Influence of surface conductivity on the apparent zeta potential of TiO2 nanoparticles

Transcription

1 Influence of urface conductvty on the apparent zeta potental of TO2 nanopartcle Phlppe Leroy, Chrtophe Tournaat, Mohamed Bz To cte th veron: Phlppe Leroy, Chrtophe Tournaat, Mohamed Bz. Influence of urface conductvty on the apparent zeta potental of TO2 nanopartcle. Journal of Collod and Interface Scence, Elever, 2011, 356 (2), pp < /j.jc >. <hal > HAL Id: hal Submtted on 14 Jun 2011 HAL a mult-dcplnary open acce archve for the depot and demnaton of centfc reearch document, whether they are publhed or not. The document may come from teachng and reearch nttuton n France or abroad, or from publc or prvate reearch center. L archve ouverte plurdcplnare HAL, et detnée au dépôt et à la dffuon de document centfque de nveau recherche, publé ou non, émanant de établement d enegnement et de recherche frança ou étranger, de laboratore publc ou prvé.

2 The nfluence of urface conductvty on the apparent zeta potental of TO 2 nanopartcle Phlppe Leroy 1*, Chrtophe Tournaat 1, Mohamed Bz 1 1 BRGM, French Geologcal Survey, 3 Avenue C. Gullemn, Orléan, France * Correpondng author and malng addre: Phlppe Leroy BRGM Water Dvon (EAU/VAP) 3 Avenue Claude Gullemn Orléan Cedex 2, France E-mal: p.leroy@brgm.fr Tel: +33 (0) Fax: +33 (0) Intended for publcaton n Journal of Collod and Interface Scence 1

3 Abtract Zeta potental a phyco-chemcal parameter of partcular mportance to decrbe on adorpton and electrotatc nteracton between charged partcle. Neverthele, th fundamental parameter ll-contraned, becaue t expermental nterpretaton complex, partcularly for very mall and charged TO 2 nanopartcle. The exce of electrcal charge at the nterface reponble for urface conductance, whch can gnfcantly lower the electrophoretc meaurement, and hence the apparent zeta potental. Conequently, the ntrnc zeta potental can have a larger ampltude, even n the cae of mple 1:1 electrolyte lke NaCl and KCl. Surface conductance of TO 2 nanopartcle mmered n a NaCl oluton etmated ung a urface complexaton model, and th parameter and partcle ze are ncorporated nto Henry' model n order to determne a contraned value of the zeta potental from electrophore. Interor conductvty of the agglomerate calculated ung a dfferental elf-content model. The ampltude of etmated zeta potental greater than that derved from the von Smoluchowk equaton and correpond to the electrc potental at the Outer Helmholtz Plane calculated by our urface complexaton model. Conequently, the hear plane may be located cloe to the OHP, contradctng the aumpton of the preence of a tagnant dffue layer at the TO 2 /water nterface. Keyword: zeta potental, electrophoretc moblty, TO 2, nanopartcle, urface conductvty, extended Stern model. 2

4 1. Introducton Ttanum doxde wdely ued a TO 2 nanopartcle and ha a large varety of potental applcaton n, for example, the bomedcal, optcal, and electronc feld [1, 2]. Due to ther mall ze, nanopartcle have a very hgh urface area to volume rato and are thu of great centfc nteret a they are a brdge between bulk materal and atomc or molecular tructure. The properte of materal change a ther ze decreae to nanocale and the proporton of urface atom become gnfcant. One of thee properte, the urface onzaton of ttanum doxde nanopartcle n contact wth an electrolytc oluton, ha been tuded extenvely [1, 3-10]. It well known that the complexaton reacton at the urface of an oxde mneral are trongly nfluenced by the development of the urface charge. The prmary urface charge determned by the reacton of proton wth the urface. Surface complexaton reacton between the urface te and the on from the bulk electrolyte at the Stern and n the dffue layer neutralze the urface charge [11]. Surface charge properte are prmarly determned ung proton ttraton data. Thee data can be modeled ung varou electrotatc model uch a the dffue double layer, bac Stern, trple layer or CD-MUSIC model [1, 3-10]. Mot ophtcated model are able to reproduce data for a wde range of expermental condton but rely on the fttng of a large number of parameter whoe phycal gnfcance not alway eay to jutfy. Moreover, the unquene of a et of parameter not alway obvou. On the other hand, le ophtcated model rely on the fttng of fewer parameter but often fal to reproduce the data under all of the expermental condton tuded. At preent, there no conenu on whch the bet model to repreent charged urface at oxde-water nterface. Of all the phyco-chemcal parameter characterzng the old/water nterface, the zeta potental partcularly mportant. It the potental at the uppoed lppng plane that 3

5 eparate the tatonary and moble phae n tangental flow of the lqud wth repect to the urface. For example, n the cae of a partcle undergong electrophore, becaue of the electrotatc nteracton between the appled electrc feld and the hydrated counter-on n the dffue layer, the nterface develop a urface of hear [12]. The electrc potental at the lppng plane of partcular nteret f we wh to etmate the crtcal coagulaton concentraton when tudyng nanopartcle agglomeraton, for example [5, 12, 13]. The zeta potental alo a key parameter for the tudy of the tranport properte of electrcally charged materal lke oxde and clay mneral [14-16]. It uually make t poble to optmze the parameter of the electrotatc urface complexaton model whle amlatng the poton of the hear-plane to the poton of the head end of the dffue layer [10, 17]. Th nterpretaton ha, however, been challenged by recent tude on ttanum doxde nanopartcle that how that ttraton data cannot be reproduced together wth zeta potental value wthout havng a hearplane poton that change a a functon of the onc trength. Bourka et al. [9] and Panagotou et al. [1] have hown that the hear plane poton (n log cale) lnearly dependant on the log value of the onc trength. Th phycal model lead, however, to a urface repreentaton n whch the hear-plane can be a far a 210 Å from the urface at an onc trength of 10-4 M (Fgure 32 of [1]), the entre volume between the oxde urface and th hear-plane beng condered to be tagnant. Moreover, th phycal repreentaton of the oxde urface rule out any ue of the zeta potental a a contrant for urface charge model. The nature of the phycal property that caue the eparaton between the tagnant dffue layer and the moble dffue layer, however, not gven. Molecular dynamc tude and X-ray meaurement on oxde and alumnolcate urface tend to how that dffue layer water properte (denty, moblty, molecule orentaton) are very mlar to thoe of bulk water [18-20]. 4

6 Electrophoretc moblty meaurement are uually ued to determne the zeta potental. However, zeta potental etmaton can be erroneou due to the uncertanty concernng the value of the converon factor ued [12]. Th could explan the oberved hft of the hear-plane poton a a functon of the onc trength n urface complexaton model. Numerou author [18-26] emphaze that the anomalou urface conductvty of partcle mght explan the low zeta potental value (n ampltude) determned from electrophore compared to value etmated by urface complexaton model at the OHP and electrcal conductvty meaurement. They alo tate that the lateral moton of adorbed counteron at the Stern layer mut not be dregarded for ome materal havng a large exce of electrcal charge at ther urface (lke clay mneral or latex upenon). Lttle work ha been done to characterze the urface conductance due to the Stern layer for ttanum doxde nanopartcle [27]. In addton, the urface conductvty of TO 2 nanopartcle may be trong due to the mall ze of the elementary partcle. The am of th work to tudy the poble nfluence of TO 2 nanopartcle urface conductvty on t electrophoretc moblty n the hope that th wll lead to zeta potental value n agreement wth thoe predcted by electrotatc urface complexaton model (wthout conderng large varaton of the dtance between the outer boundary of the compact layer and the nner boundary of the dffue layer). Surface pecaton model are needed to calculate the urface conductance of TO 2. For th reaon, the model developed recently by Panagotou et al. [1] and alternatve model are crtcally evaluated n the econd chapter of th tudy. In the thrd chapter, we dcu the theore ued to convert electrophoretc moblty to zeta potental and calculate urface conductance due to the double layer. In the fourth chapter, we decrbe our modelng 5

7 trategy and ue the propoed methodology to etmate the ntrnc zeta potental of ttanum doxde P25 n a NaCl electrolyte from electrophore. 2. Electrotatc model for ttanum oxde 2.1. Panagotou et al. model (2008) Panagotou et al. [1] propoed a Trple Plane Model (TPM, [6]) for the ttanum oxde (P25) old-oluton nterface n NaNO 3 and KNO 3 electrolyte oluton. Th model baed on a tate-of-the-art decrpton of the TO 2 urface properte wth regard to protonaton-deprotonaton procee ung the recent ab-nto calculaton and DFT development for th materal. Two man urface functonal group were found to be reponble for the urface reactvty: T 2 O H + T 2 OH log K 1, (1) TO H + TOH log K 2. (2) Total urface te denty wa fxed at a value of 5.6 te nm -2 obtaned from crytallographc conderaton. Total urface area wa 50 m 2 g -1 accordng to BET meaurement. The urface onzaton model wa then combned wth a trple plane model, and the predcton were ued together wth potentometrc ttraton, mcroelectrophore and treamng potental experment to decrbe the electrochemcal properte of the TO 2 urface. Both urface te were condered to behave mlarly wth a gven caton (Na + and K + ) or anon (NO - 3 ). Th mplfcaton enabled the author to reduce the number of adjutable parameter for ther model: T 2 O Na + T 2 O Na + log K Na, (3) 6

8 TO Na + TO Na + log K Na, (4) T 2 OH NO 3- T 2 OH NO 3 - log K NO3, (5) TOH NO 3- TOH NO 3 - log K NO3. (6) In the trple plane model, the charge of orbng caton and anon not attrbuted to only one electrotatc plane but dtrbuted over the three plane (0, 1, 2), thu addng two addtonal fttng parameter (Δz 0 and Δz 1 or Δz 1 and Δz 2 ) for each orbed pece at each urface te (Fg. 1). Agan, both urface te were condered to behave mlarly wth a gven caton or anon. Fg. 1. Schematc drawng of the bac Stern model of Bourka et al. [9], our bac Stern model, the trple plane model of Panagotou et al. [1], and our extended Stern model for a negatvely charged urface of ttanum doxde. At a gven pcture, from left to rght: metal on, urface hydroxyl, prmary and econdary water layer, compact layer, dffue layer. 7

9 The parameter of th model are gven for a NaNO 3 electrolyte background n Table 1. We reproduced ther model ung PHREEQC v2.17 [28]. We were not able to reproduce ther ttraton curve ung the parameter n the reference publcaton. The tabulated protonaton/deprotonaton contant (log K 1 and log K 2 ) had to be changed lghtly n order to obtan reult n full agreement wth the data. Hereafter, we refer to th modfed model a Reference model (Table 1). Surface charge predcton are gven n Fg. 2 together wth urface potental at the 2-plane, whch condered to be the head end of the dffue layer. Th parameter can be compared to the zeta potental. Fg. 2. TO 2 urface charge and potental at the head end of the dffue layer predcted by dfferent urface complexaton model at three onc trength I n NaNO 3. Lne depct the reference model (TPM) reult whle ymbol depct the reult of alternatve model (BSM, ESM, Bdentate, Table 1). 8

10 Panagotou et al. [1] reported zeta potental value far below the potental value of the 2-plane (n abolute value). They nterpreted th to be a conequence of a hft of the hear plane (where the zeta potental located) from th 2-plane. Whle calculatng the dtance d from the 2-plane to the hear plane, they found a log-log lnear relatonhp between the onc trength I (n M) and d (n nm): log d = log I (7) 2.2. Alternatve model Whle the data of Panagotou et al. [1] could be very well reproduced by ther TPM model, t could alo be reproduced wth the mpler Bac Stern Model (BSM). The BSM parameter (Table 1) gve urface charge reult nearly dentcal to thoe of the TPM reference model n NaNO 3 electrolyte (Fg. 2) although the BSM requre only fve fttng parameter ntead of ten for TPM. Th rae the queton of the unquene of the TPM parameter et. In prncple, th problem can be overcome by fttng a large range of data obtaned under the ame expermental condton but tetng dfferent electrolyte type. Panagotou et al. [1] provded ttraton data n a KNO 3 electrolyte background and were able to ft them adequately only by attrbutng an aocaton contant (log K K = -1.1) and a charge dtrbuton (Δz 1 K = 0, Δz 2 K = 1) for K + at the urface, whle keepng the other parameter contant. A good agreement can alo be obtaned wth the BSM model (Fg. 3) addng only an aocaton contant (log K K = - 1.7) for K +. There a dcrepancy between the ttraton predcton of the two model at M, but the dfference ncreae wth alnty, and are therefore greater and gnfcant for a alnty of 0.3 M and ph > 9.5. Conequently, the addton of fve fttng parameter for the trple plane model rele on a very retrcted ubet of expermental data pont. 9

11 Fg. 3. TO 2 urface properte predcted by dfferent urface complexaton model at three onc trength n KNO 3. Lne depct the TPM reference model reult whle ymbol depct the reult of alternatve model (Table 1). The rght de of Fg. 2 and Fg. 3 how that the choce of a gven model nfluence the predcton of the electrcal potental at the head end of the dffue layer. For thee two model, however, predcted potental are much hgher n ampltude than reported zeta potental from electrophore [3], corroboratng the hypothe of a hear plane located n the dffue layer. A an alternatve model, an Extended Stern Model (ESM) can reproduce perfectly the reference TPM for urface charge but dmnh the potental at the head end of the dffue layer. One et of th model parameter, amongt thoe teted uccefully, gven n Table 1. Correpondng ttraton and potental curve are hown n Fg. 2 and Fg.3 for NaNO 3 and KNO 3 electrolyte background, repectvely. Accordng to Panagotou et al. [1], the hgh capactance value of the reference TPM are n agreement wth theoretcal and expermental tude concernng the locaton of the 10

12 frt two water overlayer and the electrolyte counteron at the rutle urface, taken here a a good analogue of the anatae urface [29-34]. Predota and Vlcek and Predota et al. [31, 33] howed that the frt hydraton layer at the rutle urface ~1.8 Å from the urface termnal oxygen atom and that Na + caton are located at ~1 1.8 Å. Thee dtance are n agreement wth the capactance value C 1 = 3.2 F m -2 that locate 1-plane at d 1 = 1.7 Å from the 0-plane, whle conderng the followng equaton wth the mean relatve delectrc contant, ε r,1, equal to 60 (ε 0 the delectrc contant of the vacuum, C V -1 m -1 ) : C 1(2) ε 0ε r,1(2) =, d 1(2) (8) where C 2, d 2 and ε r,2 are the capactance, dtance and the mean relatve delectrc contant between the 1- and 2-plane, repectvely. The propoed ESM model ha a lower capactance C 1 = 2.5 F m -2. It could therefore be condered to not be n agreement wth above reult. However, parameter ε r,1 ll-defned. Takng a value of ε r,1 = 47 enable u to arrve at the ame dtance d 1 wth the ESM model. Conequently, avalable data at the molecular level combned wth ttraton data cannot be ued to determne whch the bet repreentatve model nce both ε r,1 value are reaonable [35]. We were, however, not able to acheve a good ft of ttraton data wth model havng a C 1 value lower than 2.5 F m -2. Th eem to confrm the preence of a layer wth a hgh capactance and Na orpton cloe to the urface. On the contrary, C 2 could be et at a value a low a 1 F m -2 n the ESM, n marked contrat wth the 4.2 F m -2 value of the reference TPM. The poton of the econd plane alo ubject to dcuon. Panagotou et al. [1] condered that the econd plane end the compact layer at a dtance of 3.4 Å from the urface. However, molecular dynamc calculaton how gnfcant water denty ocllaton up to 12 Å from the urface aocated wth 11

13 contnuou vcoty change [33]. A a conequence, the choce of a prece locaton for the econd plane cannot be ealy jutfed. Moreover, the calculaton of the capactance value for the regon between the 1- and 2-plane rele heavly on the modeler choce for ε r,2. Theoretcal calculaton ndcate that the greatet porton of orbed Na + form bdentate complexe. We tred to roughly ncorporate th nformaton by conderng the followng equlbra n another alternatve model (the Bdentate model n Table 1): 2 T 2 O Na + (T 2 O ) 2 Na + log K Na, (9) 2TO Na + (TO ) 2 Na + log K Na. (10) The charge of counter Na + wa attrbuted partly to the 0-plane and partly to the 1-plane, nce Na + wa alo hown to be partly dehydrated at the urface and engaged n nner phere complexe [34]. Fg. 2 agan how that th model can match almot perfectly the reference TPM ttraton predcton. Note alo that, dependng on the choen urface complexaton model,.e. TPM, BSM, ESM, or Bdentate, there may be dfferent locaton and then defnton of the tagnant layer (whch not necearly a monolayer of orbed counteron). For example, n the cae of a NaNO 3 electrolyte, the tagnant layer located between 1 and 2-plane for TPM, at the 1-plane for BSM, at the 1-plane for ESM and between the 0 and 1-plane for Bdentate Implcaton for mappng the "ttanum doxde/electrolyte oluton" nterface Th analy how that ttraton data combned wth urface complexaton model uch a TPM or ESM cannot yeld a unque and unambguou et of nterfacal parameter. Theoretcal calculaton and modelng at the molecular cale can help u valdate the model lkelhood but uncertante reman. 12

14 Th ugget that the relatonhp oberved between the poton of the hear plane and the onc trength (Eq. (7)) model dependant. Fg. 2 and Fg. 3 how that the reference TPM and the BSM predct nearly dentcal potental at the head end of the dffue layer, n agreement wth the fact that the reference TPM an mprovement on the BSM publhed earler by Bourka et al. [9]. However, the propoed ESM and Bdentate model predct lower potental (n abolute value) than the reference model. Predcted potental reman, however, hgher n abolute value than commonly reported zeta potental from electrophore. Conequently, the queton of the poton of the hear plane n relaton to the head end of the dffue layer reman. Zeta potental calculaton from electrophore are alo model-dependant. In the followng chapter, everal model to convert electrophoretc meaurement to zeta potental are brefly revewed. 3. From the electrophoretc moblty to the zeta potental 3.1. Electroknetc theore The mot well-known and wdely ued theory of electrophore wa developed by von Smoluchowk [36, 37]. He tuded the movement of the lqud adjacent to a flat, electrcally charged urface under the nfluence of an electrc feld appled parallel to the nterface. Von Smoluchowk [36] ued the Stoke equaton and calculated the electrcal (ung Poon equaton) and vcou force on an element of volume of the lqud to expre the electrophoretc moblty a a functon of the zeta potental. The von Smoluchowk equaton lnearly relate the electrophoretc moblty μ e (n m 2-1 V - 1 ) to the electrcal potental at the hear plane (ζ, n V): ε μ = ζ η e, (11) 13

15 where η the dynamc vcoty (n Pa ; η = Pa at T = 298 K) and ε the delectrc permttvty of water (ε = ε 0 ε r = F m -1 at T = 298 K). The von Smoluchowk equaton vald only f the thckne of the dffue layer ngnfcant compared to the ze of the partcle,.e. for thn double layer, κa >> 1, where κ the nvere of the Debye length (n m -1 ) and a the partcle radu (n m). The nvere of the Debye length gven by: κ = F 2 z εrt 2 c b, (12) κ = 2 F 2 I εrt, (13) where F the Faraday contant (96485 C mol -1 ), R the ga contant (8.314 J mol -1 K -1 ), c the on concentraton (n mol m -3 ), z ther valency, and I the onc trength. The ymbol b refer to the bulk on. For mall phercal partcle havng a thck double layer, κa << 1, the appled electrc feld not nfluenced by the preence of the partcle and the effect of the retardaton force due to the double layer on the mgraton of the partcle neglgble. Hückel [38] condered that the man retardaton force the frctonal retance of the medum. He uppoed that the electrcal conductvty of the partcle the ame a that of the urroundng medum. Therefore, the electrc feld not dtorted by the partcle. He wrote the followng equaton, whch vald for κa << 1: 2ε μ = ζ 3η e. (14) Henry [39, 40] revted Hückel theory by conderng that the conductvty of the partcle dfferent from that of the urroundng medum. In th cae, the appled electrc feld wll be dtorted o that the opotental value can be around the partcle 14

16 urface. Accordng to Henry [40], the partcle conductvty alter the hape of the potental dtrbuton of the appled feld n the lqud, modfe the flud moton wthn the electrcal double layer, and therefore change the tree of the flud exerted on the partcle. Conequently, th conductvty lead to the mutual dtorton of the appled feld and the feld of the double layer, and hence low the electrophoretc moton. For phercal partcle wth arbtrary double-layer thckne, Henry [40] wrote: 2ε μe = F ( κa, K p, K )ζ, 3η (15) [ f ( κ ) 1] F( κ a, K, K ) = 1 + 2λ a, (16) p 1 K = 2 + K p λ, p 2K + 2K (17) K p σ p =, σ b (18) K σ σ = =, b Σ aσ b (19) ( κa) 5( κa) ( κa) ( κa) f ( κ a) = a t κa κ exp( ) exp( ) dt, for κa < 1, 96 t f ( κa) = +, for κa > 1, κa 2κ a κ a (20) (21) where σ the electrcal conductvty (n S m -1 ), Σ the urface conductance of the electrcal double layer, the ubcrpt p,, b correpondng, repectvely, to the partcle nteror, the partcle urface and the urroundng medum (the bulk electrolyte). The urface conductance expree the exce of electrcal conductvty at the old urface compared to that of the bulk electrolyte. K correpond to the well- 15

17 known Dukhn number, Du (ee [18, 41] and [42] for more detal). Accordng to Eq. (15) to (17) and by replacng K wth Du, we obtan: 2ε 1 K p 2Du μ = η 2 + K p + 2Du [ f ( κa) ] ζ e. (22) In the abence of urface conductance and n the cae of an nulatng partcle, Henry theory lead to von Smoluchowk equaton for large κa value (f(κa) = 1.5) and Hückel equaton for κa << 1 (thck double layer, f(κa) = 1). For κa >> 1, f(κa) = 1.5, and Eq. (22) reduce to: 2ε 1 K μ e = 1 + 3η 2 + K p p 2Du ζ. + 2 Du (23) In the cae of nulatng partcle, K p = 0, and Eq. (23) correpond to O Bren formula [43] wthn the lmt of a DC appled electrc feld and dregardng the nertal term n h theory. O Bren [43] developed a complete pcture of the frequency dependent delectrc repone of a dlute upenon of phere wth thn double layer. Note that all of the equaton preented here, except Smoluchowk equaton, conder a Debye- Hückel onc atmophere,.e. that electrc potental n the dffue layer follow a Debye- Hückel dtrbuton. Conequently, the analytcal equaton that we ue to etmate the zeta potental from the electrophoretc moblty are vald for low zeta potental ( ζ 25.7 mv n the cae of 1:1 electrolyte at T = 298 K), but they can tll be appled for zeta potental of greater ampltude [12]. In addton, Eq. (11) to (23) do not conder pherodal partcle, partcle volume fracton, polydperty of the ample,.e. agglomerate of dfferent ze, and dffue layer overlappng. Several electroknetc model make t poble to etmate urface conductance for pherodal partcle (for example, [44]), conder polydpervty of the ample and nanopartcle agglomeraton [45], and dffue layer overlappng [46]. 16

18 Moreover, Mangeldorf and Whte numercal model, whch take nto account partcle ze effect and the adorpton of on and ther moblty n the nner part of the EDL [21], can be more accurate than the analytcal oluton we ue, epecally at low onc trength and at ph value far from the PZC where the ampltude of the urface electrc potental hgh. However, the electrcal conductvty of the partcle nteror σ p can be determned ung the o-called dfferental elf-content model [47, 48]. Th model conder mall contguty between the partcle and allow the determnaton of the electrcal conductvty of the agglomerate correpondng to the fnal concentraton of ncluon (here the elementary nanopartcle of conductvty σ e ) by addton of nfntemal porton of ncluon: σ σ b p σ + σ 1 e 2 σ = ( ) φ dω d, σ σ σ 0 e 1 Ω 3 (24) where φ the ntra-aggregate poroty, Ω = v/(v+v) the volume fracton of the elementary nanopartcle n the agglomerate wth v and V the total volume (n m -3 ) of elementary partcle and of water, repectvely. By ntegratng Eq. (24), we obtan: σ σ p b σ σ e e σ σ b p D = φ, (25) where D = 1/3 n the cae of phercal elementary partcle. Eq. (25) ha been generalzed to non-phercal partcle, but t not very practcal becaue t an equaton of the form σ = f D, φ, σ, σ, σ ). Revl [48] found an analytcal oluton of p ( b e p Eq. (25) by conderng D = 1/2 for dk-haped partcle: σ b σ = FΘ + 1 ( Θ) ( ) Θ + Θ + p FΘ F 2, (26) 17

19 σ e Θ = σ b 2Σ = a σ e b, (27) 2 F = φ. (28) Accordng to Revl [48], D = 1/2 correpond to a partcle hape uually found n mot porou meda. In addton, followng the approach of [45], the electrophoretc moblty due to the polydperty of the ample, ze, can be calculated by: μ e,.e. to agglomerate poeng dfferent N μe f ( a ) a Δa = 1 μ e = N, 3 f ( a ) a Δa = 1 3 (29) where N the number of dfferent rad, f(a ) repreent the dcretzed veron of the Partcle Sze Dtrbuton (PSD), Δa the radu nterval, and μ e the electrophoretc moblty of agglomerate wth radu a ± Δa /2. Accordng to Eq. (29), the electrophoretc moblty of the polydpere ytem the volume average of the moblte of partcle wth dfferent ze n the dtrbuton. Henry [40] and O Bren [43] condered that only the counteron n the dffue layer are reponble for the urface conductvty. They dd not conder the nfluence of the Stern layer on partcle urface conductvty [18-23]. Accordng to thee author, the zeta potental of (electrcally) charged upenon lke polytyrene lattce, determned from electrophore, are lower n ampltude than the zeta potental value etmated from electrcal conductvty meaurement. In the cae of electrophore, thee author aume that the lateral movement of on at the Stern layer n repone to the appled electrc feld can explan uch dcrepancy. The preence of moble counteron at the Stern layer and n the dffue layer lower the ampltude of zeta potental nferred from 18

20 moblty meaurement and rae thoe from conductvty meaurement, compared to the zeta potental that correpond to the ntrnc partcle charge. The anomalou urface conductance lnked to the Stern layer,.e. to on below the hear plane, can be determned for oxde mneral ung electrcal conductvty meaurement [25, 27, 42, 49] or a urface complexaton model lke a Trple Layer Model (TLM, [15, 50]) where the aocated equlbrum orpton contant are calbrated by ttraton experment. Surface complexaton model calculate the exce of counteron at the nterface and thereby make t poble to etmate the (pecfc) urface conductance of the partcle [50, 51] Determnaton of the urface conductance The dffue layer urface conductance The urface conductance of the Electrcal Double Layer (EDL) the enhanced conductvty due to the preence of a double layer at the partcle urface ([40, 49-51]; ee Fg. 4). We conder here the urface conductance due only to the preence of the dffue layer at the partcle urface. In the followng chapter, we wll conder the nfluence of the Stern layer on the urface conductance. 19

21 Fg. 4. Schematc repreentaton of the electro-chemcal properte of a upenon of phercal oxde nanopartcle. The partcle ha a local exce of electrcal conductvty at t nterface σ(χ). The urface conductance Σ etmated by ntegratng σ(χ) - σ b over the thckne of the Stern and dffue layer. Accordng to Revl and Glover [51], the urface conductance Σ (n S) can be decrbed by the ndvdual pecfc urface conductance of each on n the dffue layer, Σ = ez Σ. (30) Σ, The total urface conductance Σ made up of an electro-mgraton, Σ e, and an electro-omotc urface conductance, e o o Σ, Σ = Σ + Σ. (31) The electro-mgraton urface conductance due to the exce of Ohmc conductvty n the EDL [15], 20

22 χ [ σ ( χ σ ] dχ Σ = 0 ), e D b (32) where χ repreent the urface/oluton dtance n m and χ D the total thckne of the dffue layer (uually, χ D = 2κ -1 ). The parameter σ(χ) and σ b are the electrcal conductvty at the old/oluton nterface and n the bulk pore water defned, repectvely, by: σ ( χ) = Fz β c ( χ), b b σ b = Fzβ c, (33) (34) where β the on moblty (n m 2-1 V -1 ) at the partcle urface and n the bulk pore, b water. The electrcal conductvty of the oluton due to electro-mgraton gven by an equvalent Ohm law determned ung the generalzed Nernt-Planck equaton and the electrochemcal potental equaton for the on pece [14, 52] (electro-omo neglected). The local electrcal conductvty σ(χ) gven by analogy wth the free electrolyte conductvty ([51]). By ncorporatng Eq. (33) and (34) n Eq. (32), we obtan: b b [ β c ( χ) β c ] e D Σ = Fz dχ. χ 0 (35) Accordng to Eq. (35), by conderng the ame on moblty at the nterface (dffue b layer here) and n the bulk pore water,.e. β = β = β, we can determne e Σ [15, 53]: b [ c ( χ) c ] D zβ χ e Σ = F dχ, 0 χ e b D ± zfψ ( χ) Σ = F zβ c exp 1 dχ, 0 RT (36) (37) ( ) ψ ( χ) = ψ d exp κχ, (38) 21

23 where ψ d the electrc potental at the head end of the dffue layer. The term ± correpond to the gn of the electrcal charge aocated wth the pece ( + for caton and for anon). It relatvely eay to fnd an analytcal oluton for the on contrbuton to the urface conductance ung Eq. (30), (37)-(38) and lnearzng the exponental functon of the Boltzmann dtrbuton ([51] and Eq. (191) to (194) of [54]): Σ Σ e e 1 b ± zfψ d 2κ F zβ c exp 1, 2RT 1 b ± zfψ d 2κ β Na c exp 1, 2RT (39) (40) where Na the Avogadro Number ( te mol -1 ) and by conderng a dffue layer havng a thckne of χ = 2κ 1 D. At the urface of the ttanum doxde partcle (for condton other than the IEP and for low onc trength, typcally 0.01 M for oxde mneral, [10]), counter-on are predomnant n the dffue layer. When an electrc feld appled, t reult n a olvent convecton and, conequently, a urplu conductvty called electro-omotc conductvty. The electro-omotc contrbuton to the urface conductance decrbed by [51]: o D Σ = ρ( χ) β ( χ) dχ, χ 0 o (41) o o Σ = ez Σ, (42) ( χ ) = ± ρ F ( 1) z c ( χ), (43) ε βo( χ) = [ ψ ( χ) ψ d ], η (44) 22

24 where β o the electro-omotc moblty (n m 2-1 V -1 ) and ρ the volume charge denty n the dffue layer (n C m -3 ). Accordng to Bkerman [55]: Σ o 4εRT 1 κ Na η b ± zfψ d c exp 2RT 1. (45) The on contrbuton of the electro-omotc urface conductance determned ung Eq. (42) and (45): Σ o 4εRT κ ηez 1 Na c b ± zfψ d exp 2RT 1. (46) By combnng Eq. (40) and (46), the total on contrbuton to the urface conductance gven by: Σ B 1 b ± zfψ d = 2κ B Na c exp 1, 2RT 2εRT = β +. ηez (47) (48) In Eq. (47) and (48), we have aumed that the on moblte are the ame n the bulk electrolyte and n the dffue layer. Thee equaton correpond thoe of Bkerman [55] for characterzng the on contrbuton to the urface conductance of the partcle (due to the appled electrc feld). In h approach, the moble counteron and coon n the dffue layer are only reponble for the urface conductance. Moreover, accordng to Eq. (48), the onc electro-omotc contrbuton to the urface conductance mut not be neglected. For example, for a NaCl oluton and at T = 298 K, the Na + and Cl - on moblty value n the bulk electrolyte are and m 2-1 V -1, repectvely [15]. The econd term n Eq. (48), for z = 1, ε = F m -1, η = Pa, F = C mol -1, R = J mol -1 K -1, and T = 298 K, m 2-1 V -1, whch of the ame order of magntude a the frt term. 23

25 For a bnary ymmetrc electrolyte, z + = z - = z, and the urface conductance determned ung Eq. (30) and (47): Σ 1 b Fzψ d = 4Fzκ c B coh 1, 2RT (49) aumng that B ( + ) = B ( + ) = B, (50) and c = b b b ( + ) = c ( ) c. (51) Eq. (49) aume that both anon and caton have the ame moblty. By ung Eq. (19), (34), (49), and the Nernt-Enten relatonhp for on dffuvty, D RTβ =, Fz (52) we obtan Bkerman equaton [25, 49, 55]: 2 Fzψ d Du = coh κa 2RT 2ε ηd RT Fz 2. (53) In Eq. (53), the nfluence of ph on urface conductvty not taken nto account. Takng nto conderaton the preence of H + and OH - on, we obtan the followng equaton for urface conductance, accordng to Eq. (30), (47), and for a bnary ymmetrc background electrolyte lke NaCl or KCl [50]: Σ 2κ Fz b ph ( c B B ) 1 = ( + ) ( + ) H + Fzψ d exp 1 2RT ph p K ( c B + 10 B ) Fz exp 2RT, b ψ f d + ( ) ( ) 1000 OH 1 (54) where pk f the negatve log of the docaton contant of water (14 at T = 298K). 24

26 The Stern layer and total urface conductance For ome hghly-charged mneral lke clay, the contrbuton of the compact Stern layer to urface conductvty mut not be neglected [53]. However, t ha not been extenvely tuded for ttanum doxde nanopartcle. Accordng to everal author (e.g. [18-23]), both the dffue and Stern layer contrbute to the pecfc urface conductance: Σ = Σ Σ. (55) dffue + Stern The Stern layer contrbuton can be decrbed by [50, 53]: Stern St St Σ = zeβ Γ, (56) where the upercrpt St correpond to the Stern layer, and Γ St the urface te denty of adorbed counteron at the Stern layer (n te m -2 ). Eq. (56) decrbe the lateral movement of adorbed caton and anon [56, 57], and conequently aume that the on pece are not mmoble n the Stern layer [16]. The on moblty value at the Stern layer are tll relatvely unknown. For clay mneral and quartz, Revl and Glover [50] and Revl et al. [58] condered that the on moblte of everal caton lke Na +, K +, Ca 2+ are at leat one order of magntude maller than thoe n the bulk electrolyte. For lca mneral, Leroy et al. [15] have hypothezed that the urface on moblte are mlar to thoe of the bulk. Molecular dynamc mulaton at the old/water nterface mut be of partcular mportance to characterze the on and water moblte n the compact layer. For example, n ther work concernng the characterzaton of the urface properte of rutle, Predota et al. [32] condered a urface water moblty that 10% of the bulk on moblty at 3.7 Å from the urface. Accordng to Eq. (54) to (56), the total urface conductance of a partcle can be decrbed by: 25

27 Σ 2κ Fz b ph ( c B B ) 1 = ( + ) ( + ) + H Fzψ d exp 1 2RT b ph p K Fzψ f d ( c ) B( ) + 10 B ) exp 1 + OH 1000 zeβ 2RT St St + ( Γ. (57) A een by Eq. (57), t eay to etmate urface conductance f the electrcal potental at the head end of the dffue layer ψ d and the urface te S t Γ are known. Thee parameter can be etmated by a complete urface complexaton model of the S t TO 2 /water nterface. The on moblte of the counteron n the compact layer β are the only unknown. Below, we ue th approach to determne the zeta potental of TO 2 nanopartcle ung the partcle ze and the electrophoretc moblty data of Foy [3]. 4. Comparon wth expermental data 4.1. Modelng trategy Frtly, the functon f(κa) determned accordng to Eq. (12), (20) and (21). Secondly, ttanum doxde urface complexaton model etmate the urface conductance of the double layer Σ (Eq. (57)). The ubequent calculaton of the electrcal conductvty of the bulk electrolyte baed on t on compoton, σ b, (Eq. (34)) and the radu of an agglomerate made up of elementary partcle, a, make t poble to determne the Dukhn number, Du (Eq. 19). Thrdly, the parameter correpondng to the electrcal conductvty of the agglomerate nteror, K p, calculated thank to Eq. (26) to (28) for a gven value of the ntra-aggregate poroty φ and elementary partcle radu a e. Fnally, the zeta potental etmated from the electrophoretc meaurement ung the calculated value of f(κa), Du, K p, and Eq. (22) for the cae of Henry theory (ee Fg. 5). By comparon, the von Smoluchowk equaton (Eq. (11)) alo ued to convert 26

28 electrophoretc moblte to zeta potental value. The urface complexaton calculaton were done wth PHREEQC v2.17 [28] and the converon calculaton were done ung a MATLAB routne. Fg. 5. Modelng trategy for determnng the zeta potental from electrophore. Raw electrophoretc meaurement are neceary for thee calculaton. Unfortunately, only pre-proceed zeta potental are avalable n mot publcaton. Foy [3] provde electrophoretc meaurement for ttanum doxde P25 for a wde range of electrolyte concentraton but n a NaCl electrolyte background. Accordng to the comparatve modelng of Bourka et al. [9], NO - 3 and Cl - behave mlarly at the TO 2 urface. Th could be verfed by atfactorly reproducng the urface charge data that Rdley et al. [2] obtaned on the anatae ST-01 ung the ame model a the ESM lted n Table 1 and the ame parameter for Cl - a for NO - 3 (not hown) Input data of the electroknetc model Foy [3] dd ttraton and electrophoretc meaurement on TO 2 P25 (Degua, Germany). The crytallne form anatae (95 %). The denty meaured by helum 27

29 pycnometry after dryng 3.76 g cm -3. The etmated pecfc urface area (ung N 2 adorpton and mmeron calormetry) 54 ± 3 m 2 g -1. The ze of an elementary partcle approxmately 30 nm, but P25 nanopartcle can form aggregate meaurng everal mcrometer [1]. Foy [3] meaured electrophoretc moblte for ph value compred between 4 and 11, and alnte (NaCl) of 10-4, 10-3, and 10-2 M. Furthermore, he ued X-ray meaurement to obtan the partcle ze dtrbuton a a functon of ph at 10-3 M. At 10-3 M, we ue a normal dtrbuton n agreement wth the partcle ze dtrbuton meaured by Foy [3] and calculate the Debye length to etmate the functon f(κa). For other alnte, we alo ue a normal dtrbuton for the partcle ze. Eq. (12), (48) and (57) make t poble to calculate the pecfc urface conductance ung the value of ψ 2 (ψ 2 = ψ d here), Γ St Na, Γ St Cl determned by the ESM urface complexaton model (Fg. 6). The Na +, Cl -, H + and OH - on moblty value n the bulk and dffue layer (β ) are 5.17, 7.89, 36.20, and m 2-1 V -1, repectvely (from PHREEQC phreeqd.dat databae). The on moblte at the Stern layer reman unknown. We ued β = β, accordng to the urface conductvty model of Leroy et St al. [15] for lca mneral, to etmate urface conductance. Surface conductance ncreae gnfcantly wth alnty and wth the dfference between ph and IEP (6.5 here) (Fg. 6). Note that we retrct the nvetgated ph range at 10-4 and 10-3 M to have a contant onc trength. 28

30 Fg. 6. The electrcal potental at the 2-plane, the urface te dente of adorbed odum and chlorde at the Stern layer (1-plane), and the urface conductance veru ph for three alnte (NaCl). Accordng to Eq. (19), the Dukhn number (Du) determned ung (1) a normal dtrbuton for the partcle ze a (n agreement at 10-3 M wth the partcle ze dtrbuton meaured by Foy [3]), (2) the etmated urface conductance Σ, and (3) the electrcal conductvty of the bulk electrolyte σ b. The fttng parameter are the ntraaggregate poroty φ and the PSD at 10-4 and 10-2 M Zeta potental from electrophoretc data Accordng to the partcle ze meaurement of Foy at 10-3 M [3] (Fg. 7), ttanum doxde nanopartcle are hghly agglomerated at ph value cloe to the IEP, leadng to 20-µm agglomerate. At extreme ph value (ph = 4, ph =10) agglomerate reach ze 29

31 compred between one and three mcrometer. Thu, κa >>1 and ncreae gnfcantly at ph value cloe to the IEP. A a reult, the Dukhn number low n ampltude except for bac ph where t ncreae trongly (Fg. 7). Zeta potental nferred from electrophore ung our model are n very good agreement wth the ESM predcton aumng ψ 2 = ψ d (Fg. 7). We ue φ = 0.4, whch n accordance wth the ntraaggregate poroty value meaured recently by Xu et al. [59] for P25 (φ = 0.38) from the N 2 deorpton otherm ung the cylndrcal pore model (BJH method). The zeta potental calculaton ung the Smoluchowk equaton are alo preented by comparon. Ther ampltude are not n agreement wth the ESM predcton becaue the Smoluchowk equaton neglect urface conductvty. PSD found at other alnte eem to be phycally realtc (Fg. 8). Sze of agglomerated nanopartcle ph-and alnty-dependent, and ncreae gnfcantly cloe to the IEP and at hgh onc trength. The Dukhn number alo dmnh gnfcantly n thee phyco-chemcal condton. In addton, for 10-4 and 10-2 M, zeta potental calculated from electrophore ung our modelng approach are alo n very good agreement wth the ESM predcton (Fg. 9), except for ph value cloe to the IEP. We ue φ = 0.4. Our modelng reult ugget that the hear plane may be located cloe to the OHP, n contradcton wth the hypothe of a tagnant dffue layer havng a alnty-dependant thckne at the TO 2 water nterface [1, 9]. 30

32 Fg. 7. Input data of the model of Henry [40] and zeta potental veru ph at 10-3 M from electrophore (full trangle, our model; empty trangle, Smoluchowk equaton predcton). The lne on zeta potental fgure the ESM predcton aumng ψ 2 = ψ d. Fg. 8. Input data of the model of Henry [40] veru ph at 10-4, 10-3, and 10-2 M. 31

33 Fg. 9. Zeta potental veru ph at 10-4, 10-3, and 10-2 M from electrophore. The lne are the ESM predcton aumng ψ 2 = ψ d. 5. Concluon We have developed an extended Stern layer model to characterze the electrochemcal properte of ttanum doxde n a 1:1 electrolyte (NaCl). Th model gnfcantly lower the ampltude of electrcal potental at the OHP compared to that of other recent urface complexaton model [1, 9] wthout alterng the qualty of the ttraton data predcton. Henry model [40] ued to convert electrophoretc moblty meaurement to zeta potental value takng nto account agglomerate ze and urface conductance. Electrcal conductvty nde the agglomerate calculated ung the dfferental elfcontent model of Sen et al. [47]. The theory of partcle urface conductance due to the dffue and Stern layer decrbed n depth. By combnng the exce of electrcal charge calculated at the compact Stern layer and n the dffue layer wth Henry equaton [40], we how that the hear plane may be located cloe to the OHP, contradctng the aumpton of the preence of a tagnant dffue layer at the TO 2 /water nterface a mentoned by Bourka et al. [9] and Panagotou et al. [1]. 32

34 In the future, we wll ue a numercal model to convert electrophoretc and electrcal conductvty meaurement to zeta potental value under arbtrary condton ncludng hgh zeta potental ampltude, partcle volume fracton, polydperty of the ample, dffue layer overlappng, and urface conductance. In addton, we wll compare our modelng reult to a much more refned et of experment (phercal and monodpere/polydpere partcle) ncludng electrcal conductvty meaurement. Acknowledgment Th tudy wa done wthn the framework of the NANOSEP Project (ANR-08-ECOT- 009). The author are grateful to French Natonal Reearch Agency for fnancal upport. 33

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38 Table 1. Surface and nterface parameter for the Panagotou et al. model [1] and alternatve model. Parameter Orgnal Reference BSM ESM Bdentate Panagotou et model (TPM) al. model log K log K log K Na (K) -1.7 (-1.1) -1.7 (-1.1) -1.7 (-1.7) -1.4 (-1.1) -1 log K NO * Δz 0 Na Δz 1 Na (K) 0.7 (0) 0.7 (0) n.a. 1 (0.7) 0.45 (0.7) Δz 2 Na (K) 0.3 (1) 0.3 (1) n.a. 0 (0.3) 0 (0.3) Δz 1 NO n.a Δz 2 NO n.a. 0 0 C 1 (F m -2 ) C 2 (F m -2 ) * In the BSM, NO - 3 anon act a ndfferent anon 37

39 Fgure capton Fg. 1. Schematc drawng of the bac Stern model of Bourka et al. [9], our bac Stern model, the trple plane model of Panagotou et al. [1], and our extended Stern model for a negatvely charged urface of ttanum doxde. At a gven pcture, from left to rght: metal on, urface hydroxyl, prmary and econdary water layer, compact layer, dffue layer. Fg. 2. TO 2 urface charge and potental at the head end of the dffue layer predcted by dfferent urface complexaton model at three onc trength n NaNO 3. Lne depct the reference model (TPM) reult whle ymbol depct the reult of alternatve model (BSM, ESM, Bdentate, Table 1). Fg. 3. TO 2 urface properte predcted by dfferent urface complexaton model at three onc trength n KNO 3. Lne depct the TPM reference model reult whle ymbol depct the reult of alternatve model (Table 1). Fg. 4. Schematc repreentaton of the electro-chemcal properte of a upenon of phercal oxde nanopartcle. The partcle ha a local exce of electrcal conductvty at t nterface σ(χ). The urface conductance Σ etmated by ntegratng σ(χ) - σ b over the thckne of the Stern and dffue layer. Fg. 5. Modelng trategy for determnng the zeta potental from electrophore. Fg. 6. The electrcal potental at the 2-plane, the urface te dente of adorbed chlorde and odum at the Stern layer (1-plane), and the urface conductance veru ph for three alnte (NaCl). Fg. 7. Input data of the model of Henry [40] and zeta potental veru ph at 10-3 M from electrophore (full trangle, our model; empty trangle, Smoluchowk equaton predcton). The lne on zeta potental fgure the ESM predcton aumng ψ 2 = ψ d. 38