# Extension of effective medium theory to three-component system - two conductive particle types in the polymer matrix.

INTRODUCTION

In a previous effort, we constructed (1) a physical model that could accommodate the ideas used in the effective medium theory for electron conduction in a metal-filled polymer (2). This gave tangibility to concepts of node and adjacent node. One cube in the model could be defined as a node. The electron state in this cube is uniform throughout, or in other words, it is superconductive within the cube. A cube width away from any node surface, an adjacent node may be considered to reside. The node and the adjacent node are separated by a single nondecomposable lattice site (cube). Hence, an electron transport event involves a first node, an intervening cube through which the electron is conducted, and a terminal, adjacent node. The intervening cube may be either conductive or polymeric.

Other views of the same phenomena lead to other models. Davenport (3) (and Milewski as cited therein) adopt the basic assumption that significant conduction can take place only when there is a continuous conductive path through the resistive matrix, as when conducting spheres touch, or conducting fibers overlap and touch. Contact resistance views can be considered that include quantum mechanical tunneling (4). We have grouped that possibility as effectively incorporated with the resistance of the polymer filler, similar to the work of others (2).

Nodes in a three-dimensional cubic lattice offer several paths for conduction of electrons. One of these is through the face of an intervening cube. Geometrically, the number of the nearest face-sharing neighbors in a cubic lattice is six. Hence, the number of conductive paths from a first node to an adjacent node is three or z/2 where z is the so-called coordination number (the number of connections to the first node) (5).

Each path can be either conductive or resistive, depending on whether metal or polymer resides in the cube joining the first node and its adjacent node. In reality, materials are not representable by a cubic lattice. However, in a real material, each node will have a certain number of partners in a conduction event. Thus the coordination number, as we have considered it, can vary from node to node and is an abstract concept. Also, each cube linked to a given node pair can also be a part of several other node pairs. Thus, every cube can participate with multiple conduction partners but not all of them can be a node. The role of a given cube depends on what is considered to be the overall circuit.

The above description of the spaced lattice model rests on the following assumptions:

(a) A typical resistance path consists of both conductor (internal to a node) and resistor (contact region resistance) characteristics. Those are neglected. The single nondecomposable cube between node and adjacent node is characterized as either conductive fill or resistive polymer as the circuit element.

(b) The electron transfer is a steady process microscopically. In actuality, it is highly dynamic and unsteady.

(c) All the current that leaves the first node arrives at its adjacent node.

The heart of the effective medium theory (6) is that if [V.sub.t] is the voltage across each of the Individual i'th conductance paths, the effective voltage, [V.sub.avg], then bears the following relation

[summation of] ([V.sub.i] - [V.sub.avg]) = 0 (1)

For a macroscopic amount of material having millions of such nodes, the above expression can be changed to

[integral of] P(C)([V.sub.i](C) - [V.sub.avg]([C.sub.avg]))dC = 0 (2)

where [V.sub.i](C) and [V.sub.avg]([C.sub.avg]) are functions of the respective conductances and P(C) is the distribution function representing the probability of finding a path between a node and its adjacent node occupied by a conducting metal particle, or the polymer. (Note that in the integration over dC in Eq 2, each new C has its own associated i'th voltage drop. Thus the subscript i is a type of redundant label. We keep it because it is necessary for later manipulations).

RESISTIVITY VERSUS FILL DERIVATION

The probability of finding a conductor in a given lattice site is its volume fraction [V.sub.c] and will occur only when C = [C.sub.m]. Alternatively, the probability of finding a polymer site will be (1 - [V.sub.c]). Hence, the probability function P(C) of finding either a polymer or a metal particle in a site that is selected at random is

P(C) = ([v.sub.c]/f)[Delta](Cm) + (1 - [v.sub.c]/f)[Delta](Cp)

where f is the packing factor and its value is [Pi]/6, while [Delta] is the Dirac delta function of the conductances (2). The packing factor is known to be influenced by the length to diameter L/D ratio of the packing particles (3). In the actual materials we shall build and report on at another time, the L/D is nearly one. The influence on packing is considered negligible. The main feature of the Dirac delta is that the integral of any arbitrary function when multiplied by the Dirac delta gives the value of that function at the argument of the Dirac delta. Knowing the probability of distribution P(C) and the voltage relation, Eq 4, we can integrate Eq 2.

[integral of] (([v.sub.c]/f)[Delta](Cm) + (1 - [v.sub.c]/f)[Delta](Cp)) x (Ci-[C.sub.avg]/[Ci + (z/2 - 1) x [C.sub.avg]]) dCi = 0 (3)

We do not wish to suggest that the voltage relation

Vi(C) - [V.sub.avg]([C.sub.avg])/[V.sub.avg]([C.sub.avg]) = [C.sub.avg] - Ci/Ci + (z/2 - 1) x [C.sub.avg] (4)

is the simple relation we and others had implied in our brevity. For this reason we more carefully rederive it (Appendix A). Eq 3 is a Lebesque integral since it is highly discontinuous. It can be distinguished from the general Riemann integral in the sense that all Riemann integrands can be divided into an infinite number of small continuous intervals. Hence, the given Lebesque integral has non-zero values only when Ci = Cp and Ci = Cm. The result of the integration gives

(1 - [v.sub.c]/f)([C.sub.p] - [C.sub.avg]/[C.sub.p] + (z/2 - 1)[C.sub.avg]) + ([v.sub.c]/f)([C.sub.m]-[C.sub.avg] / [C.sub.m]+(z/2-1)[C.sub.avg]) = 0 (5)

This expression can be simplified to give a quadratic in [C.sub.avg], the

effective conductivity.

A x [([C.sub.avg]).sup.2] + B x [C.sub.avg] + Cm x Cp = 0 (6)

where

A = (2 - z)/2; B = Cm[prime] + Cp[prime]

Cm[prime] = Cm(z[v.sub.c]/2f - 1);

Cp[prime] = Cp(z/2(1 - [V.sub.c]/f) - 1)

The quadratic Eq 6 can be solved for [C.sub.avg]. This conductance becomes the equivalent with which to replace Ci in the original circuit, where nodes are separated by a length of one lattice unit. With [C.sub.avg] in hand, the average overall behavior of the material can be calculated, i.e., it can be substituted in a macroscopic material Ohm's law application.

Our previous work focused on multiple singularities in the above expression for [C.sub.avg] (equivalently, resistivity) to establish dependence on [v.sub.c] at high- and low-fill fractions. Those expressions were important because they matched more algebraically tedious derivations.

NUMERIC BRIDGE TO THE THREE COMPONENT SYSTEM

To quantify the equations obtained in the earlier section, a two-particle system consisting of poly (methyl methacrylate) (PMMA) and aluminum particles, which act as a filler, are taken as an illustration. The conductance value for both components are known. The details of how these values are obtained is shown in Appendix B (which also relates material property/model parameter and computational details). We assume that all the Al particles are uniform in size. Knowing conductance values and the packing factor ([Pi]/6), we solve the quadratic for different fill fraction values ranging from below the critical fill or the percolation threshold value, which is around 0.17, to well above the critical values. A range of values of average conductance vs. the fill fraction is obtained. The same equation is solved numerically for a range of [v.sub.c] values, and zeros of the polynomial are found that correspond to the average conductance value. The quadratic formula solution and the numerical solution are plotted together in Fig. 1. They match very well. Also in Fig. 1 is a plot derived from the experimental data of Gurland (7) (his Fig. 1 data). As explained in Appendix B, this is not a rigorous test of the theory: it is included only to show plausibility.

The critical fill 0.17 value here differs from values around 0.3 found by Gurland (7), experimentally obtained in a system of highly spherical silver particles in bakelite. As he pointed out, percolation treatments based on various cubic models can predict these high critical fill values depending on the particular cubic model. He also pointed out, by photograph and drawing, how real systems grossly disturb a cubic model. This distortion effectively yields results given by high critical fill cubic models. This tendency for real systems to require higher critical fills than predicted by ideal lattice/cubic theories follows in our work as well (to be discussed).

The reason the numerical capability has been added is that we anticipate that in the three-component case the algebra may be very discouraging for a closed-form solution. With some credibility established for the numerics in the known realm, we may then rely on them in the unknown realm.

THREE-COMPONENT SYSTEM

Having analyzed the system containing polymer and metal particles of uniform size, we propose the same foundations to hold true for the three-particle system in an analytical sense. By three particles, we mean the polymer matrix containing two different sizes (or any other label of difference, not just size) of tiller particles. For this case, the probability function can be written as

P(C) = ([v.sub.c1]/f)[Delta]([Cm.sub.1]) + ([v.sub.c2]/f)[Delta]([Cm.sub.2]) + (1 - [v.sub.c]/f)[Delta](Cp) (7)

where [v.sub.c] = [v.sub.c1] + [v.sub.c2].

Once again, the basic assumption is that the probability of finding a lattice site with a given conductor is proportional to its volume fraction. The result of the integration of Eq 3 for the above probability function yields

[Mathematical Expression Omitted]

Our anticipation that Eq 8 would involve tedious algebra appears correct. We know that for general cubic equations. depending on the nature of the coefficients, we can have one or more real and/or imaginary roots. Because of the complexity of the coefficients of the above cubic, it appears desirable that we now rely on our numberical tool. For a range of [v.sub.c] values, zeros of the above Eq 8 are found which correspond to the [C.sub.avg] value. A sample of computer code and its output illustration are available (8). Table 1 shows the values obtained for [C.sub.avg] as a function of [v.sub.c] for different particle loadings. An XY graph is constructed for vc vs. [C.sub.avg]. Two cases are taken for now:

(i) [v.sub.c1] = [v.sub.c2] = 0.5[v.sub.c] ([v.sub.c1] + [v.sub.c2] = [v.sub.c])

(ii) [v.sub.c1] = 0.1[v.sub.c]; [v.sub.c2] = 0.9[v.sub.c]

Here [Cm.sub.2] = [Cm.sub.1]/2.

It can be seen from the plot that the conductance values decrease as we go from equal volume fractions of the two kinds of conducting particles to nine times the volume of the lower conducting particle, the total conducting volume being held fixed in both cases for a given average conductance/volume fraction pair. This lower conductance can be interpreted in several ways. One is to consider that more lower conductance simply requires more total conductor volume for a given average conductance. Another interpretation would be to say that the smaller conductance represents a smaller-sized particle. The lattice size has not changed and it is still counted as a conducting site. It is just filled with the full conductor plus a little resistance. We could consider that as a way of representing the nonideality of distributed size particles. The fact that both cubic (distributed size) cases are lower than the quadratic (equivalent single size) case might be said to reflect the real world fact that, because of this nonideality (distributed size), more conductor volume is needed for a given material average conductance. This parallels the findings of Gurland (7) when he discovered that more fill was needed for a given material conductance than was predicted by some of the theories he considered. What we are showing here is that a little more faithful theory comes a little closer to describing the experiment.

Consideration of the three-component system has pointed the way to a multicomponent attack. For n conducting components, we would simply add to Eq 7 the right hand piece ... [v.sub.cn]/f)[Delta]([Cm.sub.n]), and of course [v.sub.c] = [v.sub.c1] + [v.sub.c2] + ... + [v.sub.cn]. This would have the consequences of adding to Eq 8 the righthand side term:

...([v.sub.cn]/f)([Cm.sub.n] - [C.sub.avg]/[Cm.sub.n] + (z/2 - 1)[C.sub.avg])
```Table 1. Values for Average Conductance ([C.sub.avg]) as a Function of Fill
Fraction ([V.sub.c]) (Three-Particle System).

[C.sub.avg] [C.sub.avg]

0.10 2.342 x [10.sup.-22] 2.342 x [10.sup.-22]
0.15 7.114 x [10.sup.-22] 7.114 x [10.sup.-22]
0.16 1.201 x [10.sup.-21] 1.201 x [10.sup.-21]
0.17 3.850 x [10.sup.-21] 3.850 x [10.sup.-21]
0.18 0.3704 0.2918
0.19 1.0541 0.8264
0.20 1.7448 1.3624
0.25 5.2775 4.0587
0.30 8.8985 6.7758
0.35 12.5706 9.5072
0.40 16.2751 12.2489
0.45 20.0012 14.9985
0.50 23.7425 17.7536
0.55 27.4945 20.5133
0.75 42.5668 31.5825
```

Multicomponent consequences are vast and can only be briefly touched here. Our purpose has been mainly to introduce the multicomponent attack, so that multicomponent consequences can be probed later.

APPENDIX A

The digestion of Eq 4 can perhaps best be gained through three ideas: the average conduction current-coordination number relation; its use in an equivalent first node to adjacent node circuit; and Ohm's law applied to the voltage difference in that circuit (which can be inverted for the final expression).

Average Conduction Current

If I is the current into the node and I[c.sub.avg] is the current in an average arm (there are several arms between the two nodes), then I[c.sub.avg] can be expressed as

I[c.sub.avg] = I/(z/2) (9)

since only half the connections participate in conduction away from the first node to the adjacent node; the other half are involved with conduction into the first node.

Equivalent Circuit Analysis

Consider a real circuit representation of a particle/polymer system where Ci is an individual conductance and [C.sub.n] is the parallel circuit equivalent of all other conductances between the first and the adjacent node. If I is the total current between the first node and the adjacent node then

I = I[c.sub.i] + I[c.sub.n]

where I[c.sub.i] and I[c.sub.n] are respective arm currents. Also

C = Cn + Ci

There are many such real circuits between a first node and an adjacent node with different individual conductances. All those individual conductances can be very well represented by an equivalent conductance [C.sub.avg], averaged over all the material. Now the total current for an analogous circuit between the first and adjacent node under consideration can be expressed as

I = I[c.sub.n[prime]] + I[c.sub.avg]

and the total conductance would be

C = Cn[prime] + [C.sub.avg]

An expression for (Vi - [V.sub.avg]), the difference between an individual node pair voltage drop and a whole material average node pair voltage drop at a given current, is needed.

(Vi - [V.sub.avg])

We know that V = IR = I/C (Ohm's law)

Vi(C) - [V.sub.avg]([C.sub.avg]) = (I/Cn + Ci - I/Cn[prime] + [C.sub.avg])

= I(Cn[prime] + [C.sub.avg] - Cn - Ci/Cn[prime] + [C.sub.avg])(Cn + Ci))

The conductance difference between the node pair in considering all arms but one is Cn[prime] and Cn in the two representations. This difference is small compared to the [C.sub.avg] - Ci difference. Hence,

= I/(Cn + [C.sub.avg])([C.sub.avg] - Ci/Cn + Ci)

The coefficient of the parenthetic expression can be found from application of Ohm's Law to the equivalent circuit. Rearrangement of this form gives

I/Cn + [C.sub.avg] = [V.sub.avg]

I/[V.sub.avg] = Cn + [C.sub.avg] (10)

The unknown Cn in the parenthetic expression can be given in terms of the known [C.sub.avg]. At that point, the voltage difference will be entirely expressible in terms of the single unknown Ci.

From Eq 9

[Ic.sub.avg] = I/(z/2)

Hence,

I/[V.sub.avg] = (z/2)([Ic.sub.avg]/[V.sub.avg]) = (z/2) x [C.sub.avg] (11)

Substituting in the rearranged form Eq 10, one at once finds

(z/2) x [C.sub.avg] = Cn + [C.sub.avg]

or solving for Cn

Cn = (z/2 - 1) x [C.sub.avg] (12)

This is the function of Cn in terms of the known [C.sub.avg] that we sought above.

Substituting Eq 12 in the voltage difference expression gives

Vi(C) - [V.sub.avg]([C.sub.avg])/[V.sub.avg]([C.sub.avg]) = [C.sub.avg] - Ci/Ci + (z/2 - 1) x [C.sub.avg] (13)

APPENDIX B

Sample Calculation for the Solution of the Quadratic in [C.sub.avg]

a[([C.sub.avg]).sup.2] + b([C.sub.avg]) + CmCp = 0

where

a = (2 - z)/2 = (2 - 6)/2 = -2; b = Cm[prime] + Cp[prime]

where

Cm[prime] = Cm(z[v.sub.c]/2f - 1)

Here Cm is the conductance of aluminum particles and its value is 35.36 mhos. This is obtained as follows:

We know that resistance is given by R = [Rho]1/A where [Rho] is the resistivity of A1 particles and its value is 2.828 x [10.sup.-6] [Omega] cm. We assume that the length of a cube between a node and its adjacent node is 1 [Mu] or [10.sup.-4] cm. Hence the cross-section area between the two nodes becomes [10.sup.-8] [cm.sup.2]. Therefore the resistance becomes R = 2.828 x [10.sup.-2] [Omega]. Hence the conductance is C = 1/R = 35.36 mhos. Let Cm = [Alpha] = 35.36 and the packing factor f = [Pi]/6. Hence Cm[prime] = [Alpha](18[v.sub.c]/[Pi] - 1) and

Cp[prime] = Cp{z/2(1 - [v.sub.c]/f) - 1}

Here Cp is the conductance of the polymer particles (PMMA) = 1.0 x [10.sup.-22] mhos and is obtained in the same way as the A1 particles. Resistivity of PMMA is [10.sup.18] [Omega] cm. Hence

Cp[prime] = [10.sup.-22] (2 - 18[v.sub.c]/[Pi])

Substitution of these values for the coefficients of the quadratic gives

2[([C.sub.avg]).sup.2] - ([Alpha]{18[v.sub.c]/[Pi] - 1} + [10.sup.-22]/{2 - 18[v.sub.c]/[Pi]})[C.sub.avg] - [Alpha][10.sup.-22] = 0

2[([C.sub.avg]).sup.2] - [Alpha]({18[v.sub.c]/[Pi] - 1} + [10.sup.-22]/[Alpha]{2 - 18[v.sub.c]/[Pi]})[C.sub.avg] - [Alpha][10.sup.-22] = 0

Let

[A] = ({18[v.sub.c]/[Pi] - 1} + [10.sup.-22]/[Alpha]{2 - 18[v.sub.c]/[Pi]})

Applying the quadratic formula to the above equation gives

[C.sub.avg] = [Alpha][A] + [square root of][[Alpha].sup.2][[A].sup.2] + 8.0 x [Alpha] x [10.sup.-22]/4

This expression of [C.sub.avg] is then put in the code which generates its values for a series of fill fraction ([v.sub.c]) values. It can be observed from the given expression for [C.sub.avg] that below the critical point Factor A would always be negative and that the calculated value of [C.sub.avg] is a very small number, ([Epsilon] [is greater than] 0) given by

[Epsilon] [is approximately equal to] [square root of][Alpha] x 8.0 x [10.sup.-22] = 1.682 x [10.sup.-10]

Hence, the values of [C.sub.avg] below [v.sub.c] = 0.18 are for all purposes zero.

If the Fig. 1 data of Gurland (7) could be assumed to apply to [10.sup.-3] cm-sized cubes with a factor of two-spread in size, then, since the size-relative error propagates directly as conductance-relative error, these data would appear as in Fig. 1 herein. This is not a rigorous test of the theory but a show of plausibility.

NOMENCLATURE (in approximate order of appearance)

z = Coordination number. Vi(C) = Voltage across the i'th conductance path. C = Conductance of a site. [V.sub.avg]([C.sub.avg]) = Voltage across conductance Cavg. [C.sub.avg] = Average conductance. P(C) = Probability of finding a site of conductance C. [v.sub.c] = Volume fraction of conductor in the polymer/metal matrix. Cm = Conductance of a metallic site. f = Packing fraction. [Delta] = Dirac delta function. Cp = Conductance of a polymer site. [Pi] pi = 3.14. L/D = Particle length to diameter ratio. Ci = Conductance of the i'th conductance path, or site. [v.sub.c1] = Volume fraction of conductive material 1. [v.sub.c2] = Volume fraction of conductive material 2. [Cm.sub.1] = Site conductance of conductive material 1. [Cm.sub.2] = Site conductance of conductive, material 2. [Cm.sub.n] = Site conductance of the n'th material. [v.sub.cn] = Volume fraction of the n'th material. I = Current into a node. [I.sub.cavg] = Current in an arm of average conductance. [Ic.sub.1] = Current in individual conductance. [Ic.sub.n] = Current through conductance Cn. [Ic.sub.n[prime]] = Same as above, only for material averaged case. C = First node to adjacent node conductance. Cn = Equivalent conductance of combined paths except i'th. Cn[prime] = Same as above, only for material averaged case. V = IR = I/C = Voltage current resistance conductance in Ohm's law. Cm = Conductance of aluminum particles in the example. [Alpha] = Cm, invoked for algebraic convenience. R = [Rho]1/A resistance to resistivity relationship. l = Length of lattice cube edge. A = Area of lattice cube face. [Mu] = Micron, or [10.sup.-4] cm. [A] = Group for algebraic convenience. [Epsilon] = Indicates a small number.

REFERENCES

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