Dielectrical relaxation dynamics and thermally stimulated depolarization current in polymers

Transcription

1 JOURNAL OF OPTOELECTRONICS AND ADVANCED MATERIALS Vol. 9, No. 6, June 2007, p Delecrcal relaxaon dynamcs and hermally smulaed depolarzaon curren n polymers S. MINÁRIK *, V. LABAŠ, M. BERKA Deparmen of Nonmeallc Maerals, Faculy of Maerals Scences and Technology n Trnava, Slovak Unversy of Technology n Braslava The dynamcs of delecrcal relaxaon of polymer maerals has been suded. Concep for depolarzaon process smulaon based on he prmve charge ranspor model was proposed for he reason of relaxaon mechansms nvesgaon n polymer srucure. Depolarzaon curren n polymer exposed by homogeneous elecrc feld suddenly swched-ou n a ceran nsan of me was calculaed on he bass of he model and he model parameers was dscussed o acheve conssen resuls wh delecrc relaxaon expermens. The am of our work s o recognze f he delecrcal relaxaon and depolarzaon are conneced wh he same process of spaal rearrangemen of fxed charge carrers of delecrc sysem. In hs paper he nvesgaon of polymer relaxaon dynamcs by smulaons based on he proposed prmve charge carrers ranspor model s suggesed and eorecal prncples of hermally smulaed depolarzaon currens (TSDC) are analysed n erm of he model. Expermenal resuls of TSDC measuremens n polymehyl-meacrylae (PMMA) and polyehyleneerephhalae (PET) are presened and exploaon of he measured daa n polymerc maerals research s dscussed. (Receved November 14, 2006; acceped Aprl 26, 2007) Keywords: Depolarsaon, Delecrcal relaxaon, Relaxaon me, TSDC, PMMA 1. Inroducon Delecrcal relaxaon s a process of esablshng equlbrum sae of he fxed charges n delecrc srucure. The me scale of hs process s deermned by delecrc relaxaon me. The delecrc relaxaon me s he characersc me for charge flucuaons o decay under he nfluence of he feld ha hey produce. The fxed charges sysem ha s examned a me much longer han he relaxaon me s seen o be n equlbrum. On he conrary, when he observaon me s shorer hen relaxaon me we see a snap sho of frozen n nonequlbrum sysem. The delecrc relaxaon me s relaed o he elecrcal conducvy of delecrc maerals, n he case of hgh charge carrers mobly he relaxaon me can be very small. We recognzed wheher s possble o hnk of depolarzaon as a relaxaon process of delecrc srucure. If so hen he dynamcs of boh depolarzaon and delecrcal relaxaon process can be conrolled by he same relaxaon me and daa correlaon obaned from he measuremens of boh TSDC and elecrcal conducvy can be nvesgaed. The man goal of he presen work s o beer undersand he delecrcal relaxaon of polymers and consequenly o offer he alernave bass for heorecal analyss of TSDC expermenal echnque. Many polymers buld up of connuous nework. Several relaxaon mechansms of he polymer chan have been proposed, such as repaon [1], ube lengh flucuaon [1], and hermal [2] and convecve [3] consran release [3]. We have proposed he smple model of delecrcal relaxaon mechansm based on fxed charges ranspor approxmaon. We assume ha he classcal descrpon of drf moon of he large charge clusers bounded n he polymer chans s possble and proposed model s usable for polymer maerals. The spaal rearrangemen of he charge carrers coupled n he polymer srucure was descrbed on he bass of he model and problems of relaxaon me and hermally smulaed depolarzaon curren were dscussed. 2. Theory 2.1 Depolarzaon process and hermally smulaed depolarzaon curren One way, n he me doman, o nvesgae he depolarzaon process for a delecrc maeral s o measure depolarzaon curren. Depolarzaon curren densy dep s deermned by he me-change of he amoun of he maeral polarzaon P: dp dep =. (1) d If we assume Debay model of relaxaon wh a unque relaxaon mechansm he equaon for he decay of polarzaon s: dp d P =, (2) where s delecrc relaxaon me. The soluon of he eq. (2) deermnes he me dependence of he delecrcal srucure polarzaon P()

2 Delecrcal relaxaon dynamcs and hermally smulaed depolarzaon curren n polymers 1593 ha can be wren n he form: P () P e = 0, (3) where P 0 s he nal amoun of he sysem polarzaon. As can be seen from eq.(3) he delecrc relaxaon me s a amoun of me would ake for an polarzaon of 1 sysem o decay by e n he absence of addonal forcng. The depolarzaon curren densy can be found easyly fnd by subsuon of soluon (3) o he eq.(1): dep P = 0 e. (4) The soluon (4) could be accepable n smple cases of delecrcal relaxaon wh a sngle relaxaon me f s me ndependen. Menoned condon = cons s usually sasfed a consan emperaure of maeral T. If he emperaure of delecrc maeral changes he emperaure dependence of relaxaon me mus be consdered n equaon (2) and dfferen soluon can be obaned. Consequenly we can easy elmnae me from menoned soluon f we consder: d d = β d, (5) dt where: β = dt (6) d and depolarzaon curren densy s jus emperaure dependen n ha case. If relaxaon me obeys Arrhenusype equaon he depolarzaon curren densy can be wren as [4]: ( ) Ea = Ea P0 kt 1 k T dep T e exp e (7) 0 β 0 ) Menoned expresson (7) descrbes hermally smulaed depolarzaon curren. However, he fxed charge carrers drf mobly n maeral srucure has no been fully elucdaed n he frame of menoned phenomenologcal approach. A good descrpon of depolarzaon process and analyss of TSDC echnque requres some undersandng of charges dynamcs n maeral srucure. 2.2 Process of delecrcal relaxaon Le s dscuss he delecrcal relaxaon process. To see he mporance of respecng he lmaons relaed o he delecrc relaxaon me, magne o have a polarzed delecrc sysem under he nfluence of consan homogeneous elecrc feld. Every charge carrer coupled n he polarzed maeral srucure s exposed o he elecrc feld wh srengh E 0. Consder he elecrc feld orened along y drecon. Ths carrer s defleced from equlbrum by elecrc force and we assume ha carrer`s equlbrum s y = 0. Le he elecrc feld s swched-ou n = 0 (see Fg.1a). The charge carrers are reurned o equlbrum and relaxaon s n progress nsde he sysem. There can be consdered an equlbrum resorng couner force ha s proporonal o y for small devaons. Suppose, for he sake of smplcy, ha frcon s proporonal o he carrers velocy v(). Therefore, he followng equaon has o be solved for he me evoluon of charge carrer devaon y() deermnng: 2 d y dy 2 + 2b + ε y = 0, (8) 2 d d kb k where b =, ε =. (9) 2m m m s mass of he charge carrer, k s he sprng consan of lnear oscllaor, k b s he frcon coeffcen. The sngle carrer velocy s defned as: d y () () v = (10) d and followng formula can be wren for he relaxaon elecrcal curren densy rel by means of he smples classcal model of conducvy: ( ) rel = n qv. (11) n s charge carrers number per un volume and q s elecrcal charge of he carrer. Usng he smple boundary condons: ky ( = 0 ) = qe, ( = 0 ) = 0 rel, (12) we can easy fnd followng relaon for he curren densy rel : where: rel 2b α b () σ E e snh( α ) = 0 0 nq 2 0 = k b, (13) 2 k σ, α = b. (14) m As can be easyly found he me dependency of rel deermned by (13) shows a maxmum n me m ha can be wren by: 1 b + α m = ln 2α b α (15) Relaxaon me s he mos mporan parameer for descrpon of he delecrc relaxaon process. As can be shown, he molecular relaxaon me depends upon macroscopc vscosy of maerals. Menoned fac mples

3 1594 S. Mnárk, V. Labaš, M. Berka ha delecrc relaxaon nvolves manly molecular drf moon consdered above and no jus roaon of sngle molecules. Ths argumen [5] seemed o be confrmed expermenally by Schallamach [6]. Bu s clear ha he curren densy rel deermned by eq.(13) dsagrees wh soluon (4). Fg. 1b shows he me evoluon of boh he depolarzaon curren densy dep and he calculaed curren densy rel flowng durng he process of delecrcal relaxaon. Whereas he depolarzaon curren densy dep should have o show exponenal decrease (see eq.4), rel shows maxmum n = m. From menoned resuls s no evden f he delecrcal relaxaon and depolarzaon are really conneced wh same process of spaal rearrangemen of charges fxed n maeral srucure. We sugges ha he devaon rel from he exponenal decay s a consequence of he fac ha charges move slow compared o changes n he elecrc feld. We shall now examne he possble muual relaon beween dep and rel calculaed above. Toal amoun of he maeral polarzaon P s deermned by: () nqy() P =. (16) Takng accoun of (2), (8) and (16) he followng formula for can be derved: m = { αcgh( α ) + b}. (17) k The charge carrers drf moon velocy v() canno change abruply as a consequence of carrers nera. Hence we dd no fnd unform me ndependen value for n (17). To search a possble concdence of resuls (4) and (13) le us deermne he values (17) for he me >> m. As can be easy seen from (17) n he case f he relaxaon me comes no he form: 1 lm = b α (18) The consan value (18) represens relaxaon me ha characerze he delecrc relaxaon process n he me >> m. Apparenly he relaon (13) akes he form (4) for. I s evden drecly from fg.1b ha he depolarzaon process and process of delecrcal relaxaon can be conrolled by he same relaxaon me n >> m. Tha means he rel () and dep () wll be dencal for max 0 and delecrc relaxaon me can be esmaed he model parameers α αnd b n ha case. E() () P 0 E 0 dep max rel Fg. 1. Delecrc relaxaon me depends upon emperaure. Generally, he relaxaon me for non-cooperave moon follows an Arhenus law behavour: E a kt ( T ) e = 0 a) b), (19) where he apparen acvaon energy, E a, could be n he range of real energy barrer, k B s he Bolzmann consan, T he absolue emperaure and 0 he pre-exponenal facor. If he relaxaon s deermned by "free volume effec" [7] he relaxaon me follows he Vogel equaon: C( T T ) ( T ) = e 1, (20) where s preexponenal facor, C s average hermal expanson coeffcen of free volume, T s crcal emperaure a whch he relaxaon me becomes nfne. Ths equaon s equvalen, afer parameers ransformaon o WLF (Wllams-Landel-Ferry) equaon [8]. On he oher hand, movemens of polymer chan segmens can nvolve cooperave moons. The probably of success for cooperave moons s P Z, where: 1 P (21) s he probably of sngle elemenary movemen. The Z exponen can be consdered as he number of elemenary movemens [9,10]. If he relaxaon canno be descrbed by a sngle Debye relaxaon, he dsrbuon of relaxaon mes s

4 Delecrcal relaxaon dynamcs and hermally smulaed depolarzaon curren n polymers 1595 needed n relaon (2) and depolarzaon curren densy can be dfferen from (4). Generally, he process of drf moon of charges coupled n he polymer chan can be consderably complcaed. However, consderng he superposon prncple he oal amoun of curren densy rel can be calculaed by: rel Z b = σ 0 1 E0 (22) = 1 α 2α ( b α ) () ( e ) e where σ 0, b and α are model parameers menoned above for -h ype of charge carrers fxed n he polymerc chans. Index aler from one up o Z, where Z s equal o he oal number of dsngushable ypes of charge carrers conjugaed n he polymer srucure. The emperaure dependence of curren densy rel n (22) s deermned by emperaure dependences of model parameers σ 0 (T), b (T) and α (T). Generally, can lead o non Arhenus-ype of relaxaon me. 3. Expermenal TSDC mehod s based on he measuremen of elecrcal curren overflowng across he sample n consequence of depolarzaon process resulng n he sample durng he sample conrolled heang. Frs of all he sample s exposed by srong elecrcal feld and charge carrers fxed n maeral srucure are defleced from equlbrum by elecrc force. Orened dpoles are creaed n he sample volume for ha reason. Durng conssen sample coolng here s a ncreasng of relaxaon me wha s relaed wh preservaon of creaed dpoles orenaon afer elecrcal feld dsconnecon. Relaxaon me s shorened durng he ncreasng of he sample emperaure and depolarzaon curren can be observed n a consequence of ncreased drf mobly of charge carrers. I [A] 6,00E-012 4,00E-012 2,00E-012 β, α T [ o C] Fg.2. TSDC specra of PMMA sample a hgher emperaures. Polarzaon parameers: T P = 120 o C, P = 3600 s, E 0 = 500 V.mm -1. Heang rae: ρ v o = 3 o C.mn -1. TSDC measuremens have been performed n samples from polymehyl-meacrylae (PMMA) and polyehyleneerephhalae (PET). Elecrodes based on he argenum were deposed on he surfaces of fla shape samples wh dmensons 10x10-1mm. The polarzaon emperaure was 120 o C n he case of PMMA and 140 o C n he case of PET and elecrcal feld srengh E 0 was 500 V.mm -1 (PMMA) and 1000 V.mm -1 (PET). Invesgaed samples were polarzed durng he me perod P from 3600 sec. up o 7200 sec. and lnear heang rae vared beween 2 o C.mn -1 and 10 o C.mn -1. In all cases he samples were cooled unl he emperaure -195 o C. The resuls of expermenal measuremens are shown n Fg.2 and Fg. 3. I [A] 2,50E-011 2,00E-011 1,50E-011 1,00E-011 5,00E-012 0,00E o C.mn -1 5 o C.mn -1 2 o C.mn T [ o C] Fg. 3. TSDC specra of PET sample wh varous heang raes. Polarzaon parameers: T P = 80 o C, P = 120 s, E 0 = 1000 V.mm Dscusson There s possble o dsngush a few of sharper peaks n behavour of TSDC measured on PMMA sample (see Fg.2). Three sharp maxma are evdenced. The menoned maxmums can be arbued o β` (78 o C), α (111 o C) and ρ (130 o C) relaxaons. β` relaxaon s conneced wh dpoles orenaon, α relaxaon s a combnaon of polar mechansm and glass ranson mechansm and a las ρ s space charge relaxaon [11,12]. TSDC resuls measured on PET sample show wo good dsngushable relaxaon maxmums ha can be consdered a α and β relaxaons. Dependng upon he measuremen condons he β relaxaon maxmum occurs a he emperaure nerval from -137 o C up o - 90 o C. Sharper α relaxaon shows a maxmum n he emperaure nerval from 77 o C up o 82 o C (see Fg. 3). α and β maxmums spacng changes by he measuremen condons varaon. TSDC daa measured a heang rae 2 o C.mn -1 and 10 o C.mn -1 are almos dencal. Decreasng of menoned α and β peaks spacng was observed a he ncreasng of heang rae up o 10 o C.mn -1. Relaxaon maxmums hegh ncrease wh he heang rae ncreasng. I s clear from performed heorecal analyss he me

5 1596 S. Mnárk, V. Labaš, M. Berka dependence of he curren densy caused by he delecrcal relaxaon process s deermned by (13). As can be easy by shown he decrease of depolarzaon curren densy does no occur n exreme case: α b. (23) If we consder (23) n equaon (18) we ge:. (24) We pesume ha condon (23) s dencal o even: T T (25) n equaon (20). We expec ha emperaure dependences of parameers α(t) and b(t) has o be such ha condon (23) occurs a emperaure deermned by (25). Consequenly as resuls from (13) he me dependence of depolarzaon curren densy wll ake he followng form: 2α ( ) σ E 1 e (26) dep 0 0 n he case deermned by (23). Depolarzaon curren should have permanenly flow across he sample n ha case. If we neglec he ransen phase of he process.e. a me we ge: dep σ 0 E 0, (27) where E 0 s srengh of appled elecrc feld and σ 0 s elecrcal conducvy of maeral ha s emperaure dependen: σ 0 = σ 0( T ). Aferwards he depolarzaon curren deermned by (27) s hermally smulaed (TSDC). If he approach presened n heorecal par s correc he TSDC phenomena n polymers s possble elucdae by means of delecrcal relaxaon dynamcs. In ha case s possble o look for a correlaon beween emperaure dependence of elecrcal conducvy σ 0 (T) and TSDC specra [13,14]. 5. Conclusons Amorphous polymers such as PMMA and PET exhb delecrcal and mechancal relaxaon processes denfed as α and β relaxaons. α relaxaon s mechancal or delecrcal response of "dynamcal glass ranson". β relaxaon s a response of macromolecular chan movemen [12]. Menoned relaxaons are dsngushable very well on measured TSDC daa. Menoned measuremen enables o evaluae quanavely and qualavely he occurrence of boh of menoned relaxaon n PMMA and PET. ρ relaxaon was observed oo n he case of PMMA sample. On he bass of he model presened n he conrbuon can be expeced ha f he model parameers dfference b(t) α(t) approachs o zero value a emperaures T T he me dependence of depolarzaon curren deermned by equaon (13) wll be aenuaed. Temperaure dependence of depolarzaon curren dep (T) wll be domnang n ha case and he depolarzaon curren can allowed o be hermally smulaed. The resuls show ha correlaon beween emperaure dependence of elecrcal conducvy and TSDC specra can be expeced. Acknowledgmens Ths work has been suppored by Slovak Mnsry of Educaon under Grans No. 1/20077/05, No. 1/1080/04 and projec APVT No. 20/011/307 References [1] M. Do, S. F. Edwards, The Theory of Polymer Dynamcs, (1986), Clarendon, Oxford. [2] W.. W. Gaessley, Adv Polym Sc 47, 67 (1982). [3] G. J. Marrucc, Non-Newonan Flud Mech 62, 279 (1996). [4] P. Bräunlch (ed.): Thermally Smulaed Relaxaon n Solds, Topc n Appled Physcs, Vol. 37(Sprnger, Berln, Hedelberg, New York 1979). [5] E. Bauer, M. Maga, J. Phys. Radum 9, 319 (1938). [6] A. Schallamach, Trans. Faraday Soc. 42A, 180 (1946). [7] C. Lacabanne, D. Chaan, J. Phys. Chem. 79, 283 (1975). [8] M. L. Wllams, R. F. Landel, J. D. Ferry, J. Amer. Chem. Soc. 77, 3701 (1955). [9] S. Masuoka, Journal of Research of he Naonal Insue of Sandards and Technology 102(2), 213 (1997). [10] J. Raul, Journal of Non-Crysallne Solds 271, 177 (2000). [11] M. Mudarra, A. Joumha, J. Belana, A. Tourelle: Sudy of poly(mehyl mehacrylae) relaxaons by hermally smulaed depolarzaon currens and he hermal sep mehod. In Polymer, No. 40, 1999, p [12] M. Mudarra, J. Belana: Sudy of poly(mehyl mehacrylae) space charge relaxaon by TSDC. In Polymer, No. 38, 1997, p [13] J. Kalužný, D. Ležal, M. Kublha, J. Pedlíková, E. Maran: Elecrcal and delecrc properes of TeO 2 - ZnO glasses. Ceramcs 46(4), 140 (2002). [14] J. Kalužný, D. Ležal, T. Kozík, M. Kublha, E. Maran: Ceramcs 43(3), 107 (1999).

6 Delecrcal relaxaon dynamcs and hermally smulaed depolarzaon curren n polymers 1597 * Correspondng auhor: mnark@mf.suba.sk