Supplementary Materials for - Science Advances · 2020-01-06 · The heat transfer through an...

advances.sciencemag.org/cgi/content/full/6/2/eaax0746/DC1

Supplementary Materials for

Dropwise condensation on solid hydrophilic surfaces

Hyeongyun Cha, Hamed Vahabi, Alex Wu, Shreyas Chavan, Moon-Kyung Kim, Soumyadip Sett, Stephen A. Bosch,

Wei Wang, Arun K. Kota*, Nenad Miljkovic*

*Corresponding author. Email: [email protected] (A.K.K.); [email protected] (N.M.)

Published 10 January 2020, Sci. Adv. 6, eaax0746 (2020)

DOI: 10.1126/sciadv.aax0746

The PDF file includes:

Section S1. Surface energy of PEGylated surfaces Section S2. Dropwise condensation heat transfer Section S3. Dropwise condensation model Fig. S1. Time-lapse images of droplet nucleation and departure. Fig. S2. Atomic force microscopy. Fig. S3. Condensation regime map. Table S1. Promoter coatings and their wetting properties. References (43–70)

Other Supplementary Material for this manuscript includes the following: (available at advances.sciencemag.org/cgi/content/full/6/2/eaax0746/DC1)

Movie S1 (.mp4 format). Dropwise condensation on hydrophilic surface.

Supplementary Figure

Fig. S1. Time-lapse images of (a) droplet nucleation and (b) departure during water vapor

condensation on the PEGylated surface. Upon coalescence, the three-phase contact line was able

to depin due to the low contact angle hysteresis and form discrete circular droplets.

Fig. S2. Atomic force microscopy. AFM images of the polished copper samples on three

different spatial locations showing root mean square (RMS) roughness of (a) 10.7 nm, (b)

15.4 nm, and (c) 7 nm. The direction of the roughness depends on the direction of AFM scan and

arises due to tooling marks during manufacture of the copper samples obtained from McMaster.

Section S1. Surface energy of PEGylated surfaces

Owens-Wendt approach (43) was used to estimate the surface energy of our PEGylated surfaces.

Diiodomethane (𝛾𝑙𝑣 = 50.8 mN/m) (44) was used as the non-polar liquid to estimate the

dispersive component of the solid surface energy 𝛾𝑠𝑣𝑑 and water (with dispersive component of

𝛾𝑙𝑣𝑑 = 21.1 mN/m and polar component of 𝛾𝑙𝑣

𝑝 = 51.0 mN/m) (44) was used as the polar liquid to

estimate the polar component of the solid surface energy 𝛾𝑠𝑣𝑝

. The advancing contact angles of

diiodomethane and water on our PEGylated silicon wafers are 29° and 37°, respectively. Based

on these values, we estimated the surface energy of our PEGylated surfaces to be 𝛾𝑠𝑣 ≈ 68

mN/m.

Section S2. Dropwise condensation heat transfer

The heat transfer through an individual hydrophilic droplet (𝜃a < 90°) was calculated from the

analytical solution given by Sadhal and Martin (45). For hydrophobic droplets (𝜃a > 90°), to

calculate the heat transfer through individual droplets, the numerical solution given by Chavan

et. al was used (28). The individual droplet heat transfer, characterized by the droplet Nusselt

number (Nu), is a function of the Biot number (Bi) and apparent advancing contact angle (𝜃a),

i.e. Nu = 𝑓(Bi, 𝜃a). Here, the Nusselt and Biot numbers are defined in terms of the droplet base

radius (𝑅b) as (45)

Bi =ℎi𝑅b

𝑘w (S2)

Nu = 𝑄

𝑘w𝑅b(𝑇sat − 𝑇s) (S3)

where 𝑄 is the total heat transfer through the droplet and 𝑘w is the droplet thermal conductivity.

Once Nu is obtained, 𝑄 can be calculated from equation (S5). The liquid-vapor interface heat

transfer coefficient is given by the interfacial condensation heat transfer coefficient, ℎi (46, 47)

ℎi =8𝜎

2 − 0.798𝜎

𝛾 − 1

𝛾 + 1

1

√2𝜋𝑅g𝑇sat

ℎfg2

𝜈g𝑇sat (S4)

where 𝑅g is the specific gas constant and 𝜈g is the water vapor specific volume, 𝑇sat is the water

vapor saturation temperature, ℎfg is the latent heat of condensation phase change, 𝛾 is the

Poisson constant, and 𝜎 is the condensation coefficient which is taken as, 𝜎 = 1.

To study the overall steady-state condensation heat flux, we combined the individual droplet heat

transfer 𝑄 with droplet distribution theory to account for the fraction of droplets on the surface of

a given radius 𝑅 for the surfaces undergoing gravitation shedding and jumping. For small

hydrophobic droplets (𝑅 ≤ 𝑅e), the size distribution 𝑛(𝑅) is determined by (48)

𝑛(𝑅) =1

3𝜋𝑅e3�̂�

(𝑅e

�̂�)

−23 𝑅(𝑅e − 𝑅min)

𝑅 − 𝑅min

𝐴2𝑅 + 𝐴3

𝐴2𝑅e + 𝐴3exp(𝐵1 + 𝐵2) (S5)

where �̂� is the average maximum droplet radius (departure radius), 𝑅e is the radius when

droplets begin to merge and grow by droplet coalescence, 𝑅min is the critical nucleation radius

for condensing droplets (≈10 nm for water). For large hydrophobic droplets growing due to

coalescence (𝑅 ≥ 𝑅e), the droplet distribution 𝑁(𝑅) is determined from (49)

𝑁(𝑅) =1

3𝜋𝑅e2�̂�

(𝑅e

�̂�)

−23

(S6)

The variables 𝐴1, 𝐴2, 𝐴3, 𝐵1, 𝐵2 are constants associated with droplet sweeping, defined as (50)

𝐴1 =∆𝑇

ℎfg𝜌w(1 − cos 𝜃)2(2 + cos 𝜃) (S7)

𝐴2 =𝜃

4𝑘w sin 𝜃 (S8)

𝐴3 =1

2ℎi(1 − cos 𝜃)+

1

𝑘HC sin2 𝜃[

𝑘𝑝𝜙

𝛿HC𝑘p + ℎ𝑘HC+

𝑘𝑝(1 − 𝜙)

𝛿𝐻𝐶𝑘w + ℎ𝑘HC]

−1

(S9)

𝐵1 =𝐴2

𝜏𝐴1[𝑅e

2 − 𝑅2

2+ 𝑅min(𝑅e − 𝑅) − 𝑅min

2 ln (𝑅 − 𝑅min

𝑅e − 𝑅min)] (S10)

𝐵2 =𝐴3

𝜏𝐴1[𝑅e − 𝑅 − 𝑅min ln (

𝑅 − 𝑅min

𝑅𝑒 − 𝑅min)] (S11)

𝜏 =3𝑅𝑒

2(𝐴2𝑅e + 𝐴3)2

𝐴1(11𝐴2𝑅e2 − 14𝑅𝑒𝑅min + 8𝐴3𝑅e − 11𝐴3𝑅min)

(S12)

In our case, the analysis is valid for smooth surfaces (𝜙 = 1, ℎ = 0, 𝛿HC ≈ 0) or nanostructured

superhydrophobic surfaces (ℎ ≈ 0, 𝛿HC ≈ 0), 𝐴3 is defined as

𝐴3 =1

2ℎi(1 − cos 𝜃) (S13)

For hydrophilic droplets, the definition of 𝑅𝑒 changes because of the geometry of the hydrophilic

droplet and the fact that the droplet base radius, 𝑅𝑏 will be related to the nucleation density and

not the droplet radius, 𝑅

𝑅e =1

4 sin 𝜃 √𝑁s

(S14)

We have an additional sin 𝜃 term in the denominator. Thus, for hydrophilic droplets (𝜃a < 90°),

Eq. (S5) and (S6) are modified as

𝑛(𝑅) =1

3𝜋𝑅e3�̂�(sin 𝜃e)3

(𝑅e

�̂�)

−23 𝑅(𝑅e − 𝑅min)

𝑅 − 𝑅min

𝐴2𝑅 + 𝐴3

𝐴2𝑅e + 𝐴3exp(𝐵1 + 𝐵2) (S15)

𝑁(𝑅) =1

3𝜋𝑅e2�̂�(sin 𝜃)3

(𝑅e

�̂�)

−23

(S16)

The total surface steady state condensation heat flux (𝑞") is obtained by incorporating the

individual droplet heat transfer rate obtained from literature (28, 45), with the droplet size

distributions (Eq. S5, S6 for 𝜃a > 90° and Eq. S17 and S18 for 𝜃a < 90°)

𝑞" = ∫ 𝑄(𝑅)𝑛(𝑅)𝑑𝑅𝑅e

𝑅min

+ ∫ 𝑄(𝑅)𝑁(𝑅)𝑑𝑅�̂�

𝑅e

(S19)

To calculate the condensation heat transfer coefficient, ℎ, the heat flux, 𝑞" is divided by the

temperature difference, Δ𝑇

ℎ =𝑞"

Δ𝑇 (S20)

Section S3. Dropwise condensation model

In order to take into account droplet sweeping, and to expand on the coalescence model

formulation, which clearly does not match the experimental data at elevated contact angle

hysteresis, we hypothesized that a droplet length-scale dependence must be taken into account

when considering dropwise condensation stability. Indeed, droplet shedding models can predict

unphysical droplet departure sizes exceeding diameters of 1cm (Manuscript Fig. 1c, inset), which

are not observable in real life due to rivulet or film/puddle formation. The lack of an upper length

scale on droplet shedding models motivated us to consider a second model for predicting

dropwise-to-filmwise condensation transition based on droplet to puddle transition.

We begin by addressing how a liquid puddle spreads on a solid surface. We expect a partial

wetting liquid droplet to spread initially to minimize the system free energy and to stop

spreading when it reaches equilibrium. For small liquid volumes, i.e., when the Bond number is

less than 1 (Bo ≤ 1) capillarity is the dominant driving force (33). As the volume of the liquid

increases, i.e. (Bo ≫ 1), gravity becomes the dominant driving force in the bulk while surface

tension effects remain limited to the vicinity of the moving contact line (34, 35, 51). In this

regime, the hemispherical droplet shape approximation is no longer valid. Rather, droplets will

form puddles on the surface. We hypothesize that dropwise condensation is stable while the

contact angle hysteresis mediated droplet shedding length scale is dominated by capillarity,

resulting in hemispherical droplet shapes. For surfaces having large contact angle hysteresis, the

gravitational body force required to overcome contact line pinning is so large that capillarity

ceases to govern the droplet dynamics, giving way to gravitational forces (Bo ≫ 1). The inability

to maintain hemispherical droplet shapes results in puddle or film formation on the surface,

rendering the droplet shedding model invalid, and resulting in filmwise condensation.

To test our hypothesis, we analytically calculated the Bond number for droplets immediately

prior to departure from a vertical condensing surface to determine whether the droplets reside in

the capillarity-dominated regime (Bo ≤ 1) or the gravity-dominated regime (Bo ≫ 1). We define

the capillary length as 𝑙y = √𝛾/𝜌𝑔 where 𝛾 and 𝜌 are the condensate surface tension and

density, respectively, and 𝑔 is the gravitational constant. The Bond number is defined as

Bo = 𝑅f2/𝑙y

2, where 𝑅f is the characteristic lateral length of the liquid droplet, taken to be its final

equilibrium radius immediately prior to departure. The droplet radius scales as 𝑅f ~ 𝑉1/3 in the

capillary-dominated regime, and ~ 𝑉/ℎ∗1/2

in the gravity-dominated regime, where 𝑉 is the

droplet volume, and ℎ∗ is the film of puddle height (36). For the transition studied here, the

capillary-dominated droplet radius scaling was used as the key transition is hypothesized to

occur at Bo ≈ 1 and the shedding model is valid for discrete droplets. The surface wettability

dependent droplet volume is defined as 𝑉 = 𝜋𝑅f3(2 − 3 cos 𝜃e + cos3 𝜃e)/(8 sin3 𝜃e), where

𝜃e = cos−1(0.5 cos 𝜃a + 0.5 cos 𝜃r) is defined as the equilibrium contact angle immediately

prior to shedding from the condensing surface (37, 50). The droplet departure size 𝑅f was

defined by a force balance between the gravitational body and contact line pinning forces, and is

defined as 𝑅f = [6𝛾(cos 𝜃r − cos 𝜃a) sin 𝜃e)/(𝜋𝜌𝑔(2 − 3 cos 𝜃e + cos3 𝜃e))]1/2. Taking the

dependence of the contact angle hysteresis on the droplet volume and departure radius into

account, we see a clear Bo mediated boundary separating filmwise from dropwise condensation

(fig. S3). Our model predicts Bocrit ≈ 1.4 as the boundary separating filmwise to dropwise

transition, with Bo ≤ 1.4 indicating capillary-dominated hemispherical droplet shapes at

departure, while Bo > 1.4 indicating gravitationally-dominated puddle formation and filmwise

condensation prior to discrete droplets reaching the required droplet shedding radius.

Fig. S3. Condensation regime map for the Bond number model depicting the condensation

mode as a function of the intrinsic advancing contact angle and contact angle hysteresis. The

light blue and light red shaded regions represent dropwise and filmwise condensation,

respectively. The dotted blue line bounding the two regions represents Bocrit ≈ 1.4. The black

solid line represents the inaccessible boundary (i.e. contact angle hysteresis cannot exceed the

advancing contact angle). All experimental data points are summarized in table S1.

To test our hypothesis, we compare our model with previous experimental data of solid

hydrophobic surfaces undergoing stable dropwise condensation, quasi-dropwise condensation, as

well as filmwise condensation (table S1). Note, the nature of hydrocarbon adsorption is time

dependent (13, 15, 52), hence, the advancing and receding contact angles vary depending on

when the samples are characterized. We took advantage of this fact in order to study the effect of

contact angle and contact angle hysteresis on the dropwise condensation criteria.

Table S1. Promoter coatings and their wetting properties. For full chemical names,

deposition methods, and dropwise condensation behavior, see references.

Promoter Material Sample

Geometry

Deposition

Method 𝜽𝐚 𝜽𝐫 ∆𝜽 Ref.

PTFE Tube Chemical Vapor

Deposition (CVD)

125° 115° 10° (53)

C4F8 Plate CVD 110° 102° 8° +

HTMS Plate Vapor Phase

Deposition (VPD)

112° 104° 8° (14)

Parylene Plate CVD 90° 72° 18° (54)

TFTS Tube Liquid Phase

Deposition (LPD)

122° 86° 36° (50)

p(PFDA- co-DVB) Plate Initiated CVD 132° 127° 5° (55)

THIOL Plate LPD 121° 106° 15° (50)

P2i Plate CVD 123° 114° 9° (56)

Graphene Plate Low Pressure CVD 87° 64° 23° (57)

Graphene Plate Atmospheric

Pressure CVD

93° 56° 37° (57)

Diamond Like Carbon

(DLC)*

Plate Physical Vapor

Deposition (PVD)

84° 52° 32° +

Palmitic Acid Plate VPD 101° 63° 38° (58)

Stearic Acid Plate VPD 87° 53° 34° (58)

CeO2 Plate PVD 103° 55° 48° (59)

Er2O3 Plate PVD 108° 60° 58° (59)

PEGylated Plate LPD 38° 35° 3° +

hexamethyldisiloxane

(HMDSO)

Plate CVD 120° 96° 24° (60)

HMDSO Plate CVD 120° 91° 29° (60)

HMDSO Plate CVD 120° 100° 20° (60)

Poly(r-methylstyrene) Plate CVD 102° 72° 30° (61)





Stearic Acid Plate LPD 109° 88° 21° (62)

Polyethylene Plate CVD 115° 100° 15° (63)

Polyethylene Plate CVD 112° 94° 18° (63)

Teflon FEP Plate CVD 117° 98° 19° (63)

n-octadecyl mercaptan Plate LPD 120° 101° 19° (64)

Gold Plate Hydrocarbon

Adsorption

85° 46° 39° (65)

Silver Plate Hydrocarbon

Adsorption

85° 74° 11° (65)

Silver Plate Hydrocarbon

Adsorption

89° 55° 34° (65)

Polyethylene Plate Bulk Material 111° 74° 37° (66)

PFMS/Cu Plate LPD 132° 98° 34° (67)

PFMS/Cu (HNO3) Plate LPD 133° 95° 38° (67)

C7F15CH2OCH2CH2CH2SiCl3 Plate LPD 119° 104° 15° (68)

C6F13CH2CH2SiCl3 Plate LPD 113° 94° 19° (68)



Rhodium Plate Hydrocarbon

Adsorption

82° 73° 9° (69)

Hydrocarbon Adsorption** Plate VPD 46° 23° 23° +
















+ Measured in laboratory by the authors.

* Experiment demonstrated quasi dropwise condensation (see Fig. 3 of manuscript)

** Experiment demonstrated filmwise condensation

Model Limitations

Although the Bond number model delineates the dropwise-to-filmwise transition with exquisite

precision, it does not take into account the nucleation site saturation transition mechanism of

dropwise-to-filmwise transition (70). For stable dropwise condensing surfaces, nucleation site

saturation occurs when

𝛾𝑇v

𝜌ℎfg∆𝑇< 1 nm (S1)

where 𝑇v is the steam temperature, ℎfg is the latent heat of phase change for water, and ∆𝑇 is the

steam-to-surface temperature difference.

For steam at atmospheric pressure, the critical heat flux is on the order of 5 to 10 MW/m2,

multiple orders of magnitude higher than the wettability and contact angle hysteresis governed

transitions developed in this work. Even though the transitions occur due to differing criteria,

they are related via the surface chemistry. For the classical nucleation site saturation limit, as ∆𝑇

increases, the radius of curvature of the smallest nucleated droplet (�̂� = 2𝛾𝑇v/𝜌ℎfg∆𝑇) decreases

and more nucleation sites become active. Any promoter surface has an intrinsic distribution of

potential nucleation sites which are highly dependent on the surface chemistry (11). For

example, highly wetting surfaces have higher nucleation rates, resulting in higher nucleation

density for the same ∆𝑇 as a non-wetting surface. For a given surface, when ∆𝑇 reaches a value

such that a significant proportion of the total surface has sites separated by distances of 2�̂� or

less, a significant portion of the surface becomes wetted and departure from the ideal dropwise

condensation. For wetting surfaces, this results in a lower critical condensing heat flux compared

to non-wetting surfaces, implicating that the PEGylated surfaces developed here should flood at

a lower ∆𝑇 when compared to classical non-wetting promoter coatings (2).

Supplementary Materials for - Science Advances · 2020-01-06 · The heat transfer through an...

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