Measuring dielectric properties at the nanoscale using Electrostatic Force Microscopy

Transcription

1 Masug dlctc popts at th aoscal usg Elctostatc Foc Mcoscopy R. Ao, C. Rdl,,3, G. A. Schwatz 4, G. Lévêqu, A. Algía 3,4, Ph. Todjma 5, N. E. Isaloff 6, M. Ramoda 7 ad J. Colmo,3,4 IES, UMR CNRS 54, Uvsté Motpll II, CC 83, Plac E. Batallo, 3495 Motpll Cdx, Fac Doosta Itatoal Physcs Ct (DIPC), Paso Maul d Ladzábal 4, 8 Sa Sbastá, Spa. 3 Dpatamto d Físca d Matals UP/EHU, Facultad d Químca, Apatado 7, 8 Sa Sbastá, Spa. 4 Cto d Físca d Matals CSIC-UP/EHU, Paso M. D Ladzábal 5, 8 Sa Sbastá, Spa. 5 IMFT, Uvsté d Toulous CNRS, Allé du Pofssu Camll Soula, 34 Toulous, Fac. 6 Dpatmt of Physcs, Nothast Uvsty, Bosto, Massachustts 5, USA 7 Laboato d Mcoscop Champ Poch (LMCP), Ct d Tcholog d Motpll, Uvsté Motpll II, CC 8, Plac E. Batallo, 3495 Motpll Cdx, Fac Sval lctostatc foc mcoscopy (EFM) basd mthods hav b ctly dvlopd to study th aoscal dlctc popts of th sulatg lays. Som mthods allow masug quattatvly th statc dlctc pmttvty whas som oths povd qualtatv fomato about th tmpatu-fqucy dpdc of dlctc popts. I ths chapt, all ths mthods a dscbd ad llustatd by xpmts o pu ad aostuctud polym flms. A scto s ddcatd to EFM pob sampl modls ad spcally to th Equvalt Chag Mthod (ECM). Kywods Atomc Foc Mcoscopy; Elctostatc Foc Mcoscopy; dlctc; local popts; aoscal; polym; dyamcs; glass tasto; magg.. Itoducto Boadbad Dlctc Spctoscopy (BDS) has show to b a vy usful, xtdd ad vsatl tchqu to study dlctc matals ad patcula polyms ad oth glass foms. Rct advacs hav mad t possbl to pob th dyamcs of th polym flms (wth a thckss as low as 5m), opg th aa of tfacs studs [], []. Howv, BDS has a fudamtal lmtato: t has o spatal soluto. Ths s usually ot a poblm wh homogous ad o-stuctud systms a aalyzd but t bcoms a mpotat lmtato fo studyg th local popts of htogous ad/o ao-stuctud matals. To ovcom ths costa dfft mthods hav b ctly dvlopd to masu th aoscal dlctc popts of th flms by mas of Elctostatc Foc Mcoscopy (EFM). Som mthods allow studyg quattatvly th statc dlctc popts whas som oths pmt to obta qualtatv fomato o th tmpatu-fqucy dpdt dlctc popts. All ths mthods w fst adaptd fo masumts o pot. Th passag fom sgl pot to th mappg of a htogous matal has b possbl thaks to both umcal ad xpmtal advacs. Som pvous latd woks hav b publshd dug th last yas. W ca mto fo stac th woks of Kayv t al [3], [4] latd to th study of polyms bld th fom of lay of sval mcos thckss. Ths authos showd that a lctc cotast could b obtad o EFM mags ad that such a cotast s latd to th vaatos of th latv dlctc pmttvty ε. Thy also quatfd th valu of ε th fam of a smpl sphcal capacto modl. Ufotuatly, ths modl s appopat oly ud cta codtos bcaus t maks th appoxmato that th thckss of th sampl s vy lag compad wth th tp adus ad th tp-sampl dstac. Moov, dlctc costats of two fc polyms a qud to masu a thd ukow o. O th oth had, Fumagall t al hav dvlopd th so-calld Naoscal Capactac Mcoscopy [5], wh th mcoscop s quppd wth a sub-attofaad lowfqucy capactac dtcto. Th sam goup also poposd a mthod basd o th dtcto of th lctostatc foc to mag th dlctc costat of a pupl mmba patch [6]. I ths chapt, w pst th spcfcts of som EFM basd mthods, whch ca b opatd th by dtctg th foc o th foc gadt btw th pob ad th sampl, ad by applyg th o AC voltags. A scto s ddcatd to pob-sampl lctostatc tacto modlg usg th Equvalt Chag Mthod (ECM). Compad to xstg aalytcal modls, ths umcal mthod allows calculatg th pob-sampl capactac wthout ay gomtcal stctos o th pob shap ad dlctc flm thckss. W also pst ct sults, llustatg th multpl pottal applcatos of ths dfft mthods. Amog ths applcatos w focus o th quattatv dtmato of th aoscal statc dlctc pmttvty of polym flms at dfft tmpatu [7] ad th quattatv mappg of th dlctc costat of a phas spaatd polym bld [8]. Fally, w show how t s possbl to xtd th masumt of th fqucy dpdc of th dlctc spos o sgl pot [9], [] to mag th dyamcs of a htogous polym bld [].

2 . Modlg of th pob-sampl tacto lctostatc foc mcoscopy. Bf hstocal vw of xstg modls A umb of modls dscbg pob-sampl tactos hav b poposd th two last dcads. Eal modls tatd th pob sufac as a qupottal wth a assumd dstbuto of chags, such as a sgl pot chag [] o a ufomly chagd l [3], ad th pob-sampl tacto was appoxmatd as th tacto btw th assumd chag dstbuto ad ts mag wth spct to th sampl sufac. Aoth goup of modls toducd gomtc appoxmatos to th pob shap ad solvd th pob-sampl capactac poblm th by xactly solvg th bouday valu poblm,.g., th sph modl [4] ad th hypbolod modl [5], o by toducg futh appoxmatos to th lctc fld btw th pob ad th sampl [6-8]. Ths modls povd covt aalytc xpssos of th pob-sampl tacto; howv, mo sophstcatd modls a dmadd fo studyg th latal vaato of th sampl sufac popts (.g., topogaphy ad tappd chags dstbuto) o to tak to accout th psc of a dlctc flm of vaabl thckss. Lt us cosd th AFM tp as a co of half aglθ, C z vsus th tp- wth a sphcal apx of adus R, attachd to th xtmty of a catlv, th total capactac ( ) sampl dstac z s th sum of th apx capactac C apx ( z),. th local capactac, ad th stay capactac C stay ( z), assocatd wth th tp co ad th catlv cotbutos. Fo flms thcksss lss tha m w ca f to th modl poposd by Fumagall t al [9]. It ca b xpssd as: C apx ( z) = ( sθ ) R πε R l + z + h ε () wh h ad ε a th thckss ad th latv dlctc costat of th lay, spctvly. By fttg th go wh th ffct of th local capactac s glgbl, thy foud a stay capactac of th la fom C stay ( z) = b. Δz. A scod famly of modls, also calld Equvalt Chag Mthod (ECM), placd th pob ad th sampl by a ss of pot chags ad/o l chags ad th mag chags [-3]. Basd o ths mthod, tactos btw th pob ad a coductv o dlctc sampl wth topogaphc ad/o dlctc homogts [4-6] hav b studd. Ths appoach was capabl of accommodatg dfft scaos. Th thd famly of appoachs usd umcal mthods such as th ft lmt mthod [7], th slf-cosstt tgal quato mthod [8], ad th bouday lmt mthod [9]. Th ma advatag of ths modls s th ablty to tak to accout th xact gomty of th EFM pob, whch pmts compaso of dfft pob tp shaps.. Th Equvalt Chag Mthod (ECM) I ths scto, w show how th pob-sampl foc, foc gadt ad capactac ca b calculatd usg th Equvalt Chag Mthod (ECM). Th advatag of umcal smulato compad to oth aalytcal xpsso s that th calculatd foc s xact ad allows to wok wthout ay stcto about th thckss of th sulatg flm, th tp adus ad th tp-sampl dstac. W wll fst cosd th cas of a tp fot of a mtallc plat, ad th w wll dduc th foc ad th foc gadt fo a systm composd by a tp fot of a dlctc lay ov a mtallc plat. Th cas of a systm composd by a tp fot of a coductv pla has b tatd by Blaïd t al. []. Th da of ECM s to fd a dsct chag dstbuto ( NC chag pots q at a dstac z o th axs x=) that wll cat a gv pottal at th tp sufac. Th tp gomty s pstd by a half of sph of adus R sumoutd o a co wth a chaactstc sm-agl θ = 5. Th coductv pla at a zo pottal s catd by th toducto of a lctostatc mag tp wth q chags at a dstac z o th z axs (Fg. a). Th valu of th chags q s fxd such way that th M pottal, wth =, K, M, calculatd at tst pot at th tp sufac a qual to. If w toduc D, = (wh d *, ad th pot ad th ffctv ad mag chag, spctvly) w ca xpss th pottal as: N = c D, q 4πε d, d, * d, a th dstacs btw ()

3 Th bst valu of q s obtad usg th last ma squa mthod: M ( ) = q Explctg th dvatv of th pottal, th systm to solv bcoms: M c N D, q D, 4 πε 4πε = Th, kowg th chag ad mag chag dstbutos, th total lctostatc foc actg o th tp ad th tpsampl capactac ca b calculatd. (3) (4) Fg. : a) Rpstato of th chags ( z > ), mag chags ( z < ), ad tst pots ( ) modlg th tp ov a mtallc plat; b) pottal, catd th a (z > ) ad th dlctc (z < ) by a tp (R = 3 m, θ = 5 ) fot of a dlctc lay of hght h = m wth a dlctc costat ε = 4. Th pottal s st to at th sufac of th tp. Th maxmum o o tst pot s of th od of /. Rpoducd wth pmsso fom C. Rdl t al.[7] Copyght 9 Amca Isttut of Physcs. Wh th systm s composd by a tp fot of a dlctc lay o a coductv substat, smulatos a mo complx. Ths poblm has b tatd by Sacha t al. [] toducg th G fucto fomalsm ad also by Duad [3]. W cosd o chag q th a at a dstac z of a dlctc lay of thckss h ad of dlctc costat ε. Th sulatg lay s placd ov a coductv substat. ad a spctvly th pottals catd by th chag q th a ad th dlctc. I od to satsfy th lmt codtos ( = ad ε = ε Oε at th a/dlctc tfac, ad, = at th dlctc/substat tfac), w toduc two z z ss of mag chags, o catd th coductv substat ad o th a. Th quvalt pottal calculatd by ECM th a sults fom th souc, ts mag th dlctc ad th ft ss of mag chags th coductv substat. O ca toduc th cpocal dstac, D+, btw a pot of coodat (ρ,z) ad th chag q (d. ts mag (D-), D± = ρ + ( z m cospodg to th ft ss of mag ( k A = = ρ + ( z + ( + ) h + z ) ε k = ). Th, th pottal catd th a by o chag q s xpssd as: ε + z ) ), ad th cpocal dstac A, wh th costat 3

4 q 4πε ( D + kd ( k ) A) = + Th pottal catd th dlctc s th sum of th two ft ss of mags. Itoducg th cpocal dstac fo th mags th coductv substat, B ( k B = = ρ + ( z h z ) q = πε 4 ( k)( B A) ), w obta: Th valu of ach q s th foud by solvg Eq. 3, stg th pottal calculatd aft Eq. 5, at ach tst pot pstg th tp sufac. Kowg th chag ad mag chag dstbutos, th total lctostatc foc actg o th tp ad th tp-sampl capactac ca b calculatd. I Fg. b), w pst th patto of th qupottals a ad a dlctc lay (ε = 4) fo h = m. 3. Elctostatc foc mcoscopy (EFM) xpmts (5) (6) 3. EFM basc pcpls Elctostatc foc mcoscopy (EFM) s o of th Scag Pob Mcoscopy famls, fst mplmtd by Mat t al [3] 988, whch s gally usd to mag localzd chags o sufacs [3] ad to masu th sufac pottal o smcoductg matals [3]. Lt us cosd th cas of a th dlctc flm sampl dpostd o a coductv substat. Wh a voltag app s appld to th pob (wth th sampl hold goudd), th lctostatc foc F l volvd EFM sgals ca b wtt as: ( ) C F l = app + CP (7) z CP s th cotact pottal dffc btw th pob ad th substat, cospodg to th wok fucto dffc of both matals. C s th pob-sampl-substat capactac. As show th pvous scto, t s a fucto C( R, θ, ε, h, z) of may paamts: th pob gomty, dscbd by th tp adus R ad th co smagl θ, th sampl dlctc pmttvty ε, th sampl thckss h ad th pob-sufac dstac z. ad AC voltags ca b appld smultaously o spaatly, such way that = + sω, wh ωs th agula fqucy of th lctcal xctato. Th, Fl xhbts a compot ad two oscllatg AC compots at ω ad : ω app AC t F F F ω ω C = z C = z C = 4 z ( + ) ( + ) AC CP CP AC cosω t + AC sω t (8) 4

5 Th compot s a statc attactv foc btw th lctods composg th capacto, thω compot has a sgl la dpdc o th capactv couplg C z ad th cotact pottal dffc CP, ad th ω compot s a foc ducd to th capacto oly by th AC voltag. Thω compot dsappas wh appopat bas voltag s appld to th pob to cacl CP,.., + CP =. CP ca b thfo quattatvly masud by th fdback cotol of to mata thω compot to b zo. Ths potocol s kow as Klv pob foc mcoscopy (KPFM) [3] A smla asog ca b mad wh dtctg th lctostatc foc gadt G l, qual to th fst dvatv of th foc: C G ( ) l = app + CP (9) z Gl xhbts a compot ad two oscllatg AC compots at ω ad : C AC G = ( + ) + CP z C Gω = ( + CP ) AC sωt () z C Gω = AC cosωt 4 z Th ma advatag dtctg lctostatc gadts stad of focs s to hac by about 3% th spatal soluto, as dmostatd f [33]. EFM opatos a th most of tm alzd usg dyamcs AFM mods,.. mods wh th catlv vbats a o at ts soac fqucy. Amog thm, ampltud-modulato-afm (AM-AFM), wll kow as Tappg mod, s wdly usd commcal stumts, ud oom codtos. Whl matag costat th xctato pulsato ω m, th ampltud of vbato of th catlv dcass wh appoachg th sufac du to attactv tactos. Th ampltud of vbato svs as fdback paamt fo topogaphy acqusto. EFM opatos wth AM-AFM ca b assocatd wth doubl-pass scas (lft-mod ) wh th topogaphc fomato s obtad dug th fst pass ad th lctostatc sgals a acqud dug th scod pass [7], [8]. Ths mthod has th ky advatag to avod ay couplg ffcts btw stuctual ad dlctc fomato, ad to allow sttg optmzd codtos fo lctc masumts (la gm). Th scod dyamc mod, fqucy-modulato-afm (FM- AFM), also calld o-cotact AFM, s gally usd ud vacuum wh Q-factos of catlvs a too hgh to obta a shot tast gm of th vbato. I that cas, th pob-sampl dstac s cotolld by fxg a small gatv shft of th soac fqucy causd by attactv tactos wth th sufac. Ths fqucy shft svs as fdback paamt fo th fomato of topogaphc mags. Fo stac, FM-AFM s ot assocatd wth doubl-pass scas. Th lctostatc sgal s acqud smultaously wth topogaphy [9]-[]. EFM opatos a a cass mad usg statc AFM mods [6]. ω W wll dtal th followg sctos all th dfft possblts fo dog EFM masumts of dlctcs. Fo a btt classfcato, w wll fst spaat EFM opatos catgos: Foc ad Foc gadt dtcto mthods. Fg : Expmtal st-up fo lctostatc focs dtcto Foc dtcto mthods a latvly smpl to b mplmtd. A typcal st-up s pstd Fg.. Th compot F ca b dctly obtad fom th statc dflcto sgal. ad F ω compots ca b xtactd aalysg th F ω 5

6 oscllato ampltud by a lock- amplf, wth th lctcal xctato as fc sgal. Cocg to th foc gadt dtcto mthods, f w cosd that th catlv-tp-sampl systm ca b appoxmatd by a spg mass systm havg a soac fqucy f, th latoshps btw fqucy shfts Δ f o mchacal phas shfts ΔΦ m ad lctostatc foc gadtg l (assumg G l << kc ad ta ΔΦm ΔΦ m ) Δf f m Gl kc Q G k ca b wtt as [34]: c l, () ΔΦ, () k ad Q wh c a th stffss of th catlv ad th qualty facto, spctvly. Th a two possblts to dtct th local lctostatc foc gadt. Th fst o s to masu dctly th soac fqucy shft Δ f kpg th phas shft costat. Th scod possblty s to masu th mchacal phas shft ΔΦ m at costat dvg fqucy. W oft pf to masu fqucy shfts bcaus, lato () s vald oly at low voltags (ad bcoms o la at hgh voltags), whas lato () s always vald. A typcal st-up s show Fg. 3. Δ f ad ΔΦ m sgals a povdd by most of commcal AFM cotoll lctocs ad ca b dctly lkd to G. To obta G ω adg ω compots, Δ f o ΔΦ m sgals hav to b dmodulatd by mas of a lock- amplf. Fg 3: Expmtal st-up fo lctostatc foc gadts dtcto 3. xpmts Lt us ow cosd xpmts wh oly voltags a appld. As xpctd fom latos () th cuvs Δ ( ) hav th paabolc shap ( z), wh ( z) f xpsso: f ( z) = 4kc z ( z) a Δ f a f Δ s latd to th tp-sampl capactac by th f C aδ (3) Th xpmtal potocol was pfomd o o sgl sufac posto o th bass of a doubl pass mthod ad th masumt of aδ f ( z) paabolc coffct fom th xpmtal cuvs Δ f ( ). Dug th fst sca th topogaphy s acqud th AM-AFM mod. Th tp s th tactd fom th sufac mophology by a costat hght, also calld lft hght, ad th ampltud of th tp vbato δ z s ducd H lft od to stay th la gm (ampltud << tp-sampl dstac). Dug th scod sca, whl a pottal appld to th tp (wth th sampl hold goudd) th lctc foc gadt s G s dtctd. As show Fg. 4 z s appoxmatly qual to th oscllato ampltud ( z δz ). dug th fst sca, th avag tp-sampl dstac Dug th scod sca, th dstac s th sum of th fst sca ampltud z δ ) whl th catlv oscllats wth a ampltud ofδ z. z z + H ( lft δ ad th lft hght H lft 6

7 Fg. 4: Pcpl of th doubl pass mthod. Dug th fst sca topogaphy s acqud. Th tp s th tactd by a costat hght H lft ad ampltud s ducd by a facto of about 3. Dug th scod sca, a pottal s appld to th tp ad G s dtctd. Rpoducd wth pmsso fom C. Rdl t al.[7] Copyght 9 Amca Isttut of Physcs. Th EFM xpmts a pfomd th stps: fst, od to dtm th actual valu of th tp adus R, w masu Δf ( ) cuvs at sval lft hght H lft ov a coductv sampl. Th paabolc ft gvs th xpmtal coffcts accodg to th al tp-sampl dstac. Th valu of aδ f ( z) th tp adus R s th obtad by fttg th ( z) a f Δ cuv wth xpsso (3) whch th tp-sampl capactac s calculatd usg th Equvalt Chag Modl (ECM) (s th pvous scto). Scod, th xpmt s pfomd wth a th sulatg lay of th matal ud study dpostd o th coductv substat. Δ f ( ) cuvs a codd at dfft lft hghts H lft ad a aalysd od to xtact xpmtal coffcts ( z) a f Δ fo ach lft hght. Oc R ad h (th thckss of th sampl masud by AFM) a kow fom pvous xpmts, w ca ft th ( z) a f Δ cuv usg xpsso (3) whch th capactac s calculatd by ECM, ad thby w obta th valu of th dlctc pmttvty ε. δz ad z th Fally, a thd stp, w cod a ampltud-dstac cuv to quatfy th actual valus of pvous foc gadt xpmts. It s woth to mto that th masumt of a ampltud-dstac cuv ca damag th tp ad should b do at th d. Th slop of ths cuv gvs th cospodc btw th photodtcto ms voltag ad th al oscllato ampltud. Idd, f th s o dtato of th tp to th sampl, w ca cosd that ampltud s coasly quvalt to th dstac. Th zo dstac cospods to th pot wh ampltud bcoms ull. Th tp-sampl dstac s calculatd as th dffc btw th z-posto of th actuato cospodg to th ampltud st pot ad th z-posto cospodg to th zo dstac. Nota b: Dug th cod of th ampltud-dstac cuv, th tp ca b dstoyd. It s thus commd dog t at th d of th xpmts. Cosqutly, th adjustabl paamt s th lft hght. It ca vay fom postv to gatv valus, th mmum valu cospodg to th hght wh th tp s th cotact wth th sampl. I od to mata th oscllato of th catlv a la gm, t s covt to choos a scod sca ampltud of appoxmatly 3 o 4 tms small tha δz, so δz 6m. 3.. Masumt of th aoscal dlctc pmttvty of polym flms at dfft tmpatus W tu ow to EFM xpmts pfomd o two ulta-th polym flms at dfft tmpatus [7]. PS (wght avag molcula wght M w = 795 g/mol) ad PAc (M w = 83 g/mol) w chos bcaus both th dlctc stgth ad ts tmpatu dpdc a vy dfft fo ths two polyms. Addtoally, th dlctc sposs of both polyms hav b pvously wll chaactzd th ltatu [35]-[39]. Sampls w ppad by sp coatg statg fom solutos at % (w/w) tolu. Th substat was composd of a f gold lay dpostd o a glass plat. Th pctag of polym soluto was slctd od to obta flms wth a thckss of about 5 m accodg to f [4]. I ths cas stadad EFM catlvs w usd havg a f oscllatg fqucy f = 7.4 khz ad a stffss k c = 4.4 N.m -. Th xpmts w pfomd o at PS ad PAc flms at oom tmpatu ad at 7 C (Fg. 5a ad 5b). Th masud thcksss of th flms w 5 ± m fo PS ad 5 ± 3 m fo PAc at both oom tmpatu ad 7 C. Th thcksss w dtmd by AFM, masug th hght dffc btw th polym sufac ad th gold substat aft th flms w cut usg a shap stl kf. Th accuacy of ths masumts dos ot allow dtctg ay thmal xpaso. Th xpmtal paabolc coffcts ( z) Δ obtad fo PS a show Fg. 5a. Masumts at oom tmpatu ad at 7 C a vy clos dcatg a wak tmpatu dpdc of th dlctc pmttvty as xpctd fo ths polym. I addto, th s a bg dffc btw th cuv obtad o gold ad thos obtad o PS. That mas that th pmttvty of th polym s ath low. Usg th sam potocol, t s possbl to obta th valu of th tp adus (R = 3± m) ad th dlctc pmttvty of PS at C ad 7 C: ε ( C) =.±. ad ε ( 7 C) =.6±. 3 a f. Th xpmtal paabolc coffcts obtad fo PAc a show Fg. 5b. W ca ot a sgfcat dffc btw masumts alzd at oom tmpatu ad at 7 C,.. blow ad abov th glass tasto tmpatu, Tg. At 7 C, th PAc cuv appoachs th gold cuv dcatg a mpotat cas of ε. By applyg ECM, valus of 7

8 ε ( C) =.9±. 3 ad ( 7 C) = 8.±. ε w obtad fo PAc. Th stmatd valus fo PS ad PAc a good agmt wth th macoscopc os [35]-[39]. Th vaato obsvd th dlctc pmttvty of PAc s latd wth ts stog dpol momt ad th fact that PAc cossd th glass tasto tmpatu at aoud 38 C casg th cha moblty ad thfo th dlctc pmttvty. Oppost, PS has a wak dpol momt ad ts Tg s aoud 5 C; thfo, a lttl o glgbl vaato of th dlctc pmttvty s xpctd ths cas. Fg. 5: a) a Δf (z) cuvs obtad o a 5±3 m PS th flm at C (damod) ad 7 C (squas) compaso wth th cuv obtad o a gold sampl (ccls). Th tp adus R = 3± m s obtad fom xpmts o gold usg ECM. By fttg PS paabolc coffcts usg ECM, valus of ε =. ±. at C, ad ε =.6 ±.3 at 7 C a obtad. b) a Δf (z) cuvs masud o a 5± m PAc th flm at C (damod) ad 7 C (squas) compaso wth th cuv obtad o a gold sampl (ccls). By fttg PAc paabolc coffcts usg ECM, valus of ε =.9 ±.3 at C ad ε = 8. ±. at 7 C a obtad. Rpoducd wth pmsso fom C. Rdl t al.[7] Copyght 9 Amca Isttut of Physcs. 3.. Costat dltc mappg of a aostuctud PAc/PS polym bld Mophology ad dlctc popts hav b studd [8] o a modl aostuctud soft matal costtutd by a mmscbl bld of polysty (PS) ad of poly(vyl actat) (PAc) (PS: M =669 g/mol ad M w =7 g/mol ; PAc: M =33 g/mol ad M w =93 g/mol). Th sampl flm was ppad fom a soluto of th two polyms (% w/w tolu wth 75% PS w/w ad 5% w/w PAc). Th volum facto of PAc s Φ = 6.4%. Th soluto was subsqutly sp-coatd [4] o a coductv gold substat at 3 pm. Th flm xhbts a odula mophology of PAc a cotuum phas of PS (Fg. 6a). Th oduls of PAc hav a ma hght masud by AFM aoud 5 m ad th PS a homogous thckss aoud 7 m. Stadad Pt-I coatd tp w usd fo ths masumts. Th catlv f soac fqucy was f =7.3 khz ad th stffss k c =4.5 N.m -. Th tp adus was R=9± m. Fqucy shfts w masud by mas of th succsso of two doubl pass scas at a fxd valu of th tpsampl dstac, z =8± m, ad applyg two dfft voltags of +5 ad -5. Wh th sufac s chaactzd by a zo pottal, oly a sgl doubl pass sca s cssay to mplmt ths mthod. Howv, t s commdd dog two doubl pass scas od to vfy th good accuacy of th masumts ad to chck th symmty of th paabola wth spct to th axs. Fg 6: (a) Topogaphy of th PAc/PS flm. (b) Cospodg map of th coffct a Δ f. alus of h ad a Δ f at pots M(PS) matx ad N(PAc) odul hav b tpolatd wth ECM smulatd cuvs od to xtact th cospodg valu of ε. Rpoducd wth pmsso fom C. Rdl t al.[8] Copyght Amca Physcal Socty. 8

9 I th pst study, th fqucy shft at zo voltag was foud to b aly ull th scad aa. I od to su that th two fqucy shfts cospod to th sam sampl pot, th topogaphy of th two mags at dfft bass should b as smla as possbl. Howv, wokg at 7 C a o-glgbl dft s obsvd. Taslato ffcts hav b umcally coctd by th toducto of a colato fucto. Usg ths two masumts ad assumg a zo fqucy shft fo zo voltag, th coffct ca b calculatd at ach pot of th mag. Fg. 6b psts th cospodg map of th coffct f. I th gal cas, od to tak to accout th ozo cotact pottal CP, a thd mag has to b codd at aoth appld voltag fo xampl, at. A map of th paabolc coffct could b obtad fom th fqucy shft mags usg th quato a Δ f Δf = a ( ) Δ. Fom EFM sults ad ECM umcal smulatos a ( h, ε ) f CP a Δ a Δ f Δf cuvs w calculatd at ach pot of th mag whas th sampl thckss h was dtmd by AFM (masug th hght dffc btw th polym sufac ad th gold substat). As a xampl, pots M(PS) ad N(PAc) Fg. 6a ad 6b a chaactzd by h(m)=7± m, a Δ f (M)=5.±.3 Hz/² ad h(n)=5± m, Aft succssv tpolatos btw dfft a ( h, ε ) Δf a Δ f (N)=7.8±.7 Hz/², spctvly. cuvs, valus of ε =.3±.3 fo PS ad ε =7.5 ± w foud fo PAc, valus agmt wth th macoscopc os [4]-[43]. Fgu 7a shows a quattatv map of th dlctc pmttvty of th PAc/PS flm at th aoscal. Th small asymmty obsvd o th slads of PAc o th x axs s most lkly attbutd to th scag pocss (oly tac sgal was codd). A upp lmt fo th spatal soluto of aoud 3 m has b stmatd, whch cospods to half th dstac cssay to achv th tasto btw th dlctc lvl of th slad of PAc ad th matx of PS (Fgu 7b). Ths valu s good agmt wth th thotcal o calculatd o th bass of th tp-sampl lctostatc tacto [33],[44]: Δx=(R z) / ~ m. Ths sult shows that PS ad PAc a mmscbl at scal qual o low tha 3 m. Fom th mophology mag, w foud a sufac facto of PAc clos to 4%, a valu coht wth th polym composto of th flm. Th dct cofotato of th topogaphy wth th dlctc map (Fgus 6a ad 7a) pots out that small satllt oduls aoud m a dtctd th dlctc map ad ot th topogaphy, thus showg th hgh sstvty of ths mthod. Fg 7: (a) Map of th dlctc pmttvty of th PAc/PS flm obtad by pocssg mags show Fgus 6a ad 6b. (b) Typcal pofl of th dlctc pmttvty acoss th PAc/PS tfac. Rpoducd wth pmsso fom C. Rdl t al.[8] Copyght Amca Physcal Socty. 3.3 AC xpmts 3.3. FM-EFM xpmts W tu ow to EFM xpmts wh by applyg AC xctato th tmpatu-fqucy dpdc of th dlctc spos ca m masu ad aalyzd. Expmts w cad out by FM-EFM ud Ulta Hgh acuum (UH) wth a vaabl tmpatu stag (RHK UH 35). Th tmpatu was masud wth a small thmocoupl clampd to th sampl sufac. A classcal EFM coductg catlv wth 5 m tp adus was usd. I FM-EFM, th catlv s oscllatd at ts soac fqucy, f = 7 khz, ad th soac fqucy shft Δf, du to tp-sampl tacto focs, s dtctd wth vy hgh soluto usg a Naosuf Easy Phas Lockd Loop (PLL) dtcto. Ths paamt s usd as a fdback fo cotollg th tp-sampl dstac z. Addtoal sgals a thos masud as a 9

10 sult of th lctodyamcs tacto btw th coductg tp ad th polym sampl. As th foc F/ z s popotoal to th squa of th voltag, a susodal voltag AC = s(ω t) poducs a ω compot spos. Ths compot ca b dtctd va th fqucy shft of th catlv by usg a lock- amplf (Stafod Rsach A F SR-83). Th obtad sgal ω s latd to F/ z (popotoal to G ω ) by: ω = f wh A s th 4kc z stumt ga [9]. Th foc gadt bg a fucto of th dlctc pmttvty, ay masuabl dlctc losss th matal wll b dtctd as a phas shft th ω sgal. Two dfft xpmts w pfomd to study succssvly th th polym bld pstd scto 3.. (% w/w tolu wth 75% PS w/w ad 5% w/w PAc). Th fst s dscbd fcs [9] ad [] ad cossts codg th phas o pot at dfft fqucs ad costat tmpatus. Dug th acqusto of th dlctc spctum, th fdback of th PLL dtcto (typcally Δf = 5 Hz) was tud off od to avod ay couplg wth th cospodg lctcal fqucy. A AC = s(ω t) voltag at dfft fqucs vayg fom. to Hz was appld btw th tp ad th sampl. It s woth to mto th fact that th fqucy ag s lmtd at low fqucs by th tm of acqusto ad at hgh fqucs by th badwdth of th mcoscop s lctocs. A plmay stp cossts masug th fc phas of ω, Φ f, at 9 C wh both polyms a th glassy stat ad hav o masuabl dlctc laxato. Th dffc btw ths fc stuato ad th phas masud th xpmts, ΔΦ = Φ f Φ, s usd to chaactz th dlctc dsspato. Dug th laxato, du to th loss of gy to alg th dpols th fld, a pak s obsvd ΔΦ(ω ). Usg BDS xpmts o bulk PAc, th macoscopc phas of th dlctc pmttvty, ta - (ε /ε ), paks at about 6 fo fqucs btw. ad Hz, vy clos to what was foud fo ΔΦ(ω ) of thck flms ( µm) of PAc usg th pst FM-EFM tchqu [9]. Fo th flms of PAc, t was foud that th pak ΔΦ(ω ) s cosdably ducd, to ~ 3 fo 5 m flms. Ths ca b udstood by cogzg that th foc gadt s a complx fucto of th dlctc pmttvty, flm thckss, tp adus ad tp hght that qus umcal modlg [8]. Whas th macoscopc phas s popotoal to factoal cas pmttvty, Δε/ε, th ΔΦ(ω ) masud h s popotoal to th factoal cas foc gadt, Δ( F/ z)/( F/ z). Ths facto s foud smulatos to b ducd wth ducd flm thckss du to a lag costat backgoud F/ z. Fg. 8 shows ΔΦ(ω ) ov a slad of PAc at dfft tmpatus. Th fttg ls w obtad usg a Kohlausch Wllams Watts spos fucto, xp(-t/τ) β, [45],[46] as a covt way to dscb th xpmtal sult. Maxmum valu of ΔΦ ad shap paamt (β=.4) w fxd by fttg th data masud at 4 C. Kpg ths paamts costat ad usg oly a f tmscal paamt, a ath satsfactoy dscpto of th data was obtad:, th shap of th pak dos ot dpd o tmpatu wth th volvd uctats. Th pak fqucs ad shap of ths paks a aly dtcal to thos foud pu PAc flms of smla thckss. Masumts at th sam tmpatus hav b pfomd ov th matx of PS ad th spos (ot pstd th fgu) s always flat (ΔΦ <. ). Thfo, th fqucy swp pmts to chaactz locally th popts of th sampl, good agmt wth fc [9]: th matx s composd of PS whas slads a domas of PAc. Th scod xpmt cossts scag th sampl sufac whl a 5 Hz AC voltag s appld to th tp. I od to kp a costat tp-sampl dstac, th fdback of th PLL s matad o. Fgus 9(a-f) pst mags of ΔΦ codd at dfft tmpatus. All mags hav b Fg. 8: Phas shft of G scald to th sam phas shft scal usg th WSxM ω sgals masud FM-EFM as a fucto of th fqucy ov a softwa [47]. Th phas codd o th flat matx aas s slad of PAc. Ls a smpl dscpto of always clos to zo, as xpctd fo PS. At 5 C, th mag th data usg a KWW lk dscpto (s txt). s homogous: th s o dtctabl dlctc laxato Ampltud ad shap paamts of all fttg ls occug th slads of PAc. Ths dmostats that ths w kpt costat. Th dashd l cospods to mthod s ot sstv to topogaphc ffcts. Wh th th magg fqucy of Fg. 9a. Rpoducd wth tmpatu s casd to 36 C a stog cotast s pmsso fom C. Rdl t al.[] Copyght obsvd th mags as th hgh fqucy wg of th 9 Amca Isttut of Physcs. dlctc loss pak stats cotbutg at 5 Hz th PAc gos. Th maxmum cotast s foud at 4 C. Futh casg th tmpatu ducs th cotast as th loss pak shfts to hgh fqucs ad oly th low fqucy

11 cotbutos a dtctd. As th shap of th laxato dos ot dpd sgfcatly o tmpatu, th cotast obsvd s attbutd to th passag of dyamcs assocatd wth th alpha-laxato though ou masumt wdow. Fgu 9g shows th pofl of th phas codd at 4 C. Th dffc of phas s masud to.6. Ths valu s vy smla to that s pu PAc flms of smla thckss. All valus masud o th maps usg a pofl a, wth xpmtal os, th sam as thos masud usg th fqucy swp o sgl pot at 5 Hz (Fg. 8). Th stmatd upp lmt of th spatal soluto (cospodg to half of th dstac cssay to achv th tasto btw th dlctc lvl of PAc ad PS) Δx s aoud ~4 m. Ths valu s clos to th thotcal soluto gv fc [33], [44] ad cospods to th stat-of-th-at soluto EFM wh usg stadad pobs. g) Fg. 9: a-f) FM-EFM mags of ΔΦ(ω ) (G ω phas shfts) codd at 5 Hz o th th PS / PAc flm at dfft tmpatu. Bght aas cospod to losss latd wth th sgmtal laxato of PAc. g) Pofl of th phas shft codd at 5 Hz ad 4. Th dffc of phas btw th matx of PS ad th slad of PAc s of about.6. Th latal soluto Δx s masud aoud ~ 4 m. Rpoducd wth pmsso fom C. Rdl t al.[] Copyght 9 Amca Isttut of Physcs AM-EFM xpmts Fally w pst ths scto sults obtad usg AM-EFM. Ths obsvatos a of gat lvac bcaus thy dmostat that FM-EFM ad vacuum a ot cssay to masu th dyamc dlctc spos. Ths s patculaly mpotat to study bo-matals. Such studs ca b mplmtd usg stadad commcal dvcs ud ambt codtos. Expmts cosst aalysg F ω compot by a lock- amplf followg th pocdu dscbd Fg.. Although ampltud ad phas of F ω ca b smultaously codd, oly phas sgals a volvd th h pstd xpmtal potocol. As fo FM-EFM opatos, t s fst cssay to masu a fc phas cuv Φ f (ω ), at tmpatus wh dlctc laxato of polyms flms s glgbl. Th, th phas sgal Φ(ω ) s acqud at hgh tmpatus, th dffc ΔΦ(ω ) = Φ f (ω ) Φ(ω ) bg pstatv of th loss tagt of th sampl. As a dmostato, 5 m thck pu PAc flms hav b vstgatd (alpha-laxato). ΔΦ(ω ) cuvs masud fom 4 C to 55 C at a lft hght of 5 m (z ~ 5 m) a show Fg. a. Compad wth FM-EFM xpmts, smla pak shap (fttd by a KWW fucto) ad hght (~ 3 ) w obsvd, but addto th fqucy ag has b casd by about dcads. Idd, fqucy swp has b casd up to 3 khz cospodg to half th soac fqucy of th catlv. Futhmo, th dyamcs (bta-laxato) of th flms of poly(mthyl mthacylat) (PMMA) (Fg. b) btw 8 C ad 69 C has b also studd. 5 m flms w sp-coatd statg fom 4 % soluto of PMMA tolu. Th maxmum of th pak ΔΦ(ω ) cuvs s obsvd to b hgh ad aow wh casg th tmpatu, as obsvd wh

12 vstgatg th sam matal wth stadad BDS xpmts. Cocg th compaso of laxato tm btw maco ad ao xpmts, Cd t al [9], [] hav potd out th fact that th aoscal laxato tm (.. th vs of th maxmum fqucy of ΔΦ(ω ) cuv ) s shftd to low valus wth casg tmpatu latv to th bulk valus. Ths sults ca b tptd as a small dcas th dyamcal Tg by a fw C o as a spdg up of th alpha laxato tm at Tg. A compaso of th dlctc spcta masud o th sam th sampl by EFM ad classcal BDS as potd f [] ad [] s dd od to dfftat chags comg fom th xpmtal Fg. : a) ΔΦ(ω ) cuvs (F ω sgals phas shfts) masud AM-EFM as a fucto of th lctcal fqucy o a 5 m pu PAc flm btw 4 C ad 55 C. b) ΔΦ(ω ) cuvs obtad o a 5 m pu PMMA flm btw 8 C ad 69 C. Fo both masumts, a lft hght of 5 m has b st dug th scod pass sca. Fg. : AM-EFM mags of ΔΦ(ω ) (F ω phas shfts) codd o a th PS / PAc bld sampl. I l, mags at a fxd tmpatu ad at dfft fqucs. I colum, mags at a fxd fqucy ad dfft tmpatus. alus of fqucs ad tmpatu a chos accodg to sgl pot masumts o PAc show th st. A maxmum of cotast s obtad wh cossg th alpha-tasto of PAc. EFM basd mthod fom th physcal os comg fom th dyamcs. A ss of xpmts hav b cad out o th sam PAc/PS bld sampls (4% w/w tolu wth 75% PS w/w ad 5% w/w PAc) usd scto 3.. ad amg to mag th local dyamcs. Two kd of xpmts w cad out: th fst o at a fxd fqucy of Hz ad at dfft tmpatus (4 C, 45 C ad 48 C) a pstd colum Fg.. Fqucs ad tmpatus w chos accodac wth sgl pot spcta masud ov a PAc slad (st of Fg. ). Wh scag at Hz ov a 3µm 3µm aa of th bld sampl, a maxmum of cotast of PAc slad s claly obsvd ΔΦ mags at 48 C, good agmt wth sgl pot spcta. Dfts of postog du to thmal ffcts a otd o th mags but dos ot affct th poposd tptato. Th scod kd of mags w codd at a fxd tmpatu (48 C) ad vaabl fqucy ( Hz, khz ad khz). Thy a pstd l Fg.. As xpctd th costast s maxmum at Hz ad t

13 dacass wth casg fqucy. I both cass th cotast voluto s always assocatd wth th alpha-laxato of th PAc whas th PS matx stll mas th glassy stat. Th obtad spatal soluto s about Δx = 5 m, whch s wos compad wth foc gadts mags (Fg 7 ad 9). Ths ca b patly xplad by th fact that th soluto foc mags s thotcally 3% low. Accodg to Gad t al. fc [33], Δ x = ( Rz). Nvthlss, w thk that soluto foc mags could b mpovd up to 4 m by choosg lowst possbl lft hghts ad optmal tps. Thus, th ma advatag usg AM-EFM wth foc dtcto s ts asyss to b mplmtd stadad dvcs ad th possblty of masug ud dfft vomts. 4. Coclusos I ths chapt, w hav dscbd sval EFM mthods allowg th dtmato of dlctc popts at th aoscal. Ths mthods ca b classfd dfft catgos, followg th xctato typ (AC o ) ad followg th dtcto mthod (foc o foc gadt), as summazd tabl. mthods a wll adaptd to dtm quattatvly th dlctc pmttvty ε of sulatg lays. W hav pstd a smpl umcal appoach, calld quvalt chag modl (ECM). Som xampls of polym th flms studd at dfft tmpatus hav b also pstd. I patcula, cocg PAc flms, w hav show vdc of a bg dffc of ε blow ad abov T g. AC mthods ca povd qualtatv fomatos o th tmpatu-fqucy dpdc of dlctc popts. Aft havg dvlopd aodlctc spctoscopy o sgl pot, w hav s how to xtd t to mag th dyamcs of aostuctud polyms. As a xampl, th alpha-laxato of PAc slads a matx of PS has b magd. Ths xpmts hav b fst achvd by FM-EFM ud vacuum ad lat o cad out usg AM-EFM at oom codtos. Ths last sults a of fst mpotac bcaus thy op th way to xpmts as to mplmt wth stadad AFM. I th a futu, studs of th local dlctc spos of bologcal matals o oth soft matt matals should b thus possbl. Fom th spatal soluto pot of vw, foc gadt dtcto mthods hav povdd th bst dfd mags ( to 4 m). W also thk that a compaso ca b do o th adaptablty lvl of th mthod to vstgat o spcfc aspct. Nvthlss, w hav show ths chapt that ths mthods could b adaptd to multpl poblms. Dtcto mthod Foc Foc gadt Exctato Typ AC F [6] F ω [C. Rdl t al to b publshd] Cotol of th pob-sampl dstac : Cotol of th pob-sampl dstac : Statc foc mod, Costat hght, Sgl pass AM-EFM, Doubl pass Quattatv magg of ε o SO /Au flms Naodlctc spctoscopy of PAc ad PMMA ad pupl mmba moolays th flms Imagg of th tmpatu-fqucy dpdc of th dyamcs of ao-stuctud PAc/PS flms G [7],[8] Cotol of th pob-sampl dstac : AM-EFM, Doubl pass Sgl pot masumt of ε o polyms flms (PAc, PS) at dfft tmpatus Quattatv magg of ε o ao-stuctud PAc/PS flms G ω [9]-[] Cotol of th pob-sampl dstac : FM-EFM, Sgl pass Naodlctc spctoscopy of PAc th flms Imagg of th tmpatu-fqucy dpdc of th dyamcs of ao-stuctud PAc/PS flms Tabl : Dfft EFM mthods to dtm th dlctc popts at aoscal, classfd followg th xctato typ ad th dtcto mthod. Numbs backts dcat th cospodg fcs. Ackowldgmts: Th facal suppot of Doosta Itatoal Physcs Ct DIPC s ackowldgd. Th authos would also lk to ackowldg th facal suppot povdd by th Basqu Couty Govmt Rfc No. IT-4367, Dpto. Educacó, Uvsdads Ivstgacó, th Spash Msty of Scc ad Iovato Gat No. MAT , th Euopa Commuty SOFTCOMP pogam, th PPF Rhéolog t plastcté ds matéaux mous hétéogès 7, Cotact No ad th Natoal Scc Foudato (Gat No. NSF DMR-669). Rfcs: [] A. Sgh, M. Tss, F. Km, Macomolculs 39(6), 9385 (6). [] S. Napoltao,. Lupascu, M. Wübbhost, Macomolculs 4(4), 6 (8). [3] A.. Kayv ad R.. Taloz, Polym 45, (4). 3

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