12.3.1 Some Conductivity Theory

The electrical conductivity σ of materials covers a wide range. For insulators typically σ < 10⁻¹² Ω⁻¹ m⁻¹, while semiconductors cover the range from about 10⁻¹² to 10 Ω⁻¹ m⁻¹ and for semimetals and metals σ > 10 Ω⁻¹ m⁻¹, reaching values of up to σ ≈ 10⁸ Ω⁻¹ m⁻¹. For t‐PAc, σ ≈ 10⁶ Ω⁻¹ m⁻¹.

The conductivity σ is codetermined by the mobility μ and density n of the free charge carriers with charge q and reads

12.1

The mobility is the drift (average) velocity 〈v〉 of the charge carriers per unit field strength with which the charge carriers move under the influence of an applied field E and has dimension (m² V⁻¹ s⁻¹). In a simple model, the free motion of the charge carriers is hindered by scattering events, such as with other charge carriers or impurities, and the mobility μ is determined by the average time between scattering events τ. The force qE on the mass m of the charge carrier determines μ via

12.2

Both electrons and holes may contribute in which case q is the elementary charge e. Conductivity theory tries to explain the various contributions to n and μ, and in the sequel we briefly discuss both.

With respect to the mass of the charge carrier, in practice it does not equal its physical value, and an effective mass m^* is introduced. We consider that in an applied field E the acceleration dv/dt of an electron with wave vector k is given quasiclassically by

12.3

Because the energy gain dU of the electron from the field is

12.4

substitution of dk/dt in dv/dt leads to

12.5

Because for free electrons U = ℏ²k²/2m, m^* = m. For a band model this is no longer true and m^* ≠ m, as we will see in Section 12.3.2.

The mobility μ = qτ/m is codetermined by the time τ, considered to be the time that charge carriers can move without hindrance between scattering events. In some cases the transport of charge may be hampered by traps that capture the charge carriers. If these traps have a potential well depth U_trap comparable with kT, the carriers will escape easily, and this process leads only to a reduction of the overall mobility, given by μ_trap = μ[τ_car/(τ_car + τ_trap)], where μ is the mobility between the traps and τ_car and τ_trap are the mean lifetime between the traps and mean residence time in the traps, respectively. This leads to a thermally activated trap lifetime τ_trap ∼ exp(U_trap/kT). If there are many traps, space‐charge effects arise and the current density i ∼ V²/d³, where V is the applied voltage and d the sample thickness [16]. In this regime, interpretation is less straightforward.

So far, we assumed that the presence of free charge carriers had no effect on the lattice, but in reality there is always some distortion near an uncompensated charge. For polymers this effect is particularly strong if the material has a narrow band, implying a high effective mass and low velocity, or if the charge carrier is trapped. In these situations the lattice around the charge carrier becomes polarized, and the carrier and lattice distortion move simultaneously through the lattice. This distortion can be seen as a (pseudo)particle and is called a polaron.

In a simple model, consider the interaction of an electron with two atoms [17]. Using the internal coordinate x, describing the displacement of the two atoms from their equilibrium position, the elastic energy is Ax². The electron will add an extra term −Bx, so that the total energy has a minimum when d(Ax² − Bx)/dx = 0, or x_min = B/2A. At this minimum the elastic energy increased by Ax_min² = Bx_min, while the electron energy decreased by −Bx_min, so that the overall reduction is Bx_min. Thus the electron produces a well, which will, in principle, always contain one localized state. This state becomes a bound state in which the electron is self‐trapped for a well with depth V₀ and width a when m^*V₀a²/ℏ² > 1. If a is approximately equal to the lattice constant, the polaron is a so‐called small polaron, characteristic of a strongly localized state, while if a equals several lattice constants, we speak of a large polaron. The latter has properties in between that of a small polaron and an unperturbed electron.

In Section 12.3.2 we describe band theory in some detail, explaining why intrinsically conductive polymers, in particular t‐PAc, are semiconductors. Thereafter, we discuss some effects influencing conductivity, relevant for all intrinsically conductive polymers.

12.3.2 Simple Band Theory

For the description of the conductivity along polymer chains, a simple band picture, based on the tight‐binding approximation, is useful. This approximation applies as long as the energy perturbations due to chemical bonding are small as compared with the atomic level spacing; it ignores the electron–electron interaction explicitly but takes this effect into account implicitly via the choice of its parameters. The description uses a unit lattice cell containing one or more atoms, each with one or more atomic orbitals to characterize each atom. The band model cannot describe charge transport if the band width becomes too small with a mean free path length of the electrons equal to the lattice constant. Also if the charge carriers become self‐trapped as small polarons, the concentration of impurities and traps is high, the crystalline order is disturbed too much, or by a combination of these factors, the band description fails. In that case we need to describe charge transport by hopping, briefly discussed in Section 12.3.4.

We assume that N atoms are located in a one‐dimensional (1D) lattice at positions …, –2a, –a, 0, a, 2a, …, or, generally at r_m, where a is the lattice constant. The atoms are described by a single atomic wave function, given by …, |–2a〉, |–a〉, |0〉 , |a〉 , |2a〉, …, or, generally by |m〉, where the Dirac notation is used (for the intricacies, see [18]). As all atoms are equivalent, Bloch's theorem applies, and we write, for periodic boundary conditions using the allowable wave vector k = 2πj/Na with −N/2 ≤ j ≤ N/2,

12.6

where the atomic wave functions |m〉 are assumed to be known by the solution of the atomic Schrödinger equation

12.7

with H the Hamilton operator for the energy and ε the energy eigenvalue and for which 〈m|m′〉 = δ_m,m′ applies with δ the Kronecker delta (δ_kl = 1 if k = l and δ_kl = 0 if k ≠ l). Now we want to calculate the expectation value of a quantity represented by the operator X, say, 〈k′|X|k″〉, but as all atoms are equivalent calculating 〈0|X|k〉 suffices. For this expectation value we have

12.8

For the LHS we may write

12.9

while for the RHS we have

12.10

Now we take into account only the on‐site (k″ = k′, where k″ and k′ indicate two k‐values) and nearest‐neighbor interactions (k″ = k′ ± 1) and thus apply the Hückel approximations, reading

12.11

This simplifies the solution tremendously and we are left with

12.12

describing two band of energy levels centered at ±α with respect to E_F and having a width 4β (Figure 12.5). An extended view of the band structure, showing the effect of increasing coupling, is given in Figure 12.6.

Let us return to the effective mass for a moment. Generally, for an electron in a band, the effective mass m^* = ℏ²/(d²ε/dk²) for k ≅ 0 can be estimated by expanding the cosine in ε = α + 2β cos ka to second order, yielding m^* = ℏ²/2βa². Hence, the larger β, the smaller m^*, reflecting the interaction among a set of atomic orbitals. When β → 0, the band becomes flat, the electron becomes localized, and m^* → ∞.

The next step is to allow for more than one atom per cell. In the basic equation

12.13

the index m for a cell is just a label, and we add to it the label for atom j, so that |m〉 becomes |m, j〉, and adding a coefficient c_j to be able to conserve normalization,

12.14

We have 〈k,i|l,j〉 = δ_k,lδ_i,j with δ again the Kronecker delta. The basic equation for the expectation value for the energy operator H then reads

12.15

which we can transform via

12.16

to

12.17

where . A solution is found when the determinant is zero, that is,

12.18

It appears that in this case the bands are separated by an amount depending on the values for the on‐site integrals α₁ and α₂ of atom 1 and 2 and on the values of β involved. Note that we can play the relabeling trick again, if we realize that m, j is also just a label to which we can add the label α in case we require more than one orbital at atom j, so that |m, j〉 becomes |m, j,α〉. This will lead to more bands than just a valence and conduction band. For details we refer to the literature, for example, the very readable introduction by Sutton [18].

As an example, let us apply the above to the important example of t‐PAc for which we take alternating bond lengths (Figure 12.3b) with u_n = (−1)ⁿu [19]. Using the Hückel approximations, α₁ = α₂ = α and indicating the transfer integrals for the short and long bonds by β₋ = β − 2γu and β₊ = β + 2γu, respectively, with u = (u_n+1 − u_n) the displacement from the average bond length, we obtain

12.19

or, taking the real part and using z = 2γu/β,

12.20

In Figure 12.6 this solution is plotted as a function of k for various values of γ, showing an increasing band gap with increasing value of γ. The band gap at k = π/2a is 2Δ = 8γu = 4zβ. In Figure 12.7 the results of a more sophisticated band structure calculation for t‐PAc with identical and alternating bond lengths are plotted. Here several wave functions per atom are taken into account. This figure shows not only the absence and presence of a band gap for the symmetric and distorted structure, respectively, but also a rather similar band structure for the rest of the bands [20].

The total electronic energy is given by U_elec = −2∑_kε_k, where the sum runs over the occupied levels. In view of the density of levels the sum can be replaced by an integral via Σ_k(·) → (L/π)∫(·)dk = (Na/π)∫(·)dk, so that

12.21

For z ≪ 1, the elliptic integral of the second kind E(1 − z²) can be approximated by

12.22

Now we have to realize that the σ‐bonds are also distorted, which leads to a distortion energy, approximated as an elastic energy reading U_elas = K∑_n(u_n+1 − u_n)², where K is the force constant. The total energy thus becomes

12.23

Realizing that z = 2γu/β and minimizing U via dU/du = 0, we obtain a saddle point at u = 0 and two stable minima at u₀/a = ±4 exp[−(1 + 1/2λ)], where the dimensionless electron–phonon constant λ = 2γ²/πKβ is introduced. Hence, the band gap becomes 2Δ = 4zβ = 16β exp[−(1 + 1/2λ)]. Using K = 21 eV Å⁻², as relevant for t‐PAc, and requiring that 2Δ for the ground state leads to 1.4 eV, the optimum occurs for 2γ/β = 1.65 Å and u₀ = 0.042 Å, the latter corresponding to a bond length increase of 3^1/2u₀ = 0.073 Å. Altogether the energy decrease per atom for the dimerized state as compared with the equal bond length state is ΔU/N ≅ 0.015 eV. Hence, the structure with alternating bond lengths is indeed favored. In Figure 12.8a the total energy is plotted, showing a double minimum energy well, as for any two consecutive bonds either bond could be shortened (and the other lengthened), and, hence, the ground state is doubly degenerate with configurations I and II. The model is often referred to as the Su–Schrieffer–Heeger (SSH) model, although in fact in 1955 Peierls already predicted the splitting of such a partially filled band for a linear chain (linear metal) using rather general arguments [21].

There is one more issue to consider for t‐PAc that is related to the degeneracy of its ground state, as characterized by the bond alternation parameter u/u₀ with values ±1. If two parts of the chain for some reason or another have a different bond alternation parameter, for example, due to substitution at a chain end, a defect will occur where u/u₀ changes sign (Figure 12.8b). This defect contains an unpaired electron residing at a carbon atom having two long (single) bonds, although the chain as a whole is neutral. The deviation from the regular long–short pattern is extending over several lattice spacings of size a, and for a defect centered at n₀, it can be described by u_n = u tanh[(n−n₀)/ξ] cos nπ with ξ the characteristic length. Such a defect is called a soliton. Actually for t‐PAc, ξ ≅ 7a, implying that it takes about 14a before u/u₀ has changed its value from −1 to +1. Since the energy of the chain on both sides of the defect is equal, a soliton can easily move along the chain with an activation energy estimated as 0.04 meV (≅0.0016kT at 25 °C). In contrast, if the soliton would be localized on one atom, the activation energy would be about 40 meV (≅1.6kT at 25 °C). For t‐PAc, in the band picture sketched above, for a soliton, m^* ≅ 6m. This relatively low value of the effective mass is directly related to the small value for u₀ as compared with a. The formation energy of a soliton for t‐PAc is estimated to be 0.6Δ, and since the experimental band gap of t‐PAc is ≅1.4 eV, the formation energy is ≅0.42 eV. Hence, solitons are not created thermally, and their presence rests on external factors, such as substitution. The soliton in a broader context has been reviewed by Scott, Chu, and McLaughlin [22].

12.3.3 Doping

As the electron associated with a soliton is nonbonding, a soliton is a zero energy state of the chain with its energy level lying at the center of the band gap, and its wave function can be considered to consist of an equal contribution of states at the edges of the valence and conduction band (Figure 12.9a,b). The model is said to be charge conjugate invariant since the results are independent of the type of charge carrier (electron or hole). When an electron‐donating group is attached to the chain, that is, t‐PAc is doped, a negatively charged soliton can be created with energy less than an excited electron in the conduction band. Similarly, for electron‐accepting groups a positively charged soliton can be created, needing less energy than a hole in the valence band (Figure 12.9c,d). Consequently, we can have:

• Singly occupied solitons with charge zero but spin (or − for an antisoliton).
• Doubly occupied solitons with charge −e and spin 0.
• Nonoccupied solitons with charge +e and spin 0.

As discussed in Section 12.3.1, a charged defect can form a polaron, and also a charged soliton can do so. When a neutral soliton and a charged soliton interact, their degenerate levels can form two separated levels, like a bonding and antibonding orbital. Dependent on whether the charged soliton is negatively or positively charged, the new configuration is described as a negative or positive polaron, respectively (Figure 12.9e,f).

In a theoretical description using periodic boundary conditions, the presence of solitons depends on whether the chain contains an odd or even number of atoms. As the chain end is connected to the chain beginning, both sides must have a different type of bond. This condition is fulfilled for an even number of atoms but not for an odd number. Hence, for an odd number of atoms, a periodic chain should contain at least one soliton. For an even number of carbon atoms, the chain contains obviously either no solitons or a soliton and antisoliton.

In summary, in this model doping of t‐PAc occurs via solitons, either initially present or created as a soliton/antisoliton pair. With increasing dopant levels, polarons are formed, and the soliton states broaden into a band that eventually overlaps with the valence and conduction bands, leading to a metallic state. To illustrate the effect of doping, Figure 12.10 shows the tremendous change in conductivity for t‐PAc, oxidized via exposing to halogens, like Cl₂, Br₂, or I₂, or a gas like AsF₅ [23]. This will lead to a polymer salt, such as (CH⁺(I₃⁻)_0.1)_x for 10 mol% iodine doping.

The model for t‐PAc used so far takes the deformation of the lattice into account but neglects electron–electron interaction, at least explicitly. The other extreme is to neglect the electron–lattice interaction (that is, to start with an equal bond model) but take the electron–electron interaction explicitly into account. This destroys the charge conjugation invariance and leads also to a band gap. We do not discuss that model here any further and refer to the literature [24]. In fact, for a detailed description both effects have to be taken into account, as already discussed by Salem in 1966 [25]. Another aspect that is important is the conjugation length. So far, we considered the chains as being straight. In reality this is not the case. Two models are in use to describe this. The first is the Kuhn model, in which a long chain is divided in straight parts of varying length. In this case, conjugation is maintained in every straight part. However, the local kinks in the chain have a high energy. Therefore a second model, the wormlike chain model, is considered in which a gradual deformation along the chain occurs. Here the concept of conjugation length is less clear, although eventually the deformation will destroy the conjugation.

Apart from the molecular characteristics, also the morphology plays a rather important role. Charge transfer from one molecule to another is obviously facilitated by larger chain–chain overlap, so that a material with aligned chains has a larger mobility and therefore larger conductivity. While doping can increase the electron conductivity by 5–10 orders of magnitude, aligning can lead to another 2 decades increase. These effects are illustrated for a particular example in Section 12.4, although generally the relationship is by far not so straightforward as suggested by the overlap argument.

For most other conjugated polymers, for example, for those shown in Figure 12.4, the situation is different from that for t‐PAc, because they usually do not have a degenerate ground state. Hence, in these materials solitons are not formed because such an extended defect is energetically unfavorable. It is possible, though, to form polarons, as in that case the deformation is localized and has a relatively small distortion energy. In these materials, doping usually occurs via acceptor doping with positive polarons as a result. With increasing dopant level, the polaron states start to interact and form bands that eventually overlap with the valence and conduction bands, again leading to a metallic state.

12.3.4 Hopping

The charge transport mechanism within a chain being discussed, we now turn our attention to the charge transport between chains, as this codetermines the overall conductivity. This is usually described by hopping, a mechanism that is also important for conductive particulate composites. Hopping may also occur if a band description no longer applies and mainly localized states exist as in a single conjugated molecule and in amorphous materials.

In all these cases, charge transfer can occur by moving an electron from one localized state to another. Hopping refers to transfer over a barrier, while tunneling refers to transfer through the barrier. In principle, both mechanisms can occur (also simultaneously). For the former, the electron must acquire sufficient thermal energy to overcome the barrier, while for the latter the separation between localized states must be small enough for the wave function to extend through the barrier (Figure 12.11a). In most cases, one does not discriminate between these two mechanisms. With increasing amount of disorder, the levels associated with localized states spread out into the energy region that would be the band gap for a regular lattice. Since the hopping mobility decreases rapidly with hopping distance, the mobility in this region is small and a mobility gap (Figure 12.11b) exists, even though the density of states is not zero.

The simplest approximation to such a situation is to use still the tight‐binding approximation but with a potential having random depths [26]. For a potential depth distribution width ΔV comparable with the band width 2β, the mean free path is approximately equal to the lattice spacing. In this case the wave function |Ψ〉 contains a random phase factor exp(iφ_n), as the electron is unable to imprint its phase on the next site. Moreover, it contains a localization factor exp[−(r − r₀)/ζ], in which the localization length ζ is infinite at the onset of localization but decreases as ΔV increases, so that eventually electrons are localized at position r₀. Altogether we have for the wave function

12.24

For finite length chains (oligomers), the π‐electrons, though mobile within the molecule, are obviously localized on the molecule. In this case one can use the quasi‐band structure, that is, a distribution of molecular orbital levels discretely along the dispersion curve of the infinite polymer [27]. Moreover, these molecules are usually far from perfect, which also leads to localized states. Consequently, charge transfer is dominated by hopping of charge carriers from one localized state to another.

So, there are two complementary approaches to describe the conductivity in amorphous semiconductors. In the band picture the charge carrier concentration is low but strongly temperature dependent because of the thermal activation of the electron across the gap, while the mobility is high and weakly temperature dependent. In the polaron picture the converse is true. The charge carrier concentration can be high and temperature independent since every site can form a polaron, but the mobility is low and thermally activated. In the adiabatic regime the vibrational motion of the atoms is slow as compared with that of the electrons, and electrons adjust instantaneously to the atomic positions.

In a simple model [28] for hopping, one considers two localized states, a filled one at or slightly above the Fermi level and the other one empty and above the Fermi level. Their spatial and energy separation are R and ΔE, respectively. The hopping rate χ is determined by three factors. As the two states have a different k‐value, a phonon is required, and the first factor is the probability for the phonon involved to have the energy ΔE, given by the Boltzmann factor exp(−ΔE/kT). The second factor is the probability of electron transfer. If a state is exponentially localized, that is, its wave has a tail exp(−r/ζ), the tunneling probability is given by exp(−2R/ζ). The third factor is the attempt frequency ν₀, dependent on the electron–phonon coupling constant γ and the phonon density of states. Hence, χ = ν₀ exp(−2R/ζ) exp(−ΔE/kT).

For the mobility we use the Einstein relation μ = eD/kT relating the mobility μ to the diffusion coefficient D, where the simple expression D = χR²/6 for random diffusion is used. For the charge carrier concentration, we use n = kTρ(E_F), where ρ(E_F) is the electronic density of states at the Fermi level. Hence, the conductivity σ becomes

12.25

One would expect that only the nearest neighbors participate in hopping. However, with increasing range and considering more remote neighbors, the probability for finding a low barrier increases, although the overlap between wave functions decreases. Hence, Mott argued that the exponent h = −2R/ζ−ΔE/kT should be optimal or that dh/dR = 0. As at least one state should be available at given R and ΔE, we also have 1 = (4π/3)R³ρ (E_F)ΔE. After some calculation this leads to

12.26

so that the conductivity becomes

12.27

The numerical constants should not be taken too literally, and typically the value for σ₀ is too high. Experimentally, the dependence is often observed to be T^−m with  < m <  rather than T^−1/4. The model is often addressed as variable‐range hopping, as its range of hopping depends on temperature.

12.5.1 A Glimpse of Percolation Theory

If we mix electrically conducting and insulating particles at random, using an increasing fraction of conducting particles, the mixture will be at first insulating; however, at a certain fraction a transition to a conductive mixture occurs. This change occurs over a small volume fraction change of conductive particles, and thereafter the conductivity still increases, albeit slowly. This phenomenon is called percolation, and the transition occurs at the percolation threshold. The phenomenon can be considered as a geometrical phase transition for which, at the percolation threshold, infinitely connected clusters of conductive particles appear. Percolation can take place on random networks but is mostly studied for regular networks. We first pay some attention to regular networks and thereafter make a few remarks about random networks.

Some of the regular lattices in use are shown in Figure 12.14. Essentially two types of percolation processes exist, namely, site and bond percolation. Where required, the type of percolation will be indicated by the superscript (b) or (s). Site percolation considers sites, and these sites are either occupied or empty. Two nearest‐neighbor sites are connected if they are both occupied. A (site) cluster is a group of such connected sites, and, given a site has probability p of being occupied, the percolation probability function P(p) is the probability that a site belongs to an infinite cluster. Bond percolation considers bonds between sites, and these bonds are either open or closed. Open in this respect means that transport can take place, that is, a bond connects sites. If at least one path consisting of only open bonds exists between two sites, these sites are said to be connected, and a set of bonds bounded by closed bonds is called a (bond) cluster. If a bond has probability p of being open, P(p) is the probability that the bond belongs to the infinite cluster linked by such open bonds. In both site and bond percolation, when the probability p approaches the percolation threshold p_cri, a transition from a macroscopically disconnected state to a connected state takes place. The function P(p) = 0 for p < p_cri and thereafter rises steeply until eventually P = p = 1.

The value for p_cri depends on whether site or bond percolation occurs and on the dimensionality of the lattice (Table 12.1). The values for p_cri are usually calculated via renormalization techniques and simulations, and only for the Bethe lattice and some 2D lattices, p_cri can be calculated analytically. For the Bethe lattice, for example, p_cri^(b) = p_cri^(s) = (z − 1)⁻¹, where z is the coordination number of the lattice. One can prove that p_cri^(b) ≤ p_cri^(s) for a given lattice of particular dimensionality d. Further, p_cri decreases with increasing coordination number z and, for given z, decreases with increasing d. Further, we will need the correlation length ξ, which represents for p < p_cri the typical radius of a connected cluster and the length scale over which the network is macroscopically homogeneous.

Close to the percolation threshold, physical quantities depend on p − p_cri, and for p − p_cri < 1, the probability P, conductivity σ, and coherence length ξ are given by

12.28

where P = σ ≠ 0 for p > p_cri and ξ ≠ 0 for p < p_cri. The exponents β, μ, and ν are independent of the type of lattice and depend only on the dimensionality (Table 12.1). For bond percolation zp_cri^(b) ≅ d/(d−1) is accurately obeyed empirically, the significance of which is not understood. Note that for sample size L ≫ ξ, the system is macroscopically homogeneous, but for L ≪ ξ the system is inhomogeneous and the properties depend on the value of L. For example, the mass M of clusters in this regime scales as M ∼ L^D, where the fractal dimension D = d − β/ν. For L ≫ ξ, D = d.

To illustrate some of the above for the conductivity, we consider when and how electrical conduction across a macroscopic region occurs, in terms of the microdetails of the packing of the particles. The approach is to renormalize, that is, to coarse‐grain or group particles together to a new, renormalized particle having the properties of the majority of particles participating in the formation of the renormalized particle. In principle, this process is iterated to obtain a solution of the relevant equations. As a concrete example, we use a 2D structure having a triangular lattice with renormalization by three particles (disks). For the original packing the microdescription is in terms of sites that are either occupied or unoccupied with a conductive particle. We denote by p the probability that any site is occupied, independent of whether any other site is occupied or not. The occupancy distribution of sites provides the exact microdescription of the system for which we only know the single site probability p.

When we increase the length scale by a factor of b, for example 3^1/2, we increase the area by b² = 3, we obtain a coarse‐grained or renormalized description. At this higher level of description, here thus a cell of 3 sites, we use the majority rule and consider a cell as occupied if and only if 2 or more of the sites are occupied (Figure 12.15). Let us call p₁ the probability that a cell of three sites is occupied. The probability p₁ is then the sum of probabilities of four mutually exclusive states. Using p for occupied sites and (1−p) for unoccupied sites, we have once p³ and three times p²(1−p).

The percolation threshold (critical point) occurs if the original probability p equals the renormalized probability p₁. We thus have

12.29

In this particular case, the solution of this equation does not require iteration but yields directly the solutions p^* = 0 and p^* = 1, which represent the so‐called stable fixed points, and the solution p^* = 0.5 representing the unstable fixed point. The unstable fixed point solution thus appears to be the exact solution for this 2D percolation problem, which, however, is fortuitous.

Let us now calculate the critical exponent ν, which describes the onset of the percolation phenomenon. We use the correlation length ξ for which the expression for the original packing is given by ξ = C|p − p_cri|^−ν, so that for the renormalized packing we have ξ₁ = C|p₁ − p_cri|^−ν with ξ₁ = ξ/b. Combining results in

12.30

12.31

Expanding p₁ as p₁ = p^* + λ^*(p − p^*) + ⋯ with λ^* = (dp₁/dp)_p=p* and substitution in the original equation p₁ = p³ + 3p²(1 − p) yields

12.32

so that λ^* = 3/2. Because b² = 3, we obtain

12.33

The exact value is ν = 4/3. Approaching the threshold, the coherence length ξ of the clusters increases until ξ diverges at the threshold itself. This implies that conductive clusters span the complete system, and therefore the system becomes macroscopically conductive.

For finite 3D systems, some corrections are required. For example, Fisher [51] showed that for any property

12.34

where u = L^1/ν(p − p_cri) ∼ (L/ξ)^1/ν, f(u) is an analytical function of u, L is the sample size, and ν is the correlation length exponent. If the relation X ∼ (p − p_cri)^δ near p_cri and for L → ∞ applies, then one must have x = δ/ν. There is also a shift in p_cri given by [52]

12.35

where p_cri(∞) and p_cri(L) denote the critical value for an infinite sample and of size L, respectively. In practice, for transport properties, one applies, for example [53],

12.36

so that from a range of X values, x can be estimated by fitting the results to this equation.

For coatings the thickness is the limiting factor, and for a particulate coating with particle size R_p, the effect of sample thickness is described by

12.37

where p_cri(l) is the effective percolation threshold for a sample with thickness l, p_cri(∞) the threshold for l → ∞, and c a constant. From experiments with layers of particles [54], it appeared that ν ≅ 0.85, in good agreement with the theoretical value ν ≅ 0.88. The finite thickness also affects the conductivity. Combining the above with σ ∼ (p − p_cri)^μ, one easily obtains

12.38

where σ₁ (σ₂) is the conductivity for a sample with thickness l₁ (l₂) and μ₃ (μ₂) the critical exponent for 3D (2D) percolation. This result is in good agreement with the data indicated before [54].

In reality, networks are hardly ever regular. Hence, it is important to consider the percolation on random networks. Distributing inclusions at random in an otherwise uniform system, it appears that for regular and random networks, the critical volume fraction φ_cri = p_cri^(s)f, where f is the packing fraction of the network, is invariant [55]. For 2D (i.e. for disks) φ_cri ≅ 0.45, whereas for 3D (i.e. for spheres), φ_cri ≅ 0.16. To illustrate this, consider a random close packing of spheres for which f ≅ 0.64, so that for a BCC lattice we calculate p_cri ≅ 0.25, close to the theoretical value 0.246.

Another practical aspect is the effect of a particle size distribution. A criterion for relatively weak perturbation of the particle size [56] states that, if α(d) ≡ 2 − dν_d > 0, the critical behavior can change. Here, d is the dimensionality of the system and ν_d the exponent for the correlation length ξ in d dimensions (Table 12.1). For 3D, ν_d ≅ 0.88 and thus α(3) ≅ −0.64, while for 2D, ν_d = 4/3 and thus α(2) = −2/3. Hence, in these cases the particle size distribution should not change the critical behavior [57]. However, several experiments do show an effect of the particle size distribution; see, for example, [58], which also provides a general review.

The application of percolation theory is wide and finds, for example, also application in the description of the structural, mechanical, and rheological properties of polymers and gels, fracture and fault patterns in heterogeneous rocks, the spread of diseases, transport of liquids through particulate and porous media, and the diffusion and growth of particles. A widely used introduction is the booklet by Stauffer and Aharoni [59], while Sahimi [57] provides an extensive overview over various applications. For a review of conductivity in relation to percolation, see [60].

12.5.2 Other Approaches

There are, however, other models to describe the conductivity of particulate composite materials. One of them is effective medium theory (EMT) [61] in which the property of a composite is determined by a combination of the properties of the components. If the composite is treated as a mixture of (conductive) spherical particles of varying sizes with volume fraction φ in a (poorly conductive) matrix (Figure 12.16a), the conductivity is described by the symmetric Bruggeman equation [62–64]

12.39

where σ_f, σ_m, and σ are the conductivities of the filler, matrix, and composite and A is a factor, A = 2 for spherical particles, determining the local field concentration at the conductive particles. The critical volume fraction φ_cri is related to A via A = (1 − φ_cri)/φ_cri and for spherical particles leads to φ_cri = 1/3. The main approximation is that all the domains are located in an equivalent mean field, but this is not the case close to the percolation threshold where the system is governed by the largest cluster of fillers, which is a fractal. Hence, the threshold value is far from the 16% expected from percolation theory and observed in experiments. However, in two dimensions, the model gives a threshold of 50% and has been proven to model percolation relatively well [65, 66]. In a generalized form of EMT [67], the conductivity is obtained from the asymmetric Bruggeman equation

12.40

where t and s are exponents. Here both phases are effectively a mixture of the two pure components (Figure 12.16a). Using t = s = 1, we regain the symmetric Bruggeman equations. With t, s, and A as parameters, good fits to experimental data can be obtained. An extension of the model to AC measurements is available [67]. While percolation theory is particularly useful in the concentration range where the percolation transition takes place, EMTs are more useful for high concentrations.

Frequently used is also the Maxwell Garnett model where the conductivity is governed by the equations [61, 64]

12.41

which can be solved explicitly to yield [68]

12.42

The microstructure can be visualized as built up out of a space‐filling array of coated spheres (Figure 12.16a). The Maxwell Garnett equations are equivalent to the upper and lower bounds as derived by Hashin and Shtrikman [69] for the conductivity of an isotropic two‐component mixture. Since it is assumed that the domains are spatially separated, the model is expected to be valid at low volume fractions.

As an example, we mention an EMT, taking into account the resistivity of the particles and the contact resistivity, that was used to describe the sharp increase in electrical conductivity σ at percolation for antimony‐doped tin oxide (ATO)–acrylate nanocomposite hybrid coatings [70]. The relation between σ and the volume filler fraction φ was analyzed for ATO–acrylate coatings containing ATO nanoparticles grafted with different amounts of 3‐methacryloxypropyltrimethoxysilane coupling agent. Percolation thresholds were observed at low filler fractions (1–2 vol%) for the coatings containing ATO nanoparticles with a low amount of surface grafting. A modified effective medium approximation (EMA) model was introduced for which we refer to the original papers for details. This model takes into consideration different distances between adjacent semiconductive particles in the particle network. The model elucidates how self‐arrangement of the particles influences the location of the percolation threshold in the log σ–φ plot. This modified EMA model could successfully explain the multiple transition behavior and the variable percolation thresholds found for these ATO–acrylate nanocomposite hybrid coatings (Figure 12.17).

12.5.3 The Influence of Aspect Ratio

So far we have discussed the effect of isometric particles, that is, particle with a similar dimension in all directions. However, ratio of the largest dimension over the smallest dimension – the aspect ratio − plays an enormous role in both the percolation threshold φ_cri and the saturation value of the conductivity σ_sat. Experimental studies have shown that increasing the aspect ratio typically decreases φ_cri as well as σ_sat. This behavior can also be modeled, albeit mainly with numerical simulations. As example, Figure 12.18 shows the change in φ_cri with decreasing aspect ratio [71]. Three simulation models were developed for predicting the electrical conductivity σ and the electrical percolation threshold of polymer composites. The three models are based on finite element modeling (FEM), percolation threshold modeling (PTM), and electrical networks modeling (ENM). A Monte Carlo algorithm was used to construct the geometries, with either soft‐core (overlapping) or hard‐core/soft‐shell (nonoverlapping) fibers. Conductivity measurements on carbon–fiber/PMMA composites with well‐defined fiber aspect ratios were used for experimental validation. The average fiber orientations were calculated from scanning electron micrographs. The soft‐core PTM model with experimental fiber orientations and without adjustable parameters gave accurate (R² = 0.984) predictions of the electrical percolation threshold as a function of aspect ratio. The corresponding soft‐core ENM model, with close‐contact conductivity calculated with FEM, resulted in good conductivity predictions for the longest fibers, still without the use of any adjustable parameters. The hard‐core/soft‐shell versions of the models, using the shell thickness as an adjustable parameter, gave similar but slightly poorer results.

12.5.4 Conductive Particles

As conductive materials for particles, metals and (conductive) oxides are frequently employed. Moreover, carbon materials are used.

While most metals have a decent conductivity, the most conductive ones are the metals Au, Ag, Cu, and Al. The first two are obviously usually too expensive, while the last two suffer from oxidation. In particular, for Al the oxide scale formed is nonconductive, leading easily to high contact resistance, unless plastic deformation occurs during composite formation.

Conductive oxides typically used are tin oxide, indium‐doped tin oxide (ITO), and ATO. Doping is done in order to increase the conductivity. Also in this case the surface of a particle may differ from its bulk [72, 73].

In the category of carbon materials, one distinguishes CB and the more modern variants, buckyballs, CNTs, and graphene. These allotropes can be considered as being formed from sheets of a single layer of atoms, as shown Figure 12.19 [74]. CB belongs to this class of carbonaceous materials, exists in many varieties [75, 76], is a well‐studied and by far the most used nanofiller, and is widely applied for electronic and reinforcement purposes [77]. Recent discoveries in the field of graphitic nanoparticles, in particular CNTs and graphene [78], have allowed the development of materials with exceptional electrical and mechanical properties. Graphite, the most abundant and stable form of carbon, exhibits properties that can be substantially enhanced by exfoliation of its layered structure into single or multilayer sheets [3, 4]. The abovementioned nanoscale powders are commonly incorporated into polymeric matrices to provide enhanced electrical, mechanical, and thermal properties [79–81].

Graphene can be considered as a single sheet of graphite, although in practice typically few‐layer graphene is used. The conductivity is in principle as high as for high purity graphite but depends strongly on the amount defects present in the sheet. These defects are introduced during exfoliation, that is, the separation of graphite in single and/or few‐layer sheets, often employing partial oxidation. Removal of defects on exfoliated sheets by reduction appears to be rather difficult, if not impossible. The trick, therefore, is to introduce as limited as possible or no (oxidation and other) defects during exfoliation. Nevertheless, graphene is often functionalized before further use; for example, see [74].

Folding a single sheet to a sphere, one obtains the so‐called buckyballs, also called fullerenes. In C₆₀, hexagons and pentagons of carbon link together in a coordinated fashion to form a hollow, geodesic dome with bonding strains equidistributed among 60 carbon atoms. Other variants include C₇₀, C₈₄, and fullerenes as small as C₂₈ and as large as a postulated C₂₄₀. Buckyballs were one of the first carbon nanoparticles discovered in 1985 by Smalley, Kroto, and Curl.

CNTs can be considered to be sheets of graphite folded back to become a cylinder. There are various ways to connect two edges of a sheet, and this leads to various types of tubes. In many fabrication processes the resulting CNTs are multiwalled, that is, consist of several concentric cylinders. Single‐walled CNTs are available but are considerably more difficult to fabricate and therefore much more expensive. Various pretreatments are reviewed in [82], properties and applications in [83], and conductivity mechanisms in [84].

A bit of a strange duck in the pond of conducting particles are the phthalocyanines. Although one could expect that materials based on these compounds are highly poisonous and, in fact, the synthesis of these compounds is not without safety pitfalls, the final materials appear to be rather friendly. Figure 12.20 shows as an example the molecular and crystal structure of cyanophthalocyaninato Co(III) (Phthalcon), which contains both NH₃ and H₂O as ligands. Its behavior in epoxy matrices can be well controlled [85], and the material is used in antistatic coatings [86].

Obviously, dispersing a sufficient number of conductive particles in a polymer film leads to overall conductivity. Dispersing is done by simple mixing (for rather compatible materials), by high shear mixing (for agglomerated fillers), or on laboratory scale, by sonication. However, dispersing particles will also deteriorate many other properties and, in particular, leads to increased brittleness. The trick is therefore to form strings of conductive particles, so that at low concentration still conductivity emerges. To that purpose, often surfactants are used, in particular for highly incompatible components, such as (somewhat) polar polymers and hydrophobic compounds, like the carbon allotropes [87]. A proper colloidal stability is obviously required to obtain the results desired.

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Further Reading

1. Barford, W. (2005). Electronic and Optical Properties of Conjugated Polymers. Oxford: Oxford University Press.
2. Blythe, T. and Bloor, D. (2005). Electrical Properties of Polymers, 2e. Cambridge: Cambridge University Press.
3. Chien, J.C.W. (1984). Polyacetylene: Chemistry, Physics, and Material Science. Orlando: Academic Press.
4. Farges, J.‐P. (1994). Organic Conductors. New York: Marcel Dekker.
5. Mott, N.F. and Davis, E.A. (1971). Electronic Processes in Non‐crystalline Materials, 2e. Oxford: Clarendon Press.
6. Mott, N.F. (1987). Conduction in Non‐crystalline Solids. Oxford: Oxford University Press.
7. Skotheim, T.A. and Reynolds, J.R. (2007). Handbook of Conducting Polymers, 3e. New York: Marcel Dekker.
8. Wan, M. (2008). Conducting Polymers with Micro or Nanometer Structure. Beijing: Springer Science and Business Media.
9. Zallen, R. (1983). The Physics of Amorphous Solids. New York: Wiley.