5.1 Electrostatics in matter [electrodynamics top]

Electrostatics involves the effects of stationary charges or static electric fields on a test charge, and amounts to various statements and restatements of Coulomb’s law. The electrostatic limit applies when charges are stationary and the current is zero.

We study in charges in matter, usually in an aqueous solution (we also call this an electrolyte solution) or in a metal conductor. In matter, it becomes unwieldy to keep track of all of the electrostatic interactions. To simplify things, we distinguish betweenfree charge andbound charge, and keep detailed track of only the free charge. Free charge implies a charge that is mobile over distances large as compared to atomic length scales. Free charge typically comes from electrons (in a metal) or ions (in an aqueous solution). We treat free charges specifically in the sections to follow. Bound charge implies charges of equal magnitude but opposite signs that are held in close proximity and are free to move only atomic distances (roughly 1 Å or less). Examples of bound charge include the positive charge of an atomic nucleus and the negative charge of its associated electron cloud, the uneven partial charges in a heteronuclear covalent bond, or the bound ions in a solid crystal. Rather than calculate all of the details of bound charge with detailed electrostatic equations, we replace the detailed effects of bound charge through the continuum electrical permittivity.

Since microscale flows are often driven by voltages specified by power supplies, we begin our description with the electrical potential and electric field.

5.1.1 Electrical potential and electric field

We consider a point charge q andthe force it feels owing to the presence of other charges. We define the electrical potential ϕ (also called thevoltage) at a point in space such that the electrostatic potential energy that a point charge has with respect to a reference position in given by qϕ. We define the electric field  suchthat the force on a point charge q is given by = q:

5.1.2 Coulomb’s Law, Gauss’s Law for electricity in a material, curl of electric field

Given a static source charge q embedded in a linear, instantaneously-responsive, isotropic material of electricpermittivity ε, Coulomb’s law describes the electric field induced in the material by this charge:

where  is the distance vector from the point charge to the location in question,  is the unit vector in this direction ( = ∕Δr), r is the magnitude of this vector, and ε [C∕V meter] is the electrical permittivity of the medium. We can similarly write the electrical potential caused by a point charge in matter:

(5.3)

Because these relations are for a charge embedded in matter, they take into account both the source charge and the response of all of the bound charge in the material. In treating the medium as linear and isotropic, we are neglecting nonlinear effects seen at high fields, as well as anisotropy observed, for example, in some crystals. Equation 5.2 implicitly treats the material as responding instantaneously. We address finite response times later in the chapter when we describe the frequency dependence of the permittivity.

The flux of the electric field caused by an ensemble of point charges can be integrated over a surface, deriving Gauss’s law forelectricity:

Here, the integral is performed over a closed surface S,  is a unit outward normal along this surface, dA is a differential area along this surface, ε is the electrical permittivity of the material, and ∑q is the sum of the charges [C] inside the surface. By applying the divergence theorem, this integral relation can be put in differential form:

where ρ_E is the volumetric net free charge density in the material [C∕m³]. We refer to ρ_E simply asthe charge density in solution.

By calculating the line integral of the electric field around a closed contour, we can also show that

(5.6)

where ds is a differential line element. This can be converted to differential form with Stokes’ theorem,leading to

The irrotational property of the electric field is consistent with the earlier definition of the electric field (Equation 5.1) in terms of the gradient of a scalar potential, since the curl of the gradient of a scalar is zero by definition.

Gauss’s law can also be written as

where  is the electric displacement or electric flux density [C∕m²], defined as

While  and  can be converted back and forth quite easily, their analytical purposes are different. Theelectric displacement (through ∇⋅ = ρ_E) tells us what the effects of a source charge are, while the electric field (through = q) tells us what the effects on a test charge are. In vacuum, the two properties have the same value. In matter, (especially high-ε_r materials like water), the two values are different.

5.1.3 Polarization of matter and electric permittivity

A source charge in vacuum induces an electric field, as defined by Coulomb’s law and thepermittivity of free space ε₀. The electrical permittivity of free space ε₀ is a fundamental constant, and is given byε₀ = 8.85×10^-12 C∕Vm.

When a source charge is embedded in matter, we divide the effects of the source charge into two parts: (a) thepolarization of the medium, which we treat as a continuum, and (b) the residual electric field that exists despite the polarization of the matter. Thus, we can think of all electric fields in matter (especially in electrolyte solutions, since water’s electric permittivity is large) as being residual electric fields, the small leftover field after most of the effects of the source charge are canceled by the polarization of the matter. Gauss’s law for electricity in matter lumps the two effects of the source charge (polarization and residual electric field) into one term, the electric displacement.

The electric polarization  inmatter is the dipole moment per unit volume, and measures the degree to which bound charge is polarized. The polarizability of a medium is described by theelectric susceptibility χ_e, which is a function of the atom-scale structure of the material as well as the temperature:

(5.10)

The electrical permittivity [C∕Vm] of a material reflects the sum total of the polarization response of the matter plus the residual electric field, and is given by ε = ε₀(1+χ_e). Electrical permittivities for several materials are given in Table 5.1. The relative permittivity or dielectric constant, definedas

(5.11)

is used to denote the ratio of the total response (both polarization and residual electric field) of a particular medium to the electric field alone.

The terminology here can be confusing. The electric susceptibility tells us how much a medium polarizes in response to an electric field. Free space has nothing to polarize, and its electric susceptibility is zero, whereas water has a large dipole, and its electric susceptibility is large. The electric permittivity tells us how much electric field is caused by a source charge, and is useful when we want to relate fields to charge with Gauss’s law. As compared to vacuum, the dielectric constant tells us how much smaller the electric fields are in a medium if the same source charge is used.

A common misconception is that the dielectric constant indicates how much more a medium polarizes than a vacuum—this is incorrect, because a vacuum cannot polarize. Water’s dielectric constant is 80 not because water polarizes 80 times more than space; rather, the dielectric constant is 80 because when water polarizes, the electric field that is left over is 1/80th as big as it would have been if the water remained unpolarized. The electric susceptibility, similarly, does not indicate that water polarizes 79 times more than space—it indicates that the field caused by the polarized water is 79 times bigger than the residual electric field.

For example, consider a (positively charged) sodium ion dissolved in water. As seen in Figure 5.1, the sodium ion causes the polar water molecules to orient themselves in response to the electric field, and would cause an attractive force on a negatively charged ion (for example, a chloride ion). The presence of the water molecules tends to decrease the electric field caused by the ions, because the polarized water molecules generate their own electric field in the opposite direction. The net effect is that the test charge (the chloride ion) feels an attractive force, but a much smaller one than would be felt if the ions were in a vacuum. As we have written it, Coulomb’s law treats the free charges (sodium, chloride) on an individual, atomistic level, but subsumes the effects of bound charges (water OH bonds) into a continuum with a continuum property ε.

Charge density of an electrolyte solution

We are concerned with the local netfree charge density of the fluids we are studying, which are typically aqueous solutions. Ignoring the solvent, which is typically neutral, the net free charge density ρ_E can be related to species concentrations by

(5.12)

where c_i is the molar concentration of species i, z_i is thevalence (charge normalized by the elementary charge) of species i, and F isFaraday’s constant, equal to F = eN_A = 96485 C∕mol. The free charge density thus corresponds (in liquid systems) to the density of charge of mobile ions. The bound charge has been accounted for by the continuum concept of the electrical permittivity, as discussed earlier. In the absence of external forces, the potential energy of any system of ions is minimized if the net charge is distributed evenly. Thus most media are electroneutral, i.e., the net charge density at all points within domains is zero–any charge is located only at surfaces. This is true for perfect conductors. In the fluid systems we are considering, which have finite conductivity and are also affected by diffusive forces stemming from the thermal energy of the solvent molecules, the fluid has a nonzero net charge density within a finite region near walls, specified by the Debye length λ_D, defined in Chapter 9.

5.1.4 Material, frequency, and electric field dependence of electrical permittivity

Materials experience anelectric polarization in an electric field, as described by Equation 5.10. The electric polarization is a measure of the degree to which the bound charge in a medium polarizes, with positive components being pulled in the direction of the electric field and negative components being pulled in the direction opposite to theelectric field. Multiple polarization mechanisms contribute to this phenomenon, and these mechanisms each have a different characteristic time. In the presence of an electric field, the electron cloud of an atom is displaced with respect to the nucleus, leading to electronic polarization, while atomic bonds stretch, leading to atomic polarization. At 100 V∕cm, this deflection is roughly 1 billionth of an atomic radius. These phenomena are rapid, occurring over a characteristic time approximately equal to 1×10^-15–1×10^-18 s. Molecules with permanent dipoles orient rotationally(owing to the torque, see Section 5.1.10), leading to a net orientational polarization. This is the contribution that makes water’s permittivity so large as compared to other liquids. While atomic polarization requires motion on subatomic length scales (much smaller than 1 Å), dipole orientation requires motion on atomic length scales (1 Å or higher), which takes longer. The effective rotational relaxation time for water at room temperature, for example, is about 8 ps. Orientational polarization only occurs if the frequency of the electric field is below 1×10¹² Hz. Figure 5.2 shows these modes of polarization schematically.

Debye relaxation model for frequency dependence of permittivity Equation 5.9 assumes that the response of the medium to an applied field is instantaneous. However, the polarization requires a finite time. We find it most convenient, then, to decompose the applied field into sinusoidal components and treat the permittivity as a frequency-dependent property. For applied fields whose frequency is much higher than the characteristic frequency of the polarization process, the polarization process is largely ineffective, and thus that process does not contribute to the permittivity at that frequency. For those applied fields whose frequency is much lower than the characteristic frequency of the polarization process, the polarization process contributes to the permittivity at that frequency. We thus think of ε(ω) in terms of the sum of different physical processes. Each process has a characteristictime τ that signifies the speed of the process—for example, τ for atomic polarization is approximately 1.0×10^-15 s, and τ for orientational polarization is approximately 1.2×10^-11 s. In the Debye model of permittivity, the contribution to the electric susceptibility from each polarization mechanism i takes the form

(5.13)

where ω is the frequency of the applied field, χ_e,i(0) is the contribution of mechanism i to the polarization if the applied field is steady, and τ_i is the characteristic time of polarization mechanism i. This expression is analogous to the expression for the frequency dependence of the power transmitted through a capacitor in an RC circuit. The total electric susceptibility for a material is thus given by

(5.14)

and the permittivity is given by

(5.15)

For water at room temperature, for example, χ_e,i for atomic polarization is approximately 5, while χ_e,i for orientational polarization is approximately 72, so an approximation for the permittivity of water at room temperature is

(5.16)

The Debye model assumes that the response of the medium is made up of noninteracting dipoles with a single response time for orientation. This model is a good approximate model, but typically must be replaced with a model that is more sophisticated or has more free parameters, if experimental data are to be well predicted. For orientational polarization, the Debye model does not account for interactions between molecules and the attendant spread of response times; for atomic and electronic polarization, the Debye model is inaccurate because the dipoles are created by the field rather than oriented by the field. Thus the predictions of Equation 5.16 should be considered qualitative. A graph of the dependence on the electric permittivity and the dielectric loss is shown in Figure 5.3.

Electric field dependence of permittivity The use of a permittivity as described above implies that the medium is linear, meaning that double the electric field leads to double the polarization. This is correct for electronic and atomic polarization of water. This approximation is correct for orientation of water only if the orientational energy of the inherent dipole moment of each molecule in the field is small as comparedto k_BT , meaning that the statistical orientation of the molecules is minor. Here, k_B is theBoltzmann constant (1.38×10^-23J∕K). For water, when the applied electric field is on the order of 1×10⁹ V∕m, the orientational energy pE ≃ k_BT is of the same order as the thermal energy of the system, and the water is largely aligned. Once the alignment is significant, the linear model for the response of the material is no longer accurate. An approximate description of the permittivity of water as a function of the electric field uses the Brillouin functionto approximate the statistical vector average of the dipole moment of the water molecules as a function of applied field:

(5.17)

where p is the magnitude of thedipole moment of an individual water molecule (2.95 D), denotes the statistical average dipole moment of an ensemble of water molecules, E is the magnitude of the applied electric field, and B denotes the Brillouin function for dipole orientation with two possible states, i.e., B(x) = 2coth2x-cothx. From Equation 5.17, the ensemble-averaged dipole moment of water molecules varies linearly at low fields (since B(x) ≃ (x) for small x), but approaches a constant ( = p) as the electric field E approaches infinity. Since the ensemble-averaged dipole moment cannot increase beyond p, the permittivity is proportional to 1∕E as E approaches infinity. The approximate description of the permittivity is thus given by

(5.18)

where the 6 includes atomic and electronic polarization, and the 72 corresponds to the orientational polarization. This relation is approximate, and ignores other nonlinear effects such as electrostriction.

5.1.5 Poisson and Laplace equations

The Poisson and Laplace equations are simplified forms of Gauss’s law that apply if no time-varying magnetic fields are present. From above, Gauss’s law for electricity is

(5.19)

Considering irrotational electric fields and writing theelectric field as the gradient of the electrical potential ( = -∇ϕ), we get thePoisson equation:

(5.20)

written here for spatially varying ε. For uniform ε, this can be rewritten as:

The Laplace equationis derived from the Poisson equation if ρ_E is assumed to be zero:

which is the case for electrolyte solutions far away from solid walls.

5.1.6 Classification of material types

We classify materials into several categories based on their response to electric fields. Ideal dielectrics or, equivalently, perfect insulators arematerials that have no free charge–their charge is all bound at the atomic level. These materials cannot conduct current, so their conductivity σ = 0, but they do polarize in response to electric fields and thus they have an electrical permittivity ε. Perfect conductors or ideal conductors havecharges (typically electrons) that are assumed free to move in the material. The electric field and charge density in a perfect conductor are both zero, and thus the conductor is an equipotential and is electroneutral. Regardless of any externally-applied electric field, the charge in a conductor aligns itself at the edges of the conductor to perfectly cancel out this field, and all of the net charge in a conductor is located on the surface of the conductor. Perfect conductors have σ = ∞, and, because they experience no electric field, the permittivity of perfect conductors is undefined. In between these two limits, materials that carry finite current and support a finite electric field are called weak conductors or lossy dielectrics. These materials have finite conductivity σ and finite permittivity ε. In microfluidic systems that employ electric fields, the device substrate is typically an insulator (e.g., glass or polymer) or has an insulating coating (such as a silicon oxide coating on a silicon microdevice), electrodes are made from conducting materials, and the working fluid is typically an aqueous solution that we treat as a lossy dielectric. Thus microsystem boundaries are usually perfect insulators or perfect conductors, while the governing equations are used to study weakly conducting fluids.

5.1.7 Electrostatic boundary conditions

For the electrostatic case (i.e., where all charges are static), Gauss’s lawcan be applied to the boundary between two domains to determine the boundary conditions for the normal and tangential components of the electric field. Given an interface with a charge per unit areagiven by q′′, separating two domains labeled 1 and 2 with permittivities of ε₁ and ε₂, respectively, the boundary condition for the normal component of the electric field is given by

(5.23)

The direction of  points from domain 1 to domain 2, as shown in Figure 5.5. A line integral over a loop at the boundary gives the boundary condition for any tangent component of the electric field:

(5.24)

Equation 5.24 holds for any unit vector tangent to the surface.

Boundary conditions at microdevice walls and inlets We are generally concernedwith the boundary conditions for the electric field for a weakly conducting fluid at a solid surface, either an insulating surface (such as a polymer or glass) or a conducting surface (such as a metal electrode). Because the fluid channels in a microdevice are usually connected to large reservoirs that are, in turn, connected to electrode wires, we often need to treat the inlets and outlets of a device as boundaries in an analytical or computational treatment of the electrostatic equations.

For an insulating wall that has a surface charge density q′′, Equation 5.23 is simplified because the electric field in the insulating wall (₁) normal to the surface is approximately zero. Thus the normal boundary condition is a relation between the surface charge density and the potential gradient normal to the surface:

(5.25)

where n is the coordinate normal to the surface, pointing into the weakly conducting fluid.

For a conducting wall such as a metal electrode, the metal surface provides a constant-potential boundarycondition

(5.26)

Rigorous treatment of inlets and outlets to microchannels requires specification of the entire reservoir geometry. However, since microdevice reservoirs are typically enormous as compared to the microchannels themselves, we neglect any voltage drop between the electrode and the inlet to the microchannel. Thus, in the electrostatic case,

(5.27)

and

(5.28)

Figure 5.4 depicts these boundary conditions for a sample microdevice.

5.1.8 Solution of electrostatic equations

The electrostatic equations are typically solved numerically. For systems that can be approximated as having charge only at interfaces, the Laplace equation applies and thus conformal mapping is applicable. For microchannel systems that can be approximated as having polygonal cross-sections that vary slowly along their axis, Schwarz-Christoffel transforms enable rapid analytical solution (Chapter G).

5.1.9 Maxwell stress tensor

Fundamentally, if magnetic fields are steady and if the volume is not moving quickly, the electrical forces on a particle are simply the sum of all Coulomb interactions between charges and fields. Since Maxwell’s equations relate charges and fields, the force on a control volume can be put into a form that is a function of the fields only. This is particularly useful when we know the electric fields along the surface of a controlled volume better than we know the distribution of charge inside the volume. To calculate the force on a volume, we introduce the Maxwell stress tensor:

which, in the absence of magnetic fields, is given by

With these definitions, the force on a volumecan be written in terms of the Maxwell stress tensor:

The Maxwell stress tensor is a dyadic tensor; see Section C.2.5 in the appendix. The relation between the force and the Maxwell stress tensor is useful in Chapter 17 for determining the force on a microparticle as a function of the electric fields applied to it.

5.1.10 Effects of electrostatic fields on multipoles

The multipolar theory is used to describe electricity and magnetism for both mathematical and physical reasons. Mathematically, multipolarsolutions (see Appendix F) arise naturally from use of separation of variables to analytically solve the Laplace equation. Physically, some of these solutions correspond to physical objects that play a central role in electrodynamic systems, such as point charges and magnetic dipoles. Here we summarize relations for forces and torques on monopoles (point charges) and dipoles. In all cases, the force is written in terms of the applied electric field at the location of the multipole. By applied electric field, we imply the field that would have been at that point if the multipole were absent.

Electrical force on a monopole(i.e., point charge)

The electrical force on a point charge is given by

This is shown schematically in Figure 5.6.

Electrical force on a dipole

Theelectrical force on a dipole in a nonuniform field can be found by summing the forces on its constituent monopoles. This is shown schematically in Figure 5.7. The result can be written in terms of the dipole moment  [C m]:

The dipole moment is measured in units of Coulomb-meters or Debye (where 1 D = 3.33×10^-30 C m).

Electrical torque on a dipole

The electrical torque on a dipole in a nonuniform field can be found by summing the forces on its constituent monopoles (Figure 5.7):

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