Maxwell stress

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 ... 

The double-layer force p acting between two parallel plane surfaces has two contributions—the Maxwell stress and the osmotic pressure ... 

Osmotic pressure arises because the total concentration of all ions [irrespective of their charge, i.e., c+ + c = 2n cosh (e%jkT) is always higher near the surface than in the reservoir (where c, + c = 2n). Thus the liquid in the reservoir tends to flow into the space between plates to dilute this concentration and thereby force the plates apart. So the osmotic pressure contribution is always repulsive. Of course, the sign of the net force will depend on whether the attractive Maxwell stress or the repulsive osmotic pressure is larger. 

Note that the Maxwell stress next to one surface depends only on the charge density of that surface... 

On the other hand, if the amphoteric surfaces are not identical rr0 + a, still tends to zero, but computations show that o-0 and (T tend to become equal and large in magnitude and opposite in sign. Then the osmotic pressure term in Eq, [4] remains bounded, but the Maxwell stress can overwhelm the repulsive contribution (as in Eq. [5]) so that ... 

The first term is the Maxwell stress of the electrostatic field and the second term is the momentum transfer of the ions. At the midpoint, dQ>/dx — 0, and, hence... 

The detection and compensation of the a.c. current is the classical Kelvin method however, the resulting electrostatic forces, i.e. the corresponding cantilever bending, can also be used to establish a potential sensitive feedback. If an a.c. voltage is applied between the tip and the back electrode of the sample instead of using the dither piezo, the Maxwell stress microscopy (MSM) [379-381] or the electrostatic force microscopy (EFM) [317, 382-393] can be performed. 

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] ... 

In Eq. (20) the three terms are related to the Maxwell stress (first), piezoelectric effect (second) and electrostriction (third). In order to obtain information about ferroelectricity via piezoresponse measurements, we need a link between the spontaneous polarisation and the piezoelectric constant. According to Furukawa and Damjanovic, piezoelectricity in ferroelectrics can be explained as electrostriction biased by the spontaneous polarisation if their paraelectric phase is nonpolar and centrosymmetric [461, 495, 496]. Therefore the d33 constant depends on the spontaneous polarisation P5 ... 

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 interaction force P can be calculated by integrating the excess osmotic pressure An and the Maxwell stress tensor T over an arbitrary closed surface 21 enclosing either one of the two interacting particles (Fig. 8.3), which is written as [8]... 

As shown in Chapter 8, the interaction force P can be calculated by integrating the excess osmotic pressure AH and the Maxwell stress T over an arbitrary closed surface E enclosing either one of the two interacting plates (Eq. (8.6)). As an arbitrary surface E enclosing plate 1, we choose two planes x= —oo and x = 0, since )J/(x) = d)J//dx = 0 at x=—oo (Eqs. (10.3) and (10.4)) so that the excess osmotic pressure AH and the Maxwell stress T are both zero at x = —oo. Thus, the force P(h) of the double-layer interaction per unit area between plates 1 and 2 can be expressed as... 

The electrostatic force acting between the two membranes can be obtained by integrating the Maxwell stress and the osmotic pressure over an arbitrary surface enclosing any one of the membranes. We may choose two planes x = 00 (in the solution) and x = x at an arbitrary point in the region Q[Pg.314]

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)... 

The Lorentz force coupled with the Ampere law and the Faraday law leads to the balance of electromagnetic momentum2 which involves the (divergence of) electromagnetic (Maxwell) stress. The isotropic limit of this stress is the electromagnetic pressure... 

---