Electrical conductivity, isotropically conductive

This equation is identical to the Maxwell [236,237] solution originally derived for electrical conductivity in a dilute suspension of spheres. Hashin and Shtrikman [149] using variational theory showed that Maxwell s equation is in fact an upper bound for the relative diffusion coefficients in isotropic medium for any concentration of suspended spheres and even for cases where the solid portions of the medium are not spheres. However, they also noted that a reduced upper bound may be obtained if one includes additional statistical descriptions of the medium other than the void fraction. Weissberg [419] demonstrated that this was indeed true when additional geometrical parameters are included in the calculations. Batchelor and O Brien [34] further extended the Maxwell approach. 

In isotropic media 0 and S are related by = < , where the scalar parameter a is now referred to as the permittivity. In the international (SI) system it is given by s = erso. where o is the permittivity of vacuum (see Appendix fl) and e, is a dimensionless permittivity that characterizes the medium. Furthermore, according to Ohm s law the current is given by 7 = cr< , where a is the electrical conductivity. The relation V S3 = 0 is a mathematical statement of the observation that isolated magnetic poles do not exist. 

Liquid crystal polymers (LCP) are polymers that exhibit liquid crystal characteristics either in solution (lyotropic liquid crystal) or in the melt (thermotropic liquid crystal) [Ballauf, 1989 Finkelmann, 1987 Morgan et al., 1987]. We need to define the liquid crystal state before proceeding. Crystalline solids have three-dimensional, long-range ordering of molecules. The molecules are said to be ordered or oriented with respect to their centers of mass and their molecular axes. The physical properties (e.g., refractive index, electrical conductivity, coefficient of thermal expansion) of a wide variety of crystalline substances vary in different directions. Such substances are referred to as anisotropic substances. Substances that have the same properties in all directions are referred to as isotropic substances. For example, liquids that possess no long-range molecular order in any dimension are described as isotropic. 

In general, the properties of crystals and other types of materials, such as composites, vary with direction (i.e., macroscopic materials properties such as mass diffusivity and electrical conductivity will generally be anisotropic). It is possible to generalize the isotropic relations between driving forces and fluxes to account for... 

Some important observations, which should apply de facto to many nematic systems containing dispersed nanoparticles, particularly those with metal or semiconductor cores, were reported in 2006 by Prasad et al. [297]. The authors found that gold nanoparticles stabilized with dodecanethiol decreased the isotropic to nematic phase transition of 4-pentyl-4 -cyanobiphenyl (5CB) almost linearly with increasing nanoparticle concentration (x p) and increased the overall conductivity of these mixtures by about two orders of magnitude. However, the anisotropy of the electric conductivity (Act = [Pg.349]

After reduction with dithionite, this dendrimer was cast into a film the electrical properties of which were isotropic. (This means that on the molecular and macroscopic levels there is a three-dimensional electron delocalization.) The conductivity was humidity dependent (water may take part in long-distance electron transfer). At 95% humidity, a mixed-valence film (0.55 electron per diimide) showed conduction at room temperature around 11 fl 1-cm 1 (Miller Mann 1996). As shown later, partially reduced mixed-valence materials are required for organic metals of high electrical conductivity. 

A single crystal is a homogeneous solid, which means that all parts within it have identical properties. However, it is not in general isotropic, so that physical properties such as thermal and electrical conductivity, refractive index, and non-linear optical effect generally vary in different directions. 

The cryptophanes and their complexes may thus be considered as a new family of organic donors in contrast with the flat donors, their spherical shape and large size drive the crystallization towards three-dimensional rather than unidimentional arrays. This feature may aid in the design of new materials with isotropic physical properties (e.g. electric conductivity). 

For simplicity, in this discussion of the physical basis for electrical conductivity, it shall henceforth be presumed that rr is a scalar that is, consideration will only be given to isotropic media such as cubic crystals or polycrystalline samples. Likewise, the theory of current flow in bipolar devices, such as through silicon p-n junctions, belongs in the realm of electrical engineering or semiconductor physics and is only briefly discussed in this text. 

An applied electric field can be the electric held component of an electromagnetic wave, in which case electronic excitations or other optical responses may ensue. These are the topic of the next chapter. Here, the concern is with electrostatics, specihcally, the dielectric, or insulative, properties of materials. In an electrical conductor, an applied electric held, E, produces an electric current - ions, in the case of an ionic conductor, or electrons, in the case of an electronic conductor. Electrical conductivity has already been examined in earlier chapters. In insulating solids, the topic of the current discussion, the response to an applied electric held is a static spatial displacement of the bound ions or electrons, resulting in an electrical polarization, P, or net dipole moment (charge separahon) per unit volume, which is a vector quantity. In a homogeneous linear and isotropic medium, the polarization and electric held are aligned. In an anisotropic medium, this need not be so. The fth component of the polarization is related to the jth component of the electric held by ... 

These materials are extranely attractive for their optical and electrical properties. They have a high intrinsic ionic conductivity, which varies exponentially with temperature and is also anisotropic due to their bi-layer spatial structure [64]. For example, the lyotropic metal alkanoate potassium caproate showed a higher electrical conductivity in the smectic phase than in isotropic solutions. Since they are much less solvated within the layered structure of the smectic phase than in dilute... 

An alternative representation of the volume resistivity SR is its reciprocal or electrical conductivity a= SR"1 For isotropic samples, the material resistivity... 

All physical parameters mentioned above are material specific and temperature dependent (for a detailed discussion of the material properties of nematics, see for instance [4]). Nevertheless, some general trends are characteristic for most nematics. With the increase of temperature the absolute values of the anisotropies usually decrease, until they drop to zero at the nematic-isotropic phase transition. The viscosity coefficients decrease with increasing temperature as well, while the electrical conductivities increase. If the substance has a smectic phase at lower temperatures, some pre-transitional effects may be expected already in the nematic phase. One example has already been mentioned when discussing the sign of Ua- Another example is the divergence of the elastic modulus K2 close to the nematic-smecticA transition since the incipient smectic structure with an orientation of the layers perpendicular to n impedes twist deformations. 

Electrically conductive materials can be made to be either isotropically conductive (IC) or anisotropically conductive (AIC). IC materials are the most common and conduct electricity in all directions. AIC materials conduct materials in one axis when compressed between two contact points. 

Electrically conductive adhesives may be isotropic (conduction equally in all directions) or anisotropic (conduction in the z-direction only). Both types are widely used in the assembly and packaging of electronics. 

Electrical conductivity in anisotropic adhesives occurs by a mechanism different from that of isotropic adhesives. Although metal fillers are also used, they are used in much lower amounts (0.5-5% by volume) so that the adhesive is essentially an insulator in the x-y directions. On inserting the adhesive between the electrodes (for example, the metal bumps of a flip-chip device with metal pads on a flex circuit) of two parts and applying pressure and heat, the metal particles form a z-direction electrical connection between the electrodes while the surrounding material remains insulating. The... 

Gaynes MA, Lewis RH, Saraf RE, Roldan JM. Evaluation of contact resistance for isotropic electrically conductive adhesives. IEEE Trans Comp, Packaging, Mfg Tech 1995 18(2) 299-304. Part B. 

There are two types of conductive adhesives conventional materials that conduct electricity equally in all directions (isotropic conductors) and those materials that conduct in only one direction (anisotropic conductors). Isotropically conductive materials are typically formulated by adding silver particles to an adhesive matrix such that the percolation threshold is exceeded. Electrical currents are conducted throughout the composite via an extensive network of particle-particle contacts. Anisotropically conductive adhesives are prepared by randomly dispersing electrically conductive particles in an adhesive matrix at a concentration far below the percolation threshold. A schematic illustration of an anisotropically conductive adhesive interconnection is shown in Fig. 1. The concentration of particles is controlled such that enough particles are present to assure reliable electrical contacts between the substrate and the device (Z direction), while too few particles are present to achieve conduction in the X-Y plane. The materials become conductive in one direction only after they have been processed under pressure they do not inherently conduct in a preferred direction. Applications, electrical conduction mechanisms, and formulation of both isotropic and anisotropic conductive adhesives are discussed in detail in this chapter. 

Experimental [23] as well as theoretical [24-26] studies of percolation phenomena have been reported. In random and macroscopically homogeneous materials it has been demonstrated [27-29] that at concentrations of metal particles below the percolation threshold (p < Pc) a short-range percolation coherence length, exists. Electrical conductivity is probable for length scales less than Thus even if the metal-filled composite exhibits no bulk electrical conductivity, conduction can occur within domains that are smaller than As the concentration of metal particles approaches oo and the composite becomes isotropically conductive. 

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