Capillarity and Surface Mechanics PDF

## Capillarity and the mechanics of surfaces ### Surface tension and work - Definition of surface tension - Work of extension - Contact angle, wetting, and spreading - The surface of tension - Work of adhesion and cohesion - Measurement of surface tension - Wilhelmy plate - Capillary rise...

## Capillarity and the mechanics of surfaces ### Surface tension and work - Definition of surface tension - Work of extension - Contact angle, wetting, and spreading - The surface of tension - Work of adhesion and cohesion - Measurement of surface tension - Wilhelmy plate - Capillary rise - Drop weight or volume - Maximum bubble pressure - 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 Verification of the Kelvin equation - 2.5 The surface tension of pure liquids - 2.5.1 Effect 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 containing 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 remaining in isolated drops on others. In this chapter, surface tension is defined and we see how it can be used to develop simple theories to explain these experiences. #### 2.1.1 Definition 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 sufficient to make the bristles stick to one another. However, if the brush is held completely under water the bristles separate (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. 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 stickiness 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 inflated balloon (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 forming a hole. The balloon is only an analogy, but something similar happens at the surfaces of interest to us. At any surface, if the force, F, acting tangentially to the surface and at right angles to an element, dx, of an imaginary line in the surface has a magnitude that is independent of the direction of the element, then the surface tension, $\gamma$, is: $\gamma = \frac{F}{\delta x}$ In words, the surface tension is the force per unit length acting on an imaginary line drawn in the surface. The SI units of surface tension are N m¯¹, although because the N m¹ 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. An intuitive way to understand the origin of surface tension from a molecular point of view is to consider the forces acting on a molecule at the surface of a liquid compared to those acting on one in the bulk (Figure 2.4). The attractive 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 direction. 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 film contained by a rectangular frame with one end capable of sliding along the sides (Figure 2.5) exerts a force, F, on the slide. The length of the line of contact between soap film and slide is twice the length of the slide [i.e. 2(+x) = x ] because the soap film has two sides. That is, if you think of the film 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 by then the area of surface (considering both sides of the film) increases by dA (= x dy) and the force exerted by the surface tension and resisting the extension is, from Eqn 2.1: $F = \gamma x$. The work (w) of extension is therefore: $w = F dy = \gamma x \delta y = \gamma \delta A$. 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 tension. The interfacial tension is one component of the interfacial stress, but the 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 surface 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 discussed 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). **Example 2.1 Work of extending a soap film** **Problem** Calculate the work of extending a soap film supported in a wire frame as in Figure 2.5 by 1.5 cm². The surface tension of the air-solution interface is 35 mN m¯¹. **Solution** We must remember that the soap film has two sides so the total increase in surface area is 3.0 cm². Applying Eqn 2.2 we have: $w = 35 \times 10^{-3} Nm¯¹ \times 3 \times (10^{-2}m)² = 105 \times 10^{-7} Nm$ $= 10.5 \mu Nm = 10.5 \mu J$ Note that the work (a form of energy) is positive because energy has to be put into the system. #### 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 flat solid surface (S) with air (G) as the third phase. The angle, $\theta$, 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 measured 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 tensions will be in balance and thus: $\gamma_{GS} = \gamma_{LS} + \gamma_{GL} cos\theta$. However, when the drop is initially placed on the surface the interfacial tensions will not be in equilibrium. The net force per unit length of the triple interface along the solid surface will then be: $F_1 = \gamma_{GS} - \gamma_{LS} - \gamma_{GL} cos\theta'$ where forces acting to the right are given positive signs and those acting to the left have negative signs. The angle $\theta'$ is the instantaneous contact angle and will change as the triple interface moves towards its equilibrium position where $\theta' = 0$ and $F_1 = 0$. 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 certain 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 unmeasurable. 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 surface that has primarily polar groups on the surface, such as hydroxyl groups, will have a good affinity 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 performed on a drop of increasing size (the advancing contact angle) or of diminishing size (the receding contact angle). This is normally due to roughness of the surface, and the difference between advancing and receding contact 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 difficult part of dealing with very high contact angles is keeping the drops in place long enough to be measured! The slightest tilt on the surface is enough to cause the drops to roll off. Wetting is determined by the equilibrium contact angle, $\theta$. If $\theta < 90°$, the liquid is said to wet the solid; if $\theta = 0$, there is complete or perfect wetting; if $\theta > 90°$ (cos $\theta$ < 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 $F_1$, until a value of $\theta$ is reached at which $F_1 = 0$, giving equilibrium, or, if such a position is not possible, the liquid spreads completely, and then $\theta' = 0$, cos $\theta' = 1$, and Eqn 2.4 becomes: $F_1 (\theta' = 0) = \gamma_{GS} - \gamma_{LS} - \gamma_{GL}$ It is convenient to define a spreading coefficient, $S_{LS}$, by $S_{LS} = \gamma_{GS} - \gamma_{LS} - \gamma_{GL}$. If $S_{LS} > 0$, the liquid spreads completely, whereas if $S_{LS} \leq 0$ the drop does not spread completely and it finds an equilibrium contact angle, $\thetaº$, where $F_1 = 0$. #### 2.1.4 The surface of tension It has already been pointed out that any real interface has a finite thickness, but for many purposes it is convenient to replace the real interface by a mathematical 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 finite 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 pattern 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 equations 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 (a and ẞ) 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, $w_{\alpha \beta}$. There is a contribution from each interface removed from or added to the system: $w_{\alpha \beta} = -\gamma_{\alpha \beta} + \gamma_{\alpha \omega} + \gamma_{\beta \omega}$. If, instead of two distinct phases, a column of a single liquid is pulled apart, the work of cohesion is: $w_{\alpha \alpha} = 2\gamma_{\alpha \omega}$ We note that in Eqns 2.7, 2.8, and 2.9 the definition 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: $w_{LS} = \gamma_{GL} + \gamma_{GS} - \gamma_{LS} = \gamma_{GL} (1 + cos\theta)$. This is commonly known as the equation of Young (1805) and Dupré (1869). Its significance is that it relates the work of adhesion to the readily measured quantities, $\gamma_{GL}$ and $\theta$, rather than to the inaccessible interfacial tensions involving the solid surface. **Example 2.2 Work of adhesion of a liquid on a solid** **Problem** Calculate the work of adhesion of water on four solids, where the equilibrium contact angles are 30°, 60°, 120°, and (a hypothetical) 180°. 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 first solid ($\theta = 30°$): $w^{LS} = 72 \times 10^{-3}Nm¯¹ (1 + cos30) = 134 \times 10^{-3}Nm¯¹ = 134mJm¯²$ 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 180°, but emphasize that such angles are never observed as it would imply that there is no interaction between the liquid and solid surface. ### 2.2 Measurement of surface tension Surface tension is a fundamental quantity in the investigation of fluid interfaces 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 flat plate through the surface of a liquid and measure the force acting on it. This is known as a Wilhelmy plate, and is named after the scientist who first 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: $F = \gamma 2(x + y)$. 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. If, however, the bottom edge of the plate is set level with the flat 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 filter 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 film balance (see Section 5.2.3), or in a detachment procedure where the difference 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 flat 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 oversimplification 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 position 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 ($\theta = 0$): $\gamma 2\pi r = pgh \pi r^2$ $\gamma = pghr$ #### The thorny devil (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. #### 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: $mg = 2\pi \gamma r$ 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 dipping into a liquid the bubble radius is at first 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. 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 capillary 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 flat surface (sessile drop) or hanging beneath a flat 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 pressure 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 (2r) and the maximum distance from the chord to the surface (z) as shown in Figure 2.16. Simple geometry then gives for the radius of curvature of the meniscus: $r = \frac{r_c^2 + z_c^2}{2z_c}$ #### 2.3 The Laplace equation If a fluid interface is curved, the pressures on either side must be different. For example, if we take a flat soap film stretched across a circular wire 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 demonstrate this by blowing the bubble on the end of a tube. It is necessary to blow into the tube to inflate the bubble and if the tube is left open the bubble will expel the air and shrink, eventually forming a flat film 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 interface. This is expressed in the Laplace equation: $p_a - p_b = \frac{2\gamma}{r}$ which holds for a spherical interface of radius $r$. The derivation of the Laplace equation follows. 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 (a 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 difference** For a small segment of cap of area δA the force arising from the pressure difference is (P"-PB) δA, and its component in the z direction (the central axis of the cap) is ($P_a$ - $P_b$) δA cos φ. However, dA cos & is equal to the area of the projection of DA 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: $F_a = (P_a - P_b)Σ δA = (P_a - P_b)\pi r^2$. **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 = r/r, so the component in the z direction of the surface tension force on δl is γδl r/r. The sum over the whole perimeter is therefore: $F_b = -\gamma \Sigma (2\pi r_c) / r$ $= -2\pi \gamma r_c/r$. 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: $F_a + F_b = 0$ or $(P_a - P_b)\pi r^2 -2\pi \gamma r_c/r = 0$ and so: $p_a - p_b = \frac{2\gamma}{r}$ which is the Laplace equation for a spherical surface. For a non-spherical interface two radii of curvature are needed. Figure 2.18 illustrates how the radii of curvature are defined. 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 orientation) taking a second plane containing the normal at right angles to the first plane, observing the curves formed by the intersections of these planes with the surface and then finding 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 a. The Laplace equation then becomes: $p_a - p_b = \gamma (\frac{1}{r} + \frac{1}{r''}) = \frac{2\gamma}{r_m}$ #### 2.3.1 Applications of the Laplace equation A soap film stretched across a flat wire frame is flat because the pressure is the same on both sides, but more complex shapes occur even when there is no difference in pressure. For example, if the film 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 rise The 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 $\theta$ with the capillary wall) in the capillary: $p_{atm} - P = \frac{2\gamma cos \theta}{r}$ with the pressure decrease from the weight of the liquid column: $p_{atm} - P = pgh$ gives: $\gamma = \frac{pghr}{2cos\theta}$ for a capillary of radius r. It is difficult to measure and to reproduce $\theta$ if $\theta \neq 0$ so only systems with $\theta = 0$ 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 pressure It was stated in Section 2.2.4 that the maximum bubble pressure occurs when the bubble radius is the same as the capillary radius, r. 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: $\gamma = \frac{\Delta P r}{2}$ where the pressure term is the difference between the pressure applied to the capillary and atmospheric pressure corrected for the small hydrostatic pressure arising from the depth of immersion. #### The Kelvin equation The Laplace equation forms the theoretical basis for the **Kelvin equation**, which describes the effect of surface curvature on vapour pressure. It is discussed in some detail in Section 2.4. #### 2.4 The Kelvin equation An important consequence of the Laplace equation concerns the effect of surface curvature on the vapour pressure of a liquid. This relationship is known as the **Kelvin equation**: $ln(\frac{p'}{p^\infty}) = \frac{2 \gamma V}{RT r_m}$. where $p'$ and $p^\infty$ are, respectively, the vapour pressures over the curved surface of effective radius $r_m$ (see Eqn 2.19) and a flat surface (r = ∞), and V is the partial molar volume. As can be seen from the derivation below, there is a convention of signs for 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 explanations for such phenomena as the difficulties found in self-nucleation of a new phase, the growth of large droplets at the expense of smaller ones, and condensation in capillaries. #### Derivation of the Kelvin equation At a curved interface the condition for mechanical equilibrium is given by the Laplace Eqn 2.18: $p'' - p = \frac{2\gamma}{r_m}$ where $r_m$ is the phase on the concave side. We generalize by giving r_m a positive sign if it lies within phase α, or a negative sign if it lies within phase β. For physico-chemical equilibrium: $\mu_a = \mu_ß = \mu$ If the equilibrium is shifted slightly with adjustments to p", p, r_m, and μ, we obtain: $\delta p'' - \delta p^ß = 2 \gamma \delta(1/r_m)$ and $\delta \mu_a = \delta \mu_ß = \delta \mu$ Since $\delta \mu_{L} = \delta \mu_{V}$, we have: $\delta p^V - \delta p^L = 2\gamma \delta(1/r_m)$ If ẞ is the vapour phase, $\delta p^V << \delta p^L$ and (2.22h) becomes: $2 \gamma \delta(1/r_m) = \delta p^L (V^L/V^L) = \delta \mu_L/V^L$ where $\delta \mu_L$ refers to the process where the radius is changed. Integration of Eqn 2.22i from a flat surface (1/r = 0, $µ_L$ = $µ_L$) to any selected curvature (1/r_m) yields: $2\gamma (1/r_m) = (\mu_L - \mu_L)/V^L$ or $\mu_L - \mu_L = \gamma V^L (2/r_m)$ where superscript a has been replaced by L to indicate 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 incompressible when integrating Eqn 2.22i. If the vapour behaves as an ideal gas, $\mu_L = \mu_L + RT ln(p/p^\infty)$ and $\mu_L - \mu_L = RT ln(p/p^\infty)$ Equation (2.22j) now becomes: $ln(\frac{p}{p^\infty}) = \frac{2 \gamma V^L}{RT r_m}$ which is the more usual form of the Kelvin equation. **Example 2.5 Vapour pressure of liquid droplets** **Problem** Calculate the vapour pressure of spherical water droplets of radius 10 nm and 100 nm. The temperature is 280 K and the surface tension of water at this temperature is 74.6 mN m²¹, and the molar volume is 18.01×10-6 m³. **Solution** Applying the Kelvin equation (2.22) we have for the 10 nm droplet: $ln(\frac{p}{p^\infty}) = \frac{2\gamma V}{RT r_m} = \frac{2 \times 74.6 \times 10^{-3} Nm^{-1} \times 18.01 \times 10^{-6} m^3}{8.31 JK^{-1}mol^{-1} \times 280K \times 10 \times 10^{-9}m} = 0.115$ giving $\frac{p}{p^\infty} = exp 0.115 = 1.12$.