The transient form of the electron energy balance equates the time-rate-of-change of the electron swarm’s internal energy, , to the net flow of electron enthalpy into and out of the reactor, accounting for net chemical production rates, surface losses, collisions losses, and power deposition from externally applied electromagnetic fields. This balance is stated as:

(8–139)

where is the electron mass density (equal to the product of the electron number density and the electron mass). refers to the electron enthalpy of newly created electrons in the gas-phase; when electrons are formed from the ionization of a relatively cold neutral, the electron is assumed to originate close to the neutral temperature. The energy required to thermalize new electrons is therefore taken into account. At the surface, electron losses are assumed to dominate electron emission, so that no equivalent term is included to account for new electrons coming off the surface with thermal energies equal to the surface temperature. The electron enthalpy loss at the surface is therefore calculated from the net production rate of electrons due to surface reactions on each material, , and the electron enthalpy, . The second- and third-to-last terms on the right-hand-side refer to the collision energy lost by the electrons both from elastic, momentum-transfer collisions, and from inelastic collisional processes. The inelastic collisions may include both excitation reactions, as well as chemical reactions resulting from electron-impact collisions. The source term differs from in Equation 8–30 in that it represents only that power deposited to the electrons, rather than to the plasma as a whole. In particular, some of the deposited power may contribute to heating of ions in the plasma bulk, or accelerating ions through the plasma sheath. We therefore define the electron-energy source term as:

(8–140)

The internal energy of the electron and electron specific heats are defined by:

(8–141)

If we assume , substitute Equation 8–140 , and Equation 8–141 into Equation 8–139 , and subtract Equation 8–2 multiplied by , we arrive at:

(8–142)

Here the second term on the right-hand-side represents the thermalization energy required for newly created electrons. The loss and source terms in Equation 8–142 and Equation 8–140 are defined as follows:

(8–143)

(8–144)

(8–145)

and

(8–146)

In Equation 8–143 , is the momentum-transfer collision frequency between the electrons and the k th heavy species. The plasma-reactor model calculates the momentum-transfer collision frequencies from momentum-transfer collision cross-sections specified with the input keywords XSEK and XSDF . The first term in Equation 8–144 represents the summation of electron energy loss per electron-impact reactions as specified in the Gas-phase Kinetics input file. The total number of electron-impact (that is, electron-temperature dependent) reactions , is the net rate of progress of the r th reaction, and is the net enthalpy change of the reaction. can be determined from species’ thermochemistry as available in the Ansys Chemkin thermodynamic data, or can be input directly through use of the Gas-phase Kinetics reaction auxiliary keyword EXCI ( Table 3.7: Alphabetical Listing of Gas-phase Reaction Auxiliary Keywords of the Chemkin Input Manual Input Manual). The second term on the right-hand side of Equation 8–144 represents other loss terms that the user may choose to include separately from the Gas-phase Kinetics reaction descriptions (for example, explicit user specification of electron energy loss in the User Interface). In Equation 8–145 , is the energy gained by an ion when traversing the sheath, while is the total number of ionic species. may be supplied through one of several options: direct specification of ion energy, calculation from the electron temperature and a user-specified multiplication factor (Sheath Loss Factor in the Chemkin Interface, under Materials-specific Data), or through specification of a bias power applied to the material. For the bias power option, the ion energy is determined as the power divided by the total ion current to that material. In Equation 8–146 , we introduce an ion temperature to capture the source energy that is deposited into the ions, although we are not solving an ion energy equation explicitly. The ion temperature in the plasma bulk is specified directly by the user and assumed to be constant.