Surface Tension

Surface tension is both a very complicated and simple phenomenon. While it is an integral part of the high school physics curriculum, I have not found any sources which show original work in this field. It would have been nice to have Landau or Feynman give their detailed thoughts on this. However, even books like Halliday and Resnick, avoid this topic. I have not found a book that derives the various practical parameters from basic equations.

First, the case of a drop is the simplest possible: the idea that the energy minimization principle gives a body with the least surface area for a given volume. Although mathematically difficult to prove, it is an accepted fact that this is a sphere. Another is a loop of wire placed on a soap film—given the wire is flexible enough, the surface minimization principle gives that the area of the hole has to be maximum for a given perimeter so that the area of the film itself is minimum.

Now that we have the easy stuff out of the way, let us look at the more difficult problem: namely, the angle of contact, and what that implies. Let us define the adhesive force as the force between the liquid and the material that is in contact with it—typically a glass capillary. The cohesive force is defined as the force between the molecules of the liquid. The shape of the liquid surface is determined by the interplay between the adhesive and cohesive forces. A liquid is said to wet a surface if the adhesive force is greater than the cohesive force.

Let us list the known facts about some of the effects of surface tension and the adhesive force. A capillary tube dipped in water will have water rising in it, whereas the same capillary if dipped in liquid mercury will have the liquid surface dipping inside (capillary fall rather than rise). Mercury does not wet glass.

What is the force required to separate two glass plates? The water that is trapped in between the plates will have a lower pressure compared to atmospheric, so that if the plates are to be pulled apart, a force is required.

Consider a glass capillary with some water trapped inside it. What is the shape of the meniscus for the upper and lower part? Clearly, for maximum height, the upper part should be concave and the lower part should be convex. What if the water available is smaller than this? The angle of contact cannot change if the water is in the middle of the capillary, so how can the radius of curvature change? The rise of water in a capillary of insufficient height is explained by saying that the angle of contact does not change, but the apparent angle changes due to the curving of the top of the tube. In the case of water in the middle of the tube, such an explanation is not possible.

A program for calculation of the surface tension effects is available as a free download on the Internet. This is the surface evolver program, which runs on all common platforms. The source code is also available. Apart from the general questions listed above, this can also analyze exotic mathematical shapes. Practical problems like brazing, where wetting of the braze alloy on the substrates is of concern, can also be addressed.

Mechanical Waves

Introduction

This post aims to clarify questions I have asked myself, regarding mechanical and their properties. Consistent with the current emphasis on “modern” physics, there is a lot of information out there on electromagnetic waves. Indeed, there is no question about the momentum carried by light, either in classical or in quantum physics. The classical momentum per unit area of a wave is given by I/c, where I is the intensity of the wave and c^† is the speed of light. In quantum mechanics, a photon of frequency ν has energy hν and has a momentum of h/λ^†.

Mechanical Waves

Consider the simplest possible mechanical wave, the one traveling on a stretched string of tension T. This is a transverse wave, where the individual particles travel in a direction perpendicular to that of the wave. Consider sound, which is a longitudinal wave in air. People have no problem thinking about the linear momentum of sound waves, as they think it has something to do with the motion of the particles along the direction of propagation.

However, let us get back to the waves on the stretched string. Consider a small element dl of the string. Let the angles made by the two points on either end of the string be θ₁ and θ₂ respectively. Now, T(sinθ₁ – sinθ₂) provide restoring force. What does the cos component do? It serves to accelerate the string in the horizontal direction, so that the momentum density is given by P/c², where P is the power of the wave. Thus, we find that the momentum crossing a particular position of the string carrying a mechanical wave is P/c, which in retrospect it had to be.

Let us consider the question in detail. In order to prevent longitudinal waves from being generated, we need a string of negligible Young’s modulus. If you think in terms of the spring constant k, think of a very small value, and a large initial value x₀ so that we get a finite value of T. Thus, the value of T is not affected by the shape or length of the string, to a first approximation. However, since the motion of the string is due to the different positions of the element dl, we do need to consider the angle at which it acts. Let T_x be the x-component of the tension and T_y be the y-component. We know that ds² = dx’² + dy²

Wave Reflection and Material Properties

Some materials are significantly stronger in compression rather than in tension. For instance, concrete can withstand large compressive forces, while even small tensile stresses will cause failure. This is the reason why it is typically reinforced with steel bars. Consider Alexander’s assault on Tyre—the projectiles fired by his army caused pieces of the wall to come flying out at the other end. This can be explained by the mechanical waves generated by the impact of the projectile. The compressive wave is reflected at the free surface (the other side of the wall), and there it becomes a tensile wave. This causes tensile failure, where a piece flies off. The process is called spalling.

Quantum Mechanics

Particles tunnel across a potential barrier, and can come out of a finite potential well. However, particles in a region of high potential can be reflected by the boundaries of the potential plateau, just like waves being reflected at the open ends. This is called “anti tunneling”.

Notes

Incidentally, the letter c for speed of light comes from the Latin celeritas, for speed. I guess the well educated ad people hired by Intel thought that the general public would make this association. However, they chose to go with the properties of a cheap vegetable instead. C’est la vie.

This is the principle of the “solar sail”, which is considered a viable option for interstellar travel (don’t ask how you can stop once you get there at the speed of light!).