# Rigorous bounds for the effective permittivity of a composite of two isotropic dielectrics

## Introduction

Consider a particulate composite consisting of two isotropic dielectric media with complex relative permittivities eps_a and eps_b, respectively. Under certain conditions the composite can be homogenized, i.e., replaced by a homogeneous dielectric medium with the same macroscopic electromagnetic response and a certain effective permittivity. It is the basic goal of homogenization theory to calculate or, at least, to estimate the effective permittivity of this homogenized composite medium (HCM). Based on very general physical principles, rigorous bounds have been derived by a number of authors which confine the region of physically admissible effective permittivities in the complex plane. By use of these bounds it is often possible to fix the value of the effective permittivity within a few percent or less. This makes the rigorous bounds an important tool also for practical applications. It is the purpose of the presented JAVA-applet to visualize these rigorous bounds. Additionally the estimates of the most commonly used homogenization formalisms, the Maxwell Garnett and the Bruggeman formalisms, are displayed. I hope that you enjoy playing around with the applet. Even if you have no further research interests in homogenization theory, you may find the resulting graphs appealing from an esthetical point of view. For those who want to learn more about the theory of rigorous bounds (which is also beautiful, by the way), I put some references at the bottom of this page.

## How to use the applet

• Enter the complex relative permittivities eps_a and eps_b of material a and b, respectively. The imaginary parts of the permittivities must be non-negative. At least one of the permittivities must have a positive imaginary part (in the current version of the applet).
• Enter the filling factor f_b of medium b. If you leave this field empty, only the Wiener (W) bounds will be shown. If you enter a value for f_b (0 < f_b < 1), the Hashin-Sthrikman (HS) bounds will be displayed too. Furthermore the following effective medium estimates are plotted:
• the Maxwell Garnett estimate (MG)
• the Bruggeman estimate (Br)
• the Maxwell Garnett estimate with the role of the host and inclusion interchanged (MG1).
• If the homogenized composite is isotropic, click on the checkbox Isotropic mixture. This will cause the applet to calculate the W, HS and the Bergman-Milton (BM) bounds.
• If the material b percolates, click on the checkbox b percolates. In addition to the W, HS, and BM bounds, also improved bounds are calculated which take into account the percolation of material b (PB bounds).
• After the data input, start the applet by clicking on the OK-Button. The bounds will then be displayed in a separate frame outside the browser window. Dependent on the input the W, HS, BM, or PB bounds will be shown.
• In many cases, the Wiener bounds are very coarse, whereas the HS-bounds already restrict the possible values of the effective permittivity considerably (and the BM bounds do this even more!). Use Zoom in this case to show the HS bounds in full size. Click on Zoom a second time to zoom out the BM bounds to full size. Click on OK to return to the original image.
• Click on New Frame, when you want to display results in a new frame. The frame will show up only after clicking on OK.
• Unless you use New Frame, the data will always be displayed in the frame which is currently active. If you wish to activate a certain frame, simply click on it with the mouse.

## "Guided tour"

• Use the default values for the relative permittivities eps_a and eps_b and click on the OK-button. Only the Wiener bounds are displayed (in red color), because the filling factor has not yet been specified.
• Now chose a value for the filling factor f_b, say f_b=0.2, and then click on OK. The HS bounds will appear on the graph in green color. They are allready quite narrow. Resize the frame to maximum size in order to make details visible.
• Click on the checkbox Isotropic mixture and then on OK. The BM bounds will appear in orange. They are already so narrow that they hardly can be seen.
• Click on Zoom to zoom out the HS bounds.
• Click on the checkbox b percolates and then on OK.Zoom twice to bring the BM and PB bounds to full size. Note that the region bounded by the PB bounds is fully contained in the BM bounds. The same is true for the BM in the HS, and the HS in the W bounds.
• Click on New Frame, so that the next graph will appear in a separate frame.
• For material b, enter the permittivity -2 + i*0.1 (i.e, type -2 into the left and 0.1 into the right input field). Leave eps_a and f_b unchanged. Click on OK. The allowed values for the effective permittivity now cover a large area in the complex area which indicates that the effective permittvity in this case depends sensitively on microstructural details of the composite medium.

## References

• Starters and appetizers (educational, experimental, less theoretical)
• D. E. Aspnes, "Local-field" effects and effective-medium theory: A microscopic perspective", Am. J. Phys. 50, 704 (1982).
• G. A. Niklasson and C. G. Granqvist, "Optical properties and solar selectivity of coevaporated Co-Al_2O_3 composite films", J. Appl. Phys. 55, 3382(1984).
• W. Theiss, "The Use of Effective Medium Theories in Optical Spectroscopy", in Festkörperprobleme/Advances in Solid State Physics 33, R. Helbig ed. (Vieweg, Braunschweig, 1993).
• Paper collections
• A. Lakhtakia (editor), Selected papers on linear optical composite materials (Bellingham, Washington 1996).Many of the papers listed here can be found in this book or are cited therein.
• Theory, but still educational
• D. J. Bergman, "Bulk physical properties of composite media":, in Les methodes de l'homogeneisation: theorie et applications en physique, pages 1--128 of Volume 57 of Collection de la Direction des etudes et recherches d' Electricite de France, Session qui s'est tenue au Centre du Breau-sans-Nappe, du 27 juin au 13 juillet 1983, Eyrolles, Paris, France (1985)
• Hard stuff, for theoreticians
• D. Bergman, "The dielectric constant of a composite material - a problem in classical physics", Phys. Reports 43, 377 (1978).
• G. W. Milton 1980 Bounds on the complex dielectric constant of a composite material. Appl. Phys. Lett. 37, 300--302 (1980).
• D. J. Bergman, Phys. Rev. Lett. 44, 1285 (1980).
• G. W. Milton, J. Appl. Phys. 52, 5286 (1981).
• G. W. Milton, J. Appl. Phys. 52, 5294 (1981).
• B. U. Felderhof, Physica 126A, 430 (1984).

The above list is certainly incomplete. Feel free to contact me, if you think that a certain paper should be added.