Dielectric dispersion in heterogeneous conducting materials is traditionally described by the Maxwell-Wagner theory. (Maxwell 1892, Wagner 1914). Electrical field application makes the possibly free charges of the material to move to the interfacial zones creating long dipoles in the bulk of the material (i.e. Maxwell – Wagner polarization).
Generally, the Maxwell–Wagner polarization ~also known as ‘‘interfacial polarization’’ or ‘‘space charge polarization’’!was most widely adopted to explain the high permittivity observed in the materials, including single-phase ceramics, single crystals, and composites. When applying this model, a series of barrier phenomena or even a complete series-parallel array of barrier-volume effects are assumed. This often arises in a material consisting of grains separated by more insulating intergrain barriers.This model is applied to a heterogeneous medium. On the other hand, Jonscher pointed out that although the Maxwell–Wagner polarization gives rise to satisfactory fitting in many cases, no real physical meaning can be derived.
Korea-Australia Rheology Journal Vol. 11, No. 3, September 1999 pp.169-195
2.1.2. Behaviour under ac fields:Maxwell-Wagner polarisation Consider the same isolated particle as before, but now under a step electric field of magnitude E0 applied from t=0 : E(t)=0 for t0 =E 0 for t0 (7) Immediately after the application of the field, the mobile charge carriers would have only just begun to move and not travelled any significant distance, and the electrical response of the system will be governed by the dielectric constants of the particle material and carrier fluid (ep, ec) (Davis, 1992a). ep, ec are determined for example by molecular orientations under the electric field, and this response can be assumed to be almost instantaneous (over time scales 10-6s at the longest). This initial response is independent of the conductivities of the particle or carrier fluid (sp, sc), and a similar calculation to the above gives for the initial dipole magnitude: pdielec=4peoeca3bdielecE0 (8) where (9) As time progresses, there is a migration of free charge which builds up at the carrier fluid/particle interface, and has the effect of electrically screening the field in the particle interior (Davis, 1992a). Eventually the electric field reaches a distribution determined by the conductivities of the carrier fluid and particle material, and the dipole is given by eq (5). The characteristic time tMW for charge transport, describing the transition from the dielectric-dominated to the conductivity-dominated behaviours, is given by (10) This is also sometimes called the “polarisation relaxation time”. It is easy to show that the dipole will change with time as follows (Ginder and co-workers, 1995): p(t)=4peoeca3E0 [bcond+(bdielec-bcond)e-t/tMW] (11) The model presented here is generally known as the Maxwell-Wagner model of interfacial polarisation (hence the subscript MW). This is the simplest physical model which takes into account the effects of conductive and bdielec ep ec – ep 2ec + ---------------- = tMW eo ep 2ec + ( ) sp 2sc + -------------------------- =
dielectric properties on the polarisation of a particle, and assumes ep, ec, sp, sc are constants. A typical electrorheological fluid designed for use under dc field conditions (as are many proposed devices), such as the carbonaceous particles in silicone oil suspension (Sakurai and coworkers, 1999), would have ep » 5, ec » 3, sp » 10-7mho/m with sc<<sp(at most sc » 10-10mho/m), which gives us from eq (10) tMW » 10-3s.
4.1.2. Maxwell-Wag,qer polarization All ER suspensions possess some level of conductivity. Anderson [ 551 and Davis [45,56] have pointed out that for d.c. and low-frequency a.c. electrIC fields, partICle polarization and partICle interactions will be ccntrolled not by the partICle and fluid permittivities as described above, but rather by the partICle and fluid conductivities. Conductivity in the bulk of both phases will result in free The field external to the sphere is again equivalent to that of a dipole with moment peff = $o~ca3 Re{ p *‘ei”e’}EOe,. (1% The time-averaged force on a sphere at the origin due to a second sphere at (R, 19) may be determined easily in the point-dipole limit as before, FyD(Rij,eij) =&@T2f&f( o,)E& 0 4 ; { [3 cos2 8,- l]e,+ [sin 20,]e,}, (16) II where ErmS - E,l fi, and the “effective relative polarizability” is now where Pd= t =E EP$2EC ,?NJ Oup + 2a; (17) (18) The force is essentially equivalent to the ideal case except that the effective polarizability is now a function of field frequency, as well as the permittivities and conductivities of both phases. The value of Peff, and thus the pair force, depends on the frequency relative to the polarization time constant t,,. In the limit of large frequencies, permittivities dominate the response, lim P2ff(wJ = Pi (20) oer,rnv * m while in the dc. limit, and thus conductivities control partICle polarization forces, regardless of the permittivities. The nor- malized polarization force magnitude, pzff, is plotted as a function of frequency in Fig. 4 for different values of Pd and PC, As we will see below, this likely explains, qualitatively, why high dielectrIC constant materials such as barium titanate ( eP =@( 103) ) do not show a very large ER effect in d.c. fields, and why many systems exhibit a decrease in apparent viscosity with increasing electrIC field frequency [ 3,4,8,62]. For partICles with a surface conductivity, h,, polarization and interactions in the point-dipole limit are described by the Maxwell-Wagner model as described above, provided the partICle conductivity is replaced by the apparent conductivity CT~ + 2&/a [ 9,571.
charge accumulation at the partICle/fluid interface-migration of free charges to the interface prompts the alternative names, “migration” and “interfacial” polarization. In a d.c. field, mobile charges accumulating at the interface screen the field within a partICle, and partICle polarization is completely determined by conductivities. In a high-frequency a-c. field, mobile charges have insuffICient time to respond, leading to polarization dominated solely by permittivities, unaffected by conductivities. At intermediate frequencies, both permittivity and conductivity play a role. The Maxwell-Wagner model (see, for instance, Refs. [57-611) is the simplest description of partICle polarization accounting for both the partICle and fluid bulk conductivities, as well as their permittivities. In this theory, the permittivities and conductivities of the individual phases are assumed to be constants, independent of frequency. The complex dielectrIC constants of the disperse and continuous phases are written Q* ( w,) = ek -j( Us/ ~,,a,), where IC =p,c, j = J-1, and the asterisks represent complex quantities. Consider again an isolated sphere in a uniform a.c. electrIC field, E0 = Re{&&‘e,} . The complex potential will still satisfy Laplace’s equation in the bulk phases [ 571, o”$* = 0, subject to the boundary conditions at the interface, $*i = v” (11) The solution for the complex potential resembles that for the ideal case, (12) (13) (14)
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