On complex permittivity of dilute random binary dielectric mixtures in two-dimensions
{Emre Tuncer}
{Monterey Design, 894 Ross Drive, Sunnyvale CA 94089 USA}
Influence of number of particles considered in numerical simulations on complex dielectric permittivity of binary dilute dielectric mixtures in two-dimensions are reported. In the simulations, dodecagons (polygons with $12$-sides) were used to mimic disk-shaped inclusions. Using such an approach we were able to consider $16^2$ particles in a unit-square. The effective dielectric permittivity of the mixtures were calculated using the finite element at two different frequecies which were much higher and lower than the characteristic relaxation rate of the Maxwell-Wagner-Sillars polarization. The results were compared to an analytical solution. It was found that the permittivity values at low frequencies were inside Wiener bounds however they violated the Hashin-Shtrikman bounds.\\ {Key words:} Dielectric permittivity, the finite element method, effective medium theory.
rnd1.tex
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\vol{XX}
\pyear{2002}
\received{\today}
\author[TUNCER \& TUNCER]{
       {\bf En{\.{\i}}s Tuncer} 
        \thanks{{email: {\tt enis.tuncer@physics.org}};
         Currently at Department of Physics, University of Potsdam, Potsdam 14469 Germany.}\\
	{\it Chalmers University of Technology, 
         SE-412\ 96 Gothenburg Sweden}\\ 
	{\bf Emre Tuncer}\\
        {\it Monterey Design, 894 Ross Drive, Sunnyvale CA 94089 USA}}
\title{On complex permittivity of dilute random binary dielectric mixtures in two-dimensions
}
\begin{document}
\maketitle
\begin{abstract}
  Influence of number of particles considered in numerical simulations on complex dielectric permittivity of binary dilute dielectric mixtures in two-dimensions are reported. In the simulations, dodecagons (polygons with $12$-sides) were used to mimic disk-shaped inclusions. Using such an approach we were able to consider $16^2$ particles in a unit-square. The effective dielectric permittivity of the mixtures were calculated using the finite element at two different frequecies which were much higher and lower than the characteristic relaxation rate of the Maxwell-Wagner-Sillars polarization. The results were compared to an analytical solution.  It was found that the permittivity values at low frequencies were inside Wiener bounds however they violated the Hashin-Shtrikman bounds.\\
{\bf Key words:} Dielectric permittivity, the finite element method, effective medium theory.
\end{abstract}


%\section{Introduction}
The complex permittivity of materials needs to be estimated in many applications~\cite{SihvolaBook,TuncerLic,TuncerPhD,Emre1992,Liu}. By using classical methods, {\em i.e.}, analytical techniques, effective medium approximation, {\em etc.}, the determination of the complex permittivity of dielectric materials is usually carried out for simple structures~\cite{SihvolaBook}. Moreover, the computed values should be verified by experiments or by means of other computational methods. Recently, the finite element ({\sc fe}) computer simulations~\cite{TuncerAcc1} has shown that disk-shaped inclusions in binary mixtures can be imitated by $n$-gons (polygon with $n$-sides) such that when $n\ge10$ the obtained electrical properties of the mixture were within $0.1\ \%$ error of the analytical solutions when the mixture was a dilute solution. In this paper, we investigate the electrical properties of a dilute binary mixture by considering the number of particles in the computational domain and compare the results with those obtained analytically~\cite{Emets98b}. 

%In the simulations, first lower level of inclusion concentration was assumed, since the effect of higher concentration levels was already demostrated above although for regular structures only. 

In order to obtain random structures, an algorithm based on acception and rejection method (also known as random sequential adsorption) was constructed~\cite{Gervois}. In this approach, hard-disks with radii, $r$, were generated with random center coordinates in a unit square. Overlapping of disks both with outer boundaries of the unit square and with other disks were not allowed. The coordinate-values $x$ and $y$ were picked from a uniform random number generator where $(r,r)<(x,y)<(1-r,1-r)$. The first hard-disk was fixed, then, the number of hard-disks was incrimented. In each step, the condition for overlapping of the new generated disk with the already fixed disk(s) was checked. In case of overlapping, the disk was rejected, and a new one was picked from a new series of coordinates. 
%%%%%%%%%%%RANDOM 2 BAS%%%%
%\begin{figure}[tbp]
%  \begin{center}
%    {\includegraphics[width=3in]{random_struct.eps}}\\
%  \end{center}
%    \caption{ \label{fig:random2-1}Sixteen random structures generated with four hard-disks at $q=0.01$, with their structural parameters {\em i.e.}, the number of neighbors, $\mathcal{N}_R$, and the minimum distance to the nearest neighbor, $d_n$. The lines are the values for the equivalent square lattice. $p$ is the hard-disk number.}
%\end{figure}
In the process, obviously both the concentration of the hard-disks in the background medium and the number of disks should be kept constant. As mentioned previously, instead of using exact disk shapes and consuming lot of computational power, dodecagon %dodecagon
(polygon with 12 sides) were used in the simulations.  The considered dodecagons were monodispersed.
%In Fig.~\ref{fig:random2-1}, 16 unit squares filled with 4 randomly distibuted inclusions are shown when $q=0.01$. Structural parameters {\em e.g.}, number of neighbors, $\mathcal{N_R}$, and the normalized distance to the nearest neighbor, $d_n$, are also plotted in the figure. The number of neighbors were calculated in a circle with radius, $R$, such that $R$ was the distance between hard-disk centers in an equivalent square lattice structure. Moreover, 
The boundary conditions in the {\sc fe} were such that  reflections of the computational domain in the 4 facing-directions, von Neuman neighbors~\cite{Cellular} were used which was also Rayleigh's approach~\cite{Rayleigh}. The {\sc fe} simulation procedure has been presented previously elsewhere~\cite{sar97,Tuncer2001a,TuncerAcc1}.%This approach, not only neglected the edge-effects but also, imitated the boundary conditions used in the field calculations. Artifacts of the approach could be seen in the figure in which some of the particles had more than 4 neighbors, $\mathcal{N}_R$, even though the number of hard-disks were four. In the figure, each bar represents one of the created random structures. The x-axis presents the disk number, $p$, which were numbered via pattern generation process. The distance to the nearest neighbor, $d_n$, was, therefore, calculated randomly. %The distance scale is in hard disk radii units, $r$.

In the simulations the concentration of the inclusion phase $q$ and number of inclusions in the computional domain were considered as variables. Only four concentration levels were included, $q=\{0.1,\ 0.2,\ 0.3,\ 0.4\}$. Starting with one inclusion, we have increased the number of inclusions $n$, in the computations.  The meshing of the region was arranged by the sizes of the dodecagons such that smallest meshing element was $1/4$  of the length of the  dodecagon side. This kind of procedure has allowed us to have enough number of discritization points in the inclusions. Computations with larger number of inclusions has increased the number of mesh elements, therefore, we were not able to obtain solutions for $n>16^2$ and $q\ge0.2$. For the $q=0.1$ case, the number of mesh elements were high enough when $n>12^2$. 

\begin{figure}[tp]
  \centering{
  \begin{tabular}{ccc}
    \includegraphics[width=2.1in]{lowfreq_bw1.ps}
      &\hspace{.3cm}%{lowfreq_sm.ps}
       &\includegraphics[width=2.1in]{higfreq_bw1.ps}\\%}\\%{higfreq_sm.ps}}\\
       (a)& &(b)
  \end{tabular}}
  \caption{\label{fig:high1}
Potential (horizantal lines) and electric field (vertical lines) distributions at (a) lower and (b) higher frequencies than the characteristic interfacial polarization is observed. Number of inclusions are 36 and the concentration $q$ is 0.2.}
\end{figure}

 The electrical properties of the constituent media without any intrinsic polarizations were choosen such that the relaxation rate of the interfacial polarization in the composite medium~\cite{Maxwell1954,Wagner1914,Sillars1937} was around $1\ \hertz$. The complex dielectric permittivity of a medium as a function of angular frequency, $\omega=2\pi\nu$, can be expressed as,  
\begin{eqnarray}
  \label{eq:para}
  \widetilde\varepsilon_i(\omega)=\varepsilon_i+\frac{\sigma_i}{\imath\epsilon_0\omega}
\end{eqnarray}
where $\varepsilon_i$ is the dielectric permittivity of the medium at optical frequencies and $\sigma_i$ is the conductivity of the constituent. $\epsilon_0$ is the permittivity of the free space, $\epsilon_0=1/36\pi\ \nano\farad\per\meter$ and $\imath=\sqrt{-1}$. The permittivity and conductivity of the matrix phase, $\varepsilon_1$ and $\sigma_1$, were taken $2$ and $1\ \pico\siemens\per\meter$, respectively. The permittivity and conductivity of the dodecagon inclusions, $\varepsilon_2$ and $\sigma_2$, were $10$ and $100\ \pico\siemens\per\meter$, respectively. We have estimated the electrical properties $\varepsilon_\mega$, $\varepsilon_\micro$ and $\sigma$ where subscripts $\mega$ and $\micro$ represents the frequency. 

In Fig.~\ref{fig:high1}, the electric potential and electric field distributions of one of the considered case with $n=6^2$ and $q=0.2$ are presented. The difference between the two pictures are the one on the left is at low frequencies ($\nu=1\ \micro\hertz$) where the field distribution in the material is determined by the resistive properties of the media. Similarly, the picture on the right is at $\nu=1\ \mega\hertz$ where the capacitive properties of the media set the effective properties of the mixture.

\begin{figure}[tp]
  \centering{\includegraphics[width=5.5in]{DD2.EPS}}
\caption{  \label{fig:q.vs.ratio}Ratio of electrical properties calculated and estimates of the analytical formula as a function of inclusion concentration $q$; (a) $\varepsilon_{\mega}$, (b) $\varepsilon_{\micro}$ and (c) $\sigma$. The dashed (\dashed) and solid (\full) lines represent the bounds of Wiener and Hashin-Shtrikman.}
\end{figure}
\begin{figure}[tp]
  \includegraphics[angle=-90,width=5.6in]{DD.EPS}
\caption{\label{fig:dat1}Ratio of electrical properties calculated and estimates of the analytical formula as a function of square root number of inclusions $n$; (a) $\varepsilon_{\mega}$, (b) $\varepsilon_{\micro}$ and (c) $\sigma$. Different symbols represent the considered inclusion concentrations $q$; ($\bigcirc$) 0.1, ($\square$) 0.2, ($\Diamond$) 0.3 and ($\vartriangle$) 0.4.}
\end{figure}
The Maxwell-Garnett dielectric mixture formula~\cite{Maxwell_Garnett} is valid for two-dimensional dilute mixtures when the shape of the inclusions are disks~\cite{TuncerAcc1}. The effective complex dielectric permittivity of the mixture, $\widetilde\varepsilon_{\sf e}(\omega)$ can be found from 
\begin{equation}
  \label{eq:wiener_pier}
  \frac{\widetilde\varepsilon_{\sf e}(\omega)-\widetilde\varepsilon_1(\omega)}{\widetilde\varepsilon_{\sf e}(\omega)+(n-1)\widetilde\varepsilon_1(\omega)}=q\frac{\widetilde\varepsilon_2(\omega)-\widetilde\varepsilon_1(\omega)}{\widetilde\varepsilon_2(\omega)+(n-1)\widetilde\varepsilon_1(\omega)}
\end{equation}
where $n=2$ in two-dimensions\cite{Tuncer2001a}. In Fig.~\ref{fig:q.vs.ratio}, calculated electrical properties are presented as normalized to the values obtained from Eq.~(\ref{eq:wiener_pier}). In the figure, the bounds for dielectric permittivity and conductivity obtained from Wiener~\cite{wiener} and Hashin-Shtrikman~\cite{hashin} are also displayed as dashed (\dashed) and solid (\full) lines respectively. The Wiener bounds are obtained when $n=0$ and $n=\infty$ in Eq.~(\ref{eq:wiener_pier}). In two-dimensions, the Hashin-Shtrikman bounds are expressed as
\begin{equation}
  \label{eq:hashin}
  \widetilde\varepsilon_1+\frac{q}{(\widetilde\varepsilon_2-\widetilde\varepsilon_1)^{-1}+(1-q)(2\widetilde\varepsilon_1)^{-1}} \le \widetilde\varepsilon\le \widetilde\varepsilon_2+\frac{1-q}{(\widetilde\varepsilon_1-\widetilde\varepsilon_2)^{-1}+q(2\widetilde\varepsilon_2)^{-1}}
\end{equation}
The conductivity of the mixture was calculated from the dielectric losses, $\sigma=\Im[\widetilde\varepsilon''(\omega)\epsilon_0\omega]$ at $\omega=2\pi\ \micro\hertz$. It is significant from Fig.~\ref{fig:q.vs.ratio} that the lower bound of Hashin-Shtrikman is for disks since the obtained values ratios of the bounds and Eq.~(\ref{eq:wiener_pier}) with $n=2$ are equal. This proves that the considered dilute binary mixture was isotropic. The error bars in the figure or in other words the spread of the electrical properties indicate the importance of local field effects and interaction between inclusion particles such that at high concentrations the interaction between particles started to influence the electrical properties. 

When $\varepsilon$ at low frequencies are taken into consideration (Fig.~\ref{fig:q.vs.ratio}a), the upper limits of the two bounds have shown extraordinary behavior for higher concentrations of inclusions, the validity of Eq.~(\ref{eq:hashin}) was very narrow, $0<q<0.2$. The Wiener bounds are on the other hand valid for our simulations, however, for $q>\pi/4$ the Wiener bounds also show  extraordinary behavior. It is remarkable that $q=\pi/4$ is the limiting concentration for regular arrangement of inclusions on a square net.

The permittivity values at high frequencies are displayed in Fig.~\ref{fig:q.vs.ratio}b, and they were similar to the analytical formula in Eq.~(\ref{eq:wiener_pier}), and the spread of the data was very small compared to the conductivity values presented in Fig.~\ref{fig:q.vs.ratio}c. This was due to the ratio between the permittivity values considered, $\Re[\widetilde\varepsilon_1]/\Re[\widetilde\varepsilon_2]<\Im[\widetilde\varepsilon_1]/\Im[\widetilde\varepsilon_2]$~\cite{WebmanCohen}. For high frequency permittivity and ohmic conductivity both bounds were valid.


When the influence of the number of inclusions in the computational domain are considered, it was observed that the spread of the electrical values decreased as the number of inclusions increased. This behavior is displayed in Fig.~\ref{fig:dat1} again as the ratio between the computed values and Eq.~(\ref{eq:wiener_pier}). It was clear that the error-bars decreased by increasing the number of inclusions. There was a jump between the electrical properties when one-inclusion and four-inclusions systems were compared to each other. This was due to the change in the field distributions around inclusions which has pointed out the importance of boundary conditions in the simulations. At small scale systems with less number of inclusions, the boundary conditions are crucial since each particle is not affected by the local field distributions created by the neighboring inclusions. In addition, by increasing the number of inclusions in the computation domain, the probability of having larger number of neighbors in a radius is also increased when compared with a regular arrangement of inclusions~\cite{Tuncer2001a}.  This increase in the number of neighbors should in principle enhance the polarization of individual inclusions which is more realistic to real materials. This approach was desired but not selected due to difficulties arising in the field calculations, mainly the meshing procedure in the {\sc fe} method. The distances to nearest neighbors were not influenced by the number of inclusions if scaled with the inclusion radius. One could obviously use as many hard-disks as possible, after all it would be the most realistic condition. This could be done as proposed by considering a dodecagon instead of a circle. % it was possible to increase the number of inclusions to simulate more realistic problems.
%The distances to nearest neighbors were not influenced by the number of inclusions if scaled with the inclusion radius. One could obviously use as many hard-disks as possible, after all it would be the most realistic condition. But applying {\sc fe} method to the generated geometries was not practical due to the meshing procedure. 
Finally, similar approach sould be applied to three dimensional problems to increase the number of considered particles in simulations.

%\subsubsection{Structural effects between different lattices}

%Textbooks on amorphous solids, \cite{Elliot,Zallen}  and granular materials \cite{Meakin,Bideau} give brief information on this topic, however, most of the information is for structures in three dimensions. Therefore, first radial distribution function of different regular lattice structures and later random structures discussed.   
  



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E.~Tuncer, S.~M. Guba{\'n}ski, and B.~Nettelblad,
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\bibitem{hashin}
Z.~Hashin and S.~Shtrikman,
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\bibitem{WebmanCohen}
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\end{thebibliography}

%\bibliography{../Papers/references}
%\bibliographystyle{phreport}

\end{document}





cond-mat0109170.extracted_bib
\begin{thebibliography}{10}

\bibitem{SihvolaBook}
A.~Sihvola,
\newblock {\em Electromagnetic mixing formulas and applications}, volume~47 of
  {\em IEE Electromagnetic Waves Series},
\newblock The Institute of Electrical Engineers, London, 1999.

\bibitem{TuncerLic}
E.~Tuncer,
\newblock Dielectric properties of composite structures and filled polymeric
  composite materials,
\newblock Lic\-en\-ciate thesis--Tech. rep. 338~L, Department of El\-ect\-ric
  Po\-wer Eng., Chalmers University of Technology, Gothenburg, Sweden, April
  2000.

\bibitem{TuncerPhD}
E.~Tuncer, 
\newblock Dielectric relaxation in dielectric mixtures,
\newblock PhD thesis, Chalmers University of Technology, Gothenburg, Sweden, 2001.

\bibitem{Emre1992}
E.~Tuncer and D.~P. Neikirk,
\newblock High Accurate Quasi-Static Modeling of Microstrip Lines Over Lossy
  Substrates,
\newblock IEEE Microwave and Guided Wave Letters {\bf 2}(10)  (1992), 409--411.

\bibitem{Liu}
C.~Liu and L.~C. Shen,
\newblock Dispersion characteristics of two-component two-dimensional mixtures,
\newblock Modelling and Simulation in Materials Science and Engineering {\bf
  1}(5) (1993), 723--730.

\bibitem{TuncerAcc1}
E.~Tuncer,
\newblock How round is round? On accuracy in complex dielectric permittivity
  calculations: A finite-size scaling approach,
\newblock {T}urkish Journal of Physics, accepted (2002) -- {\tt
  arXiv:cond-mat/0107384}.

\bibitem{Emets98b}
Y.~P. Emets,
\newblock Electrical characteristics of three-component dielectric media,
\newblock Journal of Experimental and Theoretical Physics {\bf 87}(3) (1998), 612--620.

\bibitem{Gervois}
A.~Gervois and D.~Bideau,
\newblock Some geometrical properties of two-dimensional hard disk packings,
\newblock in {\em Disorder and Granular Media}, edited by D.~Bideau and
  A.~Hansen, Random Materials and Processes, pages 1--31, North-Holland,
  Amsterdam, 1993.

\bibitem{Cellular}
B.~Chopard and M.~Droz,
\newblock {\em Cellular Automata Modeling of Physical Systems},
\newblock Collection Al{\'e}a-Saclay: Monographs and Texts in Statistical
  Physics, Cambridge University Press, Cambridge, 1998.

\bibitem{Rayleigh}
L.~Rayleigh,
\newblock On the influence of obstacles arranged in rectangular order upon the
  properties of a medium,
\newblock Philosophical Magazine {\bf 34}  (1892), 481--502.

\bibitem{sar97}
B.~Sareni, L.~Kr{\"a}henb{\"u}hl, A.~Beroual, and C.~Brosseau,
\newblock Effective dielectric constant of random composite materials,
\newblock Journal of Applied Physics {\bf 81}(5)  (1997), 2375--2383.

\bibitem{Tuncer2001a}
E.~Tuncer, S.~M. Guba{\'n}ski, and B.~Nettelblad,
\newblock Dielectric relaxation in dielectric mixtures: {A}pplication of the
  Finite Element Method and its comparison with mixture formulas,
\newblock Journal of Applied Physics {\bf 89}(12)  (2001), 8092--8100.

\bibitem{Maxwell1954}
J.~C. Maxwell,
\newblock {\em A Treatise on Electricity and Magnetism- Volume 1}, pages
  450--464,
\newblock Clarendon Press, Oxford, third edition, 1891,
\newblock reprint by Dover.

\bibitem{Wagner1914}
K.~W. Wagner,
\newblock Erkl{\"{a}}rung der dielekrischen {N}achwirkungsvorg{\"a}nge auf
  {G}rund {M}axwellscher {V}orstellungen,
\newblock Archiv f{\"u}r Electrotechnik {\bf II}(9)  (1914), 371--387.

\bibitem{Sillars1937}
R.~Sillars,
\newblock The properties of a dielectric containing semiconducting particles of
  various shapes,
\newblock Journal of Institution of Electrical Engineers {\bf 80}  (1937), 378--394.


\bibitem{Maxwell_Garnett}
J.~C.~M. Garnett,
\newblock Colours in Metal Glasses and in Metallic Films,
\newblock Philosphical Transactions of Royal Society of London A {\bf 203}  (1904),  385--420.

\bibitem{wiener}
O.~Wiener,
\newblock Die theorie des {M}ischkorpers f{\"u}r das {F}eld der Staton{\"a}ren
  {S}trom{\"u}ng I. Die Mittelwertsatze f{\"u}r Kraft, Polarisation und
  Energie,
\newblock Der Abhandlungen der Mathematisch-Physischen Klasse der K{\"o}nigl.
  Sachsischen Gesellschaft der Wissenschaften {\bf 32} (1912), 509--604.

\bibitem{hashin}
Z.~Hashin and S.~Shtrikman,
\newblock A variational approach to the theory of effective magnetic
  pereability of multiphase materials,
\newblock Journal of Applied Physics {\bf 33} (1962), 3125--3131.

\bibitem{WebmanCohen}
I.~Webman, J.~Jortner, and M.~H. Cohen,
\newblock Theory of optical and microwave properties of microscopically
  inhomogeneous materials,
\newblock Physical Review E {\bf 15}(12)  (1977), 5712--5723

\end{thebibliography}