Chap. 17.8(a) Solid-Vapor Interface => Surface Ten- sion (a – b Interface)

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Microscopic basis of surface tension.

force inwards tends to contract the sur- face, i.e. there is a surface tension

Force

relatively high potential energy (bulk) relatively low potential energy

Chap. 17.8(a) Solid-Vapor Interface => Surface Ten- sion (a – b Interface)

- A molecule in the bulk has a lower potential energy than one free in the gas (below the transition temperature).

- When a molecule is at the surface, its energy is intermediate between free molecule and the molecule in the bulk.

- Its potential energy would be lowered if it moved into the bulk, and so the molecules are under the influence of a force, which tends to draw them into the bulk.

- This force is attractive reducing total surface area, and we term it a tension, the sur- face tension.

Solid-Vapor Interface Solid

Vapor

page 1 =

handwave argument

Dangling Bonds

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http://bp.snu.ac.kr

The work involved in forming a surface of area dA is proportional to the area of the surface formed.

where g is a coefficient known as the surface tension of the material.

At constant pressure and temperature,

: which means that surfaces have a natural ten- dency to contract.

2

Liquids (or solids) assume a shape that minimizes their surface area because it en- ables the maximum number of molecules to the bulk rather than surface.

Surface Tension = Interfacial Gibbs-Free Energy

___________

page 2 = exact

dA dw  g

dA Vdp

SdT

dG     g

dA dG  g

0 0  

 dA

dG

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Figure 4.21

The model used for calculating the work of forming a liquid film when a wire of length l is raised and pulls the surface with it through a height h.

Force 2gl

Total area 2hl

(skip - - -)

l h

hl h

w f

hl

g g

g

2 /

distance, the

times force

the is

work the

Since

2 Work



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P_in : the pressure inside the cavity P_out : the outside pressure

The internal pressure is due to the sum of the outside pressure P_out and the surface tension.

Consider a bubble (or cavity).

The change in surface area when the radius of a sphere changes from r and r +dr is

 As force  distance is work, the force opposing stretching through the distance dr at this

radius=8prg.

dr ² is ignored

P_in

4

(skip - - -)

2 2

4 ( ) 4 8

d  p r dr  pr  prdr 8

dw g d p gr dr

  

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Laplace equation

Note: The Laplace equation shows that the pressure inside a curved surface is always greater than the pressure outside, but the difference de-

creases to zero as the radius of curvature be- comes infinite.

(skip - - -)

2 2

4pr p_in 4pr p_out 8p gr

  

2

in out

p p

r

  g

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Figure 4.22

The dependence of the pressure inside a curved surface on the radius of the surface, for two different values of the surface tension.

r

Pressure inside, p in

Radius, r p_out

Increasing surface tension, g

6

2

in out

p p

r

  g

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Assuming that the vapor is ideal,

Let the pressure on the system be increased. If the pressure on the liquid changes by dp(l), then the change in the presence of its pressure, dp(g) will be such that

Consider droplets.

At equilibrium, the chemical potentials of a vapor and its liquid are equal.

( ) ( ) ( ) ( )

m m

V g dp g V l dp l (at constant )T

m m m

d V dp S dT  d V dp

 at constant pressure

( ) .

m ( ) V g RT

 p g

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

( ) ( )

m m

V l dp l

RT dp g

dp g V l dp l

p g   p g  RT

( ) ( ) d g  d l

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One way of increasing the pressure on the liquid is to disperse it as droplets. The pressure differential across a curved surface is 2g/r.

On integration with boundary condition of p*(g)=p*(l), when the pressure exerted on the liquid is the normal pressure p*, i.e., p(g)=p* and p(l)=p*,

where p(mist) is the vapor pressure of a sample of mist with droplet radius r and p(bulk) is the pressure of a plane surface of the same material.

8

( ) ( )exp _m( ) 2

p p V l

rRT

 g 

  

  Kelvin e

mist bulk quation

 

 

^*

ln * * exp

*

m m m

V V p V p

p p p p p p

p RT RT RT

 

   

         

   

*

* *

( ) 1

( ) ( ) ( )

p p p

m

p p

dp g V l dp l p g RT









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Figure 4.23

When a capillary tube is first stood in a liquid, the latter climbs up the walls, so curving the surface. The pressure just under the meniscus is less than that arising from the at- mosphere by 2g/r. The pressure is equal at equal heights throughout the liquid provided the hydrostatic pressure (which is equal to rgh) cancles the pressure difference

The pressure exerted by a col- umnp  gh  : liquid density

2 2

p p gh gh

r r

g   g

    

2 1

, 2

h ghr

gr

g g 

  

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Figure 4.24

The variation of the surface tension of water with temperature.

Temperature, q/C Surface tension, g/(mN m-1 )

10

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Figure 4.25

The balance of forces that results in a contact angle, q_c.

g_sg > g_sl

the liquid wets (spread over)

g_sg < g_sl

g_lg

g_sl g_sg

q_c

cos cos

sg sl lg c

sg sl

c

lg

g g g q

g g

q g

 

0 q_c  90o

90o q_c

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Thermodynamic work of adhesion

The thermodynamic work of adhesion, w_ad, required to separate a unit area of a solid and a liquid phase forming an interface across which secondary forces are

acting may be related to the surface and the interfacial free energies by the Dupre equation. The re-

versible work of adhesion, w_ad, in an inert medium may be expressed by: w_ad= (sum of the surface free energies of the

solid and liquid phase – the interfacial free energy)

(2) (1)

(3) i.e.,

12

ad sg lg sl

w  g g g

sg sl lg cos c

g  g g q

cos _c ^sg ^sl

lg

g g

q g

 

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Combining eq (1) and eq (3) gives

We now see that the liquid completely “wets”

(spreads over) the surface fully, corresponding to q_c

≈ 0, when w_ad ≥ 2g_lg. The liquid does not wet the surface (corresponding to q_c > 90˚ when w_ad < g_lg).

cos _c ^ad 1 (4)

lg

q w

 g 

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Figure 4.26

The variation of contact angle (shown by the semaphore- like object) as the ratio w_ad/g_lg changes.

w_ad/g_lg cosq c

0 1 1

-1

0 2

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