THE SURFACE TENSION OF
THE SURFACE TENSION OF
PURE SUBSTANCES
PURE SUBSTANCES
INTRODUCTION
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
Our topics primarily concern on
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 As A Force:
Surface Tension As A Force:
The Wilhelmy Plate
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
l
F
2
Noted
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
two-dimensional 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
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,
The Wilhelmy Plate
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 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:
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
That is dG equals the maximum
non-pressure/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
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
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)
b
p
0
2
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
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
gz
b
p
s
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
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)
Measuring Surface Tension: Sessile Drops
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
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
1) The value of (x/b)90 for = 10
is found to be 0,60808 from the tables
2) 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
Measuring Surface Tension: Capillary Rise
A simple relationship between the height of capillary
rise, capillary radius, contact angle and surface tension can derived
2R cos = R2h g (48)
Its difficult to obtain reproducible result unless = 0o, so
the equation simplifies to
(49)
g Rh
2
The cluster constant 2/(g) is defined as the capillary constant and is given the symbol a2;
(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
(51)
(52)
2
2
a
bh
gh
b
p
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
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.
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
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
right-below 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
Schematic energy diagram for metastable states corresponding to different
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
Cross section of a sessile drop resting on a surface containing a set of concentric grooves. For both
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
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
- 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
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:
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
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
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
rLV 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
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
Notes
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:
incorrect)
e to signify the difference symbol
the use
The equation must be corrected to give
SL e
S
LV o
cos
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. Spontaneously occurring processes are characterized by negative values
of ∆G
2. Surface tension is the surface excess free energy; therefore the lowering
of with adsorption is consistent with the fact that adsorption occurs
spontaneously
3. 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
ADHESION AND COHESION
Figure illustrates the origin of surface tension at the
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= W
AA
The quantity W
AAis 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
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:
e
0 where the equality holds in the
absence of adsorption
High energy surface bind enough adsorbed
molecules to make
esignificant, example of
these are metals, metal oxides, metal sulfides
and other inorganic salts, silica, and glass
On the other hand
eis negligible for a solid
which possesses a low energy surface, most
of organic compounds, including organic
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).
The Dispersion Component of Surface Tension
1 repulsion
2 attraction
Bulk phase
Bulk phase
Interface between two phases
Interface between two phases
A A
A
A
A A
A
A A
A
A
Bulk phase
Bulk phase
Interface between two phases
Interface between two phases
A B
A
A
B A
A
A A
A
B B
B
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
TUGAS KIPER
Gunakan data tabel 6.2 untuk mem-plot profile
tetes(drop) dengan = 25. Ukur (dalam cm) jari-jari tetes yang anda gambar pada titik terjauhnya (widest point). Dengan membandingkan nilainya dengan nilai (x/b)90 dari tabel, hitung b (dalam
