Maxwells’ electric stress

First, consider the mechanical stresses developed in a material as an electric field E is applied to it. The material will deform until the Maxwell electric stresses are balanced by changes in the mechanical stresses. For an... 

For ionic surfactant solution the body force tensor, Pb, is not isotropic - it is the Maxwell electric stress tensor, i.e. Pb = f6bEE - i6jE l2, where E = -V is the electric field (Landau and Lifshitz 1960). The density of the electric force plays the role of a spatial body force, f, in the Navier-Stokes equation of motion (3). In the lubrication approximation the pressure in the continuous phase depends on the vertical coordinate, z, only through its osmotic part generated from the electric potential and the pressure in the middle plane (or the pressure, pn, corresponding to the case of zero potential) ... 

We can now identify the first term in (A.4) with Maxwell s stress tensor, which acts on any dielectric in an electric field. The magnitude of this force Pe is given by... 

Adamson (51) proposed a model for W/0 microemulsion formation in terms of a balance between Laplace pressure associated with the interfacial tension at the oil/water interface and the Donnan Osmotic pressure due to the total higher ionic concentration in the interior of aqueous droplets in oil phase. The microemulsion phase can exist in equilibrium with an essentially non-colloidal aqueous second phase provided there is an added electrolyte distributed between droplet s aqueous interior and the external aqueous medium. Both aqueous media contain some alcohol and the total ionic concentration inside the aqueous droplet exceeds that in the external aqueous phase. This model was further modified (52) for W/0 microemulsions to allow for the diffuse double layer in the interior of aqueous droplets. Levine and Robinson (52) proposed a relation governing the equilibrium of the droplet for 1-1 electrolyte, which was based on a balance between the surface tension of the film at the boundary in its charged state and the Maxwell electrostatic stress associated with the electric field in the internal diffuse double layer. 

Now, find the force acting on the conducting drop. The momentum fiux density in an electric field is defined by Maxwell s stress tensor [77]... 

The electric stress tensor reduces to what is known as the Maxwell form for the vacuum where e = 1, Equation 2.38 also can be written in the form... 

In what follows, the functions negative Helmholtz energy. Therefo on purely dimensional grounds, a Sijj, must represent surface charge d coefficient of the displacement vec represent a stress tensor. In fact electrostatic field (e.g. for p = 0 Maxwell s electric stress tensor, the next Section that Maxwell s str a part of in our colloid model,... 

Equation [19] expresses the condition of equilibrium as a balance between the electrostatic pressure on ions, where the first term is simply the Maxwell electric field stress tensor,and the osmotic pressure with respect to bulk, given by the second term. Only in special cases can the integral in Eq. [17] be evaluated analytically. Several analytical approximations to the planar PB equation for asymmetric electrolytes have been suggested. " We now present three examples possessing exact analytical solutions beginning with the classic Gouy-Chapman solution. 

We have just described the linearized theory of capillarity. In the electrostatic analogy the field M(r ) is identified with a 2D electrostatic potential ( capillary potential ) and Il(r ) with a charge density ( capillary charge ) Equation 2.8 reduces to the Poisson equation of electrostatics and Equation 2.9 relates the tensor Tn, which has the form of Maxwell s stress tensor, with the electric force exerted on the capillary charge n(rn) (also the usual boundary conditions imposed on the interface have a close electrostatic analogy [34,35]). 

So far P is only an integration constant. As we see later it has a physical meaning P corresponds to the pressure in the gap. The first term describes the osmotic pressure caused by the increased number of particles (ions) in the gap. The second term, sometimes called the Maxwell stress term, corresponds to the electrostatic force caused by the electric field of one surface which affects charges on the other surface and vice versa. 

An important case is the interaction between two identical parallel surfaces of two infinitely extended solids. It is, for instance, important to understand the coagulation of sols. We can use the resulting symmetry of the electric potential to simplify the calculation. For identical solids the surface potential -ipo on both surfaces is equal. In between, the potential decreases (Fig. 6.9). In the middle the gradient must be zero because of the symmetry, i.e. d f(f x/ 2)/df = 0. Therefore, the disjoining pressure in the center is given only by the osmotic pressure. Towards the two surfaces, the osmotic pressure increases. This increase is, however, compensated by a decrease in the Maxwell stress term. Since in equilibrium the pressure must be the same everywhere, we have ... 

In Sect. 7.3, Eqs. (18) and (19) describe the Maxwell stress forces acting on a conductive tip when a combined d.c./a.c. voltage is applied. For the PFM set-up we have to complete the total interaction force by the additional effects of piezoelectricity, electrostriction and the spontaneous polarisation. Both electromechanical effects cause an electric field-induced thickness variation and modulate the tip position. The spontaneous polarisation causes surface charges and changes the Maxwell stress force. If the voltage U(t)=U[)c+UAc sin((Ot) is applied, the resulting total force Ftotai(z) consists of three components (see also Eq. 19) Fstatic, F(0 and F2m. Fstatic is the static cantilever deflection which is kept constant by the feedback loop. F2a contains additional information on electrostriction and Maxwell stress and will not be considered in detail here (for details see, e.g. [476]). The relevant component for PFM is F(0 [476, 477] ... 

At the same time, the coulomb attraction acts between the charges on the particle surface and the counterions within the electrical double layer, which is obtained by integrating the Maxell stress tensor over an arbitrary surface surrounding the particle. The Maxwell stress tensor is given by... 

FIGURE 8.1 Electrical double layer around a charged particle exert the excess osmotic pressure AH and the Maxwell stress T on the particle. 

The effective electrical tension, transmembrane potential, Pja, is defined by the Maxwell stress tensor [59, 89, 92]... 

Many-pass techniques Electric Force Microscopy (EFM) Scanning Capacitance Microscopy (SCaM) Kelvin Probe Microscopy (SKM) DC Magnetic Force Microscopy (DC MFM) AC Magnetic Force Microscopy (AC MFM) Dissipation Force Microscopy-Scanning Surface Potential Microscopy (SSPM) Scanning Maxwell Stress Microscpy (SMMM) Magnetic Force Microscopy (MFM) Van der Waals Force Microscopy (VDWFM)... 

In certain configurations, DEs may also be operated as spring elements with variable stiffness and damping characteristics [4]. If an actuator is held at a certain displacement, changing the applied electric field will vary the Maxwell stress across the device and thus the mechanical impedance. 

The coupling between the hydrodynamics and the electric field therefore arises through the Maxwell stress tensor. The total electric force density comprises the sum of the Coulombic force arising from the presence of free charges... 

For perfectly conducting liquids, Taylor [7] showed that a conical meniscus with a half angle of 49.3° is produced by considering the static equilibrium balance between the capillary and Maxwell stresses in Eqs. 3, 6, and 17. In the perfect conducting limit, the drop is held at constant potential, and hence, the gas phase electric field at the meniscus interface is predominantly in the normal direction. It can then be shown that the normal gas phase electric field scales as in which R is the meniscus radius which then stipulates from Eq. 7 that the Maxwell pressure Pm n,g scales as HR, therefore exactly balancing the azimuthal capillary pressure pc ylR for all values of R. This exact balance, and absence of a length scale selection, is responsible for the formation of a static Taylor cone (Fig. 1) in the dominant cone-jet mode in DC electrosprays [8]. 

Li et al. [9] and Stone et al. [10] later extended Taylor s perfectly conducting limit to allow for the effect of finite liquid conductivities, showing in these cases that the tangential electric field within the slender conical liquid meniscus dominates. However, the tangential liquid phase electric field Ei also scales as MR and thus an exact balance between the Maxwell stress... 

Induced-charge and second-kind electrokinetic phenomena arise due to electrohydrodynamic effects in the electric double layer, but the term nonlinear electrokinetic phenomena is also sometimes used more broadly to include any fluid or particle motion, which depends nonlinearly on an applied electric field, fit the classical effect of dielectrophoresis mentioned above, electrostatic stresses on a polarized dielectric particle in a dielectric liquid cause dielectro-phoretic motion of particles and cells along the gradient of the field intensity (oc VE ). In electrothermal effects, an electric field induces bulk temperature gradients by Joule heating, which in turn cause gradients in the permittivity and conductivity that couple to the field to drive nonlinear flows, e.g., via Maxwell stresses oc E Ve. In cases of flexible solids and emulsions, there can also be nonlinear electromechanical effects coupling the... 

The striking difference lies in the mounting, in series for the Maxwell model and in parallel for the equivalent electric circuit. However, this is purely a question of convention. In electricity, and therefore in an equivalent circuit, the current flows through the components and the potential difference develops between their ends. In rheology, it is the opposite, stress circulates through the components and the shear rate develops across them. 

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