The molecular mechanismof

surface tension

M V Berry
H H Wills Physics Laboratory
Bristol

Introduction phenomena involving bubbles, liquid in capillary

There are often obscurities and fallacies in the teach- tubes, etc.
ing of surface tension at anelementary level and this is Similar difficulties arise in trying to account for the
perhaps the reason why the subject is no longer existence of an angle of contact, 0, between liquid and
fashionable. Many treatments (Shortley and Williams solid surfaces in the presence of a gas. The offending
1965, Nightingale 1949), present a diagramlike figure 1 diagram is shown as figure 2 (Starling and Woodall
together with the statement that the unbalanced in- 1958, Nelkon and Parker 1968). Three forces are
ward attractionon molecules in the surface layers shown; they are the surface tensions yL, ys and yLS of
causes the surface to be pulled in and to try to con- the liquid/air,solid/air,and solid/liquid interfaces.
tract. Any bright pupil or student will point out that However, it is not often made clear what the matter
there areat least two serious difficulties with this is on which these three forces act (is it a single mole-
explanation. First, the situation depicted is mechani- cule in the line of contact at A, a small mass of liquid
cally impossible if, as is often implied, it is supposed and solid at A, or what ?). Further, it is obvious that
to represent statical equilibrium. Second, and more the three forces are not in statical equilibrium, and if
important, an inward attraction on surface molecules figure 2 represented reality the matter on which the

+
does not explain the tension parallelto the surface forces act would accelerate perpendicularly away
which is invoked in the usual explanations of simple fromthe solid surface! (The equilibrium of forces
parallel to the solid surface is assumed, and indeed
0 0 0 used, to calculate e).
In the short section 2 of this article we consider

0
0
0

01
Resultant force
zero
Figure 1 Diagram sometimes found in elementary Figure 2 Diagram sometimes employed in attempt IO
textspurporting to explainsurface tension. explain angle of contact.
79
more carefully what happens to a molecule in the work, related to the positive shaded area in figure 3,
liquid surface; then, in the main part of the paper, which is called surface free energy if the surface is
section 3,we show how the tension parallel to the created reversibly and isothermally. There is a nega-
surface arises; finally in section 4 we attempt a con- tive contribution to the surface energy, related to the
sistent treatment of the angle of contact. negative shaded area in figure 3, due to the ‘breaking’
of repulsive ‘bonds’ from the very nearest neighbours,
Theinwardattractiononsurface but this must be smaller than the contribution from
molecules the attractive forces, since otherwise the liquid would
There is in fact a net inward force on a molecule in the not cohere at all.
surface, but this is only one aspect of a dynamical
equilibrium in which the repulsive forces acting on a The tension parallel to the surface
molecule from its very near neighbours (figure 3) play The argumentsjust presented in section 2, which
a key role. For a molecule in the bulk liquid the concerned themselves with the forces on, and energies
resultant repulsive force from its near neighbours and of, single molecules, can be used to explain the
the resultant attractive force from its farther neigh- tendency of liquid surfaces to contract. But the origin
bours are both zero, on the average. These resultants of the tension parallel to liquid surfaces is not explain-
are not usually zero at any instant, however, and this ed. Indeed, it has been authoritatively stated that the
causes the random heat motion. In the surface there is notion of surface tension is a ‘useful fiction’ (Cham-
an unbalanced force on a molecule; it is directed in- pion and Davy 1936) and that ‘surface tension does
wards because the decreasing density in the surface not exist as a physical reality, and is only the mathe-
layer implies that there are fewer very near neighbours matical equivalent of free surface energy’ (Adam
to give an outward repulsive force. Thus, on the aver- 1938). However, without using the concept of a tensile
age, surface molecules accelerate inwards. This inward force parallel to the surface, it is very difficult to ex-
acceleration cannotcontinue down into the bulk plain experiments involving static situations(for
because the descending molecules soon encounter the instance the support of a weighted movable wire by a
upward repulsive forces from the interior molecules. soap film on a fixed frame), where no work is being
A helpful analogy is the behaviour of a large number done. (To explain these experiments using surface
of balls (surface molecules) bouncing elastically off energy involves the principle of virtual work, which
the earth’s surface (bulk liquid). The force acting on implies the existence of forces). It comes as something
them, and the resulting acceleration, is downwards of a relief then, to read (OnoandKondo 1960,
except for the brief periods when they encounter Shoemaker et a1 1970) that in recent research work
repulsive forces from the solid ground. not only is the existence of surface tension confirmed
Similar arguments can beused to explain surface but its numerical value is also calculated from first
energy: in the bulk liquid a molecule hasmore principles - that is from the known intermolecular
attractive bonds joining it to its neighbours than one forces - and results are obtained which agree fairly
at the surface. To create new surface therefore, some well with experiment. These calculations imply a
bonds must be broken; this involves the doing of qualitative explanation of surface tension which does
not seem to have been presented before.
The basic idea is that the component of fluid pres-
sure directed parallel to the surface (we shall use the
term ‘pressure’ loosely, to denote the normal compon-
ent of the stress tensor) decreases and becomes nega-
tive - that is turns into atension - in the region near
the liquid surface. Thus surface tension is a macros-
copic quantity like density, an average over times long
compared with the mean interval between collisions,
and cannot be understood by considering the forces
on single molecules. There is no inconsistency in the
concept of macroscopic quantities whichvary sub-
stantially over intermolecular distances - the density,
for example, must change rapidly but smoothly from
Figure 3 Curve showing forceF(r) between its liquid to its gas value, and it is fairly well known
molecules whose centres are separatedby a distance r. that changes in the positions of macroscopic objects
80
can be followed over fractions of atomic distances surface, however, the tangential pressure p' and the
(Jones 1967). normal pressure p" need not be the same, since there
In a fluid in equilibrium the pressure can be defined is no longer any symmetry of direction. But the fluid
as the average normal force per unit area exerted by isin equilibrium, so that the forces on the opposite
all the molecules on one side of a small imaginary faces of a small cubemustbeequal and opposite
test surface in the liquid on all those on the other side. (figure 5). This means that (apart from a cumulative
The total pressure can be separated into two parts: downward increase due to gravity, which is negligible
the first is the kinetic contribution which arises be- near the surface) the normal pressure must have the
cause we are taking a time-averaged force and is due constant value p,, right through the surface layer. The
to the transport of momentum by molecules moving tangential pressure p', however, while being equal to
across the surface. In a perfect gas this is the only p . in the bulk liquid and vapour, can and does vary in
contribution to the pressure, but its value, givenby the surface layer.
Pk= pkT (1) To see this it is necessary first to understand how
(where p is the number density of molecules at the p" can keep its constant value p,, even though both its
point considered, k is Boltzman's constant and Tis the kinetic and static force contributions p: and pf" vary
absolute temperature) is the same in a liquid, because with 2, a coordinate perpendicular to the liquid sur-
the velocity distribution of molecules is the same for a face which increases in value into the vapour. From
liquid in equilibrium as it is for a gas. The kinetic pres- equation (l), the decrease in density with increasing 2
sure is,of course, essentially positive. The second means that p: decreases; the corresponding increase
contribution to the pressure is important in a dense in pf"from negative values to nearly zero, necessary to
gas or a liquid; it is due to the time average of the keep p" constant (figure 6), occurs because there are
static forces between molecules on opposite sides of fewer pairs of molecules attracting across the test
the test surface (figure 4). This static force pressure, surface as it nears the liquid surface. Now p' will have
p f is usually negative, since p k is very much greater the same kinetic part as p" but its static force
for a liquid than for a gas, so that attractive forces contribution p i will increase towards zero more
must dominate the repulsive forces in order to reduce slowly at first than p ; does; this is because the static
the total pressure to the value externally applied. force contribution comes mainly from molecules lying
Only when the external pressure is very great is p f along perpendiculars through the test surface, so that
positive; then the repulsive forces dominatethe the tangentially oriented test surfaces used to define
attractive ones, in order to help the kinetic pressure p" 'see' the depletion of molecules in the liquid surface
resist further compression. The totalpressure is thus before the normally oriented test surfaces used to
P = Pk + Pf (2) definep' (figure 7). If these two contributions to p' are
(usually (always (usually added(equation (2)), theresultant (figure 8) is not
positive) positive) negative) constant, but goes negative so that there is, in fact, a
In the bulk liquid and vapour the pressure has the tangential tensile stress near the liquid surface, which
same value, p o , whatever the orientation of the test even for rare gas liquids which cohere only weakly can
surface employed in its definition. Nearthe liquid amount to more than a thousand atmospheres. The

p Matter acted

Figure 4 Origin of static forcecontribution to pressure. Figure 5 Equilibrium of elementary fluidcubes in

liquid, vapour and surface.
81
 z surface layer of liquid, in contrast with the bulk, must
4* possess rigidity in order to resist the shear stress that
results from p t differing from p"; this is the basis for
the statement appearing in older textbooks that liquids
G as behave as if their surfaces are covered by an 'elastic
skin'.
The surface tension, y, defined in an elementary way
as the total force exerted between the portions of
liquid on opposite sides of a line of unit length in the
Liquid surface, is simply the integral of the underpressure
through thesurface layer, i.e.
Y= 1 -m
W

(PO- p'(Z))dZ.
Any portion of a plane surface is in equilibrium under
(3)

the action of the surface tension forces acting round

its perimeter. To observe the effects of surface tension
it is necessary either to have a curved surface as in
bubbles, menisci, etc, where the unbalanced surface
tension forces are equilibrated by excess pressure on
pr" the concave side, or to terminate the surface as in the
wireframe experiment mentioned earlier, where the
Figure 6 Contributions to normalpressure. unbalanced surface tension forces are equilibrated by
the weight mg.
The surface tension of solids arises in basically the
same way, but the details are considerably complicated
by the possibility thatthe bulk stress may include
shearcomponents.Atthe interface between two
liquids the behaviour of the tangential pressure is more
complicated but if the liquids do not mix there is
generally a net positive tension (figure 9).
The numerical calculations so far performed (Ono
Test surface Test .surface
and Kondo 1960, Shoemaker et al. 1970) have em-
for p; for P;
ployed the approximation of assuming that the fluid
Figure 7 Greater number of interactions contributing properties change suddenly from their liquid to their
to p:than top; near liquid surface. vapour values; that this procedure is unphysical is re-
vealed by the fact that the resulting calculated normal
Z pressure p" is not constant through the surface.

1 Gas
The angle of contact
Consider the forces which act on a small mass of fluid
ABC (figure 10) near the line of contact between liquid,
solid and gas, which passes through A ; BC must
exceed the thickness of the surface layers. Only the
departures from atmospheric pressure (which acts all
round) will affect the statical equilibrium. These de-
partures will be caused by the rest of the liquid and
I Liauid some solid near C, acting across BC, and by the solid
acting across AC.
The force across BC consists of two parts: first,
there is the liquid near B, which pulls with the usual
liquid surface tension yr per unit length of line of
contact. Second, there is the interfacial liquid and solid
Figure 8 Contributions to tangentialpressure. near C; inthe 'wetting' case where the liquid is
82
strongly attractedto the solid we may reasonably
expect its density to increase towards C, giving a
pressure on BC near C (figure 9), whose total force per
unit length of line of contact we call F. It is possible to
obtain an expression for F by considering the tangen-
tial forces near C in the solid as well as in the liquid
(figure 1l), and assuming (dubiously), that the stresses
in the solid are unaffected by the presence of the
liquid. This leads to theresult
F = 7s - Y u j (4)
where ys and yuj arethe solid-gas and solid-liquid
surface tensions. The third force to be considered is
that exerted by the solid on the liquid above AC.
Except for the region very near to A this will be just
p,, which acts all round and is irrelevant; near to A,
however, the density decreases and there can be a net
downward force Xdue tothe diminution of the kinetic
part of the pressure.
These three forces yL, F and X must be in equili-
brium and, if the angle of contact is B, resolving Figure 9 Contributions to tangentialpressure at
horizontally gives interface between two liquids.
yL COSB = F = ys - yLS (5)
which is the usual formula, while resolving vertically
gives
X = yL sin0 (6)
In the ‘non-wetting’ case, where the liquid is only G as
weakly attracted to the solid, similar considerations
apply but there is now a tension F near C, and the
angle of contact is obtuse (see figure 11). The result ( 5 )
can also be derived using the concept of surface energy
andthe principle of virtual work (Poynting and
Thomson 1947).

Conclusion
In this paper it has been shown that consideration of
the condition of single molecules in the surface can
provide an explanation of the tendency of liquid sur- Figure 10 Forces on fluid near line of contact in
faces to contract, but that in order to understand the ‘wetting’ case.
origin of the tensile tangential stress in the surface it is
necessary to examine the variation of averaged
quantities, namely pressure and density.
Because there really is a tensile force in liquid
surfaces, ordinary statics can validly be used in surface
tension problems: such explanations as, for example,
‘the water in a capillary tube is held up by the surface
tension forces acting round the rim of the meniscus’
are perfectly correct, and need not be prefaced by any
apologia to the effect that surface tension is merely a
‘usefulfiction’.

Acknowledgments
I would like to thank Mr R Evans, Professor F c Figure 11 Forces on fluid near line of contact in
Frank, Mr D F Gibbs, Professor J F Nye, Mr D Scott ‘non-wetting’ case.
83
and the schoolteachers of Bristol for helpful discus- Clues down
sions. 1 Ha, yell up to his faithful comet! (6)
2 Sip up theold mass - it goesto andfro (6)
References 3 In the country are the original robots and a little
Adam, N K, 1938, The Physics and Chemistry of aluminium ( 5 )
Surfaces (Oxford) 2nd Ed, p5. 4 Sicknesses froma crazy Finnishbath with two
Champion, F C and Davy, N, 1936, Properties of different directions (7)
Matter (Blackie), p99. 6 Bloody iron ore! (9)
Jones, R V, 1967, Physics Bulletin, 18, 325. 7 Bounded by two planes I’dback two-thirds of a
Nelkon, M, and Parker, D, 1968, AdvancedLevel large church (8)
Physics (Heinemann), p1 17. 8 Belligerent antics which may conceal the realities
Nightingale, E, 1949, Higher Physics(Bell), p122. of life? (3-5)
Ono, S, and Kondo,S , 1960, Molecular Theory of 11 Suddenly changing voltage turns animals upside-
Surface Tension in Liquids, Handbuch der Physik, down (4)
Vol X (Springer). 15 Give back - bounce like a ball? (9)
Poynting, J H, and Thomson, J J, 1947,Properties of 17 What 5 tends to do - sob in distress over success-
Matter (Charles Griffin) 14th Ed, p171. ful treatments (8)
Shoemaker, PD, Paul, Dand deChazal, L E, 1970, 18 Domain of the peris (8)
J Chem Phys,52,491. 20 Uninteresting, like a rainy day (4)
Shortley, G and Williams, D, 1965, Elements of 21 Employee who might be civil (7)
Physics (Prentice-Hall) 4th Ed, p367. 22 A German star at the back of the ship(6)
Starling, S G, andWoodall, A J, 1958, Physics 23 To mark the students fairly involves double secret
(Longmans Green), 2nd Ed, p105. service (6)
26 Treasure-house of electric charge is our Mother ( 5 )
E Deeson

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