contact angles young equation

8/10/2019 Contact Angles Young Equation

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CONTACT ANGLES:Laplace-Young Equation and

-R. L. Cerro

Chemical and Materials EngineeringThe University of Alabama in Huntsville

Santa Fe

16 de Abril de 2010


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Dr.Javier Fuentes (PhD-2003)-Univ. Simon Bolivar, Caracas,

VE. School of Chemical

Engineering and AnalyticalScience, The Univ. ofManchester, UK

r. ena az ar n -2004)-Depto. de Ing. Quimica &Textil, Universidad de

Salamanca, Spain.

. . ,and Astronomy Dept. LeedsUniversity, UK.


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important?

e u y- ugmente oung- ap ace quat on

Youngs relationship: static contact angles.

A 2D sessile drop = puddle.

ap ary r se

The futurebehind


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-

Super-hydrophobicity is the resultof chemistry and structure.


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-

Structure & Chemistry


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Feet of a geico and

the threads of amussel are examplesof natural adhesives.


8/41

Interfaces are:

1.Diffuse (3D)

2.Dynamic

.


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Static Contact Angles:Young relationship

- .

WETTING: The contact angle for a

THOMAS YOUNG 1805

ree-p ase reg on s e ma n

variable in Youngs equation.

cosSV SL LV =CAPILLARITY: The curvature ofinterfacial free energy of theinterface, are related to thepressure jump between the

YOUNG-LAPLACE EQUATION

inside and outside of a liquiddrop.

( )2 0

in out VLp p H =


10/41

Young-Laplace Equation in Differential Form:

inside outside

=

{ 1 21 2

2 , :principal radius of curvatureH R RR R

= +

2

;inside in outside out in o

p g z p g z g z = =

( ) ( )2

3/2 1/21/2 1/2

in out in o

d x dxg z g z z

= = +

1 1z

dx dx

+ +


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The equation of Young and Laplace:

.

Thomas Young [Phil. Trans. Roy. Soc, vol 95, pp. 65-87

Born in Milverton, Somerset (1773) youngest of 10 children Studied medicine in London, Edinburgh and physics in Gottingen Entered Emmanuel College in Cambridge and practiced medicine in London Appointe pro essor o Natura p i osop y at Roya Institution 1801 Foreign associate in French Academy of Sciences (1827) Wave theory of light, Young modulus, translated hieroglyphs, etc.

, .(1807)] Born in Normandy, 1749. Univ. of Caen (16 years old)

n v. o ar s years o . Rejected by Acad. of Sciences (22 yr old) Accepted to Berlin Acad. of Sci. (24 yr)

On the attribution of an equation of capillarity to Young and Laplace, Pujado, Huh

and Scriven, JCISvol. 38, pp 662-663, (1972).


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Surfaces in 3D space:

Surfaces in 3D space:

Orientable in space Locally have two sides

Globally have in general two sides with famousexceptions (Mobius strip)

,from an outside space with famous exceptions(Klein bottle)

hey have shape Globaly shape distinguishes a torus from a sphere


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1. Location: a point on the surface is described by

.

2. Orientation: the top and bottom are describedby a unit vector n, normal to the surface

3. Two tangents unit vectors, a1 and a2 arenormal to each other and both are normal to

the vector n.( ), ,R R x y z= 4. Points on t e sur ace can e escri e on t e

basis of a two dimensional system.

5. The rate of change of orientation (normal)

( )1 2,r R u u=

1 1 2 21/ ; 1/R R = =

correspon s o e n u ve no on o s ape.6. Curvature is defined as the inverse, 1/R, of the

radius of a circle tangent to the surface.

( )1 21 2

1 1 1 1

2 2H

R R

= + = +

7. There are two independent radius of curvature

and their directions are normal to each other.

1 2

1 2

KR R

= =


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Young-Laplace equation:simplified

. . .r n t t

F e F e F = +

2 sin 2 sinndf dl dl = +

1 2

1 12d d dl +1 2

/2

21 1 1 12d

F d d dl d

+ = +1 2 1 20 0 R R R R

( )21 2

2nP H

d R R

= = + =


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Minimal surfaces (soap films):

, ,Lagrange: Lectures on a novel method for the

determination of maxima of inte ral formulae 1762

( ) ( )22

, 1S

z zz z x y I dxdyx y

= = + +

Definition: A minimal surface is a surface whose mean curvature is zeroat every point of the surface.

all portions of surfaces bounded by the same closed curve, then thesurface is a minimal surface.

property that any portion of them bounded by a closed curve hasthe minimum area.

2

( ) ( ) ( ) ( ) ( ) ( )

1/ 2 3/ 2 1/ 22 2 22 2 2

2

1 1 1

x y xy yyxx

x y x y x y

H

z z z z z z

= +

+ + + + + +


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Static Contact Angles

SV cosSL VL o =

equilibrium of the three forces acting on the angularparticles, one on the direction of the surface of thefluid only, a second in that of the common surfaceo e so an u , an e r n a o eexposed surface of the solid. Now supposing theangle of the fluid to be obtuse, the whole superficialcohesion of the fluid bein re resented b theradius, the part of which acts in the direction of thesurface of the solid will be proportional to thecosine of the inclination; and its force added to the

,common surface of the solid and fluid, or to thedifference of their forces; consequently, the cosineadded to twice the force of the fluid; will be equal

An Essay on the

to ....

Phil. Trans. Roy. Soc.

v. 95, 65-87


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Interfaces are:

1.Diffuse (3D)

2.Dynamic

.


18/41

Macroscopic definition of contact angles

cosSV SL LV

=

Young (1805) derived relation asa balance of forces.

s Equation can be derived usingmacroscopic arguments.

are macroscopic/thermodynamicparameters.

parameters and thermodynamicfunctions.


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Surface forces of the second kind:

. ,

Derjaguin and

Obuchov (1936) ,

interaction of molecular force

fields due to the presence of ar p ase.

Forces of the second kind are the sameforces determining surface tension:

(1) Dipole-dipole, nonpolar or charge-dipole interactions.(van der Waals)

2 Electrical double la ers

(3) Structural forces induced by

molecular order.


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Augmented and Fully-augmented Young-

Static Jump-Momentum balance:Derjaguin et al. Surface

B A

Normal component:

Forces (1987)

Teletzke, Davis and Scriven(1988)

, ,VL

Tangential component:

,

L

VLg r

+

=

Miller and

,L

VL

=

The Youn -La lace e uation is valid awaRuckenstein (1974)

Jameson and del

Cerro (1976).

from the solid surface where disjoining

pressure is negligible and surface tension isconstant!


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MOTIVATION: molecular interactions

Derjaguin and Obuchov (1936)

[ ] [ ]6 6

3

,

SL LLA A =

variable surface tension, gVL

VLg

d h=

Questions!'

What is the proper definition for ?o

Where is located on the vapor/liquid interface?

o

o


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Contact Angles (Merchant and Keller, 1992)

Used the method of matched asymptoticexpans ons o va a e oung s equa on

Leading term in the outer expansion for the interfaceshape satisfies the Young-Laplace equation.

Leading term in the inner expansion satisfies an .

the slope angle of the leading term in theouter expansion is o-as given y Young s equation.


23/41

Interesting relationships for 2D systems:

. . , ,

dy= =

dx

dy

1/22

sin

1

x

dy

=

+22

2 d y dyd y

1

dx 2

1/2 3/22 2

s n

1 1

dx dxdx

dx dy dy

dx dx

= =

+ +

1/22

1 dy

+

2

2

2

d y

dx= = H2

1 dy

dx

+


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Macroscopic approach:

-

de Gennes, Brochard,

Quere (2004) the

Young-Laplace Equation: ( )2 ,L V L V CH p p p p g h h = =

ousew e pro em.

3/ 2

22

22 / 1

h hH

z z = +

cosd dh= =

Scriven, ChEn 8104 class notes (1982)

dh dz

cos Ch hd =2

L =

2h h

2Cdh L g

2 2cos . . cos 1

2

C

C C

C B C at L L

= + = =


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Integrate and introduce one

2

2 2

2

cos . . cos 1

2 C

C C

C B C at h h

L Lh

= + = =

2

2 22

2 C

C C C

L

h h h hh

=

2 2 2 22 2 2

o

C C C C L L L L

Solution to Young Laplace equationdescribes the gas-liquid interface of

Defines contact angle as theangle of intersection betweensolution of YL equation and

horizontal surface. .


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Second method:

{

2

2 2cos . . cos cos 02

C

o

C C

h h h

C B C at hL L = + =

2

2 2

cos

cos cos cos 1

o

C

o C

h h h

h h

=

= + = =C C

Solution to Young Laplace equationdescribes the gas-liquid interface ofa liquid puddle resting on a smooth,

e nes contact ang e as t eangle of intersectionbetween gas-liquid interfaceand the solid surface.

or zon a sur ace.


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Consequences:

1 Mathematical is the B.C. at the solido

definition of o

surface for solutions of the

Young-Laplace equation

w en 0o= =

2h2

2

o

CL

=

2 Measurement of

match data points to a solution of the Young-Laplace equation.

ex en e so u on o =

measure the angle.


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MOTIVATION: Questions

Young's relation is valid.macroscopically . . .

is measured intersecting YL with solid.o

o

Where is located on the vapor/liquid interface?o


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Fluid Wedge: 3-region model


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3-Region Model: Characteristic

83. 10

C

t

m

h m

1010

mh m

m

t

h

h=

3 210 10


31/41

M l l r r i n: Der a uinspecific interfacial free energy

VLg =

( ) 36

= =

VL LL SLdg A A

hdh h

2

3

VL md g h

d h h

=

6 6

2

6

LL SLm

A Ah

=

. . asVL

B C g h

2

( ) 1 2m

VLg hh

=

9Note: 0.995 when 10VL

g h m =


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Transition Region: Null Curvature Point

2 CH g h h =

2Augmented

3 2Young-Laplace

equation

m C

Cd h h L

=

2

3 21m C t

h h h =

At the null curvature point; h = ht , 2H = 0

2 2

t C C

t C = m Ct

C

h

h

Since


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2

cos m Ch h hd = 2

2cos 1 C

o

h =

0C

BC h h = =

C C

( )

2 2 2

2 2 2cos cos 2 1 cos2 2 2

m m

o o

h h h h

h h h h

= +

At andt th h = =

( )2

2

smaller

terms

cos cos 2 1 cos2

m tt o o

t C

h h

h h

= + +

2

23 3

2 2

m

t

h

h

=


34/41

Transition region: Location of

Where, on the vapor/liquid interface, is o to be found?

23cos cos 2

t o

23o t + o

Nowhere!since

for all on interface

o t

> >

( )2But: to Oo t


35/41


36/41

CAPILLARY RISE: MACROSC0PIC APPROACH

2H =

McNutt and Andes, J. of Chemical

Legendre transformation

2 22

sin2

1 sin

d yy Y y

d L Y

= =

= = =H

2 2

2 20 ; sin 1 C CL Ly =

2 2

0sin 1 1 cos / 2

2 2

o oo o o

Y Y

= = =


37/41

Capillary Rise: Molecular Approach

2cos mhd g at h h

= + = =

d h h

2 Sh

2 2cos

2

mo

s C

dhh L

=

( ) ( )/ cot / 2 tandh dh dy dy dy dy = = =

( )( ) ( )( )

1/22

2 21/22 2

2 20 0

tan / 2 / 2

/ 2Cy dy Y Y dY Y Y

LY Y

= =

( ) ( )2

2 22 2 2 2

2cos / 2 / 2

2

mo S S o o

s

hY Y Y Y

h

+ =


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( )

cosVL

dg h

d h

=

( , )VLdg hd h

=

cosln cosVL

VL VL

d g dg g C

d h d h

= = +

21 mh

2

1 mh

4

2

Determine C:

VL

L

h

h h

=

=

cos n2 h

=

cos ln mo

L

hC O

h = + +

L=


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Molecular region:910h m

contact angles young equation

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