electrical and optical properties

8/14/2019 Electrical and Optical Properties

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Chapter 9Electrical and optical properties9.1 IntroductionAs indicated in chapter 1, the first electrical property of polymers to be

valued was their high electrical resistance, which made them useful asinsulators for electrical cables and as the dielectric media forcapacitors.

They are, of course, still used extensively for these purposes. It wasrealisedlater, however, that, if electrical conduction could be added to theotheruseful properties of polymers, such as their low densities, flexibilityandoften high resistance to chemical attack, very useful materials would

beproduced. Nevertheless, with few exceptions, conducting polymershavenot in fact displaced conventional conducting materials, but novelapplicationshave been found for them, including plastic batteries,electroluminescentdevices and various kinds of sensors. There is now much emphasis onsemiconducting polymers. Figure 9.1 shows the range of conductivitiesthat

can be achieved with polymers and compares them with othermaterials.Conduction and dielectric properties are not the only electricalpropertiesthat polymers can exhibit. Some polymers, in common with certainother types of materials, can exhibit ferroelectric properties, i.e. theycanacquire a permanent electric dipole, or photoconductive properties, i.e.exposure to light can cause them to become conductors. Ferroelectricmaterials also have piezoelectric properties, i.e. there is an interaction

between their states of stress or strain and the electric field acrossthem.All of these properties have potential applications but they are notconsideredfurther in this book.

The first part of this chapter describes the electrical properties and,


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when possible, explains their origins. Unfortunately, some of thepropertiesassociated with conduction are not well understood yet; even whentheyare, the theory is often rather difficult, so that only an introduction to

theconducting properties can be given. The second part of the chapterdealswith the optical properties. The link between these and the electricalpropertiesis that the optical properties depend primarily on the interaction of theelectric field of the light wave with the polymer molecules. Thefollowing248section deals with the electrical polarisation of polymers, which

underliesboth their dielectric and their optical properties.9.2 Electrical polarisation9.2.1 The dielectric constant and the refractive indexIn an ideal insulating material there are no free electric charges thatcanmove continuously in the presence of an electric field, so no currentcanflow. In the absence of any electric field, the bound positive andnegative

electric charges within any small volume element d_ will, in thesimplestcase, be distributed in such a way that the volume element does nothaveany electric dipole moment. If, however, an electric field is applied tothisvolume element of the material, it will cause a change in thedistribution ofcharges, so that the volume element acquires an electric dipolemoment

9.2 Electrical polarisation 249Fig. 9.1 Electricalconductivities of polymerscompared with those ofother materials. All valuesare approximate.

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


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is isotropic; if this is so, the electric dipole will be parallel to the electricfield. The dipole moment d_ of the volume element will then be givenbyd_ Pd_ pEL d_ 9:1where p is the polarisability of the material and P is the polarisation

producedby the field EL acting on d_. On the molecular level a molecularpolarisability _ can be defined; and, if No is the number of moleculesperunit volume, thenP No_EL 9:2

The mechanism for the molecular polarisability is considered in section9.2.2.So far no consideration has been given to the origin of the electric fieldexperienced by the small volume element under consideration, but it

hasbeen denoted by EL to draw attention to the fact that it is the local fieldatthe volume element that is important. An expression for this field isderivedbelow.Equation (9.1) shows that P is the dipole moment per unit volume, avector quantity that has its direction parallel to EL for an isotropicmedium.Imagine a small cylindrical volume of length dl parallel to the

polarisationand of cross-sectional area dA. Let the apparent surface charges atthe two ends of the cylinder be _dq. It then follows that P dq dl=d_,where d_ is the volume of the small cylinder. However, d_ dl dA, sothatP dq=dA _, where _ is the apparent surface charge per unit areanormalto the polarisation. (Note: _ stands for conductivity in section 9.3.)

The simplest way to apply an electric field to a sample of polymer is toplace it between two parallel conducting plates and to apply a

potentialdifference V between the plates, which gives rise to an applied electricfield normal to the plates of magnitude E V=d, where d is theseparation of the plates. This field may be imagined to be produced bylayers of charge at the conducting plates. The net charges in theselayersare the differences between the free charges on the plates and the


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apparent surface charges on the polymer adjacent to the plates due toits polarisation. Imagine now that a small spherical hole is cut in thepolymer and that this can be done without changing the polarisation ofthe 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 spheremakes an angle _ with the polarisation vector, so that the apparentchargeper unit area on this element is _ cos _ P cos _. The electric fieldproducedat the centre of the sphere by all these surface elements can be shownby250 Electrical and optical propertiesintegration (see problem 9.1) to be P=3"o, where "o is thepermittivity of

free space, so that the total field at the centre of the sphere is equal toE P=3"o. The only way, however, that the polarisation couldremainundisturbed outside the imaginary spherical cavity would be for thecavityto be refilled with polymer! The total electric field at the centre of thesphere would then be increased by the field produced there by all thedipoles in this sphere of polymer. Fortunately it can be shown that thelatter field is zero if the individual molecular dipoles have randompositions

within the sphere. In this case the field at the centre of the sphereremainsE P=3"o, which is therefore the field that would be experienced byamolecule at the centre, i.e. it is the local field EL. ThusEL E P=3"o 9:3

The dielectric constant, or relative permittivity, " of the polymer isdefined by " Vo=V, where Vo is the potential difference that wouldexist between the plates if they carried a fixed free charge Qo per unitarea in the absence of the dielectric and V is the actual potential

differencebetween them when they carry the same free charge in the presenceof thedielectric. With the polymer present, the total effective charge per unitareaat the plates is Q Qo_ P when the effective surface charge on thepolymer


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is taken into account, so that" VoV Eo

E QoQ Q PQ 9:4where Eo is the field that would exist between the plates in the absenceofthe polymer. However, E Q="o (which can be proved by applying asimilar method to that of problem 9.1 to two infinite parallel plates withcharges Q and _Q per unit area) so that

" "oE P"oEor P "oE" _ 1 9:5and finally, from equation (9.3),EL " 23E 9:6A derivation of the local field in essentially this way was originally

given byH. A. Lorentz.Since, according to equation (9.2), P No_EL, equations (9.5) and (9.6)lead toNo_" 23E "oE" _ 1 or" _ 1" 2

No_3"o 9:79.2 Electrical polarisation 251

This relationship between the dielectric constant and the molecularpolarisabilityis known as the ClausiusMosotti relation. It can usefully be writtenin terms of the molar mass M and density _ of the polymer in the form


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" _ 1" 2M

_ NA_

3"o 9:8where NA is the Avogadro constant and the quantity on the RHS iscalledthe molar polarisation and has the dimensions of volume.It follows from Maxwells theory of electromagnetic radiation that" n2, where " is the dielectric constant measured at the frequencyforwhich the refractive index is n. Equation (9.8) thus leads immediatelyto theLorentzLorenz equation

n2_ 1n2 2M

_ NA_3"o 9:9relating the refractive index and the molecular polarisability. Someapplicationsof this equation are considered in sections 9.4.3 and 10.4.1. Theexpression 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-frequencydielectric constant

There are two important mechanisms that give rise to the molecularpolarisability.

The first is that the application of an electric field to a moleculecan cause the electric charge distribution within it to change and soinducean electric dipole, leading to a contribution called the distortional

polarisability.The second is that the molecules of some materials have permanentelectric dipoles even in the absence of any electric field. If an electricfield is applied to such a material the molecules tend to rotate so thatthedipoles become aligned with the field direction, which gives rise to anorientational polarisation. Thermal agitation will, however, prevent the


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molecules from aligning fully with the field, so the resultingpolarisationwill depend both on the field strength and on the temperature of thematerial. This contribution is considered first.It is relatively easy to calculate the observed polarisation of an

assemblyof non-interacting molecules for a given field EL at each molecule at atemperature T. Assume that each molecule has a permanent dipole land 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.

Theenergy u of the dipole is then given by252 Electrical and optical propertiesu _l _EL __EL cos _ 9:10If there are dn dipoles within any small solid angle d! at angle _ to the

applied field, Boltzmanns law shows thatdn Ae_u=kT d! 9:11where A depends on the total number of molecules. The total solidangled! lying between _ and _ d_ when all angles around the field directionare taken into account is 2_ sin _ d_ and the corresponding number ofdipoles is thusdn 2_Ae_u=kT sin _ d_ 9:12Each of these dipoles has a component _cos _ in the direction of thefield

and their components perpendicular to the field cancel out. They thuscontribute a total of _ dn cos _ to the total dipole moment parallel tothefield. Using the fact that dcos _ _sin _ d_ and writing cos _ _ and

_EL=kT x, the average dipole moment h_i contributed by all themoleculesat all angles to the field is given byh_i

_ _

0 2_Ae_EL cos _=kT cos _ dcos _ _0 2_Ae_EL cos _=kTdcos _ 1

_1_ex_d_ 1

_1 ex_d_ 9:13Integration by parts then leads to


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h_i_ _=xex__ 1=x2ex__ _1_1

1=xex__ _1_1

ex e_xex_ e_x_1x coth x _1x Lx9:14where Lx is the Langevin function.For attainable electric fields, _EL=kT x is very small and the

exponentialscan be expanded as e_x ffi 1 _ x x2=2 _ x3=6 Ox4.Substitution then shows that the Langevin function reducesapproximatelyto Lx x=3, so that finallyh_i ffi _2EL=3kT 9:15and the contribution to the molecular polarisability from rotation is to agood approximation _2=3kT. The total molecular polarisability istherefore

_ _ d _2=3kT, where _d is the molecular deformational

polarisability.Use of the ClausiusMosotti relation, equation (9.8), then gives" _ 1" 2M

_ NA_d3"o NA_29"okT 9:16In developing this equation, no account has been taken of any possibleinteractions between the molecular dipoles. It is therefore expected tobe9.2 Electrical polarisation 253most useful for polar gases and for solutions of polar molecules innonpolarsolvents, where the polar molecules are well separated.At optical frequencies _ _d, because the molecules cannot reorient


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sufficiently quickly in the oscillatory electric field of the light wave forthe orientational polarisability to contribute, as discussed further insection9.2.4. Setting _ _d in equation (9.9) gives _d in terms of the opticalrefractive index and, if this expression for _d is inserted, equation

(9.16)can then be rearranged as follows:3" _ n2" 2n2 2M

_ NA_29"okT 9:17Fro hlich showed that a more exact equation for condensed matter,such as

polymers, is" _ n22" n2"n2 22M

_ NAg_29"okT 9:18

The LHS of this equation differs from that of equation (9.17) because amore exact expression has been used for the internal field and the RHSnow includes a factor g, called the correlation factor, which allows for

thefact that the dipoles do not react independently to the local field. If _;g; nand _ are known for a polymer, it should therefore be possible topredictthe value of ". Of these, g is the most difficult to calculate (see section9.2.5).9.2.3 Bond polarisabilities and group dipole moments

The idea that the mass of a molecule can be calculated by addingtogether

the atomic masses of its constituent atoms is a very familiar one.Strictlyspeaking, this is an approximation, although it is an excellent one. It isanapproximation because the equivalent mass of the bonding energy isneglected; the mass of a molecule is in fact lower than the sum of the


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masses of its atoms by only about one part in 1014. In a similar way,butusually to a much lower degree of accuracy, various other properties ofmolecules can be calculated approximately by neglecting theinteractions of

the various parts of the molecule and adding together values assignedtothose individual parts. The word adding must, however, be consideredcarefully. For instance, a dipole moment is a vector quantity; if thedipolemoments of the various parts of a molecule are known, the dipolemomentof the whole molecule can be calculated only if the constituent dipolemoments 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 mole-254Electrical and optical propertiescules. Polarisability is treated there as a scalar quantity, but it is in factasecond-rank tensor quantity (see the appendix) and the correct way ofadding such quantities is somewhat more complicated than theadditionof vectors. When the refractive indices of stressed or orientedpolymers areconsidered, as in chapters 10and 11, this complication must be taken

intoaccount, as described later in the present chapter (section 9.4.3). Forthemoment attention is restricted to media in which the individualmoleculesare oriented randomly.

The high-frequency polarisation of a molecule consists almost entirelyof the slight rearrangement of the electron clouds forming the bondsbetween the atoms. It is the polarisabilities of individual bonds that areassumed to be additive, to a good approximation. If the molecules are

randomly oriented, then so are all the bonds of a particular type. Thesum of the polarisabilities of all the bonds of this type must thereforebeisotropic, so the polarisability for an individual bond can be replaced byascalar average value called the mean value or spherical part of thetensor.


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These scalar values can be added together arithmetically for thevariousbonds within the molecule to obtain the spherical part of thepolarisabilityfor 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 ofdifferent types of bond i by determining the refractive indices for alargenumber of compounds containing only these types of bond. The valueof

_d can be found for each compound by means of the LorentzLorenzequation (9.9) with _ _d and the values of _i are then chosen so thatthey add for each type of molecule to give the correct value of _d. Inorderto obtain molar refractions or molar bond refractions, _d or _i must be

multiplied by NA=3"o. The values of a number of molar bondrefractionsobtained in this way are given in table 9.1. It should be noted that thesimple additivity fails for molecules that contain a large number ofdoublebonds, which permit the electrons to be delocalised over a large regionofthe molecule (see section 9.3.4).A similar argument to the above can be used to deduce the moleculardipole for a molecule if the dipoles of its constituent parts are known. It

has been found that the additivity scheme works best for dipoles ifgroupdipoles are used, i.e. each type of molecular group in a molecule, suchas aCOOH group, tends to have approximately the same dipole momentindependently of the molecule of which it is a constituent part.Equation(9.16), which applies to dilute solutions of polar molecules in a non-polarsolvent, 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 sideofthe equation from values of " measured at various temperatures T. Thegradient of the straight line plot is NA_2=9"ok, from which _ is easily9.2 Electrical polarisation 255obtained. When this has been done for a range of molecules one canfind a


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set of group dipoles li that by vector addition predicts approximatelycorrectlythe values of _ found for all the compounds. Since the quantities liare vectors, not only their magnitudes but also their directions withinthe

molecule must be specified. This is usually done by giving the anglebetween 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 instantaneouslywhen an electric field is applied. This is the exact analogy of a factdiscussedin chapter 7, namely that the strain in a polymer takes time todevelop after the application of a stress. In fact the two phenomenaare

not simply analogous; the relaxation of strain and the rotation ofdipolesare due to the same types of molecular rearrangement. Bothviscoelastic256 Electrical and optical properties

Table 9.1. Molar bond refractions for the D line of NaaBond Refraction (10_6 m3)CH 1.676CC 1.296C

C 4.17CC (terminal) 5.87CC (non-terminal) 6.24CC (aromatic) 2.688CF 1.44CCl 6.51CBr 9.39CI 14.61CO (ether) 1.54

CO 3.32CS 4.61CN 1.54CN 3.76CN 4.82


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OH (alcohol) 1.66OH (acid) 1.80NH 1.76aReproduced from Electrical Properties of Polymers byA. R. Blythe. # Cambridge University Press 1979.

9.2 Electrical polarisation 257Table 9.2. Group dipole momentsbGroupAliphatic compounds Aromatic compoundsMoment(10_30 C m)Anglea(degrees)Moment(10_30 C m)Anglea

(degrees)CH3 0 0 1.3 0F 6.3 5.3Cl 7.0 5.7Br 6.7 5.7I 6.3 5.7OH 5.7 60 4.7 60NH2 4.0 100 5.0 142COOH 5.7 745.3 74NO2 12.3 0 14.0 0CN 13.40 14.7 0

COOCH3 6.0 70 6.0 70OCH3 4.0 55 4.3 55aThe angle denotes the direction of the dipole moment with respect to thebond joining the group to the rest of the molecule.bReproduced from Electrical Properties of Polymers by A. R. Blythe.# Cambridge University Press 1979.Example 9.1Calculate the refractive index of PVC using the bond refractions given intable 9.1. Assume that the density of PVC is 1.39 Mg m_3.SolutionFor a polymer the LorentzLorenz equation (9.9) can be rewritten

n2_ 1n2 2 NA3"o_dM_ __


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where the quantity in brackets is the molar refraction per repeat unit dividedby the molar mass of the repeat unit. The PVC repeat unit CH2CHCl ,has relative molecular mass 2 12 3 1 35:5 62:5 and molar mass6:25 10_2 kg. There are three CH bonds, one CCl bond and twoCC bonds per repeat unit, giving a refraction per repeat unit of

3 1:676 6:51 2 1:296_ 10_6 1:413 10_5 m3. Thusn2_ 1 1:413 10_5 1:39 103n2 2=6:25 10_2 0:3143n2 2,which leads to n 1:541.measurements and dielectric measurements can therefore be used tostudythese types of rearrangement. Dielectric studies have the advantageoverviscoelastic studies by virtue of the fact that a much wider range offrequenciescan be used, ranging from about 104 s per cycle (10_4 Hz) up tooptical frequencies of about 1014 Hz.In section 7.2.1 the idea of a relaxation time is made explicit throughtheassumption that, in the simplest relaxation, the material relaxes to itsequilibrium strain or stress on application of a stress or strain in such away that the rate of change of strain or stress is proportional to thedifferencebetween the fully relaxed value and the value at any instant. Asimilar assumption is made in the present section for the relaxation ofpolarisation on application of an electric field.In considering dielectric relaxation it is, however, necessary torememberthat there are two different types of contribution to the polarisation ofthe dielectric, the deformational polarisation Pd and the orientationalpolarisation Pr, so that P Pd Pr. For the sudden application of afield E that then remains constant, the deformational polarisation canbeconsidered to take place instantaneously, or more precisely in a timeofthe order of 10_14 s. When alternating fields are used the deformationalpolarisation can similarly be considered to follow the applied fieldexactly,provided that the frequency is below optical frequencies. If the limitingvalue of the dielectric constant at optical frequencies is called "1,equation(9.5) shows thatPd "o"1_ 1E 9:19Assuming that, in any applied field with instantaneous value E, Pr


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responds in such a way that its rate of change is proportional to itsdeviationfrom the value that it would have in a static field of the same value E,it follows thatdPr

dt "o"s_ "1E _ Pr9:20

where "s is the dielectric constant in a static field and is the relaxationtime. Note that, in equation (9.20), it is necessary to subtract from thetotal polarisation P "o"s_ 1E in a static field the polarisation Pd "o"1_ 1E due to deformation in order to obtain the polarisation dueto the orientation of dipoles.Consider first the instantaneous application at time t 0 of a field Ethat then remains constant. Then it follows from equation (9.20) that

Pr "o"s_ "11 _ e_t=E 9:21258 Electrical and optical propertiesNow consider an alternating applied field E Eoei!t. Because thedeformationalpolarisation follows the field with no time lag, Pd is given, accordingto equation (9.19), byPd "o"1_ 1Eoei!t 9:22Pr is now given, from equation (9.20), bydPrdt

"o"s_ "1Eoei!t_ Pr9:23

Assume that Pr Pr;oei!t, where Pr;o is complex to allow for a phase lagof Pr with respect to the field. It follows thatdPrdt i!Pr;oei!t "o"s_ "1Eo_ Pr;o

ei!t 9:24or

Pr;o "o"s_ "11 i!Eo 9:25

The total complex polarisation P is thus given, from equations (9.19)and(9.25), by


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P Pd Pr "o"1_ 1 "o"s_ "1=1 i!_E 9:26and the complex dielectric constant ", from equation (9.5), by" 1 P"oE 1 "1_ 1

"s_ "11 i! "1 "s_ "11 i! 9:27

The expression for " given on the RHS of equation (9.27) is called theDebye dispersion relation. Writing the dielectric constant " "0_ i"00,itfollows immediately from equation (9.27) that"0 "1 "s_ "1

1 !22 and "00 "s_ "1!1 !22 9:28a;b

These equations are the dielectric equivalents of the equationsdeveloped insection 7.3.2 for the real and imaginary parts of the compliance ormodulus.

Just as the imaginary part of the compliance or modulus is a measureof the energy dissipation or loss per cycle (see section 7.3.2), so is"00. Thevariations of "0 and "00 with ! are shown in fig. 9.2.

The terms relaxed and unrelaxed dielectric constant are used for thestatic and high-frequency values "s and "1, respectively. Thesequantities9.2 Electrical polarisation 259are the electrical analogues of the relaxed and unrelaxed moduli orcompliancesdefined in section 7.3.2 and a dielectric relaxation strength isdefined as "s_ "1, in exact analogy with the definition of mechanicalrelaxation 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 ofthe molecules and can be calculated from equation (9.8) (with _ _d)and values of the molar bond refractions, such as those given in table9.1. Rearrangement of this equation shows, as expected, that thegreater


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the molar refraction the greater the high-frequency dielectric constantandhence the refractive index. Table 9.1 then shows, for instance, thatpolymerswith a large fraction of CC double bonds will generally have higher

refractive indices than will those with only single CC bonds and thatpolymers containing chlorine atoms will have higher refractive indicesthanwill their analogues containing fluorine atoms. Some refractive indicesofpolymers are given in table 9.3 and these trends can be observed byreferenceto it.If a polymer consisted entirely of non-polar groups, or contained polargroups arranged in such a way that their dipoles cancelled out, it

wouldhave a low dielectric constant at all frequencies, determined only bythe260 Electrical and optical propertiesFig. 9.2 The variations of "0and "00 with ! for the simpleDebye model.

deformational polarisation, and it would not undergo dielectricrelaxation.Examples of polymers potentially in this class are polyethylene, inwhichthe >CH2 groups have very low dipole moments and are arrangedpredominantlyin opposite directions along the chain, and polytetrafluoroethylene,in which the dipole moment of the >CF2 group is moderatelylarge but the molecule is helical and the dipole moments cancel outbecausethey are approximately normal to the helix axis for each group. Suchcancelling out is never exact, however, because of conformationaldefects.An additional factor is that oxidation, even if it does not significantlyperturb the conformation, can introduce groups with different dipolemoments from those of the groups replaced. The dipoles then nolongercancel out and both a higher dielectric constant and dielectricrelaxationare observed.If a polymer contains polar groups whose moments do not cancel out,


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the actual value of the dielectric constant depends very strongly on themolecular conformation, which in turn may depend on the dipoles,becausestrong repulsion between parallel dipoles will cause the conformationto

9.2 Electrical polarisation 261 Table 9.3. Approximate refractive indices for selected polymers atroom temperaturePolymer Repeating unit Refractive indexPolytetrafluoroethylene CF2CF2 1.351.38Poly(vinlidene fluoride) CH2CF2 1.42Poly(butyl acrylate) CH2CHCOOCH23CH3 1.46Polypropylene (atactic) CH2CHCH3 1.47Polyoxymethylene CH2O 1.48Cellulose acetate See section 1.1 1.481.50Poly(methyl methacrylate) CH2CCH3COOCH3 1.49Poly(1,2-butadiene) CH2 CH(CHCH2 1.50Polyethylene CH2CH2 1.511.55 (depends on density)Polyacrylonitrile CH2CHCN 1.52Poly(vinyl chloride) CH2CHCl 1.541.55Epoxy resins Complex cross-linked ether 1.551.60Polychloroprene CH2CCl(CHCH2 1.551.56Polystyrene CH2CHC6H5 1.59Poly(ethylene terephthalate) (CH22(CO)(C6H4)(CO) 1.581.60Poly(vinylidene chloride) CH2CCl2 1.60163CH2CHPoly(vinyl carbazole) 1.68change so that the dipoles are not parallel. The effect of the molecularconformation is incorporated into the correlation factor g introduced inequation (9.18). The value of g for an amorphous polymer depends ontheangles between the dipoles within those parts of the molecule that canrotate independently when the electric field is applied. These anglesand

the sections of chain that can rotate independently are temperature-andfrequency-dependent, particularly in the region of the glass transition,so itis not generally possible to calculate g. In practice, measurements ofthe


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static dielectric constant and the refractive index are used withequation(9.18) to determine g values and so obtain information about thedipolecorrelations and how they vary with temperature.

Matters are more complicated still for semicrystalline polymers, forwhich not only are the observed dielectric constants averages overthoseof the amorphous and crystalline parts but also the movements of theamorphous chains are to some extent restricted by their interactionswiththe crystallites, which causes changes in g and hence in the staticdielectricconstant of the amorphous material. These facts are illustrated by thetemperature dependences of the dielectric constants of amorphous

and50% crystalline PET, shown in table 9.4.262 Electrical and optical properties

Table 9.4. Dielectric constants of amorphous and 50% crystallinepoly(ethylene terephthalate)aPolymer state T (K)Dielectric constantUnrelaxed, "1 Relaxed, "sAmorphous 193 3.09 3.80203 3.09 3.80213 3.12 3.80223 3.13 3.79233 3.143.783543.80 6.00Crystalline 193 3.18 3.66203 3.18 3.67213 3.19 3.66373 3.71 4.44aAdapted by permission from Boyd, R. H. and Liu, F. Dielectric spectroscopyof semicrystalline polymers in Dielectric Spectroscopy of PolymericMaterials: Fundamentals and Applications, eds. James P. Runt and John

J. Fitzgerald, American Chemical Society, Washington DC, 1997, Chap. 4,pp. 107136, Table I, p. 117.Table 9.4 shows that the unrelaxed values for these two samples areverysimilar and do not change much even at the glass transition (at about354 K). The relaxed values are different for the two samples,particularly


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above the glass-transition temperature, where the relaxed dielectricconstantof the amorphous sample is significantly greater than that of thecrystalline sample. This temperature dependence arises from the factthat, at low temperature, the C

O groups tend to be trans to each otheracross the benzene ring, so that their dipoles almost cancel out,whereas athigher temperatures rotations around the ring(CO) bonds take placemore freely in the amorphous regions so that the dipoles can orientmoreindependently and no longer cancel out. The value of the relaxedconstantfor the crystalline sample is lower than that of the non-crystalline

sampleeven at the lower temperatures because the rigid crystalline phase notonlycontributes little to the relaxed constant but also restricts theconformationsallowed in the amorphous regions.

The width of the dielectric loss peak given by equation (9.28b) can beshown to be 1.14 decades (see problem 9.3). Experimentally, losspeaks areoften much wider than this. A simple test of how well the Debye model

fitsin a particular case is to make a so-called ColeCole plot, in which "00 isplotted against "0. It is easy to show from equations (9.28) that theDebyemodel predicts that the points should lie on a semi-circle with centre at[("s "1=2; 0_ and radius "s_ "1=2. Figure 9.3 shows an example ofsuch a plot. The experimental points lie within the semi-circle,correspondingto a lower maximum loss than predicted by the Debye model and alsoto a wider loss peak. A simple explanation for this would be that, in an

amorphous polymer, the various dipoles are constrained in a widerange ofdifferent ways, each leading to a different relaxation time , so that theobserved values of "0 and "00 would be the averages of the values foreachvalue of (see problem 9.4).9.2 Electrical polarisation 263Fig. 9.3 A ColeCole plot


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for samples of poly(vinylacetate) of various molarmasses:*, 11 000 g mol_1;O, 140 000 g mol_1;&, 500 000 g mol_1; and

4, 1 500 000 g mol_1. Thedielectric constant hasbeen normalised so that"s_ "1 1. (Adapted bypermission of Carl HanserVerlag.)

Various empirical formulae have been given for fitting data thatdeviatefrom the simple semi-circular ColeCole plot. The most general of theseisthe Havriliak and Negami formula

" _ "1"s_ "1 1 i!a__b 9:29

This equation has the merit that, on changing the values of a and b, itbecomes the same as a number of the other formulae. In particular,whena b 1 it reduces, with a little rearrangement, to equation (9.27).In section 5.7 various motions that can take place in polymers and canlead to NMR, mechanical or dielectric relaxation are considered mainlythrough their effects on the NMR spectrum. Mechanical relaxation isconsidered in detail in chapter 7. Although all the motions that are

effectivein mechanical relaxation can potentially also produce dielectricrelaxation,the relative strengths of the effects due to various relaxationmechanismscan be very different for the two types of measurement. For example,ifthere is no change in dipole moment associated with a particularrelaxation,the corresponding contribution to the dielectric relaxation strength is

zero. Relaxations that would be inactive for this reason can be madeactiveby the deliberate introduction of polar groups that do not significantlyperturb the structure. An example is the replacement of a few >CH2groups per thousand carbon atoms in polyethylene by >CO groups.


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As stated in section 5.7.2, measured mechanical and dielectricrelaxationtimes are not expected to be equal to the fundamental relaxationtimes ofthe corresponding relaxing units in the polymer because of the effects

of thesurrounding medium. These effects are different for the two types ofmeasurement,so it is unlikely that the measured relaxation times will be exactlythe same. There are arguments that suggest that, in order to take thisintoaccount, the observed relaxation times for dielectric and mechanicalrelaxation should be multiplied by "1="s and Ju=Jr, respectively, beforethe comparison is made. The second of these ratios can sometimes bemuch

larger than the first, but for sub-Tg relaxations neither ratio is likely tobegreatly different from unity, so that the relaxation times determinedfromthe two techniques would then not be expected to differ significantlyforthis reason. There are, however, other reasons why the two measuredrelaxation times need not be the same.As discussed in sections 5.7.5 and 7.2.5 and above, a relaxationgenerally

corresponds not to a single well-defined relaxation time but rather to aspread of relaxation times, partly as a result of there being differentenvironmentsfor the relaxing entity within the polymer. The g values for thedifferent environments will be different, as will the contributions tomechanical strain. Suppose, for example, that the higher relaxationtimes264Electrical and optical propertiescorrespond to larger g factors than do the lower relaxation times butthat

the lower relaxation times correspond to larger contributions tomechanicalstrain than do the higher relaxation times. The mean relaxation timeobserved in dielectric relaxation will then be higher than that observedinmechanical relaxation, leading to a different position of the peaks in atemperature scan. The glass transition does not correspond to a simple


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localised motion and has a particularly wide spread of relaxation times,so that its position in mechanical and dielectric spectra is not expectedtobe 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, butthe corresponding measurement is not easy in mechanical studies. Forreasons explained in section 7.6.3, the maximum value of tan _ in anisochronaltemperature scan is frequently used as a measure of relaxationstrength for mechanical spectra. Such scans are frequently used tocomparemechanical and dielectric relaxation phenomena.Figure 9.4 shows a comparison of the dielectric and mechanical

relaxationspectra of various forms of polyethylene. The most obvious feature isthat the main relaxations, here the a, b and g relaxations, occur atapproximatelythe same temperatures in both spectra, although their relativerelaxation strengths in the two spectra are different. This is a featurecommonto the spectra of many polymers. The peaks are not, however, inexactly the same positions in the two spectra for the same type ofpolyethylene.

In addition to the possible reasons for this described above, thefrequencies of measurement are different. The dielectricmeasurementswere made at a much higher frequency than that used for themechanicalmeasurements, as is usual. Molecular motions are faster at highertemperatures,so this factor alone would lead to the expectation that the dielectricpeaks would occur at a higher temperature than the mechanical peaks.

The

g peak, which is assigned to a localised motion in the amorphousmaterialand is in approximately the same place for all samples, behaves inaccordwith this expectation.

The b relaxation in polyethylene, which is most prominent in thelowcrystallinity


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LDPE, is associated with the amorphous regions and almostcertainly corresponds to what would be a glass transition in anamorphouspolymer; a difference in its position in mechanical and dielectricspectra is

therefore not surprising. The a relaxation, as discussed in section 7.6.3,isassociated with helical jumps in the crystalline regions and, providedthatthe lamellar thickness is reasonably uniform, might be expected tocorrespondto a fairly well-defined relaxation time and to a narrow peak in therelaxation spectrum. The dielectric peak is indeed quite narrow,becausethe rotation of the dipoles in the crystalline regions is the major

contribu-9.2 Electrical polarisation 265tor to the effect. As discussed in section 7.6.3, however, themechanicalrelaxation is seen through the effect of the jumps on the freedom ofmotionof the amorphous interlamellar regions. The need for multiple jumpsforthe observation of mechanical relaxation leads to the mechanicalrelaxation

having a higher mean relaxation time and a broader peak than doesthe dielectric relaxation.

The above discussion shows that measurements of dielectric relaxationprovide very useful information about molecular motion to add to thatprovided by NMR and mechanical-relaxation spectroscopies. Aparticular266 Electrical and optical propertiesFig. 9.4 Dielectric andmechanical relaxation invarious forms ofpolyethylene. The vertical

lines are simply a guide tothe eye in comparing thepositions of the majorrelaxations. See the textfor discussion. (Adaptedby permission fromAkade miai Kiado ,


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Budapest.)

advantage of dielectric measurements is that the relaxation strengthsofcrystalline and amorphous relaxations are directly proportional to thenumbers

of dipoles in each phase (although the g values and therefore theconstants of proportionality are different for each phase).Whenconstraintson the amorphous motions due to crystallites can be neglected thisleads to asimpler dependence of dielectric relaxation strengths on crystallinitythanthat of mechanical relaxation strengths and hence to a simplerdeterminationof the separate contributions of crystalline and amorphous regions.

The problem is still not trivial, however, because the anisotropy of thecrystallites and their dielectric constants leads to a non-linearrelationshipbetween dielectric relaxation strength and crystallinity. That thevarioustechniques used to study relaxations in polymers are to some extentoverlappingand to some extent complementary in terms of the information theyprovide, while at the same time each having its own difficulties, is acommon

feature of many studies on polymers and is due to their complicatedmorphologies.9.3 Conducting polymers9.3.1 Introduction

There are three principal mechanisms whereby a polymer may exhibitelectrical conductivity:(i) metallic or other conducting particles may be incorporated into anon-conducting polymer;(ii) the polymer may contain ions derived from small-moleculeimpurities,

such as fragments of polymerisation catalysts, from ionisablegroups along the chain or from salts specially introduced to provideconductivity for a specific use; and(iii) the polymer may exhibit electronic conductivity associated withthemotion of electrons (more strictly, particle-like excitations of variouskinds) along the polymer chains.


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To produce conduction by the first method in an otherwise non-conductingpolymer, the required concentration of metallic particles is likely to begreater than about 20% by volume, or 70% by weight. This highparticle

concentration is necessary in order to achieve continuity of conductingmaterial from one side of the polymer sample to the other. Such highconcentrations tend to destroy some of the desirable mechanicalpropertiesof the polymer, but composites of this kind are used in conductingpaintsand in anti-static applications. Conductivities of order 6 103__1 m_1can be achieved in this way. Carbon black at concentrations aboveabout9.3 Conducting polymers 267

electrical and optical properties

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