure 1: Surface topography and characterization. SEM micrograph of
micro/nano textured (A) copper and (B) aluminum substrate at various
resolutions. Profilometer micrograph of (C) copper and (D) aluminum
substrates. The frequency distribution graph depicts the number of peaks and
valleys at each bin size. The schematic illustration of the randomly generated
micro/nano textures on (E) copper and (F) aluminum substrates. The scale bars
in contact angle images are 1 $mm$.

The actual surface topography of the copper and aluminum substrates can be
schematically represented based on the SEM and profilometer micrographs in
Figs. 1(E and F), respectively. The average roughness of the modified copper
and aluminum substrates was $R_{a}$ = 1.5 $\mu m$, $R_{a}$ = 5.3 $\mu m$,
respectively. The frequency distributions in Figs. 1(C and D) show that $\sim
99$ % of the peaks and valleys lie in the range of $\pm 4$ $\mu m$ and $\pm
15$ $\mu m$ for copper and aluminum substrates, respectively. These indicate
that the depth of the textures is higher on the aluminium substrate than on
the copper substrate. Prior to PFOTES functionalization, the contact angle of
the aluminium and copper substrates was nearly zero, and a water drop
dispensed on the substrate completely spread on the surface through
hemiwicking. The equilibrium contact angle of the PFOTES-coated copper surface
was $\theta$ = 161$\pm$ 2∘ and the contact angle hysteresis (CAH) was 3 $\pm$
2 ∘. The equilibrium contact angle of the superhydrophobic aluminum substrate
was $\theta$ = 156 $\pm$ 2∘ and CAH=4 $\pm$ 2∘.

### Condensation performance

Condensation experiments in the quiescent air environment were performed to
compare the water collection rate from the fabricated surfaces at a degree of
subcooling of 10 - 26 ${}^{\circ}C$ and a humidity ratio difference of 5 - 45
$g/kg$ of dry air. The different experimental conditions chosen for this study
are shown in Table 2. The condensation rate of the surfaces was evaluated by
measuring the mass of the drained condensate from the surface using a
microbalance at every experimental condition. Figure 2A depicts the amount of
water collected in a container over time for different surfaces considered in
this study. These experiments were performed at a temperature of $T_{env}=30$
${}^{\circ}C$, RH = 60 %, and a surface temperature of $T_{s}$= 10 $\pm$ 0.5
${}^{\circ}C$. On superhydrophilic surfaces, the vapor nucleated easily due to
the low energy barrier, and the condensate spread due to hemiwicking [37] and
formed a thin film that drains through the vertical surface due to gravity
(Fig. 2A (i) and (ii)). The drained condensate accumulated at the bottom of
the surface as a puddle due to the pinning resistance at the bottom edge of
the substrate. Therefrom, the condensate liquid was removed as the gravity
force on the accumulated liquid eventually exceeded the pinning resistance.
The first event of condensate drainage from superhydrophilic copper and
aluminum substrates occurred almost the same time $\sim t_{0}+3$ $minutes$ for
the size of the condensate plate considered here. The trend of the water
collection was linear with time for both superhydrophilic copper and aluminum
surfaces, but the rate of condensation of the superhydrophilic aluminum
surface was found to be higher than on the superhydrophilic copper surface
(see the plot in Fig. 2A).

| Environmental
---
temperature
$T_{env}$, ${}^{\circ}C$
| Relative
---
humidity
$\phi$, %
| Surface
---
temperature
$T_{s}$, ${}^{\circ}C$
| Dew point
---
temperature
$T_{dew}$, ${}^{\circ}C$
| Degree of
---
subcooling
$\Delta T$, ${}^{\circ}C$
| Humidity
---
ratio
difference
$\Delta\omega$, $g/kg$
20 | 75 | 6 $\pm$ 0.5 | 15.43 | 9.43 | 5.17
30 | 60 | 6 $\pm$ 0.5 | 21.39 | 15.39 | 10.26
30 | 75 | 6 $\pm$ 0.5 | 25.08 | 19.08 | 14.39
35 | 75 | 6 $\pm$ 0.5 | 29.89 | 23.89 | 21.25
40 | 75 | 8 $\pm$ 0.5 | 34.71 | 26.71 | 29.31
45 | 90 | 17 $\pm$ 1 | 42.97 | 25.97 | 45.82

Table 2: The experimental conditions considered in the present study using
environmental chamber. The chosen environmental conditions cover a wide range
of degrees of subcooling and differences in humidity ratios that can be
encountered with atmospheric water harvesting systems.

Coalescence-induced droplet jumping was observed on superhydrophobic copper
substrate [38, 39] which predominantly caused the droplet removal (Fig.
2A(iii)). The coalescence-induced droplet jumping events have been identified
as two-drop (Fig. S1a) or multi-drop jumping (Fig. S1b). Furthermore, droplet
sweeping events were also observed sporadically on it (Fig. S2a). On the other
hand, condensate drops from the superhydrophobic aluminum substrate were found
to be removed by gravity-assisted sweeping mode (Fig. 2A(iv)). The jumping
microdroplets can travel a longer distance from the substrate during their
flight. For the purpose of measurement of condensation rate, it is important
to ensure that the condensate from all the surfaces reaches the condensate
collection system without any mass loss inflicted by jumping droplets falling
outside the collector. The maximum possible velocity of the jumping droplet is
in the order of $\sim 30$ $cm/sec$ [40]. In the current experiments, the
height between the top of the surface to the microbalance was around $\sim 15$
$cm$. The horizontal distance traveled by the jumped droplet from the surface
can be theoretically computed as $U\sqrt{2h/g}$ from the equations of motion,
and the corresponding value is $\sim$5 $cm$. We used a petridish of diameter 9
$cm$ to collect the condensate from the substrate. Since the size of the
petridish is larger than the maximum possible horizontal distance a jumping
droplet can travel, we can safely assume that all the condensed droplets are
collected, and there are no mass losses associated with jumping microdroplets.
The influence of surface morphology on the microscale behavior of condensation
on superhydrophobic copper and aluminum is detailed in the Supplementary
Section S1. Coalescence of drops promptly ejected the condensate microdroplets
from the superhydrophobic copper surface even before $t_{0}$ $minutes$. The
mass of such microdroplets was difficult to measure experimentally since they
weighed less than the microbalance resolution (10 $mg$). However, with the
accumulation of multiple such droplets in the collection container,
measurement was possible from $\sim t_{0}+4$ $minutes$ onwards, as shown in
Fig. 2A. Interestingly, the onset of water collection on the superhydrophobic
aluminum substrate started at $t_{0}+19$ $minutes$ despite the surface having
a very low contact angle hysteresis (below 5∘). On the superhydrophobic
aluminum surface, condensate was removed by gravity-assisted sweeping, and the
drops exhibited a departure diameter of $\sim$1.4 $\pm$ 0.5 $mm$ above which
they were drained by gravity from the surface (Fig. S2b). Although the
drainage of condensate occurs on the superhydrophobic copper surface earlier
than the superhydrophobic aluminum surface (Fig. 2A), the water collection
from the superhydrophobic aluminum surface surpasses that from the
superhydrophobic copper substrate beyond $t_{0}+27$ $minutes$. The water
collection rate after the initial droplet drainage on the superhydrophobic
aluminum surface is found to be higher than that on the superhydrophobic
copper substrate because on the former the drainage of a single droplet causes
additional removal of multiple droplets while sweeping.

Figure 2: Water yield from different substrates. (A) The amount of water
collected from the surfaces over time at a temperature of $T_{env}=30$
${}^{\circ}C$, RH = 60 %, and a surface temperature of $T_{s}$= 10 $\pm$ 0.5
${}^{\circ}C$. The measurement was recorded at a time interval of 1 $minute$
using a microbalance of 10 $mg$ accuracy. $t_{0}$ is the time at which the
surface temperature reaches the substrate temperature of $T_{s}$. The shaded
area denotes the uncertainty in the average mass of water collected from three
repeated experiments. Inset images show the filmwise mode of condensation on
(i) copper superhydrophilic (SHL) (ii) aluminum superhydrophilic surface,
(iii) droplet jumping on superhydrophobic (SHB) copper surface, and (iv)
condensate removal from superhydrophobic aluminum surface. (B) The overall
water collection rate of different surfaces under various environmental
conditions is summarized in Table 2. (C) Percentage enhancement in
condensation mass transfer under different environmental conditions for the
surfaces between SHL Al $\sim$ SHL Cu, SHL Al $\sim$ SHB Al, SHB Al $\sim$ SHB
Cu, and SHL Cu $\sim$ SHB Cu.

A stable dropwise condensation mode was observed on superhydrophobic copper
and aluminum substrates under all experimental conditions for a prolonged
time, as shown in Fig. S4. In all cases, the superhydrophilic aluminum
substrate had a higher water collection rate than all other surfaces, as shown
in Fig. 2B. The water collection rate increases with the humidity ratio
difference and degree of subcooling [41]. The condensation rate of the
superhydrophobic aluminum surface was found to be higher than that of the
superhydrophobic copper surface at all the experimental conditions. The
coalescence-induced droplet jumping mechanism played a dominant role in
removal of the condensate from the superhydrophobic copper surface, while for
the superhydrophobic aluminum surface, the drainage mechanism was primarily
the sweeping. This suggests that the vertically oriented surfaces exhibiting
droplet jumping are less efficient for atmospheric humidity harvesting
applications [10]. Interestingly enough, the same phenomenon has been strongly
attributed to enhancing the condensation heat transfer in pure vapor
conditions [39, 42, 43]. During pure vapor condensation, the nano-engineered
superhydrophobic surface led to flooding at higher subcooling due to an
uncontrolled nucleation rate [39, 44, 27]. However, the event of flooding was
not observed on the superhydrophobic surfaces during the humid air
condensation experiments at higher subcooling conditions even after 6 hours.
This indicates that the presence of NCG hampers the nucleation of fresh
embryos on the subcooled substrate and helps in the prevention of flooding on
superhydrophobic surfaces at higher subcooling conditions. Figure 2C compares
the percentage enhancement of condensation mass transfer between different
surfaces under a wide range of environmental conditions. The superhydrophobic
copper surface yielded 50 - 190 % less condensate collection than the
superhydrophobic aluminum surface since the surfaces exhibiting droplet
jumping are less effective than the surfaces exhibiting sweeping mode of
drainage in the atmospheric humid air environment [10]. The filmwise mode on
the superhydrophilic aluminum surface yielded 16 - 90 % more condensate
collection than the dropwise mode on the superhydrophobic aluminum surface
with gravity-assisted drainage. The dropwise condensation on the
superhydrophobic copper substrate yielded 57 - 333 % lower condensation rate
than the superhydrophilic copper surface. A reduction in condensation rate by
333 % on superhydrophobic copper surface at low vapor content
($\Delta\omega=5.2$ $g/kg$) indicates that the superhydrophobic surfaces,
which can show droplet jumping during condensation, are the least effective
surfaces for humidity harvesting.

### Condensation on superhydrophilic surfaces: Role of rivulets

Another important observation from this study is that the condensation rate is
higher on the superhydrophilic aluminium surface than the superhydrophilic
copper surface, although both surfaces had a contact angle of almost 0∘. The
condensation rate of the superhydrophilic aluminum surface has shown an
enhancement of 9 - 26 % over the superhydrophilic copper surface. Importantly,
at low vapor content ($\Delta\omega=$5.2 $g/kg$), the superhydrophilic
aluminum surface showed an enhancement of nearly 26 %, which is essential for
the humidity harvesting system because getting a higher condensation rate at a
low humidity ratio situation is critical [45]. The influence of surface
textures of superhydrophobic surfaces on condensation was extensively
investigated in the past [24, 25, 26, 27, 28, 29, 30, 31] because they
exhibited enhanced performance under pure vapor conditions. However, such
influences of superhydrophilic or hydrophilic surfaces were not reported.
These results imply that the significance of the micro/nano features on the
superhydrophilic surface has an impact on condensation performance in the
humid air.

Figure 3: Rivulets on superhydrophilic surfaces. Time-lapse snapshots of
condensation on superhydrophilic (SHL) (A) copper and (B) aluminum substrates
at $T_{env}$= 27 ${}^{\circ}C$, $RH$= 60 %, $T_{dew}$= 18.6 ${}^{\circ}C$ and
$T_{s}$= 6 $\pm$ 0.5 ${}^{\circ}C$. $t_{0}$ is the time at which the surface
temperature reaches the set value. Here, the green and red dashed regions
represent the rivulet and puddle regimes, respectively. The Yellow dashed line
represents the bottom edge of the substrate. For clear visualization of
different regimes on superhydrophilic surfaces during condensation, we used IR
imaging (FLIR-A655sc) to produce the above image. The scale bar is 3 $mm$.

At the initial stage of condensation on superhydrophilic vertical surfaces,
the nucleated drops spread over the surface and form the rivulet regime in
which condensate is found to be drained downward by gravity. The condensate,
drained through rivulets, accumulates at the bottom of the substrate as a
puddle (marked as a red dashed line in Fig. 3) due to the pinning force
offered by the bottom edge. The puddle volume grows over time, and it
eventually drains off the surface when gravity overcomes the pinning force at
the bottom-edge. The size of the puddle reduces right after the drainage and
then again increases with time until the next shedding happens. The nucleated
condensate observed in the thin-film regime (the region outside of the puddle
and rivulet regimes) spreads and merges with the rivulet or puddle. Gravity
effects are negligible in the thin-film region. The rivulet regime has a
thicker condensate film than the thin film regime. The shape of the rivulet
depends on the surface topography and heterogeneity. The observed rivulet
shape differed between copper and aluminum superhydrophilic substrates, and
did not vary with time once they reached a steady-state (Fig. 3). The rivulets
on the superhydrophilic aluminum substrate are mostly vertical channels,
whereas the rivulets on the superhydrophilic copper surface consist of
multiple branches that are inclined or horizontal and connected to the
vertical channel. The condensate present in the vertical rivulet channel is
smoothly drained due to gravity. The condensate accumulated in the rivulet
branches is either stagnant or flowing with a lower velocity towards the
central vertical channel due to the flow resistance offered from the
accumulated condensate in the central channel. Thus, the overall drainage rate
on the superhydrophilic copper substrate is reduced.

The above observations clearly elucidate that condensation performance of an
superhydrophilic surface under humid air condition is significantly influenced
by rivulet formation and the dynamics of the drainage mechanism through the
rivulets. The spreading of condensate in the thin film region leads to the
continuous transport of condensate from the thin film regime to the rivulet
regime, which promotes homogeneous nucleation in the thin-film regime. As a
result, at steady state, the rate of condensate formation in the thin-film
regime is higher than that in the rivulet regime. Overall, it is clear that an
optimal superhydrophilic surface with vertically shaped rivulet channels can
improve the condensation rate for humidity harvesting.

Notably, under all experimental conditions considered in this work,
condensation mass transfer in the filmwise mode of condensation on the
superhydrophilic surface was greater than that in the dropwise mode of
condensation on the superhydrophobic surface (Fig. 2B). This observation
contradicts the general perception of condensation under pure steam conditions
[11, 12, 13]. Although multiple research works summarized in Table 1 showed
observations similar to our foregoing experimental results, the reason for
this observation was not explained in any of these studies. A detailed
comparison between condensation phenomena in the pure vapor and NCG
environments is necessary and presented next.

### Influence of non-condensable gases

To understand the influence of NCG during condensation, a theoretical analysis
of a single condensing drop [32, 33] and a condensate film [46] was carried
out for a wide range of NCG concentrations. This section compares the
nucleation performance of different surfaces and the individual contribution
of temperature drop due to different thermal resistances at a moderate and
exemplar subcooling of 10 $K$.

The nucleation rate of a substrate is defined as the probability of the liquid
embryo that is continuing to grow on a nucleation site without evaporation
during vapor-liquid phase conversion. The heterogeneous nucleation rate per
unit area ($dN/dt$) can be expressed as [47],

$\frac{dN}{dt}=\frac{R_{0}}{3}Ae^{\frac{-W}{k_{B}T_{s}}}$ (1)

where $R_{0}$ is the radius of the water molecule, $R_{0}\approx 1.375\times
10^{-10}$ $m$, $A$ is the Arrhenius prefactor, $W$ is the total energy
required for the formation of a liquid nuclei, $k_{B}$ is the Boltzmann
constant and $T_{s}$ is the substrate temperature.

Figure 4: Nucleation rate and temperature drops in different modes of
condensation. (A) The influence of NCG concentration and surface wettability
on heterogeneous nucleation rate for a fixed supersaturation ratio,
$S=P_{v}/P_{s}=2$. Schematic of heat transfer model with thermal resistance
network across (B) a single droplet during dropwise condensation and (C) a
film during filmwise condensation in the presence of NCG. The influence of NCG
concentration on temperature drop across the liquid (film or droplet) due to
conduction ($\Delta T_{drop/film}$) and temperature drop across the vapor
region ($\Delta T_{vapor}$) during (D) dropwise mode, and (E) filmwise mode.
The thermal conductivity ($k_{coat}$) and the thickness of the hydrophobic
layer coating ($\delta_{coat}$) are approximated as 0.2 $W/mK$ and 1 $\mu m$
respectively [33]. $\delta_{f}$ is the average condensate film thickness and
it is approximated as 25 $\mu m$ [48]. Refer Supplementary section S2 for more
details.

The vapor pressure is significantly reduced with the increase in NCG
concentration (Figure 4A). It also reduces the heterogeneous nucleation rate
[47]. The accumulation of NCG near the subcooled surface can further reduce
water vapor diffusion from the bulk region to the vapor-liquid interface, and
thus, hinders the nucleation rate. Figure 4A compares the influence of NCG
concentration on the nucleation rate for extreme wettability (such as
superhydrophilic and superhydrophobic) surfaces at a supersaturation ratio
($S$) of 2. Supersaturation ratio is defined as partial pressure of the water
vapor ($P_{v}$) to the saturation pressure corresponding to the substrate
temperature $T_{s}$ ($P_{s}$). As can be seen from Fig. 4A, the nucleation
rate decreases with NCG concentration; at 90 % NCG concentration the
nucleation rate of the superhydrophilic surface reaches a value that is one
order of magnitude lower than that for the pure vapor condition. On the other
hand, the nucleation rate on the superhydrophobic surface reaches a value that
is eleven orders of magnitude lower than that of the pure vapor condition, as
shown in Fig. 4A. As a result, the nucleation occurs much slowly from humid
air than from pure vapour on superhydrophobic surfaces. The interplay between
the phase transition of vapor molecules and vapor diffusion from the bulk
fluid makes the nucleation phenomena complex in the presence of NCG. The
probability of activating a nucleation site in the NCG environment depends
significantly on the localised vapor content and vapor diffusion rate across
the NCG diffusion layer.

Figures 4(B and C) show a schematic representation of the thermal resistance
network model in the presence of NCG across a single droplet for the dropwise
condensation and across the film for the filmwise condensation. The schematics
consist of three regions, namely hydrophobic coating region, the condensed
drop/film region, and the vapor region. The vapor region consists of
interfacial region called Knudsen layer, the diffusion layer and the bulk
vapor-NCG mixture region. In the Knudsen layer, the kinetic theory of gases
governs the transfer of vapor molecules to the liquid drop interface. The
region outside the Knudsen layer is known as the diffusion layer, and during
condensation, a concentration gradient of vapour exists in this region. The
thickness of the diffusion layer depends both on the hydrodynamic parameter
(the free stream velocity for a forced flow and the Grashof number in case of
a thermo-gravitational flow), and the thermophysical properties (e.g., thermal
and mass diffusivities). For the present cases, the hydrodynamic influence on
the vapor-NCG boundary layer may be neglected since the pertinent thermo-
gravitational Grashof number, $Gr=g\beta\Delta TL_{c}^{3}/\nu^{2}$ is small
($\sim 10^{5}-10^{6}$), where $g$ is the acceleration due to gravity, $\beta$
is the co-efficient of thermal expansion, $L_{c}$ is the characteristic length
scale, and $\nu$ is the kinematic viscosity. On the other hand, a change in
the degree of subcooling and the bulk vapor concentration alters the average
mass and thermal diffusivities, thereby altering the diffusion layer
thickness. The thickness of the diffusion layer increases as the vapour
concentration of the mixture decreases. Outside the Knudsen outer interface,
combined mass diffusion and energy conservation laws govern the transport. For
condensation under pure steam scenario, the vapour in the bulk region remains
in an equilibrium state, and the interface temperature of the drop is always
maintained at the saturation temperature corresponding to the vapour pressure.
However, presence of NCGs adds heat and mass transfer resistances, thus
rendering the actual prediction of interface temperature difficult. It can
only be predicted by solving coupled mass and energy conservation equations
[49, 6, 50]. The total temperature difference ($\Delta T$) between the cold
surface and the bulk vapor-NCG mixture is expressed as follows:

$\Delta T=\Delta T_{coat}+\Delta T_{drop/film}+\Delta T_{vapor}$ (2)

where $\Delta T_{coat}$, $\Delta T_{drop/film}$, $\Delta T_{vapor}$ are the
temperature drops across the promoter layer coating, liquid drop/film due to
conduction, vapor region respectively. It is important to note that the
$\Delta T_{vapor}$ includes the temperature drop due to curvature resistance
($\Delta T_{c}$, see Eq. S11) and interfacial temperature drop ($\Delta
T_{int}$, see Eq. S12). The estimation of each temperature drop during
dropwise and filmwise mode of condensation is calculated from the numerical
iteration of analytical expressions or an existing correlation (Supplementary
Section S2).

The amount of temperature drop over the condensate drop and vapor region for a
drop of radius 100 $nm$ \- 1 $mm$ during dropwise mode condensation of pure
vapour and the vapor-NCG mixture is shown in Figure 4D. In the dropwise mode
of condensation, the temperature drop across the vapor region is significantly
higher for the droplets of radius below 0.1 $\mu m$ irrespective of the NCG
concentration. This is because the curvature component ($\Delta T_{c}$) offers
greater thermal resistance for smaller droplets. The contribution of these
thermal resistances to the overall temperature drop decreases as drop size
increases. In the case of pure vapor condensation, the temperature drop across
the droplet is significant except for the case of smaller droplets of radius
below 0.1 $\mu m$ and it reaches above 90 % of the total subcooling for
droplets with a radius greater than 200 $\mu m$. With a rise in NCG
concentration, the temperature drop across the drop due to conduction
decreases. With 95 % NCG (for example, atmospheric humid air), the temperature
drop within the condensate droplet is entirely negligible for any droplet size
(Fig. 4D), and the magnitude of temperature drop across the coating is
negligible (Fig. S3), even at a coating thickness of 10 $\mu m$. For filmwise
condensation, variation in the temperature drop across the film and vapor
regions with different NCG concentrations is shown in Fig. 4E. Both the
temperature drops (across the film and vapor region) are significant, and they
occur in the same order as for pure vapour condensation. However, the
temperature drop across the film decreases with NCG concentration and becomes
negligible at concentration greater than 50 %. Moreover, the drop occurs
primarily at the vapor region.

Therefore, regardless of the mode of condensation, the heat transfer
performance during humid air condensation is primarily controlled by the rate
of diffusion of the vapour molecules through the NCG diffusion layer, and the
thermal resistance offered by the condensate drop or the coating is
insignificant. The heat transfer performance during the filmwise mode is poor
in a pure vapor environment due to the significant thermal resistance across
the condensate film. In humid air condensation however, a similar conclusion
is incorrect because the thermal resistance across the film/drop is negligible
and the interface temperature of the film/drop is close to the substrate
temperature [6]. In addition to this, the nucleation rate of the
superhydrophilic surface is twenty seven orders of magnitude higher than the
superhydrophobic surface (Fig. 4A). As a result, rejuvenation of heterogeneous
nucleation sites after the drop removal is energetically expensive for
superhydrophobic surfaces in humid air conditions. Based on these
understandings and observations from the experiments, we can interpret that
superhydrophilic surfaces can show better condensation heat transfer
performance over the superhydrophobic surface in the humid air environment
with more than 95 % NCG, and they can outperform in atmospheric humidity
harvesting applications.

## Discussion

The major challenge in humidity harvesting devices is obtaining the required
amount of water yield from the atmospheric humid air on a daily basis. The air
temperature and relative humidity vary throughout the day, and the water yield
from the surface can be drastically reduced, particularly when the humidity
ratio is low. As a result, increasing water yield under low humidity
conditions is critical for obtaining the desired water yield from the humidity
harvesting system. The superior condensation rates on superhydrophilic
surfaces compared to the superhydrophobic surface at low humidity ratios is
appealing for humidity harvesting systems. From other aspects also, such as
the cost associated with surface fabrication, scalability of the surface
fabrication technique, and chemical contamination possibilities of condensate
with hydrophobic coating material, superhydrophilic surface appears to be more
suitable than the superhydrophobic surfaces for humidity harvesting.

In summary, this study investigated the condensation performance of vertically
oriented superhydrophilic and superhydrophobic surfaces in a wide range of
humid air environments. Interestingly, at a vapor concentration of below 5 %,
we found that the filmwise mode of condensation on a superhydrophilic surface
shows superior water collection than the dropwise mode of condensation
observed on a superhydrophobic surface. Theoretical analyses indicate that
under humid air conditions, the conduction thermal resistance across the
condensate liquid or the surface coatings is negligible, and the condensation
performance is driven by diffusion and drainage rates. Hence, film formation
does not significantly affect the overall condensation performance unlike what
usually happens in the pure steam environment [11, 12]. Besides, the very high
nucleation rate on a superhydrophilic surface can substantially increase the
condensation rate. However a wettability-patterned design with optimum
nucleation rate and drainage rate may produce higher water yield than
homogeneous superhydrophilic surface. In effect, our study provides a new
mechanistic understanding of why the filmwise mode of condensation is better
for humidity harvesting, as well as give an insights into exploiting new
physics of rivulets to maximise this effectiveness. The key findings of this
study can usher in a new pathway to the development of efficiently engineered
surfaces for humidity harvesting systems.

## Methods

### Surface preparation and characterization

Aluminum alloy (Al-6061) and copper (99.9 % purity) plates of size 80 $mm$
$\times$ 40 $mm$ $\times$ 2 $mm$ were used as the substrate for the
condensation experiments. First, the substrates were mechanically polished
with sand paper of grade 220 to 2000 sequentially. Then the polished substrate
was washed with DI water before being ultrasonically cleaned for 10 minutes in
a solution of DI water, ethanol, and acetone. The clean copper substrate was
kept in a 3$M$ $HNO_{3}$ (Merck, Emplura grade) solution for 10 minutes to
remove the native oxides, and further rinsed with DI water and dried with
nitrogen gas. Thereafter, the substrate was oxidised in an aqueous solution of
2$M$ $NaOH$ (Sigma Aldrich, ACS reagent grade pellets) and 0.1$M$
$K_{2}S_{2}O_{8}$ (Sigma Aldrich, ACS reagent grade) for 1 hour [25], yielding
a superhydrophilic surfaces. Next, the substrate was rinsed with DI water, and
dried with nitrogen gas, and kept in the hot air oven for 30 minutes at a
temperature of 150 ${}^{\circ}C$. To achieve superhydrophobicity, the oxidised
copper substrate was dip-coated for 4 hours in an ethanol solution with 0.5 %
perfluorooctyl-triethoxy silane (Sigma Aldrich) and dried in the hot air oven
at 150 ${}^{\circ}C$ for 1 hour.

The aluminium substrate was microtextured by dipping it in a 3$M$ solution of
$HCl$ (Merck, Emplura grade) for 5 minutes and then rinsing it with DI water.
The chemically etched aluminum substrate was further immersed in a hot DI
water bath at a temperature of 100 ${}^{\circ}C$ for 30 minutes to generate
the nanostructures above the microstructures [51]. The aluminum substrate
became superhydrophilic after those fabrication steps. Next, the substrate was
functionalized with 0.5 % perfluorooctyl-triethoxy silane (PFOTES) in ethanol
solution for 4 hours and then dried in the laboratory environment for 12 hours
for achieving superhydrophobicity.

A high-resolution scanning electron microscope (Inspect F-FEI) was used to
examine the surface morphology. The roughness of the modified substrate was
measured using a non-contact type profilometer (NT1100-Wyco). The static
contact angle of water on the substrate was measured with a goniometer
(Holmarc) by placing a sessile drop of volume 5 $\mu L$ using a micro-pipette.
The dynamic contact angles were determined by injecting/drawing DI water
into/from the sessile drop with a syringe pump at a creeping flow rate of 0.1
$\mu L/sec$ [52]. Each reported contact angle comprises of the average of six
measurements taken at various locations on the substrate.

### Experimental procedures

The condensation experiments for a vertically oriented surface were performed
in an environmental chamber (PR2J-Espec) of size 500 $\times$ 750 $\times$ 600
$mm^{3}$ at different environmental conditions as shown in Fig. S5. The
condensing substrate was placed at the centre of the environmental chamber to
avoid the boundary wall effects. The fluctuations in air temperature and
relative humidity were $\pm 0.5$ ${}^{\circ}C$ and $\pm 3$ %, respectively.
The substrates under test were attached to a Peltier (CP061HT-Tetech) element
with thermal conductive tape and the remaining area of the Peltier element was
covered with nitrile foam to avoid condensation on the Peltier cooling plate.
The temperature of the Peltier element was controlled using a PID controller.
Four holes of size 0.8 $mm$ were drilled on the lateral side of the substrate
for the insertion of K-type thermocouples (Omega, 0.13 $mm$ bead diameter) for
surface temperature measurement. The temperature from the thermocouples was
logged onto a data acquisition system (DAQ970A-Keysight) at every minute. The
condensation experiments were performed for 2-3 hours, and the condensed water
was collected in a Petri dish with a diameter of 9 $cm$. This ensured that the
condensate from all the surfaces reached the container while jumping or
sweeping. The mass of the water from the Petri dish was measured in a
microbalance (SPX622-Ohaus) with a precision of 10 $mg$. At least three
experiments were performed to calculate the average condensation rate on a
surface. The condensation images were captured with a DSLR camera (D750-Nikon)
and a 105 $mm$ macro lens at 30 $fps$. An LED spotlight was used to illuminate
the imaging area (CW8-Moritex).

## Data Availability

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

## Author Contributions

T.M.T and P.S.M conceived the initial idea of the research. T.M.T performed
the fabrication and characterization of the materials, designed the
experimental setup, performed the experiments, analyzed the results, and wrote
the Matlab code for the numerical model. R.G conceived the idea of numerical
modelling. All authors contributed in writing the manuscript. P.S.M, R.G and
M.K.T guided this research.

## Acknowledgements

P.S.M acknowledges the Science and Engineering Research Board (SERB) of
Government of India (Project number ECR/2018/001806) for funding the work.
P.S.M also acknowledges the partial funding through the Indian Institute of
Technology Madras to the Micro Nano Bio Fluidics Group for Institutions of
Eminence scheme of Ministry of Education, Government of India (Sanction. No:
$11/9/2019-U.3(A)$). M.K.T. also acknowledges support by the Wellcome/EPSRC
Centre for Interventional and Surgical Sciences (WEISS) (203145Z/16/Z),
Wolfson Foundation and Royal Society for his Royal Society Wolfson Fellowship,
and the NICEDROPS project supported by the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
under grant agreement No. 714712. Authors also acknowledge Dr. Arvind
Pattamatta, IIT Madras for providing the access of IR camera facility.

## References

  * [1] Mesfin M Mekonnen and Arjen Y Hoekstra.  Four billion people facing severe water scarcity.  Science Advances, 2(2):e1500323, 2016.
  * [2] Yaodong Tu, Ruzhu Wang, Yannan Zhang, and Jiayun Wang.  Progress and expectation of atmospheric water harvesting.  Joule, 2(8):1452–1475, 2018.
  * [3] Ankit Nagar and Thalappil Pradeep.  Clean water through nanotechnology: Needs, gaps, and fulfillment.  ACS Nano, 14(6):6420–6435, 2020.
  * [4] Iwan Haechler, Hyunchul Park, Gabriel Schnoering, Tobias Gulich, Mathieu Rohner, Abinash Tripathy, Athanasios Milionis, Thomas M Schutzius, and Dimos Poulikakos.  Exploiting radiative cooling for uninterrupted 24-hour water harvesting from the atmosphere.  Science Advances, 7(26):eabf3978, 2021.
  * [5] Detlev G Kroger and Warren M Rohsenow.  Condensation heat transfer in the presence of a non-condensable gas.  International Journal of Heat and Mass Transfer, 11(1):15–26, 1968\.
  * [6] Shaofei Zheng, Ferdinand Eimann, Christian Philipp, Tobias Fieback, and Ulrich Gross.  Modeling of heat and mass transfer for dropwise condensation of moist air and the experimental validation.  International Journal of Heat and Mass Transfer, 120:879–894, 2018\.
  * [7] Donald Frederick Othmer.  The condensation of steam.  Industrial & Engineering Chemistry, 21(6):576–583, 1929.
  * [8] L Slegers and RA Seban.  Laminar film condensation of steam containing small concentrations of air.  International Journal of Heat and Mass Transfer, 13(12):1941–1947, 1970.
  * [9] JL Oestreich, CWM Van Der Geld, JL Goes Oliveira, and AK Da Silva.  Experimental condensation study of vertical superhydrophobic surfaces assisted by hydrophilic constructal-like patterns.  International Journal of Thermal Sciences, 135:319–330, 2019.
  * [10] Tibin M Thomas and Pallab Sinha Mahapatra.  Condensation of humid air on superhydrophobic surfaces: Effect of nanocoatings on a hierarchical interface.  Langmuir, 37(44):12767–12780, 2021.
  * [11] E Schmidt, W Schurig, and W Sellschopp.  Condensation of water vapour in film-and drop form.  Technische Mechanik und Thermodynamik, 1:53–63, 1930.
  * [12] EJ Le Fevre and JW Rose.  An experimental study of heat transfer by dropwise condensation.  International Journal of Heat and Mass Transfer, 8(8):1117–1133, 1965.
  * [13] Hyeongyun Cha, Hamed Vahabi, Alex Wu, Shreyas Chavan, Moon-Kyung Kim, Soumyadip Sett, Stephen A Bosch, Wei Wang, Arun K Kota, and Nenad Miljkovic.  Dropwise condensation on solid hydrophilic surfaces.  Science Advances, 6(2):eaax0746, 2020.
  * [14] Soyoung Choo, Hak-Jong Choi, and Heon Lee.  Water-collecting behavior of nanostructured surfaces with special wettability.  Applied Surface Science, 324:563–568, 2015.
  * [15] Pallab Sinha Mahapatra, Aritra Ghosh, Ranjan Ganguly, and Constantine M Megaridis.  Key design and operating parameters for enhancing dropwise condensation through wettability patterning.  International Journal of Heat and Mass Transfer, 92:877–883, 2016\.
  * [16] Donghyun Seo, Junghun Lee, Choongyeop Lee, and Youngsuk Nam.  The effects of surface wettability on the fog and dew moisture harvesting performance on tubular surfaces.  Scientific Reports, 6(1):1–11, 2016.
  * [17] Kongyang Hou, Xiaoyang Li, Qiang Li, and Xuemei Chen.  Tunable wetting patterns on superhydrophilic/superhydrophobic hybrid surfaces for enhanced dew-harvesting efficacy.  Advanced Materials Interfaces, 7(2):1901683, 2020.
  * [18] RA Pinheiro, DD Damm, AA Silva, RM Volu, KF Almeida, FM Rosa, VJ Trava-Airoldi, and EJ Corat.  Water vapor condensation from atmospheric air by super-hydrophobic vacnts growth on stainless steel pipes.  MRS Advances, 4(35):1929–1936, 2019.
  * [19] Romario Araujo Pinheiro, Filipe Menezes Rosa, Renê Martins Volú, Getúlio de Vasconcelos, Vladimir Jesus Trava-Airoldi, and Evaldo José Corat.  Vertically aligned carbon nanotubes (vacnt) surfaces coated with polyethylene for enhanced dew harvesting.  Diamond and Related Materials, 107:107837, 2020.
  * [20] Rohit Gupta, Chayan Das, Amitava Datta, and Ranjan Ganguly.  Background oriented schlieren (bos) imaging of condensation from humid air on wettability-engineered surfaces.  Experimental Thermal and Fluid Science, 109:109859, 2019.
  * [21] Junbeom Lee, Seunghwan Lee, and Jaeseon Lee.  Improved humid air condensation heat transfer through promoting condensate drainage on vertically stripe patterned bi-philic surfaces.  International Journal of Heat and Mass Transfer, 160:120206, 2020\.
  * [22] Anna Lee, Myoung-Woon Moon, Hyuneui Lim, Wan-Doo Kim, and Ho-Young Kim.  Water harvest via dewing.  Langmuir, 28(27):10183–10191, 2012.
  * [23] Ankit Nagar, Ramesh Kumar, Pillalamarri Srikrishnarka, Tiju Thomas, and Thalappil Pradeep.  Scalable drop-to-film condensation on a nanostructured hierarchical surface for enhanced humidity harvesting.  ACS Applied Nano Materials, 4(2):1540–1550, 2021.
  * [24] Chander Shekhar Sharma, Juliette Combe, Markus Giger, Theo Emmerich, and Dimos Poulikakos.  Growth rates and spontaneous navigation of condensate droplets through randomly structured textures.  ACS Nano, 11(2):1673–1682, 2017.
  * [25] Xuemei Chen, Justin A Weibel, and Suresh V Garimella.  Exploiting microscale roughness on hierarchical superhydrophobic copper surfaces for enhanced dropwise condensation.  Advanced Materials Interfaces, 2(3):1400480, 2015.
  * [26] Konrad Rykaczewski, Adam T Paxson, Sushant Anand, Xuemei Chen, Zuankai Wang, and Kripa K Varanasi.  Multimode multidrop serial coalescence effects during condensation on hierarchical superhydrophobic surfaces.  Langmuir, 29(3):881–891, 2013.
  * [27] Emre Olceroglu and Matthew McCarthy.  Self-organization of microscale condensate for delayed flooding of nanostructured superhydrophobic surfaces.  ACS Applied Materials & Interfaces, 8(8):5729–5736, 2016.
  * [28] Daniel Orejon, Alexandros Askounis, Yasuyuki Takata, and Daniel Attinger.  Dropwise condensation on multiscale bioinspired metallic surfaces with nanofeatures.  ACS Applied Materials & Interfaces, 11(27):24735–24750, 2019.
  * [29] Rebecca L Winter and Matthew McCarthy.  Dewetting from amphiphilic minichannel surfaces during condensation.  ACS Applied Materials & Interfaces, 12(6):7815–7825, 2020.
  * [30] Ye Zhao, Yuting Luo, Juan Li, Fei Yin, Jie Zhu, and Xuefeng Gao.  Condensate microdrop self-propelling aluminum surfaces based on controllable fabrication of alumina rod-capped nanopores.  ACS Applied Materials & Interfaces, 7(21):11079–11082, 2015.
  * [31] Chander Shekhar Sharma, Cheuk Wing Edmond Lam, Athanasios Milionis, Hadi Eghlidi, and Dimos Poulikakos.  Self-sustained cascading coalescence in surface condensation.  ACS Applied Materials & Interfaces, 11(30):27435–27442, 2019.
  * [32] EJ Le Fevre and John W Rose.  A theory of heat transfer by dropwise condensation.  In International Heat Transfer Conference Digital Library. Begel House Inc., 1966.
  * [33] Sunwoo Kim and Kwang J. Kim.  Dropwise condensation modeling suitable for superhydrophobic surfaces.  Journal of Heat Transfer, 133(8), 05 2011.
  * [34] Qingfeng Yang and Anzhong Gu.  Dropwise condensation on sam and electroless composite coating surfaces.  Journal of Chemical Engineering of Japan, 39(8):826–830, 2006.
  * [35] Weixin Zhang, Xiaogang Wen, Shihe Yang, Yolande Berta, and Z Lin Wang.  Single-crystalline scroll-type nanotube arrays of copper hydroxide synthesized at room temperature.  Advanced Materials, 15(10):822–825, 2003.
  * [36] Peng Zhang, Yota Maeda, Fengyong Lv, Yasuyuki Takata, and Daniel Orejon.  Enhanced coalescence-induced droplet-jumping on nanostructured superhydrophobic surfaces in the absence of microstructures.  ACS Applied Materials & Interfaces, 9(40):35391–35403, 2017.
  * [37] Jungchul Kim, Myoung-Woon Moon, and Ho-Young Kim.  Dynamics of hemiwicking.  Journal of Fluid Mechanics, 800:57–71, 2016.
  * [38] Jonathan B Boreyko and Chuan-Hua Chen.  Self-propelled dropwise condensate on superhydrophobic surfaces.  Physical Review Letters, 103(18):184501, 2009.
  * [39] Nenad Miljkovic, Ryan Enright, Youngsuk Nam, Ken Lopez, Nicholas Dou, Jean Sack, and Evelyn N Wang.  Jumping-droplet-enhanced condensation on scalable superhydrophobic nanostructured surfaces.  Nano Letters, 13(1):179–187, 2013.
  * [40] Soumei Baba, Kenichiro Sawada, Kohsuke Tanaka, and Atsushi Okamoto.  Dropwise condensation on a hierarchical nanopillar structured surface.  Langmuir, 36(34):10033–10042, 2020.
  * [41] Chayan Das, Rohit Gupta, Saikat Halder, Amitava Datta, and Ranjan Ganguly.  Filmwise condensation from humid air on a vertical superhydrophilic surface: Explicit roles of the humidity ratio difference and the degree of subcooling.  Journal of Heat Transfer, 143(6):061601, 2021.
  * [42] Rongfu Wen, Shanshan Xu, Xuehu Ma, Yung-Cheng Lee, and Ronggui Yang.  Three-dimensional superhydrophobic nanowire networks for enhancing condensation heat transfer.  Joule, 2(2):269–279, 2018.
  * [43] Chuntian Liu, Meirong Zhao, Yelong Zheng, Luya Cheng, Jiale Zhang, and Clarence Augustine TH Tee.  Coalescence-induced droplet jumping.  Langmuir, 37(3):983–1000, 2021.
  * [44] Rui Wang, Feifei Wu, Dandan Xing, Fanfei Yu, and Xuefeng Gao.  Density maximization of one-step electrodeposited copper nanocones and dropwise condensation heat-transfer performance evaluation.  ACS Applied Materials & Interfaces, 12(21):24512–24520, 2020.
  * [45] Konstantinos Gerasopoulos, William L Luedeman, Emre Olceroglu, Matthew McCarthy, and Jason J Benkoski.  Effects of engineered wettability on the efficiency of dew collection.  ACS Applied Materials & Interfaces, 10(4):4066–4076, 2018.
  * [46] W Nusselt.  The surface condensation of water vapour.  Zeitschrift Des Vereines Deutscher Ingenieure, 60:541–546, 1916\.
  * [47] Daniel Beysens.  The Physics of Dew, Breath Figures and Dropwise Condensation, volume 994.  Springer, 2022.
  * [48] Wei Zhang, Suilin Wang, and Mu Lianbo.  Analytical modeling for vapor condensation in the presence of noncondensable gas and experimental validation.  Journal of Heat Transfer, 143(1):011601, 2021.
  * [49] Xisheng Luo, Yu Fan, Fenghua Qin, Huaqiao Gui, and Jianguo Liu.  A new model for the processes of droplet condensation and evaporation on solid surface.  International Journal of Heat and Mass Transfer, 100:208–214, 2016\.
  * [50] Weihong Liu and Xiang Ling.  Heat transfer model based on diffusion layer theory for dropwise condensation with high non-condensable gas.  AIP Advances, 10(12):125305, 2020.
  * [51] W Vedder and DA Vermilyea.  Aluminum+ water reaction.  Transactions of the Faraday Society, 65:561–584, 1969.
  * [52] Chaoyi Peng, Zhuyang Chen, and Manish K Tiwari.  All-organic superhydrophobic coatings with mechanochemical robustness and liquid impalement resistance.  Nature Materials, 17(4):355–360, 2018.