Experimental Study of Condensation of Water on Polydimethylsiloxane-Coated
Copper Surfaces T. Pfeiffer, S. Li, M. Kappl, H. Butt, P. Stephan & T.
Gambaryan-Roisman [1]Till Pfeiffer<EMAIL_ADDRESS>Institute for
Technical Thermodynamics, Technical University of Darmstadt, Darmstadt,
Germany, 64287

# Experimental Study of Condensation of Water on Polydimethylsiloxane-Coated
Copper Surfaces

Shuai Li  Michael Kappl  Hans-Jürgen Butt  Peter Stephan  Tatiana Gambaryan-
Roisman  Institute for Technical Thermodynamics, Technical University of
Darmstadt, Darmstadt, Germany, 64287  Physics at Interfaces, Max-Planck-
Institute for Polymer Research, Mainz, Germany, 55128

###### Abstract

Modification of surfaces to enable dropwise condensation is a promising
approach for achieving high condensation rates. In this work, we present an
experimental study on condensation of water on copper surfaces coated with an
ultrathin,
$$5\text{\,}\mathrm{n}\mathrm{m}$-$10\text{\,}\mathrm{n}\mathrm{m}$$ thick
polydimethylsiloxane (PDMS) layer. This hydrophobic coating possesses a very
low thermal resistance, which in combination with copper substrate enables
achieving high condensation rates in heat transfer applications. The PDMS-
coated copper substrates have been fabricated with a newly developed method,
which involves turning, sanding, polishing, oxidation, and polymer coating
steps. The measured static contact angle was
$$110\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$,
and a the contact angle hysteresis was
$2\text{\,}\mathrm{\SIUnitSymbolDegree}$. The achieved very low hysteresis is
advantageous for promoting dropwise condensation. The surface showed no ageing
effects during 100 repetitions of advancing and receding contact angle (ARCA)
measurements. Condensation heat transfer on uncoated and PDMS-coated copper
surfaces surfaces has been studied experimentally in a saturated water vapor
atmosphere at $60\text{\,}\mathrm{\SIUnitSymbolCelsius}$. An enhancement
factor for heat flux and heat transfer coefficient of up to 1.6 was found on
PDMS-coated copper surfaces compared to uncoated surfaces, which decreased to
1.1 on second and third day of condensation operation.
Images of the condensation surface were recorded while conducting condensation
experiments and post processed to evaluate drop departure diameter and
frequency of drop sweeping events.
It has been shown that the behavior of the heat transfer coefficient
correlates with the frequency of the sweeping events.

###### keywords:

Dropwise condensation; Heat transfer; Surface fabrication

††volume: Volume x, Issue x, 2023

##  1 Introduction

Water condensation is critical in various applications, such as atmospheric
fresh water harvesting Bhushan (2020); Hou et al. (2018); Jarimi et al.
(2020); Khalil et al. (2016), HVAC applications Cavallini et al. (2003) and
power conversion Beér (2007); Pan et al. (2018); Zhao and Burnside (1994),
where high condensation rates are essential. Dropwise condensation (DWC) is
characterized by significantly higher condensation rates and higher heat
transfer coefficients than filmwise condensation (FWC) Schmidt et al. (1930),
since the continuous condensate film forming during FWC, due to its high
thermal resistance, isolates the cool substrate from the vapor. In contrast,
the thermal resistance of drops is lower and decreases with decreasing drop
size. In the case of condensation on vertical or inclined surfaces, the drops
roll off by gravity as they reach a sufficient size, known as drop departure
diameter, providing free space for nucleation of new drops. The efficiency of
DWC obviously increases with decreasing drop departure diameter and the
related increase of the frequency of drop sweeping events.
The common materials used in condensers, such as copper, possess high thermal
conductivity. These materials are usually hydrophilic and therefore promote
the filmwise condensation. One promising approach to enhance the condensation
rate is modifying the surface to achieve hydrophobic behavior with low contact
angle hysteresis and thereby to facilitate DWC. However, the available
hydrophobic coatings have shortcomings, either being thermally insulating,
non-durable or both Ahlers et al. (2019).
Numerous studies on surface modifications to enable DWC have been conducted
and different approaches with corresponding advantages and disadvantages have
been reported in literature Attinger et al. (2014); El Fil et al. (2020);
Goswami et al. (2021); Hu et al. (2021); Xu et al. (2021). The most commonly
investigated surface coating techniques to enable DWC are polymer coatings,
self-assembled monolayers (SAMs), ion-implantation and lubricant infused
surfaces (LIS).
Polymer coatings have already been thoroughly studied Orkan Uçar (2013). They
combine advantages of hydrophobic wetting behavior with high coating
durability Holden et al. (1987) and well established fabrication methods
Goswami et al. (2021). Common polymers used to coat surfaces in condensation
heat transfer applications are PTFE Ma et al. (2002), PFDA Paxson et al.
(2014) and polymer composites Damle et al. (2015). Nevertheless, since the
thermal conductivity of common polymers is low, the polymer coatings
constitute a thermal resistance between a subcooled wall and vapor phase.
Therefore, a trade-off between the stability of coating and heat transfer
enhancement must be found when choosing the thickness of the polymer coatings
Holden et al. (1987).
Self-assembled monolayers (SAMs) consist of a monomolecular layer of organic
polar chemical compound, which adsorbes to a metallic substrate and shows
hydrophobic wetting behavior. Due to their thickness in the range of tens of
angstroms, SAMs thermal resistance is low compared to that of coatings
fabricated by other techniques Das et al. (2000); Yang and Gu (2006).
Nevertheless, the low coating thickness makes SAMs susceptiple to coating
defects, ageing effects and with that strong decline of condensation heat
transfer enhancement in a matter of hours of operation Vemuri et al. (2006).
For ion-implanted surfaces, the promotion of dropwise condensation takes place
without deterioration of thermal contact between the subcooled substrate
surface and the vapor. During ion-implantation, foreign element ions are
inserted into metal surfaces and, by that, reduce the surface energy leading
to a change in wetting behavior from hydrophilic to hydrophobic. Studies
conducted on various substrate materials with different foreign element ions
prove the versatility of this method, yet its high costs make the process
economically unattractive Leipertz and Fröba (2008); Qi et al. (1991).
Lubricant-infused surfaces (LISs) consist of a surface microstructure, which
is flooded with a lubricant in order to promote drop mobility and thereby
facilitate DWC. Current works aim at minimizing the lubricant’s thermal
resistance and continuous draining during operation Tripathy et al. (2021);
Weisensee et al. (2017).
One polymer coating, which has recently gained attention in DWC research, is
trimethylsiloxy terminated linear (poly)dimethylsiloxane, PDMS Eduok et al.
(2017); Zhang et al. (2020). In addition to hydrophobicity, PDMS has shown a
potential for self healing Xue et al. (2016), which has been applied to
fabrics to maintain self-cleaning properties and water repellency over
multiple washing and abrasion cycles. Teisala et al. Teisala et al. (2020)
developed ultrathin PDMS brush coatings, which combine multiple advantageous
characteristics for condensation heat transfer applications. The low coating
layer thickness of several nanometers ensures a significantly reduced thermal
resistance compared to other polymer coatings. The low thermal resistance of
PDMS brushes and favorable wetting properties (hydrophobicity and low contact
angle hysteresis), resulting in DWC mode promise great potential to enhance
condensation heat transfer. The coating procedure has been developed for glass
slides and polished silicon wafers as substrates, taking advantage of the fact
that the methyl-terminated polydimethylsiloxane spontaneously bonds on
materials with silicon oxide surface chemistry at moderate temperatures. These
types of substrates, however, have only limited practical applications Lo et
al. (2019).
Fazle Rabbi et al. Fazle Rabbi et al. (2022) investigated PDMS to promote DWC
on copper, aluminum and stainless steel during ethanol, hexane and pentane
condensation. In order to enable DWC and prevent condensate penetration of the
PDMS coating, a coupling of flexible PDMS with low energy silane was
investigated. The developed coating was applied to the aforementioned metal
substrates, and condensation heat transfer enhancement of 274% for ethanol,
347% for hexane and 636% for pentane compared to uncoated surfaces was
observed on coated horizontal tubes. The DWC mode was sustained for 15 days of
steady operation without signs of coating degradation. While in this study the
results of detailed condensation heat transfer measurements for an extended
experimental time scale are reported, no quantitative characteristics of
wetting behavior during the condensation experiment (such as the drop
departure diameter or frequency of sweeping events) have been determined in
the study.
Li et al. Li et al. (2022) investigated wetting and heat transfer during
condensation of water-ethanol mixtures of different compositions on PDMS-
coated substrates. By introducing a small water fraction of 10% into the vapor
phase, an increase in heat transfer coefficient of 1800% from
$3\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}\mathrm{K}$ for pure
ethanol to $57\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}\mathrm{K}$
for the mixture was observed. In addition, the coalescence velocity for
neighboring condensate droplets increased by 170% by introducing the
aforementioned water fraction, which led to a quicker re-exposure of the
subcooled wall to the vapor phase and thus to a heat transfer enhancement.
While this study lead to deeper insight into coupled surface wetting and heat
transfer enhancement, the effects responsible for the higher condensation
efficiency lay in the mixture-nature of the water ethanol composition. No
investigations on pure water as condensing fluid have been conducted in that
framework.
In spite of the increasing interest in the scientific community for heat and
mass transfer during dropwise condensation, the simultaneous quantitative
investigations of heat transfer characteristics and the characteristics of
drop dynamics during condensation, such as drop coalescence, drop departure
diameter and frequency of drop sweeping events, are very rare Li et al.
(2022); Tripathy et al. (2021). While Rose Rose (1976) investigated surface
wetting behavior during DWC, characterized by sweeping frequency, maximum drop
size and nucleation site density, the heat transfer data is only measured for
varying maximum drop sizes. No statement on the relation to heat transfer
coefficient is made for the other wetting characteristics. Merte et al. Merte
and Yamali (1983) investigated drop profile, defined by advancing and receding
contact angle, and departure diameter during DWC, but no quantitative
deductions on the heat transfer coefficient could be made. The lack of those
combined data limits the understanding of mechanisms determining the
condensation rates and, therefore, the methods for optimization of
condensation efficiency. In this work, condensation of water vapor on copper
surfaces coated with ultrathin PDMS layers is studied experimentally. The
fabrication method of PDMS layers previously developed for glass and silicon
substrates Teisala et al. (2020) has been extended and adapted to copper
surfaces which are commonly used in industrial applications. The heat transfer
characteristics during condensation are measured simultaneously with capturing
the dynamics of condensate on the substrate. This allows establishing a direct
relationship between the measured heat flux and the heat transfer coefficient,
on one side, and the wetting behaviour and dynamics of drops on the other
side. The heat transfer enhancement in comparison to uncoated copper surfaces
is reported. The ageing effects of PDMS-coated copper substrates during the
condensation experiment, which are of fundamental interest for heat exchanger
applications and their use in industry, are investigated by comparison of
contact angles before and after the condensation operation, as well as by
following the temporal evolution of characteristics of drop dynamics, such as
the drop departure dynamics and frequency of the sweeping events, during the
condensation.

##  2 Experimental methods

###  2.1 Surface fabrication and characterization

The PDMS-coating protocol involves surface preparation, coating, and
characterization. These steps are depicted in Figure 1.

(1)(2)(3)(4)(5)(6)

Figure 1: Schematic steps of the copper substrate fabrication and coating
procedure

Initially, the copper body is turned from the bar stock (1). The condensation
surface is then wet-sanded with increasing grit ranging from 400 to 2500 (2),
until a surface roughness value of $S_{a}=$31\text{\,}\mathrm{n}\mathrm{m}$$
is achieved, where $S_{a}$ represents the arithmetic mean height of the
surface. The roughness values are assessed using a 3D confocal microscope
($\mu$surf expert, NanoFocus AG). Subsequently, polishing paste is applied to
attain a mirror-like finish (3), and confocal microscopy analysis shows a
surface roughness value of $S_{a}=$11\text{\,}\mathrm{n}\mathrm{m}$$ after the
polishing process. The copper sample is then cleaned in an ultrasonic acetone
bath for 10 minutes to eliminate organic residues and flushed with DI water
afterwards. The substrate is subsequently oxidized in a ventilated muffle
furnace at $190\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 18 hours to
generate a uniform copper oxide layer on the condensation surface (4). The
substrate is again placed in an ultrasonic acetone bath for 10 minutes
following the oxidation and cleaning process, and a slight increase in surface
roughness to $S_{a}=$22\text{\,}\mathrm{n}\mathrm{m}$$ is observed.
To coat the surface with PDMS, the copper surface is first cleaned with oxygen
plasma (Diener Electronic Femto, $120\text{\,}\mathrm{W}$). Subsequently a
drop of silicone oil (Sigma-Aldrich,
$M=$11\,740\text{\,}\mathrm{g}\mathrm{/}\mathrm{m}\mathrm{o}\mathrm{l}$$) is
allowed to spread and left for $24\text{\,}\mathrm{h}$ at
$100\text{\,}\mathrm{\SIUnitSymbolCelsius}$ (5). Afterwards, excess PDMS is
removed from the surface by sonication of the sample in toluene, ethanol and
water for 10 minutes in each liquid, as described by Teisala et al. Teisala et
al. (2020), resulting in a
$$5\text{\,}\mathrm{n}\mathrm{m}$-$10\text{\,}\mathrm{n}\mathrm{m}$$ thick
hydrophobic coating.
The wetting behavior of the PDMS-coated copper surface is evaluated by
measuring the contact angle using a dataphysics® OCA 25 optical contact angle
measurement device (6). For the measurement of advancing and receding contact
angles (ARCA), a sessile water droplet is deposited on the surface and its
volume is increased from $1.7\text{\,}\mu\mathrm{l}$ to
$3\text{\,}\mu\mathrm{l}$ and then decreased back to its initial value. The
variation in droplet volume is carried out using a cannula that penetrates the
drop cap. The procedure is displayed in Figure 2. The advancing contact angle
is determined as the three-phase contact line advances during the increasing
drop volume phase, while the receding contact angle is determined as the
three-phase contact line recedes during the decreasing drop volume phase.

$V_{Drop}=$$1.7\text{\,}\mu\mathrm{l}$$V_{Drop}=$$3\text{\,}\mu\mathrm{l}$$V_{Drop}=$$1.7\text{\,}\mu\mathrm{l}$

Figure 2: Drop expansion and reduction sequence during ARCA investigation

###  2.2 Setup and procedure for the investigation of the condensation process

The experimental setup used for this study is schematically illustrated in
Figure 3. The test cell (I) is custom-made from vacuum tubing parts and filled
with water vapor at saturation conditions. The saturation conditions are
continuously monitored using temperature and pressure sensors and maintained
at a steady state by an externally controlled evaporator (II). The condensate
generated from the hydrophobic, PDMS-coated copper substrate (III) under
investigation is re-evaporated by the heater (II).

(I)(IV)(III)(II)(V)

Figure 3: Working scheme of the experimental setup

The PDMS-coated substrate (III) is sealed tightly on the subcooling subsystem
with a coolant cycle (IV), including a Coriolis mass flow sensor (TMU S008,
Heinrichs GmbH) and a chiller (Ministat 230, Huber GmbH), attached to lower
the wall temperature and promote condensation. The coated condensation surface
is monitored visually by a black and white camera (V) to record and analyze
the wetting behavior. The data is recorded with an Andor Zyla 5.5 black and
white camera with a modular Navitar Zoom 6000® lens system.
The wall subcooling subsystem, as shown in Figure 5, comprises a central
coolant feed line (1), which directs the coolant liquid to the backside of the
coated substrate, where cooling fins (2) are located. The temperature of the
coolant feed line is measured using a PT100 temperature sensor (3), while
another PT100 temperature sensor (both $3\text{\,}\mathrm{m}\mathrm{m}$
diameter, 1/10 Class B, THM GmbH) is used to measure the temperature increase
of the coolant caused by the latent heat of condensation, as it passes through
the coolant return line (4).

(1)(2)(3)(4)

Figure 4: Cut view of the wall subcooling subsystem

(A)(B)(C)(D)(E)

Figure 5: Cut view of the condensation substrate CAD model

To investigate the wetting behavior and heat transfer performance of coated
substrates during the condensation process, the copper substrate shown in
Figure 5 is utilized. The substrate has a plane face equipped with an O-ring
and a centering ring (A), which effectively seals the coolant cycle against
the pure water vapor atmosphere at saturation conditions inside the test cell.
The substrate is designed with cooling fins (2 in Figure 5, B in Figure 5) on
the backside, which enhance the single phase heat transfer between the coolant
and the copper substrate. The substrate features a visually accessible PDMS-
coated condensation surface (C). Parasitic heat fluxes through surfaces other
than the PDMS-coated condensation surface are minimized by press fitting the
copper substrate into a PEEK insulation (D). The substrate has holes for four
thermocouple type T sensors ($1\text{\,}\mathrm{m}\mathrm{m}$ diameter, Type
T, Class 1, THM GmbH), with two of them labeled (E) and the other two placed
outside of the cut plane.
The following procedure is applied during the condensation experiments: First,
the coated substrate is fixed on the coolant flange as shown in Figure 5,
tightly sealing the coolant cycle against the atmosphere inside the test cell.
Next, the vacuum pump is connected to the test cell and the attached tubing to
remove non-condensable gases (NCGs). The evaporator (II in Figure 3) is used
to generate pure saturated water vapor, which fills the inside of the test
cell and connected tubing. Subsequently, the coolant liquid temperature is
reduced using a thermostat. The wall subcooling is stepwise increased to
examine the heat transport at different levels of subcooling. In each step,
when steady state is reached, 40 measurement values are accumulated and
averaged. These averaged values are subsequently used to compare the
condensation performance for different wall subcoolings and substrates.
Saturation conditions of $60\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and
$199\text{\,}\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$ were selected for the
test campaign. The saturation conditions remain constant throughout the
experiment, while the coolant liquid temperature on the back side of the
substrate is gradually reduced in increments of $7.5\text{\,}\mathrm{K}$. The
accumulation of measurement values to be averaged starts 5 minutes after
reaching steady state. No significant temporal changes take place during the
period of data accumulation.
The coolant mass flow rate $\dot{m}_{cool}$ is measured using a Coriolis mass
flow sensor. The inlet and outlet coolant temperatures are denoted as $T_{in}$
and $T_{out}$, respectively, and are measured at locations (3) and (4) as
indicated in Figure 5 by PT100 sensors ($3\text{\,}\mathrm{m}\mathrm{m}$
diameter, uncertainty 1/10 Class B, THM GmbH). The wall temperature,
$T_{wall}$, is determined by taking an average of readings from three
thermocouples ($1\text{\,}\mathrm{m}\mathrm{m}$ diameter, Type T, TMH GmbH)
located at positions $3\text{\,}\mathrm{m}\mathrm{m}$ below the condensation
surface, as indicated by (E) in Figure 5. The vapor temperature, $T_{vap}$, is
measured using a PT100 sensor ($1.5\text{\,}\mathrm{m}\mathrm{m}$ diameter,
Class A, THM GmbH) located approximately $50\text{\,}\mathrm{m}\mathrm{m}$
away from the condenser surface. All temperature sensors are calibrated to a
reference with a measurement uncertainty of $30\text{\,}\mathrm{m}\mathrm{K}$.
The measurement frequency for all sensor signals is
$0.36\text{\,}\mathrm{H}\mathrm{z}$.
The heat flux is determined through an energy balance for the coolant liquid
on the back side of the substrate:

$\dot{q}=\dot{m}_{cool}c_{cool}(T_{in}-T_{out})/A_{cond}.$ (1)

In equation (1), the condensation surface area of the PDMS-coated copper
substrate is represented by $A_{cond}$, and $c_{cool}$ denotes the heat
capacity of the coolant liquid (Lauda Kryo 51), which is determined from the
data sheet as a function of temperature. The heat transfer coefficient,
$\alpha$, is calculated according to the following expression:

$\alpha=\dot{q}/(T_{vap}-T_{wall}),$ (2)

where the heat flux $\dot{q}$ is determined using equation (1).
In addition, images of the vertical condensation surface were captured and
analyzed to examine the surface wetting behavior during condensation. For this
purpose, a series of 80 pictures of the entire substrate is taken with a
recording frequency of $36\text{\,}\mathrm{H}\mathrm{z}$ at each subcooling
step, with a spatial resolution of 678 pixels per centimeter, and then
processed to determine the droplet departure diameter and frequency of the
roll off and sweeping events.
The drop departure diameter is a crucial parameter for assessing the mobility
of droplets and the ability of the surface to sustain DWC and renew nucleation
sites. The departure diameter is determined by measuring the diameter of a
drop about to be accelerated downwards due to gravity. This means that the
recorded image sequence is evaluated with regard to departing droplets. A
droplet is considered to be a departing droplet if it fulfills the following
criteria: First, as soon as a droplet starts to move downward, it does not
stop again due to reaching a pinning site or surface defect. Second, the
departure event is initiated by apparent continuous growth and not by
coalescence with a neighboring droplet of similar or comparable size or
sweeping by a passing droplet from the top. To obtain a representative value
for each subcooling step, three different departing drops are measured and the
diameters are averaged.
Furthermore, the frequency of sweeping events in the central rectangular
region of the substrate for each subcooling step is examined. It should be
noted that the frequency of sweeping events is highly dependent on the
location of the field of view on the substrate, since the presence of multiple
droplets above a growing drop increases the likelihood of the drop being
located in a sweeping path and therefore being removed from the surface. Here,
a region of interest of
$$8\text{\,}\mathrm{m}\mathrm{m}$\times$9.5\text{\,}\mathrm{m}\mathrm{m}$$ in
the substrate center is evaluated. Each time a fully visible droplet rolls
through the frame, the motion is registered as a sweeping event. Taking the
duration of the recording and area into account, the frequency of sweeping
events is then normalized in space and time.
In order to comprehensively assess changes in wetting behavior originating
from the condensation operation on the PDMS-coated copper surface, static,
advancing and receding contact angles, as well as the roll off angle are
measured, both, before and after consecutive experimental runs. The static
contact angle is measured by depositing a $2\text{\,}\mu\mathrm{l}$ water drop
on the substrate. To evaluate the advancing and receding contact angles, as
well as the roll off angle, a $30\text{\,}\mu\mathrm{l}$ drop is deposited on
the surface, and the surface is gradually tilted starting from the horizontal
state . Contrary to the presented ARCA procedure used to characterize the
wetting behavior on freshly PDMS-coated substrates in Figure 2, this procedure
reliably quantifies surface wetting characteristics even on strongly aged
surfaces, defined by high contact angle hysteresis and contact line pinning.
When investigating these characteristics on the strongly aged surfaces with
the aforementioned ARCA-procedure, no significant measurement values could be
obtained due to strong contact line pinning. The aforementioned parameters
allow insights in drop mobility on the PDMS-coated surface Extrand and Gent
(1990); Nan Gao et al. (2018).

##  3 Results and Discussion

###  3.1 Wetting of sessile droplets on PDMS-coated copper substrates

The results of the advancing and receding contact angle (ARCA) investigations
on freshly prepared PDMS-coated copper substrates are displayed in Figures 7
and 7.

Figure 6: Contact angle and drop volume in single ARCA investigation

Figure 7: Contact angle and drop volume in 100x repeated ARCA
investigation

Both figures depict the behavior of contact angle and drop volume during ARCA
investigation. Specifically, Figure 7 illustrates the relationship between the
contact angle and the drop volume over time for a single ARCA measurement.
During the drop expansion phase, as the drop volume increases from
$1.7\text{\,}\mu\mathrm{l}$ to $3\text{\,}\mu\mathrm{l}$, the contact angle
value rises to a peak of
$$111\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$.
Subsequently, as the drop volume remains constant at
$3\text{\,}\mu\mathrm{l}$, the contact angle value stabilizes at
$$110\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$.
During the receding phase, as the drop volume decreases back to its initial
value of $1.7\text{\,}\mu\mathrm{l}$, the contact angle value reduces to
$$109\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$,
resulting in a contact angle hysteresis of
$2\text{\,}\mathrm{\SIUnitSymbolDegree}$ for water on PDMS-coated copper
surfaces. These low hysteresis values, which are comparable with those
measured on PDMS brush coatings on glass and silicon wafers Teisala et al.
(2020) are very promising for condensation applications. Figure 7 shows the
behavior of contact angle and drop volume over 100 repeated ARCA cycles. As
drop volumes smaller than $1.5\text{\,}\mu\mathrm{l}$ tend to cause
significant drop shape distortion due to the piercing canula, resulting in
apparent contact angle deviations, the initial drop volume for repeated ARCA
measurements is increased to $3\text{\,}\mu\mathrm{l}$ and periodically
refilled after every 25 repetitions. This approach ensures that a sufficient
drop volume is maintained throughout the measurement, counteracting drop
evaporation that leads to a drop volume decrease, thus ensuring reliable and
accurate results regarding the wetting behavior of water droplets on PDMS-
coated copper surfaces.
During the first expansion phase, the drop volume is increased by
$1.5\text{\,}\mu\mathrm{l}$ to a maximum value of $4.5\text{\,}\mu\mathrm{l}$
and an advancing contact angle of
$$111\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$
is recorded. During the first reduction phase, a receding contact angle of
$$108\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$
is recorded. Until the first water drop refill after 25 repetitions, the
advancing contact angle value decreases to
$$110\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$
and the receding contact angle value decreases to
$$106\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$,
resulting in a contact angle hysteresis of
$4\text{\,}\mathrm{\SIUnitSymbolDegree}$.
During the three subsequent measurement blocks of 25 ARCA repetitions, the
advancing contact angle value reduces from
$$110\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$
to
$$109\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$,
while the receding contact angle value stays around
$$108\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$
to
$$107\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$,
resulting in a contact angle hysteresis of
$2\text{\,}\mathrm{\SIUnitSymbolDegree}$. The value obtained for the contact
angle hysteresis after 100 cycles of passing three phase contact lines is
close to the hysteresis value obtained from the single ARCA measurement,
thereby confirming the durability of the PDMS-coating on the copper surface.
These results are consistent with the wetting characteristics of PDMS-coated
glass substrates, as reported by Teisala et al. Teisala et al. (2020), who
observed advancing contact angles of
$$107\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$2\text{\,}\mathrm{\SIUnitSymbolDegree}$$,
receding contact angles of
$$105\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$2\text{\,}\mathrm{\SIUnitSymbolDegree}$$,
and a contact angle hysteresis of $2\text{\,}\mathrm{\SIUnitSymbolDegree}$.
These findings provide further evidence for the effectiveness of PDMS-coatings
in controlling surface wetting behavior.

###  3.2 Condensate wetting on PDMS-coated copper substrates and ageing
effects

In Figure 8, the surface wetting behavior of PDMS-coated and uncoated copper
surfaces during condensation operation is displayed.

$10\text{\,}\mathrm{m}\mathrm{m}$

Figure 8: Surface wetting behavior during condensation.
First row: uncoated copper substrate at wall subcoolings (left to right)
$1.6\text{\,}\mathrm{K}$, $2.9\text{\,}\mathrm{K}$, $4.9\text{\,}\mathrm{K}$,
$6.5\text{\,}\mathrm{K}$, $10.6\text{\,}\mathrm{K}$
Second row: PDMS-coated substrate on first day of condensation experiments at
wall subcoolings (left to right) $2.3\text{\,}\mathrm{K}$,
$2.8\text{\,}\mathrm{K}$, $3.7\text{\,}\mathrm{K}$, $4.9\text{\,}\mathrm{K}$,
$6.2\text{\,}\mathrm{K}$
Third row: PDMS-coated substrate on second day of condensation experiments at
wall subcoolings (left to right) $2.1\text{\,}\mathrm{K}$,
$3.6\text{\,}\mathrm{K}$, $5.0\text{\,}\mathrm{K}$, $6.4\text{\,}\mathrm{K}$,
$7.5\text{\,}\mathrm{K}$
Fourth row: PDMS-coated substrate on third day of condensation experiments at
wall subcoolings (left to right) $2.4\text{\,}\mathrm{K}$,
$3.6\text{\,}\mathrm{K}$, $5.2\text{\,}\mathrm{K}$, $6.4\text{\,}\mathrm{K}$,
$8.0\text{\,}\mathrm{K}$

In the first row, the surface wetting behavior on the uncoated surface is
shown for various wall subcoolings. Initially, even on the uncoated copper
surface, DWC is observed. The visible droplets show a strongly pinning wetting
behavior and are irregular in shape. At increasing wall subcoolings, a
transition regime is reached where some areas are covered with rivulets while
other areas are covered by a continuous liquid film. At maximum wall
subcooling, nearly all the surface is covered by the liquid film with an
exception of a small area on right side, upper part of the surface. In the
second row, the surface wetting behavior on the freshly PDMS-coated surface is
shown for various wall subcoolings. With the increasing wall subcooling (from
left to right), dropwise condensation is observed with an increasing number of
visible drops on the surface. At the highest subcooling step in the end of the
first measurement run, the drop sizes are increased compared to earlier
recordings. The increased drop size indicates surface coating ageing, leading
to reduced drop mobility and thus increased drop size before roll off. This
trend of increased drop sizes before departure continues on the second day of
condensation experiments, as illustrated in images displayed in the third row.
In addition, the formation of filmwise condensation areas on the substrate
starts at high values of subcooling.On the third day of condensation
experiments, displayed in the fourth row of Figure 8, the filmwise
condensation area grows in size until the large part of the substrate is
covered by a liquid film at the last subcooling step.
In Figure 9, the droplet sweeping events on a
$$8\text{\,}\mathrm{m}\mathrm{m}$\times$9.5\text{\,}\mathrm{m}\mathrm{m}$$
area in the substrate center on the first, the second and the third day of
experiment (top to bottom row) at increasing values of wall subcooling (from
left to right) are displayed. On all images, apart from the second image from
the left in third row, the sweeping event is displayed, and the sweeping drop
is highlighted. It can be seen that the average size of sessile condensate
droplets at the moment of sweeping changes with increasing subcooling and with
increasing surface age. On the first day of the experiment, the drops grow
comparatively little before being swept by bigger drops rolling off. On the
second day, a bigger population of larger drops can be observed on the surface
at all subcooling steps. This trend continues to even larger drops on the
third day of condensation operation. If the subcooling is fixed, the average
size of the sessile condensate drops decreases with increasing frequency of
the sweeping events. Therefore, the increasing size of the drops after the
first day of the condensation experiment is consistent with the decreasing
frequency of sweeping events with the surface ageing, which is discussed
below. It has been also found that for all three days of condensation
experiments, the number of downward moving droplets within the field of view
per time interval increases with wall subcooling, leading to a higher removal
rate of sessile condensate droplets.

$2\text{\,}\mathrm{m}\mathrm{m}$

Figure 9: Zoomed central rectangular region of substrate to detect droplet
sweeping events on PDMS-coated substrate. The sweeping drops are highlighted
with white dashed ovals.
First row: First day of condensation experiments at wall subcoolings (from
left to right) $2.3\text{\,}\mathrm{K}$, $2.8\text{\,}\mathrm{K}$,
$3.7\text{\,}\mathrm{K}$, $4.9\text{\,}\mathrm{K}$, $6.2\text{\,}\mathrm{K}$
Second row: PDMS-coated substrate on second day of condensation experiments at
wall subcoolings (left to right) $2.1\text{\,}\mathrm{K}$,
$3.6\text{\,}\mathrm{K}$, $5.0\text{\,}\mathrm{K}$, $6.4\text{\,}\mathrm{K}$,
$7.5\text{\,}\mathrm{K}$
Third row: PDMS-coated substrate on third day of condensation experiments at
wall subcoolings (left to right) $2.4\text{\,}\mathrm{K}$,
$3.6\text{\,}\mathrm{K}$, $5.2\text{\,}\mathrm{K}$, $6.4\text{\,}\mathrm{K}$,
$8.0\text{\,}\mathrm{K}$

In Figure 11, the drop departure diameter on the PDMS-coated surface is
plotted over the wall subcooling. At the beginning of the first day of
condensation experiments, droplets start to accelerate downwards due to
gravity at a diameter of $0.8\text{\,}\mathrm{m}\mathrm{m}$ during the first
three subcooling steps of $2.3\text{\,}\mathrm{K}$, $2.8\text{\,}\mathrm{K}$
and $3.7\text{\,}\mathrm{K}$. For the last two subcooling steps
($4.9\text{\,}\mathrm{K}$ and $6.2\text{\,}\mathrm{K}$), the drop departure
diameter increases to $1.3\text{\,}\mathrm{m}\mathrm{m}$. On the second day of
condensation experiments, the drop departure diameter for the first two
subcooling steps ($2.1\text{\,}\mathrm{K}$ and $3.6\text{\,}\mathrm{K}$) is
increased to approximately $3.5\text{\,}\mathrm{m}\mathrm{m}$ and later
increases further to $4.2\text{\,}\mathrm{m}\mathrm{m}$ for the wall
subcooling steps of $5.0\text{\,}\mathrm{K}$ and $6.4\text{\,}\mathrm{K}$. For
the second experiment day’s last of the five subcooling steps, no roll off
event happened during the 80 frame long recording. During the third day of the
experiment, only during the second subcooling step, a roll off event happened
during the recording time, showing an even further increased drop departure
diameter of $5.6\text{\,}\mathrm{m}\mathrm{m}$ at $3.6\text{\,}\mathrm{K}$
wall subcooling. In general, the longer the PDMS-coated substrate was used for
condensation operation, the stronger is the droplets’ mobility decreased as an
increasingly higher single drop volume becomes necessary to start the
acceleration process.
The reduction of mobility of drops can also be observed in Figure 11 when
comparing the frequency of sweeping events at different wall subcoolings for
three subsequent days of condensation operation. During the first day of
operation, the frequency of sweeping events in the substrate center is
initially rising with wall subcooling, which is expected as higher wall
subcooling causes more vapor to condense and subsequently to be drained. The
increasing number of droplets leads to a more frequent sweeping of the
surface. The sweeping is caused by draining droplets, which are accelerated
downwards due to gravity. On their way down, they merge with sessile droplets
and by that remove those from the condensation surface. The more droplets form
on the surface and accelerate downwards with gravity, the higher the frequency
of sweeping of surface area beneath the moving droplet. With a higher
frequency of sweeping events for increased wall subcooling, the time for
sessile droplets to grow and to increase in diameter reduces, which leads
firstly to smaller droplets sitting on the surface and secondly to higher
cleansing rate of nucleation sites. Both effects affect heat transfer
positively Tripathy et al. (2021); Weisensee et al. (2017).
For the last subcooling step of first day’s experiment, a decrease in
frequency of sweeping events is observed. This decrease can be attributed to
surface ageing effects. This suggestion is supported by increasing of
condensate drops in size, as can be seen in Figure 8, first picture from the
right in the second row. The trend towards reduced frequencies of sweeping
events continues after the first day of the condensation experiment. During
second and third day of the condensation experiment the overall values of
sweeping events is strongly decreased compared to the first day’s values. On
second and third day, the frequency of sweeping events increases with wall
subcooling, but follows similar trends and values during both days, indicating
no more changes in condensate-surface wetting behavior in the investigated
central substrate region after the first day of condensation operation.

Figure 10: Drop departure diameter over wall subcooling

Figure 11: Frequency of sweeping events over wall subcooling

To further characterize the condensation operation induced changes in wetting
behavior, contact angle measurements have been conducted on the PDMS-coated
surface prior to and after each day of experiment. The results of measurements
of the static contact angle as well as of the roll off angles are shown in
Figure 12.

Figure 12: Static contact angle, contact angle hysteresis and roll off angle
over time of operation

The initial values for static contact angle of
$$108\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$
and contact angle hysteresis of
$$2\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$
prior to the first experimental campaign match the values displayed in Figures
7 and 7. In addition, a roll off angle of
$$8\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$4\text{\,}\mathrm{\SIUnitSymbolDegree}$$
is determined for a $2\text{\,}\mu\mathrm{l}$ drop.
After 8 hours of operation, the static contact angle decreases to
$$103\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$2\text{\,}\mathrm{\SIUnitSymbolDegree}$$,
while a contact angle hysteresis of
$$41\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$3\text{\,}\mathrm{\SIUnitSymbolDegree}$$
is determined. Here, a drop volume of $30\text{\,}\mu\mathrm{l}$ is used as no
roll off was observed for smaller drops. A roll off angle of
$$56\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$5\text{\,}\mathrm{\SIUnitSymbolDegree}$$
is determined.
While the trend for decreasing static contact angles with increasing hours of
operation continues to
$$92\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$14\text{\,}\mathrm{\SIUnitSymbolDegree}$$
after 24 hours, contact angle hysteresis cannot be determined for higher
experimental durations as no drop roll off was observed anymore for
$90\text{\,}\mathrm{\SIUnitSymbolDegree}$ surface inclination and
$30\text{\,}\mu\mathrm{l}$ drops.
In this study, three different measures have been used to quantify the surface
wetting behavior during and after the condensation experiment and to
characterize the surface ageing. The drop departure diameter is nearly
independent of subcooling, but strongly changes from day to day of
condensation operation. On the first day of condensation operation, the
frequency of sweeping events initially strongly increases with increasing wall
subcooling, reaches a maximum and decreases at the last subcooling step. On
the second and third day, the increase of the frequency of the sweeping events
with the subcooling is much weaker. Moreover, no substantial change of the
behavior of the sweeping frequency from the second to the third day has been
observed. Contact angle measurements show a decrease in static CA and an
increase in CA hysteresis, indicating a reduction in the PDMS coating’s
hydrophobicity. The results of all three characterization methods prove a
strong change of the surface after the first day of the condensation
experiment. The contact angle, contact angle hysteresis and drop departure
diameter indicate a further change of the surface quality after the second day
of the experiment. However, these changes did not affect the frequency of the
sweeping events. The relationship between the reported characteristics will be
investigated in the future.
To determine the relevance of the different characteristics of wetting
behavior for condensation heat transfer, in the next section we compare the
trends observed for wetting characteristics with the trends of the
condensation heat transfer parameters.

###  3.3 Heat transfer

In order to assess the PDMS-coating’s impact on condensation heat transfer
performance, the heat flux and heat transfer coefficient of both PDMS-coated
and uncoated substrates are investigated. The uncoated substrates are
fabricated using a process similar to that used for the PDMS-coated
substrates, with the exception of the final PDMS-coating step. In Figure 14,
the heat flux over wall subcooling on both investigated surfaces is displayed.
For the PDMS-coated surface, the heat flux for three consecutive days of
experiments are displayed. On the polished and oxidized copper surface,
represented by purple markers, an increase of heat flux over wall subcooling
until a maximum of $247\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}$
at $10.7\text{\,}\mathrm{K}$ was observed. On the PDMS-coated copper
substrate, increasing wall subcooling leads to a rise in heat flux on each
measurement day. On the first day of experiment (blue markers), a
significantly higher heat flux has been measured compared to the uncoated
surface. On this day, a maximum heat flux of
$327\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}$ was reached at a
wall subcooling of $6.2\text{\,}\mathrm{K}$. On the second day of experiment
(red markers), the transferred heat fluxes are reduced compared to the first
day of experiment. The maximum value of
$314\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}$ was reached at a
wall subcooling of $7.5\text{\,}\mathrm{K}$. On the third day of operation
(black markers) the heat no significant changes of the heat flux compared to
the second day of operation were observed, and the maximum heat flux value of
$307\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}$ was reached at a
wall subcooling of $8.0\text{\,}\mathrm{K}$. Compared to an uncoated surface,
the heat flux on a freshly coated substrate is increased by a factor of 1.6 at
$5.0\text{\,}\mathrm{K}$ wall subcooling and a factor of 1.5 at
$6.2\text{\,}\mathrm{K}$ wall subcooling. On second and third day of
operation, the transferred heat flux is increased by a factor of only 1.1 at
$5.0\text{\,}\mathrm{K}$ wall subcooling and at $6.2\text{\,}\mathrm{K}$ wall
subcooling.

Figure 13: condensation heat flux over wall subcooling
for PDMS-coated and uncoated copper surfaces

Figure 14: condensation heat transfer coefficient over wall
subcooling for PDMS-coated and uncoated copper surfaces

In Figure 14, the heat transfer coefficient over wall subcooling on coated and
uncoated surfaces is displayed together with the result of the Nusselt model
for the filmwise condensation heat transfer. On the polished and oxidized
copper surface, represented by purple markers, an initial rise in heat
transfer coefficient from
$39\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}\mathrm{K}$ at
$1.5\text{\,}\mathrm{K}$ up to
$43\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}\mathrm{K}$ at
$2.8\text{\,}\mathrm{K}$ is followed by a decrease down to
$23\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}\mathrm{K}$ at
$10.7\text{\,}\mathrm{K}$.
On the freshly PDMS-coated copper substrate, represented by blue markers, an
initial rise in heat transfer coefficient from
$26\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}\mathrm{K}$ at
$2.3\text{\,}\mathrm{K}$ up to
$55\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}\mathrm{K}$ at
$3.7\text{\,}\mathrm{K}$ is followed by a decrease down to
$53\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}\mathrm{K}$ at
$6.2\text{\,}\mathrm{K}$. On the second day of operation, represented by red
markers, the heat transfer coefficient continuously rises from
$28\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}\mathrm{K}$ at
$2.1\text{\,}\mathrm{K}$ up to
$42\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}\mathrm{K}$ at
$7.5\text{\,}\mathrm{K}$. On the third day of operation, represented by black
markers, the heat transfer coefficient results match those of second day of
operation.
The heat transfer coefficient enhancement factors are equal to those presented
on heat flux above. Namely, the enhancement factor of 1.5-1.6 has been
observed for the freshly coated surface and an enhancement factor of 1.1
during the second and the third day of condensation operation. Indicated in
green, the analytical solution of heat transfer coefficient during filmwise
condensation by Nusselt is displayed Nusselt (1916). The heat transfer
coefficient as computed by the Nusselt model weakly decreases with increasing
of subcooling. The deviations of heat transfer coefficient measured on an
uncoated copper surface from the predictions of Nusselt’s model can be
explained by analysing the wetting behavior during the condensation of water
vapor on this surface. The model of Nusselt has been developed under an
assumtion that the surface is fully covered by a condensate film, which drains
down the surface in a laminar flow. These assumptions are obviously not valid
for the flow pattern on the uncoated copper surface, as can be seen in Figure
8, top row.
When comparing the heat transfer performance of the PDMS-coated surface to the
uncoated one, the initial similarity at low $\Delta T$ can by explained by
comparing both surfaces’ wetting behavior at low subcoolings. Both surfaces
show DWC mode. Even though condensate droplets on the uncoated surface are
comparatively large, the absence of hydrophobic coating’s additional thermal
resistance leads to an increased heat transfer and compensates the decrease of
heat transfer due to increased drop size.
At higher values of the wall subcooling, the uncoated surface is increasingly
covered by condensate, forming a liquid film separating large parts of the
substrate from the vapor phase, resulting in a decreasing of the heat transfer
coefficient with increasing of wall subcooling. Concerning the PDMS-covered
surface, the continuously increasing heat transfer coefficient with the wall
subcooling can be explained by considering the sustained DWC mode at all
subcooling levels and only limited increasing of the surface coverage with
condensate, except for the highest values of the wall subcooling at the third
day of condensation experiment (see Figure 8, bottom row). At higher wall
subcoolings, the sweeping frequency is increased due to increased mass flow of
condensate, which leads to less time needed for clearing drop nucleation sites
and resetting the droplet nucleation process. The frequent clearing of the
surface and resetting the nucleation process promotes the improvement of heat
transfer Weisensee et al. (2017); Sablowski et al. (2022).
Indeed, one can observe that the measured heat transfer coefficient follows
the trend of the frequency of the sweeping events (see Figure 15). In
particular, the dependence of the heat transfer coefficient and of the
frequency of sweeping events on the wall subcooling is similar during the
first day of experiment. Followed by a strong initial increase, both the heat
transfer coefficient and the sweeping frequency reach a maximum value at
approximately the same value of subcooling and decrease with the further
increasing of subcooling. During the second and the third day of the
condensation experiment, both the wetting behavior and the heat transfer are
affected by the ageing of the surface coating. The heat transfer coefficient
and frequency of sweeping events show no significant changes from the second
to the third day of experiment. The heat transfer coefficient and the
frequency of the sweeping events stay the same as in the first day for the
lowest value of the subcooling. The ageing effect manifests itself in a much
slower increasing of the heat transfer coefficient and of the frequency of the
sweeping events with the subcooling. It can be seen that at the subcooling of
around 6.4 K, the increasing trend of heat transfer coefficient and of the
frequency of the sweeping events terminates.
The results presented in Figure 15 indicate that the heat transfer coefficient
during the dropwise condensation correlates with the frequency of sweeping
events which can be determined from optical data. The conditions and
limitations of this interrelation will be determined in future experiments in
an increased range of experimental parameters. Although the above correlation
can be rationalized on the basis of principles of the condensation heat
transfer, quantitative prediction of the interrelation requires a dedicated
theoretical and numerical model.
The reasons of the ageing of the PDMS-coating, as well as the measures for the
improvement of its robustness are presently under investigation. The
simultaneous analysis of the heat transfer and wetting behavior
characteristics, in particular the frequency of sweeping events, as a function
of the subcooling and the duration of condensation operation, as proposed in
the present work, is a valuable tool for monitoring the quality of the coating
and its suitability for heat transfer applications.

Figure 15: Heat transfer coefficient and frequency of sweeping events over
wall subcooling

##  4 Summary and conclusions

In this study, condensation of water on polydimethylsiloxane (PDMS)-coated
copper surfaces is investigated. The presented surface fabrication method
results in a contact angle value of
$$111\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$
during drop expansion and a stable value of
$$110\text{\,}\mathrm{\SIUnitSymbolDegree}$\pm$1\text{\,}\mathrm{\SIUnitSymbolDegree}$$
during constant drop volume. The contact angle hysteresis of
$2\text{\,}\mathrm{\SIUnitSymbolDegree}$ and its ultrathin coating thickness
of $$5\text{\,}\mathrm{n}\mathrm{m}$-$10\text{\,}\mathrm{n}\mathrm{m}$$
indicate the PDMS-coating’s suitability for dropwise condensation (DWC)
applications. Repeated advancing and receding contact angle (ARCA)
investigations support these findings, with consistent contact angle
hysteresis of $2\text{\,}\mathrm{\SIUnitSymbolDegree}$ over 100 repetitions.
These results align with previous studies on PDMS-coated glass substrates,
reinforcing the effectiveness of PDMS coatings in promoting the dropwise
condensation.
The PDMS-coated surface’s wetting behavior during and after condensation
experiments is investigated and compared to an uncoated surface. On the
uncoated surface, DWC at low subcooling values is observed, followed by a
transition regime with rivulets and liquid film coverage at higher
subcoolings. On the PDMS-coated surface, initial DWC behavior is observed in a
wide range of the subcooling values. Drop departure diameter remains constant,
independent of subcooling, but increases on the second day and again on third
day, limiting the droplet mobility. The frequency of sweeping events depends
on the wall subcooling and is strongly affected by the surface ageing. On the
first day of the experiment, higher subcoolings lead to more frequent sweeping
events, until a maximum of the sweeping frequency is reached. After that the
frequency decreases with a further increase of the wall subcooling. On the
second day of the condensation experiments, the sweeping frequency is reduced.
No substantial changes of the frequency have been observed on the third day in
comparison with the first day. Static contact angle investigations and drop
departure diameter reveal a decreasing surface wettability during the second
and third day of operation.
Finally, the condensation heat transfer performances of PDMS-coated and
uncoated substrates are compared with each other by evaluating the heat flux
and heat transfer coefficient. On the first day of the condensation
experiment, the heat flux on the PDMS-coated surface reaches a maximum of
$327\text{\,}\mathrm{k}\mathrm{W}\mathrm{/}\mathrm{m}^{2}$ at a wall
subcooling of $6.2\text{\,}\mathrm{K}$, 1.5-times that of the uncoated
surface. On the second and third day of condensation experiments, the heat
transfer enhancement factor reduces to a constant 1.1 compared to the uncoated
surface. It has been found that the behavior of heat transfer coefficient
correlates with the behavior of the frequency of the sweeping events on all
three days of the condensation experiment.

###### Acknowledgements.

The authors acknowledge the financial support by the German Research
Foundation - Project number 265191195 - subproject C03.

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where Eq. (<ref>) follows from $\pixsc^{\top} \ths \leqslant 1$, $r_l \leqslant \rbti \leqslant 1$, and $ - \zeta_{\ti}^{\top} \ths \leqslant |\zeta_{\ti}^{\top} \ths| \leqslant \|\zeta_{\ti}\| \|\ths\| \leqslant 1$. Eq. (<ref>) is derived through the analysis of the Agents' term in Eq. (<ref>), which is given in Appendix <ref>. Given that $|N_T| = T - |N_T^c|$, the remaining section of our proof focuses on determining the upper bound and lower bound for $|N_T^c|$.

§.§ Proof of Theorem <ref>

In Algorithm <ref>, the upper bound and lower bound of $|N_T^c|$ are given by

\begin{align*}
\scalemath{0.95}{|N_T^c|} &\scalemath{0.95}{\geqslant \frac{4}{M} \log(\frac{d}{\delta}) - \frac{(\frac{2 (2 - \alpha) \beta_{0, i}}{\alpha \rbtaui})^2 - (\lambda + M T)}{2 M \rho (1 - \rho)}} \scalemath{0.95}{- \sqrt{\frac{16}{M^2} \log^2(\frac{d}{\delta}) - \frac{4 ((\frac{2 (2 - \alpha) \beta_{0, i}}{\alpha \rbtaui})^2 - (\lambda + M T)) \log(\frac{d}{\delta})}{M^2 \rho (1 - \rho)}}} \\
\scalemath{0.95}{|N_T^c|} &\scalemath{0.95}{\leqslant \sqrt{(\frac{4 \sqrt{6}}{M} \log(\frac{d}{\delta}))^2 - \frac{16 ((\frac{2 (2 - \alpha) \beta_{0, i}}{\alpha \rbtaui})^2 - (\lambda + M T)) \log(\frac{d}{\delta})}{M^2 \rho (1 - \rho)}}}.
\end{align*}

By using the same approach as shown in Lemma <ref>, Lemma <ref>, and Theorem <ref>, we can derive the upper and lower bounds for $|N_T^c|$, given by

\begin{align*}
|N_T^c| &\geqslant \frac{4}{M} \log(\frac{d}{\delta}) - \frac{(\frac{2 (2 - \alpha)}{\alpha \rbtaui} (\sigma \sqrt{2 \log \frac{1}{\delta}} + \lambda^{\frac{1}{2}}))^2 - (\lambda + M T)}{2 M \rho (1 - \rho)}\\
&- \sqrt{\frac{16}{M^2} \log^2(\frac{d}{\delta}) - \frac{4 ((\frac{2 (2 - \alpha)}{\alpha \rbtaui} (\sigma \sqrt{2 \log \frac{1}{\delta}} + \lambda^{\frac{1}{2}}))^2 - (\lambda + M T)) \log(\frac{d}{\delta})}{M^2 \rho (1 - \rho)}} \\
|N_T^c| &\leqslant \sqrt{(\frac{4 \sqrt{6}}{M} \log(\frac{d}{\delta}))^2 - \frac{16 ((\frac{2 (2 - \alpha)}{\alpha \rbtaui} (\sigma \sqrt{2 \log \frac{1}{\delta}} + \lambda^{\frac{1}{2}}))^2 - (\lambda + M T)) \log(\frac{d}{\delta})}{M^2 \rho (1 - \rho)}}.
\end{align*}

In the following analysis, we will combine the findings of Proposition <ref> and Theorem <ref> to establish the cumulative regret for Algorithm <ref>. Determining the communication cost follows the same approach described in Theorem <ref>.

The cumulative regret of Algorithm <ref> with $\beta_{\ti} = \beta_{\ti}(\sqrt{1 + \sigma^2}, \delta / 2)$ is bounded at round $T$ with probability at least $1 - M \delta$ by

\begin{align*}
\R (T) &\leqslant 4 \beta_T \sqrt{d \bar{c}^{\prime} \log (MT)} + 4\beta_T \sqrt{MTd \log MT} \log(MT) + \sqrt{2 \bar{c}^{\prime} \log (\frac{2}{\delta})} + \bar{c}^{\prime \prime} (2 \rho + 1 - r_l),
\end{align*}

where $\bar{c}', \bar{c}'' >0$ is given by

\begin{align*}
&\bar{c}^{\prime} = M T - 4 \log(\frac{d}{\delta}) + \frac{(\frac{2 (2 - \alpha)}{\alpha \rbtaui} (\sigma \sqrt{2 \log \frac{1}{\delta}} + \lambda^{\frac{1}{2}}))^2 - (\lambda + M T)}{2 \rho (1 - \rho)}\\
&+ \sqrt{16 \log^2(\frac{d}{\delta}) - \frac{4 ((\frac{2 (2 - \alpha)}{\alpha \rbtaui} (\sigma \sqrt{2 \log \frac{1}{\delta}} + \lambda^{\frac{1}{2}}))^2 - (\lambda + M T)) \log(\frac{d}{\delta})}{\rho (1 - \rho)}} \\
&\bar{c}^{\prime \prime} = \sqrt{(4 \sqrt{6} \log(\frac{d}{\delta}))^2 - \frac{16 ((\frac{2 (2 - \alpha)}{\alpha \rbtaui} (\sigma \sqrt{2 \log \frac{1}{\delta}} + \lambda^{\frac{1}{2}}))^2 - (\lambda + M T)) \log(\frac{d}{\delta})}{\rho (1 - \rho)}}.
\end{align*}

Further, for $\delta = \frac{1}{M^2 T}$, Algorithm <ref> achieves a regret of $O(d \sqrt{MT} \log^2 T)$ with $O(M^{1.5} d^3)$ communication cost.

The proof methodology used to prove this theorem is  the same as that applied for Theorem <ref> in terms of both cumulative regret and communication cost.

$M=1, \alpha=0.5$

Movielens-100K dataset: In Figure <ref>, we set the parameters as $R=0.1$, $K=50$, $M=1$, $\ths = \frac{1}{\sqrt{3}}[1, 0, 0, 0, 1, 0, 0, 0, 1]$, and noise variance $=10^{-2}$, and the baseline action is set by the 40$^{\rm th}$ best action . In  Figure <ref>, the parameters are set as $R=0.1$, $K=50$, $M=3$, the reward parameters for the different tasks lie in $\Theta=\{\frac{1}{\sqrt{3}}[1, 0, 0, 0, 1, 0, 0, 0, 1], \frac{1}{\sqrt{3}}[0, 0, 0, 0, 1, 0, 0, 0, 1], \frac{1}{\sqrt{3}}[1, 0, 0, 0, 0, 0, 0, 0, 1], \frac{1}{\sqrt{3}}[1, 0, 0, 0, 1, 0, 0, 0, 0]\}$, noise variance $=10^{-2}$, and the baseline action of a particular round is set as the 40$^{\rm th}$ best action of that round.The $\alpha$ values are varied as $\alpha = \{0.1, 0.3, 0.5\}$.

Jiabin edit: Synthetic data: In our experiments, we consistently set the parameters $R=1$, $d=2$, and the number of contexts $|\C|=100$. Feature vectors $\phi$ are randomly generated using a standard normal distribution, ensuring that the rewards lie between 0 and 1. We considered noise with a mean of $0$ to obtain $\psi$ from $\phi$. To compare different numbers of agents, the parameters are $\lambda=1$, $K=90$, $M=\{3,5,10\}$, $\ths = [1, 1]$, the baseline action is set by the $30$-th best action, rewards parameters lie in $\Theta=\{[1, 1], [1, 0],[0, 1]\}$ and a noise variance of $0.01$ to generate $\psi$. To compare different alpha values, the settings are $\lambda=0.1$, $K=90$, $M=3$, $\alpha = \{0.1, 0.3, 0.5\}$, $\ths = [1, 1]$, the baseline action is set by the $80$-th best action, and noise variance of $0.0001$ to generate $\psi$. To compare different algorithms, we set $\lambda=0.1$, $K=40$, $M=1$, $\ths = [0.9, 0.4]$, the baseline action is set by the $10-$th best action, and a noise variance of $0.000025$ to generate $\psi$. Experiments with $M=1$ are averaged over 100 trials, while those with $M>1$ are averaged over 25 trials.

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