Electrical And Optical Properties

Chapter 9

Electrical and optical properties 9.1 Introduction

As indicated in chapter 1, the first electrical property of polymers to be valued was their high electrical resistance, which made them useful as insulators for electrical cables and as the dielectric media for capacitors. They are, of course, still used extensively for these purposes. It was realised later, however, that, if electrical conduction could be added to the other useful properties of polymers, such as their low densities, flexibility and often high resistance to chemical attack, very useful materials would be produced. Nevertheless, with few exceptions, conducting polymers have not in fact displaced conventional conducting materials, but novel applications have been found for them, including plastic batteries, electroluminescent devices and various kinds of sensors. There is now much emphasis on semiconducting polymers. Figure 9.1 shows the range of conductivities that can be achieved with polymers and compares them with other materials. Conduction and dielectric properties are not the only electrical properties that polymers can exhibit. Some polymers, in common with certain other types of materials, can exhibit ferroelectric properties, i.e. they can acquire a permanent electric dipole, or photoconductive properties, i.e. exposure to light can cause them to become conductors. Ferroelectric materials also have piezoelectric properties, i.e. there is an interaction between their states of stress or strain and the electric field across them. All of these properties have potential applications but they are not considered further in this book. The first part of this chapter describes the electrical properties and,

when possible, explains their origins. Unfortunately, some of the properties associated with conduction are not well understood yet; even when they are, the theory is often rather difficult, so that only an introduction to the conducting properties can be given. The second part of the chapter deals with the optical properties. The link between these and the electrical properties is that the optical properties depend primarily on the interaction of the electric field of the light wave with the polymer molecules. The following 248

section deals with the electrical polarisation of polymers, which underlies both their dielectric and their optical properties.

9.2 Electrical polarisation 9.2.1 The dielectric constant and the refractive index In an ideal insulating material there are no free electric charges that can move continuously in the presence of an electric field, so no current can flow. In the absence of any electric field, the bound positive and negative electric charges within any small volume element d_ will, in the simplest case, be distributed in such a way that the volume element does not have any electric dipole moment. If, however, an electric field is applied to this volume element of the material, it will cause a change in the distribution of charges, so that the volume element acquires an electric dipole moment 9.2 Electrical polarisation 249 Fig. 9.1 Electrical conductivities of polymers compared with those of other materials. All values are approximate.

proportional to the field. It will be assumed for simplicity that the material

is isotropic; if this is so, the electric dipole will be parallel to the electric field. The dipole moment d_ of the volume element will then be given by d_ ¼ Pd_ ¼ pEL d_ ð9:1Þ where p is the polarisability of the material and P is the polarisation produced by the field EL acting on d_. On the molecular level a molecular polarisability _ can be defined; and, if No is the number of molecules per unit volume, then P ¼ No_EL ð9:2Þ The mechanism for the molecular polarisability is considered in section 9.2.2. So far no consideration has been given to the origin of the electric field experienced by the small volume element under consideration, but it has been denoted by EL to draw attention to the fact that it is the local field at the volume element that is important. An expression for this field is derived below. Equation (9.1) shows that P is the dipole moment per unit volume, a vector quantity that has its direction parallel to EL for an isotropic medium. Imagine a small cylindrical volume of length dl parallel to the polarisation and of cross-sectional area dA. Let the apparent surface charges at the two ends of the cylinder be _dq. It then follows that P ¼ dq dl=d_, where d_ is the volume of the small cylinder. However, d_ ¼ dl dA, so that P ¼ dq=dA ¼ _, where _ is the apparent surface charge per unit area normal to the polarisation. (Note: _ stands for conductivity in section 9.3.) The simplest way to apply an electric field to a sample of polymer is to place it between two parallel conducting plates and to apply a potential difference V between the plates, which gives rise to an applied electric field normal to the plates of magnitude E ¼ V=d, where d is the separation of the plates. This field may be imagined to be produced by layers of charge at the conducting plates. The net charges in these layers are the differences between the free charges on the plates and the

apparent surface charges on the polymer adjacent to the plates due to its polarisation. Imagine now that a small spherical hole is cut in the polymer and that this can be done without changing the polarisation of the remaining polymer surrounding it. New apparent surface charges will be produced at the surface of the spherical hole. The normal to any small surface element of the sphere makes an angle _ with the polarisation vector, so that the apparent charge per unit area on this element is _ cos _ ¼ P cos _. The electric field produced at the centre of the sphere by all these surface elements can be shown by 250 Electrical and optical properties

integration (see problem 9.1) to be P=ð3"oÞ, where "o is the permittivity of free space, so that the total field at the centre of the sphere is equal to E þ P=ð3"oÞ. The only way, however, that the polarisation could remain undisturbed outside the imaginary spherical cavity would be for the cavity to be refilled with polymer! The total electric field at the centre of the sphere would then be increased by the field produced there by all the dipoles in this sphere of polymer. Fortunately it can be shown that the latter field is zero if the individual molecular dipoles have random positions within the sphere. In this case the field at the centre of the sphere remains E þ P=ð3"oÞ, which is therefore the field that would be experienced by a molecule at the centre, i.e. it is the local field EL. Thus EL ¼ E þ P=ð3"oÞ ð9:3Þ The dielectric constant, or relative permittivity, " of the polymer is defined by " ¼ Vo=V, where Vo is the potential difference that would exist between the plates if they carried a fixed free charge Qo per unit area in the absence of the dielectric and V is the actual potential difference between them when they carry the same free charge in the presence of the dielectric. With the polymer present, the total effective charge per unit area at the plates is Q ¼ Qo _ P when the effective surface charge on the polymer

is taken into account, so that "¼ Vo V¼ Eo E¼ Qo Q¼ QþP Q ð9:4Þ where Eo is the field that would exist between the plates in the absence of the polymer. However, E ¼ Q="o (which can be proved by applying a similar method to that of problem 9.1 to two infinite parallel plates with charges þQ and _Q per unit area) so that "¼ "oE þ P "oE or P ¼ "oEð" _ 1Þ ð9:5Þ and finally, from equation (9.3), EL ¼ "þ2 3 E ð9:6Þ A derivation of the local field in essentially this way was originally given by H. A. Lorentz. Since, according to equation (9.2), P ¼ No_EL, equations (9.5) and (9.6) lead to No_ "þ2 3 E ¼ "oEð" _ 1Þ or "_1 "þ2¼ No_ 3"o ð9:7Þ 9.2 Electrical polarisation 251

This relationship between the dielectric constant and the molecular polarisability is known as the Clausius–Mosotti relation. It can usefully be written in terms of the molar mass M and density _ of the polymer in the form

"_1 "þ2 M _¼ NA_ 3"o ð9:8Þ where NA is the Avogadro constant and the quantity on the RHS is called the molar polarisation and has the dimensions of volume. It follows from Maxwell’s theory of electromagnetic radiation that " ¼ n2, where " is the dielectric constant measured at the frequency for which the refractive index is n. Equation (9.8) thus leads immediately to the Lorentz–Lorenz equation n2 _ 1 n2 þ 2 M _¼ NA_ 3"o ð9:9Þ relating the refractive index and the molecular polarisability. Some applications of this equation are considered in sections 9.4.3 and 10.4.1. The expression on the LHS is usually called the molar refraction of the material, which is thus equal to the molar polarisation at optical frequencies.

9.2.2 Molecular polarisability and the low-frequency dielectric constant

There are two important mechanisms that give rise to the molecular polarisability. The first is that the application of an electric field to a molecule can cause the electric charge distribution within it to change and so induce an electric dipole, leading to a contribution called the distortional polarisability. The second is that the molecules of some materials have permanent electric dipoles even in the absence of any electric field. If an electric field is applied to such a material the molecules tend to rotate so that the dipoles become aligned with the field direction, which gives rise to an orientational polarisation. Thermal agitation will, however, prevent the

molecules from aligning fully with the field, so the resulting polarisation will depend both on the field strength and on the temperature of the material. This contribution is considered first. It is relatively easy to calculate the observed polarisation of an assembly of non-interacting molecules for a given field EL at each molecule at a temperature T. Assume that each molecule has a permanent dipole l and that there is no change in its magnitude on application of the field. Consider a molecule for which the dipole makes the angle _ with EL. The energy u of the dipole is then given by 252 Electrical and optical properties

u ¼ _l _EL ¼ __EL cos _ ð9:10Þ If there are dn dipoles within any small solid angle d! at angle _ to the applied field, Boltzmann’s law shows that dn ¼ Ae_u=ðkTÞ d! ð9:11Þ where A depends on the total number of molecules. The total solid angle d! lying between _ and _ þ d_ when all angles around the field direction are taken into account is 2_ sin _ d_ and the corresponding number of dipoles is thus dn ¼ 2_Ae_u=ðkTÞ sin _ d_ ð9:12Þ Each of these dipoles has a component _cos _ in the direction of the field and their components perpendicular to the field cancel out. They thus contribute a total of _ dn cos _ to the total dipole moment parallel to the field. Using the fact that dðcos _Þ ¼ _sin _ d_ and writing cos _ ¼ _ and _EL=ðkTÞ ¼ x, the average dipole moment h_i contributed by all the molecules at all angles to the field is given by h_i _¼ Ð_ 0 2_Ae_EL cos _=ðkTÞ Ð cos _ dðcos _Þ _ 0 2_Ae_EL cos _=ðkTÞdðcos _Þ ¼ Ð1 _1 _ex_d_ Ð1 _1 ex_d_ ð9:13Þ Integration by parts then leads to

h_i _ ¼ ð_=xÞex_ _ ð1=x2Þex_ _ _1 _1

ð1=xÞex_ _ _1 _1

¼ ex þ e_x ex _ e_x _ 1 x ¼ coth x _ 1 x ¼ LðxÞ ð9:14Þ where LðxÞ is the Langevin function. For attainable electric fields, _EL=ðkTÞ ¼ x is very small and the exponentials can be expanded as e_x ffi 1 _ x þ x2=2 _ x3=6 þ Oðx4Þ. Substitution then shows that the Langevin function reduces approximately to LðxÞ ¼ x=3, so that finally h_i ffi _2EL=ð3kTÞ ð9:15Þ and the contribution to the molecular polarisability from rotation is to a good approximation _2=ð3kTÞ. The total molecular polarisability is therefore _ ¼ _d þ _2=ð3kTÞ, where _d is the molecular deformational polarisability. Use of the Clausius–Mosotti relation, equation (9.8), then gives "_1 "þ2 M _¼ NA_d 3"o þ NA_2 9"okT ð9:16Þ In developing this equation, no account has been taken of any possible interactions between the molecular dipoles. It is therefore expected to be 9.2 Electrical polarisation 253

most useful for polar gases and for solutions of polar molecules in nonpolar solvents, where the polar molecules are well separated. At optical frequencies _ ¼ _d, because the molecules cannot reorient

sufficiently quickly in the oscillatory electric field of the light wave for the orientational polarisability to contribute, as discussed further in section 9.2.4. Setting _ ¼ _d in equation (9.9) gives _d in terms of the optical refractive index and, if this expression for _d is inserted, equation (9.16) can then be rearranged as follows: 3ð" _ n2Þ ð" þ 2Þðn2 þ 2Þ M _¼ NA_2 9"okT ð9:17Þ Fro¨ hlich showed that a more exact equation for condensed matter, such as polymers, is ð" _ n2Þð2" þ n2Þ "ðn2 þ 2Þ2 M _¼ NAg_2 9"okT ð9:18Þ The LHS of this equation differs from that of equation (9.17) because a more exact expression has been used for the internal field and the RHS now includes a factor g, called the correlation factor, which allows for the fact that the dipoles do not react independently to the local field. If _; g; n and _ are known for a polymer, it should therefore be possible to predict the value of ". Of these, g is the most difficult to calculate (see section 9.2.5).

9.2.3 Bond polarisabilities and group dipole moments The idea that the mass of a molecule can be calculated by adding together the atomic masses of its constituent atoms is a very familiar one. Strictly speaking, this is an approximation, although it is an excellent one. It is an approximation because the equivalent mass of the bonding energy is neglected; the mass of a molecule is in fact lower than the sum of the

masses of its atoms by only about one part in 10 14. In a similar way, but usually to a much lower degree of accuracy, various other properties of molecules can be calculated approximately by neglecting the interactions of the various parts of the molecule and adding together values assigned to those individual parts. The word ‘adding’ must, however, be considered carefully. For instance, a dipole moment is a vector quantity; if the dipole moments of the various parts of a molecule are known, the dipole moment of the whole molecule can be calculated only if the constituent dipole moments are added vectorially. In section 9.2.2 above it is shown that the refractive index of a medium is related to the high-frequency deformational polarisability of its mole254Electrical and optical properties

cules. Polarisability is treated there as a scalar quantity, but it is in fact a second-rank tensor quantity (see the appendix) and the correct way of adding such quantities is somewhat more complicated than the addition of vectors. When the refractive indices of stressed or oriented polymers are considered, as in chapters 10and 11, this complication must be taken into account, as described later in the present chapter (section 9.4.3). For the moment attention is restricted to media in which the individual molecules are oriented randomly. The high-frequency polarisation of a molecule consists almost entirely of the slight rearrangement of the electron clouds forming the bonds between the atoms. It is the polarisabilities of individual bonds that are assumed to be additive, to a good approximation. If the molecules are randomly oriented, then so are all the bonds of a particular type. The sum of the polarisabilities of all the bonds of this type must therefore be isotropic, so the polarisability for an individual bond can be replaced by a scalar average value called the mean value or spherical part of the tensor.

These scalar values can be added together arithmetically for the various bonds within the molecule to obtain the spherical part of the polarisability for the molecule. This is what has so far been called _d: It is possible to find values of the mean polarisabilities _i for a set of different types of bond i by determining the refractive indices for a large number of compounds containing only these types of bond. The value of _d can be found for each compound by means of the Lorentz–Lorenz equation (9.9) with _ ¼ _d and the values of _i are then chosen so that they add for each type of molecule to give the correct value of _d. In order to obtain molar refractions or molar bond refractions, _d or _i must be multiplied by NA=ð3"oÞ. The values of a number of molar bond refractions obtained in this way are given in table 9.1. It should be noted that the simple additivity fails for molecules that contain a large number of double bonds, which permit the electrons to be delocalised over a large region of the molecule (see section 9.3.4). A similar argument to the above can be used to deduce the molecular dipole for a molecule if the dipoles of its constituent parts are known. It has been found that the additivity scheme works best for dipoles if group dipoles are used, i.e. each type of molecular group in a molecule, such as a —COOH group, tends to have approximately the same dipole moment independently of the molecule of which it is a constituent part. Equation (9.16), which applies to dilute solutions of polar molecules in a nonpolar solvent, can be used to find _ for a range of different molecules. This is done by plotting against 1/T the values calculated for the left-hand side of the equation from values of " measured at various temperatures T. The gradient of the straight line plot is NA_2=ð9"okÞ, from which _ is easily 9.2 Electrical polarisation 255

obtained. When this has been done for a range of molecules one can find a

set of group dipoles li that by vector addition predicts approximately correctly the values of _ found for all the compounds. Since the quantities li are vectors, not only their magnitudes but also their directions within the molecule must be specified. This is usually done by giving the angle between li and the bond joining the group i to the rest of the molecule. Table 9.2 gives dipoles for some important groups in polymers.

9.2.4Dielectric relaxation The orientation of molecular dipoles cannot take place instantaneously when an electric field is applied. This is the exact analogy of a fact discussed in chapter 7, namely that the strain in a polymer takes time to develop after the application of a stress. In fact the two phenomena are not simply analogous; the relaxation of strain and the rotation of dipoles are due to the same types of molecular rearrangement. Both viscoelastic 256 Electrical and optical properties

Table 9.1. Molar bond refractions for the D line of Naa Bond Refraction (10_6 m3) C—H 1.676 C—C 1.296 C—— C 4.17 C——— C (terminal) 5.87 C——— C (non-terminal) 6.24 C—C (aromatic) 2.688 C—F 1.44 C—Cl 6.51 C—Br 9.39 C—I 14.61 C—O (ether) 1.54 C—— O 3.32 C—S 4.61 C—N 1.54 C—— N 3.76 C——— N 4.82

O—H (alcohol) 1.66 O—H (acid) 1.80 N—H 1.76 aReproduced from Electrical Properties of Polymers by A. R. Blythe. # Cambridge University Press 1979. 9.2 Electrical polarisation 257

Table 9.2. Group dipole momentsb Group Aliphatic compounds Aromatic compounds Moment (10_30 C m) Anglea (degrees) Moment (10_30 C m) Anglea (degrees) —CH3 0 0 1.3 0 —F 6.3 5.3 —Cl 7.0 5.7 —Br 6.7 5.7 —I 6.3 5.7 —OH 5.7 60 4.7 60 —NH2 4.0 100 5.0 142 —COOH 5.7 745.3 74 —NO2 12.3 0 14.0 0 —CN 13.40 14.7 0 —COOCH3 6.0 70 6.0 70 —OCH3 4.0 55 4.3 55 aThe angle denotes the direction of the dipole moment with respect to the bond joining the group to the rest of the molecule. bReproduced from Electrical Properties of Polymers by A. R. Blythe. # Cambridge University Press 1979. Example 9.1 Calculate the refractive index of PVC using the bond refractions given in table 9.1. Assume that the density of PVC is 1.39 Mg m_3. Solution For a polymer the Lorentz–Lorenz equation (9.9) can be rewritten n2 _ 1 n2 þ 2 ¼ NA 3"o _d M __ _

where the quantity in brackets is the molar refraction per repeat unit divided by the molar mass of the repeat unit. The PVC repeat unit —ðCH2—CHCl—Þ , has relative molecular mass ð2 12Þ þ ð3 1Þ þ 35:5 ¼ 62:5 and molar mass 6:25 10_2 kg. There are three C—H bonds, one C—Cl bond and two C—C bonds per repeat unit, giving a refraction per repeat unit of ½ð3 1:676Þ þ 6:51 þ ð2 1:296Þ_ 10_6 ¼ 1:413 10_5 m3. Thus ðn2 _ 1Þ ¼ 1:413 10_5 1:39 103ðn2 þ 2Þ=ð6:25 10_2Þ ¼ 0:3143ðn2 þ 2Þ, which leads to n ¼ 1:541.

measurements and dielectric measurements can therefore be used to study these types of rearrangement. Dielectric studies have the advantage over viscoelastic studies by virtue of the fact that a much wider range of frequencies can be used, ranging from about 104 s per cycle (10_4 Hz) up to optical frequencies of about 1014 Hz. In section 7.2.1 the idea of a relaxation time is made explicit through the assumption that, in the simplest relaxation, the material relaxes to its equilibrium strain or stress on application of a stress or strain in such a way that the rate of change of strain or stress is proportional to the difference between the fully relaxed value and the value at any instant. A similar assumption is made in the present section for the relaxation of polarisation on application of an electric field. In considering dielectric relaxation it is, however, necessary to remember that there are two different types of contribution to the polarisation of the dielectric, the deformational polarisation Pd and the orientational polarisation Pr, so that P ¼ Pd þ Pr. For the sudden application of a field E that then remains constant, the deformational polarisation can be considered to take place ‘instantaneously’, or more precisely in a time of the order of 10_14 s. When alternating fields are used the deformational polarisation can similarly be considered to follow the applied field exactly, provided that the frequency is below optical frequencies. If the limiting value of the dielectric constant at optical frequencies is called "1, equation (9.5) shows that Pd ¼ "oð"1 _ 1ÞE ð9:19Þ Assuming that, in any applied field with instantaneous value E, Pr

responds in such a way that its rate of change is proportional to its deviation from the value that it would have in a static field of the same value E, it follows that dPr dt ¼ "oð"s _ "1ÞE _ Pr ð9:20Þ where "s is the dielectric constant in a static field and is the relaxation time. Note that, in equation (9.20), it is necessary to subtract from the total polarisation P ¼ "oð"s _ 1ÞE in a static field the polarisation Pd ¼ "oð"1 _ 1ÞE due to deformation in order to obtain the polarisation due to the orientation of dipoles. Consider first the ‘instantaneous’ application at time t ¼ 0 of a field E that then remains constant. Then it follows from equation (9.20) that Pr ¼ "oð"s _ "1Þð1 _ e_t=ÞE ð9:21Þ 258 Electrical and optical properties

Now consider an alternating applied field E ¼ Eoei!t. Because the deformational polarisation follows the field with no time lag, Pd is given, according to equation (9.19), by Pd ¼ "oð"1 _ 1ÞEoei!t ð9:22Þ Pr is now given, from equation (9.20), by dPr dt ¼ "oð"s _ "1ÞEoei!t _ Pr ð9:23Þ Assume that Pr ¼ Pr;oei!t, where Pr;o is complex to allow for a phase lag of Pr with respect to the field. It follows that dPr dt ¼ i!Pr;oei!t ¼ "oð"s _ "1ÞEo _ Pr;o ei!t ð9:24Þ or Pr;o ¼ "oð"s _ "1Þ 1 þ i! Eo ð9:25Þ The total complex polarisation P is thus given, from equations (9.19) and (9.25), by

P ¼ Pd þ Pr ¼ ½"oð"1 _ 1Þ þ "oð"s _ "1Þ=ð1 þ i!Þ_E ð9:26Þ and the complex dielectric constant ", from equation (9.5), by "¼1þ P "oE ¼ 1 þ ð"1 _ 1Þ þ "s _ "1 1 þ i! ¼ "1 þ "s _ "1 1 þ i! ð9:27Þ The expression for " given on the RHS of equation (9.27) is called the Debye dispersion relation. Writing the dielectric constant " ¼ "0 _ i"00, it follows immediately from equation (9.27) that "0 ¼ "1 þ "s _ "1 1 þ !22 and "00 ¼ ð"s _ "1Þ! 1 þ !22 ð9:28a;bÞ These equations are the dielectric equivalents of the equations developed in section 7.3.2 for the real and imaginary parts of the compliance or modulus. Just as the imaginary part of the compliance or modulus is a measure of the energy dissipation or ‘loss’ per cycle (see section 7.3.2), so is "00. The variations of "0 and "00 with ! are shown in fig. 9.2. The terms relaxed and unrelaxed dielectric constant are used for the static and high-frequency values "s and "1, respectively. These quantities 9.2 Electrical polarisation 259

are the electrical analogues of the relaxed and unrelaxed moduli or compliances defined in section 7.3.2 and a dielectric relaxation strength is defined as "s _ "1, in exact analogy with the definition of mechanical relaxation strength.

9.2.5 The dielectric constants and relaxations of polymers

As discussed in sections 9.2.2 and 9.2.3, the high-frequency, or optical, dielectric constant depends only on the deformational polarisability of the molecules and can be calculated from equation (9.8) (with _ ¼ _d) and values of the molar bond refractions, such as those given in table 9.1. Rearrangement of this equation shows, as expected, that the greater

the molar refraction the greater the high-frequency dielectric constant and hence the refractive index. Table 9.1 then shows, for instance, that polymers with a large fraction of C——C double bonds will generally have higher refractive indices than will those with only single C—C bonds and that polymers containing chlorine atoms will have higher refractive indices than will their analogues containing fluorine atoms. Some refractive indices of polymers are given in table 9.3 and these trends can be observed by reference to it. If a polymer consisted entirely of non-polar groups, or contained polar groups arranged in such a way that their dipoles cancelled out, it would have a low dielectric constant at all frequencies, determined only by the 260 Electrical and optical properties Fig. 9.2 The variations of "0 and "00 with ! for the simple Debye model.

deformational polarisation, and it would not undergo dielectric relaxation. Examples of polymers potentially in this class are polyethylene, in which the >CH2 groups have very low dipole moments and are arranged predominantly in opposite directions along the chain, and polytetrafluoroethylene, in which the dipole moment of the >CF2 group is moderately large but the molecule is helical and the dipole moments cancel out because they are approximately normal to the helix axis for each group. Such cancelling out is never exact, however, because of conformational defects. An additional factor is that oxidation, even if it does not significantly perturb the conformation, can introduce groups with different dipole moments from those of the groups replaced. The dipoles then no longer cancel out and both a higher dielectric constant and dielectric relaxation are observed. If a polymer contains polar groups whose moments do not cancel out,

the actual value of the dielectric constant depends very strongly on the molecular conformation, which in turn may depend on the dipoles, because strong repulsion between parallel dipoles will cause the conformation to 9.2 Electrical polarisation 261

Table 9.3. Approximate refractive indices for selected polymers at room temperature Polymer Repeating unit Refractive index Polytetrafluoroethylene —ð CF2—CF2—Þ 1.35–1.38 Poly(vinlidene fluoride) —ðCH2—CF2—Þ 1.42 Poly(butyl acrylate) —ðCH2—CHðCOOðCH2Þ3CH3Þ—Þ 1.46 Polypropylene (atactic) —ðCH2—CHðCH3Þ—Þ 1.47 Polyoxymethylene —ðCH2—O—Þ 1.48 Cellulose acetate See section 1.1 1.48–1.50 Poly(methyl methacrylate) —ðCH2—CðCH3ÞðCOOCH3Þ—Þ 1.49 Poly(1,2-butadiene) —ðCH2— CH(CH—— CH2Þ—Þ 1.50 Polyethylene —ðCH2—CH2—Þ 1.51–1.55 (depends on density) Polyacrylonitrile —ðCH2—CHðCNÞ—Þ 1.52 Poly(vinyl chloride) —ðCH2—CHCl—Þ 1.54–1.55 Epoxy resins Complex cross-linked ether 1.55–1.60 Polychloroprene —ðCH2—CCl(CH——CH2Þ—Þ 1.55–1.56 Polystyrene —ðCH2—CHðC6H5Þ—Þ 1.59 Poly(ethylene terephthalate) —ð (CH2Þ2—(C—— O)—(C6H4)—(C—— O)—Þ 1.58–1.60 Poly(vinylidene chloride) —ðCH2—CCl2—Þ 1.60–163 —ðCH2—CH—Þ Poly(vinyl carbazole) 1.68

change so that the dipoles are not parallel. The effect of the molecular conformation is incorporated into the correlation factor g introduced in equation (9.18). The value of g for an amorphous polymer depends on the angles between the dipoles within those parts of the molecule that can rotate independently when the electric field is applied. These angles and the sections of chain that can rotate independently are temperatureand frequency-dependent, particularly in the region of the glass transition, so it is not generally possible to calculate g. In practice, measurements of the

static dielectric constant and the refractive index are used with equation (9.18) to determine g values and so obtain information about the dipole correlations and how they vary with temperature. Matters are more complicated still for semicrystalline polymers, for which not only are the observed dielectric constants averages over those of the amorphous and crystalline parts but also the movements of the amorphous chains are to some extent restricted by their interactions with the crystallites, which causes changes in g and hence in the static dielectric constant of the amorphous material. These facts are illustrated by the temperature dependences of the dielectric constants of amorphous and 50% crystalline PET, shown in table 9.4. 262 Electrical and optical properties

Table 9.4. Dielectric constants of amorphous and 50% crystalline poly(ethylene terephthalate)a Polymer state T (K) Dielectric constant Unrelaxed, "1 Relaxed, "s Amorphous 193 3.09 3.80 203 3.09 3.80 213 3.12 3.80 223 3.13 3.79 233 3.143.78 3543.80 6.00 Crystalline 193 3.18 3.66 203 3.18 3.67 213 3.19 3.66 373 3.71 4.44 aAdapted by permission from Boyd, R. H. and Liu, F. ‘Dielectric spectroscopy of semicrystalline polymers’ in Dielectric Spectroscopy of Polymeric Materials: Fundamentals and Applications, eds. James P. Runt and John J. Fitzgerald, American Chemical Society, Washington DC, 1997, Chap. 4, pp. 107–136, Table I, p. 117.

Table 9.4 shows that the unrelaxed values for these two samples are very similar and do not change much even at the glass transition (at about 354 K). The relaxed values are different for the two samples, particularly

above the glass-transition temperature, where the relaxed dielectric constant of the amorphous sample is significantly greater than that of the crystalline sample. This temperature dependence arises from the fact that, at low temperature, the C—— O groups tend to be trans to each other across the benzene ring, so that their dipoles almost cancel out, whereas at higher temperatures rotations around the ring—(C—— O) bonds take place more freely in the amorphous regions so that the dipoles can orient more independently and no longer cancel out. The value of the relaxed constant for the crystalline sample is lower than that of the non-crystalline sample even at the lower temperatures because the rigid crystalline phase not only contributes little to the relaxed constant but also restricts the conformations allowed in the amorphous regions. The width of the dielectric loss peak given by equation (9.28b) can be shown to be 1.14 decades (see problem 9.3). Experimentally, loss peaks are often much wider than this. A simple test of how well the Debye model fits in a particular case is to make a so-called Cole–Cole plot, in which "00 is plotted against "0. It is easy to show from equations (9.28) that the Debye model predicts that the points should lie on a semi-circle with centre at [("s þ "1Þ=2; 0_ and radius ð"s _ "1Þ=2. Figure 9.3 shows an example of such a plot. The experimental points lie within the semi-circle, corresponding to a lower maximum loss than predicted by the Debye model and also to a wider loss peak. A simple explanation for this would be that, in an amorphous polymer, the various dipoles are constrained in a wide range of different ways, each leading to a different relaxation time , so that the observed values of "0 and "00 would be the averages of the values for each value of (see problem 9.4). 9.2 Electrical polarisation 263 Fig. 9.3 A Cole–Cole plot

for samples of poly(vinyl acetate) of various molar masses: *, 11 000 g mol_1; O, 140 000 g mol_1; &, 500 000 g mol_1; and 4, 1 500 000 g mol_1. The dielectric constant has been normalised so that "s _ "1 ¼ 1. (Adapted by permission of Carl Hanser Verlag.)

Various empirical formulae have been given for fitting data that deviate from the simple semi-circular Cole–Cole plot. The most general of these is the Havriliak and Negami formula " _ "1 "s _ "1 ¼ ½1 þ ði!Þa__b ð9:29Þ This equation has the merit that, on changing the values of a and b, it becomes the same as a number of the other formulae. In particular, when a ¼ b ¼ 1 it reduces, with a little rearrangement, to equation (9.27). In section 5.7 various motions that can take place in polymers and can lead to NMR, mechanical or dielectric relaxation are considered mainly through their effects on the NMR spectrum. Mechanical relaxation is considered in detail in chapter 7. Although all the motions that are effective in mechanical relaxation can potentially also produce dielectric relaxation, the relative strengths of the effects due to various relaxation mechanisms can be very different for the two types of measurement. For example, if there is no change in dipole moment associated with a particular relaxation, the corresponding contribution to the dielectric relaxation strength is zero. Relaxations that would be inactive for this reason can be made active by the deliberate introduction of polar groups that do not significantly perturb the structure. An example is the replacement of a few >CH2 groups per thousand carbon atoms in polyethylene by >C—— O groups.

As stated in section 5.7.2, measured mechanical and dielectric relaxation times are not expected to be equal to the fundamental relaxation times of the corresponding relaxing units in the polymer because of the effects of the surrounding medium. These effects are different for the two types of measurement, so it is unlikely that the measured relaxation times will be exactly the same. There are arguments that suggest that, in order to take this into account, the observed relaxation times for dielectric and mechanical relaxation should be multiplied by "1="s and Ju=Jr, respectively, before the comparison is made. The second of these ratios can sometimes be much larger than the first, but for sub-Tg relaxations neither ratio is likely to be greatly different from unity, so that the relaxation times determined from the two techniques would then not be expected to differ significantly for this reason. There are, however, other reasons why the two measured relaxation times need not be the same. As discussed in sections 5.7.5 and 7.2.5 and above, a relaxation generally corresponds not to a single well-defined relaxation time but rather to a spread of relaxation times, partly as a result of there being different environments for the relaxing entity within the polymer. The g values for the different environments will be different, as will the contributions to mechanical strain. Suppose, for example, that the higher relaxation times 264Electrical and optical properties

correspond to larger g factors than do the lower relaxation times but that the lower relaxation times correspond to larger contributions to mechanical strain than do the higher relaxation times. The mean relaxation time observed in dielectric relaxation will then be higher than that observed in mechanical relaxation, leading to a different position of the peaks in a temperature scan. The glass transition does not correspond to a simple

localised motion and has a particularly wide spread of relaxation times, so that its position in mechanical and dielectric spectra is not expected to be closely the same even allowing for differences of frequency. Dielectric measurements can easily be made over a wide range of frequencies, allowing true relaxation strengths "s _ "1 to be determined, but the corresponding measurement is not easy in mechanical studies. For reasons explained in section 7.6.3, the maximum value of tan _ in an isochronal temperature scan is frequently used as a measure of relaxation strength for mechanical spectra. Such scans are frequently used to compare mechanical and dielectric relaxation phenomena. Figure 9.4 shows a comparison of the dielectric and mechanical relaxation spectra of various forms of polyethylene. The most obvious feature is that the main relaxations, here the a, b and g relaxations, occur at approximately the same temperatures in both spectra, although their relative relaxation strengths in the two spectra are different. This is a feature common to the spectra of many polymers. The peaks are not, however, in exactly the same positions in the two spectra for the same type of polyethylene. In addition to the possible reasons for this described above, the frequencies of measurement are different. The dielectric measurements were made at a much higher frequency than that used for the mechanical measurements, as is usual. Molecular motions are faster at higher temperatures, so this factor alone would lead to the expectation that the dielectric peaks would occur at a higher temperature than the mechanical peaks. The g peak, which is assigned to a localised motion in the amorphous material and is in approximately the same place for all samples, behaves in accord with this expectation. The b relaxation in polyethylene, which is most prominent in the lowcrystallinity

LDPE, is associated with the amorphous regions and almost certainly corresponds to what would be a glass transition in an amorphous polymer; a difference in its position in mechanical and dielectric spectra is therefore not surprising. The a relaxation, as discussed in section 7.6.3, is associated with helical ‘jumps’ in the crystalline regions and, provided that the lamellar thickness is reasonably uniform, might be expected to correspond to a fairly well-defined relaxation time and to a narrow peak in the relaxation spectrum. The dielectric peak is indeed quite narrow, because the rotation of the dipoles in the crystalline regions is the major contribu9.2 Electrical polarisation 265

tor to the effect. As discussed in section 7.6.3, however, the mechanical relaxation is seen through the effect of the jumps on the freedom of motion of the amorphous interlamellar regions. The need for multiple jumps for the observation of mechanical relaxation leads to the mechanical relaxation having a higher mean relaxation time and a broader peak than does the dielectric relaxation. The above discussion shows that measurements of dielectric relaxation provide very useful information about molecular motion to add to that provided by NMR and mechanical-relaxation spectroscopies. A particular 266 Electrical and optical properties Fig. 9.4 Dielectric and mechanical relaxation in various forms of polyethylene. The vertical lines are simply a guide to the eye in comparing the positions of the major relaxations. See the text for discussion. (Adapted by permission from A´ kade´ miai Kiado´ ,

Budapest.)

advantage of dielectric measurements is that the relaxation strengths of crystalline and amorphous relaxations are directly proportional to the numbers of dipoles in each phase (although the g values and therefore the constants of proportionality are different for each phase).When constraints on the amorphous motions due to crystallites can be neglected this leads to a simpler dependence of dielectric relaxation strengths on crystallinity than that of mechanical relaxation strengths and hence to a simpler determination of the separate contributions of crystalline and amorphous regions. The problem is still not trivial, however, because the anisotropy of the crystallites and their dielectric constants leads to a non-linear relationship between dielectric relaxation strength and crystallinity. That the various techniques used to study relaxations in polymers are to some extent overlapping and to some extent complementary in terms of the information they provide, while at the same time each having its own difficulties, is a common feature of many studies on polymers and is due to their complicated morphologies.

9.3 Conducting polymers 9.3.1 Introduction There are three principal mechanisms whereby a polymer may exhibit electrical conductivity: (i) metallic or other conducting particles may be incorporated into a non-conducting polymer; (ii) the polymer may contain ions derived from small-molecule impurities, such as fragments of polymerisation catalysts, from ionisable groups along the chain or from salts specially introduced to provide conductivity for a specific use; and (iii) the polymer may exhibit ‘electronic’ conductivity associated with the motion of electrons (more strictly, particle-like excitations of various kinds) along the polymer chains.

To produce conduction by the first method in an otherwise nonconducting polymer, the required concentration of metallic particles is likely to be greater than about 20% by volume, or 70% by weight. This high particle concentration is necessary in order to achieve continuity of conducting material from one side of the polymer sample to the other. Such high concentrations tend to destroy some of the desirable mechanical properties of the polymer, but composites of this kind are used in conducting paints and in anti-static applications. Conductivities of order 6 103 __1 m_1 can be achieved in this way. Carbon black at concentrations above about 9.3 Conducting polymers 267