out-class reading: levine p. 387-390 13.2 curved interfaces

§8.2 Surface phenomenon of liquid

Out-class reading:

Levine p. 387-390

13.2 Curved interfaces

8.2.2 Curved surface and additional pressure

drop

1. Curved liquid surface

In graduated cylinderConvex surface Concave surface


p additional pressure

Convex surfaceConcave surface

For convex surface: p>0 For concave surface: p < 0

in exp p p= +

pex



( )p dp dV dA + =

pex

Δin exp p p= +

To increase the volume (dV) of liquid at pex =

p + dp

dAp

dV

=

3

3

4rV =

24 rA =

2p

r

=

2

8 2

4

rdrp

rr dr

= =

Laplace equation

pin



1 2

1 1p

r r

= +

For curved surface:

Laplace-Young equation

r is the radius of curvature.

For convex surface, r > 0, p > 0, point to

the interior of liquid;

For concave surface, r<0, p < 0, point to

the gaseous phase;

For plane surface, r →, p → 0, pex = pin,

2p

r

=



For bubble



4p

r

=

Δ Δr m mV dp V p = − = =

*lnRT p = +

For liquid with plane surface:

For liquid in droplet:

lnr rRT p = +

*

2Δ ln Δr

m

p MRT V p

rp

= = =

*

2ln rp M

RT rp

= Kelvin equation

For droplet or bubble

The droplets gradually disappear and the

water level in the beaker increases.

8.2.3 Vapor pressure under curved surface


*

2ln rp M

RT rp

=

r / m 10-6 10-7 10-8 10-9

pr / p* 1.001 1.011 1.111 2.95

The change in vapor pressure is not

large enough to be of any concern in

the case of macroscopic systems, such

as d > 10-7 m, or 0.1 m.





If a vapor is cooled or compressed to a

pressure equal to the vapor pressure of the

bulk liquid, condensation should occur.

(1) supersaturated vapor / supercooling The difficulty is that the first few molecules

condensing can only form a minute drop and

the vapor pressure of such a drop will be

much higher than the regular vapor pressure.

p = p*

pr = 2.95p*



Artificial rainfallln

vap mHp k

RT

= − +

2) Increase the initial radius of the embryo: dust,

AgCl particles



Relative humidity (RH): 30 oC, 100%, 4.242 kPa

70%, 2.969 kPa

For condensation: At 30 oC, in order to form a droplet

of 1 nm in radius from the air with RH of 70%, the

vapor pressure should be as low as 1.438 kPa

corresponding to ca. _______ oC

Is embryo of a new phase possible?

Droplet can not form from the pure

saturated vapor spontaneously. Therefore, in

clean systems, large degrees of

supersaturation or super-cooling are possible.

Microscopic fluctuation plays important

role in formation of new phase.

fluctuation



Boil

2) superheated liquid:

Δin ex lp p p p= + +

0,Δr p→ →

pex

pin

pl

p



Evaporate

Superheating:

When temperature is over boiling point,

liquid does not boil.

0 vap 0

1 1 2ln 1

R

T T H rP

− = − −

The smaller the bubble, the higher the

boiling temperature.

For water with air bubble with diameter of

10-6 meter as seed, it boils at 123 oC.



2) superheated liquid:

Once the bubble of relative large diameter

formed, the evaporation would proceed in an

explosion manner—explosive boiling.



3) condensation in capillary:

When liquid forms concave surface

in capillary, r < 0

*

2ln rp M

RT rp

=

pr < p*, it is easy for vapor to condense

in capillary.

Constant-temperature

evaporation

vapor

liquidPorous

materials



沐雾甲虫（Onymacris unguicularis），



3) supersaturated solution and ageing of crystal

By simple modification of the above

analysis, the same equations apply to

the supercooling / supersaturated

liquid or solution.

rRT

M

S

Sr

2ln =

ageing of crystal

Decrease in diameter of solid will

increase surface area and thus specific

surface energy of the system and lower

melting point, increase solubility of the

solid.



8.3 Interaction between two phases--Wetting and spreading

Superhydrophobic, superhydrobicity

§8.3 Interaction between liquid and solid surface

g

S

l

g-l + g-s → s-l

G = s-l – (g-l + g-s) = -Wa

Work of Adhesion

8.3.1 concepts--Adhesion

Wa = g-l + g-s – s-l

Wa > 0

The solid can be wetted by the liquid.


8.3.1 concepts--Immersion

g-s → s-l

G = s-l - g-s = -Wi

Wi = g-s - s-l > 0

Work of immersion

8.3.1 concepts--Spreading

g-s → s-l + l-g

G = s-l + l-g - s-g = -S

S = s-g - s-l - l-g > 0

The liquid spreads over the solid

spontaneously.

spreading coefficient


g-l

g-s

s-l

Under equilibrium:

g-l cos + s-l = g-s

Young equation

lg

slgs

−

−− −=

cos

When :g-s - s-l = g-l , cos =1, = 0 o,

Complete wettable.

When :g-s-s-l< g-l , 0<cos <1, <90 o,

wettable.

When :g-s < s-l , cos < 0, > 90 o,

nonwettable.

Define the direction of surface tension

8.3.2 Contact angle ()


Hydrophobicity of conversion layer on Mg alloy goniometer

The contact angle () is the angle measured

through the liquid, where a liquid/vapor

interface meets a solid surface

8.3.2 Contact angle ()


http://en.wikipedia.org/wiki/File:Rame-hart_goniometer.jpg

(3) Lyophobic and lyophilic solids

The greater the specific

energy, the easier the

spreading of liquid over solid.

g-s – g-l – s-l > 0

g-s > g-l + s-l

g-s > g-l

g-l

g-s

s-l


g-s > 100 mN m-1, high-energy surface:

Metals, oxides, chlorides, inorganic

salts. g-s 500 ~ 5000 mN m-1

g-s < 100 mN m-1, low-energy surface:

organic solids, polymers. PTFE: g-s

18 mN m-1

Nonstick cooker

(3) Lyophobic and lyophilic solids


How can we judge the cleanness of

the glass surface?

8.3.3 Lyophobic and lyophilic solids


Blurry Clean

8.3.4 Spreading over liquid

SO/W = - G = W - O - W/O

SO/W > 0, oil can spread over water

SO/W < 0, oil floats in shape of lens.

Liquids Iso-C5H12O C6H6 C6H12 CS2 CH2I2

SO/W 44.0 8.8 3.4 -8.2 -26.5


Clapham Common (2000 m2)

1774 Benjamin Franklin (2.4 nm)

The film formed over water is of one molecule

thick. (proved by Pockels and Rayleigh):

Unimolecular film, monolayer, Insolvable film



wreck of a tanker

Spreading of oil over seawater



8.3.5 Capillarity

Capillary rise / depression


1 2

2( )

lp p

h gr

=

= −

grh

)(

2

21

−=

Rr =cos

gRh

)(

cos2

21

−=

Discussion

8.3.5 Capillarity


This relation can be used to

determine the surface tension of

liquids – capillary rise method

gRh

)(

cos2

21

−=

Measurement of porosity distribution:

Mercury method

p

8.3.5 Capillarity


8.3.5 Capillarity


out-class reading: levine p. 387-390 13.2 curved interfaces

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