Fresnel Diffraction

In Section 13.7 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the medical uses of ultrasound. One important problem we analyze is the pressure distribution produced by a piezoelectric transducer.

There are some important features of the radiation pattern from a transducer which we review next. Consider a circular transducer, the surface of which is oscillating back and forth in a fluid… Each small element of the vibrating fluid creates a wave that travels radially outward, the points of constant phase being expanding hemispheres. The amplitude of each spherical wave decreases as 1/r, the intensity falling as 1/r². We want the pressure at a point z on the axis of the transducer. It is obtained by summing up the effect of all the spherical waves emanating from the face of the transducer….

The [average intensity] is plotted in Fig. 13.13 for a fairly typical but small transducer (a = 0.5 cm, f = 2 MHz)... Close to the transducer there are large oscillations in intensity along the axis: there are corresponding oscillations perpendicular to the axis, as shown in Fig. 13.14. The maxima and minima form circular rings. This is called the near field or Fresnel zone… The depth of the Fresnel zone is approximately a²/λ [where a is the radius of the transducer, λ is the wavelength, and f is the frequency].

Fig, 13.13.

The calculated intensity along the axis, as shown in our Fig. 13.13, is interesting. In the Fresnel zone, the intensity has many points where it is zero. In Intermediate Physics for Medicine and Biology we calculate why this happens mathematically, but it is illuminating to describe what is happening physically. Basically, this is a result of wave interference. Our statement that “each small element of the vibrating fluid creates a wave that travels radially outward” is often called Huygens principle. Each point on the face of the transducer produces such a wavelet. To understand the pressure distribution, we must examine the phase relationship among these various wavelets. Very near the face of the transducer, the waves that contribute significantly to the pressure are in phase; they all interfere constructively and you get a maximum (evaluate Eq. 13.39 at z = 0 and you get a nonzero constant). However, as you move away, more distant points on the transducer face contribute to the pressure on the axis, and these points may be out of phase with the pressure produced by the point at the center. For some value of z the in-phase and out-of-phase wavelets interfere destructively, resulting in zero intensity. Increase z a little more, and not only do the in-phase points at the center and the out-of-phase points just away from the center contribute to the pressure, but so do some in-phase points even farther from the center. When you add it all up, you get a net constructive interference and a non-zero intensity. And so it goes, as you move out farther and farther along the z axis.

Fig. 13.14.

The radial distribution of the intensity is surprisingly rich and complex, given the rather simple integral that underlies the behavior. If you want to explore the radial distribution in more detail, go to the excellent website http://wyant.optics.arizona.edu/fresnelZones/fresnelZones.htm, where you can perform these calculations yourself. You can adjust the parameters as you wish and create plots such as those in Fig. 13.14, and also produce grayscale images of the full intensity distribution that provide much insight. The website was produced with optics in mind, so you have to put in strange looking parameters to model ultrasound. To reproduce the middle panel of Fig 13.14, input 770,000 for the wavelength in nm, 10,000 for the aperture diameter in microns, and 15.75 for the observation distance in mm. To my eye, the agreement between the website’s calculation and Fig. 13.14 is impressive. At small values of z the plots get very complex and beautiful. For the same wavelength and aperture, I like the richness of z = 5 mm, and for z = 4 mm you get a fairly uniform brightness except for a dramatic dark spot right at the center. It reminds me of Poisson’s spot, which I discussed in the September 17, 2010 entry in this blog, about Augustin-Jean Fresnel. Indeed, the physics behind the calculations in Fig. 13.14 and Poisson’s spot in optics are nearly identical. The circular aperture is a classic problem in Fresnel diffraction. You can find a more detailed discussion of this topic in the textbook Optics (4th edition), by Eugene Hecht. (My bookshelf contains the first edition, by Hecht and Zajac, that I used in my undergraduate optics class at the University of Kansas).

If you want to be clever, you could make the ultrasound transducer vibrate only at those radii that result in constructive interference along the axis, and have it remain stationary at radii that cause destructive interference. (Of course, this would mean you would have to design your transducer face cleverly so concentric rings vibrate, separated by rings that do not, which might make constructing the transducer more difficult.) Using such a trick eliminates the dark spots along the z axis, increasing the intensity there. This method is commonly used to focus light waves, and is called a zone plate. It has been used occasionally with ultrasound.

The Nonlinear Poisson-Boltzmann Equation

Last week’s blog entry was about the Gouy-Chapman model for a charged double layer at an electrode surface. The model is based on the Poisson-Boltzmann equation (Eq. 9.10 in the 4th edition of Intermediate Physics for Medicine and Biology). One interesting feature of the Poisson-Boltzmann equation is that it is nonlinear. In applications when the thermal energy of ions in solution is much greater than the energy of the ions in an electrical potential, the equation can be linearized (Eq. 9.13). That is not always the case.

Homework problem 9 in Chapter 9 of Intermediate Physics for Medicine and Biology was added in the 4th edition. It begins

Problem 9 Analytical solutions to the nonlinear Poisson-Boltzmann equation are rare but not unknown. Consider the case when the potential varies in one dimension (x), the potential goes to zero at large x, and there exists equal concentrations of monovalent cations and anions. Chandler et al. (1965) showed that the solution to the 1-d Poisson-Boltzmann equation, d²ζ/dx²=sinh(ζ), is…

You will need to get a copy of the book to see this lovely solution. It is a bit too complicated to write in this blog, but it involves the exponential function, the hyperbolic tangent function, and the inverse hyperbolic tangent function. I like this homework problem, because you can solve both the nonlinear and linear equations exactly, with the same boundary conditions, and compare them to get a good intuitive feel for the impact of the nonlinearity. I admit, the problem is a bit advanced for an intermediate-level book, but upper-level undergraduates or graduate students studying from our text should be up to the challenge.

The full citation to the paper by Knox Chandler, Alan Hodgkin, and Hans Meves mentioned in the problem is

Chandler, W. K., A. L. Hodgkin, and H. Meves (1965) “The Effect of Changing the Internal Solution on Sodium Inactivation and Related Phenomena in Giant Axons,” Journal of Physiology, Volume 180, Pages 821–836.

I always thought it odd that one finds a really elegant analytical solution to the nonlinear Poisson-Boltzmann equation in a paper about sodium channel inactivation in a squid nerve axon (with Nobel Prize-winning physiologist Alan Hodgkin as a coauthor). The solution is buried in the discussion (in a section set of in a smaller font than the rest of the paper). The reason for its appearance is that Chandler et al. found changes in membrane behavior with intracellular ion concentration, and postulated that the measured voltage drop between the inside and outside of the axon consisted of a voltage drop across the membrane itself (which affects the ion channel behavior) and a voltage drop within a double layer adjacent to the membrane. It is the double layer voltage that they model using the Poisson-Boltzmann equation.

Nowadays, the nonlinear Poisson-Boltzmann equation is typically solved using numerical methods. See, for example, the paper that Russ Hobbie and I cite in Intermediate Physics for Medicine and Biology, written by Barry Honig and Anthony Nicholls: “Classical Electrostatics in Biology and Chemistry,” Science, Volume 268, Pages 1144–1149, 1995 (it now has over 1500 citations in the literature). Their abstract states

A major revival in the use of classical electrostatics as an approach to the study of charged and polar molecules in aqueous solution has been made possible through the development of fast numerical and computational methods to solve the Poisson-Boltzmann equation for solute molecules that have complex shapes and charge distributions. Graphical visualization of the calculated electrostatic potentials generated by proteins and nucleic acids has revealed insights into the role of electrostatic interactions in a wide range of biological phenomena. Classical electrostatics has also proved to be a successful quantitative tool yielding accurate descriptions of electrical potentials, diffusion limited processes, pH-dependent properties of proteins, ionic strength-dependent phenomena, and the solvation free energies of organic molecules.

Such calculations continue to be an active area of research. See, for example, “The Role of DNA Shape in Protein-DNA Recognition” by Remo Rohs, Sean West, Alona Sosinsky, Peng Liu, Richard Mann and Barry Honig (Nature, Volume 461, Pages 1248–1253, 2009).

Gouy and Chapman

Sometimes when I am studying physics, I run across a model or equation with names attached to it, and I wonder “just who are these people?” For example, in Chapter 9 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the Gouy-Chapman model.

In this section we study one model for how ions are distributed at the interface in Donnan equilibrium. The model was used independently by Gouy and by Chapman to study the interface between a metal electrode and an ionic solution. They investigated the potential changes along the x axis perpendicular to a large plane electrode. The same model is used to study the charge distribution in a semiconductor.

So who are Gouy and Chapman? I can tell they lived a long time ago, because in the next section we write “In an ionic solution, ions of opposite charge attract one another. A model of this neutralization was developed by Debye and Huckel a few years after Gouy and Chapman developed [their] model.” I know Peter Debye worked in the early days of quantum mechanics, so Gouy and Chapman had to be a bit earlier.

As if often the case with scientific developments in the 19th century, a Frenchman and an Englishman share credit for the discovery. Louis Georges Gouy (1854–1926) was a French experimental physicist from the University of Lyon. He is best known as the inventor of the Gouy balance, a device for measuring magnetic susceptibility. Gouy had an interest in Brownian motion at a time when the atomic nature of matter was still an open question. Russ and I discuss Brownian motion in Section 4.3 of our book.

[The] movement of microscopic-sized particles, resulting from bombardment by much smaller invisible atoms, was first observed by the English botanist Robert Brown in 1827 and is called Brownian motion.

Albert Einstein wrote a fundamental paper on Brownian motion in 1905, his miraculous year. In a subsequent paper on the same topic, Einstein began (Annalen der Physik, Volume 19, Pages 371–381, 1906)

Soon after the appearance of my paper on the movements of particles suspended in liquids demanded by the molecular theory of heat, Siedentopf (of Jena) informed me that he and other physicists—in the first instance, Prof. Gouy (of Lyons)—had been convinced by direct observation that the so-called Brownian motion is caused by the irregular thermal movements of the molecules of the liquid.

Gouy’s model sought to explain the electrical potential around small Brownian particles, or colloids. He described the double layer of charge that develops at the surface, with one charge layer bound to the surface of the particle, and a layer of counterions in the surrounding fluid.

David Leonard Chapman (1869–1958) was an English physical chemist at Oxford. He was interested in the theory of detonation in gasses, and developed the Chapman-Jouget condition describing their behavior. About three years after Gouy, Chapman derived a model describing the double layer at a charged surface. The Gouy-Chapman model is an application of what is now known as the Poisson-Boltzmann equation (Eq. 9.13 in our book).

In 1924, physicist Otto Stern extended the Gouy-Chapman model by noting that ions can’t be represented as point charges when they are within a few ion radii of the surface. This leads to the Stern layer of immobile counter-ions right next to the surface, and a diffuse layer of counter-ions whose concentration decays exponentially.

Fisher-Kolmogorov equation

Mathematical Biology,
by James Murray.

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss many of the important partial differential equations of physics, such as Laplace’s equation, the diffusion equation, and the wave equation. One lesser-known PDE that we don’t discuss is the Fisher-Kolmogorov equation. However, our book supplies most of what you need to understand this equation.

In Section 2.10, we examine the logistic equation, an ordinary differential equation governing population growth,

du/dt = b u (1 − u) .

For u much less than one, the population grows exponentially with rate b. As u approaches one, the population levels off near a steady state value of u = 1. Our Eq. 2.28 gives an analytical solution to this nonlinear equation.

In Section 4.8, we drive the diffusion equation, which for one dimension is

du/dt = D d²u/dx² .

This linear partial differential equation is one of the most famous in physics. It describes diffusion of particles, and also the flow of heat by conduction. D is the diffusion constant.

To get the Fisher-Kolmogorov equation, just put the logistic equation and the diffusion equation together:

du/dt = D d²u/dx² + b u (1 − u) .

The Fisher-Kolmogorov equation is an example of a “reaction-diffusion equation.” Russ and I discuss a similar reaction-diffusion equation in Homework Problem 24 of Chapter 4, when modeling intracellular calcium waves. The only difference is that we use a slightly more complicated reaction term rather than the logistic equation.

Mathematical Biology,
by James Murray.

In his book Mathematical Biology, James Murray discusses the Fisher-Kolmogorov equation in detail. He states

The classic simplest case of a nonlinear reaction diffusion equation … is [The Fisher-Kolmogorov equation]… It was suggested by Fisher (1937) as a deterministic version of a stochastic model for the spatial spread of a favoured gene in a population. It is also the natural extension of the logistic growth population model discussed in Chapter 11 when the population disperses via linear diffusion. This equation and its travelling wave solutions have been widely studied, as has been the more general form with an appropriate class of functions f(u) replacing ku(1−u). The seminal and now classical paper is that by Kolmogoroff et al. (1937)…. We discuss this model equation in the following section in some detail, not because in itself it has such wide applicability but because it is the prototype equation which admits travelling wavefront solutions. It is also a convenient equation from which to develop many of the standard techniques for analyzing single-species models with diffusive dispersal.”

The Fisher-Kolmogorov equation was derived independently by Ronald Fisher (1890–1962), an English biologist, and Andrey Kolmogorov (1903–1987), a Russian mathematician. The key original papers are

Fisher, R. A. (1937) “The Wave of Advance of Advantageous Genes.” Annals of Eugenics, Volume 7, Pages 353–369.

Kolmogoroff, A., I. Petrovsky, and N. Piscounoff (1937) “Etude de l’equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique.” Moscow University Mathematics Bulletin, Volume 1, Pages 1–25.