5.2: Charge Conservation with Material Convection

With the objective of deriving a law obeyed by each species of charge carrier in its self-consistent evolution, consider a volume \(V\) of the deforming material having a fixed identity. That is,in a macroscopic sense, the surface \(S\) enclosing this volume is always composed of the same material particles: \(S = S(t)\). The ith species of charge carrier is defined as having a number density \(n_i\) (particles per unit volume), charge magnitude \(q_i\) (per particle) and hence a magnitude of charge density \(\rho_i = n_i q_i\). Positive and negative charge or charge density will be denoted explicitly by upper and lower signs respectively.

A statement that the total charge of the ith species is lost from \(V\) at a rate determined by the net outward current flux and accrued at a rate determined by the net effect of volumetric processes is

\[ \frac{d}{dt} \int_V \pm \rho_i dV = - \oint_S \overrightarrow{J}_i' \cdot \overrightarrow{n} da \pm \int_V (G-R) dV \label{1} \]

Generation and recombination of the carriers within the volume are represented by \(G\) and \(R\), respectively,which have the units of charge/unit volume/sec. Because \(S\) is fixed relative to the media, \(\overrightarrow{J}_i'\) is defined as the ith species current density measured in the materials frame of reference.

The generalized Leibnitz rule for differentiation of an integral over a time-varying volume, Eq. 2.6.5, makes it possible to take the time derivative inside the integral on the left in Equation \ref{1}. In using Eq. 2.6.5 for this purpose, note that the velocity of the surface \(S\) is the material velocity \(\overrightarrow{v}\). Thus Equation \ref{1} is converted to

\[ int_v \frac{\partial{\pm \rho_i}}{\partial{t}} dV + \oint_S \pm \rho_i \overrightarrow{v} \cdot \overrightarrow{n} da = - \oint_S \overrightarrow{J}_i' \cdot \overrightarrow{n} da \pm \int_V (G-R) dV \label{2} \]

By Gauss' theorem, Eq. 2.6.2, the surface integrations are converted to volume integrations. Because the volume \(V\) is arbitrary, it follows that

\[ \frac{\partial{\rho_i}}{\partial{t}} + \nabla \cdot [ \rho_i \overrightarrow{v} \pm \overrightarrow{J}_i'] = G - R \label{3} \]

To make use of this differential law, the current density must be related to the charge density, and the rates of generation and recombination must be specified.

Carriers, dominated by collisions in their motion through a neutral medium, are usually described by the current density

\[ \overrightarrow{J}_i' = n_i b_i q_i \overrightarrow{E} \pm K_{Di} \nabla (q_i n_i) \equiv b_i \rho_i \overrightarrow{E} \pm K_{Di} \nabla \rho_i \label{4} \]

The term proportional to \(q_i \overrightarrow{E}\) represents migration and is familiar from Sec. 3.2. Because of the electric field, a charged particle sustains a net migration as it undergoes frequent thermally induced collisions with neutral particles. These collisions are so frequent that on the time scale of interest there is an instantaneous equilibrium between the electrical force and an effective drag force. In terms of a friction coefficient \((m_i \nu_i)\), this force equilibrium is expressed by

\[ \pm q_i \overrightarrow{E} = (m_i \nu_i) \overrightarrow{v}_i \label{5} \]

The particle velocity \(\overrightarrow{v}_i\) relative to the neutral medium is expressed in terms of the mobility \(b_i\) as

\[ \overrightarrow{v}_i = \pm b_i \overrightarrow{E} \label{6} \]

where \(b_i \equiv q_i/m_i \nu_i\). Thus, the first term in Equation \ref{4} is the product of the charge density \(\pm \rho_i\) and the particle velocity \(\overrightarrow{v}_i\). Large molecules and macroscopic particles in gases\(^1\) and liquids\(^2\) are often modeled as being spherical and obeying Stokes's law (Sec. 7.21), in which case the friction factor is \(m_i \nu_i = 6 \pi \mu a\), where \(\mu\) and \(a\) are the fluid viscosity and particle radius respectively. For such particles, the mobility is

\[ b_i = \frac{q_i}{6 \pi \mu a} \label{7} \]

The second term in Equation \ref{4} recognizes that because of the thermally induced motions of the particles,on the average there will be a particle flux away from regions of high concentration. This flux is proportional to the spatial rate of change of concentration.

As might be expected from their common origins in the thermal particle motions, the diffusion coefficient \(K_{Di}\) and the mobility are related properties of the medium through which given particles migrate and diffuse. For ideal gases and liquids, \(K_{Di}\) and \(b_i\) are linked by the Einstein relation

\[ K_{Di} = (\frac{kT}{q_i}) b_i ; \, \frac{kT}{e} = 26.6x10^{-3} \, \text{volts at} \, T = 20^o C \label{8} \]

where \(k\) is the Boltzmann constant, \(T\) is the absolute temperature in degrees Kelvin and \(q_i\) is the particle charge. The quantity \(kT/q\) is measured in volts and at room temperature for \(q\) equal to the electron charge, \(e\), has the value given with Equation \ref{8}.

Physical examples to which Equation \ref{4} applies are given in Table 5.2.1, together with typical values for the mobility and diffusion coefficient.

In inserting Equation \ref{4} into the charge conservation equation, Equation \ref{3} it is now assumed that the mate-rial deformations of interest are incompressible in the sense that \(\nabla cdot = 0\), so that

\[ \frac{\partial{\rho_i}}{\partial{t}} + ( \overrightarrow{v} \pm b_i \overrightarrow{E}) \cdot \nabla \rho_i = \nabla \cdot (K_{Di} \nabla \rho_i) \pm \rho_i \nabla \cdot b_i \overrightarrow{E} + G - R \label{9} \]

Each of \(n\) species contributing to the transfer of charge is described by an expression of the form of Equation \ref{9}. The evolution of one species is linked to the others through Gauss' law, which recognizes that the net charge from all of the species is the source for the electric field:

\[ \nabla \cdot \varepsilon \overrightarrow{E} = \Sigma_{i=1}^{n} \pm \rho_i \label{10} \]

Of course, in the electroquasistatic approximation \(\overrightarrow{E}\) is irrotational, a condition that is automatically met by requiring that

\[ \overrightarrow{E} = - \nabla \phi \label{11} \]

Given appropriate source and recombination functions \(G\) and \(R\), and the material velocity distribution \(\overrightarrow{v}(r,t)\), Eqs. \ref{9}-\ref{11} constitute \(n + 1\) scalar expressions and one vector equation describing \(n\) charge densities, \(\phi\) and the vector \(\overrightarrow{E}\).

In the remainder of this chapter, certain of the physical implications of these relations are explored, with emphasis on the interplay of the material convection and the charge transport processes. Approximations are necessary if practical use is to be made of these relations. In this regard, the relative importance of the migration and diffusion contributions to the current density, Eq.\ref{4}, is important. To approximate the ratio of diffusion and migration terms for a given species, the charge density gradient is characterized by \(\rho_i/t\), where \(l\) is a typical length. For media described by the Einstein relation, Equation \ref{8},

\[ \frac{\text{diffusion current density}}{\text{migration current desnity}} = \frac{kT/q_i}{l |\overrightarrow{E}|} \label{12} \]

Suppose that each carrier supports one electronic charge. Then if \(|\overrightarrow{E}| = 1 \, V/m\), the influence of diffusion equals or exceeds that of migration for length scales shorter than about 2.5 cm. But, for fields of the order of \(10^4 \, V/m\), the length scale must be shorter than 2.5 um for this to be true. In relatively conducting materials, such as electrolytes, fields of interest might be no more than \(1 \, V/m\). But, motions of ions in insulating liquids and gases, with fields typically exceeding \(10^4 \, V/m\), are not influenced by diffusion except in accounting for certain processes in the immediate vicinity of boundaries.

Table 5.2.1. Typical mobilities of various charged particles.

Macroscopic Particles in Fluids
Charged to saturation by ion impact, the particle charge is given by Eq. 5.5.1. Introduced into Equation \ref{7}, this charge implies the mobility \[ b = \frac{2 \varepsilon_o a E}{\mu} \tag{a} \] where \(a\) is the particle radius,\(E\) is the electric field in which the charging occurs, and \(\mu\) is the viscosity of the gas or liquid. In air under standard conditions this expression is valid for radii down to about 0.5 wm, below which the finite mean free path of air molecules and diffusional charging become important\(^3\). For air, this expression becomes \(8.8 x 10^{-7} aE\), so that for \(a = 1 \mu m\) and \(E =10^6 \, V/m\) the mobility is \(10^{-7} \, (m/sec)/(V/m)\).
Ions in Gases
At atmospheric pressure, ions are typically generated by a corona discharge. Ions drawn from the discharge by an electric field are usually not distinguished. Reported ion mobilities distinguish among various gases, but do not specify the type of ion. Some published values, unless otherwise indicated for atmospheric pressure and \(20^oC\), are:
Gas		\(CCl_4\)	\(CO_2\)	\(H_2\)	\(H_2O \\ (100^{o} C)\)	\(H_2 S\)	\(N_2\)	\(N_2 \\\text{Very pure}\)	\(O_2\)	\(SO_2\)
\(b_+\) (units of \(10^{-4} m^2/V \, sec\))		0.30	0.84	5.9	1.1	0.62	1.27	1.28	1.31	0.41
\(b_-\) (units of \(10^{-4} m^2/V \, sec\))		0.31	0.98	8.15	0.95	0.56	1.84	145	1.8	0.41
Low mobilities in impure gases are thought to result from formation of "clusters," while extremely high negative mobilities are attributed to an "ion" spending part of its time as a free electron\(^4\).
Ions in Highly Insulating Liquids
Approximate formulas relate mobility to the viscosity, \[ b_+ \simeq 1.5 x 10^{-11}/ \mu; \, b_- \simeq 3x10^{-11}/ \mu \tag{b} \] Thus, for a liquid having the viscosity of water, \(\mu \simeq 10^{-3}\), mobilities are \(1.5 x 10^{-8}\) and \(3 x 10^{-8}\) respectively. For a careful evaluation with liquid and type of ion specified see Adamczewski\(^5\).
Ions in Water at \(25^oC\) Forming an Electrolyte at Infinite Dilution\(^6\)
Ion		\(Na^+\)	\(K^+\)	\(H^+\)	\(Cl^-\)	\(I^-\)	\(OH^-\)	\(Ca^{2+}\)	\(SO_4^{2-}\)	\(NO^-_3\)
\(b_+\) (units of \(10^{-4} m^2/V \, sec\))		5.20	7.62	36.3	7.90	7.96	20.5	6.16	8.27	7.40

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1. C. Orr, Jr.,. Particle Technology, Macmillan Company, New York, 1966, p. 296.

2. F. Daniels and R. A. Alberty, Physical Chemistry, 3rd ed., John Wiley & Sons, New York, 1967,pp. 405-406.I

3. H. J. White, Industrial Electrostatic Precipitation, Addison-Wesley Publishing Company, Reading,Mass., 1963, p. 137.

4. Handbook of Physics, E. U. Condon and H. Odishaw, Eds., McGraw-Hill Book Company, New York, 1958,pp. 4-161.

5. I. Adamczewski, Ionization, Conductivity and Breakdown in Dielectric Liquids, Taylor & Francis,London, 1969, pp. 224-225.

6. Ref. 2, p. 395.