Electrohydrodynamics is a general term describing phenomena that involve the interaction between solid surfaces, ionic solutions, and applied electric and magnetic fields. Electrohydrodynamics is frequently employed in microfluidic devices to manipulate fluids and move particles for sample handling and chemical separation.

Electrokinetics refers to a range of fluid flow phenomena involving electric fields. These include electroosmosis, electrothermal effects, electrophoresis, and dielectrophoresis. Electroosmosis describes the motion of fluids induced by the forces on the charged EDLs at the surfaces of the fluid. Electrothermal effects occur in a conductive fluid where the temperature is modified by Joule heating from an AC electric field. This creates variations in conductivity and permittivity and thus Coulomb and dielectric body forces. Electrophoresis and dielectrophoresis describe the motion of charged and polarized particles in a nonuniform AC or DC applied field.

Magnetohydrodynamics refers to fluid flow phenomena involving magnetic fields. Magnetophoresis is the motion of diamagnetic particles in a nonuniform magnetic field and is commonly used for magnetic bead separation.

Table 2-7 summarizes these categories. Although these examples describe specific multiphysics couplings, COMSOL Multiphysics is not limited to these cases — for example, it is possible to include both magnetic and electric fields to simulate electromagnetophoresis.

When a polar liquid (such as water) and a solid surface (such as glass or a polymer-based substrate) come into contact, charge transfer occurs between the surface and the electrolytic solution. At finite temperature the charges on the surface are not screened perfectly by the ions in the liquid and a finite thickness electric double layer (EDL) or Debye layer develops. Electroosmosis is the process by which motion is induced in a liquid due to the body force acting on the EDL in an electric field.

The Poisson–Boltzmann equation is sometimes linearized — this is referred to as the Debye–Hückel approximation which applies when , where ζ is the potential at the surface of the moving volume of fluid (the zeta potential). At room temperature this corresponds to the limit  mV. Note that there is a layer of immobile ions trapped adjacent to the surface (the Stern layer, which is of order one hydrated ion radius thick) with an associated volume of immobile fluid; this means that ζ is not simply the wall potential. ζ is usually determined experimentally from electroosmotic flow measurements.

The Poisson–Boltzmann equation has a characteristic length scale — the Debye length, λD:

where kB is Boltzmann’s constant, T is the temperature, z is the ion’s valence number, e is the electron charge, and c∞ is the ion’s molar concentration in the bulk solution. When the Debye length is small compared to the channel thickness, the electroosmotic flow velocity can be modeled by the Helmholtz–Smoluchowski equation:

where E is the applied electric field and μ is the liquid’s dynamic viscosity. From this equation, the electroosmotic mobility is naturally defined as:

Further details of the theory of electroosmosis can be found in Ref. 1 and Ref. 2.

where a is the radius of the particle, E is the electric field, and f(a/λD) is known as Henry’s function (note that f(a/λD) is often written as f(κa) where κ=1/λD). Henry’s function takes the value of 1 in the limiting case where (often known as the Hückel approximation) and 3/2 in the case where (this is equivalent to Equation 2-1 and is referred to as the Smoluchowski approximation). Note that if λD is large compared to the fluid volume, the charge is effectively unscreened and the electric force is simply that due to the charge on the molecule. The following result is obtained from the force balance between the Stokes drag and the electric force for an unscreened charged particle:

where q is the charge on the molecule and μ is the fluid viscosity. Further details of the theory of electroosmosis can be found in Ref. 2.

Dielectrophoresis (DEP) describes the motion of polarizable particles suspended in a fluid with an applied nonuniform electric field. The electric field induces a dipole moment on the particles, which in turn produces a net force. In the following discussion the general case of AC electrophoresis is treated, which is more commonly used for practical reasons (see Ref. 1 for further details).

The time-dependent dielectrophoretic force acting on a particle in an applied field is given by (Ref. 1):

where εm and εp are the complex permittivities of the medium and the particle, respectively; a is the radius of the particle’s equivalent homogeneous sphere; and K(εm,εp) is the Clausius–Mossotti function. The complex permittivity, ε*, for an isotropic homogeneous dielectric is

where ε is the electric permittivity, σ is the electric conductivity, and ω is the angular field frequency. The Clausius–Mossotti function is given by:

which depends on the particle’s complex permittivity, εp, and that of the medium, εm.

Taking the time average of Equation 2-2 gives the time-averaged force:

where μ is the dynamic viscosity.

The value of Erms can be computed from an Electrostatics interface in which the root mean square (RMS) voltages are applied as boundary conditions. The gradient of the field can then be computed term by term from the spatial derivatives of an expression for the dot product of the electric field with itself. Finally the dielectrophoretic velocity term can be added to the velocity field to compute the particle velocity field.

Particles with an induced or permanent magnetization can be moved relative to a fluid flow by the application of an external, inhomogeneous magnetic field. This process is analogous to dielectrophoresis and is usually termed magnetophoresis. In the case of a nonconducting particle, in a static, irrotational external applied field, Hext, the magnetophoretic force is given by (Ref. 1):

where μm and μp are the permeabilities of the medium and the particle, respectively (usually μm = μ0); a is the radius of the particle’s equivalent homogeneous sphere; and K(μm,μp) is the Clausius–Mossotti function. Note that the external magnetic field, Hext, should be distinguished from the local field H which includes contributions from the particle itself. In this case the Clausius–Mossotti function is given by:

where μ is the dynamic viscosity.

The value of Hext can be computed from a Magnetic Fields or a Magnetic Fields, No Currents interface.

where σ is the conductivity, ε is the fluid’s permittivity, ω is the angular frequency of the electric field, and τ = ε/σ is the fluid’s charge-relaxation time. The electric-field vector E contains the amplitude and direction of the AC electric field but not its instantaneous value.

Because of the heating, ε and σ are temperature dependent, and their gradients are functions of the temperature gradient: ∇ε = (∂ε/∂T)∇T and ∇σ = (∂σ/∂T)∇T. With water, for example, the relative change rates for the permittivity and the conductivity are (1/ε)(∂ε/∂T) = −0.004 1/K and (1/σ)(∂σ/∂T) = 0.02 1/K, respectively.

The contact angle of a two-fluid interface with a solid surface is determined by the balance of the forces at the contact point. The equilibrium contact angle, θ0, is given by Young’s equation:

Here γs1 is the surface energy per unit area between fluid 1 and the solid surface, γs2 is the surface energy per unit area between fluid 2 and the solid surface, and σ12 is the surface tension at the interface between the two fluids.

In electrowetting the balance of forces at the contact point is modified by the application of a voltage between a conducting fluid and the solid surface. For many applications the solid surface consists of a thin dielectric deposited onto a conducting layer; this is often referred to as electrowetting on dielectric (EWOD). In this case, the capacitance of the dielectric layer dominates over the double layer capacitance at the solid-liquid interface (Ref. 3). The energy stored in the capacitor formed between the conducting liquid and the conducting layer in the solid reduces the effective surface energy of the liquid to which the voltage is applied. For the case when a voltage difference occurs between fluid 1 and the conductor beyond the dielectric Young’s equation is modified as follows:

Here ε is the permittivity of the dielectric, V is the potential difference applied, and df is the dielectric thickness. This equation can be rewritten as

The EWOD applications can be modeled by using Equation 2-3 as an expression for the contact angle. This can simply be entered as the value of the contact angle in the Two-Phase Flow interface that is being used. Note that the software automatically handles a time varying voltage in this expression.