# Principles of Soil and Plant Water Relations || Static Water in Soil

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• 6

Static Water in Soil

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We now look at how water interacts with the solid system of the soil. Inparticular, we shall study surface tension, and then see how it is relatedto the rise and fall of water in soil pores, which, in turn, explains hys-teresis.

I. SURFACE TENSION

We first recall the definitions of some terms from elementary physics(Kirkham and Powers, 1972, p. 11). An object is under tension if a pull isbeing exerted on it. In Fig. 6.1, the cross section A of the cylinder is undertension due to the forces F. Tension is a pull or stretching force per unitarea. Pressure implies a push and is a compression force per unit area. If wereverse the directions of the arrows in Fig. 6.1, the cylinder will be underpressure. In talking about soil water and plant water, we sometimes saythat the water is under stress. Stress may be either a pull or push, tension,or compression. So stress may properly be expressed as a pull or push perunit area.

The term surface tension should not be confused with tension(Kirkham and Powers, 1972, pp. 1112). Surface tension, or more specifi-cally, the surface tension coefficient, an energy per unit area, is equivalentlya force per unit length, whereas tension is a force per unit area. We abbrevi-ate the surface tension coefficient using the Greek letter sigma (s).

s = energy/area = (force) (distance)/area (6.1)

or

s = force/length. (6.2)

Surface tension may be compared with the force that develops in a sheet ofpaper when we pull it on opposite edges. In Figure 6.2 the force F when

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• divided by the length AB gives a surface tension coefficient s, which maybe denoted by

s = F/(2AB), (6.3)

where the 2 in the denominator is used because the sheet of paper has anupper and a lower surface even though the paper is thin.

Laplace (17491827), a French mathematician and astronomer,explained surface tension. (See the Appendix, Section IV, for a history ofsurface tension and the Appendix, Section V, for a biography ofLaplace.) A molecule in the body of a fluid (Fig. 6.3) is attracted equallyfrom all sides. But a molecule at the surface undergoes a resultant inward

68 6. STATIC WATER IN SOIL

FIG. 6.1 Illustration of tension. (From Advanced Soil Physics by Kirkham, D., and Powers,W.L., p. 11, 1972, John Wiley & Sons, Inc., New York. This material is used by permissionof John Wiley & Sons, Inc. and William L. Powers.)

FIG. 6.2 Surface tension in a sheet. (From Advanced Soil Physics by Kirkham, D., andPowers, W.L., p. 12, 1972, John Wiley & Sons, Inc.: New York. This material is used by per-mission of John Wiley & Sons, Inc. and William L. Powers.)

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• pull, because there are no molecules outside the liquid causing attraction.Hence, molecules in the surface have a stronger tendency to move to theinterior of the liquid than molecules in the interior have to move to thesurface. What results is a tendency for any body of liquid to minimize itssurface area. A molecule at the surface of a liquid is acted on by a netinward cohesive force that is perpendicular to the surface. So it requireswork to move molecules to the surface against this opposing force, andthe surface molecules have more energy than interior ones (Schaum,1961, p. 108).

This tendency to minimize surface area is often opposed by externalforces acting on the body of liquid, as gravity acting on a water drop rest-ing on a flat surface, or as adhesive forces between water and other materi-als. Thus, the actual surface may not be an absolute minimum, but rather aminimum depending on the conditions in which the body of liquid is found(Kirkham and Powers, 1972, pp. 1213).

The surface tension coefficient has been expressed as a force per unitlength. If a wire is pulled horizontally from beneath a liquid, as illustratedin Fig. 6.4, the force required to pull it out depends on the length of wire.Let the symbols in Fig. 6.4 be defined as follows:

F = upward pull required to balance surface tension forces (gravitationalforces neglected)

L = length of wire s = surface tension (units of force per unit length) d = distance wire is raised.

Then we have

F = 2(sL), (6.4)

SURFACE TENSION 69

FIG. 6.3 Laplaces surface tension theory. (From Advanced Soil Physics by Kirkham, D.,and Powers, W.L., p. 12, 1972, John Wiley & Sons, Inc.: New York. This material is used bypermission of John Wiley & Sons, Inc. and William L. Powers.)

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• where the 2 is used because the force to be overcome by surface tension isdeveloped on the two sides of the wire. (Note that the wire is circular, butwater adheres to two sides.) Now the work W required to pull the wireagainst surface tension forces through the distance d is

W = Fd. (6.5)

That is, using the relation F = 2(sL), we have

W = s(2Ld) (6.6)

or

s = W/(2Ld) (6.7)

or

s = W/(increased area of surface). (6.8)

That is, s is the energy stored in the surface per unit increase in its area. Soby pulling the wire out of a liquid, we can see that the coefficient of surfacetension may be expressed as the energy stored per unit area of increase inthe surface.

Surface tension causes the rise or fall of a liquid in a capillary tube.We are going to relate the rise of water in soil to the rise of water in capil-lary tubes, so we need to understand the rise of water in capillary tubes.The equation for the height of rise in a capillary tube, h, is (Schaum, 1961,p. 108)

h = (2s cos )/(rrg), (6.9)

70 6. STATIC WATER IN SOIL

FIG. 6.4 Wire being pulled from water with adhering water film. (From Advanced SoilPhysics by Kirkham, D., and Powers, W.L., p. 13, 1972, John Wiley & Sons, Inc.: NewYork. This material is used by permission of John Wiley & Sons, Inc. and William L.Powers.)

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• where

s = surface tension of the liquid r = radius of the tube r = density of the liquid a = contact angle between the liquid and tube (called the wetting angle in

terminology used in soil science) g = acceleration due to gravity.

We shall now prove this equation in a simple manner, recognizing thatmore complicated proofs exist using calculus (Porter, 1971).

Let us look at Fig. 6.5 (Schaum, 1961, p. 109). Consider the body ofliquid inside the tube and above the outside level. The vertical (downwardand upward) forces acting on it must balance. The downward force is itsweight. Remember from Chapter 2 that weight, w, is a force and w = mg,where m is mass and g is defined above.

weight of liquid inside tube = volume weight per unit volume

= r2h mg/V = r2h (m/V)g (6.10)

= r2hrg acting downward. (6.11)

The upward force is due to surface tension. Remember that surface tension,s, is a force/length and the length of the tube is its circumference, 2pr. Theforce is the perpendicular force, so to get the normal component we must

SURFACE TENSION 71

FIG. 6.5 Height of rise of a liquid in a capillary tube. (From Schaum, D., Theory andProblems of College Physics, p. 109, 1961, Schaum Publishing Co.: New York. This mate-rial is reproduced with permission of The McGraw-Hill Companies.)

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• multiply 2pr by cos a. So the upward force is 2prscosa, and this is theforce due to surface tension. For vertical equilibrium:

upward force = downward force

2rs cos a = r2hrg (6.12)

or

h = (2s cos)/(rrg). (6.13)

The meniscus in the capillary tube can be either convex or concave. (Inphysics, meniscus is defined at the curved upper surface of a column of liq-uid. It comes from the Greek word meniskos, which is a diminutive ofmene, the moon.) A convex meniscus is illustrated by mercury on glassand a concave meniscus is illustrated by water on glass (Fig. 6.6). Mercuryin contact with glass has an angle of contact of 130 to 140 degrees (Porter,1971, p. 447). For water in contact with most soil minerals the wettingangle, a (also abbreviated q), is close to zero (Linford, 1930), so in theequation for the height of rise in a capillary tube, we can take cos 0 = 1 orcos a = 1. However, in highly repellent soils, the contact angle is large.

Leon Linford (1930) developed a clever way to measure the wettingangle in soil by using mirrors and the well-known laws of reflection inphysics. The angle of incidence is the angle between the incident ray andthe normal to the reflecting surface at the point of incidence (Schaum,1961, p. 214) (Fig. 6.7). The angle of reflection is the angle between thereflected ray and the normal to the surface. The laws of reflection are: 1) The incident ray, reflected ray, and normal to the reflecting surface lie inthe same plane. 2) The angle of incidence equals the angle of reflection.Concave mirrors form real and inverted images of objects located outsideof the principal focus; if the object is between the principal focus and the

72 6. STATIC WATER IN SOIL

FIG. 6.6 Angles of contact. (From Advanced Soil Physics by Kirkham, D., and Powers,W.L., p. 22, 1972, John Wiley & Sons, Inc.: New York. This material is used by permissionof John Wiley & Sons, Inc. and William L. Powers.)

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• mirror, the imagine is virtual, erect, and enlarged. Convex mirrors produceonly virtual, erect, and smaller images. Linford (1930) beamed light downinto soil and said,

Any point of the [water] meniscus, acting as a cylindrical concave mirrorwould reflect the light back and upwards at an angle such that it is twice theangle between the tangent to the meniscus at the point in question and the verti-cal. If the angle of