# Prospects for the Development of High Energy Density Dielectric Capacitors
## 3 Analysis of Dielectric Cells Using Nanoparticles Embedded in a Host Polymer  
The general background for this modeling can be found in [46; 47]. The polymer composite capacitor is modeled as an assembly of nm scale micro-capacitors consisting of carbon elements. Uniformly distributed in the host polymer. It is assumed that the carbon elements form a highly regular structure functioning as micro-capacitors that store the electrical charge displaced from the host polymer molecules that are electronically insulating having a dielectric constant \(\varepsilon_{1}\). A schematic of the polymer composite structure is shown in Figure 5. The microcaps are shown using both platelet and spherical shaped carbon elements. The characteristic dimensions \(\mathrm{r_{p}}\) and \(\mathrm{r_{h}}\) of the spherical and platelet elements are defined in the figure. The elements are of nm scale. The loading of the conducting elements is given in terms of its volume fraction \(\mathrm{f_{v}}\). The weight fraction \(\mathrm{f_{w}}\) can be calculated from the following relation. \[\mathrm{f_{w}}=\mathrm{f_{v}}/(\varepsilon_{1}/\varepsilon_{\mathrm{p}}\ (1-\mathrm{f_{v}})+\mathrm{f_{v}})\]  
Considering the case of the spherical elements, the number \(\mathrm{N_{p}}\) of elements is given by \(\mathrm{N_{p}}=(\mathrm{VL})\ \mathrm{f_{v}}/(4/3\Pi\ \mathrm{r_{p}}^{3})\), where \(\mathrm{VL}\) is the volume of the dielectric space. The distance \(\Delta\) between the centers of the elements is  
\[\mathrm{L}/\Delta\mathrm{x}\ \mathrm{W}/\Delta\times\delta/\Delta=\mathrm{N_{p}}\]  
where \(\mathrm{L}\), \(\mathrm{W}\), \(\delta\) are the dimensions of the dielectric space. Solving for \(\Delta\),  
\[\Delta=(4/3\Pi/\mathrm{f_{v}})^{1/3}\ \mathrm{r_{p}}\]  
The length between the elements \(\mathrm{\delta_{c}}\), which is the separation distance between the electrodes in the microcaps, is given by  
\[\mathrm{\delta_{c}}=\Delta-2\mathrm{r_{p}}=\mathrm{k_{1}}\ \mathrm{r_{p}},\ \mathrm{where}\ \mathrm{k_{1}}=(4/3\Pi/\mathrm{f_{v}})^{1/3}-2\]  
The capacitance \(\mathrm{C_{c}}\) of individual microcaps is given by  
\[\mathrm{C_{c}}=\mathrm{k_{c}}\ \varepsilon_{1}\varepsilon_{0}\Pi\mathrm{r_{p}} ^{2}/\delta_{c}\]  
where \(\mathrm{k_{c}}\) is shape factor for the capacitor  
\[\varepsilon_{0}=8.85\times 10^{-12}\ (\mathrm{coul})^{2}/\mathrm{newton }\mathrm{-m}^{2}\]  
The charge on the microcap is \(\mathrm{Q_{c}}=\mathrm{C_{c}}\ \Delta\mathrm{V}=\mathrm{C_{c}}\ (\mathrm{V}/\delta)\ \Delta\), where \(\mathrm{V}\) is the voltage across the dielectric space (see Figure 1) assuming \(\mathrm{V}\) is linear and 2D uniform. \begin{table}
\begin{tabular}{c c c c c} \hline
**Dielectric Cell** & **Breakdown** & **Cell Voltage** & **Dielectric Constant** & **Dielectric Film** \\
**Wh/L** & **Strength V/\(\mu\)** & & **Needed** & **Thickness \(\mu\)** \\ \hline
200 & 400 & 300 & 1020 & 0.75 \\ \hline  & & 800 & 1020 & 2.0 \\ \hline
100 & 400 & 300 & 510 & 0.75 \\ \hline  & & 800 & 510 & 2.0 \\ \hline
200 & 250 & 300 & 2610 & 1.2 \\ \hline  & & 800 & 2610 & 3.2 \\ \hline
100 & 250 & 300 & 13 Substituting for \(C_{c}\),  
\[Q_{c}=(k_{c}k_{2}/k_{1})\ \varepsilon_{1}\varepsilon_{0}\Pi{r_{p}}^{2}\ (V/\delta),\ k_{2}=(4/3\Pi/f_{v})^{1/3}\]  
The total charge displaced in the dielectric space is \(Q_{\rm total}\), which is given by  
\[Q_{\rm total}=N_{p}\ Q_{c}=(k_{c}k_{2}/k_{1})\ \varepsilon_{1}\varepsilon_{0} \ (V/\delta)\ f_{v}\ (WL)/4/3\ r_{p}\]  
For the dielectric cell,  
\[Q_{\rm total}=\varepsilon_{2}\varepsilon_{0}\ (A/\delta)\ V\]  
The effective dielectric constant \(\varepsilon_{2}\) of the composite dielectric space is  
\[\mathrm{f_{w}}=\mathrm{f_{v}}/(\varepsilon_{1}/\varepsilon_{\mathrm{p}}\ (1-\mathrm{f_{v}})+\mathrm{f_{v}})\]0  
where \(k_{1}=(4/3\Pi)^{1/3}-2\), \(k_{2}=(4/3\Pi)^{1/3}\). A maximum value for \(f_{v}=(4/3\Pi)/8=0.52\).