5.3: Migration in Imposed Fields and Flows

In this section, the spatial scale of interest is such that the diffusion current can be considered negligible compared to the migration current. In addition, the medium is one in which generation and recombination of the charged species is negligible. Hence, the first and last two terms in Eq. 5.2.9 can be dropped. For carriers having a constant mobility, what remains on the right in Eq. 5.2.9 is proportional to the divergence of the electric field. By Gauss' law, this term is there-fore proportiopal to the net space charge. If the density of carriers is small, Gauss' law, Eq. 5.2.4,requires that E be solenoidal:

\[ \nabla \cdot \overrightarrow{E} = 0 \label{1} \]

and Eq. 5.2.9 therefore reduces to

\[ \frac{\partial{\rho_i}}{\partial{t}} + (\overrightarrow{v} \pm b_i \overrightarrow{E}) \cdot \nabla \rho_i = 0 \label{2} \]

In this "imposed field" approximation, the electric field is essentially determined by charges outside of the region of interest. Typically, these charges reside on boundaries and, in terms of the potential, \(\overrightarrow{E}\) is governed by Laplace's equation. Thus, as an example, if the potentials of all boundaries were constrained, \(\overrightarrow{E}\) would be determined by solving Laplaces equation subject to these boundary conditions, and that value of \(\overrightarrow{E}\) "imposed" in Equation \ref{2}. For such a physical situation, each species migrates independently of the others, as is evident from the fact that the coupling between species afforded by Gauss' law is now absent.

The assumption that the electric field distribution is not appreciably affected by the migrating species says that the net charge density is small but not necessarily zero. In general there is an electrical force density acting throughout the moving medium. As in all of this chapter, it is assumed that the effect of this force density on the relative velocity distribution \(\overrightarrow{v}(r,t)\) is negligible. In this sense, the flow is also-"imposed."

The imposed field and flow approximation gives the opportunity to study the effect of convection on the migration of charged particles. As San be seen from Table 5.2.1, ions moving in a field of \(10^5 \, V/m\) through air have a migration velocity \(b_i \overrightarrow{E}\) on the order of \(20 \, m/sec\). Thus, an air velocity on this order could have a large influence on an ion trajectory. Macroscopic charged particles, such as dust in an electrostatic precipitator, typically have a considerably lesser mobility, and are therefore strongly influenced by modest motions of the gas. Although typical velocities of a liquid are likely to be less than for a gas, because of the relatively lower mobilities of ions and macroscopic particles in highly insulating liquids, the effects of convection can again be appreciable.

With the replacement of the velocity by the ion velocity \(\overrightarrow{v} \pm b_i \overrightarrow{E}\), Equation \ref{2} takes the form of a convective derivative. It states that the time rate of change of the species charge density as viewed by a charged particle of fixed identity is zero (see Sec. 2.4 for a discussion of the physical significance of the convective derivative):

\[ \frac{d \rho_i}{dt} = 0 \label{3} \]

on

\[ \frac{d \overrightarrow{r}}{dt} = \overrightarrow{v} \pm b_i \overrightarrow{E} \label{4} \]

In what amounts to a rederivation of the convective derivative, consider the transition from Equation \ref{2} to the representation of Eqs. \ref{3} and \ref{4} in a somewhat more formal way. The three spatial coordinates and time constitute a four-dimensional space. Each set of coordinates \((\overrightarrow{r},t)\) in this space has an associated solution \(\rho_i(\overrightarrow{r},t)\). An incremental change in the coordinates therefore leads to a change in \(\rho_i\) given by

\[ d \rho_i = dt \frac{ \partial{\rho}_i}{\partial{t}} + dx \frac{ \partial{\rho}_i}{\partial{x}} + dy \frac{ \partial{\rho}_i}{\partial{y}} + dz \frac{ \partial{\rho}_i}{\partial{z}} \label{5} \]

As it stands, this expression is nothing more than a prescription for computing \(d\rho_i\) for a given change \((d\overrightarrow{r},dt)\) in the coordinates of the \((\overrightarrow{r},t)\) space. But, can these incremental changes be specified so that Equation \ref{2} reduces to an ordinary differential equation? Division of Equation \ref{5} by dt and comparison to Equation \ref{2} shows that the desired specification is Equation \ref{4}. Along a given characteristic line, represented by Equation \ref{4}, Equation \ref{2} becomes Equation \ref{3}. These lines have the physical significance of being the trajectories of the carriers.

If the evolution of the charge species is to be determined within a given volume \(V\), then the charge density of each species must be specified where the associated characteristic line "enters" the volume of interest. The "direction" of a characteristic line is one of increasing time. Formally, with \(\overrightarrow{n}\) taken as positive if directed outward from the volume of interest, the boundary condition is imposed on the ith species wherever

\[ \overrightarrow{n} \cdot (\overrightarrow{v} \pm b_i \overrightarrow{E}) < 0 \label{6} \]

Boundary conditions consistent with causality seem obvious in the transient case, but Eqs. \ref{4} and \ref{5} pertain also to steady flows in which rates of change with respect to time for an observer at a fixed location are zero.

Steady Migration with Convection

In the laboratory frame of reference, \(\overrightarrow{v},\overrightarrow{E}\) and the boundary conditions represented by Equation \ref{6} are all invariant. Even so, the time rate of change for the particle, as expressed by Equation \ref{4}, is finite. Explicit expressions for the particle trajectories can be found in a wide class of physically interesting situations, following the approach now illustrated.

Both \(\overrightarrow{v}\) and \(\overrightarrow{E}\) are solenoidal, and hence can be represented in terms of vector potentials. The discussion of Sec. 2.18 centers around four common configurations in which only a single component of these vector potentials is required to describe the vector functions. By way of illustration, the polar and axisymmetric spherical configurations are now considered, with the results applied to specific problems in the next two sections.

In polar coordinates, define vector potentials such that

\[ \begin{bmatrix} \overrightarrow{E} \\ \overrightarrow{v} \end{bmatrix} = \begin{bmatrix} \overrightarrow{i}_r \frac{1}{r} \frac{\partial}{\partial{\theta}} - \overrightarrow{i}_{\theta} \frac{\partial}{\partial{r}} \end{bmatrix} \begin{bmatrix} A_E \\ A_v \end{bmatrix} \label{7} \]

as suggested by Table 2.18,1. Similarly, in spherical coordinates

\[ \begin{bmatrix} \overrightarrow{E} \\ \overrightarrow{v} \end{bmatrix} = \frac{1}{r \, sin \, \theta} \begin{bmatrix} \overrightarrow{i}_r \frac{1}{r} \frac{\partial}{\partial{\theta}} - \overrightarrow{i}_{\theta} \frac{\partial}{\partial{r}} \end{bmatrix} \begin{bmatrix} \Lambda_E \\ \Lambda_v \end{bmatrix} \label{8} \]

In terms of these functions, in the respective configurations, Equation \ref{4} becomes

\[ \begin{align} &\quad \quad \quad \quad \text{Polar} & &\quad \quad \text{Axisymmetric spherical} \nonumber \\ &\frac{dr}{dt} = \frac{1}{r} \frac{\partial}{\partial{\theta}} (A_v \pm b_i A_E) & &\frac{dr}{dt} = \frac{1}{r \, sin \, \theta} [\frac{1}{r} \frac{\partial}{\partial{\theta}}] (\Lambda_v \pm b_i \Lambda_E) \label{9} \\ &r \frac{d \theta}{dt} = - \frac{\partial}{\partial{r}} (A_v \pm b_i A_E) & &r \frac{d \theta}{dt} = - \frac{1}{r \, sin \, \theta} \frac{\partial}{\partial{r}} (\Lambda_v \pm b_i \Lambda_E) \label{10} \end{align} \]

Remember that steady-state conditions prevail, so that the quantities on the right are independent of time. Time is therefore eliminated as a parameter by solving each of these expressions for \(dt\) and setting the respective equations equal to each other

\[ \begin{align} &\frac{\partial}{\partial{r}} (A_v \pm b_i A_E) dr + \frac{\partial}{\partial{\theta}} (A_v \pm b_i A_E) d \theta = 0 & &\frac{\partial}{\partial{r}} (\Lambda_E \pm b_i \Lambda_v) dr + \frac{\partial}{\partial{\theta}} (\Lambda_v \pm b_i \Lambda_E) d \theta = 0 \label{11} \end{align} \]

Because there is no time dependence to the potential functions, these expressions constitute total derivatives, and can be just as well written as

\[ \begin{align} d(A_v \pm b_i A_E) = 0 & d(\Lambda_v \pm b_i \Lambda_E) = 0 \label{12} \end{align} \]

The lines along which a species charge density is constant are implicitly given by

\[ \begin{align} A_v \pm b_i A_E = \text{constant} & \Lambda_v \pm b_i \Lambda_E = \text{constant} \label{13} \end{align} \]

Quasistationary Migration with Convection

To integrate the particle equations of motion, and thus arrive at Eqs. \ref{13}, it is necessary to require that the particles be in essentially the same field and flow distribution throughout their motions through the volume of interest. In that sense, the motions are steady. But the particle transit times may be brief compared to a dynamical time of interest, perhaps that required for a surface upon which the particles impinge to charge, and hence change the electric field intensity. Thus, over a longer time scale, the flow and field distribution, hence the functions \((A_E,A_v)\) and \((\Lambda_E,\Lambda_v)\),may be functions of time. This is often the situation during impact charging of macroscopic particles, discussed in Sec. 5.5.

For unipolar migration, the assumption that the electric field is solenoidal (that space charge has a negligible effect on the electric field distribution) is equivalent to the postulate of quasistationary migration (that the transit time for a particle through the volume of interest is short com-pared to the time required to charge a boundary). This point is best made in Sec. 5.6 after a quasi-stationary process is considered in Sec. 5.5.