Surface Tension Concepts

2 Capillarity and the mechanics of surfaces

2.1 Surface tension and work

2.1.1 Defi nition of surface tension

2.1.2 Work of extension

2.1.3 Contact angle, wetting, and spreading

2.1.4 The surface of tension

2.1.5 Work of adhesion and cohesion

2.2 Measurement of surface tension

2.2.1 Wilhelmy plate

2.2.2 Capillary rise

2.2.3 Drop weight or volume

2.2.4 Maximum bubble pressure

2.2.5 Sessile and pendant drops

2.2.6 Micropipette

2.2.7 Indicator oils

2.3 The Laplace equation

2.3.1 Applications of the Laplace equation

2.4 The Kelvin equation

2.4.1 Consequences of the Kelvin equation

2.4.2 Verifi cation of the Kelvin equation

2.5 The surface tension of pure liquids

2.5.1 Eff ect of temperature on surface tension

2.5.2 The surface tension of liquids

2.5.3 The hydrophobic–hydrophilic interaction

2.1 Surface tension and work

Many commonly occurring phenomena that are observed in systems con-taining an interface in which one of the phases is a liquid can be understood through the concept of surface tension. Some examples are the rise of liquid in a narrow tube, the fact that drops of liquid tend to be spherical, and the observation that water spreads evenly on some surfaces, while remain-ing in isolated drops on others. In this chapter, surface tension is defi ned and we see how it can be used to develop simple theories to explain these experiences.

2.1.1 Defi nition of surface tension

The bristles in a wet paint brush tend to stick together, and we might be tempted to think that the presence of water is suffi cient to make the bristles stick to one another. However, if the brush is held completely under water the bristles sepa-rate (Figure 2.1), so it is not the fact that the bristles are wet, but the presence of the air–water interface that causes them to stick together.

2

2.1 SURFACE TENSION AND WORK | 11

Another example is that it is easy to make sandcastles with damp sand, but if the sand is dry or very wet it doesn’t hold together. In both cases, the sticki-ness depends on the presence of the air–water interface, and the phenomena can be explained by realizing that the interface acts as though it were under tension. That is, it experiences forces that pull the bristles or sand particles together.

Yet another common example is that a drop of water tends to assume a spherical shape (distorted perhaps by gravity or by air resistance if falling, Figure 2.2). Now a spherical shape has the lowest surface area for a given volume of liquid, so what we observe in these and other examples is that the area of an interface tends to a minimum. The force that causes this to happen is called the surface tension or sometimes the interfacial tension.

If an imaginary line is drawn on a surface it will be pulled by the surfaces on either side towards those surfaces. For example, consider a partly infl ated bal-loon (Figure 2.3). If a line is drawn on the surface, and then more air is added to the balloon, we observe that the line broadens as the rubber expands, and if the balloon were to be cut along the line the two sides would separate form-ing a hole.

The balloon is only an analogy, but something similar happens at the sur-faces of interest to us. At any surface, if the force, F, acting tangentially to the surface and at right angles to an element, δx, of an imaginary line in the surface has a magnitude that is independent of the direction of the element, then the surface tension, γ, is:

Fx

γ = ⋅δ

(2.1)

In words, the surface tension is the force per unit length acting on an imagi-nary line drawn in the surface. The SI units of surface tension are N m−1, although because the N m−1 is rather large (the surface tension of the air–water interface at room temperature is about 0.072 N m−1), surface tension is more commonly quoted in mN m−1.1

An intuitive way to understand the origin of surface tension from a molecu-lar point of view is to consider the forces acting on a molecule at the surface of

1 The CGS unit for surface tension is the dyn cm−1, where 1 dyn cm−1 = 1 mN m−1. This unit is seen in the older literature and is rarely used in current work.

Dry

Wet

Immersed

Fig. 2.1 Effects of water on the fi bres of a brush.

Fig. 2.2 Spherical drops falling from a round tube.

12 | 2 CAPILLARITY AND THE MECHANICS OF SURFACES

a liquid compared to those acting on one in the bulk (Figure 2.4). The attrac-tive forces acting on one molecule in the bulk are, when averaged over time, isotropic. That is, there is no net force pulling the molecule in any given direc-tion. A molecule at the surface, however, will feel an unbalanced force due to the relative scarcity of near neighbours in the direction of the gas phase. The result is that there is a tendency for that molecule to be pulled into the bulk, as is the case for every other molecule at the surface. Hence, the origin of the tendency to minimize the area of the surface is clear.

2.1.2 Work of extension

As the area of an interface tends to a minimum, energy must be brought into the system to extend the interface.

A soap fi lm contained by a rectangular frame with one end capable of slid-ing along the sides (Figure 2.5) exerts a force, F, on the slide. The length of the line of contact between soap fi lm and slide is twice the length of the slide [i.e.

( )122 x x= ] because the soap fi lm has two sides. That is, if you think of the

fi lm as being like this page, there are two interfaces, one on the reader’s side of the page, and one on the reverse side of the page. If the surface is extended by moving the slide through a distance δy then the area of surface (considering both sides of the fi lm) increases by δA (= x δy) and the force exerted by the surface tension and resisting the extension is, from Eqn 2.1:

.F xγ=

The work (ws) of extension is therefore:

sw F y x y A= δ = δ = δγ γ . (2.2)

This relationship is the basis for developing the thermodynamics of surfaces, a topic that will be explored in the next chapter.

In this argument it has been assumed that the new interface created when the surface area is increased has the same composition as the original. This requires new material to be brought from the bulk phases. If, instead, the original interface is stretched with no new material brought to the surface, additional work would be needed to increase the molecular separation.

If the surface of a solid is extended, then in general the work required is performed against the interfacial stress, rather than against the surface ten-sion. The interfacial tension is one component of the interfacial stress, but the

FF

Fig. 2.3 The tension in the rubber acts on a line in the surface, causing it to stretch.

Air

Surface

Bulkliquid

Fig. 2.4 Forces acting on molecules near a surface.


relationship between these quantities is complex and depends on the nature of the solid, with such factors as the isotropic or non-isotropic nature of the surface, the presence of defects, and so on. Methods for measuring the sur-face tensions of solids are at best approximate and often involve questionable assumptions. For example, the measured increase in vapour pressure for small spherical particles of solid can be used with the Kelvin equation (to be dis-cussed later in this chapter (Eqn 2.22), but the assumptions of sphericity and uniformity of the particles are usually questionable. Further consideration of this topic is outside the scope of this book. More information can be obtained from Lyklema (2000) and Adamson and Gast (1997, p. 259).

F

Wire frame

Soapfilm(double-sided)

Slidewire

12X

δy

Fig. 2.5 Wire frame for extending the surface of a (double-sided) soap fi lm.

Problem Calculate the work of extending a soap fi lm supported in a wireframe as in Figure 2.5 by 1.5 cm2. The surface tension of the air–solutioninterface is 35 mN m–1.

Solution We must remember that the soap fi lm has two sides so the totalincrease in surface area is 3.0 cm2.

Applying Eqn 2.2 we have:

3 1 2 2 735 10 N m 3 (10 m) 105 10 N m3 1 2 2 71 2 2

10.5 N m 10.5 Jsw 3 1 2 21 2 21 2 2= 35 10 N m 3 (10 m) 10510 N m 3 (10 m) 1053 1 2 21 2 22 2

= μ = μ10.5 N m 10.5 JN m 10.5 J

Note that the work (a form of energy) is positive because energy has to be putinto the system.

Example 2.1 Work of extending a soap fi lmExample 2 1 Work of extending a soap film

2.1.3 Contact angle, wetting, and spreading

When a drop of liquid is placed on a solid surface the triple interface, formed between solid, liquid, and gas will move in response to the forces arising from the three interfacial tensions until an equilibrium position is established. The situation is illustrated in Figure 2.6, which shows a drop of liquid (L) on a fl at solid surface (S) with air (G) as the third phase. The angle, θ, between the solid surface and the tangent to the liquid surface at the line of contact with the solid is known as the contact angle. By convention the contact angle is meas-ured in the liquid phase. The largest observed value for water on a smooth solid surface at equilibrium is about 120°.


The position of the triple interface will change in response to the horizontal components of the interfacial tensions acting on it. At equilibrium these ten-sions will be in balance and thus:

GS LS GL cosγ γ γ θ= + . (2.3)

However, when the drop is initially placed on the surface the interfacial ten-sions will not be in equilibrium. The net force per unit length of the triple interface along the solid surface will then be:

GS LS GLh cosF γ γ γ θ= − − ′ (2.4)

where forces acting to the right are given positive signs and those acting to the left have negative signs. The angle θ′ is the instantaneous contact angle and will change as the triple interface moves towards its equilibrium position where θ′ = θ and Fh = 0.

Although Eqn 2.3 describes the equilibrium contact angle in terms of the interfacial tensions involved, it gives no real insight into the reason that a cer-tain value of contact angle is reached. We observe that some surfaces have a very high contact angle for water, while for others it is so low as to be unmeas-urable. An understanding of the origin of contact angle requires knowledge of the balance of forces between molecules in the liquid drop (cohesive forces), and those between the liquid molecules and the surface (adhesive forces). A sur-face that has primarily polar groups on the surface, such as hydroxyl groups, will have a good affi nity for water and, therefore, strong adhesive forces and a low contact angle. Such a surface is called hydrophilic. If the surface is made up of non-polar groups, which is common for polymer surfaces or surfaces covered by an organic layer, we say that the surface is hydrophobic, and the contact angle will be large. It follows that measurements of contact angle are frequently used as a quick and simple method to gain qualitative information about the chemical nature of the surface. More generally, the terms appropriate for all liquids are lyophilic and lyophobic (solvent loving and solvent hating).

Many surfaces show an apparent hysteresis, where different values of the contact angle are measured depending on whether the measurement is per-formed on a drop of increasing size (the advancing contact angle) or of dimin-ishing size (the receding contact angle). This is normally due to roughness of the surface, and the difference between advancing and receding contact

G

L

S

GS

GL

ES ug

g

g

Fig. 2.6 The forces acting at the triple interface for a drop of liquid on a fl at solid surface.


angles is another useful and quick method of gaining qualitative information about the nature of the surface. Although the macroscopic observation of angles shows hysteresis, it is likely that the microscopic angles, if they could be observed, would show no hysteresis. The advancing contact angle (liquid moving over an apparently dry surface) is greater than the receding angle, thus when a contact angle close to zero is needed, as for example when surface tension is to be measured by a Wilhelmy plate (see Section 2.2.1), the surface of the solid is often roughened and a receding contact angle used.

Contact angle measurements are made by placing a drop of the liquid on a surface and viewing it with some type of magnifying lens (Figure 2.7). The angle can then be measured optically, usually by taking a digital image and using software to determine the angle. The most diffi cult part of dealing with very high contact angles is keeping the drops in place long enough to be meas-ured! The slightest tilt on the surface is enough to cause the drops to roll off.

Wetting is determined by the equilibrium contact angle, θ. If θ < 90°, the liquid is said to wet the solid; if θ = 0, there is complete or perfect wetting; if θ > 90° (cos θ < 0), the liquid does not wet the solid. Contact angles of 180° are not found, as there is always some interaction between the liquid and the solid.

The spreading of a liquid over a solid depends on the components of the interfacial tensions acting parallel to the solid surface at the line of contact, as in Eqn 2.4. The triple interface will move in the direction dictated by Fh until a value of θ is reached at which Fh = 0, giving equilibrium, or, if such a position is not possible, the liquid spreads completely, and then θ’= 0, cos θ’ = 1, and Eqn 2.4 becomes:

GS LS GLh( 0)F θ γ γ γ= = − −′ . (2.5)

It is convenient to defi ne a spreading coeffi cient, SLS, by

LS GS LS GLS γ γ γ= − − . (2.6)

If SLS > 0, the liquid spreads completely, whereas if SLS ≤ 0 the drop does not spread completely and it fi nds an equilibrium contact angle, θeq, where Fh = 0.

Fig. 2.7 A contact angle image showing a drop of water on a PTFE surface. The needle used to deposit the drop is visible at the top.


The leaves of the lotus plant (Nelumbo) display a remarkable ability to remain clean even in the muddy environment in which they grow. This is due to the complex nanostructure of the surface, giving extreme hydrophobicity, which results in droplets of water rolling off the surface and taking surface contami-nation with them. Biomimetics is the name given to the practice of mimicking the properties of biological systems and their application to technological problems. The lotus leaf has inspired researchers to emulate its amazing prop-erties in synthetic surfaces.

The contact angle of water on surfaces varies greatly, depending on the chemi-cal nature of the surface. Hydrocarbon surfaces (like the paint surface of a car after it has been polished with a wax polish) have high contact angles, as do fl uoropolymers like Tefl onTM. Superhydrophobicity generally refers to surfaces with water contact angle θ > 150°. Fabrication of superhydrophobic coat-ings has become a rapidly expanding area of research over the last few years, due mainly to the useful property of self-cleaning that such surfaces (and the lotus leaf) exhibit. The two key attributes to be considered in the design of such interfaces are the chemical composition and the micro- and nanostructure. Nanostructural roughness is manipulated to create superhydrophobicity from (chemically) low (θ < 90°) contact angle materials. Typically, the surface areas of such materials are 2.5 times greater than would be expected from an equivalent fl at surface. This minimizes the use of commonly used, but expensive fl uorinated compounds that also achieve high contact angles by exploiting chemical inter-actions. Such architecture can be applied to surfaces as varied as fabric, paint, and building materials leading to buildings that wash themselves when it rains.

The image above shows a three-dimensional view of a superhydrophobic surface viewed with an atomic force microscope. Although the contact angle of the underlying poly(dimethylsiloxane) is about 75°, with the rough coat-ing of nanoparticles the contact angle is 160–170° (courtesy of R. Lamb and H. Zhang).

How lotus leaves help keep buildings cleanHow lotus leaves help keep buildings clean

12

3

4

μm


2.1.4 The surface of tension

It has already been pointed out that any real interface has a fi nite thickness, but for many purposes it is convenient to replace the real interface by a math-ematical surface of zero thickness that is mechanically equivalent to the real interface. This mathematical interface is called the surface of tension.

If it were possible to observe the pattern of forces acting on a plane cutting through an interface it would be complex with the complexity extending over a region of fi nite thickness. Possibly a pattern such as that in Figure 2.8(a) might be found, but as the means for observing such a pattern do not exist some alternative is needed. In 1805, Young suggested that this complex pat-tern could be replaced by a mathematical surface under tension with a pattern of uniform forces in each of the bulk phases extending unchanged right up to the mathematical surface (Figure 2.8b).

This model surface is the surface of tension. Proper location of the surface of tension is important when the interface is curved. In essence, two equa-tions are obtained from the requirements that both the resultant forces and the resultant moments of those forces must be the same in the model system as in the real system. Details of the procedure have been described by Defay, et al. (1966).

2.1.5 Work of adhesion and cohesion

If two phases (α and β) in contact are pulled apart inside a third phase ω, the original interface is destroyed and two new interfaces are formed (see Figure 2.9).

The work energy per unit area in performing this operation is called the work of adhesion, wαβ. There is a contribution from each interface removed from or added to the system:

.w γ γ γαβ αβ αω βω= − + + (2.7)

If, instead of two distinct phases, a column of a single liquid is pulled apart, the work of cohesion is:

2wαα αω= γ (2.8)

Phase β

Phase α

Surfaceregion

Surface oftension

Model interfaceReal interface

(a) (b)

γ

Fig. 2.8 Possible pattern of forces in the real interface and the model interface.


We note that in Eqns 2.7, 2.8, and 2.9 the defi nition of work (and symbol w) is different from that usually used as it is work per unit area (Everett, 1972, p. 597). The units are therefore J m–2 (= N m–1) compared with J.

When one of the phases is a solid, the expression for work of adhesion (Eqn 2.7) can be combined with the equation for the contact angle (Eqn 2.3) to give:

LS GL GS LS GL(1 cos ).w γ γ γ γ θ= + − = + (2.9)

This is commonly known as the equation of Young (1805) and Dupré (1869). Its signifi cance is that it relates the work of adhesion to the readily meas-ured quantities, γGL and θ, rather than to the inaccessible interfacial tensions involving the solid surface.

Phase ωβ

β

α α

Interface β/α Interface ω/α

Interface ω/β

Fig. 2.9 The separation of two phases.

Problem Calculate the work of adhesion of water on four solids, where the equilibrium contact angles are 30o, 60o, 120o, and (a hypothetical) 180o. The surface tension of the air/water interface is 72 mN m–1.

Solution We can use the Young–Dupré equation to calculate the work of adhesion per unit area of contact between water and the solid. For the fi rst solid (θ = 30o):

( )LS 3 1 3 1 272 10 N m (( 134 10 N3 m 134mJmw −3 3 −= ×72 =) =110 N3 m

Using the same procedure for the other solids we obtain for the four cases: 134, 108, 36, and 0 mN m–1.

We note that the work of adhesion falls to zero when the contact angle is 180o, but emphasize that such angles are never observed as it would imply that there is no interaction between the liquid and solid surface.

Example 2.2 Work of adhesion of a liquid on a solidExample 2 2 Work of adhesion of a liquid on a solid

2.2 MEASUREMENT OF SURFACE TENSION | 19

2.2 Measurement of surface tension

Surface tension is a fundamental quantity in the investigation of fl uid inter-faces so its measurement is of great importance. Many methods exist for measuring the value of the surface tension of an interface, and the choice of method depends on the given system. The surface tension of the interface between a static, aqueous solution, and air can be measured by a number of methods, but if, for example, the solution is viscous or the surface tension is changing rapidly, a careful choice of techniques must be made.

2.2.1 Wilhelmy plate

Possibly the easiest way to demonstrate the force arising from surface tension is to dip a fl at plate through the surface of a liquid and measure the force act-ing on it. This is known as a Wilhelmy plate, and is named after the scientist who fi rst used such a device to measure surface tension (Wilhelmy, 1863).

If the plate is perfectly wetted by the liquid a meniscus will form where it passes through the liquid surface giving a contact angle of 0°. The situation is shown in Figure 2.10.

If the plate is hanging vertically, the meniscus will contact the plate along a line of length 2(x + y), where x and y are, respectively, the horizontal length and thickness of the plate. Along the line of contact, the liquid surface will be vertical so the surface tension along this line will, from Eqn 2.1, exert a downward force on the plate of:

2( )F x yγ= + ⋅ (2.10)

As F, x, and y can all be measured, the surface tension can be determined.Lane and Jordan (1970) have shown that despite the complex geometry

of the meniscus at the edges of the plate, as shown in Figure 2.11, the simple formula of Eqn 2.10 holds. Thus, the only correction that is needed arises from the buoyancy of the plate and that depends on the depth of immersion.

Torsionwire

Lightbeam

MirrorBeam

Paper plate

Plate(end-on)Liquid

surface

Fig. 2.10 Wilhelmy plate for measuring surface tension. Here, the force is measured by the twist of a simple torsion wire amplifi ed by an optical beam. Electronic devices are normally used.

Fig. 2.11 Photograph of the water meniscus on a Wilhelmy plate of roughened mica.


If, however, the bottom edge of the plate is set level with the fl at surface of the liquid the buoyancy correction is zero.

In the past, Wilhelmy plates have been made from a variety of materials, the most common being roughened mica, etched glass, and platinum. Scrupulous and elaborate procedures were needed with these materials to ensure that the surfaces were clean and that the contact angle was zero. In 1977 Gaines suggested the use of paper plates. With high quality fi lter or chromatography paper the liquid saturates the plate and essentially forms a liquid surface over the paper ensuring that the contact angle is zero.

Wilhelmy plates may be used in a static mode, as, for example, in a surface fi lm balance (see Section 5.2.3), or in a detachment procedure where the dif-ference in force between the plate hanging (wet) in air and at the moment of detachment when withdrawn from the liquid is measured. Detachment data require a negative correction for buoyancy and there is usually some excess liquid clinging to the bottom edge of the plate. To some extent, these two sources of error cancel, but nevertheless the Wilhelmy plate in detachment mode is only approximate.

The du Noüy ring is a variant of the Wilhelmy plate detachment technique. Instead of the fl at plate a horizontal ring with a diameter of about 10 mm is used. However, the assumption that the force attributable to surface tension is simply twice the circumference of the ring multiplied by the surface tension is an oversim-plifi cation because of the complex geometry of the surface (see Figure 2.12) so a correction procedure must be followed (Adamson and Gast, 1997, Chapter II).

2.2.2 Capillary rise

If a narrow capillary tube is dipped into a liquid the level of liquid in the tube is usually different from that in the larger vessel. With a clean glass capillary and a liquid that wets it, the liquid rises up the tube until an equilibrium posi-tion is attained (Figure 2.13).

At this point it can be considered that the liquid column in the capillary is supported by the surface tension. For the situation where the liquid wets the capillary wall perfectly (θ = 0):

2c c

1c2

2 r gh r

ghr

==

γ π ρ πγ ρ

(2.11)

Ring support

Ring

Water surface

Fig. 2.12 Cross-section of the surface for the du Noüy ring method for measuring surface tension.


where ρ is the density of the liquid (more accurately, it should be the difference in density between the liquid and the surrounding gas), g is the acceleration due to gravity, rc is the radius of the capillary, and h is the measured capillary rise. As these quantities are known or measurable the surface tension can be determined. The capillary rise is measured from the fl at surface of the liquid in the large container to the height of the lowest part of the meniscus in the capillary (Figure 2.13). An alternative is to measure the difference in height between the rises of liquid in two capillaries of different internal diameters. This removes the diffi culty, sometimes experienced, of referring the height to a fl at liquid surface (see Exercise 2.4).

LiquidSurfacetension

Weight of liquidin capillary

Capillary

Beaker

Rise, h

meniscus

h

Fig. 2.13 The phenomenon of capillary rise.

(Moloch horridus), a small lizard found in Australian deserts, uses a fascinating application of capillary rise to stay alive in extremely dry conditions. It is about 20 cm long, is slow moving, and lives mainly on ants. It is covered by grooved thorns, somewhat like rose thorns, which form capillaries through which water is able to move over its body to the corners of its mouth from where it is able to drink. The water can come from the ground or from dew condensing on its body in the cold desert nights. Water is not absorbed through the skin.

The thorny devilThe thorny devil

The measurement of capillary rise and the use of Eqn 2.11 provide a simple and accurate means for determining the surface tension of a liquid. However, it is essential that the capillary be scrupulously clean and that the contact angle be zero, as the accurate determination of θ in a capillary is extremely diffi cult.


It is usually possible to meet these requirements with pure liquids, but with solutions there may be adsorption of the solute on the capillary walls leading to θ ≠ 0.

Problem Ethanol has a surface tension of 22.4 mN m–1 and has a contactangle on glass of 0o. Its density is 0.79 g cm–3. Calculate the rise of ethanol upa capillary of radius 0.1 mm.

Solution Using Eqn 2.11 we have:

3 13

3 6 3 2 3

2 2 22.4 10 N m3

57.8 10 m3

0.79 10 10 kg m 9.81 ms 0.1 10 m3 6 3 2 36 3 2c

hgrc

γρ

3

3 6 3 26 3 23 2

22.422.4= = =3 6 3 2 3 57.8γ

10 10 kg m 9.81 ms 0.110 10 kg m 9.81 ms 0.13 6 3 26 3 26 3 2

Thus, the expected rise is 57.8 mm above the fl at liquid surface. This empha-sizes the fact that narrow capillaries of especially uniform bore are required forsuch measurements.

Example 2.3 Capillary riseExample 2 3 Capillary rise

A more rigorous derivation of Eqn 2.11 is based on the Laplace equation (Section 2.3.1). It shows more clearly that for very precise measurements it is necessary to correct for the deviations of the liquid surface from a spherical shape. The procedures are too complex to be described here, but a detailed treatment can be found in Adamson and Gast (1997, Chapter II). An itera-tive procedure is used to refi ne the value of the capillary constant, a, which is defi ned by:

22.a rh

gγ

ρ= =

(2.12)

Here, r is the radius at the lowest part of the meniscus and is always slightly larger than the capillary radius, rc. Actually, the term capillary constant is misleading as its value depends on properties of the liquid, as well as the capillary.

2.2.3 Drop weight or volume

A drop of liquid hanging from the tip of a capillary is supported by the surface tension of the liquid. If we assume that the downward force due to the weight of the drop immediately before detachment is balanced by the upward force due to surface tension at the line of contact with the capillary tip we have:

2mg rπ γ= (2.13)

where m is the mass of the drop and g is the acceleration due to gravity. The experimental procedure is to allow a known number of drops to form slowly and detach. The total weight of the detached drops is then measured. However, there are serious errors in the assumption that the weight measured


is the total weight of the drops, as a sizable fraction of the liquid hanging beneath the capillary remains attached to the capillary after each drop has fallen (Figure 2.14). Thus, a correction factor, that depends on the capillary radius and the drop volume, has to be applied to Eqn 2.13. Nevertheless, the method is fast and easy to use.

The drop volume method is a closely related one in which the drops are formed from the tip of a calibrated syringe. It is one of the few techniques that can be used at the liquid–liquid interface.

2.2.4 Maximum bubble pressure

During the formation of a bubble of an inert gas beneath a capillary dip-ping into a liquid the bubble radius is at fi rst large, decreases to a minimum when the radius is the same as that of the capillary, and then increases again (Figure 2.15). As a consequence of the Laplace equation (2.17), discussed below, the pressure of gas in the bubble increases, passes through a maximum, and then decreases.

(a) (b) (c) (d)

Fig. 2.14 When a drop falls from a capillary, a signifi cant amount of liquid remains (d). This necessitates a correction to be applied to be simple formula of Eqn 2.13.

Liquid phase

(a) (b) (c) (d) (e)

Fig. 2.15 Shapes of a bubble in a liquid during development. Note that the radius of the bubble, indicated by the arrows, decreases to a minimum for case c, where the radius is equal to that of the capillary, and then increases again.


Because the bubble radius at the measured maximum pressure is known (equal to the radius of the capillary) the surface tension can be calculated from the Laplace equation.

However, because the pressure that the liquid exerts on the bubble varies with the depth of immersion the bubble shape is not exactly spherical and corrections similar to those used in capillary rise measurements are needed for accurate work.

Great care must also be exercised in the design and construction of the cap-illary tip. Usually, a very sharp edge is used so that there is no ambiguity about the capillary radius on which the bubble is being formed.

2.2.5 Sessile and pendant drops

The shape of a drop sitting on a fl at surface (sessile drop) or hanging beneath a fl at solid surface (pendant drop) is determined by the size of the drop and the surface tension of the liquid. Usually, photographs of the drop are taken and measurements made on these. The methods are useful when only small volumes of the liquid are available.

2.2.6 Micropipette

Liquid in the tip of a tapered micropipette and subject to a small excess pres-sure in the pipette will form a curved meniscus similar to that formed in a capillary tube (Figure 2.13) except for the taper of the pipette walls. If we can assume that the interface forms a spherical cap, the radius of the sphere (r) can be calculated from measurements of a chord (2rc) and the maximum distance from the chord to the surface (zc) as shown in Figure 2.16. Simple geometry then gives for the radius of curvature of the meniscus:

2 2c c

c2r z

rz+= . (2.14)

rc

zc

Tapered micropipette

rc

Testliquid

Air or otherimmiscible liquid

r

Fig. 2.16 Test liquid in a micropipette showing the measurements (rc and zc) required for calculating the radius, r, of the meniscus.


Measurement of the pressure difference across the interface and combination of Eqn 2.14 with the Laplace equation (2.17) then gives the interfacial tension (Lee et al., 2001). Improved accuracy can be attained by using several pressure differences and corresponding meniscus radii. The technique is particularly useful when only minute quantities of liquid (≤ 400 μL) are available. Note, however, that optical distortions arising from the curvature of the capillary glass can make accurate measurements diffi cult.

2.2.7 Indicator oils

For measurements in situations where the use of laboratory instrumentation is not feasible, on reservoir surfaces for example, it is possible to estimate the surface tension by observing the spreading or otherwise of drops of various indicator oils. A drop of the oil is placed on the surface and its spreading behaviour observed. Usually, this procedure is used to detect the presence of a surface fi lm and to estimate the change in surface tension caused by the fi lm.

For a drop of indicator oil placed on the surface of a liquid (the subphase), Eqn 2.6 shows that spreading will occur when:

af os aoγ γ γ≥ + (2.15)

where a, o, s, and f refer to air, indicator oil, subphase, and subphase with fi lm, respectively. For a given oil γao is fi xed and known, and if we can assume that the oil drop displaces any surface fi lm on the subphase γos is also fi xed and known. Consequently the equality in Eqn 2.15 gives the minimum value of

the surface tension of the subphase, γaf. Usually, however, it is the spreading coeffi cient, So, (Eqn 2.6) that is found in tables of data. This can be regarded as a spreading or surface pressure (see Eqn 1.2) exerted by the oil.

Prove that these two quantities are the same.

Solution Let superscripts a, o, s, and f indicate air, indicator oil, subphase, and subphase with fi lm, respectively. Thus, we have from Eqns 1.2 and 2.6:

as afΠ γ γasγγ

( )os asS γ ((as= γ

where Sos is the tabulated value of the spreading coeffi cient for clean subphase surface. When the oil just spreads. Sof = 0 and:f

af os aoγ γaf γ= +osγ

so:

os as afS γ γ Πas afas= γ γγ (2.16)

The tabulated values of the spreading coeffi cient are often referred to as the spreading pressures of the oils.

Example 2.4 Relation between surface pressure and spreading coeffi cientExample 2 4 Relation between surface pressure and spreading coefficient


By using a set of indicator oils with different spreading coeffi cients (relative to a clean subphase surface) the surface tension of the subphase covered by a surface fi lm can be estimated. Table 2.1 shows some potentially useful indica-tor oils for studying water surfaces. However, it is important to note that some of the liquids shown in Table 2.1 have a slight solubility in water and some can dissolve small amounts of water. Such solubility can alter the surface tension values and, hence, the spreading coeffi cient, so it is the initial spreading behav-iour that should be observed, before appreciable dissolution has occurred.

Timblin et al. (1962) developed a set of indicator oils consisting of solutions of dodecanol in a light mineral oil. The spreading pressure is dependent on the dodecanol concentration and is approximately proportional to the logarithm of the concentration. Details of the mineral oil were not given so calibration is needed before this procedure can be used with a different oil.

2.3 The Laplace equation

If a fl uid interface is curved, the pressures on either side must be different. For example, if we take a fl at soap fi lm stretched across a circular frame the pressures on either side are the same (usually atmospheric pressure). However, in a soap bubble the pressure inside must be greater than that outside and we can dem-onstrate this by blowing the bubble on the end of a tube. It is necessary to blow into the tube to infl ate the bubble and if the tube is left open the bubble will expel the air and shrink, eventually forming a fl at fi lm over the end of the tube.

When the system is at equilibrium, every part of the interface must be in mechanical equilibrium. For a curved interface, the forces of surface tension are exactly balanced by the difference in pressure on the two sides of the inter-face. This is expressed in the Laplace equation:

2P P

rα β− = γ

(2.17)

which holds for a spherical interface of radius r.The derivation of the Laplace equation follows.

Table 2.1 Indicator oils for estimating the surface tension of fi lm-covered surfaces of water at 20°C

Indicator oil Minimum spreading tension, γ

min/mN m−1

Initial spreading coeffi cient, So/mN m−1

n-Octane 72.6 0.15

Benzene 63.9 8.9

n-Propyl palmitate 57.2 15.5

Aniline 48.8 24.0

Oleic acid 48.1 24.6

n-Octanol 36.0 35.7

n-Butanol 26.2 46.5

iso-Butanol 22.8 49.9

2.3 THE LAPLACE EQUATION | 27

Consider a spherical cap, symmetrical about the z-axis and part of a spherical interface. The pressures exerted on the interface by the two bulk phases (α and β) will be different if the interface is curved and this difference (ΔP) will give rise to a force acting along the normal to the interface at each point. The cap will also be subject to a force arising from surface tension acting tangentially at all points around the perimeter of the cap. These forces are shown in Figure 2.17(a).

Force arising from the pressure diff erence

For a small segment of cap of area δA the force arising from the pressure dif-ference is (Pα−Pβ) δA, and its component in the z direction (the central axis of the cap) is (Pα−Pβ) δA cos ϕ. However, δA cos ϕ is equal to the area of the projection of δA on to the plane containing the perimeter of the cap. The sum of the resolved forces in the z direction for the entire area of the cap is thus:

2c( ) ( )P

zF P P A P P rΔ α β α β= − δ = − ⋅∑ π (2.17a)

Force arising from surface tension

The surface tension will exert a force, γδl, on each element, δl, of the perimeter and acting tangentially to the surface at δl. The component in the z direction is γδl cos θ. But cos θ = rc/r, so the component in the z direction of the surface tension force on δl is γδlrc/r. The sum over the whole perimeter is therefore:

c

c c

2c

( ) /

(2 ) /

2 / .

zF l r r

r r r

r r

γ γ

γ π

π γ

= − δ

= −

= −

∑

(2.17b)

The negative sign is used because this force acts in the opposite direction to the force arising from the pressure difference.

Mechanical equilibrium

If the system is in mechanical equilibrium the forces in the z direction must sum to zero:

0Pz zF FγΔ + =

or

Derivation of the Laplace equationDerivation of the Laplace equation

Phase β

zF (ΔP)

δA

δA cos fPhase α

(a) (b)

g g

Pβ

Pα

fuu

F (g)r

rc

Fig. 2.17 (a) Forces on a spherical cap; (b) resolution of forces.


For a non-spherical interface two radii of curvature are needed. Figure 2.18 illustrates how the radii of curvature are defi ned. The two radii of curvature at a selected point (r′ and r″) are obtained by taking the normal to the surface at that point, drawing a plane containing the normal (in an arbitrary orienta-tion) taking a second plane containing the normal at right angles to the fi rst plane, observing the curves formed by the intersections of these planes with the surface and then fi nding the radii of the two circles drawn in these planes that have the same curvatures as these lines at the selected point.

By convention, positive values are assigned to the radii of curvature, r, r′, or r″, if they lie in phase α.

The Laplace equation then becomes:

m

1 1 2P P

r r rα β ⎛ ⎞− = + =⎜ ⎟′ ′′⎝ ⎠

γγ (2.18)

2 2c c( ) 2 / 0P P r r rα β− − =π π γ (2.17c)

and so:

2P P

rα β− = γ (2.17)

which is the Laplace equation for a spherical surface.

Plane throughnormal: black

Line on surfacein black plane

Line on surfacein blue plane

Circles in thetwo planes

Normal tosurface

Plane throughnormal:blue

r ′

r ′′

Fig. 2.18 Schematic illustration of the procedure for obtaining the radii of curvature at a selected point on a curved surface.

2.3 THE LAPLACE EQUATION | 29

where 1/rm is the mean curvature (inverse of the radius) defi ned by:

m

1 1 1 12r r r

⎛ ⎞= + ⋅⎜ ⎟⎝ ⎠′ ′′ (2.19)

2.3.1 Applications of the Laplace equation

The shapes of soap filmsA soap fi lm stretched across a fl at wire frame is fl at because the pressure is the same on both sides, but more complex shapes occur even when there is no dif-ference in pressure. For example, if the fi lm is formed as an open cylinder (as in Figure 2.19) a simple cylindrical shape is not possible as there is no pressure difference and this would therefore violate the Laplace equation. Instead, it will form into a ‘wicker basket’ shape where r′ = −r″ at all points on the surface.

An interesting application of the Laplace equation is seen when two bubbles of different size are connected by a tube, as in Figure 2.20. It is often surprising that the small bubble shrinks while the larger bubble grows, but is perfectly understandable in the light of the Laplace equation, which predicts that the pressure inside the smaller bubble is greater than that in the larger one.

This and other experiments with soap bubbles have been described by Boys (1911).

Capillary riseThe capillary rise phenomenon can also be treated using the Laplace equation. Balancing the pressure decrease under the curved liquid surface (assumed to be spherical and making a contact angle of θ with the capillary wall) in the capillary:

atm

c

2 2 cosP P

r rγ γ θ− = =

with the pressure decrease from the weight of the liquid column:atmP P ghρ− =

Wire frame

r ′′r ′

Soap film

Wire frame

Fig. 2.19 Sketch of a soap fi lm stretched between two circular wire frames. If the top and bottom ends are open, the air pressure will be the same on both sides of the fi lm so Eqn 2.17 requires that at any point on the surface r´ and r´´ must be equal, but with opposite signs.

Three-waytap

Fig. 2.20 Illustration of the Laplace equation. When the tap is turned to connect the two bubbles (as in the lower fi gure), the smaller bubble shrinks and the larger bubble grows until the curvatures of both bubbles are the same. That is, the fi lm over the end of the left tube has the same curvature as the large bubble.


gives:

c

2cosghrργ

θ= (2.20)

for a capillary of radius rc. It is diffi cult to measure and to reproduce θ if θ ≠ 0 so only systems with θ = 0 are used in practice. Deviations from the spherical shape are a complication for accurate measurements of surface tension and several correction procedures have been devised. These are described in detail by Adamson and Gast (1997).

Measuring surface tension by maximum bubble pressureIt was stated in Section 2.2.4 that the maximum bubble pressure occurs when the bubble radius is the same as the capillary radius, rc. At this point the radius of curvature of the bubble has its minimum value so according to the Laplace equation the pressure must be the maximum. For this situation we have:

c

2P rγ Δ= (2.21)

where the pressure term is the difference between the pressure applied to the capillary and atmospheric pressure corrected for the small hydrostatic pres-sure arising from the depth of immersion.

The Kelvin equationThe Laplace equation forms the theoretical basis for the Kelvin equation, which describes the effect of surface curvature on vapour pressure. It is dis-cussed in some detail in Section 2.4.

2.4 The Kelvin equation

An important consequence of the Laplace equation concerns the effect of sur-face curvature on the vapour pressure of a liquid. This relationship is known as the Kelvin equation:

c L

m

2ln

p Vp RT r

γ∞

⎛ ⎞ ⎛ ⎞⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ (2.22)

where pc and p∞ are, respectively, the vapour pressures over the curved sur-face of effective radius rm (see Eqn 2.19) and a fl at surface (r = ∞), and V is the partial molar volume. As can be seen from the derivation below, there is a convention of signs for rm that assigns a positive sign to the radius when it lies in the liquid phase and a negative sign when it lies in the vapour phase.

2.4.1 Consequences of the Kelvin equation

The Kelvin equation has many important consequences as it provides expla-nations for such phenomena as the diffi culties found in self-nucleation of a new phase, the growth of large droplets at the expense of smaller ones, and condensation in capillaries.

2.4 THE KELVIN EQUATION | 31

At a curved interface the condition for mechanical equilibrium is given by the Laplace Eqn 2.18:

m

2p p

rα β− = γ

(2.22a)

where α is the phase on the concave side. We general-ize by giving r a positive sign if it lies within phase α, or a negative sign if it lies within phase β.

For physico-chemical equilibrium:

.i i iμ μ μα β= = (2.22b)

If the equilibrium is shifted slightly with adjustments to pα, pβ, rm, and μi, we obtain:

m2 (1/ )p p rα βδ − δ = δγ (2.22c)

and

.i i iμ μ μα βδ = δ = δ (2.22d)

Since

, ,T n Ai

Vpμ⎛ ⎞∂

=⎜ ⎟∂⎝ ⎠ (2.22e)

we have:

, ,

, .T n T ni i

V Vp pμ μα β

α βα β

βα

⎛ ⎞ ⎛ ⎞∂ ∂= =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠ (2.22f)

If the process is carried out at constant temperature and without transfer of material across the interface then Eqns 2.22d and 2.22f give:

.i V p V pμ α α β βδ = δ = δ (2.22g)

Substitution of (2.22g) into (2.22c) yields:

m2 (1/ ) ( ) /r p V V Vβ β β αδ = δ − ⋅γ (2.22h)

If β is the vapour phase,

V Vβ α>>

and (2.22h) becomes:

m2 (1/ ) ( / ) /ir p V V Vβ β α αδ = δ = δγ μ (2.22i)

where δμi refers to the process where the radius is changed. Integration of Eqn 2.22i from a fl at surface

m(1/ 0, )i ir μ μ∞= = to any selected curvature cm(1/ , )ir μ

yields:

c Lm2 (1/ ) ( ) /i ir Vγ μ μ∞= −

orc L

m(2 / )i i V rμ μ γ∞− = (2.22j)

where superscript α has been replaced by L to indi-cate the liquid phase. Equation 2.22j is the general form of the Kelvin equation.

Note the approximation between Eqns 2.22h and 2.22i, and the assumption that the liquid is incom-pressible when integrating Eqn 2.22i.

If the vapour behaves as an ideal gas,

μi = μi⦵ + RT ln(p/p⦵)

and

μic − μi

∞ = RT ln(pc/p⦵) (2.22k)

Equation (2.22j) now becomes:

c L

m

2ln

p Vp RT r

γ∞

⎛ ⎞ ⎛ ⎞⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ (2.22)

which is the more usual form of the Kelvin equation.

Derivation of the Kelvin equationDerivation of the Kelvin equation

Droplet growthFor a spherical droplet in contact with its vapour, the two principal radii are equal, lie within the liquid phase and, thus, have positive signs. Hence, accord-ing to the Kelvin equation (2.22) the vapour pressure of the droplet will be greater than that of the same liquid with a fl at surface. There is, for exam-ple, approximately a 10% increase in vapour pressure for a water droplet of radius 10 nm (see Example 2.5). The smaller the radius, the higher the vapour pressure so that if there are droplets of various sizes present the smaller ones will tend to evaporate, while the larger ones will tend to grow. An important example occurs in clouds where the larger droplets grow until they are heavy enough to fall as rain.


Problem Calculate the vapour pressure of spherical water droplets of radius10 nm and 100 nm. The temperature is 280 K and the surface tension of waterat this temperature is 74.6 mN m–1, and the molar volume is 18.01×10–6 m3.

Solution Applying the Kelvin equation (2.22) we have for the 10 nm droplet:

L 3 1 6 3

1 1

74.6 10 N m 18.01 10 m3 1 6 33 1 6

ln 8.31JK mol 280K1 11

0.115

RT

3 113 1

8 31JK 1 11RT

⎛ ⎞LLV ⎛ ⎞2⎛ ⎞cpc ⎛ ⎞210 N m 18.0110 N m 18.013 13 13 1

= =⎜ ⎟⎞⎞

⎜ ⎟⎛ ⎞⎛ ⎞

⎜ ⎟p⎛⎛ p

⎜ ⎟⎛ ⎞⎛ ⎞

280K ⎝ ⎠10 10 m⎜ ⎟⎜ ⎟910 m91010⎝ ⎠p⎜ ⎟⎜ ⎟p ⎝ ⎠mrm⎟⎟r⎜ ⎟⎜ ⎟⎝ ⎠RTRTRTRTRTRT⎜ ⎟⎜ ⎟RTRTRTRTRTRTRTRT

=

giving

c / exp0.115 1.12p pc / =exp0 115 .

Similarly, for the 100 nm droplet:

c / exp0.0115p pc / =exp0 0115 1.012.

There is clearly only a small increase in vapour pressure once the droplet radiusexceeds 100 nm.

Example 2.5 Vapour pressure of liquid dropletsExample 2 5 Vapour pressure of liquid droplets

A similar mechanism is thought to exist for crystals in a solution: smaller crystals having higher solubilities than larger ones. Consequently the larger crystals tend to grow at the expense of smaller ones leading to the process of digestion or Ostwald ripening, which is used to increase effi ciency in analyti-cal and industrial fi ltration.

It is worth noting that the equilibrium between a small liquid droplet and its vapour is an unstable one. If the drop, originally at equilibrium, loses a few molecules and decreases in size its vapour pressure will increase above the actual vapour pressure, and the droplet will evaporate and continue to evaporate. If, on the other hand, a few more molecules collide with the droplet increasing its size, its vapour pressure will drop below the actual vapour pres-sure so that condensation will continue and the droplet will grow.

Self-nucleation of a new phaseIn self-nucleation or homogeneous nucleation of a new phase, the fi rst step is the formation of very small nuclei or embryos of the new phase inside the old phase. Examples include the formation of minute liquid clusters in a supersaturated vapour and the formation of minute bubbles of vapour in a superheated liquid. However, if the experimental conditions are only marginally in favour of the new phase (e.g. if the supersaturation or super-heating is only slight) the Kelvin equation shows that stable embryos would need to be large and unlikely to form. Smaller embryos might form, but they would be unstable and tend to disappear. When the conditions more strongly favour the new phase the critical size given by the Kelvin equation will be smaller and the probability that some embryos may exceed that size will increase. They are then termed nuclei and will be stable and capable of further growth.


The formation of embryos of the critical size (critical nuclei) is thus a ques-tion of probability, rather than thermodynamics. The treatment is complex and beyond the scope of this book, but an extended discussion is given by Defay et al. (1966, Chapter 18). So although the Kelvin equation gives the critical droplet size for a given value of pc/p∞ it does not indicate whether such a droplet is likely to form under the prevailing conditions.

William Thomson, the fi rst Baron Kelvin, was born in Belfast in 1824. His father, Dr. James Thomson, taught mathematics and engineering and in 1832 was appointed Professor of Mathematics at Glasgow University. William studied fi rst at Glasgow and then at Peterhouse, Cambridge, and at age 22 was appointed Professor of Natural Philosophy at Glasgow, a post he held for more than 50 years. His interests were broad, encompassing particularly electricity, thermodynam-ics, marine, and engineering. He received many hon-ours, published more than 600 scientifi c papers and fi led 70 patents. He died in 1907 and because there was no heir the title died with him.

He was knighted for his work on the design and lay-ing of the fi rst telegraph cable across the Atlantic. A keen yachtsman, his marine interests led to improvements in ships compasses, timekeepers, and depth sounding.

The relationship between work and heat was a contentious topic and Thomson was a major participant, eventually working with Joule on the problem. Thus, he contributed greatly to the development of the fi rst and second laws of ther-modynamics. For this and other work on thermo-dynamics he was created Baron and chose as his title the name of the river that fl owed past Glasgow University, the Kelvin.

He was dissatisfi ed with the measurement of tem-perature and considered that the scale should be such that the transfer of heat from one body to another body at a temperature one unit less should do the same amount of work irrespective of the temperatures of the experiment. This led ultimately to the concept of absolute zero and to his name being attached to the unit of temperature.

Lord KelvinLord Kelvin

Problem Calculate the minimum size for a bubble of water vapour forming 10 cm beneath the surface of water at 375 K that would remain stable. At this temperature the density of water is 957 kg m–3; the surface tension is 58.5 mN m–1; and the vapour pressure is 108.8 kPa.

Solution We fi rst need to calculate the applied pressure at the depth of the bubble. This is the atmospheric pressure plus the hydrostatic pressure (ρgh) from the water. It equals the vapour pressure that the curved surface of the bubble would have to develop, i.e. pc:

c 3 2 2101325Pa (957 kg m 9.81ms 10 10 m) 102263Pa3 2 23 2

102.26kPa.

p 3 223 2= 101325Pa (957 kg m 9.81ms 10 10 m)(957 kg m 9.81ms 10 10 m)3 2 23 23 2

=

Rearranging the Kelvin equation (2.22), and noting that /V M ρ, gives:

c

m

3 1 1

3 1 1

2ln

2 58.5 10 N3 m 01 .018kg mol 102.3ln

957 kg m 83 .31JK m1 ol 375K 108.8

M plnrm RT p

γρ ∞

3 −

3 −

=

× 58.5=×18.31JK m1 ol

Example 2.6 Bubble nucleation in boiling waterExample 2 6 Bubble nucleation in boiling water


Capillary condensationCapillary condensation occurs when the adsorption of a vapour in a capillary forms a liquid surface with a very small radius of curvature. The radii then lie in the vapour phase and, consequently, the vapour pressure of the liquid is lower than that of the same liquid with a fl at surface. Condensation, rather than adsorption will occur if the actual vapour pressure is higher than the vapour pressure calculated from the Kelvin equation for the curved surface. This actual vapour pressure may well be lower than the saturation vapour pressure for a fl at surface. This aspect will be discussed in more detail in Section 8.6.

Heterogeneous nucleationIn most practical situations, solid surfaces, such as the vessel walls, dust par-ticles, boiling chips, and so on, provide sites where heterogeneous nucleation of a new phase can occur under less extreme conditions than those required for homogeneous nucleation.

giving:

rm = –11.5 nm.

The negative sign arises because the logarithmic term is negative and indicates correctly that the radii lie in the vapour phase. We note also that it is unlikely that a bubble of this size would form. A higher temperature would be needed before self-nucleated boiling could occur.

Water droplets in clouds are often below the freezing point of water, but are liquid nevertheless. It has been estimated that the self-nucleation of ice in these supercooled droplets will not occur unless the temperature is below about –40oC. However, if at somewhat higher temperatures, some of the droplets do freeze, their vapour pressures are then lower than those of the remaining liquid droplets and, consequently, water moves through the vapour phase from the liquid droplets to the ice crystals. Hence, these crystals grow until they fall out of the cloud, often melting in warmer air into rain drops.

Freezing of supercooled droplets may be induced naturally by dust parti-cles, but it can also be initiated artifi cially. Silver iodide has a crystal structure that is reasonably similar to that of ice and can act as a nucleating material (Vonnegut, 1949). It has been shown that water adsorbed on silver iodide crystals freezes at only –4oC. Thus, rain can be generated by introducing silver iodide particles into supercooled clouds. The silver iodide particles can be formed by burning a solution of the salt in a fl ammable solvent and this should occur near the site of action, either on an aircraft fl ying through the cloud or from ground-based generators situated where there is a strong updraft (as at the foot of a mountain range).

Making rainMaking rain


Problem A particular ‘ink bottle’ pore in a solid consists of a cylindrical tube of radius 16 nm leading to a spherical cavity of radius 41 nm. Calculate the relative pressure at which capillary condensation of nitrogen at 77 K (the boil-ing temperature at atmospheric pressure) would occur in each section. In this experiment, the gas pressure is being gradually increased, so we will assume that there has been prior adsorption covering the surfaces of both sections with a 1 nm layer of nitrogen (approximately 2 layers and suffi cient to give a liquid-like surface). Describe the process of capillary condensation in the pore. The surface tension of nitrogen liquid at this temperature is 8.85 mN m–1 and the molar volume is 34.7 cm3 mol–1; atmospheric pressure is 101.3 kPa.

Solution Applying the Kelvin equation to the cylindrical section we fi rst adjust the radius for the thickness of the adsorbed gas layer. We then have a hollow cylinder of liquid, and for the purposes of the Kelvin equation one radius will be the internal radius of this cylinder and the other will be the radius in the plane of the cylinder axis, and thus infi nite. Hence:

3 1 6 3 1

1 1

ln

8.85 10 N m 34.7 10 m mol3 1 6 3 11 6 3

8.31J K mol 77K1 11

0.0320.

RT3 1 6 31 6 31 6 3

1 11

⎛ ⎞LV ⎛ ⎞1 1⎛ ⎞cpc

=⎞⎞⎞⎞

⎟⎛ ⎞⎛ ⎞

+⎛⎛

⎜ ⎟p⎛⎛ p

⎝ ⎠p⎜ ⎟⎜ ⎟p ⎝ ⎠1 2r r1 2⎜ ⎟⎜ ⎟r r

+⎝ ⎠RTRT⎜ ⎟⎜ ⎟RTRTRTRT

⎛ ⎞1 110 N m 34.710 N m 34.73 11

= 1 1 ⎟⎞⎞

+⎛⎛

77K ⎝ ⎠15 10 m m⎟⎟9 m m9 +915 10 m15 10 m15 10 m10 m9

= −

Although we are given the atmospheric pressure (= p∞), we are mainly inter-ested in the relative pressure (= pc/p/ ∞) [see Figure 8.3 (po ≡ p∞)]:

c / 0.726p pc / .

For the spherical cavity the two radii are the same, so:

3 1 6 3 1

1 1 9 9

8.85 10 N m 34.7 10 m6 mol 11 1ln

8.31J K mol 71 7K 40 10 m 40 10 m9

0.0240

−3 N 36

−8 31J K −40 1071 7K

⎛ ⎞cp ⎛ ⎞× ×3 110 N m +9

⎛= 1 1⎜ ⎟

p⎛ ⎞⎛ ⎞p ⎛⎛⎛⎛⎟⎞⎞

−77K 9× 10 m⎝⎜⎜⎜⎜ ⎠⎟⎟⎝ ⎠p⎜ ⎟⎜ ⎟p∞

= −

giving:

c / 0.976p pc / .

Thus, as the gas pressure is increased, capillary condensation will occur fi rst in the narrow cylindrical neck of the pore. Once this process has begun the radius of the liquid surface within the pore will decrease, so that capillary condensa-tion will continue rapidly. This will lead to fi lling of the neck and blocking of access to the spherical cavity.

Example 2.7 Capillary condensationExample 2 7 Capillary condensation


2.4.2 Verifi cation of the Kelvin equation

Not surprisingly, an equation such as the Kelvin equation is diffi cult to verify experimentally. In order to do so, it is necessary to measure both the vapour pressure and the surface curvature of a liquid with a highly curved surface. Convincing data have been obtained by La Mer and Gruen (1952) using monodisperse aerosols. Details of this procedure are given in Section 4.11.4.

Another approach uses the surface forces apparatus described in more detail in Section 9.9.6 (Fisher and Israelachvili, 1981). In this apparatus, the forces of interaction between two crossed cylinders of molecularly smooth mica can be measured as a function of separation. If the cylinders in contact with one another are exposed to the vapour of a test liquid, capillary con-densation may occur at the junction if the vapour pressure is high enough. This vapour pressure can be controlled by setting the temperature of the bulk liquid or adding a non-volatile solute. With an annulus of condensed liquid at the junction, the cylinders are slowly separated so that a bridge of liquid forms between the mica surfaces. When this bridge suddenly evaporates the separation is measured and from this rm can be calculated. It was found that the Kelvin equation held down to values of rm as low as 2.5 nm for several organic liquids. This result also indicates that macroscopic values of γ and ρ hold at such low dimensions.

2.5 The surface tension of pure liquids

2.5.1 Eff ect of temperature on surface tension

The chief factor affecting the surface tension of a pure liquid is the tempera-ture (for example, Figure 2.21). Experimentally it is found that the surface tension of a pure liquid decreases nearly linearly with temperature. It must, of course, be zero at the critical temperature, where the interface disappears.

80

75

70

65

Sur

face

tens

ion,

g/m

N m

−1

60

55260 280 300 320

Temperature, T/K

340 360

Air–waterinterface

380

Fig. 2.21 The surface tension of the air–water interface as a function of temperature. (Data from Vargaftik et al., 1983).

2.5 THE SURFACE TENSION OF PURE LIQUIDS | 37

For some liquids the empirical Eötvös equation describes the dependence of surface tension on temperature:

( )2 / 3

7 2 / 3 1d ( / )

2.12 10 J mol Kd

M

T

γ ρ− − −= − × (2.23)

where M is the molar mass and ρ is the density.

The effect of temperature on surface tension can be nicely demonstrated by spreading a thin fi lm of water on a metal plate and holding a piece of ice against the underside of the plate. Water migrates to the area above the ice, producing an observable thickening of the water layer and if the ice is moved slowly around the thick region moves with it. This is an example of the Marangoni effect, attributed by Thomson (1855) and Marangoni (1871, 1878) to gradi-ents in surface tension, in this instance arising from the temperature gradient. A related phenomenon is discussed in Section 4.10.

The Marangoni eff ectThe Marangoni effect

2.5.2 The surface tension of liquids

Values for the surface tensions of some pure liquids are given in Table 2.2.The very high surface tension of mercury, typical of values for metals, is

attributable to short-range electron exchange interactions, the origin of so-called metallic bonds.

Table 2.2 The surface tensions of some common pure liquids at the vapour–liquid interface at 20°C (from Jasper, 1972)

Liquid Surface tension, γ/mN m−1

Liquid Surface tension, γ/mN m−1

Acetic acid 27.6 n-Hexane 18.4

Acetone 25.1 Isobutyl alcohol 22.9

Benzene 28.9 Methanol 22.5

n-Butyl alcohol 25.4 Mercury 486.5

Carbon tetrachloride 27.0 n-Octane 21.6

Chloroform 27.2 Oleic acid 32.5

Cyclohexane 25.2 Propanoic acid 26.7

Ethyl acetate 24.0 n-Propyl alcohol 23.7

Ethanol 22.4 Pyridine 37.2

Di-ethyl ether 17.0 Toluene 28.5

Glycerol 63.4 Vinyl acetate 24.0

Ethylene glycol 48.4 Water 72.8


It is notable that the surface tension of water is signifi cantly higher than the surface tensions of all of the organic liquids. Only glycerol has a surface ten-sion approaching that of water. Water and glycerol (plus some other liquids not shown in Table 2.2) are highly polar hydrogen-bonding liquids and it is this feature that explains the high surface tensions. As water is a major com-ponent in most aspects of interface science, we will discuss it in detail below.

2.5.3 The hydrophobic–hydrophilic interaction

The structure of the water molecule is well known, but it is important to con-sider here the distribution of electric charge in the molecule as this is the key to understanding its interactions with other molecules. The basic confi guration is shown in Figure 2.22.

Water molecules tend to form hydrogen bonds with four other water molecules, giving a tetrahedral arrangement, with each molecule providing the hydrogen atoms for two of the bonds and with the hydrogen atoms in the other two bonds coming from neighbouring water molecules. All four hydrogen bonds are approximately linear. In ice, this structure dominates, but although it tends to be retained in liquid water it is somewhat disordered and labile. Nevertheless, the hydrogen bonds in liquid water have a strong tendency for directionality and linearity. It is also important to note that the strength of hydrogen bonds (10–40 kJ mol−1) is appreciably greater than for typical van der Waals attractions (~1 kJ mol−1), but much less than for cova-lent and ionic bonds (~500 kJ mol−1) (Israelachvili, 1991, p. 127).

The charges on water molecules and their strong tendency to form hydro-gen bonds govern their interactions not only with other water molecules, but with other materials as well.

If the other material is an ion, a polar compound, or a charged surface the neighbouring water molecules are able to arrange themselves with their charges of opposite sign towards the other material. This is the process of hydration. Energetically, this is a very favourable arrangement. Solid surfaces at which there is extensive hydration are described as hydrophilic. In addition to these charged or polar entities, there are certain atoms that readily form hydrogen bonds when in appropriate compounds. Oxygen in alcohols and polyethylene oxide chains, and nitrogen in amines are important examples. These can interact with water by forming hydrogen bonds.

H + q = 0.24 e

H

O

−q

−q

+ q

l = 0.1 nm

= 109°

l' = 0.08 nm

Fig. 2.22 Sketch of water molecule showing the tetrahedral distribution of charge with bond lengths and angles. The charges (q) on the left represent the charges attributable to the unshared electron pairs. e is the electronic charge. Based on the model of Stillinger and Rahman (1974).


If, conversely, the other material is non-polar and incapable of forming hydrogen bonds the water molecules arrange themselves to minimize the number of unused potential hydrogen bonds. When the other molecule is small, the water molecules are able to arrange themselves into a cage-like structure around that molecule and so utilize all four potential hydrogen bonds per water molecule, forming a clathrate-like compound. This results in a decrease in entropy, as the water molecules are more ordered than in the bulk liquid, an effect that restricts the solubility of the solute. Important examples for the present context are the hydrocarbons and some long-chain hydrocarbon derivatives. For the hydrocarbons the entropy decrease is roughly proportional to the area of surface exposed to the water. Such sub-stances are described as hydrophobic.

When two hydrophobic molecules encounter one another in an aqueous solution there would obviously be a decreased area of exposed hydropho-bic surface if the two molecules came together and, consequently, a smaller entropy decrease. There is, therefore, an entropically driven tendency for such molecules to aggregate: a process known as hydrophobic interaction. The description of this attraction as a hydrophobic bond is not appropriate as no actual bond is formed and the attraction may occur over longer distances than for typical bonds.

At a solid or liquid surface formed from hydrophobic materials the water molecules must organize themselves to minimize the number of unused hydro-gen bonds. Thus, a drop of water on a hydrophobic surface tends to roll into a ball with a high contact angle (see Section 2.1.3) thereby minimizing contact with the surface and maximizing the formation of hydrogen bonds with other water molecules. The high surface tension of water can also be attributed to this effect as the air essentially forms a hydrophobic surface.

When a hydrophilic group and a hydrophobic group are combined in the one molecule we have a class of compounds now commonly referred to as amphiphiles (from Greek, meaning loves both, referring to water and to oil). In these molecules the presence of the hydrophilic group modifi es the hydro-phobic nature of the attached hydrophobic group. In long-chain alcohols, for example, the hydrophilic OH group signifi cantly disrupts the tendency for a clathrate structure to form around the alkyl chain (Israelachvili, 1991, p. 135). Thus, while insoluble monolayers can be formed with tetradecanol and longer chain alcohols, indicating negligibly low solubility, the shorter alcohols, methanol, ethanol, and propanol are fully miscible with water, with the solubility then decreasing rapidly with chain length. A detailed discussion of the behaviour of amphiphilic molecules is given in Section 4.6.

The concept of surface or interfacial tension has been introduced to explain the observa-tion that surfaces and interfaces tend to contract to the minimum area. Surface tension is defi ned as the force per unit length acting on an imaginary line drawn in the surface in a direction away from the line and tangential to the surface (Eqn 2.1). Surface tension can also be equated to the work of extending the surface by unit area, a defi nition that will lead to the thermodynamics of surfaces as discussed in Chapter 3.

SUMMARY


Such tensions control the wetting of a solid surface by a drop of liquid, the adhesion between two phases, the pull of a liquid surface on a wetted solid dipping into it, the rise of liquid up a capillary tube, and other phenomena. Methods for measuring surface and interfacial tension are described.

When a surface is curved there is a diff erence in pressure on the two sides of the sur-face that is described by the Laplace equation (Eqn 2.17). The Laplace equation is used to describe such phenomena as the shapes of soap fi lms and capillary rise. It also leads to the Kelvin equation (Eqn 2.22), which describes how the curvature of a liquid surface changes the equilibrium vapour pressure of the liquid. The consequences of the Kelvin equation are profound: it governs such processes as the growth of droplets and crystals, the self-nucleation of a new phase, and capillary condensation.

Finally, the surface tension of pure liquids is discussed. Water has a signifi cantly higher surface tension than most organic liquids and this is attributable to the formation of hydrogen bonds. Furthermore, it is the possibilities or otherwise for hydrogen bonding that determine the nature and strength of the interactions of liquid water with solutes and with surfaces.

Adamson, A.W. and Gast, A.P. (1997). Physical Chemistry of Surfaces, 6th edn. Wiley, New York.

Boys, C.V. (1911). Soap Bubbles: Their Colours and the Forces Which Mould Them. Republished by Dover Publications, New York, 1959. Describes many simple experiments with soap fi lms designed to generate interest and at the same time impart some science.

Defay, R., Prigogine, I., Bellemans, A., and Everett, D.H. (1966). Surface Tension and Adsorption. Longmans, London.

Fletcher, N.H. (1962). The Physics of Rain Clouds. University Press, Cambridge.

Israelachvili, J.N. (1991). Intermolecular and Surface Forces, with Applications to Colloidal and Biological Systems, 2nd edn. Academic Press, London. Provides a somewhat different perspective on surface phenomena.

Lewis, G.N., Randall, M., Pitzer, K.S., and Brewer, L. (1961). Thermodynamics, 2nd edn. McGraw Hill, New York.

Lyklema, J. (2000). Fundamentals of Interface and Colloid Science, III, Liquid–Fluid Interfaces. Academic Press, San Diego.

Everett, D.H. (1972). Pure Appl. Chem. 31, 597.

Fisher, L.R. and Israelachvili, J.N. (1981). J. Colloid Interface Sci. 80, 528.

Gaines, G.L. (1977). J. Colloid Interface Sci., 62, 191.

Jasper, J.J. (1972). J. Phys. Chem. Ref. Data, 1, 841.

Lamb, R.N., Zhang, H., and Raston, C.L. (1997). Hydrophobic Coatings. WO 9842452.

La Mer, V.K. and Gruen, R. (1952). Trans. Faraday Soc. 48, 410.

FURTHER READING

REFERENCES


Lane, J. and Jordan, D.O. (1970). Aust. J. Chem., 17, 7.

Lee, S., Kim, D.H. and Needham, D. (2001). Langmuir 17, 5537.

Marangoni, C. (1871). Nuovo Cimento [2], 16, 239.

Marangoni, C. (1878). Nuovo Cimento [3], 2, 97.

Stillinger, F.H. and Rahman A. (1974). J. Chem. Phys., 60, 1545.

Thomson, J. (1855). Phil. Mag., 10, 330.

Timblin, L.O., Florey, Q.L., and Garstka, W.U. (1962). In V. K. La Mer, (ed.), Retardation of Evaporation by Monolayers: Transport Processes. Academic Press, New York.

Vargaftik, N.B., Volkov, B.N., and Voljak, L.D. (1983). J. Phys. Chem. Ref. Data, 12, 817.

Vonnegut, B. (1949). Chem. Rev. 44, 277.

Wilhelmy, L. (1863). Ann. Physik., 119, 177.

Zhang, H., Jones, A.W., and Lamb, R.N. (2000) Hydrophobic coating material containing modifi ed gels. WO 2001014497.

EXERCISES

2.1 Calculate the work required to stretch a soap fi lm so that its area increases by 4.0 cm2. The surface tension of the air–soap solution interface is 28 mN m−1. [w = 22.4 μJ].

2.2 Describe the spreading behaviour you would expect when a drop of benzene is placed on a clean water surface. For the pure liquids the interfacial tensions (in mN m−1) are:

• Air–water 72.8• Air–benzene 28.9• Benzene–water 35.0

However small amounts of each liquid slowly dissolve in the other and this affects the interfacial tensions which then become:

• Air–water 62.2• Air–benzene 28.8• Benzene–water 35.0

[S0 = 8.1 mN m–1; Sf = –1.6 mN m–1]

2.3 A spherical drop of oil of radius 1.0 cm is dispersed as an emulsion in water. The oil particles in the fi nal emulsion are spherical and have a radius of 0.1 mm. Calculate the work required for the hypothetically reversible dispersion process assuming that the interfacial tension remains constant at 30.0 mN m−1 throughout. [w = 3.78 J]

2.4 Derive an expression for the surface tension when the difference in capillary rise is measured with two capillaries of different radii.

2.5 A syringe needle (circular in section) with a sharp tip of radius r is used to form air bubbles beneath the surface of a solution of surface tension γ. Derive an expression for the maximum pressure, p, observed during the slow formation of an air bubble at the tip of the needle. Assume that the bubble is a partial sphere.

EXERCISES


Show by a qualitative argument that the formula does indeed correspond to the maximum pressure.

2.6 Use the Kelvin equation to calculate the radius of an open-ended tube within which capillary condensation of nitrogen at 77 K and a relative pressure of 0.75 might be expected. Assume that prior adsorption of nitrogen has formed a layer 0.9 nm thick coating the inside of the tube. For nitrogen at 77 K the surface tension is 8.85 mN m−1 and the molar volume is 34.7 cm3 mol−1. [rc = 2.57 nm]

2.7 A jet aircraft is fl ying through a region where the air is 10% supersaturated with water vapour (i.e. the relative humidity is 110%). After cooling, the solid smoke particles emitted by the jet engines adsorb water vapour and can then be considered as minute spherical droplets. What is the minimum radius of these droplets if condensation is to occur on them and a ‘vapour trail’ form? Data:

γ (H2O) = 75.2 mN m−1;

M(H2O) = 0.018 kg mol−1;

ρ(H2O) = 1030 kg m−3;

T = 275 K. [r = 12.0 nm].

2.8 Following the procedure presented in Example 2.6, calculate the minimum mean radii of curvature for bubbles forming at a depth of 5 cm in water at 378 K and 383 K. The relevant data are:

T for 378 K: ρ = 955 kg m–3; γ = 57.9 mN m–1; p∞ = 120.8 kPa; [- 4.06 nm]

T for 383 K: ρ = 951 kg m–3; γ = 57.0 mN m–1; p∞ = 143.2 kPa. [- 1.99 nm]

2.9 In deriving Eqn 2.20, the radius of curvature of the liquid surface in the capillary, r, is equated to rc/cos θ. Draw a diagram of this situation and prove this relationship.

2.10 Draw an imaginary profi le of the roughened surface of a solid and show by a series of sketches how a drop sitting on the surface can exhibit hysteresis in its macroscopic contact angle while retaining the same microscopic contact angle.

Surface Tension Concepts

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