# A Comparison of Electrical Breakdown Models for Polyethylene Nanocomposites
## 2 Electrical Breakdown Models
### Charge Injection and Charge Transport in Nanocomposites
The time-dependent change in the charge densities of mobile electrons and mobile holes per unit volume is related to the current density flowing into and out of the control volume. Additionally, trap capturing and recombination lead to a decrease in mobile electrons and mobile holes, while the detrapping of trapped charges results in an increase in mobile electrons and mobile holes. The time-dependent changes in trapped electrons and trapped holes per unit volume are related to charge trapping, charge detrapping, and recombination. Charge detrapping and recombination lead to a decrease in deep trapped electrons and deep trapped holes, while charge trapping leads to an increase in deep trapped electrons and deep trapped holes. Four partial differential equations are needed to describe the dynamic processes of mobile electrons, deep trap electrons, mobile holes, and deep trap holes in nanocomposites, respectively [25,29,30]:  
\[\frac{\partial n_{M(e)}}{\partial t}+\frac{\partial\Big{(}n_{M(e)}n_{0(e)}E \Big{)}}{\partial x}=-P_{T(e)}n_{M(e)}\Bigg{(}1-\frac{n_{T(e)}}{N_{T(e)}} \Bigg{)}+P_{D(e)}n_{T(e)}-R_{Me,Mh}n_{M(e)}n_{M(h)}-R_{Me,Th}n_{M(e)}n_{T(h)} \tag{3}\]  
\[\frac{\partial n_{T(e)}}{\partial t}=P_{T(e)}n_{M(e)}\Bigg{(}1-\frac{n_{T(e)}} {N_{T(e)}}\Bigg{)}-P_{D(e)}n_{T(e)}-R_{Te,Mh}n_{T(e)}n_{M(h)} \tag{4}\]  
\[\frac{\partial n_{M(h)}}{\partial t}+\frac{\partial\Big{(}n_{M(h)}n_{0(h)}E \Big{)}}{\partial x}=-P_{T(h)}n_{M(h)}\Bigg{(}1-\frac{n_{T(h)}}{N_{T(h)}} \Bigg{)}+P_{D(h)}n_{T(h)}-R_{Me,Mh}n_{M(e)}n_{M(h)}-R_{Te,Mh}n_{T(e)}n_{M(h)} \tag{5}\]  
\[\frac{\partial n_{T(h)}}{\partial t}=P_{T(h)}n_{M(h)}\Bigg{(}1-\frac{n_{T(h)}} {N_{T(h)}}\Bigg{)}-P_{D(h)}n_{T(h)}-R_{Me,Th}n_{M(e)}n_{T(h)} \tag{6}\]  
The detrapping probabilities \(P_{D(e)}\) and \(P_{D(h)}\) of the deep trapped charges in the nanocomposite are related to the trap energy levels \(u_{T(e)}\) and \(u_{T(h)}\) as \(P_{D(e,h)}=\
u_{0}\)exp \((-u_{T(e,h)}/k_{B}T)\). Here, \(\
u_{0}\) is the attempt-to-escape frequency. \(R_{Me,Mh}\) is the recombination coefficient between mobile electrons and mobile holes. According to the Langevin recombination model, the recombination coefficient \(R_{Me,Mh}\) is determined by \(e(\mu_{0(e)}+\mu_{0(h)})/\varepsilon_{0}\varepsilon_{r}\)[31]. \(R_{Me,Th}\) and \(R_{Te,Mh}\) are the recombination coefficient between mobile electrons and trapped holes, and that between trapped electrons and mobile holes, respectively. According to the Shockley-Read-Hall model, the recombination rates \(R_{Me,Th}\) and \(R_{Te,Mh}\) are determined by \(e\mu_{0(e)}/\varepsilon_{0}\varepsilon_{r}\) and \(e\mu_{0(h)}/\varepsilon_{0}\varepsilon_{r}\), respectively [32]. When space charges accumulate inside the nanocomposite, the space charges can distort the electric potential and the electric field. The potential \(\varphi\) inside the nanocomposite can be calculated by [PERSON]'s equation:  
\[\partial^{2}\varphi/\partial x^{2}=-e\Big{(}n_{M(h)}+n_{T(h)}-n_{M(e)}-n_{T(e )}\Big{)}/\varepsilon_{0}\varepsilon_{r} \tag{7}\]  
Then, the electric field distribution inside the nanocomposite can be calculated through the negative gradient of the electric potential, namely \(E=-\
abla\phi\).