12
PM2
SURFACE TENSION
The swan can swim while sitting down,
For pure conceit he takes the crown,
He looks in the mirror over and over,
and claims to have never heard of Pavlova.
OBJECTIVES
Aims
In this chapter you will look at the behaviour of liquid surfaces and the explanation of that behaviour
both in terms of forces and in terms of energy. The principle of minimum potential energy can be
invoked to explain many surface phenomena
Minimum Learning Goals
When you have finished studying this chapter you should be able to do all of the following.
1.
Explain, interpret and use the terms
intermolecular forces, capillarity, angle of contact, wetting.
2
(i) Describe an experimental determination of the surface tension of a liquid by the
measurement of the force on a glass slide in contact with the liquid.
(ii)
3
Perform simple numerical calculations associated with such a determination.
(i) Use a model of the microscopic structure of liquids to explain the phenomenon of
surface tension in terms of potential energy.
(ii) Extend this argument to explain why liquids tend to assume a shape which minimises the
surface area of the liquid.
(iii) Do simple numerical calculations associated with energy per area.
4
(i) Explain how the surface tension of a liquid can be measured either in terms of force per
length or of energy per area.
(ii)
Demonstrate that these two descriptions are dimensionally equivalent.
5
(i) Explain how the phenomenon of capillarity results from forces between solid (e.g. glass)
and liquid (e.g. water) molecules.
2T
(ii) Recall, explain and use the relationship h = rgr for capillary rise.
6
Give examples of how the wetting characteristics of surfaces can be altered.
7
Explain, by identifying the relevant forces and using scaling arguments, why insects can walk
on water but larger animals cannot.
8
Recall that the surface tension of water has a magnitude of 0.1 N.m-1.
PM2: Surface Tension
13
PRE-LECTURE
Recall from earlier lectures, particularly chapters FE2 and FE3 the following facts about the general
nature of forces.
(i) The molecules of any substance - solid, liquid or gas - attract one another if they are far apart;
at short distances, the intermolecular force is repulsive. There is a crossover point where the
force is zero - neither attractive nor repulsive.
(ii) When a system is in equilibrium, then the sum of all the forces acting on the system is zero. In
particular, the molecules of a substance tend to come together (pulled by the intermolecular
attraction) until on the average their distances apart correspond to the cross over point between
attraction and repulsion. This means the normal state of a substance is an average kind of
equilibrium.
(iii) Equilibrium can be discussed in terms of potential energy. The equilibrium configuration is
one in which the potential energy is least.
For a simple two body system you can see this by considering the diagrams on pages 17 and
59 of the Forces and Energy book.
LECTURE
2-1 PHENOMENON OF SURFACE TENSION
The surface of any liquid behaves as though it is covered by a stretched membrane.
Small insects can walk on water without getting wet.
Demonstration
The membrane used is obviously quite strong: it will support dense objects, provided they are small and of
the right shape:
a needle,
a small square of aluminium sheet (weighted),
a container made of fine wire gauze.
The strength of the membrane varies for different liquids, e.g. it is much less for soapy water than pure
water.
Demonstration
Ducks swim on water without getting very wet. However, they cannot swim on soapy water. [There are
cases on record where ducks have drowned in farmyard ponds into which washing water was emptied, or in
streams polluted with non degradable detergents.]
2-2 MEASUREMENT AND DEFINITION OF SURFACE TENSION
The strength of the surface membrane can be imagined to arise from a set of forces acting on each
point of the surface, parallel to the surface, like the skin of a drum.
14
PM2: Surface Tension
Demonstration
The easiest way to measure these forces is with the following apparatus
BALANCE
ADJUSTABLE
WEIGHT
AND
SAND
GLASS
SLIDE
FIXED
COUNTER
WEIGHT
WATER
Fig 2.1 Experimental measurement of surface tension
Note that because the water surface curves up near the glass slide the surface tension forces between the
glass and the water are vertical rather than horizontal.
SLIDE
MENISCUS
WATER
Fig 2.2 Shape of liquid meniscus
A first experiment yielded this result:
A certain amount of sand (weight, W) was needed to keep slide just in contact with water; when the water
was removed this amount of sand plus a 0.55 g (extra weight 5.4 mN) was needed to have the slide in the same
position
The difference, 5.4 mN, is a measure of the force due to the pull of the water on the slide.
A second experiment tested whether the force depended on the length of the slide (recall that on the surface
of a drum, a bigger cut is harder to repair than a smaller one).
Length of slide used in first experiment:
38 mm
Length of slide used in second experiment: 76 mm
Result of second experiment: the force due to the pull of surface increases to 10 mN
Deduction: The force which a liquid surface exerts on any body with which it is in intimate
contact (as described above) is directly proportional to the length of the line of contact.
Force = T ¥ length.
The constant of proportionality, T, is called the surface tension of the liquid.
PM2: Surface Tension
15
Demonstration
In the second experiment the width of the slide was 1 mm, so the total length of the line of contact
between the glass and the water was (76 + 1 + 76 + 1)mm. These values give a value for the surface tension
of water of 0.06 N.m-1.
[Most books of tables quote 0.07 N.m-1.]
Other liquids have different surface tensions (see post lecture material).
Demonstration
A little detergent added to the water lowers it surface tension considerably.
As defined here the dimensions of surface tension are force per length. Its units in the S.I.
system are N.m-1.
2-3 MICROSCOPIC EXPLANATION AND SURFACE ENERGY
To understand why the phenomenon of surface tension arises, you must think of intermolecular
attraction as recalled in the pre-lecture material.
Molecules of any substance want to pack together so that their average separation is low.
In solids, this separation is fixed, whereas in gases, the random motion due to heat
predominates. In liquids, there is some random motion but, on the average, the molecular separation
is low.
Consider a fixed number of liquid molecules. If they are packed so that they have a large
surface area, their average intermolecular separation is relatively high. If they have small surface
area, the average intermolecular separation is relatively low. Their total potential energy is lower in
the latter case.
A logical conclusion from this is that energy has to be added in order to increase the surface
area of a liquid. The bigger the change in surface area, the more energy has to be put in. Associated
with the surface there is a potential energy that depends on the area of the surface. This means that
an alternative approach is to consider surface tension as an energy per surface area.
Since the equilibrium configuration of any system is that in which the potential energy is least,
a liquid left to itself will assume a shape which minimises surface area, thereby minimising the total
surface potential energy.
Demonstration
Drops of water are spherical
Loop of thread on water; detergent added inside loop; loop takes a circular shape.
LOOP OF
THREAD
CONTAINER
PURE WATER
WATER AND
DETERGENT
Shaded area here is greater than shaded area here
Fig 2.3 Effect of placing a drop of detergent inside a loop of string that is floating on the
surface of water
(Surface tension of detergent and water is much lower than that of water.)
PM2: Surface Tension
energy
force
The dimensions of energy are force ¥ length, so area has the same dimensions as length .
Sometimes it is easiest to explain surface phenomena in terms of energy considerations,
sometimes in terms of force considerations
Demonstration
Three matches on water:
CONTAINER
MATCHES
DETERGENT
ADDED
becomes
PURE
WATER
Fig 2.4 Effect of placing a drop of detergent inside a triangle of matches that are floating
on the surface of water
This is basically the same as the loop of thread demonstration, but it is easier to explain why
each match moved in terms of forces as thus for the match at the top of the diagram:
larger force
(water: higher
surface tension)
smaller force
(detergent: lower
surface tension)
Fig 2.5 The net force acting on the match pushes it away from the detergent
2-4 CAPILLARITY
A consequence of the phenomenon of surface tension is that many liquids will "creep up" tubes, an
observation made readily with glass tubes of very narrow bore.
h
WATER (DYED)
Fig 2.6 Capillary rise
The height of the water in the capillary above the level of the liquid in the surrounding liquid,
as indicated by h in the diagram, is called the capillary rise.
16
17
PM2: Surface Tension
Demonstration
Glass tube of narrow bore in water.
It can be demonstrated that:
(i) the capillary rise is larger for liquids of higher surface tension than of lower surface tension (e.g. larger
for pure water than for water and detergent) ;
(ii) the height increases as the radius of the bore of the tube gets smaller.
In fact, the height varies inversely as r.
Demonstration
Glass wedge in water:
ELEVATION
PLAN
HYPERBOLIC
!! !SHAPE
TWO GLASS SHEETS
WIDE
END
NARROW
END
RUBBER BAND
WATER (DYED)
Fig 2.7 The rise of water in a wedge between two flat glass sheets
(iii)
We would like to have shown that height decreased with increasing density, but we could not
find two common liquids with roughly the same surface tension and vastly different densities.
The relation between capillary rise, surface tension and density (see post lecture) is
2T
h
= rgr
The tube used in the demonstration had a bore of radius 0.50 mm and the measured rise was
28!mm. For a tube of this radius, the calculated rise is
2!¥ !0.06!N.m-1
h
=
1!¥ !103 !kg.m-3!¥ !9.8!m.s-2!¥ !0.50!¥ !10-3!m
= 2 cm.
Specific Applications:
(i) Rise of water through soils.
Demonstration
Although water rising in a column of soil is not rising through a tube of uniform bore it is moving
through spaces roughly the same size as the soil grains. So the same kind of capillarity formula will apply.
A consequence is that water rises highest in column with finest grains.
[Note water rises fastest in column with largest grains. We return to this in the post lecture of chapter
PM4.]
(ii)
Chromatography.
Demonstration
This is a method of chemical analysis which can be done by eye. See post lecture material for a more
careful description.
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PM2: Surface Tension
2-5 WETTING
A question we have skimmed over is: why is there an attractive force between water and glass
causing the rise of water in a glass capillary tube? This is a question about intermolecular forces
which only chemists can answer properly. But certainly different liquids are attracted to different
solids in different degrees. For example, the level of mercury will fall in a glass capillary tube.
Demonstration
Drops on solid surfaces.
WATER
MERCURY
WATER
GLASS
MERCURY
LEAD
Fig 2.8 Water and mercury drops on glass and lead surfaces
Laboratory workers measure the intersurface forces in terms of the angle of contact defined
as follows.
tangent
ANGLE OF f
line
CONTACT
Fig 2.9 Definition of f, the angle of contact between a liquid and a solid surface
The concept of angle of contact is treated further in the post lecture.
This phenomenon is called wetting. Water is said to wet glass completely (the angle of contact
is virtually zero).
The wetting characteristics of surfaces can be changed by putting a layer of a different material
on the surface.
Demonstration
Oil on glass will repel water.
WATER
OIL
GLASS
Fig 2.10 The presence of oil results in the water forming a drop rather than spreading over
the glass surface
Demonstration
Waterproofing of material (this usually involves coating fibres with oil or polymers).
Demonstration
Preening of birds.
Water birds spread oil on their feathers to make them water resistant.
Demonstration
Water resistant sands.
Some West Australian sands are virtually impervious to water as a result of fibrous material
between the grains making them water resistant. This leads to bad run off conditions in vast areas of
the state.
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PM2: Surface Tension
Detergents
The properties of detergents arise from their complicated molecular structure. This can be illustrated
schematically thus:
This end is repelled by water
molecules [hydrophobic] and is
This end is attracted to water
attracted to oils, fats [lipiphilic]
molecules [hydrophilic]
H
(i)
H
H
H
H
H
H
H
H
H
H
H
C
C
C
C
C
C
C
C
C
C
C
H
H
H
H
H
H
H
H
H
H
H
O
C
O-
Fig 2.11 A detergent molecule
When detergent is put into water this happens:
Fig 2.12 Detergent molecules in water (schematic)
Note that along the surface there are water molecules and hydrophobic ends. The surface
tension is lower than that of pure water. It is easier to pull this surface apart than it is to pull a
surface of pure water apart
(ii)
In washing up water the following sequence occurs as the water is stirred up.
grease
water
DETERGENT ADDED
Fig 2.13 Stirring of soapy water during "washing up"
STIRRED
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PM2: Surface Tension
The particles of organic matter are rendered soluble by being coated with detergent molecules:
lipophilic ends stick to the particles and hydrophilic ends point outwards.
Emulsification.
Many organic substances which are insoluble in water (DDT is a good example) can be mixed into
an emulsion with water by the addition of a little detergent.
Demonstration
Oil and water.
POST-LECTURE
2-6 UNITS AND DIMENSIONS
A couple of statements were made (or implied) in 2-3 above, which may not be all that obvious.
Q2.1
The loop of thread changed its shape to a circle because a circle is the geometrical shape which has
maximum area for a fixed circumference.
This is not easy to prove in general but consider the following concrete example: assume that the length of
thread in the loop was 0.l!m and work out which, of the following possible shapes the loop could have, has
the largest area.
2.5 cm
1 cm
2.5 cm
3.3 cm
3.3 cm
4 cm
3.3 cm
3.2 cm
Fig 2.14 Diagram for Q2.1
Q2.2
Energy/area is the same as force/length. The following example illustrates this fact.
Imagine you are increasing the area of a rectangular soap film; as indicated the original dimensions of the film are a
and Ú. The surface tension of the soapy water is T.
a
Ú
F
Fig 2.15 Diagram for Q2.2
PM2: Surface Tension
21
Suppose that to stretch the film at a constant speed a uniform force F equal (and opposite) to the force associated with
surface tension is applied.
Since the film has two surfaces, the relation between F and T is
F
=
2Ú T .
Calculate the total work done in increasing the distance a by an amount b, and show it is proportional to the change
in area of the soap film.
2-7 MORE ON CAPILLARITY
The law quoted in 2-4 can be derived theoretically as follows. Ask yourself first, why should water
rise up inside the tube? It is an effect of the surface tension at the top of the water column,
particularly where it meets the glass wall.
Glass
molecules
Water molecules
Fig 2.16 Interaction of water and glass molecules
Each water surface molecule exerts forces on those near it Since there is equilibrium the last
water molecule must also have a force exerted on it by the glass molecule near it. Therefore, all
around the top of the water, the glass is exerting a force on the water. Because is so happens that
water wets glass so well, this force is a vertical force.
So that it why the water rises in the tube: because the glass is pulling it up. The length of the
line of contact between the water and the glass is 2p times the radius of tube, so the magnitude of the
upward force is:
=
T ¥ (2p radius of tube)
=
2 p rT.
The next question is: why does not the water keep rising indefinitely?
The answer is that the higher the column the more the weight of the water in the column pulls it
back. Thus there is a downward force equal to r (pr2h) g.
The two forces are in equilibrium so
2prT
=
rπr2hg
and, therefore, for this situation, where the water wets the glass completely, the final height of the
water column can be written
2T
h
=
rgr
Q2.3
In the experiment with soil, we found that for the coarse grained soils (radius of soil grains ~ 0.3!mm) after
a long time the water finally stopped rising at a height of ~ 150 mm.
Although soil is by no means a series of uniform bore capillary tubes, it cannot be too bad an approximation to
apply the above relation. Apply the relation and find how much error is in fact introduced.
PM2: Surface Tension
2-8 ANGLE OF CONTACT
The angle of contact is defined to be the angle between the surface of the liquid and the solid
surface at the point of contact.
tangent
ANGLE OF f
line
CONTACT
Fig 2.17 Angle of contact for a liquid that does not "wet" the solid surface
You will observe that for a water-glass contact, as in the next diagram, the angle of contact is
much smaller;
angle of
tangent
contact
line
small
f
Fig 2.18 Angle of contact for water-glass contact
for mercury-glass, as in the next diagram, it is almost 180°.
angle of
tangent
contact
line
large
f
Fig 2.19 Angle of contact for mercury-glass contact
When the angle of contact is less than 90°, the liquid is said to wet the solid surface, while it is
said not to wet the surface if the angle of contact is greater than 90°.
When the angle of contact is not 0° or 180°, the angle explicitly enters those equations which
directly or indirectly involve the force exerted by a solid on a liquid due to surface tension.
22
PM2: Surface Tension
23
Forces associated
with surface tension
Angle
f
Fig 2.20 Close up of part of Fig 2.16
Redrawing an earlier diagram in a more general way, we note that the force the liquid exerts on
the wall (and vice versa) is not vertical. There is a horizontal component, T sin f (which for f equal
to 0˚ or 180˚ is zero), which results in a usually imperceptible distortion of the wall. There is a
vertical component, T cos f (which for f equal to 0˚ or 180˚ is T), which causes the liquid in a
capillary tube to rise.
So the equation for capillary rise that we wrote is not complete. The general form is
2Tcosf
h =
rgr
For clean glass-water contacts f ª 0 and cos f ª 1. So the equation was suitable for water in
a clean glass tube.
Q2.4
For mercury-glass we saw f ª 180° and we know that cos 180° = -1. The formula for capillary height will
therefore have a minus sign in it. Does this mean that if you put a glass tube in mercury the level of the
surface would be lower inside the tube?
PM2: Surface Tension
2-9 SCALING QUESTIONS
Q2.5
Why can insects walk on water, but larger animals (no matter how much water repellent material they put on
themselves) cannot?
Similarly, why will a needle float on water, but a much larger piece of metal of exactly the same shape will not?
Try to answer this question as follows:
(i)
Consider a nice simple geometric shape for the needle, say a rectangular bar. Take the length to be 40 mm
and the width 0.50 mm.
(ii)
Calculate its weight (the density of iron is 7.8 ¥ 103 kg.m-3).
(iii)
Now assume it is on top of the water with an angle, f, as shown.
Needle
f
Fig 2.21 Needle "floating" on water
Calculate the total upward force (remember the force associated with surface tension acts right around the contact line
between the needle and the water).
(iv)
Can the weight of the needle be supported?
(v)
How does the angle of contact depend on the weight?
(vi)
Now assume the "needle" is 4 m in length and 5 cm thick.
Will its weight be supported by surface tension?
(vii)
See if you can use the kind of scaling argument which was employed in chapter FE8 to answer the original
question succinctly.
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25
PM2: Surface Tension
<< 2-10
CHROMATOGRAPHY
Chromatography is a technique for separating out the chemical constituents of mixtures.
useful in biological contexts. There are two commonly used forms.
It is particularly
Paper Chromatography: Here a few drops of the chemical mixture are put onto a piece of filter paper and
allowed to dry. Next the paper is touched to a reservoir of some solvent which will dissolve the chemical substance
you hope to detect. The solvent is sucked up into the filter paper (by capillary action), and as it flows past the dried
mixture, it dissolves out the chemical constituents and carries them along. However, different chemical substances
adhere more or less strongly to the paper (i.e. the surface tension between the surface of the solution and the fibres of
the paper differs) and so different chemical substances are carried along at different rates. So if you remove the paper
from the solvent after a while the various chemical constituents of the original mixture will be at different positions
on the filter paper.
Colour Chromatography
(This is the experiment we filmed.) Here the solvent is put on top of the
mixture, and allowed to flow through a plug composed of grains of cellulose. Again, the adhesion between the
chemical constituents of the sample (spinach leaf) and the cellulose grains is different and they all sink at different
rates. In our experiment (which we filmed in the Department of Agricultural Chemistry with the help of Dr Bob
Caldwell) the final order of chemical constituents is
TOP:
Flavonoid
(Yellow)
Chlorophyll B
(Green)
Xanthophyll
(Yellow)
Chlorophyll S
(Green)
Pheophytin
(Purple)
BOTTOM
Carotenoids
(Yellow)
Only the two chlorophyll bands show up well on the TV screen.
>>
2-11 VALUES OF SURFACE TENSION
Here are the values of surface tension of some common liquids. They are listed here
merely for the purpose of showing you what range the values of surface tension can
have
.
Liquid
Surface Tension/N.m-1
water (20°C)
0.073
water (100°C)
0.059
alcohol
0.022
glycerine
0.063
turpentine
0.027
mercury
0.513
2-12 REFERENCES
"Surface tension in the lungs"
Scientific American, p 120, Dec 1962.
"Synthetic detergents"
Kushner & Hoffman, Scientific American, p 26, Oct 1951.