Capillary forces

Type of physical science: Fluid dynamics, Liquids, Classical physics

Field of study: Fluids

Forces acting at the boundary between two fluids as a result of the curvature of the boundary are referred to as "capillary forces." Such forces can become very large and can control fluid transport in tubes of small cross-sectional area. Capillary forces are an integral component of numerous natural, biological, and human-made processes, including subsurface transport of water, oil, and gas; transport of water from roots, plants, and trees to their branches and leaves; nutrient transport in all living beings; detergency; adhesive technology; and separation and purification.

Overview

The existence and nature of capillary forces are most easily illustrated by studying the rise of water in a capillary tube. Consider inserting clean, empty glass tubes of small but varying radius vertically in a container full of water. Water will rise up each tube. The level to which the water rises in each tube will be observed to depend on the radius of the specified tube; the smaller the tube radius, the higher the level to which water will spontaneously rise inside the tube. Based on the force of gravity alone, however, there should not be any rise of water in the tubes. This provides evidence that there is an additional force acting on the boundary between air and water inside the tubes; such tubes of small radius are commonly referred to as "capillaries" after the Latin word capillus, meaning "hair."

Such additional forces acting on the fluid boundary inside the capillary tubes and on any other curved fluid-fluid interface are referred to as "capillary forces"; these forces cause the water to rise against the force of gravity. (In the above example, the air in the tubes acts as one of the fluids in the interface, but capillary forces can also exist at the interface of two liquids.) The height of the water rise at equilibrium represents the magnitude of the capillary force; the smaller the tube diameter, the larger the magnitude of the capillary force.

Capillary force is directly proportional to the interfacial tension of the fluid pair under consideration. It is also directly proportional to the mean curvature of the interface. The capillary force acting per unit area of the fluid-fluid boundary at equilibrium can be calculated as two times the product of interfacial tension and the interface curvature. Capillary force per unit area is usually referred to as "capillary pressure."

Both radii of curvature for a spherical interface are the same as the radius of the sphere. Spherical interfaces are commonly encountered in bubbles, dewdrops, raindrops, and meniscis in cylindrical capillary tubes. In the special case of spherical interfaces, the capillary force per area, or the capillary pressure, is inversely proportional to the radius of the sphere. Because of capillary forces, pressure inside a raindrop or bubble at equilibrium must be greater than the pressure outside. The difference between pressure inside and outside a drop or bubble is the capillary pressure, the magnitude of which is inversely proportional to the radius of the drop or bubble. This means that the smaller the size of a drop or bubble, the greater the pressure difference between inside and outside the drop or bubble.

With this understanding, it is possible to explain the behavior of the interface in the cylindrical tubes and the phenomenon of capillary rise. The glass surface has greater affinity for water than for air. When a clean, dry tube is inserted in water, the water from the container is pulled into the tube at the tube walls. As water rises, the weight of the resulting column of water inside the tube, supported by the line of contact between glass and water, resists any further rise. At the equilibrium, the pull at the contact line causes the air-water interface to achieve a radius of curvature that is proportional to the tube radius. This results in a suction capillary force at the interface that is inversely proportional to the radius of the interface. This capillary force is able to support a column of water of height inversely proportional to the tube radius.

Capillary forces are important at the boundary between two fluids that are different from each other and do not dissolve in each other. Such fluid combinations may consist of gas-liquid and liquid-liquid systems. For gas-liquid systems, the most obvious difference is in the density of the two fluids. For example, in the fluid pairs consisting of air-water, steam-water, nitrogen-water, or methane-oil, the gaseous fluid has much lower density than the liquid involved. At the boundary of the fluid pair, a step change in the density and other fluid properties is observed. Such step change in refractive index of the fluid at the boundary allows one to detect the boundary, which is often referred to as the "interface."

For liquid-liquid systems, the density contrast is usually not as significant as it is for gas-liquid systems. A more important characteristic for a liquid-liquid system is the nature of bonding in the molecules of the two liquids in the pair. For example, for many oil-water systems, the density contrast is not very large, but oil and water are very different as a consequence of the nature of bonding in oil and water molecules, respectively. Water molecules are held together by ionic bonds, causing water molecules to be very polar. On the other hand, molecules in most oils are held together by covalent bonds, causing the oil molecules to be nonpolar.

Usually, polar liquids do not mix with nonpolar liquids, resulting in a boundary with an interfacial tension. Interfacial tension is a result of the fact that the molecules at or near the boundary are at a higher energy level and therefore in an unstable state. When a larger boundary is created, additional energy must be added to the system to account for the greater number of higher energy molecules at the interface.

For a given radius of curvature, capillary force and pressure are directly proportional to the interfacial tension between the fluids in contact. The interfacial tension, in general, is inversely proportional to the temperature of the fluids. As the temperature increases, the interfacial tension for a fluid pair gradually declines. This, in turn, causes the capillary forces for a system to diminish in magnitude with increasing temperature.

For many fluid pairs, the interfacial tension decreases as the pressure increases. Such fluid pairs include water-water vapor, oil-nitrogen, oil-carbon dioxide, and oil-methane systems. The reduction in interfacial tension and capillary forces with pressure is related to the changes in the properties of fluids, in the pair, with pressure. As the pressure increases, the properties of the two fluids in the pair become more similar to each other; as a result, larger number of molecules of each fluid in the pair can dissolve in the other fluid. This is especially true for gas-liquid interfaces. At some very high pressure (referred to as the "critical pressure"), all properties of the two fluids in the pair are identical, and all molecules of one fluid can readily dissolve into the other fluid, resulting in only one fluid. At this pressure, by definition, there is only one fluid in the system, and the boundary disappears. Stated in the context of capillary forces, the interfacial tension for any fluid pair and capillary force is zero at the critical pressure. At pressures above the critical pressure, the two fluids are said to be "completely miscible," and the mixture of the two fluids is referred to as a "supercritical" fluid.

Below the critical pressure, the two fluids in a pair can be distinguished by the presence of a definable boundary or interface and are referred to as "immiscible" fluids. It is then reasonable to conclude that the concepts of interfacial tension, curvature, and capillary force are definable only at pressure values below the critical pressure.

For two immiscible liquids, the interfacial tension is much less sensitive to pressure, and the critical pressure may not exist. Two liquids are usually immiscible because of significant differences in their molecular composition. For example, water and most oils are immiscible because water molecules are polar and oil molecules are usually nonpolar. Water and alcohols of lower molecular weights are miscible because the polar character of smaller alcohol molecules controls their behavior. On the other hand, water is not miscible with alcohols of larger molecular weights because, in such alcohols, the very long covalently bonded organic chain controls their behavior.

Certain chemicals known as "surfactants" or "detergents" are capable of reducing the interfacial tension and capillary forces between two otherwise immiscible liquids. These detergents usually consist of molecules that have a polar character for part of the molecules and a nonpolar character for the remaining part. When a small concentration of such a chemical is added to a system of two immiscible fluids, the polar head of the molecule attaches to the polar liquid (such as water) and the nonpolar tail of the molecule attaches to the nonpolar liquid. It is easy to see that such configuration of stability for the detergent molecule is only available at the boundary between the two liquids. This means that the detergent molecules build up a layer at the boundary, which has the effect of significantly lowering the interfacial tension between the two liquids. The resulting reduction in capillary forces allows greater mixing between fluids.

Applications

Capillary forces form an integral part of everyday life. Many critical functions in the human body depend upon the presence of capillary forces.

As an example of the significance of capillary forces, consider that the growth of plants and the production of all plant-based foods are dependent on capillary forces. Capillary forces are responsible for transporting water and dissolved nutrients from the soil to the roots, to the branches via the stem, and then to the leaves and all other parts of the plants. The fibrous composition of the core of roots, stem, and branches causes the space between the fibers to act as fine capillaries.

Because capillary force is inversely proportional to the effective radius of the capillary, a very strong capillary suction force is created that is capable of pulling water from the soil all the way to the highest branches and leaves in the plants. To get an estimate of the magnitude of such natural capillary force, consider that although some trees are hundreds of feet tall, the leaves on the highest branches still get water and nutrients from the roots of the tree.

To some extent, the capillary forces in plants and trees are self-correcting. For example, in times of extremely dry weather, if enough water is not being fed to the branches, a net loss in moisture results, causing the spacing between fibers in the core to decrease. This has the effect of reducing the effective radius of capillaries and increasing the capillary suction. It is important to appreciate that, in the absence of the capillary forces, no grains, vegetables, or fruits would be produced. As all living beings directly or indirectly depend on plant products for their survival, capillary forces are equally important for animal-based food supply.

Most personal-hygiene paper and other fiber-based products depend upon the presence of capillary forces for their effectiveness. Soaps, shampoos, and detergents all perform by reducing interfacial tension, which, in turn, results in a reduction in capillary forces between the water used as cleansing fluid and the oils that attach dirt and contaminants to skin and clothes. The reduction in capillary forces allows oils to be dissolved in water, thereby detaching dirt and soil and rendering the surfaces clean.

When one uses cloth or paper towels to dry objects after rinsing with water, capillary-force-driven suction of water into the dry towels is critical to the towels' ability to suck moisture from the wet surface. Numerous other products are based on the ability of capillaries to absorb water as a result of capillary forces. Most absorbent materials used to contain spills are based upon the action of capillary forces.

Another manifestation of capillary forces is in the flow of water and other fluids in soils and other subsurface sediments. Rainwater tends to move deeper into the subsurface based on gravity forces. However, capillary suction forces retain water in pore spaces between soil grains, which, in this application, act as capillaries. In this context, the connected fluid-flow channels formed in the space between soil grains constitute fine capillaries. The size of such capillaries is controlled by the grain size and the level of compaction of the soil. When the effective radius of such capillaries is smaller, the capillary force and amount of water retained becomes greater. The net amount of water retained in the soil is determined by a balance between the gravity force and the capillary force acting on water molecules. As a result, a significant volume fraction of pore space in the soils above the water table may be occupied by water as a consequence of capillary forces. Therefore, the roots of plants and trees that do not reach the water table may still contact the capillary-held water. If the capillary suction forces in the roots are stronger than in the soil, such water can be transferred to the roots.

Another common manifestation of capillary forces is in the process of writing with pens. Capillary forces are necessary for the ink to be absorbed on the surface of the paper. If these forces are too strong, the ink may spread too much, as when one tries to write on a "blotting" paper. On the other hand, if the surface has no capillary forces, the ink may not adhere to the surface, as when one tries to write on a plastic film with an ordinary ink pen. Further, as the ink from the tip of the pen is transferred to the surface of paper, ink must be replenished to the tip from an ink reservoir. This process is also driven by the capillary forces acting in the capillaries surrounding the tip of a pen. A related application can be observed in the printing of documents, a process that requires the ink from the printer to be absorbed on the surface of the paper.

Context

Capillary forces surround all forms of matter and affect virtually every aspect of life. Observations of liquid rise in a narrow capillary and other similar manifestations of capillary forces, as recorded in history, can be traced back to ancient times. Recorded explanations for capillary forces can be found at least as early as the times of Leonardo de Vinci. In 1712, Brook Taylor, in a letter to Hans Sloane of the Royal Society of London, reported his observations concerning the rise of water in the capillary channel formed between two glass plates. He reported that the rise pattern was hyperbolic in nature and that the location of the asymptote in the hyperbola formed by the boundary between water and air depended on the angle between the glass plates. In 1805, Thomas Young, in a famous paper presented to the Royal Society of London, discussed the concept of mean curvature of a surface. In 1806, Pierre Simon Laplace presented an expression for the curvature of a surface analytically derived from fundamental principles. Laplace was among the first credited with pointing out that the phenomenon of surface tension results from the necessity that the intermolecular attraction is unequally distributed among the molecules near the surface of the liquid. In 1830, Carl Friedrich Gauss explained capillary phenomena on the basis of the principle of virtual work.

Capillary forces and other manifestations of capillarity are so ubiquitous that they are often not noticed. The importance of these phenomena can be realized by observing that many critical functions in the life process of all plants and living beings depend on capillary forces. In addition, behaviors and properties of physical systems such as soils, paints, inks, lubricants, and adhesives, among many others, depend upon capillary forces. The history of science, or any other subject, literally could not be written or printed without the existence of capillary forces. This phenomenon became the subject of many detailed investigations and analysis by some of the most noted scientists of the nineteenth century. After a period of diminished activity during the first half of the twentieth century, demands in medicine and industry rekindled research activities into capillary phenomena in the later part of the twentieth century. The discovery and exploitation of major subsurface oil and gas reservoirs during the same period led to an in-depth investigation of the application of capillary forces. Some of the most successful technologies for improving the production of oil and gas depend on the successful manipulation and control of capillary forces in reservoir rocks to produce increased quantities of oil and gas more efficiently.

Advances in the understanding and applications of capillary phenomena cross disciplinary boundaries and have truly matured into a rigorous discipline that unifies the principles of physics, chemistry, mathematics, and biology. As the frontiers of science are pushed further, and as miniaturization and nanotechnology become more important, an enhanced understanding of capillary phenomena will make feasible their application toward obtaining solutions to a new set of challenging problems.

Principal terms

capillary tube: a tube of cross section so small that capillary forces play an important role in fluid transport through it; capillary tubes may have circular or noncircular cross sections

contact angle: the angle between a reference surface and a tangent to the boundary between two fluids in contact with the surface, measured through the fluid that has greater relative affinity for the surface (wetting fluid); the contact angle is a manifestation of the wetting characteristics of a surface

curvature: a measure of the deviation of a curve from a straight line, or a measure of the deviation of a curved surface from a plane

interfacial tension: the force per unit length of a boundary between two fluids; also defined as energy per unit area of the interface (boundary) between two fluids

wettability: the relative affinity of a surface to a fluid among the two fluids in contact with the surface

Bibliography

Adamson, Arthur W. Physical Chemistry of Surfaces. New York: John Wiley & Sons, 1982. This exceptionally well-written book provides a wealth of information on physical and mathematical aspects of capillary forces. It also presents various physical and chemical processes that determine the magnitude of capillary forces and various applications in which capillary forces are important. Excellent illustrations, references, clear explanations and examples, and a detailed index.

Bikerman, J. J. Physical Surface. New York: Academic Press, 1970. Volume 20 of a monograph series in physical chemistry edited by Ernest M. Loebl. Addresses physics and chemistry of the surfaces and capillary phenomena. Begins with an easy introduction and builds up concepts gradually, including the mathematical treatment of capillary phenomena, supported by examples. Good illustrations, references, author and subject index.

Finn, Robert. Equilibrium Capillary Surfaces. New York: Springer-Verlag, 1986. Gives detailed information on the concepts, physics, and mathematics of capillary surfaces and forces. Also provides good historical understanding of the study of the phenomena of capillary forces, including a copy of a 1712 letter from Brook Taylor documenting the capillary rise of water between two glass plates. Illustrations, clear explanations, and a bibliography.

Israelachvili, Jacob N. Intermolecular and Surface Forces. San Diego: Academic Press, 1992. Presents excellent historical and modern views of capillary forces and of short-range and long-range forces that constitute capillary and other forces of nature. Discusses fundamental concepts as well as advanced ones. Assumes a basic knowledge of physics, chemistry, and mathematics, but the analysis is at a simple level. Most equations are derived from first principles and are followed by examples of how they apply to specific situations. Excellent illustrations, references, clear explanations and examples, references, and an index.

Lake, Larry W. Enhanced Oil Recovery. Englewood Cliffs, N.J.: Prentice-Hall, 1989. A comprehensive treatment of the applications of capillary forces in improving oil and gas recovery from subsurface reservoirs. Presents manipulation of capillary forces as one of the important ways of mobilizing oil held by capillary action in the interstices of porous rocks. Excellent presentation of applications, illustrations, reference, author and subject index.

Taylor, Wilson. A New View of Surface Forces. Toronto: University of Toronto Press, 1925. A collection of scientific papers that provide deep, almost poetic insight into capillary and other interfacial force phenomena. The book also provides a historical context for capillary forces and contains a paper on the poetry of mathematics. Good illustrations, poetry, crystal-clear explanations, and references.

Van Oss, Carol J. Interfacial Forces in Aqueous Media. New York: Marcel Dekker, 1994. Includes treatment of a surface and solution chemistry. Bibliographical references and index.