· j = 0In this appendix, more mathematical detail is given to describe the phenomena measured in the single fiber experiments, and an analytical example of a coated sphere is given. The dilute limit (defined below) is considered because that is the only situation where exact closed-form analytical results are known in 3-D.
Consider a composite material in the dilute limit: a uniform matrix containing a small volume fraction of inclusions. By dilute limit is meant that the inclusions are far enough apart so that the effect that each has on the overall properties is independent of the other particles. Only the linear conductivity problem is considered, where the conductivity of the matrix phase is σ1, and the conductivity of the inclusions is σ2. These conductivities are complex. The equation to be solved, for steady-state current flow at a single frequency, is the time independent complex current continuity equation,
|
· j = 0 |
(A.1) |
where 
,
and the
potential V and the normal current are continuous at the boundaries of the
inclusions, and are in general complex quantities
[19]. In the dilute limit, only the fields around
a single inclusion need to be solved. This has been
done for many particle shapes analytically and numerically
[17]. From these solutions, the
frequency-dependent intrinsic conductivity, [σ ( δ )], can be computed,
where δ = σ2
/ σ1
is the contrast between the phases. The fields can be
isotropically averaged, to simulate a random orientation of the particle, or
the intrinsic conductivity can be defined for a single orientation. The
conductivity of the composite, with respect to the conductivity of the matrix,
is given exactly by
 | (A.2) |
where c is the inclusion volume fraction.
Most of the known solutions for σ ( δ ) have been for the special cases of
δ 
1,
where [σ ( δ )] is denoted as [ σ ]¥
, or δ 
1, in which [σ (
δ )] =
[σ ]o
[17]. The intrinsic conductivity is negative when
δ < 1, and positive when
δ > 1. Generally, the intrinsic
conductivity depends on the shape of the inclusion particle and the value of
δ. However, in the two limits mentioned,
the intrinsic conductivity becomes a function of shape only.
Now consider the case of long, thin inclusions. It has been shown that [σ]¥ M increases with the aspect ratio (length to width ratio) of the fiber, so its value can be extremely large for thin objects. For the case of equal width fibers, then the intrinsic conductivity in this limit will increase with the length of the fiber. However, the value of [σ]o goes to a limit of about −5/3 for long thin objects [17]. Therefore, in the DC limit, a small volume fraction (less than 1%) of insulating fibers will have almost no effect on the DC conductivity, while the same amount of highly conducting fibers will have a great effect on the overall conductivity. This effect was seen in the simulation results shown in Figs. 5 and 6. In the dilute limit, the diameter of the arc at low frequency (righthand arc) in the Nyquist plot is then the difference between and R DC ≈ {s 1+ [σ ( δ ) ]o c}−1 and Rint ≈ {σ1 + [σ ( δ ) ]¥ c}−1.
For inclusions having a uniform thickness coating of a third phase, so that solving the dilute limit means solving a three-phase problem, the only known analytical results are for spheres, so that the following analytical example is for the dilute limit of coated spherical inclusions. That is why the needle cases looked at in this paper required computer simulation, since no analytical treatment was available. The sphere example is also relevant to a cement and concrete audience [20].
For a sphere of radius b and conductivity σ2, surrounded by a layer of thickness h and conductivity σ3, in a matrix of conductivity σ1, the intrinsic conductivity in terms of the volume fraction, c, of the inner particle is given by:
 | (A.3) |
where β = [(b + h) / b]3 [20].
When σ3, the layer conductivity, is zero, eq. (A.3) reduces to −3 β /2, the intrinsic conductivity for an insulating sphere of radius b+h [17]. This result is independent of the value of σ2, so that the object acts as if it were entirely made of insulating material. This is true for any thickness h (disregarding any quantum tunneling phenomena), because no current can get through the insulating outer layer.
When σ3, the layer conductivity, is much bigger than σ1 or σ2, eq. (A.3) reduces to 3 β, the intrinsic conductivity for a highly conducting sphere of radius b+h [17]. As long as σ3 is much bigger than σ1 or σ2, this result is independent of the value of σ 2, so that the object acts as if it were highly conducting, for any thickness h. This is because the highly conducting surface layer channels all the current around the object. When the inner object is also highly conducting, so that both σ3 and σ2 are large, then this limit is also reached.
For the complex case, as long as the time constant of the shell is well-separated from that of the matrix, then the layer conductivity will effectively go from insulating to highly conducting before the matrix changes very much, so the conductivity will go from σ / σ 1 = 1 +[σ] o c < 1 to σ / σ 1 = 1 + [σ]¥ c > 1. The resistance will of course have the opposite behavior. Figure 9 shows the result for an spherical inclusion volume fraction of 5%. The units are such that the conductivity of the matrix is unity. Note the real axis intercepts at RDC = 1 / (1 − 3 β c/2) and Rint = 1 / (1 + 3 β c), for c = 0.05 and β = 1.16.
Figure 9: The theoretical Nyquist plot of eq. (A.3), in units where RDC for the matrix is unity, for a coated sphere. The coating thickness is slightly more than 5% of the radius of the sphere, and the volume fraction of dilute spheres was 5%.