Waves in composites and metamaterials/Anisotropic mass and generalization

The content of these notes is based on the lectures by Prof. Graeme W. Milton (University of Utah) given in a course on metamaterials in Spring 2007.

Rigid Bar with Frequency Dependent Mass edit

Recall the a rigid bar with   cavities, each containing a spring-mass system with mass   and complex spring constant   (see Figure 1).

Figure 1. A rigid bar containing   voids. Each void contains a spherical ball that is attached to the bar by springs.

The momentum of the bar ( ) is related to its velocity ( ) by the relation


where   is the frequency and   is the effective mass which is given by


where   is the mass of the rigid bar~.

Let us now consider a specific model for the springs. A simple model is the one-dimensional Maxwell model shown in Figure 2. In this case instead of a spring with a complex spring constant we have an elastic spring (with real spring constant  ) and a dashpot (with a real viscosity  ) in series. Let   be the displacement of the elastic spring and let   be that of the dashpot under the action of a force  .

Figure 2. The Maxwell model for a spring with complex spring constant.

For the elastic spring, we have


For the dashpot, we have


Once again, assuming that the functions  ,  , and   can be expressed as harmonic functions, we have


Plugging equations (3) into equations (1) and (2), we get


Recall that the displacement   of the sphere of mass   inside the cavity is related to the applied force   by the relation


Now,  . Hence,




So, with the Maxwell model, the effective mass is




This model is remarkably similar to a simple model for the frequency dependent dielectric constant  .

Comparison with a simple model for   edit

Consider the following simple model of an electron bound to an atom by a harmonic force under the influence of a slowly varying electric field (a detailed description can in found in Jackson75, Sec. 7.5). A schematic of the situation is shown in Figure 3.

Figure 3. An electron bound by a harmonic force.

Let the electric field be   and assume that it varies slowly in space over the distance that the electron moves, i.e.,  . Let   be the mass of the electron and let   be its charge. Let   be the frequency of the harmonic force binding the electron to the atom and let   be a damping coefficient due to interactions with obstacles.

Then, the equation of motion of the electron is given by




Plugging equations (7) into equation (6), we get




Let   be the charge density of the electron-atom system, i.e.,


where   is the Dirac delta function and   is the position of the atom. Then, the dipole moment (the first moment of the charge) contributed by the electron-atom system is given by




Suppose now that instead of a binding frequency   for all the atoms and all the electrons in a volume, there are   atoms per unit volume with binding frequencies   and damping constants   and. Then, we can write the polarization as


Recall that the electric displacement is related to the electric field and the polarization by


Therefore, from equations (8) and (9) we get


Comparing equation (5) with equation (10) we observe that they have the same form which implies that the effective permittivity is analogous to the effective mass. This also implies that the electric field is analogous to the velocity. Similarly, the polarization is analogous to the momentum.

More on frequency dependence of the mass edit

More generally, we get a frequency dependent density if all the constituents do not move in lock step. Lock step motion almost never occurs because there are thermal vibrations at the microscale. There are also many macroscopic situations in which lock step motion does not occur. Consider the example of a porous rock containing some water (see Figure 4). Both the rock grains and the water are connected, though this is not obvious from the figure

Figure 4. A porous rock containing some water.

In this case, the water will move with a different frequency that the rock and the density of the composite will be dependent on the frequency.

At a molecular level, we can have a crystal with lead atoms attached by single bonds to the structure (see Figure 5). Presumably, the resonant frequency of such molecules is very high. so we will see the frequency dependence of the mass only at very high frequencies.

Figure 5. A crystal lattice containing lead atoms.

In fact, Sheng et al. Sheng03 have shown in experiments that materials indeed have frequency dependent masses. An example of such a material is shown in Figure 6.

Figure 6. A composite material with frequency dependent mass.

Generalizations of the Rigid Bar Model edit

Consider the rigid body containing cavities shown in Figure~7. This is just a two-dimensional extension of the model shown in Figure~1. Here   and   are complex spring constants in the   and   directions.

Figure 7. Schematic of a material with an anisotropic mass density. In this case the springs in each cavity are parallel to each other.

In this case, the effective mass along the  -direction is given by


while that along the  -direction is given by


In matrix form, we then have


Hence, the effective mass is clearly anisotropic. Note however that from a macroscopic perspective it is not the average velocity in the matrix which is important. In fact, such a quantity does not even make sense because the velocity is not defined in the void phase. Rather it is the velocity of the matrix that is the relevant quantity in this model.

One can generalize the model one step further by having the springs be oriented at different angles to each other as shown in Figure 8. Also let the springs in each cavity have different spring constants and the masses in each cavity are different.

Figure 8. Schematic of a material in which the springs in each cavity are oriented at various angles to each other.

In this case, let   be the rotation that is needed to orient each set of springs with the   and   axes. Then, if   is the rotation matrix for cavity   containing a mass  , and if   and   are the complex spring constants for that cavity, the effective mass can be written as


The eigenvalues of   can therefore depend on  .

General form of edit

Take any body with a rigid matrix. Suppose, instead of applying harmonically varying velocities, we apply a time varying velocity   and observe what the momentum   is. There will be some linear constitutive relation


The kernel   is second-order tensor valued and may possibly be singular. Also, since both   and   are physical and real,   must be real. Causality implies that   when   (or  ) since the inertial force cannot depend on velocities in the future.

Taking the Fourier transform of equation (13) and using the convolution theorem, we get




The quantity   can be shown to satisfy the Cauchy-Riemann equations only if  . This is a consequence of causality   and the fact that the integral in equation (15) only converges in the upper half of the complex   plane. \footnote{ Note the positive sign in the power of   in equation (15). This occurs because we have chosen the inverse Fourier transform to be of the form


The effect of such a choice is that the imaginary part of   is positive instead of negative.} Hence,   is analytic in   when  . The quantity   is real.

Now, the complex conjugate of   is given by


where   denotes the complex conjugate of   (for any complex  ).

Also assume that, for large enough frequencies, the dynamic mass tends toward the static mass, i.e.,


Equation (15) can be used to establish the Kramers-Kronig equations for the material. To do that, recall Cauchy's formula for a function   which is analytic on a domain that is enclosed in a piecewise smooth curve  :


Since the function   is analytic on the upper-half   plane, for any point   in a closed contour   in the upper-half   plane, we have


Let us now choose the contour   such that it consists of the real   axis and a great semicircle at infinity in the upper half plane (see Figure~9).

Figure 9. The closed curve   that is used to evaluate the integral in equation (18).

Also, from equation (17) we observe that


Hence, there is no contribution to the integral in equation (18) due to the semicircular part of the contour and we just have to perform an integration only over the real line:


Now, let us consider the integral


There is a pole at the point   (in the figure, the contour is shown as a semicircle of radius   centered at the pole). In the limit  , this integral may be interpreted to mean the Cauchy principal value (not the same as principal values in complex analysis)


From Figure~9 we observe that this integral can be broken up into the integral over the path   minus the sum of the integrals over the paths   and  . From the Cauchy-Goursat theorem, the integral over the closed path   is zero. We have also seen that since   as  , the integral over   is zero. The integral over the path   around the pole is obtained from the Residue Theorem (where the value is divided by two because the integral is over a semicircle), i.e.,


The negative sign arises because the curve is traversed in the counter-clockwise direction.





Letting   and taking the limit as  , we get


Note that, in the above equation, both   and   are real. Expanding equation (20) into real and imaginary parts and collecting terms, we get the first form of the Kramers-Kronig relations


Therefore, the real part of the frequency dependent mass can be determined if we know the imaginary part and vice versa.

We can also eliminate the negative frequencies from equations (21). Recall from equation (16) that


Since, in equations (21),   and   are real, we have


This implies that


Consider the first of equations (21). We can write this relation as


Similarly, the second of equations (21) may be written as


Therefore, the alternative form of the Kramers-Kronig relations is


Significance of   edit

Consider harmonically varying   and   given by


Alternatively, we may write these as




which implies that


The average rate of work done on the system in a cycle of oscillation will be


This quadratic form will be non-negative for all choices of   if and only if   is positive semidefinite for all real  . Note that the quadratic form does not contain  . Since the work done in a cycle should be zero in the absence of dissipation, this implies that the imaginary part of the mass is connected to the energy dissipation (for instance, into heat).

References edit

J. D. Jackson. Classical Electrodynamics: 2nd Edition. John Wiley and Sons, New York, 1975.

P. Sheng, X. X. Zhang, Z. Liu, and C. T. Chan. Locally resonant sonic materials. Physica B. Condensed Matter, 338:201--205, 2003.