The nfluence of Stern layer conductance on the delectrophoretc behavour of latex nanophere Mchael Pycraft Hughe* Bomedcal Engneerng Group, Unverty of Surrey, Guldford, GU2 7XH, UK Ncola Gavn Green Boelectronc Reearch Centre, Dept Electronc and Electrcal Engneerng, Unverty of Glagow, Glagow G12 8LT, UK *Correpondng author: Tel +44 1483 686775 Fax +44 1483 689395 Emal m.hughe@urrey.ac.uk Runnng ttle: Stern layer conductance n delectrophore 1
Abtract: The nfluence of the Stern layer conductance on the delectrophoretc behavour of ub-mcrometre zed latex phere examned. The delectrophoretc repone of the partcle meaured and analyed n term of a model of urface conductance dvded nto dcrete component related to the tructure of the double layer. The effect of both co- and counter-on n the bulk oluton on the Stern layer conductance demontrated. Key Word: delectrophore; electroknetc, conductance, electrcal double layer 2
A polarable partcle upended n a non-unform electrc feld experence a force due to the nteracton of the feld and the nduced dpole moment. The reultng movement of the partcle referred to a Delectrophore (1,2). The delectrophoretc force potve (partcle attracted to regon of hgh feld trength) or negatve (partcle repelled) when the partcle more or le polarable than the medum repectvely. An example of delectrophore hown n fgure 1, wth fluorecent latex phere upended n KCl on a quadrupole electrode tructure (3,4). At low frequence, the phere are attracted to regon of hgh feld at the edge of electrode and between adjacent electrode. At hgh frequence the partcle are repelled to the local low feld regon at the centre of the electrode array (3). Th type of latex phere, upended n an electrolytc oluton, undergoe a ngle nterfacal relaxaton proce at a frequency related to the properte of partcle and medum (2). The partcle have a hgh urface charge denty and therefore a polarablty greater than the aqueou upendng medum at low frequence. A a reult, the partcle experence potve delectrophore at low frequence. At hgh frequence the permttvty of the partcle gnfcantly lower than that of water, the partcle are therefore le polarable and experence negatve delectrophore. The delectrophoretc force change gn at a charactertc frequency, generally referred to a the croover frequency. Analy of the croover frequency a a functon of medum conductvty can be ued to charactere the polarablty and a a reult the delectrc properte of a partcle. Th method of delectrophoretc analy ha been demontrated on ub-mcrometre partcle uch a latex bead (5,6) and vrue (7-9). 3
Prevou work (5,10) ha howed that the delectrophoretc behavour of ubmcrometre latex phere domnated by the effect of charge movement acro the partcle urface. In th paper we demontrate that conductance n the double layer nfluenced not only by the movement of counteron at the partcle urface, but alo by the net moblte of both counteron and coon n the bulk medum. The delectrophoretc force FDEP actng on a homogeneou, otropc delectrc phere gven by (2): F DEP 3 2 2 r Re f E [1] m CM where a the partcle radu, m the permttvty of the upendng medum, E the local RMS electrc feld and Re f CM the real part of the Clauu-Moott factor, gven by: f CM * p * p * m * 2 m [2] where m * and p* are the complex permttvte of the medum and partcle repectvely and a general complex permttvty gven by *, wth the angular frequency of the appled feld, the conductvty and the permttvty. Th factor decrbe the frequency dependence of the dpole moment of the partcle and the reultng delectrophoretc behavour of the partcle. 4
Expermental meaurement of the delectrophore of collodal latex partcle ha demontrated that for very mall partcle (le than 500nm n dameter) the effectve partcle conductvty vare a a functon of the medum conductvty (5,10). Recent work by Hughe et al. (10) ha demontrated that the effectve conductvty of a latex partcle can be decrbed a the um of three component: the bulk conductvty of the partcle, pbulk ; and urface component due to the movement of charge n the dffue double layer (conductance K ) and the Stern layer (conductance d K ). The total conductvty of the partcle then wrtten a d 2K 2K p pbulk [3] r r For latex phere, the bulk conductvty neglgble and the effectve conductvty of the partcle domnated by the urface conductance. The conductance of the dffue layer can be expreed a (11): K d 2 2 d 2 4F cz D (1 3m / z ) zf coh 1 RT 2RT [4] where D d the on dffuon coeffcent, z the valency of the counteron, F the Faraday contant, R the molar ga contant and T the temperature. the recprocal Debye length, gven by 2czF RT 2,c the electrolyte concentraton (mol m -3 ), and the electrotatc potental at the boundary of the lp plane. The 5
dmenonle parameter m decrbe the electroomotc contrbuton to d K and gven by (11) 2 RT 2 m m [5] d F 3 D where the vcoty. In prevou work (10) the conductance of the Stern layer wa gven by the expreon: K u [6] where u the urface charge denty and the moblty of the counteron. However, analy of expermental reult ung latex phere upended n dfferent electrolyte wth dentcal counteron but dfferent coon (5), how that equaton [6] doe not decrbe thee expermental reult correctly. Returnng to the defnton of the Stern layer conductance, the Dukhn number Du (11) for a Stern layer contanng one on pece gven by u Du [7] 2rz Fc m where μ and μm are the moblte of the on pece n the Stern layer and bulk medum repectvely. The Dukhn number decrbe the rato between urface and bulk conductance and gven by (11) 6
K Du [8] r m where σm the conductvty of the bulk electrolyte. Combnng equaton [7] and [8] and rearrangng we obtan the followng expreon for K : K u [9] 2z Fc m m If the electrolyte ymmetrcal, t poble to replace the conductvty term and concentraton c wth molar conductvty Λ (S m 2 mol -1 ): K u. [10] 2z F m Value of Λ are contant for gven electrolyte (a table of value gven by Bockr and Reddy (12)). Equaton [10] reduce to [6] only f the value of the moblte of the co- and counteron are equal, n whch cae the value of 2z Fm goe to 1. The valdty of equaton [10] wa teted by meaurng the delectrophoretc croover frequency for latex phere n dfferent electrolyte. The latex phere (Molecular Probe, Oregon, USA) were 216nm n dameter and loaded wth fluorecent dye. The upendng electrolyte were aqueou oluton of KCl and K2HPO4-KH2PO4 (a 50:50 mxture of monobac and dbac potaum phophate) wth conductvte n the 7
range 10 4 to 10 2 Sm -1. The ph of the two oluton wa meaured at 6.5 and 7.1 for the KCl and K2HPO4-KH2PO4 repectvely; nce the ph of the meda are mlar, we aume that the number of docated carboxyl group on the bead mlar for the bead n both meda. The upenon of partcle were placed on mcrofabrcated electrode tructure energed ung a gnal generator (3-5). The partcle movement wa oberved ung a Photonc Scence Coolvew HS lght-ntenfyng camera and a fluorecence mcrocope. The croover frequence of the bead n the two meda are hown n fgure 2. Three key feature are ued n the bet-ft analy of undecrbed homogeneou phere; the value of the croover frequency n low conductvty meda, the gradent of the re n croover a a functon of ncreang medum conductvty, and the conductvty at whch the croover drop by a decade or more. However, where the nternal properte of the partcle are known and we are ntereted n determnng the urface properte only, we are ntereted only n the croover behavour below the charactertc drop n conductvty. In th regon, the delectrophoretc behavour dctated by the urface charge denty whch determne the low-conductvty croover, (or y-ntercept n fgure 2) and the potental affect the apparent ncreae n partcle conductvty wth medum conductvty (the lope of the graph n fgure 2). The conductvty at whch the frequency drop a factor of the two urface conductance component and provde no extra nformaton; where the nternal properte of the partcle are known, we need only ue two parameter the y-ntercept and lope to determne the urface conductance by bet-ft method. 8
Bet-ft lne of equaton [1], ung equaton [3] and [4] to decrbe the partcle conductvty are alo hown. Relatve permttvte for the upendng electrolyte and partcle, of 78 and 2.55 repectvely, were aumed. The bet value were K = 0.85nS (±0.15nS) and = 110mV (±5mV) for KCl and K = 1.25nS (±0.1nS) and = 120mV (±10mV) for K2HPO4-KH2PO4. We can analye thee value to determne the valdty of equaton [10]. Snce carboxylated bead are negatvely charged, the on pece nvolved n urface conducton K +. We aume n modellng the data that the parameter for potental and K are contant over the range of conductvte tuded; n realty, both wll reduce lghtly a conductvty ncreaed, but the magntude of th change over the range of molarte of the electrolyte ued hould be relatvely mall (ee [11] ecton 4.6e). For latex phere upended n KCl and K2HPO4-KH2PO4, equaton [6] ndcate that the Stern layer conductance the ame and that the nfluence of th conductance on the delectrophoretc behavour hould alo be the ame. However, the bet-ft value from the expermental data gve the rato between K n K2HPO4-KH2PO4 and KCl a 1.47. If we now conder the defnton of the Stern layer conductance gven by equaton [10], we would expect that the rato between K n K2HPO4-KH2PO4 and KCl to be gven by the rato of the molar conductvte of the two electrolyte, all other factor beng equal. The molar conductvte of K2HPO4-KH2PO4 and KCl are 21.2mSm 2 mol -1 and 14mSm 2 mol -1 repectvely (12), gvng an expected rato between the Stern layer conductance of 1.51 n good agreement wth the bet ft value. In order to enure that the effect beng oberved not merely the reult of the varaton n on concentraton for the two electrolyte, the croover data wa analyed 9
a a functon of K + concentraton rather than conductvty; th graph (not hown) demontrate trend mlar to the frequency-conductvty plot hown n fgure 2, ndcatng that the oberved effect are not due to the dfference n K + concentraton n the two electrolyte. Applcaton of the ame analy to the data preented n Green and Morgan (5) for mlar latex bead n oluton of KCl, NaCl and K2HPO4-KH2PO4 (not hown) ndcate mlar reult to thoe preented here. Th addtonal data ndcate the ame relatonhp between KCl and K2HPO4-KH2PO4 a decrbed above.e. matchng gve dfferent value for K, for the ame counteron but dfferent coon. In addton, the data alo ndcate that NaCl alo can be ftted ung the argument preented n th paper,.e. that the value of K for NaCl and KCl are mlar, for the ame counteron but dfferent coon. Meaurement of the electrophoretc moblty of latex phere of varou ze (5) demontrated that the urface charge denty of latex phere n the range 22-40mCm -2, wth 33mCm -1 fallng wthn the tolerance range of all bead ze. Conderng the above expermental data for bead n KCl and the deduced value of K, t poble to etmate the value of the Stern layer moblty. Conder the data for latex bead n KCl preented above. KCl ha a molar conductvty of 14.1mSm 2 mol -1 and the bulk moblty of the K + counteron 7.69 x 10-8 m 2 V -1-1. Ung thee value wth the etmated value for K, we calculate from equaton [9] a value for on moblty wthn the Stern layer of 2.73x10-8 m 2 V -1-1, or approxmately 35% of the bulk on moblty. Th reult n keepng wth Lyklema aerton (11) that the moblty of counteron n the Stern layer hould be cloe to, or 10
omewhat lower than, the correpondng bulk value. If we calculate the value of K u ung the above data we fnd of 2.575 x 10-8 m 2 v -1-1,.e. the value of 2z Fm approxmately 0.95 and the approxmaton n equaton 6 hold. Th reult mportant n that t demontrate the mechanm by whch the co-on n oluton affect the Stern layer moblty, and hence the urface conductance, n lne wth oberved relt. Acknowledgement Th work wa funded by a Royal Socety reearch grant to MPH and a Royal Academy of Engneerng Potdoctoral Reearch fellowhp to NGG. The author wh to thank Davd Lamb for the ue of the fluorecence mcrocope. 11
Reference 1. Pohl H.A., Delectrophore, Cambrdge Unverty Pre, Cambrdge 1978 2. Jone T.B., Electromechanc of partcle, Cambrdge Unverty Pre, Cambrdge 1995 3. Green N.G., Ramo A. and Morgan H., J. Phy. D: Appl. Phy. 33, 632-641 (2000) 4. Hughe M.P. and Morgan H J. Phy.D: Appl. Phy. 31, 2205-2210 (1998) 5. Green N.G. and Morgan H., J. Phy. Chem. 103, 41-50 (1999) 6. Hughe M.P. and Morgan H., Anal. Chem 71, 3441-3445 (1999) 7. Morgan H. and Green N.G., J. Electrotatc 42, 279-293 (1997) 8. Hughe M.P., Morgan H., Rxon F.J., Burt J.P.H. and Pethg R., Bochm. Bophy. Acta 1425, 119-126 (1998) 9. Hughe M.P., Morgan H. and Rxon F.J. Eur. Bophy. J. 30, 268-272 (1998) 10. Hughe M.P., Morgan H. and Flynn M.F. J. Collod Interface Sc. 220, 454-457 (1999) 11. Lyklema J., Fundamental of Interface and Collod Scence, Academc Pre, London, 1995 12. Bockr J.O M., Reddy A.K.N., Modern Electrochemtry, Plenum, New York, 1970 12
Fgure Legend Fgure 1. Fluorecence mcrograph howng potve and negatve delectrophore of 216nm dameter latex phere. Four gold electrode (vble a four barhaped, paler regon termnatng n pont) are energed uch that oppong electrode hare the ame phae, whlt adjacent electrode are n antphae. Fgure 1a how potve delectrophore (partcle attracted to regon of hgh electrc feld; that, n the regon nbetween adjacent electrode), whch occur at lower frequence; fgure 1b how negatve delectrophore (repulon from regon of hgh electrc feld, caung collecton at a feld null at the centre of the electrode) whch occur at hgher frequence. The tranton frequency between thee mode of behavour called the croover frequency, and related to the delectrc properte of partcle and medum. The bead were upended n a 1mSm -1 KCl oluton, at frequence of 1MHz and 10MHz, and appled potental 5Vpk-pk, repectvely. Scale bar: 20m. 13
Fgure 2. Expermental DEP croover pectra for 216nm dameter latex bead. The vertcal lne how the range of frequence n whch delectrophoretc croover wa oberved for bead upended n KCl (flled quare) and K2HPO4-KH2PO4 (empty quare) oluton of varyng conductvty. The dagonal curve ndcate the bet ft model to thee data. The value for bet ft ndcate that the value of K and are 0.85nS and 100mV for the bead upended n KCl, and 1.25nS and 120mV n K2HPO4-KH2PO4 repectvely. In both cae the bead permttvty wa 2.250 and nternal conductvty wa neglgble. 14