1

wetting of complex surfaces

lv

slsv

σσσ

θ−

=cos

so far:sessile drops on flat, homogeneous surfaces

.11

21

constRR

pp lvlvL ==

+==∆ κσσ

∑ ==∆−=i

iii svsllvipVAAG ,,min!

][ σ

equilibrium morphology is determined by minimum of free energy

functional variation yields two equationsa) capillary equation

b) Young equation

• spherical cap-shape • contact angle: Young-eq.

lecture 1:

(Bo<<1)

2

two types of complexity

stripe with a different wettability

b) complex surface topographya) complex chemical patterns

)(// rsvslsvsl σσ =?

3

∑ ∆−=i

iii pVAAG σ][

energy minimization still applicable, but now: )(// rsvslsvsl σσ =

{ } min!

)()(][ =∆−−+= ∫ pVrrdAAAG svslsllvlvi σσσ

lv

slsv rrrσ

σσθ

)()()(cos −=.11

21

constRR

pp lvlvL ==

+==∆ κσσ

functional minimization yields two equations:a) capillary equation (unchanged) b) local Young equation

θ is measured w.r.t. local surface normal

min!~)(cos][~=∆−+= ∫ VprdAAAG sllvi θ lvGG σ/~

= lvpp σ/~ =

energy minimization on heterogeneous surfaces

(Swain, Lipowsky 1998)

4 Seemann et al. PNAS 2004

5

wetting on a striped surface

microcontactprinting; period: 1µm

x

y

50 µm

HexaethyleneglycolPTCS:Si

stripes with alternating wettability

• drops are stretched along the pattern direction• contact line is partially pinned along stripe edges

6

local structure at contact line

Pompe, Herminghaus PRL 2000

AFM 500 nm 0 1 2

0.0

0.1

0.2

0.3

z [µ

m]

x [µm]

hydrophilic hydrophobic

θphil

θphob

• contact line is corrugated• corrugation amplitude decreases with

increasing height• local contact angle is low on hydrophilic and

high on hydrophobic stripes• intermediate macroscopic c.a.

7

3 questions

n how is the drop surface disturbed due to the contact line distortion?

n what is the contact angle perpendicular to the stripes? (pinning, metastability, morphological transitions, c.a. hysteresis)

n what is the macroscopic average contact angle on a heterogeneous surface?

8

1. surface perturbation

0=+ zzyy ζζ

θ=90°; R=∞ à capillary equation:

qyz cos)0( 0ζζ ==:B.C.

ζ : surface distortion

contact line contour:

solution: )exp(cos),( 0 qzqyzy −= ζζ

[ ] 20

20

22

21||

41

21

ζζσζζσ surflvyzlv kqdzdyF ==+=∆ ∫additional surface energy:

surface stiffness: qk lvsurf σ21

=

flat surface at large distance

à exponentially decaying perturbation

9

2. contact angle along domain boundary

∆θ

hydrophilic

hydrophobic

striped surface:

θphil

θphob

αθθαθ +≤≤−

surface roughness:∆θ

α

à for surface roughness with slope angle α:

à contact angle at domain boundaryies is not uniquely defined: phobphil θθθ ≤≤

crucial for super-hydrophobic surfaces!

θ

10

consequence: existence of metastable states

• configurations A and B have same volume

• both configurations fulfill phobphil θθθ ≤≤

à both configurations are allowed and mechanically stable

à surface heterogeneity (as well as roughness) gives rise to metastable statesà characteristic energy barriers: O(size x surface tension) >>kBT

θphobθphil

A

B

11

metastable states at isolated defect

• mechanically equilibrated shape: balance of wetting & additionalliquid-vapor surface area

• history-dependence of shape: à hysteresis• depinning: sudden release of stored surface energy à dissipation

12

two-dimensional surfaces

TkJnmmJE B61522 1010)100(/1.0 ≈=×≈∆ −

typical energy barriers (e.g. 100nm heterogeneity):

à even for nanoscale heterogeneity: TkE B>∆

13

morphological transitions

philic

Lenz, Lipowsky PRL 1998

phobic

condensation from vapor:

2 configurations with equal pressure!

v < half sphere: dP/dV > 0v > half sphere: dP/dV < 0

à instability of ensemble of cylinders (maximum bubble pressure method)

02

2

>dV

Fd required for a minimum is not fulfilled for ensemble of cylinders with side angle >π/2

14

morphological transitions (II)

0 100 200 300 400 5000,0

0,5

1,0

θs = 39° 32°

a / a

0

U [V]

V=4.2~

V=1.8~

droplet height

FM, J. Appl. Phys. 2004

water in silicone oil(AF 1600)

)(// rsvslsvsl σσ =

15

a single hydrophilic stripe

Brinkmann, Lipowsky J. Appl. Phys. 2002

free energy(around II/III transition)

increasing volume possible morphologies

θstripe=38°

16

comparison to theory

25 30 35 40 450

2

4

6

8

10

~

V

θs [°]

θcr=39.2°Vcr/L3=2.85

17

contact angle vs volume on striped surface

Iwam

atsu

et

al. J.

Col

l. In

terf

. Sc

i. 20

06Jo

hnso

n, D

ettr

e, J

. Ph

ys. Che

m. 19

64

18

stability of contact line at domain boundary

hydrophilic

hydrophobic

θphil

θphob

σsv σsl∆σsol. ∆σcol.

à on a striped surface only configurations with liquid sitting on the hydrophilic side of a boundary are stable

stable interface positionunstable interface position

19

contact angle vs volume on striped surface

Vol.>>stripe width3: transition from discrete morphological transitions to appearance of well-defined advancing and receding contact angle (equal to θA and θB)

20

large drops: contact angle hysteresis

AYR θθθ <<advancing contact angle θA

receding contact angle θR

RA θθθ −=∆contact angle hysteresis

OTS-hydrophobized Si wafer: ∆θ < 5°

deGennes, Brochart, QuéréCapillarity and Wetting Phenomena

Springer

21

measuring contact angle hysteresis

water on Teflon AF in air; ∆θ ≈ 10°; Ca << 1

0 100 200

110

120

θ

time [s]

∆θ

θA

θR

22

qualitative correlation: hysteresis & pinning forces

θR

θA

θY

σlv

σsv

σsl

AR

psvlvsl f

θθθ

σθσσ

<<→

±=+ 2/||cosforce balance:

fp

|coscos|cos~RApf θθθ −=∆=random pinning force

similarly: chemical heterogeneity

simplistic picture:

23

overcoming contact angle hysteresisfree energy vs. contact angle

(striped surface)

Johnson, Dettre, J. Chem. Phys. 1964Della Volpe et al. Coll. Surf. A 2002

θ

idea: provide energy to drop to overcome energy barriers

θadv

θrec

F

Wilhelmibalance + loudspeaker

experiment: mechanical shaking via acoustic excitation

forc

e

displacement

24

θ

“equilibrium” angle on a heterogeneous surfaces

drop >> length scale of heterogeneity à averaging over heterogen.

1dim. surface: θadv=θB; θrec=θA

25

heterogeneous surfaces (I)

BBAA ff θθθ coscoscos * +=

Cassie Baxter equation

θA θBθ ∗

dx cos θ ∗

dx

energy minimization

{ } { } 0cos *,,,, =+−+−= dxdxfdxfW lvBBsvBslAAsvAsl θσσσσσδ

fA , fB: fractional coverage of material A, B

chemical heterogeneity:

)cos(coscoscos)1(coscos *BAABBAAA fff θθθθθθ −+=−+=

26

heterogeneous surfaces (II)

θ ∗dx cos θ ∗

dx

energy minimization { } 0cos =+⋅−= dxdxrW Ylvsvsl θσσσδ

., appsl

sl

AAr =roughness-induced interface enlargement

r

cos θ

-1

1

Yr θθ coscos * =Wenzel equation

i.e. θ >(<) 90°: roughness in(de)creases θ

surface roughness:

27

à drops must be large compared to characteristic length scale of roughness

28

summary

n on heterogeneous surfaces Young eq. has to be fulfilled locally

n heterogeneity leads to pinning of contact line at domain boundaries and existence of metastable states

n energy barriers are typically >> kBTn metastable states lead to morphological transitions and

to contact angle hysteresisn averaging over small scale roughness and chemical

heterogeneity leads to Wenzel and Cassie-Baxter prediction of contact angle on heterogeneous surface