THE SURFACE TENSION OF PURE SUBSTANCES

THE SURFACE TENSION OF
PURE SUBSTANCES
INTRODUCTION

Introduction
 Surface

tension  is the contractile force
which always exists in the boundary
between two phases at equilibrium
 Its actually the analysis of the physical
phenomena involving surface tension
which interests us

Our topics primarily concern on
 Surface

tension as a force
 Surface tension as surface free energy
 Surface tension and the shape of mobile
interfaces

 Surface tension and capillarity
 Surface tension and intermolecular forces

Surface Tension As A Force:
The Wilhelmy Plate
 The

surface of a liquid appears to be
stretched by the liquid it encloses
 Example of this are:
 the

beading of water drops on certain surfaces;
 the climbing of most liquids in glass capillaries
 The

force acts on the surface and
operates perpendicular and inward from
the boundaries of the surface, tending to
decrease the area of the surface

F

2l

Noted
 Equation

above defines the units of surface
tension to be those of force per length or dynes
per centimeter in the cgs system
 The apparatus shown resembles a twodimensional cylinder/piston arrangement, so its
analogous to a two dimensional pressure
 A gas in the frictionless, three-dimensional
equivalent to the apparatus of the figure would
tend to expand spontaneously. For a film
however the direction of spontaneous change is
contraction

 A quantity

that is closely related to surface
tension is the contact angle , defined as
the angle (measured in the liquid) that is
formed at the junction of three phases, as
shown in figure 6.1b
 Although the surface tension is a property
of two phases which form the interface, 
requires that three phases be specified for
its characterization

The Wilhelmy Plate
a



b

Figure 6.2 The Wilhelmy plate method for measuring .
In (a) the base of the plate does not extend below the

horizontal liquid surface. In (b) the plate is partially
submerged to buoyancy must be considered

 Figure

6.2 represent a thin vertical plate
suspended at a liquid surface from the arm of
tarred balance
 The manifestation of surface tension and contact
angle in this situation is the entrainment of a
meniscus around the perimeter of the
suspended plate
 Assuming the apparatus is balanced before the
liquid surface is raised to the contact position,
the imbalance that occurs on contact is due to
the weight of the entrained meniscus
 Since the meniscus is held up by the tension on
the liquid surface, the weight measured by the
apparatus can be analyzed to yield a value for 

 The

observed weight of the meniscus w, must
equal the upward force provided by the surface
w = 2(l+t) cos 
  is the contact angle, l and t are the length and
thickness of the plate. Because of the difficulties
in measuring , the Wilhelmy plate method is
most frequently used for system in which  = 0
so
w = 2(l + t)
 Since the thickness of the plate used is generally
negligible compared to their length (t 0) shape

Measuring Surface Tension: Sessile Drops









The Bashfort and Adams tables provide an alternate way
of evaluating  by observing the profile of a sessile drop
of the liquid under investigation
Once  known for a particular profile, the Bashfort Adams
tables may be used further to evaluate b
For the appropriate  value, the value of x/b at  = 90o is
read from the tables. This gives the maximum radius of
the drop in units of b
From the photographic image of the drop, this radius may
be measured since the magnification of the photograph is
known
Comparing the actual maximum radius with the value of
(x/b)90 permits the evaluation of b





1)

2)





The figure can use for
example of the procedure
described
Theoretically its shown to
correspond to a  value of
10,0 then b is evaluated as
follows
The value of (x/b)90 for  = 10
is found to be 0,60808 from
the tables

Assume the radius of the
actual drops is 0,500 cm at
its widest point
Item (1) and (2) describe the
same point; therefore b =
0,500/0,60808 = 0,822 cm
Assuming  to be 1,00 g
cm-1 and taking g = 980 cm s2 gives for 
gb 2 {1,00(980)0,822}2



10,0
66,3 ergs cm -2

Measuring Surface Tension: Capillary Rise




A simple relationship between the height of capillary
rise, capillary radius, contact angle and surface tension
can derived
2R cos  = R2h g (48)
Its difficult to obtain reproducible result unless  = 0o, so
the equation simplifies to

2
Rh 
g



(49)

The cluster constant 2/(g) is defined as the capillary constant and is given the symbol a 2;
(50)

2
a 

g
2

 The

apex of the curved surface is identified as
the point from which h is measured. As we have
seen before, both radii of curvature are equal to
b at this point
 At the apex of the meniscus, the equilibrium
force balance leads to the result

2
p  gh
b
2
bh a

(51)
(52)










Equation (48) is valid only when R = b, that is for a
hemispherical meniscus.
In general this is not the case and b is not readily meaured
so we have not yet arrived at a practical method of
evaluating γ from the height of capillary rise. Again the tables
of Bashfort and Adams provide the necessary information
For liquid to make an angle of 0o with the supporting walls,
the walls must be tangent to the profile of the surface at its
widest point
Accodingly (x/b)90o in the Bashfort and Adams tables must
correspond to R/b. since the radius of the capillary is
measurable, this information permits the determination of b
for a meniscus in which θ = 0
However there is a catch. Use of the Bashfort and Adams
tables depends on knowing the shape factor β. It is not
feasible to match the profile of a meniscus with theoritical
contours, so we must find a way of circumventing the
problem










The procedure calls for using successive approximation
to evaluate β. Like any iterative procedure, some initial
values are fed into a computational loop and recycle until
no further change results from additional cycles of
calculation
In this instance, initial estimates of a and b (a1 and b1)
are combined with Eqs. (46) and (50) to yield a first
approximation to β (β1)
The value of (x/b)90o for β1 is read or interpolated from
the tables
This value and R are used to generate a second
approximation to b (b2). By Eq.(52) a second
approximation of a (a2) is also obtained and –starting
from a2 and b2 – a second round of calculation is
conducted.
The following table shows an example of this procedure







It is sometimes troublesome to find a starting point for these iterative
calculations. The following estimates are helpful for the capillary rise
problem:
From Eqs (49) and (50) a1  Rh
Treating the menicsus as hemisphere b 1  R
The initial value of table 6.3 assuming R = 0.25 cm and h = 0.40 cm

Measuring Contact Angle








The experimental methods used to evaluate θ are not
particularly difficult, but the result obtained may be quite
confusing
The situation is best introduced by refering figure rightbelow which shows a sessile drop on a tilted plane
It is conventional to call the larger value the advancing
angle θa and the smaller one the receding angle θr
With the sessile drop, the advancing angle is observed
when the drop is emerging from a syringe or pipet at the
solid surface
The receding angle is obtained by removing
liquid from the drop

Weight of a meniscus in a Wilhelmy plate experiment versus
depth of immersion of the plate. In (a) both advancing and
receding contact angles are equal. In (b) a > r

The general requirement for hysteresis is the existence of a large
number of metastable states which differ slightly in energy and
are separated from each other by small energy barriers
 The metastable states are generaly attributed to either the
roughness of the solid surface or its chemical heterogeneity, or
both.


Schematic energy diagram for metastable states corresponding to different
contact angles

Cross section of a sessile drop resting on a surface
containing a set of concentric grooves. For both
profiles, the contact angle is identical microscopically,
although macroscopically different

Kelvin Equation
 Another

result of pressure difference is the effect
it has on the free energy of the material
possessing the curved surface
 Suppose we consider the process of transferring
molecules of a liquid from a bulk phase with a
vast horizontal surface to a small spherical drop
of radius r
 Assuming the liquid to be incompressible and the
vapor to be ideal, ∆G for the process of
increasing the pressure from po to po + ∆p is as
follows:

1. For the liquid :
po  p

2VL
G  VL dp VL dp 
r
po
Where VLis the molar volume of the liquid
2. For the vapor :
po  p
p
G RT ln
RT ln
po
po
When liquid and vapor are at equilibrium, these two
values of G are equal :
p 2VL 2 M
RT ln


po
r
r





The Kelvin equation enables us to evaluate the actual
pressure above a spherical surface and not just the
pressure difference across the interface, as was the case
with the Laplace equation
Using the surface tension of water at 20oC, 72,8 ergs cm2, the ratio p/p is seen to be
o




p
2 18,0  72,8
1,08 x10  7
exp 
exp

7
po
r
  0,998 8,31x10  293 r 








Or 1,0011; 1,0184; 1,1139; and 2,9404 for drops of
radius 10-4, 10-5, 10-6 and 10-7 cm respectively.
Thus for a small drops the vapor pressure may be
considerably larger than for flat surfaces

The Kelvin equation may also be applied to the
equilibrium solubility of a solid in a liquid
 In this case the ratio p/po in equation is replaced by
the ratio a/ao where ao is the activity of dissolved
solute in equilibrium with flat surface and a is the
analogous quantity for a spherical surface
 For an ionic compound having the general formula
MmXn the activity of a dilute solution is related to the
molar solubility A as follows:


a  mS   nS 
Therefore for a solid sphere
2 M
a
S
RT ln  m  n  RT ln
r
ao
So
m

n

 The

equation provides a thermodynamically valid
way to determine SL, for example the value of SL for
the SrSO4-water surface has been found to be 85
ergs cm-2 and for NaCl-alcohol surface to be 171
ergs cm-2 by this method
 The increase in solubility of small particles and using
it as a means of evaluating SL is fraught with
difficulties:
The difference in solubility between small particles and
larger one will probably differ by less than 10%
 Solid particles are not likely to be uniform spheres even
if the sample is carefully fractionated
 The radius of curvature of sharp points or protuberances
on the particles has a larger effect on the solubility of
irregular particles than the equivalent radius of the
particles themselves.


The Young Equation
 Suppose

a drop of liquid is placed on a perfectly
smooth solid surface and these phases are allowed
to come to equilibrium with the surrounding vapor
phase
 Viewing the surface tension as forces acting along
the perimeter of the drop enables us to write
equation which describes the equilibrium force
balance
LV cos  = SV - SL

First Objections










Real solid surfaces may be quite different from the idealized
one in this derivation
Real solid surface are apt to be rough and even chemically
heterogeneous
If a surface is rough a correction factor r is traditionally
introduced as weighting factor for cos , where r > 1
The factor cos  enters equation by projecting LV onto the
solid surface
If the solid is rough a larger area will be overshadowed by
the projection than if the surface were smooth
Young’s equation becomes
rLV cos  = SV - SL
A surface may also be chemically heterogeneous. Assuming
for simplicity that the surface is divided into fractions f 1 and f2
of chemical types 1 and 2 we may write
LV cos  = f1(S1V - S1L) + f2(S2V - S2L)
Where f1 + f2 = 1

Second Objection








The issue of whether the surface is in a true state of
thermodynamically equilibrium, it may be argued that the
liquid surface exerts a force perpendicular to the solid
surface, LV sin 
On deformable solids a ridge is produced at the
perimeter of a drop; on harder solids the stress is not
sufficient to cause deformation of the surface
Is it correct to assume that a surface under this stress is
thermodynamically the same as the idealized surface
which is free from stress?
The stress component is absent only when  = 0 in
which case the liquid spreads freely over the surface and
the concept of the sessile drop becomes meaningless

Notes
LV cos  S o  SL (incorrect)


We must assume that SV and S may be different

Let us consider what occurs when the vapor of a volatile liquid
is added to an evacuated sample of a non volatile solid
 This closely related to the observation that the interface
between a solution and another phase will differ from the
corresponding interface for the pure solvent due to the
adsorption of solute from solution
 For now we may anticipate a result to note that adsorption
always leads to decrease in , therefore:


S o SV
we shall use the symbol e to signify the difference

So  SV  e



The equation must be corrected to give

LV cos  S o   e  SL


Figure shows relationship between terms write at the right hand side, it
also suggest that the shape of the drop might be quite different in
equilibrium and non equilibrium situations depending on the magnitude of
e



There are several concepts which will assist us in anticipating the range
of e values:
1.

2.

3.

4.

Spontaneously occurring processes are characterized by negative values
of ∆G
Surface tension is the surface excess free energy; therefore the lowering
of  with adsorption is consistent with the fact that adsorption occurs
spontaneously
Surfaces which initially posses the higher free energies have the most to
gain in terms of decreasing the free energy of their surfaces by adsorption
A surface energy value in the neighborhood of 100 ergs cm-2 is generally
considered the cutoff value between ‘high energy’ and ‘low energy’
surfaces

ADHESION AND COHESION


Figure illustrates the origin of surface tension at the
molecular level

 In

(a) which applies to a pure liquid, the
process consists of producing two new
interfaces, each of unit cross section,
therefore for the separation process:
∆G = 2A = WAA

 The

quantity WAA is known as the work of
cohesion since it equals the work required
to pull a column of liquid A apart
 It measures the attraction between the
molecules of the two portion

G = WAB = final - initial = A + B - AB
 This quantity is known as the work of adhesion
and measures the attraction between the two
different phases
 The work of adhesion between a solid and a
liquid phase may be define analog:
WSL = S + LV - SL
 By means of previous equation S may eliminated
to gives
WSL = SV + e + LV - SL
 Finally Young’s equation may be used to
eliminate the difference:
WSL = LV(1 + cos ) + e

 0 where the equality holds in the
absence of adsorption
 High energy surface bind enough adsorbed
molecules to make e significant, example of
these are metals, metal oxides, metal sulfides
and other inorganic salts, silica, and glass
 On the other hand e is negligible for a solid
which possesses a low energy surface, most
of organic compounds, including organic
polymers are in this criteria.
 e

 The

difference between the work of adhesion and
the work of cohesion of two substances defines
as quantity known as the spreading coefficient of
B on A, SB/A:
SB/A = WAB – WBB
 If WAB > WBB the A-B interaction is sufficiently
strong to promote the wetting of A by B (positive
spreading). Conversely no wetting occurs if WBB >
WAB since the work required to overcome the
attraction between two molecules B is not
compensated by the attraction between A and B
(negative spreading).
SB/A = A - B - AB = A – (B + AB)

The Dispersion Component of Surface Tension

1  repulsion
2  attraction
3  specific Interaction
4  resultant

Interface between two phases

A

A

A

A

A Bulk phase
A
A

A

A

A

A
A

Interface between two phases

A

B

B

A
A

A Bulk phase
B
B
A
B
A
B

A
B
A

 F.

Fowkes has proposed that any
interfacial tension may be written as the
summation of contributions arising from
the various types of interactions which
would operate in the material under
consideration, in general then:
  = d + h + m +  + i = d + sp
 Superscripts refer to dispersion forces (d),
hydrogen bonds (h), metallic bonds (m),
electron interactions () and ionic
interactions (i).

d

E s   A B

d



(Work ) A  A  E s



A

 A 

d

 A B

d

Total work of forming the AB interface is the sum of them all
d
d
d
d
d
d
 AB   A   A B    B   A B   A  B  2  A B

 

Comparing two equations yield



d

WAB 2  A B



d 1/ 2

TUGAS KIPER
 Gunakan

data tabel 6.2 untuk mem-plot profile
tetes(drop) dengan  = 25. Ukur (dalam cm) jarijari tetes yang anda gambar pada titik terjauhnya
(widest point). Dengan membandingkan nilainya
dengan nilai (x/b)90 dari tabel, hitung b (dalam
cm) untuk tetes yang anda gambar. Andaikan
tetes sesungguhnya memiliki nilai  ini (25), jika
jari-jari pada widest point 0,25 cm dan  = 0,50
g cm-3, berapa  untuk antarmuka tetes tsb.