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- Herausgeber: Wiley-VCH GmbH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Physics and Chemistry of Interfaces
Comprehensive textbook on the interdisciplinary field of interface science, fully updated with new content on wetting, spectroscopy, and coatings
Physics and Chemistry of Interfaces provides a comprehensive introduction to the field of surface and interface science, focusing on essential concepts rather than specific details, and on intuitive understanding rather than convoluted math. Numerous high-end applications from surface technology, biotechnology, and microelectronics are included to illustrate and help readers easily comprehend basic concepts.
The new edition contains an increased number of problems with detailed, worked solutions, making it ideal as a self-study resource. In topic coverage, the highly qualified authors take a balanced approach, discussing advanced interface phenomena in detail while remaining comprehensible. Chapter summaries with the most important equations, facts, and phenomena are included to aid the reader in information retention.
A few of the sample topics included in Physics and Chemistry of Interfaces are as follows:
- Liquid surfaces, covering microscopic picture of a liquid surface, surface tension, the equation of Young and Laplace, and curved liquid surfaces
- Thermodynamics of interfaces, covering surface excess, internal energy and Helmholtz energy, equilibrium conditions, and interfacial excess energies
- Charged interfaces and the electric double layer, covering planar surfaces, the Grahame equation, and limitations of the Poisson-Boltzmann theory
- Surface forces, covering Van der Waals forces between molecules, macroscopic calculations, the Derjaguin approximation, and disjoining pressure
Physics and Chemistry of Interfaces is a complete reference on the subject, aimed at advanced students (and their instructors) in physics, material science, chemistry, and engineering. Researchers requiring background knowledge on surface and interface science will also benefit from the accessible yet in-depth coverage of the text.
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Veröffentlichungsjahr: 2023
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Physics and Chemistry of Interfaces
Fourth Edition
Hans‐Jürgen Butt
Karlheinz Graf
Michael Kappl
Authors
Prof. Hans‐Jürgen ButtMPI for Polymer ResearchAckermannweg 1055128 MainzGermany
Prof. Dr. Karlheinz GrafHochschule NiederrheinPhysikalische ChemieAdlerstr. 3247798 KrefeldGermany
Dr. Michael KapplMPI for Polymer ResearchAckermannweg 1055128 MainzGermany
Cover image: © Getty Images
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Cover Design Adam Design, Weinheim, Germany
Preface
This textbook serves as a general introduction to surface and interface science. It focuses on basic concepts rather than specific details, on understanding rather than on learning facts. The most important techniques and methods are introduced. The book reflects that interfacial science is a diverse and interdisciplinary field of research. Several classic scientific or engineering disciplines are involved. It contains basic science and applied topics such as wetting, friction, and lubrication. Many textbooks concentrate on certain aspects of surface science such as techniques involving ultrahigh vacuum or classic “wet” colloid chemistry. We tried to include all aspects because we feel that for a good understanding of interfaces, a comprehensive introduction is helpful.
Our book is based on lectures given at the universities of Siegen and Mainz. It addresses advanced students of engineering, chemistry, physics, biology, and related subjects and scientists in academia or industry who are not yet specialists in surface science but desire solid background knowledge of the subject. The level is introductory for scientists and engineers who have a basic knowledge of the natural sciences and mathematics. An advanced level of mathematics is not required.
When looking through the pages of this book, you will see a substantial number of equations. Please do not be scared. We preferred to give all transformations explicitly rather than writing “as can easily be seen” and stating the result. Chapter 3 is the only exception; to appreciate that chapter, a basic knowledge of thermodynamics is required. However, you can skip it and still be able to follow most of the rest of the book. If you do decide to skip it, please at least read and try to gain an intuitive grasp of surface excess (Section 3.2) and the Gibbs adsorption equation (Section 3.5.2).
A number of problems with solutions are included to enable for self‐study. Unless noted otherwise, the temperature was assumed to be 25 °C. At the end of each chapter, the most important equations, facts, and phenomena are summarized.
One of the main problems with writing a textbook is trying to limit its content. We tried hard to keep the volume within the scope of one advanced course lasting roughly 15 weeks, two days per week. Facing the growth in knowledge, this meant cutting short or leaving out altogether certain topics. Statistical mechanics, heterogeneous catalysis, and polymers at surfaces are issues that could be expanded.
This book no doubt contains errors. Even after several readings by various people, this is unavoidable. If you find any mistakes, please let us know about them by sending an e‐mail ([email protected]) so that they can be corrected and do not confuse more readers.
We are indebted to several people who helped us in collecting information and preparing and critically reading the manuscript. In particular, we would like to thank K. Amann‐Winkel, E. Backus, T. Balgar, C. Bayer, K. Beneke, T. Blake, J. Blum, M. Böhm, E. Bonaccurso, P. Broekmann, R. de Hoogh, A. del Campo, A. de los Santos, M. Deserno, W. Drenckhan, K. Drese, J. Elliott, G. Ertl, R. Förch, S. Geiter, G. Glasser, G. Gompper, M. Grunze, J. Gutmann, L. Heim, M. Hietschold, M. Hillebrand, T. Jenkins, X. Jiang, U. Jonas, J. Krägel, R. Jordan, Krüss GmbH, J. Laven, I. Lieberwirth, G. Liger‐Belair, C. Lorenz, M. Lösche, S. Luding, T. Makulik, E. Meyer, R. Miller, A. Müller, P. Müller‐Buschbaum, T. Nagel, D. Quéré, J. Rabe, H. Schäfer, P. Schmiedel, J. Schreiber, M. Stamm, M. Steinhart, C. Stubenrauch, G. Subklew, F. Thielmann, J. Tomas, K. Vasilev, J. Venzmer, D. Vollmer, R. von Klitzing, K. Wandelt, B. Wenclawiak, R. Wepf, R. Wiesendanger, J. Wintterlein, G. de With, J. Wölk, D.Y. Yoon, M. Zharnikov, and U. Zimmermann.
Hans‐Jürgen Butt, Karlheinz Graf, and Michael Kappl
Mainz, June 2022
1Introduction
An interface is an area that separates two phases from each other. If we consider the solid, liquid, and gas phases, we immediately get three combinations of interfaces: solid–liquid, solid–gas, and liquid–gas interfaces. The term surface is often used synonymously with interface, although interface is preferred to indicate the boundary between two condensed phases and in cases where the two phases are named explicitly. For example, we talk about a solid–gas interface and a solid surface. Surface is used for a condensed phase in contact with a gas or a vacuum. Interfaces can also separate two immiscible liquids such as water and oil. These are called liquid–liquid interfaces. Interfaces may even separate two different phases within one component. In a liquid crystal, for example an ordered phase may coexist with an isotropic phase. Solid–solid interfaces separate two solid phases. They are important for the mechanical behavior of solid materials such as concrete. Gas–gas interfaces do not exist because gases mix.
Often, interfaces and colloids are discussed together. Colloid is an abbreviated synonym for colloidal system. Colloidal systems are disperse systems where one phase has dimensions on the order of 1 nm to 1 (Figure 1.1). The word colloid comes from the Greek word for glue and was first used in 1861 by Thomas Graham.1 He used the word to refer to materials that seemed to dissolve but were unable to penetrate membranes such as albumin, starch, and dextrin. A colloidal dispersion is a two‐phase system that is uniform on the macroscopic but not on the microscopic scale. It consists of grains or droplets of one phase in a matrix of the other phase.
Different kinds of dispersions can be formed. Most have important applications and have special names (Table 1.1). While there are only five types of interfaces, we can distinguish ten types of disperse systems because we must discriminate between the continuous, dispersing (external) phase and the dispersed (inner) phase. In some cases, this distinction is obvious. Nobody will, for instance, confuse fog with foam, although in both cases, a liquid and a gas are involved. In other cases, the distinction between continuous and inner phase cannot be made because both phases might form connected networks. Some emulsions, for instance, tend to form a bicontinuous phase, in which both phases form an interwoven network.
Figure 1.1 Schematic of a dispersion.
Table 1.1 Types of dispersions.
Continuous phase
Dispersed phase
Term
Example
Gas
Liquid
Aerosol
Clouds, fog, smog, hairspray
Solid
Aerosol
Smoke, dust, pollen
Liquid
Gas
Foam
Lather, whipped cream, foam on beer
Liquid
Emulsion
Milk
Solid
Suspension
Ink, muddy water, dispersion paint
Solid
Gas
Porous solids
Solid foam
Styrofoam, soufflé
Liquid
Solid emulsion
Butter
Solid
Solid suspension
Concrete
Colloids and interfaces are closely related. This is a direct consequence of the enormous specific surface area of colloids. Their interface‐to‐volume ratio is so large that their behavior is determined mainly by interfacial properties. Gravity and inertia are negligible in the majority of cases. For this reason, colloidal systems are often dominated by interfacial effects rather than bulk properties. For the same reason, interfacial science is essential in nanoscience and technology.
Colloidal systems are often influenced by thermal fluctuation and colloidal particles move due to Brownian motion. This distinguishes them from granular matter, a material composed of macroscopic elements. The constituents of granular matter must be large enough so that they are not subject to thermal motion. Usually, the structure of granular matter depends on its history; it is generally far from being in thermodynamic equilibrium, and gravitation can play a significant role.
Example 1.1 A granular system that is dominated by surface effects is shown in Figure 1.2. A scanning electron microscope (SEM) image shows aggregates of particles (diameter 0.9 ). These particles were blown as dust into a chamber filled with gas. While sedimenting, they formed fractal aggregates due to attractive van der Waals forces and collected on the bottom. These aggregates are stable for weeks or months, and even shaking does not change their structure. Thermal fluctuations are completely insignificant, and the material would count as granular matter. Still, it is not a typical granular material because once it has formed, gravity and inertia, which rule the macroscopic world, are not able to bend down the particle chains. Surface forces are much stronger.
Figure 1.2 Agglomerate of silicon oxide particles.
Sometimes a distinction is made between colloidal particles, which are in the size range of 1–1000 nm, and nanoparticles, which are 1–100 nm in diameter. For particles smaller than 100 nm, sedimentation is usually negligible.
It is useful to introduce the characteristic length scale of a system. The characteristic length scale can often be given intuitively. For example, for a spherical particle, one would use the radius or the diameter. For more complex systems, however, intuition leads to ambiguous results. We suggest using the ratio of the total volume divided by the total interfacial area of a system as the characteristic length scale: . For a sphere of radius , the characteristic length scale is . For a thin film is equal to the thickness. For a dispersion of spherical particles with a volume fraction , the total volume of the system is , where is the number of particles. With a total surface area of , we get a characteristic length scale of .
Why is there an interest in interfaces and colloids? First, to gain a better understanding of natural processes, for example, in biology. The interfacial tension between water and lipids allows for the formation of lipid membranes. This is a prerequisite for the formation of compartments and, thus, any form of life. In geology, the swelling of clay or soil in the presence of water is an important process. The formation of clouds and rain due to the nucleation of water around small dust particles is dominated by surface effects. Many foods, such as butter, milk, or mayonnaise, are emulsions; their properties are determined by the liquid–liquid interface. Second, interfaces and colloids have many technological applications. An example is flotation in mineral processing or the bleaching of scrap paper. Washing and detergency are examples of applications encountered in everyday life. Often, the production of new materials such as composite materials involves intensive processes at interfaces. Thin films on surfaces are often dominated by surface effects. Examples include latex films, coatings, and paints. The flow behavior of powders and granular media is determined by surface forces. In tribology, wear is reduced by lubrication, which, again, is a surface phenomenon.
Typical of many industrial applications is a very refined and highly developed technology, but only a limited understanding of the underlying fundamental processes of that technology. A better understanding is, however, required to further improve the efficiency or reduce dangers to the environment.
Introductory books on interface science are [1–3]. For a deeper understanding, we recommend the series of books by Lyklema [4–7].
Note
1
Thomas Graham, 1805–1869; British chemist, professor in Glasgow and London.
2Liquid Surfaces
2.1 Microscopic Picture of a Liquid Surface
A liquid surface is not an infinitesimal sharp boundary in the direction of its normal; it has a certain thickness. For example, if we consider the density normal to a surface (Figure 2.1), we can observe that, within a few molecules, the density decreases from that of the bulk liquid to that of its vapor [8].
The density is only one criterion by which one may define the thickness of an interface. Another possible parameter is the orientation of the molecules. For example, water molecules at the surface prefer to be oriented with their hydrogen atoms “out” toward the vapor phase. This orientation fades with increasing distance from the surface. At a distance of 0.5‐1 nm, the molecules are again randomly oriented like in the bulk.
Which thickness do we have to use? This depends on the relevant parameter. If we are interested, for instance, in the density of a water surface, a realistic thickness is on the order of 1 nm. Let us assume that a salt is dissolved in water. Then the concentration of ions might vary over a larger distance (characterized by the Debye length, Section 4.2.2). With respect to the ion concentration, the thickness is thus much larger. When in doubt, it is safer to choose a large value for the thickness.
The surface of a liquid is a very turbulent place. Molecules may evaporate from the liquid into the vapor phase and vice versa. In addition, they diffuse into the bulk phase and molecules from the bulk diffuse to the surface.
Example 2.1 To estimate the number of gas molecules hitting a liquid surface per second, we recall the kinetic theory of ideal gases. In textbooks on physical chemistry, the rate of effusion of an ideal gas through a small hole is given by
Here, is the cross‐sectional area of the hole and is the molecular mass. This is equal to the number of water molecules hitting the surface area per second. Water at 25 °C has a vapor pressure of 3168 Pa. With a molecular mass of kg, water molecules per second hit a surface area of 10 . In equilibrium, the same number of molecules escapes from the liquid phase. The area covered by one water molecule is approximately 10 . Thus, the average time a water molecule remains on the surface is in the order of 0.1 .
Figure 2.1 Snapshot of molecular structure of water as obtained by computer simulation and density of water versus coordinate normal to its surface [9]. The density in water vapor at saturation and 25 °C is only 0.02 g/. Therefore, it is negligible on the scale plotted (kindly provided by D. Horinek).
2.2 Surface Tension
The following experiment helps us to define the most fundamental quantity in surface science: the surface tension. A liquid film is spanned over a frame with a mobile slider (Figure 2.2). The film is relatively thick, say 1 , so that the distance between the back and front surfaces is large enough to avoid overlapping of the two interfacial regions. Practically, this experiment might be tricky even in the absence of gravity, but it violates no physical laws, so in principle, it is feasible. If we increase the surface area by moving the slider a distance to the right, work must be done. This work is proportional to the increase in surface area . The surface area increases by twice because the film has a front and back side. Introducing the proportionality constant , we get
The constant is called surface tension.
Equation (2.2) is an empirical law and a definition at the same time. The empirical law states that the work is proportional to the change in surface area. This is not only true for infinitesimally small changes in (which is trivial) but also for significant increases in the surface area: . In general, the proportionality constant depends on the composition of the liquid and the vapor, temperature, and pressure, but it is independent of the area. By definition, we call the proportionality constant surface tension.
Figure 2.2 Schematic setup to verify Eq. (2.2) and define the surface tension.
The surface tension can also be defined by the force required to hold the slider in place and to balance the surface tensional force:
Both forms of the law are equivalent, provided that the process is reversible. Then we can write
The force is directed to the left, while increases to the right. Therefore, we have a negative sign.
The unit of surface tension is either joule per square meter or newton per meter. Surface tensions of liquids are on the order of 0.02–0.08 N/m (Table 2.1). For convenience, they are usually given in millinewtons per meter (or ).
Empirically one finds that the surface tension of liquids decreases linearly with temperature. Thus, if we know the surface tension at a given temperature , then we can approximate the surface tension at a temperature according to
The coefficient is negative. As we will see in Chapter 3, is equal to the surface entropy.
Example 2.2 If a water film is formed on a frame with a slider length of 1 cm, then the film pulls on the slider with a force of
That corresponds to a weight of 0.15 g.
Example 2.3 Calculate the surface tension of water at . With and /(K m) at (Table 2.1), we get
This is close to the experimental value of 67.9 mN/m.
The term surface tension is tied to the concept that the surface stays under a tension. In a way, this is similar to a rubber balloon, where a force is required as well to increase the surface area of its rubber membrane against a tension. There is, however, a difference: while the expansion of a liquid surface is a plastic process and the surface tension remains constant, the stretching of a rubber membrane is usually elastic, and the tension increases with increasing surface area.
Table 2.1 Surface tensions and of some liquids at different temperatures [10].
Substance
Water
25
71.99
−15.6 × 10
−5
Methanol
25
22.07
−7.73 × 10
−5
Ethanol
25
21.97
−8.33 × 10
−5
1‐Propanol
25
23.32
−7.75 × 10
−5
1‐Butanol
25
24.93
−8.98 × 10
−5
2‐Butanol
25
22.54
−7.95 × 10
−5
Phenol
50
38.20
−10.7 × 10
−5
Glycerol
25
63.70
−5.98 × 10
−5
Cyclohexane
25
24.65
−11.9 × 10
−5
Benzene
25
28.22
−12.9 × 10
−5
Toluene
25
27.93
−11.8 × 10
−5
‐Pentane
25
15.49
−11.1 × 10
−5
‐Hexane
25
17.89
−10.2 × 10
−5
‐Heptane
25
19.65
−9.80 × 10
−5
‐Octane
25
21.14
−9.51 × 10
−5
‐Nonane
25
22.38
−9.36 × 10
−5
‐Decane
25
23.37
−9.20 × 10
−5
Acetone
25
23.46
−11.2 × 10
−5
Formamide
25
57.03
−8.44 × 10
−5
Dichloromethane
25
27.20
−12.8 × 10
−5
Chloroform
25
26.67
−12.9 × 10
−5
Decaline
25
31.0
−10.3 × 10
−5
PDMS
25
19.0–20.4
−3.65 × 10
−5
Hexamethyldisiloxane
25
15.70
−8.77 × 10
−5
Octamethylcyclotetrasiloxane
25
17.61
−6.60 × 10
−5
Perfluorohexane
25
12.03
−10.5 × 10
−5
Perfluoroheptane
25
12.70
−9.51 × 10
−5
Perfluorooctane
25
13.83
−8.94 × 10
−5
25
45.9
−7.83 × 10
−5
25
40.6
−5.72 × 10
−5
Sodium chloride
NaCl
801
192
−7.2 × 10
−5
Potassium chloride
KCl
771
176
−7.4 × 10
−5
Calcium chloride
775
196
−4.5 × 10
−5
Argon
Ar
−186
11.90
−25.1 × 10
−5
Nitrogen
−196
8.85
−22.5 × 10
−5
Mercury
Hg
25
485.48
−20.5 × 10
−5
Gallium
Ga
29.8
715.3
−9.0 × 10
−5
Silver
Ag
961
966
−24.5 × 10
−5
Gold
Au
1064
1120
−14 × 10
−5
Notes: PDMS stands for poly(dimethylsiloxane) or . The surface tension of polymer melts increases slightly with the number of monomers and saturates for long chains [11]. 1‐‐3‐ (bmin) with or are ionic liquids.
Figure 2.3 Schematic molecular structure of a liquid–vapor interface.
How can we interpret the concept of surface tension on the molecular level? For molecules, it is energetically favorable to be surrounded by other molecules. Molecules attract each other by different interactions such as van der Waals forces or hydrogen bonds (for details see Chapter 5). Without this attraction, there would not be a condensed phase at all, there would only be a vapor phase. The sheer existence of a condensed phase is evidence for attractive interactions between molecules. At the surface, molecules are only partially surrounded by other molecules, and the number of adjacent molecules is smaller than in the bulk (Figure 2.3). This is energetically unfavorable. To bring a molecule from the bulk to the surface, work must be carried out. With this view, can be interpreted as the energy required to bring molecules from inside the liquid to the surface and to create new surface area.
With this interpretation of surface tension in mind, we immediately realize that must be positive. Otherwise, the Gibbs free energy of interaction between molecules would be repulsive, and all molecules would immediately evaporate into the gas phase.
Example 2.4 Estimate the surface tension of cyclohexane from the energy of vaporization at . The density of cyclohexane is , and its molecular weight is .
For a rough estimate, we picture the liquid as being arranged in a cubic structure. Each molecule is surrounded by its six nearest neighbors. Thus, each bond contributes roughly . At the surface, one neighbor – and hence one bond – is missing. Per mole, we therefore miss an energy of 5.08 kJ/mol.
To estimate the surface tension, we need to know the surface area occupied by one molecule. If molecules form a cubic structure, then the volume of one unit cell is , where is the distance between nearest neighbors. This distance can be calculated from the density:
The surface area per molecule is . For the surface energy, we estimate
For such a rough estimate, the result is surprisingly close to the experimental value of 0.0247 J/.
The concept of surface tension can be generalized to liquid–liquid interfaces. For example, ‐octane and water form two phases that are separated by an interface. The associated interfacial tension is 51.2 N/m at C. This is higher than the surface tension of octane.
Example 2.5 The range of interfacial tensions is large. At room temperature, mercury has the highest surface tension with 485 mN/m. Above C, gallium melts and has an even higher surface tension of 708 mN/m. At even higher temperatures, other molten metals have higher surface tensions. Examples are nickel, with 1780 mN/m (at a melting temperature of C), and iron, with 1940 mN/m C) [12]. Very low interfacial tensions are measured between the different phases of liquid crystals. For example, at high temperature, 4‐octyl‐4′‐cyanobiphenyl (8CB, ) forms an isotropic liquid. When it is slowly cooled to , the material becomes a nematic liquid crystal, where the molecules align and show a preferred orientation. The tension of the interface between the isotropic and the nematic phase is only 9.5 N/m [13]. Very low interfacial tensions were measured between phase‐separated polymer mixtures in solutions. For example, the interfacial tension between aqueous gelatine and dextrane solutions can be as low as 0.5 N/m [14].
2.3 Equation of Young and Laplace
2.3.1 Curved Liquid Surfaces
If in equilibrium a liquid surface is curved, then there is a pressure difference across it. To illustrate this, let us consider a circular part of the surface. The surface tension tends to minimize the area. This results in a planar geometry of the surface. To curve the surface, the pressure on one side must be larger than on the other side. The situation is much like that of a rubber membrane. If we, for instance, take a tube and close one end with a rubber membrane, the membrane will be planar (provided the membrane is under some tension) (Figure 2.4). It will remain planar as long as the tube is open at the other end, and the pressure inside the tube is equal to the outside pressure. If we now blow carefully into the tube, the membrane bulges out and becomes curved due to the increased pressure inside the tube. If we suck on the tube, the membrane bulges inside the tube because now the outside pressure is higher than the pressure inside the tube.
Figure 2.4 Rubber membrane at the end of a cylindrical tube to illustrate the Laplace pressure. An inner pressure that is different than the outside pressure can be applied.
The Young1 –Laplace2 equation, also simply called Laplace equation, relates the pressure difference between the two phases and the curvature of the surface [15, 16]:
Here, and are the two principal radii of curvature. is called the Laplace or capillary pressure. Equation (2.6) is also referred to as Laplace equation. It is valid in the absence of gravitation.
The curvature at a point on an arbitrarily curved surface is obtained as follows: at the point of interest, we draw a normal through the surface and then pass a plane through this line and the intersection of this line with the surface. One angle of orientation of this plane is not defined and can be chosen conveniently. The line of intersection will, in general, be curved at the point of interest. The radius of curvature is the radius of a circle inscribed in the intersection at the point of interest. The second radius of curvature is obtained by passing a second plane through the surface; this second plane also contains the normal but is perpendicular to the first plane. This gives the second intersection and leads to the second radius of curvature . So the planes defining the radii of curvature must be perpendicular to each other and contain the surface normal. Otherwise, their orientation is arbitrary. A law of differential geometry says that the value for an arbitrary surface is invariant and does not depend on the orientation as long as the radii are determined in perpendicular directions.
Let us illustrate the curvature for two examples. For a cylinder of radius , a convenient choice is and , so that the curvature is . For a sphere with radius , we have , and the curvature is (Figure 2.5).
Figure 2.5 Illustration of the curvature of a cylinder and a sphere.
Example 2.6 How large is the pressure in a spherical bubble with a diameter of 2 mm and a bubble 20 nm in diameter in pure water compared with the pressure outside?
The curvature of a bubble is identical to that of a sphere: . Therefore,
With mm, we get
With nm, the pressure is . The pressure inside the bubbles is therefore 144 Pa and Pa, respectively, higher than the outside pressure.
The Young–Laplace equation has several fundamental implications:
If we know the shape of a liquid surface, then we know its curvature and we can calculate the pressure difference.
In the absence of external fields (e.g. gravity), the pressure is the same everywhere in the liquid; otherwise, there would be a flow of liquid to regions of low pressure. Thus, is constant, and the Young–Laplace equation tells us that in this case, the surface of the liquid has the same curvature everywhere.
With the help of the Young–Laplace
equation (2.6)
, it is possible to calculate the equilibrium shape of a liquid surface [
17
,
18
]. If we know the pressure difference and some boundary conditions (such as the volume of the liquid and its contact line), then we can calculate the geometry of the liquid surface. For axially symmetric situations, interfaces of constant curvature are spheres, cylinders, nodoids, or unduloids.
In practice, it is usually not trivial to calculate the geometry of a liquid surface with Eq. (2.6). The shape of the liquid surface can mathematically be described by a function . The ‐coordinate of the surface is given as a function of its ‐ and ‐coordinates. The curvature involves the second derivative. As a result, calculating the shape of a liquid surface involves solving a partial differential equation of second order, which is usually not an easy task.
In many cases, we deal with rotational symmetric structures. Assuming that the axis of symmetry is identical to the ‐axis, describes a liquid surface, where is the radial coordinate.
Let us assume the ‐axis is vertical and in the plane of the paper. Then it is convenient to put one radius of curvature also in the plane of the paper. This radius is given by [19]
The other principal radius of curvature is in a plane perpendicular to the plane of the paper and oriented parallel to . It is given by
where and are the first and second derivatives with respect to , respectively. Mathematically, surfaces of constant curvature are spheres, cylinders, or parts of nodoids or unduloids [18].
2.3.2 Derivation of Young–Laplace Equation
To derive the equation of Young and Laplace, we consider a small part of a liquid surface. This part should be so small that the curvature does not change significantly. First, we pick a point X and draw a line around it that is characterized by the fact that all points on that line are the same distance away from X (Figure 2.6). This line is the cross section that the studied surface will have with a sphere with its center in X and with a radius . If the liquid surface is planar, the line would be a flat circle. On this line, we take two cuts that are perpendicular to each other (AXB and CXD). Consider in B a small segment on a line of length . The surface tension pulls with a force . The vertical force on that segment is . For small surface areas and small , we have , where is the radius of curvature along AXB. The vertical force component is
Figure 2.6 Diagram used to derive the Young–Laplace equation.
The sum of the four vertical components of the forces related to the small line segments at points A, B, C, and D is
The value of is independent of the particular orientation. This means that, although it was developed for the ABCD orientation, Eq. (2.10) is valid for any orientation of the cross‐sectional planes as long as they are orthogonal to the surface at X and mutually orthogonal. Integration over the borderline (only rotation of the four segments) gives the total vertical force, caused by the surface tension:
In equilibrium, this downward force must be compensated by an equal force in the opposite direction. This upward force is caused by an increased pressure difference on the concave side of . Equating both forces leads to
These considerations are valid for any small part of the liquid surface. Since the part is arbitrary, the Young–Laplace equation must be valid everywhere.
2.3.3 Applying the Young–Laplace Equation
When applying the equation of Young and Laplace to simple geometries, it is usually obvious at which side the pressure is higher. For example, both inside a bubble and inside a drop, the pressure is higher than outside (Figure 2.7
