International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
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International Journal of Heat and Mass Transfer
journal homepage: www.elsevier.com/locate/ijhmt
Microdroplet evaporation on superheated surfaces
Shawn A. Putnam a,b,⇑, Alejandro M. Briones c,d, Larry W. Byrd c, Jamie S. Ervin c,d, Michael S. Hanchak c,d,
Ashley White c,d, John G. Jones a
a
Air Force Research Laboratory, Materials and Manufacturing Directorate, Thermal Sciences and Materials Branch, Wright-Patterson AFB, OH 45433, United States
Universal Technology Corporation, 1270 N. Fairfield Rd., Dayton, OH 45432, United States
c
Air Force Research Laboratory, Propulsion Directorate, Thermal and Electrochemical Branch, Wright-Patterson AFB, OH 45433, United States
d
University of Dayton Research Institute, University of Dayton, Dayton, OH 45410, United States
b
a r t i c l e
i n f o
Article history:
Received 13 December 2011
Received in revised form 21 May 2012
Accepted 22 May 2012
Available online 30 June 2012
Keywords:
Evaporation
Water
Microdroplet
Numerical simulation
Structured surfaces
Depinning
a b s t r a c t
A comprehensive experimental investigation on water microdroplet evaporation is presented from the
standpoint of phase-change cooling technologies. The study investigates microdroplet evaporation on a
variety of different surface materials at surface temperatures ranging from 25 °C to 250 °C. The temporal
evolution of the droplet profile and contact line dynamics are measured using high-speed photography
and image analysis methods. The material systems tested consist of (1) thin-film surface coatings on glass
substrates (e.g., Al, Ti, Cu, and SAMs) and (2) Cu substrates with different surface morphologies (e.g., mirror polished, 800-grit sanded, and micron-sized Cu pillar arrays). As expected, changes in surface energy
influences the contact line dynamics. Changes in surface energy, however, showed no systematic influ_ LG Þ scales linearly with
ence on the evaporation efficiency. For all systems studied, the evaporation rate ðm
_ LG / R) and stick-slip contact line dynamics are observed. For
the microdroplets contact radius (i.e., m
example, the evaporation efficiency reduces after contact line depinning due to a reduction in the total
_ LG is a constant and directly
length of the solid-liquid-vapor contact line, whereas, before depinning, m
proportional to the surface temperature, contact radius, and substrate thermal conductivity. For micro_ max
droplet evaporation on thin-film surfaces, maximum evaporation rates ðm
LG ffi 32 8 lg/s) and evaporative heat fluxes (qmax ffi 600 ± 100 W/cm2) are observed at superheats of 70 °C [ DT [ 130 °C. These
maxima in the evaporative heat transfer performance signify a transition in the heat transfer process
from a purely microdroplet evaporation regime to a droplet/film boiling regime (which is analogous to
the critical heat flux observed in pool boiling). For microdroplet evaporation on Cu substrates, droplet/film
boiling occurs at much lower superheats (e.g., 10 °C [ DT [ 25 °C); yet, comparable maximum evapora2
_ max
tion rates (m
LG ffi 36 8 lg/s) and evaporative heat fluxes (qmax ffi 750 ± 150 W/cm ) are observed. In
_ max
ffi
35 5 lg/s during
short, this work suggests a reliable upper limit for the evaporation efficiency of m
LG
water microdroplet evaporation on superheated surfaces (which turns out to be independent of substrate
thermal conductivity, surface structure, and surface hydrophobicity).
Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Many of the today’s cooling technologies have stringent
requirements to accomplish one of two challenges: (1) reliably
avoid critical heat flux (CHF) while efficiently dissipating enormous
amounts of heat from very small areas [1] (e.g., heat fluxes beyond
2000 W/cm2) or (2) regulate system temperatures within sometimes fractions of a degree over a broad range of operation loads/
conditions (e.g., active, not passive, temperature control is imperative during operation because of irregularities in the environment
and changing power loads). These requirements are especially
stringent for the new generation of high power electronics, lasers,
⇑ Corresponding author.
E-mail address: [email protected] (S.A. Putnam).
0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.076
propulsion, and energy conversion systems, where the cooling efficiency is dictated by both the cooling configuration and the coolant
itself. To date, spray cooling with water is arguably the most effective cooling process due to its ability to: (1) uniformly remove
large heat fluxes, (2) use small fluid volumes, (3) take advantage
of large heat of vaporization, (4) use low droplet impact velocity,
and (5) provide optimal control and regulation of system
temperatures[2].
Not only the liquid coolant, but also the surface properties and
surface-liquid interactions are important. This is especially true at
micro- and nano-scale dimensions, where interactions at liquid
interfaces can have a dramatic influence on the macroscopic properties of the system, dictating physical properties like the fluid viscosity, thermal conductivity, and surface tension. Over the past
several years, several groups have shown that micro- and nano-
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S.A. Putnam et al. / International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
Nomenclature
A
cp
Dh2 oair
D
F
G
h
h
kB
LV
Mw
_ LG
m
NA
p
P
q
r
R
RH
t
T
*
v
V
We
z
surface or contact area (m2)
specific heat capacity at constant pressure (J/kg/K)
binary diffusivity of water vapor to air (m2/s)
droplet diameter & contact diameter (lm)
surface tension force per unit length of contact line
(mN/m)
interfacial Gibbs free energy (J)
heat transfer coefficient (kW/m2/K)
droplet height at centerline (lm)
Boltzmann’s constant (1.3806503 10-23 m2 kg/s2/K)
latent heat of vaporization for water (2.26 106 J/kg)
molecular weight (g/mol)
evaporation rate or liquid-to-gas mass transfer rate (lg/
s)
Avogadro’s number (6.022137 106 J/kg)
gauge pressure (Pa)
pressure (Pa)
heat flux (W/cm2)
radial direction at centerline along the surface (lm)
droplet radius & contact radius (lm)
relative humidity (%)
time & droplet lifetime (s)
temperature (K or °C)
velocity (m/s)
droplet volume (pL)
Weber number
z-direction at centerline (lm)
Acronyms
CHF
critical heat flux (W/cm2)
FS
fused silica
VOF
volume of fluid
Superscript
exp
experiment
max
maximum
sim
simulation
structures can enhance the overall heat transfer coefficient in pool
boiling [3–5], flow boiling [6,7], spray cooling [8–10], heat pipes,
and post-CHF film boiling [11]. Nanowire-like structures have also
been predicted to enhance the long-range van der Waals interactions at solid interfaces [12,13], which, in turn, dictate the formation of specific liquid-wetting [14] and gas-spreading [15] states
at a liquid-solid interface. However, a cogent explanation of the
operative mechanisms that cause enhancements in heat transfer
performance has not been established.
Insight into the fundamental mechanisms responsible for wetting, fluid dynamics, and two-phase heat transfer can be obtained
from single droplet studies. In this regard, single droplet studies
are related to all mainstream multi-phase cooling processes (e.g.,
evaporation, spray and jet impingement cooling, flow and pool boiling, heat pipes, and thermosyphons). Thermal transport during single droplet impingement and evaporation on a heated surface is,
however, a very complex phenomenon involving a multitude of
physical-chemical events such deposition, spreading, receding, rebound, jetting, splash, contact line pinning, depinning, fluid and vapor convection, air and vapor bubble entrapment, and vapor bubble
nucleation. There is a relatively vast literature on these topics, and
we refer the reader to Refs. [16–30] and the citations therein.
This work focuses on surface or interface cooling during evaporation with small, water microdroplets – e.g., microdroplets with
Greek symbols
|raG|
droplet interfacial surface area density (m2)
evaporation ratio
bLG
DT
superheat (K)
dF
depinning force (mN/m)
dG
excess free energy (J)
dGl
excess free energy per unit length of contact line (J/m)
dGA
excess free energy per unit solid-liquid surface area (J/
m2)
dT
Tolman length (nm)
e
accommodation coefficient or evaporation efficiency
c
surface tension (mN/m) or surface energy density (J/m2)
j
curvature (m-1)
k
mean-free-path (m)
K
thermal conductivity (W/m/K)
l
dynamic viscosity (kg/m/s)
q
density (g/cm3 or kg/m3)
p_ LG
evaporation rate per unit length of contact line (kg/s/m)
h
contact angle (deg)
sD
depinning time (s)
sE
total evaporation time or droplet lifetime (s)
/
volume fraction
w
mass fraction
Subscripts
0
initial condition & contact condition
air
air/gas
h2o
water
D
depinning
im
impingement
Eq
equilibrium
G
gaseous-phase
L
liquid-phase
LF
Leidenfrost
S
surface & solid-phase
SC
spherical cap
V
vapor-phase
initial volumes (radii) less than 2500 pL (85 lm). To minimize
the role of surface structure, microdroplets evaporating on smooth,
metal thin-film surfaces are studied. Evaporation studies on microstructured and micro-roughened surfaces are also conducted for
comparison. We find, for these small water droplets, that the key
mechanisms/properties influencing the heat transfer performance
are the droplet contact line dynamics (i.e., pinning and depinning),
substrate thermal conductivity, and air/vapor bubble entrapment
and nucleation.
The goal of this work is to characterize how the interfacial heat
transfer performance is coupled to the contact line dynamics during microdroplet evaporation on surfaces heated up to the Leidenfrost temperature. With this said, the aim is to improve single
droplet impact dynamics and evaporation models for later
enhancement of spray cooling models. To accomplish this, we
combine numerical simulations with high-speed photography
measurements. High-speed photography facilitates characterization of droplet wetting behavior, which is coupled to the evaporation efficiency. The experiments and simulations are conducted in
parallel with small water microdroplets to facilitate later advancements in modeling methods in a reasonable time frame.
The remainder of the manuscript is organized as follows: In Section 2, the experimental apparatus, measurement technique, and
data analysis methods are described. Section 3 presents the
S.A. Putnam et al. / International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
numerical model used to simulate microdroplet evaporation on an
isothermal surface. Section 4.1 covers, in detail, microdroplet evaporation on Al thin-films with surface temperatures up to 250 °C.
Section 4.2 overviews the heat transfer performance in terms of
the heat flux and evaporation rate for a variety of different thinfilm surfaces (e.g., Al, Cu, Ti, and SAM thin-film coatings on glass
substrates). Section 4.3 overviews the heat transfer performance
during microdroplet evaporation on solid Cu substrates with different surface structures. In Section 4.4, the role of contact line
depinning is discussed in terms of its influence on the evaporation
efficiency, where calculations of the depinning force and free energy barriers for contact line depinning are made for all the material systems tested. The conclusions of this work are provided in
Section 5.
2. Experimental methods
2.1. Materials and sample fabrication methods
The materials systems studied in this work consists of micronsized water droplets in contact with (1) Al thin-films, (2) Ti
thin-films, (3) Cu thin-films, (4) self-assembled monolayer (SAM)
coatings, (5) polished Cu substrates, and (6) micro-structured Cu
substrates. For all experiments, deionized (DI) water purified and
filtered with a Barnstead Nanopure Water Purification System is
used. All metal thin films are deposited by magnetron sputtering
on flat fused silica (FS) glass windows [31]. The metal thin-film
thicknesses ranged between 60 and 80 nm with surface roughness
(Ra) less than Ra [ 0.01 lm. The polished Cu substrates are
2 mm thick oxygen-free Cu discs (1’’ in diameter) with surface
roughness of either Ra ffi 0.05 lm (800-grit finish) or Ra [ 0.01 lm
(mirror finish). The micro-structured Cu sample [32] consists of a
simple-cubic array of circular-pillars separated by 20 lm with a
height and diameter of 20 lm and 8.5 lm, respectively. A transmission-electron-micrograph (TEM) of the pillared Cu surface is
provided in Fig. 13. The SAM surface consists of an octadecyl-trimethoxysilane monolayer chemically grafted to an Al thin-film
on a glass substrate. The SAM surface is hydrophobic. For preparation, an Al/glass sample is immersed in a toluene solution containing 0.01 M octadecyl-trimethoxysilane (SAM layer) and 0.01 M
triethylamine (catalyst) and then left overnight to self-assemble.
The silane reacted and covalently bonded to hydroxyls on the Al
surface while the 18-carbon chains drove self assembly and formed
the hydrophobic SAM layer.
2.2. Microdroplet impingement and evaporation measurements
Data is acquired with a standard dispensing/imaging system
shown in Fig. 1. The micron-sized water droplets are generated
with an ink-jet dispensing unit [33]. In short, the ink-jet consists
5795
of a cylindrical piezo element surrounding a glass capillary with
a specific inner nozzle diameter. Three different ink-jets with inner
nozzle diameters of either 120 lm, 80 lm, or 50 lm are used. In
practice, the diameter of a spherical water droplet dispensed from
the ink-jet equals the inner nozzle diameter (within a few microns). Experiments comprise of dispensing single microdroplets
on a sample having a controlled surface temperature. Data for multiple droplets impinging on top of each other are omitted from
analysis (unless stated otherwise).
Video recordings during microdroplet impingement and evaporation are captured with a high-speed camera [34]. The high-speed
camera is coupled with imaging optics resembling a standard compound microscope setup (i.e., eye-piece lens and microscope objective). The camera is equipped with an internal 8 Gbit buffer,
1280 800 megapixel CMOS sensor, and programmable electronic
shutter with a minimum exposure time of 300 ns. Evaporation and
impingement data is acquired with video acquisition rates ranging
between 1200 fps and 120,192 fps. An exposure time (electronic
shutter setting) of 10 ls is used in all experiments. The compound
microscope optics consists of an achromatic eye-piece lens
(120 mm focal length and 30 mm lens diameter) and a Mitutoyo long working distance microscope objective. Image resolution
is then based on the magnification of the microscope objective selected. For example, both 20X and 50X objectives are used, facilitating image resolutions on the megapixel senor of 1.29 lm/
pixel and 0.667 lm/pixel, respectively.
External TTL triggers from a computer controlled pulse generator are used to coordinate droplet dispensing and video recordings.
The data acquisition procedure for measuring a single evaporation
event follows: First, a digital TTL trigger initiates simultaneously
(1) high-speed video recording and (2) microdroplet dispensing.
Then, a 2nd TTL trigger initiates saving the acquired data. A final
TTL trigger then initiates rotation of the Al-glass sample on the
sample stage for the next droplet evaporation event.
The surface temperature (TS) of the sample is regulated with an
Omega temperature controller using a platinum RTD (resistancetemperature-device) and a 50 O resistive heating element/wire
contained within a custom X, Y, Z, h sample-stage assembly. This
custom sample-stage assembly is shown in Fig. 1b, where the
RTD and resistive heating wire are part of the black sample-base
located between the sample-stage (upper-arrow) and the sample.
A temperature calibration was performed (and routinely checked)
to correlate the surface temperature with the set-point temperature of the temperature controller. For this calibration, we used
both IR thermometry and a custom procedure using another RTD
in contact with the top surface of different sample substrates
(e.g., Cu discs, Al discs, stainless steel discs, and Al thin-films on
glass). For the IR thermometry measurements, the sample surfaces
were painted with black (emissive) paint. We note that the doubleRTD method yields more systematic temperature calibration
Fig. 1. (a) Schematic diagram of the microdroplet impingement and evaporation apparatus. (b) Close-up image of the temperature controlled X, Y, Z, h sample stage, Al-glass
sample, microdrop dispenser (ink-jet), and microscope objective. (c) Schematic representation of droplet evaporation in a pinned contact line mode of evaporation. In a stickto-slip evaporation mode, the contact line (solid-liquid-vapor interface) recedes after a critical contact angle (hD) is reached (resulting in a sudden decrease in the contact
radius (R) and increase in the droplet height (h)).
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S.A. Putnam et al. / International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
results. Nevertheless, the error in surface temperature is denoted
with errors bars throughout the rest of this manuscript.
2.3. Image and data analysis
A custom image analysis program written in LabView is used to
measure the microdroplet’s contact radius (R), height (h), and contact angle (h). The image analysis program (based on edge-contrast
algorithms) defines the best edge/contour of the microdroplet and
then fits a circle to the edge contour. The volume (V) of the microdroplet is then calculated based on the volume equation for a
spherical cap
VSC ¼
ph
6
pR
2
ð3R2 þ h Þ ¼
3 sin h
ð2 3 cos h þ cos3 hÞ;
ð1Þ
where the droplet contact radius (R), droplet height (h), and the surface location are measured in each image. For a spherical cap, the
interfacial liquid/vapor surface area (A) is
2
ASC ¼ pðR2 þ h Þ ¼
2p R
2
2
sin h
*
@ð/L qL Þ
_ LG
þ r ð/L qL v Þ ¼ m
@t
ð3Þ
Fluid zone momentum equation
*
3
3
fields for all variables and material properties are shared by the
phases and represent volume-averaged values. The liquid phase
is composed of only water, whereas the gaseous phase is composed
of water vapor and air. The axisymmetric governing equations of
continuity, momentum, energy, and species are solved using the
segregated pressure-based solver [38]. The governing equations
in differential notation for the fluid zones are as follows.
Fluid zone VOF equation
ð1 cos hÞ;
ð2Þ
where the contact angle (h) is measured directly or calculated from
R and h (of course, the right-hand sides of Eqs. (1) and (2) are only
valid for h < 180° and/or R > 0). In Section 4.1, an example of the image analysis process is provided (see, Fig. 3). For small, stationary
water droplets these equations work well since gravity effects do
not distort the curvature of the droplet. In some cases, the microdroplet surface reflection is used to improve/refine the edge contrast prior to edge detection. Typically, this is not necessary.
However, it useful in some experiments at high temperatures
where a vapor bubble would nucleate, grow and oscillate within
the interior of the water microdroplet. The role of vapor bubbles inside evaporating droplets will be discussed in subsequent study
[35].
For improved data analysis statistics, data for multiple droplet
evaporation events are recorded. The variations in impingement
time (dtim) and impact location (drim) are small (e.g., at most, dtim =
±100 ls and drim = ±4 lm). Again, the sample stage is rotated after
each individual microdroplet impingement and evaporation event
to ensure that footprint effects are minimized. These footprints are
presumably due to the accumulation of small dust particles from
the laboratory air at the three-phase contact line. The effects, however, only have a small influence on the impingement dynamics
and initial contact conditions. For example, ten or more droplets
usually need to impact the same surface location before dust accumulation is visible and begins to influence the droplet contact line
dynamics (e.g., spreading, receding, and depinning). Nevertheless,
the sample stage is rotated to minimize footprint effects.
**
*
*
@ðq v Þ
2cjL q
þ r ðq v v Þ ¼ rpþ r ½lðr v þr v T Þþ
r /L
qL þ qG
@t
ð4Þ
Fluid zone energy equation
*
@ðqcp TÞ
1*
_ LG
þ r v qcp T þ v 2 þ p
¼ r ðKr TÞ LV m
@t
2
ð5Þ
Fluid zone species equation
@ð/G qG wh2 o Þ
*
_ LG
þ r ð/G qG wh2 o v Þ ¼ r ð/G qG Dh2 oair r wh2 o Þ þ m
@t
ð6Þ
The volume fraction of gas (/G) and mass fraction of air (wair) are
obtained from /G = 1 - /L and wair ¼ 1 wh2 o , respectively. The density (q), dynamic viscosity (l), and thermal conductivity (K) in Eqs.
(4) and (5) are obtained from q = /LqL + /GqG, l = /LlL + /GlG, and
K = /LKL + /GKG [16,17]. The thermodynamic and transport properties for water, water-vapor, and air are exactly those used in Refs.
[16] and [17]. The calculation of the vapor-air mixture properties
also used the procedure in Refs. [16] and [17].
3.2. Computational domain
In Fig. 2, a subset of the computational domain is shown for a
water microdroplet evaporating after impact. Only a subset of
3. Numerical methods
To simulate microdroplet evaporation, an explicit volume of
fluid (VOF) model in FLUENT is used. Simulations are preformed
for water microdroplets evaporating on an isothermal wall as a
function of accommodation coefficient (e). A thorough description
of our numerical approach is provided in Refs. [16,17], and [18].
In the following, the numerical approach is summarized in terms
of (1) the physical fields and governing equations, (2) the computational domain, (3) evaporation using Schrage’s mass flux equation [36]. and 4) the initial conditions of the simulations presented.
3.1. Physical fields and governing equations
The model tracks the time-dependent volume fractions of liquid
(/L) and gas (/G) throughout the computational domain [37]. The
Fig. 2. Subset of the computational domain for a numerical simulation of a water
microdroplet evaporating on an isothermal domain wall at TS = 100 °C. The figure is
color coded to show the liquid volume fraction (/L), where red indicates liquid,
white indicates gas/air, and the colors between indicate the liquid/vapor interface.
The velocity vectors show the convection patterns in both the liquid- and gasphases. The solid black lines indicated the temperature isotherms in units of K. The
time stamp indicates the time elapsed during sessile droplet evaporation. (For
interpretation of the references to colour in this figure legend, the reader is referred
to the web version of this article)
S.A. Putnam et al. / International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
the computational domain is provided because the domain actually extends into both the z– and r–directions a distance 10 times
the microdroplet radius. The figure presents the instantaneous
droplet with velocity vectors and temperature isotherms. No solid
zone is needed for modeling on an isothermal wall. The governing
equations above correspond to the fluid zone which contains both
the liquid and gas phases. The fluid zone is enclosed by the wall
interface, centerline, surroundings, and enclosing walls for the
gas phases. The enclosing walls for the gas phases have adiabatic
boundary conditions. The centerlines and surroundings are modeled with axis of symmetry and pressure-outlet boundary conditions, respectively.
3.3. Simulating the liquid-vapor mass transfer rate
To model evaporation or the volumetric mass transfer rate
_ LG Þ, the expression
ðm
_ LG ¼ jraG j
m
"
!#
1=2
2e
Mw
pL
pG
ð2 eÞ 2pNA kB
T L1=2 T 1=2
G
ð7Þ
is used, which is the product of the droplet interfacial surface area
density (|raG|) and the Schrage’s mass flux (bracketed term) [36],
where e is the evaporation efficiency or accommodation coefficient,
Mw is the molecular weight, NA is Avogadro’s number, kB is Boltzmann’s constant, pG (pL) is the gauge pressure of the gas (liquid),
and TL (TG) is temperature of the liquid (gas). It is assumed at the
liquid-vapor interface that the temperature of a simulation cell is
T = TG = TL, where PL is the saturation pressure at T (i.e., Psat(T) = PL)
and PG is the absolute vapor partial pressure. In this context, if PL
(the saturation pressure at T) is greater than wh2 o Pcell ðTÞ, then
_ LG is positive. Condensation occurs
the liquid evaporates and m
5797
when the converse is true. Pcell refers to the absolute pressure at
the interface cell. The value of Pcell is calculated from the equation
of state (i.e., Pcell is a function of T), noting that Pcell can also be obtained by summation of the absolute pressure (i.e., operating pressure) and the gauge pressure at that cell. Likewise, the water
saturation pressure is also computed to obtain the pressure of the
liquid at the interface cell, where PL = Psat(T). During a simulation,
all variables on the right-hand side of Eq. (7) evolve in time except
for e (the accommodation coefficient). To date, the accommodation
coefficient is not well established for a given fluid. In fact, in some
cases, e has shown to differ between experiments and/or theory
by up to three orders of magnitude [39]. We assume, as others have,
that e is a constant, using either e = 0.1 or e = 1.0 in the simulations.
Modeling evaporation with the Schrage mass flux (i.e., bracketed term in Eq. (7)) has been done by others. For example, Dhavaleswarapu et al. [40] recently implemented the Schrage mass
flux equation to simulate steady-state meniscus evaporation in a
channel, using the heat flux and meniscus profile as model inputs.
In this study, unsteady evaporation is simulated, using the ‘‘initial’’
droplet shape and ‘‘initial’’ wall temperature as model inputs (thus,
everything is evolving in time). How Eq. (7) is implemented is also
different. Unlike Ref. [40], this study uses the Schrage equation as a
_ LG
source/sink term in the fluid governing equations, where m
serves as a sink term in the liquid volume of fluid (VOF) transport
equation (Eq. (3)), as a source term in the water vapor species
equation (Eq. (6)), and as a sink term in the energy equation
(Eq. (5)).
3.4. Details of simulations presented
Two simulations were conducted for the evaporation dynamics.
Impingement simulations were not carried out because droplet
impingement on an isothermal wall was studied previously [16].
For both simulations, the temperature of isothermal wall was set
to Ts = 373.15 K and the initial temperatures of the surroundings
and the microdroplet were set to ambient conditions (i.e.,
TG = 300 K, RH = 30%). The droplet was placed with an initial contact angle of h0 = 90° and contact radius of R0=35.05 lm with no
*
impingement velocity ðv im ¼ 0m=sÞ. For the first simulation the
accommodation coefficient was fixed at e = 0.1, whereas, for the
second, e = 1.0.
The numerical simulations were conducted in parallel using
three processors in conjunction with dynamic meshing, Non-iterative time advancement (NITA) [41] and variable time-stepping. The
model is first-order accurate in time and second-order accurate in
space. Eq. (7) is implemented in FLUENT via C programming language subroutines. Previously, we showed that the flow fields
(i.e., liquid and vapor convection patterns in Fig. 2) are accurately
resolved with our grid size and meshing schemes [16,17].
4. Results and discussion
Fig. 3. (a) Dimensionless droplet-profile data as a function of time for a water
microdroplet evaporating on an Al/glass substrate with a surface temperature of
TS ffi 175 °C, where R/R0 (black-symbols), h/h0 (red-symbols), h/h0 (green-symbols),
and V/V0 (blue-symbols) are the dimensionless contact radius, droplet height,
contact angle and droplet volume, respectively. (b) Microdroplet image acquired
with high-speed camera at t ffi 40 ms. (c) Same droplet image provided in (b) with a
circle fit (yellow-line) to the edge contour (red-lines). Experimental Details: TS
175 ± 6.5 °C, RH ffi 30%, R0 ffi 52 lm, h0 ffi 96 lm, h0 ffi 117, and V0 ffi 871 pL. (For
interpretation of the references to colour in this figure legend, the reader is referred
to the web version of this article)
The study focuses on water microdroplets with small impinge*
ment-diameters (Dim) and small impingement-velocities ðv im Þ
varying within typically (50 lm[ Dim [ 170 lm) and
*
ð0:4m=s K v im K 1:1 m=sÞ, respectively. Thus, the Weber numbers
*
2
ðWe ¼ q v im Dim =cÞ are also small (We [ 3.0). In this regard, phenomena such as droplet recoil, jetting, splash, break-up and rebound do not take place during the short-time impingement
dynamics (t [ 500 ls). Oscillations in the droplet’s contact angle
and height do take place after impact; however, during these
short-time impingement oscillations, the contact line is pinned at
the maximum spreading diameter. In the following, the impingement dynamics are not discussed. Instead, the study focuses on
the longer-time evaporation kinetics (t [ 500 ls).
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S.A. Putnam et al. / International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
4.1. Microdroplet evaporation on Al thin-film surfaces
4.1.1. Changes in droplet size and shape during evaporation
Evaporation experiments with isothermal Al/glass surfaces covered surface temperatures ranging from room temperature up to
superheated temperatures of TS ffi 250 °C. Fig. 3 shows a raw data
set acquired with the high-speed camera during a single evaporation experiment at Ts ffi 175 °C. Fig. 3 actually plots the dimensionless microdroplet-profiles (R/R0, h/h0, h/h0, and V/V0) as a function
of time. As shown, the droplet’s contact radius (black-lines) is relatively constant until 33 ms into the evaporation process. After
that, the contact radius rapidly decreases to another fixed value
for 3 ms, and then, subsequent stepwise reductions in the contact
radius take place until the entire droplet has evaporated. The
microdroplet’s contact angle (green-lines) changes accordingly
during this stick-slip contact line behavior. The blue-lines correspond to the droplet volume calculated with Eq. (1) using the measured contact radius (R) and droplet height (h). As shown in Fig. 3,
the rate of change in the microdroplet volume is not constant (i.e.,
the evaporation rate is not constant).
Fig. 4 provides droplet-profile data for another experiment at
TS ffi 118.5 °C. The data is similar to that provided in Fig. 3, showing
(in dimensionless form) the changes in the microdroplet’s radius,
height, contact angle, and volume as a function of time. Like the
data in Fig. 3, stick-slip contact line behavior is observed. For example, the droplet’s contact radius is fixed (pinned) until 100 ms into
the evaporation process. After that, the contact radius rapidly decreases to another fixed value for 12 ms, and then, subsequent
stepwise reductions in the contact radius take place until the entire
droplet has evaporated. The contact angle changes accordingly during this pinning and depinning of the contact line. Again, the rate of
change in droplet volume is not constant. For example, after the
droplet has evaporated for T ffi 120 ms the slope of V(t) changes
from, arguably, a constant value of dV/dt ffi -6.33 pL/ms to a smaller
value of dV/dt ffi -2.89 pL/ms. Hence, at t ffi 120 ms the average
_ LG ffi 6:2 lg/s to
evaporation rate decreases in magnitude from m
_ LG ffi 2:8 lg/s (i.e., a reduction in m
_ LG greater than 50%). A nearly
m
identical V(t) dependence is shown in Fig. 3.
4.1.2. Comparisons between simulation and experiment
Fig. 5 compares experimental data with simulation results.
Fig. 5a shows V(t) data for small water microdroplets
ðR0 35 lmÞ evaporating at TS ffi 100 °C. Fig. 5b provides the same
data in dimensionless form (i.e., V(t/sE)/V0, where sE is droplet lifetime). In both experiments, contact line depinning occurs at t/sE ffi
0.55. The simulations do not account for contact line depinning
(i.e., the contact radius is held constant, R(t) = R0, throughout the
entire simulation). Simulation results are provided for two different accommodation coefficients (i.e., e = 0.1 and e = 1.0). As shown,
the simulations under predict the evaporation rate when an
accommodation coefficient of e = 0.1 is used. Better agreement is
found for e = 1.0. Similar V(t) predictions are obtained for accommodation coefficients within the range 0.4 6 e 6 1.0. This is because the predicted value of sE is not sensitive to changes in e for
0.46 e 6 1.0. This is discussed in more detail in Refs. [17] and
[18], where comparisons are made between simulation data as a
function of accommodation coefficient, droplet size, and droplet
curvature.
Fig. 6 compares the experimental and simulated droplet-profile
data. For these small water microdroplets, the measured evapora_ LG / dVðtÞ=dt) is relatively constant until contact
tion rate (i.e., m
line depinning occurs (indicated by the arrow symbols in Fig. 5
and the change in R/R0 and h/h0 in Fig. 6). The simulated evaporation rate also decreases in magnitude as the droplet evaporates.
However, this is not due to depinning because the contact line is
pinned in the simulations (i.e., R(t)/R0 = 1 for the simulation data
Fig. 4. Dimensionless droplet-profiles as a function of droplet lifetime during
evaporation on Al/glass at TS ffi 118.5 °C. Droplet images are provided at two
different droplet lifetimes, where the 100 lm scale-bar applies to both images.
Experimental Details: TS = 118.5 ± 4.6 °C, RH ffi 30%, V0 ffi 977.5 pL, R0 ffi 64.2 lm,
h0 ffi 91.3 lm, and h0 ffi 111°.
in Fig. 6). Instead, the simulated evaporation rate decreases in
magnitude as the droplet height decreases. As discussed in Ref.
[18], in a pinned contact line mode of evaporation, reductions in
the droplet height coincide with reductions in the velocity of circulative fluid motion (i.e. vorticity), which, in turn, reduces the magnitude of the simulated evaporating rate. Based on these results,
the simulations are over estimating the influence of fluid convection on the evaporation rate. An improved simulation model is currently under development which can accurately account for (1) the
constant evaporation rate observed during the pinned mode of
microdroplet evaporation and (2) stick-slip contact line motion.
4.1.3. Microdroplet evaporation at different surface temperatures
Fig. 7 shows temporal volume data for microdroplets evaporating on Al/glass at three different surface temperatures. The data is
plotted in dimensionless form. The initial droplet volumes are not
the same in each experiment. However, this does not change the
functional form of the dimensionless data presented. For example,
contact line depinning occurs at t/sE ffi 0.5 in all three experiments,
and, in result, V(t/sE)/V0 is nearly identical for the three experiments. Again, the evaporation rate is constant before contact line
depinning and then decreases in magnitude as the microdroplet
gets smaller. Typically, the average evaporation rate decreases by
more than 50% after contact line depinning. Similar results were
obtained by Mollaret et al. [42] with millimeter-sized water droplets evaporating on both Al and Teflon at surface temperatures
within 25 °C[ DTS [ 100 °C.
It is interesting that V/V0 as a function t/sE is independent of
surface temperature. Non-dimensional relationships have shown
to be powerful in predicting the functional form of the hydrodynamic boundary layer in fluid dynamics. Although, care should
be taken in interpreting that V/V0 as a function of t/sE is truly independent of the surface temperature. This will only be true if the
temporal depinning dynamics are also temperature independent
(i.e., the change in the contact radius relative to the droplet lifetime, R(t/sE), is temperature independent).
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S.A. Putnam et al. / International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
droplet-profile (dimensionless)
1.0
0.8
0.6
0.4
0.2
0
0
15
30
time (ms)
45
60
Fig. 6. Predicted (lines) and measured (symbols) dimensionless droplet-profiles for
water microdroplets evaporating on Al/glass at TS ffi 100 °C. Simulation Details:
e = 1.0, RH = 20%, TS = 100 °C, V0 = 89.8 pL, R0 = 35 lm, h0 = 90°, and sE ffi 46.3 ms.
Experimental details: RH = 30 ± 3%, TS = 100 ± 2 °C, R0 ffi 34.6 ± 0.5 lm, 91 ± 1°,
sE ffi 58.5 ms, sD ffi 32 ms.
Fig. 5. (a) Predicted (lines) and measured (symbols) droplet volume as a function of
time for microdroplets evaporating on Al/glass at TS ffi 100°C. (b) Volume ratio (V/
V0) as a function of dimensionless droplet lifetime (t/sE). Simulation results are
provided using accommodation coefficients of e = 0.1 (black lines) and e = 1.0 (red
lines). Experimental data are provided for two experiments with initial droplet
volumes of V0 ffi 100 pL (black circles) and V0 ffi 90 pL (blue triangles), where the
arrows signify when depinning occurs (sD). Simulation Details: RH = 20%, TS = 100 °C,
V0 = 89.8 pL, R0 = 35 lm, h0 = 90°; black lines – (e = 0.1, sE ffi 155 ms); and red lines –
(e = 1.0, sE ffi 46.3 ms). Experimental Details: RH = 30 ± 3%, TS = 100 ± 2 °C; black
circles – (R0 ffi 35.7 ± 0.5 lm, h0 = 91 ± 1°, sE ffi 64.5 ms, sD ffi 36 ms); and blue
triangles – (R0 ffi 34.6 ± 0.5 lm, h0 ffi 91 ± 1°, sE ffi , 58.5 ms, sD ffi 32 ms). (For
interpretation of the references to colour in this figure legend, the reader is
referred to the web version of this article)
We point out that this ‘‘temperature-independent’’ transition in
_ LG occurs when the microdroplets are still relatively large. That is,
m
the droplets are still several micrometers in size (R dT, h dT),
where dT, the Tolman length, is on the order of nanometers. There_ LG is not attributed with size-dependent
fore, the transition in m
deviations in the droplet’s surface tension due to disjoining pressures, which are important for liquid thin-films and droplets with
sizes comparable to dT [43,44].
Fig. 8 summarizes the results for water microdroplets evaporating on Al/glass at surface temperatures within 25 °C [ DTS [
230 °C. Each data point at a given surface temperature is based
on the average of four or more ‘‘independent’’ experiments (usually
more). Data at surface temperatures beyond 230 °C is omitted
because the droplets start bouncing, exploding, and flipping after
Fig. 7. Dimensionless-droplet-volume ratio (V/V0) plotted as a function of dimensionless-droplet-lifetime ratio (t/sE) for microdroplet evaporation on Al/glass at
three different surface temperatures. Experimental details: RH = 30 ± 3%; red-square
data – (TS ffi 175 °C, V0 ffi 923 pL, h0 ffi 117°, sE ffi 85 ms, sD ffi 42 ms); black-circle
data – (TS ffi 119 °C, V0 ffi 981 pL, 111, sE ffi 203 ms, sD ffi 102 ms); and blue-triangle
data – (TS ffi 48 °C, V0 ffi 265 pL, h0 ffi 107°, sE ffi 770 ms, sD ffi 380 ms). (For interpretation of the references to colour in this figure legend, the reader is referred to the
web version of this article)
impingement. Three different evaporation rates are provided at
each surface temperature because the evaporation rate decreases
in magnitude after contact line depinning (i.e., the temporal form
sD
_ t<
of V(t) follows that discussed previously). For example, m
LG
(open-circles) corresponds to the average evaporation rate measD
_ t<
sured before contact line depinning (i.e., m
¼ qdV=dt, where
LG
dV/dt is the average rate of change of the droplet volume before
_ sLGE (filled-circles) corresponds to
depinning). On the other hand, m
the average evaporation rate measured for complete dry out (i.e.,
_ sLGE ¼ qV0 =sE ). The evaporation rate before depinning is always
m
the greatest. All evaporation rates are provided because they are
all important in an evaporative cooling technology.
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S.A. Putnam et al. / International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
The remainder of this manuscript focuses on the evaporation
rate measured before contact line depinning; stressing again that
sD
_ t<
m
is a constant, whereas for t > sD the evaporation rate graduLG
ally decreases in magnitude during the stepwise reductions in
the contact radius. The reason why a change in the contact radius
(length of the contact line) influences the measured evaporation
rate is discussed in the following.
4.1.4. Influence of Al surface quality and changes in the contact line
length on the evaporation rate
Fig. 9 summarizes all the evaporation data measured with
water microdroplets on different Al thin-films. Experimental data
for three different Al/glass samples are reported. Two of the data
sets are for samples that have been used in multiple experiments
prior to these measurements. For example, these samples have
been wiped clean of dust several times, rinsed with water, alcohol,
and acetone, and went through several heating and cooling cycles.
Hence, these Al thin-film samples (filled-circles and red-star-circles) are referred to as ‘‘practical’’ Al thin-films because they better
represent an Al thin-film surface in a cooling system. The ‘‘cleanroom’’ data (open-circles), on the other hand, is for an Al/glass
sample with an ideal Al thin-film surface (e.g., the ‘‘cleanroom’’
sample was only exposed to the laboratory air prior to testing).
In principle, it is difficult to maintain and produce ideal surfaces. Thus, in many cases, it is impractical to base system-level
cooling designs on ‘‘cleanroom’’ quality material systems. The data
for multiple Al/glass samples is presented because system reliability is an important parameter in two-phase cooling [45].
At first glance it appears that the evaporation efficiency is improved with the cleanroom quality Al surface. However, for the
data in Fig. 9, the initial droplet volume (V0) also contributes to
_ LG measured. For example, the ‘‘cleanroom’’
the magnitude of m
data are for the slightly larger droplets (relative to the ‘‘practical’’
data) and, in result, have the largest evaporation rates. Many
experiments have been conducted with microdroplet volumes
ranging from 50 to 42,000 pL. The results of these experiments
have consistently shown that the evaporation rate increases with
increasing droplet size. However, it is not actually the droplet
Fig. 8. Temperature dependent evaporation rates for water microdroplets evaporating on an Al /glass substrate. Three evaporation rates are provided for each set of
experiments at a given surface temperature. The symbols correspond to the average
evaporation rate measured (1) before contact line depinning (open circles), (2) after
contact line depinning (open triangles), and (3) for complete dry out (filled circles).
Experimental Details: 5% [ RH [ 30%, 40 lm [ R0 [ 56 lm, 103°[ h0 [ 125°, and
260 pL [ V0 [ 1150 pL.
volume or droplet contact area that influences the magnitude of
the evaporation rate measured. Instead, it is the total length of
the contact line that influences the evaporation rate.
Fig. 10 shows the evaporation rate (before contact line depinning) per unit length of contact line:
sD
sD
_ t<
p_ t<
¼m
LG
LG =ð2pR0 Þ:
ð8Þ
As shown, all evaporation data collapses onto a single evaporation
sD
_ t<
curve by scaling m
by the length of the contact line (2pR0). This
LG
not surprising, our simulations show significant increases in the
evaporative mass flux near the contact line (e.g., the simulation data
in Fig. 2 shows the significant difference between the magnitude of
the mass flux near the contact line and the mass flux at the droplet
sD
_ t<
apex). Others have also shown that the m
scales linearly with
LG
the droplets contact radius [46]. A systematic trend like that shown
sD
_ t<
in Fig. 10 is not found by scaling m
by other powers of R0 (e.g.,
LG
2
R0 ), supporting that phase-change heat transfer is dominated by
the liquid-vapor phase transformations at the solid-liquid-vapor
(a.k.a., three-phase) interface.
4.1.5. Maximum evaporation rates at high surface temperatures
_ max
A maximum (critical) evaporation rate of m
LG ¼ 32 3 lg/s is
found for water droplet evaporation on Al thin-films. This is signi_ LG measured at
fied in Fig. 9 by the maximum values of m
sD
TS ffi 180 °C. Based on the p_ t<
data in Fig. 10, the maximum evapLG
oration rate per unit length of contact line is p_ max
LG ¼ 65 5 ng/s/
lm.
For the data discussed thus far, the highest temperature data
points correspond to the surface temperature at which the droplet’s evaporation dynamics become non-sessile (perhaps chaotic
in nature [47]). Data analysis at surface temperatures beyond the
highest temperature data point yields results inconsistent with
the lower temperature data. This transition in impingement and
evaporation dynamics at large superheats can be attributed to
the Leidenfrost effect [48]. which is prevalent in both boiling and
sessile droplet evaporation (beyond the CHF) where a thermallyresistive vapor layer forms between the droplet and a heated
Fig. 9. Temperature dependent evaporation rates measured before contact line
depinning for water microdroplets on three different Al thin-films on glass.
Experimental Details: open circles – (largest droplets, V0 = 1650 ± 580 pL,
R0 = 88 ± 13 lm, h0 = 95 ± 9°, RH = 48 ± 3%); red-star circles – (smallest droplets,
V0 = 340 ± 110 pL, R0 = 54 ± 5 lm, h0 = 91 ± 10°, RH = 23 ± 7%); and filled circles –
(intermediate droplet volumes, V0 = 610 ± 308 pL, R0 = 50 ± 5 lm, h0 = 115 ± 8°,
sD
_ t<
RH = 18 ± 12%, note: same m
data provided in Fig. 8). (For interpretation of the
LG
references to colour in this figure legend, the reader is referred to the web version of
this article.)
S.A. Putnam et al. / International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
5801
conducted with this practical thin-film at TS ffi 200 °C. In these
experiments a majority of the microdroplets either elastically
bounced after impact or exploded during the evaporation process.
The data at TS ffi 200 °C represents the few small microdroplets that
actually wet the Al surface and did not explode. As shown, the heat
flux starts to decrease at surface temperatures beyond 180 °C, sig2
nifying a critical evaporative heat flux of qCHF
Al ¼ 600 100 W/cm .
4.2. Microdroplet evaporation on different thin-film surfaces
Fig. 10. Evaporation rate per unit length of contact line measured before contact
sD
sD
_ t<
is calculated with Eq. (8) using R0 and m
measured at a
line depinning. p_ t<
LG
LG
given surface temperature. Experimental details: provided in Fig. 9 caption.
surface [45]. In general, a transition from a purely microdroplet
evaporation regime to a droplet/film boiling regime is observed,
where this transition is coupled to both (1) a critical surface tem_ max
perature — T max
and (2) a critical evaporation rate — m
LG ). Thus,
S
this transition is analogous to that observed in pool boiling at
CHF, where, at CHF, the boiling process is switching from a nucleate-boiling regime to a film-boiling regime.
As shown in Fig. 10, the maximum surface temperatures measured with microdroplets on Al thin-films ranges within 170 °C
[ DTmax [ 230 °C, which corresponds to maximum superheats
of DTmax = 100 ± 30 °C. In fact, this measured value of Tmax for
the filled-circle data corresponds very well with the Leidenfrost
temperatures (TLF) measured by Mudawar et al.[45] during water
droplet impingement on polished Al surfaces (e.g., see Table 4 in
Ref. [45]). However, Tmax is not equated with TLF becasue Tmax is
really a surface temperature somewhere between the surface temperature at critical heat flux (TCHF) and the true Leidenfrost temperature of the system (i.e., TCHF [ DTmax [ DTLF).
The size of the droplet also influences the surface temperature
_ max
at which the maximum evaporation rate ðm
LG Þ is observed. As
_ max
shown in Fig. 9, m
is
reached
by
increasing
either 1) the surface
LG
_ max
temperature or 2) the droplet size. Thus, m
LG is coupled both DT
and R0 (similar to the correlation between qCHF and the vapor-bubble release size (RVB) observed in boiling).
In addition to Al thin-films, several other thin-film surfaces are
studied to understand how changes in thin-film thermal conductivity and thin-film surface energy influence the heat transfer performance. Fig. 12 shows the heat transfer performance at different
surface temperatures for all the thin-film surface materials tested.
We note that the Cu thin-film (blue-triangles), Ti thin-film (redstars), and SAM surface (yellow-diamonds) are all ‘‘cleanroom’’
quality surfaces. Fig. 12a provides the average evaporation rate
measured per unit length of contact line, whereas Fig. 12b provides
the average evaporative heat flux (q) measured based on Eq. (9)
(i.e., q before contact line depinning).
As shown in Fig. 12, the maximum surface temperatures range
within 170 °C[ DTmax [ 230 °C, corresponding to maximum
superheats of DTmax = 100 ± 30 °C. The data also shows that the
thin-film thermal conductivity has no significant influence on the
heat transfer performance. For example, the thermal conductivity
(K) of these thin-film samples ranged from that of Cu
(KCu ffi 400 W/m/K) to Al (KAl ffi 200 W/m/K) to Ti (KTi ffi 20 W/
m/K) and then to presumably the smallest value for the SAM
sD
(KSAM [ 5 W/m/K); yet, both the heat flux (q) and p_ t<
fall within
LG
the scatter of the data. This null result may be due to several factors, such as (1) the presence of the oxide layer on the metal surfaces, (2) the low thermal conductivity of the glass substrate,
and/or (3) that largest resistance to heat transfer is at the liquidvapor interface (not the solid-liquid interface) [18].
The thermal conductivity of an oxide surface layer is estimated
to range between that of aluminum oxide ðKAl2 O3 ffi 45 W=m=KÞ
and that of titanium oxide ðKTiO2 ffi 8 W=m=KÞ. In this case, the variation in the thermal conductivity of an oxide surface layer is within a factor of 5; yet, a small change in both the evaporation rate and
4.1.6. Evaporative heat flux before contact line depinning
Fig. 11 shows the heat flux measured as a function of surface
temperature with water microdroplets on heated Al/glass substrates. The heat flux is calculated based on
sD
2
_ t<
q ¼ Lv m
LG =ðpR0 Þ;
ð9Þ
sD
_ t<
m
LG
where
is the average evaporation rate measured before depinning, R0 is the measured contact radius, and Lv is the latent heat of
vaporization of water. As shown in Fig. 11, the smallest droplets
(red-star-circles) have the largest heat flux and biggest droplets
(open-circles) have the smallest heat flux. Thus, larger droplets have
increased evaporation rates but reduced heat fluxes (as clearly
shown by comparing the heat flux data in Fig. 11 with the evaporation data in Fig. 9).
The maximum heat flux measured with water microdroplets on
superheated Al thin-films is qmax ffi 700 W/cm2. This was measured
in one experiment of the four conducted at TS ffi 180 °C with a practical Al thin-film (red-star-circle data). Several experiments were
Fig. 11. Heat flux as a function of surface temperature for water microdroplets on
three different Al thin-film samples. The heat flux is calculated with Eq. (9) using R0
sD
_ t<
and m
measured. Experimental details: provided in Fig. 9 caption, open circles –
LG
(largest droplets, V0 = 1650 ± 580 pL), red-star circles – (smallest droplets,
V0 = 340 ± 110 pL), and filled circles – (intermediate droplets, V0 = 610 ± 308 pL).
(For interpretation of the references to colour in this figure legend, the reader is
referred to the web version of this article)
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S.A. Putnam et al. / International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
heat flux is observed. Others have shown that heat conduction
through the substrate can be the rate-limiting mechanism for
droplet evaporation [46,49,50] (i.e., the substrate thermal conductivity limits the heat flux, which, in turn, regulates the evaporation
rate). In this regard, a small change in the heat transfer performance would be expected for these different thin-film surfaces because the thermal conductivity of the glass substrate is regulating
the heat flux (which, in turn, dictates the heat transfer coefficient
in a purely evaporative heat transfer regime).
Interestingly, the change in surface energy also has a small influence on the heat transfer performance (i.e., q(DT)). The surface energy of these thin-film samples presumably ranged from the most
hydrophilic Cu, to neutrally hydrophobic Al and Ti surfaces, and
then, to the hydrophobic SAM surface. We note that just measuring
the static contact angle after impingement is not a reliable metric
for qualitatively estimating the changes in surface energy. For
example, static contact angle measurements for all thin-film samples at 25 °C [ TS [ 95 °C suggested that the Ti surface was the
most hydrophilic; yet the receding contact angles were the smallest
with the Cu and ‘‘cleanroom’’ Al thin-films [51]. Nevertheless,
changing the surface material did influence the depinning dynamics. For example, the microdroplets evaporating on the SAM surface
started to depin much earlier relative to all the metal thin-film surfaces. However, the evaporation rates and heat fluxes for all samples are within error of each other (especially at high surface
temperatures approaching T max
where the maximum evaporation
S
rates are observed). The small changes in the evaporation rate
and heat flux for the SAM surface relative to the metal thin-films
also follows previous evaporation studies with monomolecular surfaces [52,53], where in Ref. [52] only a 30% reduction in the evaporation rate was observed by changing surface from highly
hydrophilic (hEq ffi 20°) to hydrophobic (hEq ffi 110°).
These results showing the small influence on the thin-film thermal conductivity and surface energy suggests that the greatest
resistance to evaporative heat transfer exists at the liquid-vapor
interface. Yet, not only the liquid-vapor interface but the interface
in the proximity of the solid-liquid-vapor contact line. Larger
_ LG Þ relative
microdroplets will have increased evaporation rates ðm
to smaller ones; yet, the heat flux is reduced by the corresponding
_ LG is
increase in the wetted surface area. Moreover, even though m
greater with larger microdroplets, the evaporation rate per unit
length of contact line ðp_ LG Þ is arguably constant and independent
of the surface material.
Traditionally, it is expected that hydrophilic surfaces will provide the best heat transfer performance. This is not true in regards
to the evaporative heat flux with water microdroplets. As shown in
Fig. 12b, the heat flux performance with the hydrophobic SAM surface is the best of the ‘‘cleanroom’’ quality samples. This suggests
that thin-film surfaces that yield microdroplet contact angles between 95 and 115° will, in turn, remove the most heat. Moreover,
regardless of the thin-film surface material to be used in a spray
cooling technology, evaporation rates and evaporative heat fluxes
CHF
_ max
_ max
beyond m
= 600
LG ¼ 32 5 lg/s, pLG ¼ 63 7 ng/s/lm, and q
± 100 W/cm2 should not be expected with water microdroplets of
volumes within 100 pL[ V0 [ 2500 pL. Thus, maximizing the contact line length per unit surface area is the most important design
parameter to consider in a spray cooling system (which can be
accomplished by simply spraying small water microdroplets).
4.3. Microdroplet Evaporation on Cu Surfaces
Fig. 12. Heat transfer performance for water microdroplets evaporating on a
variety of different thin-film surfaces. (a) Evaporation rate per unit length of contact
line measured before contact line depinning. (b) Evaporative heat flux before
contact line depinning. For clarity, the error bars are removed from most of the
‘‘practical’’ Al thin-film data. Experimental details: Cu thin-film – (V0 = 1631 ±
1000 pL, R0 = 82 ± 18 lm, h0 = 97 ± 7°); Ti thin-film – (V0 = 1460 ± 723 pL, R0 = 92 ±
15 lm, h0 = 84 ± 8°); hydrophobic SAM – (V0 = 1404 ± 618 pL, R0 = 68 ± 11 lm,
h0 = 113 ± 5°); ‘‘cleanroom’’ Al thin-film – (V0 = 1650 ± 580 pL, R0 = 88 ± 13 lm,
h0 = 95 ± 9°); and ‘‘practical’’ Al thin-films – (red-star circles: V0 = 340 ± 110 pL,
R0 = 54 ± 5 lm, h0 = 91 ± 10°; open circles: V0 = 610 ± 308 pL, R0 = 50 ± 5 lm,
h0 = 115 ± 8°).
A variety of Cu material systems are also studied to explore the
role of surface microstructure and substrate thermal conductivity
on microdroplet evaporation. The Cu systems tested are Cu
thin-films on glass and solid Cu substrates with different surface
structure/roughness (i.e., mirror-polished, 800-grit sanded, and
micron-sized Cu pillars).
Fig. 13 shows droplet profile data during evaporation on the Cu
pillar surface at TS ffi 90 °C. For this data, multiple 1000 pL microdroplets were used to create/deposit a 42,000 pL microdroplet on
the pillar surface (e.g., t=0 sec corresponds to the droplet after the
last impingement event with a 1000 pL microdroplet). As shown,
the evaporation kinetics with this big microdroplet on the pillar Cu
surface is analogous to the thin-film data discussed previously (i.e.,
the evaporation rate decreases after contact line depinning). In
fact, all surfaces tested in this study show stick-slip contact line
behavior and constant evaporation rates before depinning.
Fig. 14 summarizes the heat transfer performance for microdroplets evaporating on the different Cu surfaces. The data focuses on
microdroplets with initial volumes within the range
1000 pL[ V0 [ 2000 pL. Evaporation data with bigger microdroplets like that shown in Fig. 13 is only provided for the Cu pillar
surface. Again, the highest temperature data points correspond to
surface temperatures approaching the Leidenfrost temperature,
where T max
is proprotional to R0 and signifies the surface temperS
ature for transition from a purely evaporative heat transfer regime
S.A. Putnam et al. / International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
5803
Fig. 13. Dimensionless droplet-profiles as a function of microdroplet lifetime
during microdroplet evaporation on a Cu surface structured with Cu pillars at
TS ffi 90 °C. Droplet images are provided at two different droplet lifetimes, where the
160 lm scale-bar applies to both images. The inset image is a TEM micrograph of
the Cu pillar surface. Experimental details: TS = 90 ± 2 °C, RH ffi 25%, V0 ffi 41,700 pL,
R0 ffi 123.5 lm, h0 ffi 395.5 lm, and h0 ffi 146°.
to a droplet/film boiling regime. Data at TS [ 90 °C for the bigger
microdroplets on the Cu pillars is not provided because boiling
takes place within these bigger microdroplets at surface temperatures slightly beyond 100 °C (i.e. the saturation temperature).
The microdroplets on all solid Cu substrates (Cu mirror, Cu
800-grit and Cu pillars) start to boil at much lower surface temperatures relative to that for microdroplets evaporating on a Cu thinfilm. For all the solid Cu substrate materials tested, the maximum
superheats observed fall within the range 3 °C [ DTmax [ 30 °C, in
comparison to 70 °C [ DTmax [ 130 °C for the thin-film surface
coatings on glass substrates. Again, this is due the difference in
the thermal conductivity substrate, which dictates the maximum
allowable heat flux during evaporation (i.e., the Cu thin-film is only
70 nm thick so the heat flux through the thin-film on the glass
substrate is limited by the thermal conductivity of glass).
Comparison of the evaporation data suggests maximum achiev_ max
_ max
able evaporation rates of m
LG ¼ 36 4 lg/s and pLG ¼ 80 4 ng/
s/lm (which is comparable to that observed with the thin-film
materials discussed in Section 4.2). Comparison of the evaporative
heat flux suggests a critical heat flux of qCHF = 750 ± 300 W/cm2
(also comparable to that observed with the thin-film materials).
Moreover, as before, larger droplets have increased evaporation
rates; yet, reduced heat fluxes (as clearly shown in Fig. 14 by comparing the Cu pillar data of different droplet volumes).
In regards to the heat transfer performance, the microstructured
Cu pillar sample yields the best heat transfer coefficient (h) of all
substrate materials tested, assuming h = q/DT and that the average
surface temperature underneath the microdroplet is within the error of the surface temperature calibration. For example, the heat
transfer coefficients measured at a superheat of DT 10 °C are hpil2
2
lar = 650 ± 200 kW/m /K, h800-grit = 260 ± 50 kW/m /K, hmirror = 240
± 90 kW/m2/K, and hthin-film = 50 ± 25 kW/m2/K. Thus, a 2.5
increase in the heat transfer performance is observed with the
Fig. 14. Heat transfer performance for water microdroplets evaporating on
different Cu substrates. (a) Evaporation rate per unit length of contact line before
depinning (Eq. (8)). (b) hD/h0 Evaporative heat flux before depinning (Eq. (9)).
Experimental details: RH = 48 ± 3%; Cu thin-film – (V0 = 1631 ± 1000 pL, R0 =
82 ± 18 lm, h0 = 97 ± 7°); Cu mirror – (V0 = 1405 ± 539 pL, R0 = 87 ± 12 lm, h0 = 89
± 12°); Cu 800grit – (V0 = 1200 ± 610 pL, R0 = 74 ± 17 lm, h0 = 100 ± 13°); Cu pillars
– (V0 = 1000 ± 410 pL, R0 = 45 ± 6 lm, h0 = 130 ± 7°); and Cu pillars with bigger
droplets – (V0 = 6000 ± 1200 pL, R0 = 105 ± 10 lm, h0 = 125 ± 10°).
Cu-pillars relative to the other Cu substrates with either a mirror
or 800-grit surface finish. This suggests that the Cu pillar surface
could be promising for enhancing performance in current and future evaporative heat exchangers. This enhanced heat transfer performance is attributed to the significant increase in the contact line
length per unit surface area. The water microdroplets on the pillared Cu surface are effectively hydrophobic and are in either (1)
a partial wetting or (2) nonwetting (Cassie) state [54,24,55]. In this
case, vapor convection and evaporation is possible underneath the
microdroplet (i.e., within the pillar region that separates the microdroplet from the Cu solid substrate). Hence, an increased heat
transfer coefficient is obtained because there is an increase in contact line length per unit area (i.e., the region where the majority of
evaporative heat transfer takes place).
4.4. Contact Line Depinning and Surface-Liquid Intermolecular Forces
4.4.1. Changes in the evaporation efficiency after depinning
Fig. 15 shows the effects of contact line depinning on the evaporation efficiency. As shown, changes in the surface material
changes the depinning angle, but does not systematically change
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S.A. Putnam et al. / International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
sD _ t>sD
Fig. 15. Measured evaporation ratio ðbLG ¼ p_ t<
LG =pLG Þ as a function of the
depinning contact angle (hD). Data is based on all water microdroplet experiments
at surface temperatures within 25 °C[ DTS [ 125 °C (260 pL[ V0 [ 2200 pL). The
symbols for each surface material are same as that used previously (see the legends
in Figs. 12 and 14).
the ratio of the evaporation rates (per unit length of contact line)
sD _ t<sD
before and after contact line depinning (i.e., bLG ¼ p_ t>
LG =pLG ). A
direct correlation between hydration forces and the evaporation
efficiency is not observed in this study. In general, the measured
evaporation ratio scatters around bLG ffi 1 for all surface materials
tested. However, the data suggests that bLG may start to increase
beyond unity for highly hydrophilic surfaces (i.e., hD ? 0°). This result is reasonable given that not all the evaporation takes place at
the contact line. Some evaporation takes place at the droplet apex.
So, for thin-film like droplets the evaporation rate at the apex may
start to approach that at the contact line as the droplet height reduces. Yet, this is surprising because reduced evaporation rates are
expected for small water microdroplets in contact with a hydrophilic surface in comparison to a hydrophobic surface. For example, a transition in the evaporation kinetics due to water
hydration has recently been suggested by Golovko et al.[44] How_ LG Þ was
ever, in Ref. [44], the reduction in the evaporation rate ðm
observed with submicron-sized water droplets on hydrophilic Si
surfaces (not micron-sized water droplets, as studied here). More_ LG Þ is reported (not, p_ LG ,
over, in Ref. [44], the evaporation rate ðm
the evaporation rate per unit length of contact line, as studied
here). Thus, our results are limited to micron-sized water droplets
(i.e. V0 [ 5 pL) and are not expected to hold when the droplet size
(or water film thickness) is reduced to a length scale within an order of magnitude of either the Tolman length (R0 10 dT) or the
mean-free-path of the vapor phase ðR0 10 kV Þ. Nevertheless,
even though surface/liquid interactions directly influence the
depinning process, the relative change in the evaporation efficiency before and after contact line depinning is not systematically
dependent on the surface material. Further studies of p_ LG with
smaller water droplets (R0 [ 2 lm) on surfaces approaching the
super–hydrophilic and super–hydrophobic wetting regimes are
needed.
4.4.2. Surface tension force required for contact line depinning
It is instructive to estimate the energy barriers and surface-liquid forces associated with contact line depinning. Fig. 16 provides
the depinning force (dF) associated with contact line depinning.
The depinning force is calculated based on the unbalanced
Young–Laplace equation: [56]
Fig. 16. Depinning force (dF) for water droplets on different surfaces p lotted as a
function of (a) the dynamic contact angle for depinning (hD/h0) and (b) the
depinning contact angle (hD). dF is calculated with Eq. (11) using measured and
data. The symbols with error bars correspond to the surface materials tested in this
work with water microdroplets at 25 °C[ DTS [ 125 °C (260 pL[ V0 [ 2200 pL),
where the symbol descriptions are provided in Figs. 12 and 14. The symbols
without error bars correspond to data taken from Ref. [56] for millimeter-sized
water droplets on different surface materials at room temperature
(3.0 lL[ V0 [ 3.5 lL): glass/Teflon (), glass/Al (asterisks), glass/Si (purple-diamond), Si (hexagon), Paryelene (bronze-star), and Teflon (six-point-star).
F ¼ ðcSL cSV Þ þ cLV cos h;
ð10Þ
where F is the surface tension force per unit length of contact line
and cSL, cSV, and cLV are the surface tensions (energy densities) at
the solid-liquid, solid-vapor, and liquid-vapor interfaces, respectively. At equilibrium, the contact angle equals the equilibrium contact angle (i.e., h = hEq) and surface tension force is zero. However, as
the water microdroplet evaporates in a pinned contact line mode of
evaporation, the surface tension force at the contact line builds up
in direction toward the bulk liquid. At the threshold of contact line
depinning, the depinning force per unit length of contact line is
dF ¼ cLV ðcos hD cos h0 Þ;
ð11Þ
assuming that the difference between h0 and the equilibrium
tact angle is small (i.e., h0 ffi hEq) and that cSL, cSV, and cLV are
stants during evaporation. The effects of the contact
curvature [57] (i.e., j0 = 1/R0) on dF are neglected because the
tact radius is constant (pinned) during evaporation.
conconline
con-
S.A. Putnam et al. / International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
Fig. 16 shows the depinning force calculated for each surface
material tested in this study. The predicted values of dF are based
on experimental data at surface temperatures within
25 °C [ DTS [ 125°C. For TS > 100 °C, a few data sets were omitted
from analysis because vapor bubbles nucleated and oscillated
within the evaporating microdroplets. Data analysis included the
temperature dependence of cLV to account for the measurements
at different surface temperatures. Also included in Fig. 16 are recent data by Shanahan et al.[56] for millimeter-sized water droplets evaporating on different surfaces at room temperature. As
shown, dF is the greatest for the most hydrophilic surfaces and approaches zero as the surfaces become more and more hydrophobic
(i.e., dF? 0 as hD/h0? 1). Error bars are provided in a ‘‘slope format’’
because the scatter in the data for multiple experiments is not uniformly distributed around the average value reported. Instead, the
data scatter follows the slope of the error bars. For example, the
depinning contact angles measured with water microdroplets on
Ti/glass (red-stars) ranged within 30°[ hD[ 60°; for droplets
depinning around hD = 56 ± 4° the corresponding depinning force
is dFTi = 26 ± 5 mN/m, whereas for hD = 34 ± 4° the depinning force
is dFTi = 42 ± 5 mN/m. Error bars are not provided for the data from
Ref. [56] because multiple experiments on each surface material
were not reported. Nevertheless, dF increases with increasing surface energy (hydrophobicity), supporting that dF is expected to be
greater than the surface tension of water for superhydrophilic surfaces. In this regard, larger deviations are also expected between h0
measured and hEq with high surface energy surfaces because droplet spreading during impingement is also inhibited by the intrinsic
energy barrier for contact line motion. dF appears to be greater for
the water microdroplets in comparison to millimeter-sized water
droplets. The difference in the contact line tension [57] (curvature)
between millimeter-sized and micron-sized water droplets may be
contributing to this trend (which is not included in the calculations
of dF using Eq. (11)).
4.4.3. Excess free energy barrier for contact line depinning
It is also instructive to estimate the change in the free energy
required for contact line depinning. Fig. 17 provides the excess free
energy per unit solid-liquid contact area (dGA) associated with
5805
contact line depinning. The free energy barrier for contact line
depinning is also calculated based on Young’s equation: [58]
G ffi pR2 ðcSL cSV Þ þ AcLV þ ;
ð12Þ
where G is the total excess free energy of a solid-liquid-vapor system, R is the droplet’s contact radius, and A is the droplet’s interfacial liquid/vapor surface area. The ellipsis term in Eq. (12)
represents higher order contributions to the free energy. For small
droplets (not affected by gravity) the interfacial surface area follows
that for a spherical cap (see, Eq. (2)). If equilibrium is reached after
droplet impingement, then the free energy is a minimum. In this
case, the excess free energy at the threshold of contact line depinning is
dG ffi cLV ðAD A0 Þ;
ð13Þ
where AD and A0 are the surface areas of the liquid-vapor interface
at the threshold of depinning and at equilibrium (before evaporation), respectively. Eq. (13) describes the total excess free energy,
assuming that h0 = hEq, cSL, cSV, and cLV are constant during evaporation, and that the higher order contributions to G (e.g., contact line
curvature and viscous dissipation due to fluid motion at the droplet
interface) can be neglected. At the threshold of depinning, the corresponding excess free energies per unit length of contact line and
per unit solid-liquid surface area are
dGl ¼ dG=lCL ¼ cLV ðAD A0 Þ=ð2pR0 Þ;
ð14Þ
dGA ¼ dG=ASL ¼ cLV ðAD A0 Þ=ðpR20 Þ;
ð15Þ
where the contact line length (lCL = 2pR0) and surface-liquid contact
area ðASL ¼ pR20 Þ are constants until depinning.
Fig. 17 shows dGA calculated for (1) each material system tested
in this work (i.e., 25 °C [ DTS [ 125 °C) and (2) the millimetersized water droplets studied at room temperature in Ref. [56]. In
general, dGA scatters within -5 mJ/m2 [ dGA [ -30 mJ/m2 for all
material systems studied. In this regard, we estimate dGA ffi 5 mJ/m2 as a minimum value for solid-liquid-vapor systems that
exhibit stick-slip contact line behavior. The magnitude of dGA
may be slightly over estimated with the microdroplets because
mechanical equilibrium is probably not achieved by the start of
the evaporation process (e.g., only advancing contact line motion
takes place after impact, so the measured values of h0 are expected
to be slightly greater than hEq). Nevertheless, the contact line is initially pinned during all these evaporation experiments, so dGA is
expected to be within a factor of two of that reported. The greatest
variation in dGA is observed with the microdroplets evaporating on
the Cu-pillar surface. This is reasonable because ASL –pR20 for the
pillar surface. Future studies with different pillar geometries and
surfaces approaching the super-hydrophilic and super-hydrophobic wetting regimes are needed to confirm the applicability of this
result.
5. Conclusions
Fig. 17. Excess free energy barrier per unit solid-liquid surface area, dGA, plotted as
a function of hD/h. dGA is calculated with Eq. (15) using measured droplet-profile
data (i.e., R0, AD, and A0). The symbols with error bars are for the surface materials
tested with microdroplets in this work (260 pL[ V0 [ 2200 pL). The symbols
without error bars are for the surface materials tested with millimeter-sized water
droplets in Ref. [56] (3.0 lL[ V0 [ 3.5 lL). The symbols for each surface material
are the same as that used previously. (see, the caption in Fig. 16).
A comprehensive experimental investigation of water microdroplet evaporation on superheated surfaces is provided. Measurements are made with microdroplets on a variety of different
surface materials. The main experimental properties/parameters
measured with each material system at different temperatures
_ LG Þ, evaporation rate per unit length
are the evaporation rate ðm
of solid-liquid-vapor contact line ðp_ LG Þ, and evaporative heat flux
(q). The key findings in this work include:
(1) For microdroplet evaporation, a maximum (critical) surface
Þ exists that represents the transition from
temperature ðT max
S
a purely evaporative heat transfer regime to a droplet/film
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S.A. Putnam et al. / International Journal of Heat and Mass Transfer 55 (2012) 5793–5807
boiling regime (analogous to the superheat at CHF in boiling). This critical surface temperature is inversely proportional to both the droplet’s contact radius (R0) and the
substrate thermal conductivity (K), where increasing R0
and/or K will reduce T max
toward the saturation temperaS
ture of water.
_ max
(2) At T max
, a maximum (critical) evaporation rate ðm
LG Þ is
S
_ max
found. m
is
independent
of
droplet
size
and
substrate
LG
thermal conductivity. In fact, a universal value of
_ max
m
LG ffi 35 5 lg/s is found for all surface materials tested.
For example, regardless of the properties of surface material
(i.e., thermal conductivity, surface hydrophobicity, and surface microstructure), the maximum evaporation rate mea_ max
_ max does
sured at transition is m
LG ffi 35 5 lg/s. Thus, mLG
not change by increasing the substrate thermal conductivity.
_ max
Instead, m
LG ffi 35 5 lg/s is just reached at a lower surface
temperature (superheat) on a higher thermal conductivity
substrate.
_ max
(3) Coupled to T max
and m
LG are the corresponding evaporative
S
heat flux (q) and evaporative heat transfer coefficient (h). In
this work, the largest evaporative heat flux measured with
water microdroplets is q = 1100 ± 100 W/cm2. This was measured with small (140 pL) microdroplets on roughened Cu
substrates. The largest heat transfer coefficient, on the other
hand, was measured with microdroplets evaporating on Cu
substrates having a Cu-pillar surface structure, where hpil2
lar = 650 ± 200 kW/m /K at a superheat of DT 10 °C (in
comparison to 50 ± 25 kW/m2/K (for different thin-films on
glass substrates) and 230 ± 90 kW/m2/K (for smooth and
800-grit sanded Cu substrates)). These results support that
maximizing the solid-liquid-vapor contact line length per
unit surface area is most important parameter for enhancing
the evaporative heat transfer performance (e.g., q>1000 W/
cm2 and h>600 kW/m2/K), which can be achieved by simply
spraying tiny water microdroplets on microstructured
surfaces.
(4) For all droplet-surface systems studied, stick-slip contact
line behavior is observed during evaporation. As expected,
the surface energy controls the contact line dynamics, where
the magnitude of the contact line depinning force is directly
proportional to surface energy. This is supported by comparison of data with free energy and contact line depinning
force calculations for all material systems studied, where fair
agreement is found between the microdroplets studied here
and millimeter-sized water droplets studied by others. Due
to stick-slip contact line motion, droplets evaporate in a pinned contact line mode until a critical contact angle (depinning force) is reached for contact line motion. Before
_ LG is a constant and is
contact line depinning (slippage), m
directly proportional to the surface temperature, contact
radius, and substrate thermal conductivity, whereas, after
depinning, the evaporation rate reduces in magnitude due
to a reduction in the contact line circumference (i.e., 2pR).
Thus, even though changes in surface energy can dramatically change the depinning dynamics, changes in surface
energy, however, has no systematic influence on the ratio
of the evaporation efficiency before and after contact line
depinning, suggesting that phase-change heat transfer is
dominated by the evaporation process at the solid-liquidvapor contact line.
Acknowledgements
This material is based on research sponsored by U.S. Air Force
Office of Scientific Research under Grant No. 2303BR5P. The
authors are thankful to Mike Check, Chris Murtatore, John Ferguson, Andrey Voevodin, and Chad Hunter for their help and frequent
technical discussions. The authors thank Arthur Safriet for technical contributions during design and construction of the experimental apparatus. The authors are also grateful to HPCMO for the
computing resources at the AFRL/DSRC. The views and conclusions
contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of U.S. Air Force Office of
Scientific Research or the U.S. Government.
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Cu
Al
SAM
hTi
ffi 113 20 (in order from
0 ffi 79 60; h0 ffi 91 30; h0 ffi 95 50, and h0
most hydrophilic to most hydrophobic). Average receding (depinning) contact
angles
measured
(25 °C [ TS
[ 95 °C):
AlðcÞ
AlðpÞ
hCu
ffi 38 190; hTi
ffi 66 160,
and
0 ffi 29 70; h0
D ffi 45 130; h0
SAM
h0 ffi 101 50.
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