More data about dielectric and electret properties of poly(methyl methacrylate)

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1997 J. Phys. D: Appl. Phys. 30 1383

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J. Phys. D: Appl. Phys. 30 (1997) 1383–1398. Printed in the UK PII: S0022-3727(97)76910-9

More data about dielectric andelectret properties of poly(methylmethacrylate)

Karol Mazur

Department of Physics, Technical University, 65-246 Zielona Gora, Poland

Received 5 August 1996, in final form 13 January 1997

Abstract. We have used the probe technique together with the thermally stimulateddischarge (TSD) method in order to determine the potential distribution V (x) inpoly(methyl methacrylate) (PMMA) thermoelectrets polarized at temperatures Tpbelow the glass transition temperature Tg . We have obtained current–voltage (j –V )characteristics for PMMA at temperatures Tp < Tg on the basis of the V (x)distribution and some theoretical considerations. This characteristic is of sub-ohmicshape, namely j = gV n , where n < 1 and g = constant. On the basis of the aboverelation, a two-layer condensor (PMMA with one-sided metallized and an air gap)was considered as a model for the formation of PMMA thermoelectrets. It followsfrom this model that the interfacial charge density changes nonlinearly with theapplied polarizing voltage (Vp). The theoretical results have been compared withexperimental data of PMMA thermoelectrets. The problems of the mean realcharge depth and the charge decay in these electrets have also been considered.In particular we can state that the isothermal absorption current, current–voltagecharacteristic and, moreover, charge decay after poling of PMMA and of thePMMA/BaTiO3 composite are interrelated.

1. Introduction

Much experimental and theoretical work has been devotedto the potential distribution, charge transport and permanentpolarization in polymeric thermoelectrets. The earlyliterature on these electrets has been reviewed by severalauthors [1–13], notably by van Turnhout [5], Lushchejkin[7], Sessler [12] and Hilczer and Malecki [13]; themore recent works have been discussed in a number ofpublications of different scope (concerning charge storage,charge transport and charge distribution in polymers), in theproceedings of several conferences [14–19] and in Nalwa’snew publication on ferroelectric polymers [20].

The problems of the determination of the spatialvariation of the potential, the internal electric field, thedipolar polarization and the space-charge distribution inpolymer electrets are important both in fundamental and inapplied studies. A means of direct measurement of spacecharge or polarization in the materials would contribute toour understanding of the physical processes involved [21].

The first fundamental studies of permanent dielectricpolarization concerned thermoelectrets formed from poly-mers with molecular dipole moments such as polymethylmethacrylate (PMMA) [22]. The structure of the PMMAmacromolecule is linear with respect to the main carbon

chain [5, 12, 13]: H x| |— C —C—| |

H y

n

where x = CH3. The dipole moment of PMMA isassociated with the ester groupy = COOCH3

O H|| |— C — O — C —H|

H

and the orientation of this group in the electric field whilethe thermomoelectret is being formed is responsible for theappearance of a heterocharge. Such electrets are still usedin laboratory basic research [12–20].

It is known [5, 23–25] that the current thermogram of ashorted PMMA heteroelectret shows a number of maximawhich reflect the relaxation processes occurring in thispolymer. By succesive applications of the thermal cleaningmethod [26] it was possible to isolate three different dipolarpeaks: theα and β peaks and another one,β ′, locatedbetween them [5, 24, 25]. Theβ peak, which occurs at222 K, is ascribed to the reorientation of the polar ester sidegroups (—COOCH3) by local motions around the C—Cbond [5]. Theα peak, found at 375 K, is due to the

0022-3727/97/091383+16$19.50 c© 1997 IOP Publishing Ltd 1383

K Mazur

collective reorientation of the side groups with adjacentmain chain segments —C—CH2— [5]. The temperaturelocation of this peak corresponds to the glass transitiontemperature,Tg [5].

A process similar to theβ ′ one was observed by meansof AC measurements [27] and the dilatometric technique[28]. It has been attributed to the presence of heterotacticsequences in the conventional PMMA. Moreover, forsamples charged above the glass transition temperature(Tg), the current thermogram response consists of dipolar-reorientation peaks and a space-charge peak (theρ peak)at a temperature higher thanTg [5].

Owing to the presence of the hydrophilic ester groups,PMMA can absorb up to about 2% of its weight ofwater [29]. In connection with this fact, we used thethermally stimulated discharge (TSD), differential scanningcalorimetry (DSC) and infrared spectroscopy (IRS) methodsto determine the effect of absorbed water on theβ ′,α and ρ relaxation processes in PMMA [30, 31]. Itfollows from the TSD analysis for this polymer that suchparameters as the intensityjM , TSD peak areaσ , maximumtemperatureTM , activation energyU and timeτ of the so-called β ′, α and ρ relaxations undergo evident changesdepending on the degree of swelling of the polymer inwater. In addition to the plasticizing effect of H2O inlowering the temperature of theα relaxation, the obtainedresults lead to the conclusion that the space-chargeρ peakis enlarged. Moreover, the parameters ofβ ′ relaxationchange, which arises probably from the formation ofclusters of water–ester groups. The last conclusion isconfirmed by the fact that water changes the infraredspectrum of PMMA. We note that the similar effects ofabsorbed water on the TSD of charged Mylar and polyvinylacetate thin films were discussed earlier by Perlman andCreswell [32] and Mahendruet al [33], respectively.Doubtless, the above-mentioned molecular processes ofdielectric relaxation determine some phenomenologicalrelations concerning such properties of PMMA as theelectric, electret, piezoelectric and pyroelectric properties.

In this paper we will show how knowledge of thepotential distribution, determined by the probe techniquetogether with the TSD method, can be applied forevaluation of the shape of the current–voltage characteristicin PMMA polymer polarized at temperature below theglass transition temperatureTg of this material. TheTgwas evaluated from DSC thermograms [31]. Finally, theeffect of the nonlinear current–voltage characteristic onthe relation between the surface charge density of thePMMA thermoelectret and the polarizing voltage, will beconsidered both analytically and experimentally.

Moreover, the purpose of this paper is to show that,by using both electrostatic and TSD methods and utilizingsome boundary condition for the potential distribution, wecan roughly separate the induction charge density from thedipolar and real charge density of the PMMA thermoelectret[34]. On this basis we can also evaluate the mean realcharge depth in a dipolar polymer such as PMMA by theadoption of Sessler’s formula [35]

The paper is based on three posters presented bythe author at international meetings in Heidelberg [34],

Erlangen [36] and Braunschweig [37]. However, the datafrom these posters have been extended by some studies onthe dielectric behaviour (AC measurements) and dielectricabsorption (DC measurements) of PMMA at differenttemperatures.

The organization of this paper is as follows. Sections 2and 3 review some aspects of the dielectric behaviourof PMMA and of the temperature dependence of theheterocharge in PMMA thermoelectrets. The investigationof the potential distribution in PMMA thermoelectrets isthe topic of section 4. Section 5 presents an examinationof isothermal dielectric absorption at temperatures belowthe glass transition temperature of PMMA. The current–voltage characteristic and the relation between the surfacecharge density of PMMA thermoelectrets and the polarizingvoltage are discussed in sections 6 and 7. The chargedecay in PMMA thermoelectrets is considered in section 8.Finally, a summary and conclusions are presented insection 9.

2. The dielectric behaviour of PMMA

2.1. Introduction

Studies on the dielectric behaviour of a series ofpolyalkylmethacrylates were presented in detail by Ishidaand Yamafuji [29, 38]. In this section we describe onlysome dielectric properties of PMMA obtained in ourlaboratory during free-radical polymerization of methylmethacrylate (MMA). In particular, the values ofεs −ε∞ and ε(Tp) − ε(Tf ) have to be established in orderto determine the saturation dipole polarizationPs or thepersistent polarizationPp of the polymer. Here,εs andε∞ are the static and optical dielectric constants; however,ε(Tp) and ε(Tf ) are the static dielectric constants at thepolarizing(Tp) and final(Tf ) temperatures, respectively.

2.2. Experimental details

In order to obtain the PMMA, the polymerization processwas initiated in chemically pure MMA using benzoylperoxide as an initiator. After the polymerization had beenconcluded at the higher temperature, the samples of PMMAwere formed into discs 3 cm in diameter and 0.25 cm thick.We used electrodes of du Pont silver paste deposited onthe plane surfaces of the samples. The real part of thepermittivity ε′ and tanδ were measured by means of thecapacitance BM 484 Tesla bridge in the frequency range200 Hz to 10 kHz.

2.3. Results and discussion

It is known that the relaxation strengthεs − ε∞ depends onthe temperature [5, 12, 13] and it is given approximately by

εs − ε∞ ∼= Nµ2/(3ε0kT ) (1)

where N is the number of permanent dipoles per unitvolume, µ is their dipole moment,k is Boltzmann’sconstant, andε0 is the permittivity of vacuum. Thesaturation polarizationPs expected after a sufficiently

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Dielectric and electret properties of PMMA

Figure 1. The frequency dependences of ε′ at variousfixed temperatures for PMMA.

Figure 2. The frequency dependences of tan δ at variousfixed temperatures for PMMA.

long polarizing time depends on the polarizing fieldEp[5, 6, 12, 13] as

Ps = ε0(εs − ε∞)Ep = Nµ2Ep/(3kTp). (2)

In order to evaluate theεs − ε∞ values, the frequencydependences of the real part of the dielectric constantε′, tanδ and the dielectric loss factorε′′ = ε′ tanδwere measured at various fixed temperatures for PMMA(figures 1–3), then Debye’s semicircle was drawn (forexample, figure 4). As we know [13], Debye’s equationis frequently represented graphically in the complex planeε′′–ε′ as a semicircle. This semicircle, known also as theCole–Cole plot, permits easy extrapolation of the results tothe values ofε∞ andεs even when few experimental pointsare available [39].

We also know that, if the experimental results forma Debye semicircle, the effects responsible for dielectric

Figure 3. The frequency dependences of ε′′ at variousfixed temperatures for PMMA.

Figure 4. A Cole–Cole plot for PMMA at 353 K.

relaxation can be described in terms of the simple Debyemodel with one relaxation time. However, particularlylarge divergences from this simple mechanism are observedin the case of polymeric materials such as PMMA, forexample, since a number of processes (β, β ′, α and ρ)with different relaxation times are involved [5, 12, 23–25, 30, 31]. (For discussion on this subject, see alsosection 3).

We note that, in the case of polymers, the usefulapproach is the empirical approach of Fuoss and Kirkwood[40] who showed that the dependence of arccosh(ε′′max/ε

′′)on the logarithm of the frequency (lnω) gives a straightline which intersects the frequency axis atω = ωmax withits slopeβ related to the static permittivity in accordancewith [40]

β = 2ε′′max/(εs − ε∞). (3)

From this plot (figure 5) for PMMA we haveεs − ε∞ ∼= 2at T = 353 K.

On the basis of experimental results similar to the onespresented in figure 5 and those from the Fuoss–Kirkwoodequation we can obtain the temperature dependence of theεs−ε∞. Contrary to the Langevin equation (equation 2), theexperimental relation (figure 6) shows an increase inεs−ε∞with increasing temperature. This discrepancy is probably

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K Mazur

Figure 5. The dependence of arccosh (ε′′max/ε′′) on the

logarithm of the frequency ω for PMMA at 353 K. (τ is therelaxation time.)

Figure 6. The temperature dependences of 1ε′ (curve 1),εs − ε∞ (curves 2 and 3 for β and β plus α relaxation,respectively) and Pp/(ε0Ep) (curve 4).

due to the changes in mobility of the structural elementsof PMMA when temperature increases. On the other hand,the persistent polarizationPp of polymer electrets expectedafter a sufficiently long polarizing time depends on thepolarizing temperature as [7]

Pp = ε0[ε′(Tp)− ε′(Tf )]Ep = ε01ε′Ep (4)

both for polar and for nonpolar materials.For PMMA, 1ε′ can be evaluated from the data

presented in figure 7. The result of this evaluation is shownin figure 8. In figure 6 the temperature dependences of theεs − ε∞ (curves 2 and 3 forβ andβ + α relaxation) and1ε′ values (curve 1) are shown. Moreover, in figure 6,the temperature dependence of the so-called reduced charge

Figure 7. The temperature dependences of ε′ at variousfixed frequencies for PMMA.

Figure 8. The temperature dependences of 1ε′ at variousfixed frequencies for PMMA.

density1ε = Pp/(ε0Ep) (curve 4) is also shown. Here,Pp is the TSD charge density obtained from discharge ofthe PMMA thermoelectret.

A comparison of the reduced released TSD hetero-charge1ε = Pp/(ε0Ep) calculated from theα peak withthe 1ε′ increment indicates that there is good agreementbetween these values at polarizing temperaturesTp near350 K. It is also interesting that at this temperature, thedipolar strengthεs − ε∞ for the β peak is near1ε (forfurther details see also section 3).

3. The temperature dependence of theheterocharge in PMMA thermoelectrets

On the basis of Debye’s theory and the TSD spectrumanalysis the dependence of the heterocharge on the

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Dielectric and electret properties of PMMA

Figure 9. The theoretical fit to the β ′, α and ρ peaks usingArrhenius (◦) and Eyring (×) relaxations:(——), experimental and sum of TSC spectra.

polarizing temperatureTp can be made more clear, asfollows. Let us assume a time dependence of the dipolepolarizationP(t) with a single relaxation timeτ(T ). Itbuilds up

dP(t)

dt+ P(t)

τ (Tp)= P0

τ(Tp)(5)

whereP0 equalsNµ2Ep/(3kTp). Integrating equation (5)yields [41]

P(tp) = P0{1− exp[−tp/τ (Tp)]} (6)

where τ(Tp) = τ0 exp[U/(kTp)]. Approximatingequation (6) for the polarizing timetp � τ(Tp) gives

P(tp) = P0(Tp). (7)

We note that the inequalitytp � τ(Tp) is likely tohold in our experiments only forTp � Tg. However,approximating equation (6) for the polarizing timetp <

τ(Tp) gives

P(tp) = P0tp

τ (Tp)= ε0(εs − ε∞)Eptp

τ0exp

(−UkTp

). (8)

Thus, in the absence of a homocharge, we have for theinitial heterocharge density

σ0 = P(tp) = C exp[−U/(kTp)] (9)

where C = ε0(εs − ε∞)Eptp/τ0. Obviously, C inequation (9) is temperature dependent in accordance withequation (6).

We note that the inequalitytp < τ(Tp) is likely to holdin our experiments when the polarization temperatureTp issufficiently below the glass transition temperatureTg (about105◦C) of the PMMA. However, when the polarizing

Figure 10. The Tp dependence of the surface chargedensity σ for each peak (β ′, α and ρ) and for the effective(sum) charge density: full lines are theoretical curveswhereas symbols represent experimental data.

temperature increases from room temperatureTr to the finaltemperatureTf , the relation betweentp and τ(TP ) willchange fromtp < τ(Tr) via tp = τ(TM) to tp � τ(Tf ).Under this condition the dependence on the polarizingtemperature of the dipole polarization satisfies equation (7),obviously. However, the relation betweenP(tp) and Tpis more complicated than that indicated by equation (6),because the heterocharge of a PMMA thermoelectret isdetermined by dipole polarization and the space-chargepolarization with a distribution of the relaxation timeτi oractivation energyUi , wherei denotesβ ′, α or ρ relaxationat temperatures above room temperature. If we assumein the simplest case three relaxation processes in PMMAwith discrete relaxation timesτi , theTp dependence of thecharge densityσ can be expressed as follows:

σ(Tp) =∑i

P0i

[1− exp

(− tp

τi(Tp)

)](10)

whereτi(Tp) = τ0i exp[Ui/(kTp)].In order to compare the theory (equation 10)

with experiments, one should calculate some relaxationparameters on the basis of the TSD method for the materialsunder consideration. From the TSC thermogram of PMMA(figure 9) and its analysis, it is possible to determineUi ,P0i andτ0i (see table 1).

The Tp dependence of the surface charge densityσ

for each (β ′, α and ρ) peak and the effective (sum)charge density from having applied equation (10) and theseparameters are presented in figure 10 in comparison withthe experimental data. It can be seen in figure 10 thatthe maximal effective charge density can be obtained inPMMA for Tp > Tg, where the glass transition temperaturefor PMMA, Tg, is 378 K [31].

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Table 1. Parameters determined for the various peaks.

U TM τ0(0) τ0(1) P0 jMPeak (eV) (K) (s) (sK) (µC m−2) (nA m−2)

β ′ 0.50 331 1.00× 10−5 2.88× 10−3 2.72 2.61α 1.20 375 1.5× 10−14 5.49× 10−12 5.15 9.87ρ 2.10 400 4.35× 10−25 1.79× 10−22 2.24 6.49

4. The potential distribution in PMMAthermoelectrets

4.1. Introduction

We used the probe technique together with the TSD methodin order to determine the potential distribution in PMMAthermoelectrets. The probe technique is widely used inpractice [11, 42, 43] but in its original form it cannotbe applied to determine the potential distribution in theelectrets because the measuring apparatus has too low animpedance. Therefore we used the probe technique incombination with the TSD method. The method is based onthe Gross postulate of charge invariance [44]: the amountof charge which can be released by re-heating of an electretcontaining a ‘frozen-in’ charge is a constant which dependsonly on the state of the systems at the time when there-heating begins, not on the subsequent heating rate andtemperature.

4.2. The experimental technique

The electret formation and the TSD current measurementwere carried out for a PMMA sample placed betweentwo nickel electrodes using the standard electrical systempresented in figures 11(a) and (b). Before thethermoelectret formation, six cylindrical silver probes oflength lp = 1 cm and diameter 2rp = 0.2 mm wereinserted into a disc-shaped PMMA sample of thicknessl = 0.525 cm at various distancesx from the cathode(figure 12).

After this procedure the sample was polarized in theelectric field of Ep = 0.5 MV m−1 at temperatureTp = 353 K for the time tp = 4 h. The changes intemperature and field during the consecutive stages of theexperiment (polarization, storage and TSD measurements)are presented in figure 11(b).

The TSD current was monitored with a sensitiveelectrometer by re-heating the sample at a constant heatingrate ofb = 3 K min−1. The above conditions were appliedin order to obtain the TSD thermograms for the wholesample and for each of the six probes.

4.3. Results and discussions

Thermally stimulated discharge currents of PMMAthermoelectrets have been measured by a number of authors[5, 24, 25]. For samples charged above the glass transitiontemperature,Tg = 378 K, the TSC response consists ofa dipole-reorientation peak aroundTg (the α peak) and aspace-charge peak at a higher temperature (theρ peak).

(a)

(b)

Figure 11. (a) A block diagram of the electrical systemused for the electret formation and TSD currentmeasurement: S, sample; C, cathode; AA′, moving anode;Pi , probe; Th, thermostat; H, heater; Pt, temperaturesensor; TR, temperature recorder; TC, temperaturecontroller; TP, temperature programmer; HV, high-voltagesource; E, electrometer; CR, current recorder; and K andK′, keys. (b) The field E (t) and temperature T (t)programme for the electret formation and TSD currentmeasurement.

These peaks are opposite and the same in sign for blockingand open electrodes, respectively [5]. For samples chargedbelow Tg, only the dipole-reorientation peak is observed[24]. In our experiments we chose just these polingconditions(Tp < Tg) in order to obtain one peak in the TSCthermogram from the total sample. The TSC thermogramof the investigated total sample under these conditions ispresented in figure 13. The current densityj (T ) exhibits amaximum (theα peak) at the temperatureTM of 377 K. Thisresult is in agreement with the data from Vanderschueren’spaper [24].

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Dielectric and electret properties of PMMA

(a)

(b)

(c)

(d )

Figure 12. The arrangement of probes in the sample (a)and the way of their insertion (b)–(d ).

According to the well-known equations

σ =∫ ∞

0j (t) dt σ =

∫ Tf

T0

j [T (t)] dT (11)

whereT (t) = T0+bt , we can calculate the effective chargedensityσe from the TSC peak area.

The amount of this charge released during TSD fromthe total sample is equal to 16µC m−2 (the room andfinal temperatures,T0 andTf , are not shown in figure 13).We assume that this charge is independent of the heatingrate b and of the functionT (t) during the discharge, butis dependent on the state of the system which depends onsuch parameters asEp, Tp and tp. In contrast to the resultin figure 13, the TSC thermograms monitored by particularprobes are more complicated (figures 14(a)–(c)). First ofall, these thermograms show two peaks. Secondly, thepeaks are the same in sign for the probes located nearer thecathode side and opposite in sign for the remaining probes(figures 14(b) and (c)). Moreover, such parameters as thetemperature positionTM , heightiM , width 1T and area ofthese TSC peaks change depending on the relative distancexi/ l. By a graphical integration of the curvesi(T ) whichare presented in figures 14(a)–(c) we can obtain the dipolechargeQα′ , the real chargeQρ and the effective chargeQe

as functions of the relative distancex/l.The above data are presented in table 2. We note that,

for probes 3–6, the dipole and space-charge evaluation ispossible because two clearly separate peaks are found inthe TSC diagrams (figures 14(b) and (c)).

Figure 13. The current-density thermogram j (T ) of aPMMA thermoelectret 0.525 cm in thickness and 4.4 cm indiameter. The charging and discharging conditions wereEp = 0.5 MV m−1, Tp = 353 K, tp = 4 h, t0 = 0.25 h andb = 3 K min−1.

On the other hand, we can evaluate the dipole chargeas follows:

σα ∼= 2∫ TM

T0

i(T ) dT

for the first and second probes. The results of suchintegrations are presented in figure 15 (for the realcharge) and 16 (both for dipole and for effective charges).Generally, we can notice from the data in table 2 and alsoin figures 15 and 16 that the outer regions of the PMMAthermoelectret are charged more highly than is the middleregion. This fact agrees with the opinion of Van Turnhout[5], who, by sectioning technique measurements, obtainedclear indications of a space-charge distribution in PMMAelectrets. However, it will be seen in figure 15 that thereal charge at the cathode side is higher than that at theanode. Moreover, the thermograms of the outer regions(figures 14(a) and (c)) differ perceptibly, which impliesthat the numbers and mobilities of cations and anionsstored in them are not equal. It also confirms the factthat the maximum temperaturesTM of ρ peaks (table 2and figures 14(a) and (c)) are lower at the cathode side(404 K) and higher at the anode side (410 K) of the electret.Moreover, we can draw conclusions about the nature of thespace-charge transport on the basis of the signs of theα

and ρ peaks. The latter problem was considered by VanTurnhout [5]. Besides, measurements based on TSD forthe total sample gave only an average of what was takingplace in the sample under study. Thus measurements byprobes together with the TSD method would be one ofthe ways of understanding the physical processes involved.Afterwards we can develop our considerations in order todetermine the fieldE(x) and potentialV (x) distributionsin PMMA thermoelectrets.

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K Mazur

Table 2. The position TM , height iM and area of TSC Qi/Q1 at each probe xi/l .

Probe number

Quantity 1 2 3 4 5 6 Total sample

xi/l 0.17 0.27 0.35 0.53 0.73 0.83 1TM (K), α peak 373 373 371 377 373 373 377TM (K), ρ peak 402 402 400 408 410 410iM (pA), α peak 0.57 0.39 0.50 0.50 0.75 1.20 40iM (pA), ρ peak 0.34 0.15 0.09 0.08 0.14 0.25Qi/Q1, α peak 1 0.70 0.64 0.57 1.00 1.55Qi/Q1, ρ peak 1 0.25 0.17 0.22 0.26 0.65Qi/Q1, overall TSC 1 0.49 0.37 0.21 0.43 0.57

The following assumptions are made implicitly [43] formeasurements by means of the probe technique.

(i) The probe does not disturb the potential distribution.(ii) The probe assumes a potential near that of the

dielectric in contact with it. These two conditions, ingeneral, are not true.

In their study of electrical conduction in ionic crystals,de Mey and de Wilde [45] showed that the electric field atthe electrodes will beεs/ε∞ times greater than the meanelectric fieldEp in the sample(εs and ε∞ are the staticand high-frequency dielectric constants respectively). Theirapproach can be applied also to dipolar polymers in whichεs andε∞ differ [34].

The disturbance at the contact between the insertedprobes and the polymer can be roughly eliminated inour consideration by formal introduction of the ‘effective’surfaceSe. It can be proved that this fictitious surface isalsoεs/ε∞ times greater than the trueSp probe surface. Wecan describe the surface charge densityσe as a function ofthe x distance on this basis and from figure 16:

σe(x) = Qe(x)/Se. (12)

On the other hand, the effective charge density isnumerically equal to the electric displacementD.

Neglecting an instantaneously varying polarization[46], from the definition of electric displacement we canwrite

D(x) = ε0E(x)+ Ps(x) (13)

where Ps(x) is a slowly varying internal polarization(persistent polarization) andε0 is the permittivity ofvacuum. At not too high an electric field the Langevinfunction is linear; hence the persistant polarization can beexpressed as

Ps(x) = ε0(εs − ε∞)E(x). (14)

The combination of equations (12)–(14) gives the relationbetween the internal electric field and the effective chargeat the probes

E(x) = (ε0χSe)−1Qe(x) (15)

where

χ = 1+ (εs − ε∞) Se = (εs/ε∞)2πrplp.

From equation (15) and figure 16 we can calculate thevalues of E(x) and σ(x) for particular probes. Forexample, for the first and fourth probes we haveE(x1) =6.1 kV cm−1, E(x4) = 1.3 kV cm−1, σ(x1) = 2 nC cm−2

and σ(x4) = 0.4 nC cm−2. The calculations wereperformed usingχ = 3.7 andεs/ε∞ = 2 from literaturedata [38] and our experimental results (section 3). It wasto be expected that the values of the surface charge densitywould be higher in the outer and lower in the middle regionsthan those of the total sample, namely 1.6 nC cm−2.

By extrapolation ofQe(x) to x = l, we can roughlyevaluate the value ofE(l) at the anode side of the sample.This value is close to(εs/ε∞)Ep. It was also to beexpected, considering the short time between finishing thecharging and starting the TSD (t0 = 0.25 h).

Using (εs/ε∞)Ep as a boundary condition, byintegration of theQe(x) curve from figure 16, finallywe could obtain the potential distributionV (x) in thePMMA thermoelectret (figure 17). Under quasi-steady-state conditions, with one-dimensional planar geometryand temperaturesT below the glass transition temperatureTg of PMMA, this potential distribution can be writtenapproximately as [36]

V (x,∞) ∼= Vp − E(0,∞)x + [ρ0/(2ε0εs)]x2 (16)

for values ofx that are small compared with the thicknessl of the sample. In equation (16),x is the distance fromthe anode side of the sample,Vp is the applied voltage,E(0,∞) is the field atx = 0 after a long polarizing time(tp = ∞), εs is the static dielectric constant of PMMA,ε0 is the permittivity of free space andρ0 is the volumecharge density within some layer. For further discussion ofthis subject see section 6.

This curve (figure 17), the analytical expressiondescribed by the curve of equation (16) and that obtainedby Gupkin [11], who used the probe technique under anapplied electric field, are similar in shape. The exactconsideration of boundary conditions at different contactsbetween the electrodes and the dielectric material duringTSD is an open problem. Some remarks on this subjecthave been made by Jaffe [47] and Van Turnhout [5], forexample.

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Dielectric and electret properties of PMMA

(a)

(b)

(c)

Figure 14. Current thermograms i (T ) of the same sampleunder the same conditions (as in figure 13) measured byusing probes located nearer to the cathode. (a) The valuesof the relative distance (x/l) between the probes and thecathode side of the sample were 0.17 (1) and 0.27 (2).(b) The x/l relative distances were 0.35 (3) and 0.55 (4).(c) The x/l distances were 0.73 (5) and 0.83 (6).

Figure 15. The spatial distributions of the positive andnegative real charge Qρ in PMMA heteroelectrets.

Figure 16. The spatial distributions of the dipole (Qα) andeffective (Qe) charges in PMMA heteroelectrets (absolutevalues).

5. Isothermal dielectric absorption(DC measurements)

5.1. Introduction

It is known that the absorption current(i) which flows inorganic polymers and amorphous materials on applicationof an external electric field(E) depends on dipolarorientation in the direction of the field, free chargeaccumulation at structural or electrical inhomogeneities andthe space charge near the electrodes [48]. In many casesthe absorption current responds over a wide interval of time

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K Mazur

Figure 17. The potential distribution V (x) in the PMMAthermoelectret. The charging and discharging conditionswere Ep = 0.5 MV m−1, Tp = 353 K, tp = 4 h, t0 = 0.25 h,b = 3 K min−1 and Vp = Epl .

according to the von Schweidler law [49]

i ∝ t−k (17)

where 0< k < 1. The time domain and the frequency(ω)-domain responses are related, thus [50, 51]

t−k � ωk−1.

This law can be represented in terms of distributionactivation energies or natural frequencies [49, 52]. Mostauthors analysed this problem using the conception ofGaussian (non-dispersive) transport. However, the carriertransport in amorphous dielectrics is known to have a non-Gaussian (or dispersive) character.

The experimental manifestations of dispersive transienttransport in disordered solids have been discussed bymany authors [53–61] and compared with the predictionsof theoretical treatments. The general result of non-Gaussian transient transport of carriers in disordered solidsis summarized in the following equations [53–56]:

i(t) ∝ t−(1−α) for t < tT (18)

i(t) ∝ t−(1+β) for t > tT (19)

wheretT is interpreted as being the transit time of the carrierfront whereasα andβ are parameters describing the degreeof dispersion.

The field–time dependence of the absorption current orinjection current is often described by [12, 62–64]

i = A(T )[E(t)]nt−k (20)

where A(T ) is a temperature-dependent factor. Theexponentk is generally less than unity, but approachesthis value as the trapping becomes increasingly effective,

Figure 18. The general features of the current i versustime t characteristics shown as log–log plots for PMMA atTp = 355 K.

whereasn is expected to exceed unity [12]. However,under some conditionsn can be less than unity for PMMA[34, 36, 62, 64].

Generally, typical values ofk for a polymer are in therange 0.1–1, whereas those forn in the range 0.5–3. TheparametersA, k andn are interrelated.

5.2. Results and discussion

The electric current decay after the application of a stepvoltage in PMMA at Tp = 353 K is described in thissection. Figure 18 represents the general feature of thecurrent i versus timet characteristic in log− log plots.It can be seen from figure 18 that the logi versus logtrelationship can be expressed approximately by two straightlines with the break pointq:

i(t) = c1t−k1 for t < tq (21)

i(t) = c2t−k2 for t > tq (22)

where the timetq

(i) decreases with increasing polarizing field(Ep) inthe range 0–2 MV m−1 (region I),

(ii) is practically independent ofEp while Ep increasesfrom 2 to 6 MV m−1 (region II) and

(iii) increases sharply above 6 MV m−1 (region III).

The values ofk1, k2 and tq at differentEp are listed intable 3. We note that equations (21) and (22) are formallysimilar to equations (18) and (19).

The change in DC conductivity duringtq ≤ t may bedue to an accumulation of hetero-space charges near theelectrodes, at low electric field, but also to the neutralizationof these charges by the homocharges when the electric fieldis higher (E > 6 MV m−1). Apart from that, we shouldmention that the current decrease duringt > tq may berelated to the physical ageing processes of PMMA. Theinfluence of these processes on the electric properties of thepolymers has been investigated by many authors ([65–67],for example). From Van Turnhout’s paper [67] it is known

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Dielectric and electret properties of PMMA

Table 3. The dependences of k1, k2 and tq on Ep .

Ep (MV m−1) k1 k2 tq (min) Region

0.5 0.53 0.19 85 I1.0 0.44 0.16 25 I1.5 0.33 0.13 15 I2.0 0.29 0.12 10 I, II3.0 0.38 0.12 10 II4.0 0.44 0.11 10.5 II6.0 0.47 0.13 12 II, III8.0 0.41 0.05 15 III

10.0 0.24 0.006 23 III

that the decrease in free volume, which accompanies ageingof the polymer, lowers the mobility of charge carriers suchas ions, so that a decrease in DC conductivity can beexpected. Then the application of a step-voltage functionto the polymer results normally in a long-term absorptioncurrent which decays in part according to the equations (21)and (22), as has been pointed out by other workers [68].This would mean that a long wait might be necessary beforea steady-state condition is reached. According to VanTurnhoutet al [67] the timet∞ required for the attainmentof equilibrium atTp < Tg reaches a value of 100 years.It is difficult to check this conclusion experimentally. Itcan be seen in figure 18 and table 3 that the experimentalresults obtained for values oftq and the results calculatedfor the k2 exponents are practically constants in region IIof Ep changes; thus the current–voltage charcteristic isindependent of the ageing time. Therefore, we assume thatcurrent beyondtq is the so-called quasi-steady-state current,and thattq = t∞.

6. The current–voltage characteristic of PMMAthermoelectrets

Now we will show how the potential distribution in PMMAexpressed analytically by equation (16) can be appliedfor the determination of the current–voltage characteristic.Equation (16) is formally similar to the expression for thepotential distribution within a Schottky barrier [36, 69] ofthicknessλ:

V (x) = Vp − [ρ0/(2ε0ε∞)](2λx − x2). (23)

Camposet al [70], assuming the validity of total ionizationor the depleted model, indicated the possibility of thecreation of such a barrier during the electret formationprocess. On the other hand, de Mey and de Wilde [45]have shown that the electric field in ionic crystals at theelectrodes should beεs/ε∞ times greater than the meanfield of Ep = Vp/l; therefore we have

E(0,∞) = −dV/dx|x=0 = (εs/ε∞)Vp/l (24)

where ε∞ is the high-frequency dielectric constant. Thisrelation is also satisfied for dipolar polymers.

On the basis of experimental data shown in figures 4and 5 the ratioεs/ε∞ for PMMA is about 2. This slope ofV (x)/x is indeed observed at the anode side of the sample

Figure 19. The current–voltage characteristic (j − Vp) ofPMMA at the polarizing temperature Tp = 353 K.

(figure 17, straight broken line). The slope at the cathodeis greater thanεs/ε∞ and is caused by the high mobility ofthe positive ions. Atx � l the total current densityjn isroughly the sum of drift and diffusion current densities fornegative carriers:

jn = eDn{[−en/(kBT )] dV/dx + dn/dx} (25)

where n(x) is the carrier concentration and the othersymbols have their usual meanings.

Since the current is independent ofx, equation (25)can be integrated, using exp[−eV/(kBT )] as an integratingfactor [69]. On the basis of equations (23) and (25) andthe boundary condition from equation (24) we can derivethe following sub-ohmic current–voltage characteristic:

jn = gnV 1/2p (26)

in which gn = µnn(λ)[ρ0/ε0εs)]1/2. While solvingequation (25) Einstein’s relation(µn/Dn = e/(kBT ))

was taken into account and the terms containingexp[−eVp/(kBT )] were neglected.

The resulting current–voltage characteristic for PMMAat T < Tg shows the ohmic region (figure 19, region I)preceding the non-ohmic region within which the currentbecomes proportional toV np , where n < 1. In theexperiment we used samples 0.04 cm thick, of diameter4 cm, which were metallized on both sides by du Pontsilver paste.

At the temperatureT = 353 K, the sub-ohmicregion of PMMA, according to equation (26), exhibitedan approximately square-root dependence of the currentdensity on the mean fieldEp = Vp/l (figure 19,region II). At this temperature the factorgn = 2.9 ×10−11 A (V 1/2 m−2).

Roberts has shown [71] that evidence for such a sub-ohmic region may be found in an analytical solution tothe same problem in which trapping effects and diffusionare neglected and equal hole and electron mobilities areassumed. The diffusion transport model in the polymersis not fully reasonable due to the low carrier mobility.However, for large concentration gradients in the immediate

1393

K Mazur

vicinity of x = 0, which may be found particularly inheteroelectrets made from polar polymers, the neglect ofdiffusion is not a consistent approximation [5, 72]. We notethat the non-ohmic behaviour can have a particular meaningin thin polymer films, which has previously been analysedby Hilczer and Ma lecki [13, 73].

7. The relation between the surface chargedensity of PMMA thermoelectrets and thepolarizing voltage

7.1. Theoretical considerations

van Turnhout [5] considered the Maxwell–Wagner effect innon-polar laminates for estimation of the relation betweenthe interfacial charge magnitude and the applied voltage.Similarly, we consider a two-layer system in which one ofthe layers is a dipolar polymer, for example a PMMA sam-ple metallized on one side, with an air gap as the other layer.

Our considerations are limited to isothermal and steady-state conditions. Besides, we assume for the voltageV1(∞)across the air layer up toV1(∞) ≤ Vb that the currentdensityj1(∞) is given by Ohm’s law:

j1(∞) = γ1E1(∞) = (γ1/l1)V1(∞) (27)

whereγ1 is the electrical conductivity of the air,l1 is thethickness of the air gap,Vb is the breakdown voltage andE1(∞) is the electric field after a long polarizing time(tp = ∞).

On the basis of figure 9 (curve 2, region II) we assumein addition that the current densityj (∞) in the one-sidedmetallized PMMA film can be described by an equationsimilar to equation (26), namely

j (∞) = gV 1/2. (27a)

We note that Adamec [74] has proved the similarity of theelectrical conductivity characters of PMMA metallized ontwo sides and PMMA with contactless electrodes (at lowand intermediate electric fields).

By using Kirchhoff’s equation(V + V1− Vp = 0) andthe continuity equation(j − j1 = 0) we can make thevoltage (V1 and/orV ) across individual layers dependenton the applied voltage(Vp), namely

V1(∞) = 2c1/2[(c + Vp)1/2− c1/2] (28)

wherec = [gl1/(2γ1)]2. For the relation between the twoelectric fieldsE1(∞) andE(0,∞) and the surface chargedensityσ(∞) we have Gauss’s law (at the non-metallizedanode,x = 0)

σ(∞) = σr(0,∞)− Ps(0,∞)= ε0ε∞E(0,∞)− ε0ε1E1(∞) (29)

whereσr is the real charge density andPs is the persistentpolarization. By combining equations (28) and (29) withthe simplest boundary conditionE(0,∞) = (εs/ε∞)V/lwe obtain

σ(∞) = a{bVp − 2c1/2[(c + Vp)1/2− c1/2]} (30)

wherea = ε0(εs/ l + ε1/l1) andb = [1+ (lε1)/(l1εs)]−1.Thus, we assume from equation (30) that the interfacial

chargeσ(∞) changes nonlinearly with the applied voltageVp. It is this charge that can be frozen in.

Figure 20. The induction charge σi (circles) and theinterfacial charge σ∞ (dashed line) calculated fromequation (30) as a functions of Ep .

7.2. Experimental results and discussion

The PMMA thermoelectrets metallized on one side,l =0.04 cm in thickness and 2 cm in diameter were preparedusing the generally known method. The following chargingconditions were applied: t ′p = 3 h at Tp = 353 K,with the cooling timet ′′p = 0.25 h and variousEp =Vp/l. The surface charge density(σi) was measured byan electrostatic induction method, besides which the TSDcharge density was calculated from the TSC peak areaobtained at 3 K min−1 heating rate.

The relations amongσi, σ (∞) charges and thepolarizing field Ep are shown in figure 20. We notethat the constantsa, b and c of equation 30 weredetermined experimentally. The troublesome ratio ofl1/l

was determined by measuring the permittivities for samplesmetallized on one and two-sides. Taking into accountε1, εs, γ1, g, l1 and l, the relation betweenσ(∞) andVp/l(equation (30)) was plotted for the following constants:a = 2.4× 10−7 F m−2, b = 0.48 andc = 177 V.

It can be seen in figure 20 that a plot ofσ(∞) = f (Ep)represents the experimental relation betweenσi and Epwell up to 6 MV m−1. It is difficult to find any relationbetweenσi andEp above this value. One should noticethat, forEp = 6 MV m−1 the voltage across the air gapcalculated from equation (28) is equal toV1(∞) = 1 kV. Itis nearing the breakdown voltage for the air-gap thicknessof l1 = 0.01 cm [12].

It is useful to compare the data presented in figure 20with representative TSC thermograms (figure 21) andTSD charges calculated from TSC peak areas. The TSCthermogram atEp = 1.5 MV m−1 has two peaks withthe same sign (figure 21, curve 1). Whereas that TSCat Ep = 4 MV m−1 exhibits the two well-knownα andρ peaks of opposite signs (figure 21, curve 2). Theserelations change when the polarizing field strength changes.TSD charges are shown invidually as a function of the

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Dielectric and electret properties of PMMA

Figure 21. Representative TSC thermograms of PMMAthermoelectrets metallized on one side for(1) Ep = 1.5 MV m−1 and (2) Ep = 4 MV m−1.

Figure 22. The dependences on the TSD charge of theapplied field (Ep). σα and σρ concern the α and ρ TSCpeak areas.

applied field in figure 22 forα and ρ peaks (the brokenline indicates the effective chargeσeff ). It will be seenthat the curves ofσ(∞) = f (Ep) and σeff = f (Ep) areof similar character but the maximum(EM) and zero(E0)

positions are different. This fact may be connected withthe change in location of the free charge at differentEpvalues.

The shift of the maximum and zero positions may becaused also by the inequality between the local polarizationPs(0,∞) and the dipolar TSD chargeσα. Using thesectioning technique elaborated by Gross and de Moraes[75], Van Turnhout [5] showed that the outer sections ofa PMMA electret are charged more highly than is themiddle section. Then, the dipolar TSD chargeσα is amean magnitude over all of the sample volume and for

Figure 23. The mean real relative charge depth as afunction of the applied field (Ep).

a monorelaxation process it can be expressed as

σα = ε0χl−1∫ l

r

E(x) dx = ε0χV (∞)/ l (31)

whereχ is the dielectric susceptibility.The polarization Ps(0,∞) is a local magnitude;

with regard to the boundary conditionE(0,∞) =(εs/ε∞)V (∞)/ l it can be written as

Ps(0,∞) = ε0χE(0,∞) = ε0χ(εs/ε∞)V (∞)/ l. (32)

From equations (31) and (32) we have

σi = ±σr(0,∞)− Ps(0,∞) = ±σr(0,∞)− (εs/ε∞)σα.(33)

Such a separation ofσi betweenPs andσr encourages usto determine the mean real charge depth by the adoption ofthe formula of Sessler and West [76]

x/ l = (1+ |σr/σρ |)−1. (34)

This approach, although none too well substantiated, ispresented in figure 23.

Our assumptions thatγ1 is constant,j = gV 1/2 andE(0,∞) = (εs/ε∞)V (∞)/ l are only an idealization. Thisidealization has been effected in order to show that notonly the changes inγ1 but also the nonlinear behavioursof electret materials affect the form of the relation betweenthe induction charge density and the applied polarizing field(Ep). The real electric properties of both dielectric layersobviously determine the electret effect [77].

8. The charge decay in PMMA and PMMA/BaTiO 3

thermoelectrets

In phenomenological theories of electrets [12, 13, 73, 78–81] it is assumed that the dipole component of the

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K Mazur

Figure 24. The current–voltage j –Vp characterestics ofPMMA/BaTiO3 composites at the polarizing temperatureTp = 293 K. For PMMA (2) Tp = 353 K. The polarizing timewas tp = 3 h. The exponent n and the factor a are,respectively: n = 0.16 and a = 1 for PMMA (1), n = 0.51and a = 1 for PMMA (2), n = 0.79 and a = 1 for 10%BaTiO3 in the composite, n = 1.10 and a = 1 for 39%BaTiO3 in the composite, n = 1.95 and a = 100 for 50%BaTiO3 in the composite and n = 2.05 and a = 1000 forBaTiO3 ceramics.

heterochargeσf (fictitious charge) decays due to heatdisorientation according to the rule

σf = σof exp(−αt) (35)

where α = 1/τf is the reciprocal of the relaxation timeof the ‘frozen’ polarization. However, the ionic (real)component of the electret chargeσr changes with theconductivity according to Ohm’s law:

−(dσr/dt) = γEi = βσ (36)

whereEi is the internal field of the electret,σ = σf +σr isthe effective value of the surface charge on the electret andβ = 1/τM is the reciprocal of the space charge relaxationtime.

The assumption defined by equation (36) withreference to a composite dielectric in which therecoexist dipole polarization, space-charge polarization atinterfaces between different phases and charge injectedby the polarizing electrodes is a barely justifiedassumption. In general, the current–voltage characteristicsof PMMA/BaTiO3 composites (figure 24) can be expressedby

j = gV np (37)

where g is independent of the polarizing voltageVp;however, this factor and the exponentn are dependent onthe BaTiO3 content in the composites. These characteristics(figure 24) show the ohmic region preceeding thenon-ohmic region within which the current becomesproportional to the polarizing voltageV np , wheren < 1for a low volume fraction BaTiO3 (about 0.3) andn > 1for a higher one. Forn < 1, we have to deal withdiffusion charge transport [36]. On the other hand, in

Figure 25. The dependence of the electrical conductivity γon the BaTiO3 fraction in PMMA/BaTiO3 composites.

BaTiO3 ceramics we have to deal with space-charge-limitedcurrents (SCLC), which also are not linear in relation to thevoltage(n > 1) [10, 11, 62, 64, 82].

In connection with these facts, the charge decay ofelectrets made from the PMMA/BaTiO3 composite can beexpressed by

−(dσr/dt) = β ′σn (38)

whence

σ ∼= σ0

[1+ (n− 1)β ′σ (n−1)0 t ]1/(n−1)

− σof (1− e−αt ) (39)

whereβ ′ = γ /[(ε0ε)nEn−1

q ], γ is the electrical conductivityat a low polarizing field (figure 25),ε0 is the electricalpermittivity of free space,ε is the dielectric constant of thecomposite,Eq is the polarizing field at the break point onthe j -Vp plot (figure 24) andσ is the initial charge densityof the electret.

A comparision of this theoretical result with the experi-mental dependenceσ(t) for the composite PMMA/BaTiO3is shown in figure 26. It can be seen from figure 26 thatthese relations are in good agreement for the initial stageof the electret’s ‘life’.

9. Summary and conclusions

The problem of measurement of the internal spatialdistribution of immobilized charges and permanentlyordered dipoles in polymer electrets is always importantboth in fundamental and in applied investigations. On thebasis of our experiments and theoretical analysis it has beenshown that a combination of the probe technique and theTSD method gives good results in this context for thickPMMA thermoelectrets.

Using this method, the space charge and persistent po-larizations were measured in order to determine the poten-tial distribution within a PMMA thermoelectret. Knowl-edge of that is necessary in the charge-transport analysis.

It was stated that the outer regions are charged morehighly than the middle region of the investigated electrets.It was shown that the current–voltage characteristic of

1396

Dielectric and electret properties of PMMA

Figure 26. The dependence of the surface charge densityon time for PMMA/BaTiO3 composites: symbols areexperimental data; the broken lines are curves resultingfrom equation (39) at n = 0.2 for PMMA, n = 1.5 for 45%BaTiO3 and n = 1.95 for 50% BaTiO3. The conditions wereEp = 0.8 MV m−1, Tp = 353 K and tp = 3 h.

PMMA at T < Tg has a sub-ohmic shape (j = gV n, wheren < 1).

By using the above data, a two-layered condensorwas considered as the model of the formation of PMMAthermoelectrets. It follows from this model that theinterfacial charge densityσ(∞) changes nonlinearly withthe applied voltageVp. The relation betweenσ(∞) andVprepresents experimental results for PMMA thermoelectretswell up toVp/l = 6 MV m−1.

By using both the electrostatic and the TSD methodand utilizing the simplest boundary condition we canroughly separate the induction charge density into thedipolar and real charge densities. On this basis we canroughly determine the mean real charge depth in a dipolarpolymer such as PMMA by the adoption of the formulaof Sessler and West. In particular, we can state thatthe isothermal absorption currenti(t), the current–voltagecharacteristic and the charge decay(−dσ/dt) after polingof the PMMA/BaTiO3 composite are interrelated.

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