DIELECTRICS MICROSCOPIC VIEW

Transcription

1 HYS22 M_ DILCTRICS MICROSCOIC VIW DILCTRIC MATRIALS Th tm dilctic coms fom th Gk dia lctic, wh dia mans though, thus dilctic matials a thos in which a stady lctic fild can st up without causing an appcial cunt. Matt is usually nutal with an qual nums of ngativ and positiv chags. In dilctic matials, ths chags a not f to mov fa und th influnc of an applid xtnal lctic fild, as a conduction lctons in a mtal conducto. Howv, th focs du to an xtnal fild do caus small lativ displacmnts (on an atomic scal) of th chags of ach sign. Th xtnt of such displacmnts dpnds upon th tightnss with which th chags a hld fixd. Also, pola molculs otat in th xtnal lctic fild ty to align thmslvs with th xtnal lctic fild. This displacmnt of th chags and otation of molculs sulting fom an applid xtnal lctic fild is calld polaization of th matial. Th dilctic constant is a masu of th xtnt of th polaization. Th paamt that dictly lats th polaization of th matial to th lctic fild is calld th lctic suscptiility wh W can stat with a vy cud modl to xplain th haviou of dilctic matials. W assum a continuum of two unifom chag distiutions of opposit signs. In th asnc of an applid lctic fild, th positiv and ngativ chag distiutions a xactly supimposd. Whn an xtnal lctic fild is applid, th positiv distiution is displacd in th diction of th xtnal lctic fild and th ngativ distiution is displacd in th opposit diction. This sults in a cancllation of th chags in th intio of th dilctic matial, ound chag dnsity [C.m ] and sults in inducd ound (polaizd) sufac chags q and q at th nd sufacs of th dilctic matial. Th ound sufac chag dnsity [C.m 2 ] is dq da In this cas, th ound sufac chag dnsity dtmins th polaization nˆ wh ˆn is a unit nomal vcto pointing away fom th dilctic. On th atomic lvl, th is anoth possil point of viw. In th dilctic matial ach atom o molcul is distotd to poduc an lctic dipol and a dilctic matial can though of as consisting of lag num of lctic dipols. W can attiut th ffct of polaization of th matial to th sum of all th filds of all th dipols. Spaation of chag lctic dipol polaization mp_.doc Oct.

2 A dipol consists of two qual and opposit chags q and q spaatd y a vcto distanc d d lctic dipol momnt p = p = q d p q d q q points fom ngativ to positiv p Inducd dipol momnt hlium atom 2 2 Zo lctic fild hlium atom symmtic zo dipol momnt A d ffctivly chag 2 at A and 2 at B B dipol momnt p = 2 d p mp_.doc Oct.2

3 otntial and lctic fild fom an lctic dipol q q q 2 V( ) 4 4 d 2 2 qdcos V( ) 2 2 d 2 4 cos 4 q d cos p cos p xtnds fom th cnt of th dipol to th point Th adial and tangntial componnts of th fild at point a V 2 pcos 4 V psin 4 ˆ ˆ (d/2)cos 2 (d/2)cos q q d (d/2)cos along th axis of th dipol = = along th ight iscto of th dipol = /2 = lctic fild appoachs zo much mo quickly than a point chag.??? Why? LCTRIC DIOL LOT MATLAB Why is difficult to plot th potntial in a plan passing though th axis of th dipol? otntial: lctic Dipol mp_.doc Oct.

4 % lctic_dipol.m % Ian Coop School of hysics,univsity of Sydny clos all cla all clc % mconstants c =.8; % spd of light =.629; % lmntay chag ps = 8.852; % pmittivity of f spac NA = 6.22; % Avogado constant m = 9.; % lcton st mass mp =.6727; % poton st mass mn =.67527; % nuton st mass h = ; % lanck's constant kb =.82; % Boltzmann's constant kc = ; % Coulom constant mu = 4*pi*7; % pmaility of f spac amu =.6627; % atomic mass unit % Stup q = ; % dipol chag d =.67958; % dipol spaation distanc q = q; q2 = q; % spaatd chags kc = /(4*pi*ps); % constant in Coulom's Law x = d/2; x2 = d/2; % position of dipol y = ; y2 = ; scal =.25; % plotting gion xmax = scal * d; ymax = xmax; xmin = xmax; ymin = ymax; % plan aov diopl num = ; x = linspac(xmin,xmax,num); y = x; [xx yy] = mshgid(x,y); = sqt((xxx).^2 (yyy).^2); 2 = sqt((xxx2).^2 (yyy2).^2); % distanc fom chags % to tst point to calc. potntial V = kc.* q./ (); % potntial fom ach chag V2 = kc.* q2./ (2); Vtot = V V2; Vmax = max(max(vtot)); sat =.5; % satuat th potntial Vtot(Vtot > sat*vmax) = sat * Vmax; % potntial na a chag % is xtmly lag Vtot(Vtot <.5*Vmax) = sat * Vmax; Vtot = Vtot/(max(max(Vtot))); figu(2); % [D] plot suf(xx/d,yy/d,vtot,'faccolo','intp',... 'dgcolo','non',... 'FacLighting','phong') daspct([ ]) axis tight; viw(45,2) camlight lft; colomap(jt) gid off; axis off coloa titl('otntial: lctic Dipol') mp_.doc Oct.4

5 OLARIZATION Th quantity of al intst is not an individual dipol momnt ut th lctic dipol momnt p unit volum. In a gion of unifom polaization, th polaization is thn n p wh p is th inducd atomic dipol momnt and n is th num of lctic dipols p unit volum. Th wod polaization has two manings: a qualitativ on fing to any lativ displacmnts of positiv and ngativ chag and th quantitativ on, giving th sulting vcto dipol momnt p unit volum,. Th lins of connct ound chags (ngativ to positiv). Th polaization dscis th xtnt to which pmannt o inducd dipols com alignd. Th polaization givs is to a sufac ound chag dnsity ound and a volum ound chag dnsity ound. Consid th polaization of th dilctic twn th plats of a chagd paalll plat capacito. Thoughout th ody of th dilctic, th chags on adjacnt nds of th pola molculs nutaliz on anoth. At oth th top and ottom of th dilctic th chags do not nutaliz ach oth ound sufac chags. Fo a cylind of th dilctic of cosssctional aa da xtnding fom on plat to th oth lctic dipol momnt p = q d dp ( da) d dp dp ( da) d d dad dad nˆ wh ˆn is th nomal outwad pointing unit vcto. f f d da Thus, th polaization quals th magnitud of th ound (inducd) chag p unit aa on th sufac of th dilctic matial. Also, th polaization can otaind though th lationship (no poof) wh is th volum dnsity of th ound chags. mp_.doc Oct.5

6 Homognous dilctic unif omly polaizd ˆn xtnal dp dilctic nˆ ˆn cos Dilctic is nutal q q Th lctical f ild is ducd in th dilctic matial How is th macoscopic masual quantity, th dilctic constant latd to quantitis at an atomic lvl? Conductos Contain chags that a f to mov and in th psnc of an lctic fild, distiut thmslvs on th sufac of th conducto so that th lctic fild is zo in th intio. Dilctics Inducd dipols (lctonic) in an xtnal lctic fild, positiv chag (nuclus) and ngativ chags (lcton cloud) pushd in opposit dictions inducd lctic dipol momnt. lctic Fild ola molculs g H 2 O, N 2 O nutal ut a lopsidd chag distiution on sid xcss positiv and th oth xcss ngativ chag pmannt lctic dipol momnt zo lctic fild, andom ointation of molculs in gass and liquids nt lctic dipol momnt is zo. In an xtnal lctic fild dipols xpinc a toqu to ointat thm with th lctic fild thmal agitation of th molculs opposs th alignmnt align is not pfct. Ionic contiution to lctic dipol momnt in a molcul som of th atoms hav an xcss positiv o ngativ chag sulting fom th ionic natu of th ond: in an lctic fild, th ions and ions a shiftd in opposit dictions. Nt lctic dipol momnt = (inducd dipols lctonic pmannt dipols ointation ionic dipols ionic) p p po pi p p mp_.doc Oct.6

7 Nt polaization = (lctonic polaization ointation polaization ionic polaization) n p o i Th nt polaization is latd to a sufac chag dnsity ound and th dnsity of ound chags ound ˆ ound n ˆn is th nomal pointing out of th volum ound z x y z x y th dimnsional vaiation of th polaization If th polaization dpnds on tim, thn w may xpct that th ffct is simila to that of a cunt polaization cunt dnsity J t and nds to addd to possil cunts associatd with f chags. Consid a mdium whn an applid lctic fild is tund on. As a consqunc th atoms o molculs fom small dipols wh non xistd fo th alignmnt of th molculs constituts a cunt. DOLARIZATION FACTOR Whn a dilctic matial is placd in an lctic fild, th inducd polaization chag always acts to dcas th avag lctic fild within th dilctic fom its valu it had fo th dilctic was instd. In gnal, th polaizd chag poducs a nonunifom lctic fild, so th oiginal lctic fild is modifid diffntly at diffnt gions in th dilctic matial. Howv, w will only consid th application of a unifom xtnal lctic fild xt and dilctics in which th polaization is also unifom. Th inducd fild in th dilctic is calld th dpolaizing lctic fild dp. Dilctics which hav an llipsoidal shap satisfy this citia (unifom polaization within th dilctic whn an xtnal lctic fild is applid). In cass wh th shap appoximats an llipsoidal shap, w can than f to th avag lctic fild within th dilctic dilctic xt dp dilctic Th dpolaizing lctic fild dpnds upon a gomtic facto L and th magnitud of th polaization dp L Th gomtical facto L is calld th dpolaizing facto and can tak valus fom to, dpnding on th shap of th dilctic. mp_.doc Oct.7

8 Flat plat with its plan ppndicula to th xtnal lctic fild (~ flat oad llipsoid): simila to a paalll plat capacito, th avag lctic fild in th dilctic is ducd y th facto. Th avag lctic fild within th flat plat dilctic is ( ) plat xt dp xt plat dp plat xt plat plat L plat dp plat xt This flat shap givs th maximum valu fo th dpolaizing lctic fild fo a givn polaization of th matial. Long thin od with its axis paalll to th xtnal lctic fild (~ long thin llipsoid): if th od is long and thin nough, th inducd chags at th nds a small thn th dpolaizing lctic fild is ssntially zo dp and th dpolaizing facto has its smallst valu, L. Th avag lctic fild in th od (ignoing th nds) is od xt Sph: th valu fo th dpolaizing facto fo a sph is L = /. Th sufac chag dnsity is givn y cos wh is masud with spct to th diction of th lctic fild and th polaization. Th dpolaizing lctic fild dp and th avag lctic fild within in th sph sph a dp sph xt dp Flat plat L = Max polaization dp plat xt xt dp Thin long od L = Zo polaization dp od xt dp Sph L = / Concntation of chags At suf ac givn y cos dp sph xt Th xtnal lctic fild can ais caus of a distiution of f chags. Histoically, a nw vcto was intoducd, th lctic displacmnt D such that D = f wh f is th sufac chag dnsity that givs th xtnal lctical fild. Hnc, dilctic D mp_.doc Oct.8

9 RSONS OF A MOLCUL TO AN LCTRIC FILD Th lctic suscptiility tlls us aout th polaizaility of th atoms in matt. Fom a macoscopic viw (only consid cass in which th lctic fild and polaization a unifom), th polaization dpnds upon th lctic fild within th dilctic dilctic dilctic dilctic Howv, th macoscopic o avag lctic fild dilctic is not a satisfactoy masu of th local lctic fild loc poducing th polaization of ach atom. W can assum th lctic dipol momnt p and polaization a popotional to th local lctic fild loc xpincd y th molcul. Taking into account th th contiutions lading to th polaization of atoms o molculs lctic dipol momnt p ( ) o i loc loc olaization n p n( ) n o i atomic polaizaility lctonic o molcula polaizaility ointation polaizaility ionic polaizaility o i loc loc W can now lat th macoscopic quantitis th lctic suscptiility and th dilctic constant to a popty of th molculs, calld th atomic polaizaility. Th atomic polaizaility lats to th as in which lctic dipols momnts can fomd giving is to th polaization of th matial and hnc to th dilctic constant of th matial. 7 What is th local (inn) fild loc that acts upon an individual molcul within th dilctic? loc th local (inn) fild lctic fild acting upon a molcul within a dilctic Dilctic not continuous composd of molculs S Consid dilctic twn th plats of a paalll plat capacito, wh th avag lctic fild with in th dilctic is = dilctic f Th local lctic fild at th point O consists of 4 pats: O f mp_.doc Oct.9

10 Fild at O du only to th chagd plats f 2 olaization of th chags on th sufac of th dilctic 2 olaization of chags on th sufac of S which would fomd if th sphical sction of th dilctic was movd s following poof 4 olaization fom th pola molculs within th sphical sction, 4 4 K4 K 4 som constant, usually, 4 can t calculatd xactly. Hnc, th local lctic fild loc at O is loc loc loc 2 4 K f 4 D K 4 loc K 4 lctic fild insid dilctic loc K K is som positiv constant. D This quation givs th lctic fild loc that acts upon a singl molcul of th dilctic. Fo dilctics with 4, K 4 and K = /, th total local lctic fild at O is loc This is a usful sult, as this quation is applical to cuic cystals, dilut solutions and gass. mp_.doc Oct.

11 Calculation of Assum a sphical sction is movd fom th dilctic. is found y summing th contiutions to th fild of all ing lmnts of polaization chag on th sufac S. sufac S Width of ing d Radius of ing sin Aa of th shadd ing twn and d 2 sin d d sin cos Th chag dnsity s on th sphical sufac is givn y th componnt of th polaization nomal to S ˆ s n cos wh is masud w..t. Sufac aa of th ing lmnt is ds 2 sin d Th chag on th sufac lmnt ds that lis twn and d is dq cos 2 sin d By symmty, all componnts of th lctic fild that a not nomal to th capacito plats cancl. Only th lctic fild du to lmnts of chag is in th vtical diction contiut. Hnc, th lctic fild du to an lmnt of chag dq (Coulom s Law) iis dq cos cos 2 4 lctic f ild at O du to chag dq cos lmnt of chag dq mp_.doc Oct.

12 Th lctic fild du to th ing with chag dq is d cos 2 sin d 2 dq cos cos cos sin d Th sultant fild at th cnt of th sph is otaind y intgating ov = 2 (2)() (2)() 2 cos sin d cos Th local lctic fild is gat than th lctic fild within th dilctic caus of th contiution of mp_.doc Oct.2

13 NONOLAR DILCTRICS Molculs without any intinsic dipol momnt will acqui an inducd dipol momnt in an xtnal lctic fild, and so such molculs hav a dilctic constant. Th lctons shift position slightly insid thi molculs and do so vy quickly, in aound 5 s, so that tmpatu and fquncy hav littl ffct. Monatomic gass (nonpola) W will consid th a gass such as hlium and agon caus of th simpl thotical modl that can usd, although fo most pactical puposs it is not vy usful. Simpl modl of a singl atom (givs sults that a coct to an od of magnitud) ositiv nuclus Z and lctons Z Atomic nuclus: diamt ~ m nucla diamt ~ 5 m Nuclus point chag and lcton cloud of chag Z distiutd homognously thoughout a sph of adius a m Whn th atom of adius a placd into xtnal lctic fild xt ( xt = loc ) nuclus and lcton cloud mov in opposit dictions to cat an inducd lctic dipol quiliium stalishd with th nuclus shiftd slightly lativ to th cnt of th lcton cloud y a distanc d. a Z a d Z d << a Th nuclus will xpinc a foc in th diction of th lctic fild F F = Z xt and an opposing foc F c du to th lctic fild of th ngativ chag of th lcton cloud which is assumd to acts as a point chag at th cnt of th cloud Th lctic fild c xpincd y th nuclus at a distanc d fom th cnt of th ngativly chagd lcton cloud is dtmind y th application of Gauss s Law 2 Z ( d / a ) c 4 d wh th chag nclosd is (Z)(d 2 2 /a ) Z d Fc Z c 4 a mp_.doc Oct.

14 4 a F F d Z c xt Th displacmnt distanc d is popotional to th xtnal lctic fild xt. Fo th singl atom loc = xt, th molcula (lctonic) polaizaility of a monatomic gas is 4 a p Z d Z Z loc xt ( ) ( ) xt 4 a Th lctonic (molcula) polaizaility is popotional to th volum of th lcton cloud ( a ) th lag th atom, th gat th chag spaation and th gat th inducd dipol momnt: igg th atom th lag 4 F.m 2 H N A K X Now, w consid a a gas containing n molculs.m in an lctic fild. W can nglct any intactions twn th inducd dipols in th atoms (good appoximation fo a gas). Micoscopic viw: th polaization of th gas is n p n loc = Macoscopic viw: th polaization of th gas is dilctic = Thfo w can lat th micoscopic molcula polaizaility with th macoscopic dilctic constant n 4na W hav otaind a lationship twn th masual quantity and th micoscopic quantitis and a. How good is ou simpl modl? Hlium gas: tmpatu, T = 29 K and pssu, p g = atm =. 5 a Dilctic constant =.684 pg V N kbt n= N / V = atoms.m a 4 n Coct od of magnitud!!! ou simpl modl not too ad / 6 m W can stimat th lativ shift d twn th nuclus and th cnt of th lcton cloud ~ 5 V.m a ~ m Z ~ 2 d ~ 7 m d is vy small vy slight ptuing influnc of th applid lctic fild on th atom mp_.doc Oct.4

15 Num dnsity n Th num dnsity fo a gas is otaind fom th idal gas quation N pg pg V N kb T pg kb T nkb T n V kb T Fo a gas at atmosphic pssu and 2 o C, th num dnsity n is p g = atm =. 5 a T = 2 o C = 29 K k B =.8 2 J.K n = molculs.m Fo a solid o liquid (dnsity, molcula mass m, num of molculs N, mass of sampl m sampl ) th num dnsity n is otaind as follows m N m M N M M N m m n n V V N V N sampl A A A N A n M Avogado s num N A = molculs.mol Mola mass M (in kilogams) Fo copp = 8.9 kg.m M = 6.5 g = 6.5 kg n = atoms.m Not: th num dnsity of solids is much gat than that of gass. 4 mp_.doc Oct.5

16 Gass, dilut solutions and simpl solids (nonpola) Gass, dilut solutions, solids on kind of atom g diamond, phosphous (cuic cystals), No pmannt dipol momnts o ions olaization du to lativ displacmnt of lcton clouds and nucli Local lctic fild sam fo all atoms K = / (no pola molculs 4 = K 4 = ) = n p n dilctic = loc loc Comining ths th quations givs th ClausiusMossotti lationships n 2 n 2 Th distanc twn atoms in a solid is affctd only slightly y tmpatu and thfo, n,, K and a in a fist appoximation indpndnt of th tmpatu. Fo a solid, a typical valu fo th num dnsity is n ~ 5 28 m. Th dilctic constant fo th solids with a diamond stuctu a: (C) = 5.68 (Si) = 2 (G) = 6 Th dilctic constant fo th gass a vy clos to g (H 2 ) =.2. Why is th dilctic constant fo a solid much gat than fo a gas? n If vy clos to 2 th sam quation as fo monatomic gass mp_.doc Oct.6

17 OLAR DILCTRICS Consid th dilctic matial containing n molculs.m. Assum that ach molcul has a pmannt lctic dipol momnt p. Wat is a typical pola liquid. Its constitunt molculs hav a pmannt dipol momnt. Th polaization is du to th lctonic polaization (nuclus shiftd slightly lativ to th cnt of th lcton cloud) and th ionic polaization i (ionic natu of ond twn atoms) and th ointation polaization o (otation and alignmnt of th pola molculs in th xtnal lctic fild). and i a ssntially indpndnt of th tmpatu ut o is vy tmpatu dpndnt. At a tmpatu T and zo xtnal lctic fild, th molculs will andomly ointd zo polaization. Whn th is an xtnal lctic fild, th molculs will ty to align with th fild. ach pola molcul can considd to a simpl dipol. Th foc on th dipol povids th toqu to otat th molcul so that thy will in th lowst ngy stat wh thy a paalll to th fild. If th w no thmal motion, all dipols would lin up along th xtnal fild diction. p d Q F F Q Th lctic foc on th dipol poducs a coupl and th toqu acting to otat th dipol aout its cnt is d d F sin F sin Q d sin p sin p 2 2 St th potntial ngy U( ) of th dipol to zo whn = 9 o. Th potntial ngy of th dipol fo an aitay angl is thn givn y U( ) p sin d p cos p o 9 U = p Lowst ngy stat = U = = 9 o U = p highst ngy stat = 8 o mp_.doc Oct.7

18 Th dipol has th lowst potntial ngy whn th dipol is paalll to th lctic fild and th highst ngy whn antipaalll to th fild small angls a pfd ov lag ons. And if thy w no thmal motion, all dipols would lin along th diction of th xtnal lctic fild. Th gat th tmpatu, th gat th thmal motion ducd alignmnt of th dipols with th fild. U p p Th ointation polaization o is givn y th Langvin function (95) π/2 π o p n pcoth k p B T kb T o p k T B o n p Complt alignmnt this dos not occu in gass T p k T B p o n p kbt Most pactical cas: o p T 2 o o np slop = / p / k T Th total polaization of a polyatomic gas is givn y wh loc = 2 p i o n i kb T Maco viw is latd to th molcula poptis y 2 n p i kb T How wll dos this pdiction ag with xpimnt? mp_.doc Oct.8

19 If dilctic constant plottd against /T staight lin Slop n p 2 / k B masumnt of p Intcpt n ( i ) masumnt of ( i ) intcpt n i 2 np slop k /T B Dipol momnts of gass in dy units (. C.m) NO. CO. HCl.4 HB.79 HI.8 NO 2.4 CO 2 CH 4 H 2 O.84 H 2 A NH.4 Dilctic constant masumnts hav playd an impotant pat in dtmining molcula stuctu: CO 2 has zo sultant dipol momnt, whas ach CO ond dos hav a nonzo dipol momnt O=C=O. H 2 O molcul must hav a tiangula stuctu. At odinay tmpatus and lctic filds, th avag dipol momnt and hnc polaization a gatly ducd y th thmal agitation th molculs point in vy way with only a slight alignmnt with th xtnal lctic fild. In th low lctic fild appoximation p p o n p kbt kbt k T is th thmal ngy and p is th ngy of th dipol molcul whn alignd with th ffctiv lctic fild. Th polaization is ducd y th facto (p / k B T) and th gat th tmpatu th gat th duction in th polaization and th polaization incass with incasing lctic fild stngth. mp_.doc Oct.9

20 Th dilctic constant of wat is 8 and th molcul dipol momnt of a wat molcul is 6.2 C.m. Find th lctic fild quid to maximiz th polaization of wat. Wat: M = 8 kg = kg.m N A = mol p = 6.2 C.m To maximiz th polaization, all th wat molculs nd to alignd. Th polaization and lctic fild a givn y Num dnsity n.4 molcul.m M olaization n p =.2 C.m 2 lctic fild N A V.m. V.m 8 This mans that th wat molculs a not all alignd up until th lctic fild achs MV.m, which is nough to ak down wat and poduc an lctic ac mp_.doc Oct.2