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 <<< l)
equation may approximated:
w = 2l 
Surface Tension As Surface Excess
Free Energy
 The work done on the system of figure 6.1 is given by
Work = F dx = 2l dx =  dA
 This supplies a second definition of surface tension, it equals
the work per unit area required to produce new surface
 If the quantity w’ is defined to be the work done by the
system when its area is changed, then equation becomes
w’ = -γdA according to the first law
 dE = q - w in which w is the work done by the system and
q is the heat absorbed by the system. It relates to Gibbs free
energy by following equation :
dG = TdS – pdV - wnon-pV + pdV + Vdp – TdS – SdT for a
constant temperature, constant pressure and reversible
process
dG = -wnon-pV
 That is dG equals the maximum nonpressure/volume work derivable from such a
process since maximum work is associated with
reversible process
 We already seen that changes in surface area
entail non-pV work, therefore we identify w’ as
wnon-pV and write
dG = γdA
 Even better in view of the stipulations we write
γ = (G/A)T,p
 This relationship identifies the surface tension as
the increment in Gibbs free energy per unit
increment in area
The Laplace Equation
 1
1 

 p   


R
R
2 
 1
For spherical
For a cylindrica
surface R1  R 2  R
l surface R 1  
For a planar surface R1  R 2  
p 
2
R
p 

R2
p  0
 Our attention will focus on those specific
surfaces with most readily allow the
experimental determination of 
 The shape assumed by a meniscus in a
cylindrical capillary and the shape assumed by a
drop resting on a planar surface (called a sessile
drop) are most useful in this regard
 Figure 6.4 may be regarded as a portion of the
surface of either of these cases
 As can be seen the curve represent the profile of
a sessile drop; inverted, the solid portion
represent the profile of a meniscus
 The actual surfaces are generated by rotating
these profiles around the axis of symmetry
 Because the symmetry of the surface, both
values R must be equal at the apex of the drop
 The value of the radius of curvature at this
location is symbolized b, therefore, at the apex
(subscript 0)
2
  p 0 
b
 Next, let us calculate the pressure at point S. At
S the value of p equals the difference between
the pressure at S in each of the phases
 These may be expressed relative to the
pressure at the reference plane through the
apex (subscript 0) as follows
 In phase A:
pA = (pA)0 + Agz
 In phase B:
pB = (pB)0 + Bgz
 Therefore, p at S equals
(p)S = pA – pB = (pA)0 – (pB)0 + (A - B)gz
= (p)0 + gz
Where  = A - B and we can write it
 p s

2
b
   gz
Notes
 If A > B,  will be positive and the drop will be
oblate in shape since the weight of the fluid
tends to flatten the surface
 If A < B, a prolate drop is formed since the
larger buoyant force leads to a surface with
much greater vertical elongation. In this case 
is negative
 A  value of zero correspond to a spherical drop
and in a gravitational field is expected only when
p = 0
 Positive values of  correspond to a sessile
drops of liquid in gaseous environment
 Negative  values correspond to sessile bubbles
extending into a liquid
Notes
The previous statement imply that the drop
is resting on a supporting surface
If instead the drop is suspended from a
support (called pendant drops or bubbles),
g becomes negative, and it is the liquid
drop that will have the prolate ( < 0)
shape and the gas bubbles the oblate
( >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 }
 66 , 3 ergs cm
10 , 0
-2
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
Rh 
2
g
(49)
 The cluster constant 2/(g) is defined as the capillary
constant and is given the symbol a2;

a 
2
2
g
(50)
 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
p 
2
   gh
(51)
b
bh  a
2
(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 b1  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
G 
V
L
dp  V L dp 
2V L 
po
Where V L is the molar volu
r
me of the liquid
2. For the vapor :
 G  RT ln
po  p
 RT ln
po
When liquid
po
and vapor are at equilibriu
values of  G are equal :
RT ln
p
po

p
2V L 
r

2M
r
m, these two
 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
cm-2, the ratio p/po is seen to be


2 18 , 0 72 ,8 
1, 08 x10
 exp 
  exp
7
po
r
 0 ,998  8 ,31 x10  293 r 
p


 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
7
 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
Therefore
2M
r
  nS 
m
n
for a solid sphere
 RT ln
a
ao
  m  n  RT ln
S
So
 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 f1 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   SL ( incorrect)
o
 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   SV
o
we shall use the symbol  e to signify th
 S   SV   e
o
e difference
 The equation must be corrected to give
 LV cos    S   e   SL
o


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
 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.
 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).
E 
s
A B
d
(Work ) A  
d

 E
A
s

A
Total work of forming
 AB    A 

Comparing

W AB  2 

d
d
 A  B     B 

A
B

d 1/ 2
A B
d
the AB interface
two equations
d
A


yield
d
is the sum of them all
d
d
d
d
 A  B    A   B  2  A  B

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.