In the Classroom
On Capillary Rise and Nucleation R. Prasad Department of Applied Mechanics, Indian Institute of Technology, Hauz Khas, New Delhi, 110016 India;
[email protected] The change of one phase into another, known as phase transformation, is important in physical chemistry (1). There are many examples such as melting (solid to liquid), solidification (liquid to solid), evaporation (liquid to vapor), condensation (vapor to liquid), and so forth. Many natural phenomena and industrial processes depend upon such transformations. An important example of phase transformation in nature is condensation of water vapor in the atmosphere to form clouds (2). An example from technology is solidification of liquid metals, the basis of the foundry industry (3). Many phase transformations, of which condensation and solidification are examples, take place via nucleation and growth (4). Nucleation is the first appearance of a stable particle of the product phase in the metastable phase. Subsequent increase in the size of the particle is called growth. Control of nucleation is an important step in the control of microstructure and properties of the final product. An increase in nucleation rate during solidification results in a microstructure with fine grains and hence enhanced mechanical properties (5). On the other hand, suppressing nucleation altogether leads to an amorphous or glassy product (6). Capillary rise is another important topic in physical chemistry (1). It is a good practical example demonstrating the effect of surface energy. The two concepts, capillary rise and homogeneous nucleation, are usually discussed separately. In this article we present a pedagogic comparison of nucleation and capillary rise. We show that both phenomena result from a competition between two opposing energy factors: a volume energy and a surface energy. In capillary rise the surface energy factor drives the process whereas the volume energy factor opposes it. The roles of the two energy factors are reversed in nucleation: surface energy opposes it whereas the volume energy drives it. We first present derivations of equilibrium height of capillary rise and critical radius for homogeneous nucleation. These form the A
B
Hsv
h
R Hlv
basis for the comparison suggested above. Finally we offer some suggestions on how to include this material in teaching. Capillary Rise The rise of a liquid against gravitational forces in a vertical tube of small diameter is well-known; the formula for the height of the liquid column is derived in most physical chemistry textbooks (1). The most common method for doing this is through a force balance: the surface tension force balances the weight of the liquid column. However, to compare this phenomenon with that of nucleation, it is instructive to use the energy approach proposed by Markworth (7). Despite its utility, most textbooks do not use or mention this approach. Consider a liquid column of height h as shown in Figure 1. Following Markworth (7), the change in the energy ΔEh of the system due to the rise of this column can be given by h %Eh 2 Qr h H sl H sv Q r 2 h S g (1) 2 where r is the inner radius of the tube; γsl and γsv are the solid– liquid and solid–vapor interfacial energies per unit area, respectively; ρ is the density of the liquid; and g is the acceleration due to gravity. The first term on the right-hand side of eq 1 represents the decrease in Gibbs energy due to replacement of higher energy solid–vapor interface with lower energy solid–liquid interface (for systems showing capillary rise). This decrease in Gibbs energy is what drives the capillary rise and it can be called its “driving force”. However, the rise of the liquid column in the capillary also leads to the inevitable rise in the gravitational potential energy of the system. The potential energy is the product of the mass (πr 2hρ) of the liquid column, the gravitational acceleration ( g) and the height (h/2) of the center of mass of the liquid column. Thus the second term on the right-hand side of eq 1 opposes the capillary rise and hence acts as a barrier. This energy term depends upon the volume of the column and can be called the volume energy. As can be seen from eq 1 that the volume energy term is proportional to h2 whereas the surface energy term is proportional to h, hence their ratio goes as h. This means that the driving force represented by surface energy will be dominant at small h but the opposing term of gravitational potential energy will eventually become significant at large h. The total energy ΔEh, which is the sum of surface energy and volume energy, initially decreases as a function of h, reaches a minimum at h*, and then begins to increase. Thus the liquid column will rise initially but will face increasing opposition later and so will finally stop at an equilibrium height h*. The energy changes are shown in Figure 2. The equilibrium height h* is given by setting ∂ΔEh/∂h = 0:
Hsl
Figure 1. (A) The rise of a liquid in a capillary, h is the height of the liquid in the capillary. (B) Surface tensions acting at the junction of the meniscus and the capillary wall. γlv, γsv, and γsl are the surface tensions of the liquid–vapor, solid–vapor, and the solid–liquid interfaces, respectively. θ is the contact angle.
h*
2 H sl H sv
Sg r
(2)
Note that since (γsl – γsv) is negative for a system showing capillary rise, h* is positive.
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 10 October 2008 • Journal of Chemical Education
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In the Classroom
More commonly, h* is expressed in terms of the liquid– vapor interfacial energy γlv and the contact angle θ, see Figure 1B. This can be done through the balance of surface tension forces leading to the well-known Young–Dupré equation: H sl H lv cos R H sv
(3)
Substituting γsv from eq 3 into eq 2, we get the familiar expression for capillary rise: 2 H lv cos R Sg r
h*
(4)
We would like to point out that the energy-based derivation of capillary rise used here as given originally by Markworth (7) helps in comparison of this phenomenon to that of nucleation. However, the derivation has some limitation, for example, Henriksson and Eriksson (8) have shown that the energy-based derivation cannot account for the meniscus curvature.
V
Energy
%Eh h*
S
Height Figure 2. The energy terms involved in a capillary rise. ΔEh is the change in the energy of the system and is the sum of two terms: the surface energy term S and the volume energy term V. The equilibrium height h* to which the liquid finally rises correspond to the minimum of ΔEh curve.
Energy
S
In this section, we present a derivation for the critical radius for homogeneous nucleation. For simplicity, we consider the case of condensation of liquid droplets in an undercooled vapor. As mentioned in the introduction this has relevance to atmospheric chemistry and formation of clouds (2). The Gibbs energy change for the formation of a spherical liquid droplet of radius R in an undercooled vapor below its condensation point is given by (e.g., see ref 9)
% dG
R*
%dG
Radius Figure 3. The energy terms involved in a homogeneous nucleation resulting in a critical radius. The surface energy term S and the volume energy term V combine to give the free energy of formation ΔdG of a liquid droplet in a supersaturated vapour below its condensation point. The maximum in the ΔdG curve correspond to the critical radius R*.
4 Q R 3 % vG 4 Q R 2 H lv 3
(5)
where Δ vG is the Gibbs energy difference Gv − Gl between liquid and vapor per unit volume and γlv is the surface tension of the vapor–liquid interface. The two terms on the right-hand side of eq 5 and their sum, ΔdG, are shown in Figure 3. The first term on the right-hand side of eq 5 is the volume term representing the decrease in Gibbs energy due to formation of a stable liquid droplet of lower energy in the higher energy metastable vapor. This decrease in Gibbs energy is what drives the nucleation and is appropriately called the “driving force”. However, the droplet cannot form without an inevitable vapor–liquid interface. This interface has a surface energy associated with it that will tend to increase the energy of the system. Thus the second term in eq 5 representing the surface term opposes the nucleation. Since the surface energy term is proportional to R2 whereas the volume energy term is proportional R3, their ratio goes by 1/R. This means that the surface energy term will be dominant at small R while the volume energy term will become important at larger R. For liquid droplets of very small radius, the increase in surface energy more than offsets any decrease in volume energy. Any droplet of such radii, if formed, will vaporize. On the other hand for droplets of reasonably large radii the decrease in volume energy more than compensates for any increase in the surface energy. Such droplets can grow in size. Thus the total energy Δ dG, which is the sum of surface energy and volume energy, initially increases as a function of R, reaches a maximum and then begins to decrease. The radius corresponding to the maximum in Δ dG is called the critical radius R* and is given by setting ∂Δ dG∙∂R = 0:
V
1390
Homogeneous Nucleation
R*
2 H lv % vG
(6)
A droplet of radius R* is at the maximum of energy and can be considered to be in a state of unstable equilibrium. If it shrinks slightly by vaporization it will continue to shrink until it disappears. On the other hand, if it grows slightly it will grow into a stable phase. Substituting R* from eq 6 into eq 5 gives ΔG*, the value of Δ dG at R*. This is the so called barrier to nucleation and is given by
%G *
16Q Hsl 3 3 % vG 2
(7)
Comparison The derivation for the capillary rise based on energy approach of Markworth (7) brings out the similarity to and differences from the critical radius for homogeneous nucleation. This
Journal of Chemical Education • Vol. 85 No. 10 October 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
In the Classroom Table 1. A Comparison of Homogeneous Nucleation and Capillary Rise Property
Homogeneous Nucleation
Capillary Rise
Volume energy
Chemical Gibbs energy
Gravitational potential energy
Cubic in R
Quadratic in h
Assists nucleation
Opposes rise
γlv
γsl – γsv
Quadratic in R
Linear in h
Opposes nucleation
Assists rise
Unstable
Stable
Surface energy
Nature of equilibrium
is summarized in Table 1. As can be seen, both homogeneous nucleation and capillary rise are a result of balance between two opposing factors: a surface energy and a volume energy. In the case of homogeneous nucleation the volume factor is the chemical Gibbs energy change and it drives the transformation. On the other hand, in the case of capillary rise, the gravitational potential energy, playing the role of volume factor, opposes the rise. Similarly the role of surface energy is also reversed in the two cases: it opposes the nucleation but drives the capillary rise. The critical radius corresponds to the maximum Gibbs energy and thus gives rise to an unstable equilibrium: both smaller and larger droplets move away from the critical radius, the smaller by vaporizing and the larger by growing larger still. In contrast, the equilibrium height of the liquid column corresponds to the minimum of the potential energy. This gives a stable equilibrium: a column of smaller height will rise to h* whereas one of larger height will fall to the same value. Inclusion into Curriculum We now make some suggestions regarding the inclusion of this material into a curriculum. Nucleation is a kinetic process. However, the rate of nucleation depends upon the thermodynamic barrier ΔG*. As has been shown in eq 7, this barrier can be estimated in terms of equilibrium thermodynamics. Thus the topic of nucleation can be approached soon after the concept of equilibrium thermodynamics and phase equilibria have been discussed. We describe two examples from standard physical chemistry textbooks to illustrate how our suggestions can be practically incorporated in classroom teaching. In their textbook Atkins’ Physical Chemistry the authors Atkins and de Paula (1) introduce nucleation in section 6.9(b). However they do not develop it to the point of introducing the concept of critical size of the nucleus. In the very next section, 6.10(a), they introduce capillary rise and derive the equation for equilibrium height. Thus there is a clear opportunity here to relate the two concepts in the manner shown in our work and thus introduce the concept of critical nucleus. Our next example is in the textbook Physical Chemistry by Silbey and Alberty (10). They briefly mention nucleation in
section 6.4: Effect of Surface Tension on Vapor Pressure. In the same section capillary rise appears only in the list of methods to determine surface tension. But neither of the topics has been developed further. Thus again it is possible to introduce nucleation and critical size through comparison with the capillary rise as has been discussed in the present work. Acknowledgments I wish to thank Charusita Chakravarty, N. D. Kurur, and A. Ramanan of the Department of Chemistry, Indian Institute of Technology, Delhi, India, and A. L. Greer of the Department of Materials Science and Metallurgy, University of Cambridge, for helpful discussions and comments on the manuscript. Literature Cited 1. Atkins, P.; de Paula, J. Atkins’ Physical Chemistry, 7th ed.; Oxford University Press: Oxford, 2002; Chapter 6. 2. Wayne, R. P. Chemistry of Atmospheres, 3rd ed.; Oxford University Press: Oxford, 2000; Section 2.5. 3. Kurz, W.; Fisher, D. J. Fundamentals of Solidification; Trans Tech Publications: Aedermannsdorf, Switzerland, 1989. 4. Oxtoby, D. W. Acc. Chem. Res. 1998, 31, 91–97. 5. Callister, W. D., Jr. Materials Science and Engineering: An Introduction, 6th ed.; John Wiley: New York, 2003; pp 174–176. 6. Debenedetti, P. G. Metastable Liquids: Concepts and Principles; Princeton University Press: Princeton, 1996. 7. Markworth, A. J. J. Chem. Educ. 1971, 48, 528. 8. Henriksson, U.; Eriksson, C. J. Chem. Educ. 2004, 81, 150–154. 9. Talanquer, V. J. Chem. Educ. 2002, 79, 877–883. 10. Silbey, R. J.; Alberty, R. A. Physical Chemistry, 3rd ed.; John Wiley: New York, 1995; Chapter 6.
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