ENS 06 Paris, France, December 2006

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1 ENS 6 Paris, France, 4-5 December 6 Ming-Chun Chien,Feng-Cheng Yang,an Gou-Jen Wang *, Deparmen of Mechanica Engineering Naiona Chung-Hsing Universiy Taichung 4, Taiwan Insiue of Inusria Engineering Naiona Taiwan Universiy Taipei 6, Taiwan E-mai giwang@ragon.nchu.eu.w ABSTRACT In his arice, a heoreica moe ha akes ino consieraion boh he iffusion an eecric osmosis fuxes o esimae he iffusion curren hrough a nanochanne is propose. Equivaen impeance of a nanochanne is moee as a series connecion of finie RC circuis. The impeances of he eecrica oube ayer an buffer souion in each infiniesima RC circui are represene by a capacior an a resiser, respecivey. Due o he eecrica oube ayer effec, he iffusion inuce curren in he propose moe is posiion epenen. Compuer simuaions an experimens using a nanoporous anoic auminum oxie (AAO) hin fim as he fier o separae eecroysis wih ion concenraion graien are conuce. A high egree coincience beween he heoreica an experimena aa is observe.. INTRODUCTION Diffusion process ue o concenraion graien aows parices o rave from higher concenraion area o ower concenraion area. I is a phenomenon ha is frequeny presene in our ay-o-ay ife. Many usefu appicaions such as iffuser, mixer, reacor, oping in soi sae using iffusion process have been impemene [-4]. Fick s aw an is erivaives [5-7] are use as he basis for mahemaica moeing of iffusion processes. When a hin fim wih porous nanochannes is impemene o separae souions wih ion concenraion graien, iffusion inuce curren can be eece. For ions wih reaivey ow iffusiviy, he consan graien approximaion (CGA) approach [8] can be impemene o moe he inear ecreasing concenraion profie. Base on he CGA moe, he ion iffusion eecrica poenia across a membrane can be esimae. When a membrane wih hrough nanochannes is uiize as he separaing fier o prouce concenraion graien, he simpe CGA moe is no aequae o we escribe he ion iffusion properies. The main reason is ha boh he concenraion graien an osmosis effec affec he ions behaviors insie a nanochanne [9-]. In nano omain, i seems ineresing o have eeper examinaions on he ions movemen insie he nanochannes. Furhermore, he iffusion inuce curren is more aracive when he iffusion mechanism is impemene o he eveopmen of a phoovoaic sysem such as he ye-sensiiviy soar ce [-4]. In his research, a mahemaica moe ha consiers boh he iffusion an eecric osmosis effecs o escribe he propery of he iffusion curren hrough a nanochanne is invesigae. Equivaen impeance of a nanochanne is moee as a series connecion of finie RC circuis. The impeance of he eecrica oube ayer ha causes he eecric osmosis fux is characerize by a capacior an he buffer souion is moee by a resiser. Compuer simuaions an experimens using a nanoporous anoic auminum oxie (AAO) [5, 6] hin fim as he fier o separae eecroysis wih ion concenraion graien are conuce o verify he propose moe.. DIFFUSION GRADIENT THROUGH A NANO POROUS MEMBRANE. Diffusion fux of a membrane Diffusion of ions across a membrane can be escribe by he Fick s aw as, PhD suen Associae Professor Professor Emai: gjwang@ragon.nchu.eu.w jane@ragon.ccu.eu.w TIMA Eiions/ENS6 - - ISBN

2 Ming-Chun Chien, Feng-Cheng Yang an Gou-Jen Wang c ψ D x () ψ enoes he fux of ions ue o iffusion, c is he ion concenraion, x is he hickness of he membrane, D is he iffusion coefficien (m /s). For ions wih sufficieny ow iffusiviy, he concenraion profie across he membrane can be moee by he consan graien approximaion (CGA) [8] as schemaicay iusrae in Figure. Assume he area of he membrane is A, he voume of each vesse is v an v, an he ion concenraion in each vesse is c an c, respecivey. Uner he CGA, he iffusion coefficien of he membrane is consan. The rae of change of he fux can be escribe by, ψ DA ( + ) ψ v v () Combining Equaion () an (), he ion concenraion ifference beween vesse an can be cacuae as, DA c ( ) c()exp( ( + ) ) c()exp( / τ ) v v () Where c()c ()- c () an τ DA [ ( + )] v v. c > c c L D v v c Figure. Schemaic iusraion of he consan graien approximaion (CGA). Ion iffusion inuce eecrica poenia Base on he CGA moe, he ion iffusion inuce eecrica poenia across a membrane wih hrough nanochannes can be cacuae as, κt c V V V n( ) q c (4) Simiar o Equaion (), he above expression can ony escribe he ion iffusion inuce eecrica poenia across a membrane. Insie a nanochanne, boh he concenraion graien an osmosis effec affec he ions behaviors. The fux of ions ha couns boh he concenraion graien an osmosis effec [9-] can be escribe by, Vx µκt c ψ czµ q x (5) The firs iem is he eecric osmosis fux an secon iem enoes he ion iffusion fux. Where c is he ion concenraion, z is he vaence of ions, µ enoes he mobiiy of ions, V x is he eecrica poenia a a isance x from he enrance of a nanochanne, q is he charge on each ion, T is absoue emperaure, κ is he Bozman consan.. DIFFUSION INDUCED CURRENT INSIDE A NANOCHANNEL. Impeance of a Nanochanne The ion iffusion inuce eecrica poenia across a membrane wih hrough nanochannes can be escribe by Equaion (4). To furher ienify he ion iffusion inuce curren hrough he nanoporous membrane, he impeance of he membrane has o be moee. As aforemenione, iffusion ions raveing aong a nanochanne are infuence by boh he iffusion fux an he eecric osmosis fux ha resus from he eecrica oube ayer as schemaicay epice in Figure. The eecrica oube ayer consising of he fixe sern ayer an he iffuse Gouy-Chapman ayer [7], gives o an eecrica poenia (zea poenia) beween he wa of he nanochanne an buk of he buffer souion ha carries he ions [8]. Equivaen impeances of a nanochanne can be moee as a series connecion of finie RC circuis as shown in Figure 4. The capacior an resiser r in each infiniesima RC circui enoe he impeance of he eecrica oube ayer an he buffer souion, respecivey. V c Buffer c Vesse Diffusion inuce curren souion Vesse Nanochanne of membrane Figure. Nanochanne oube ayer r r r Figure 4. Nanochanne impeance moe. Ion iffusion inuce curren r As epice in Figure, he eecric oube ayer inuce poenia a he wa of a nanochanne wi rap caions ha rave aong he axia irecion, resuing in reucing of he ion iffusion curren sage by sage. The V TIMA Eiions/ENS6 - - ISBN

3 Ming-Chun Chien, Feng-Cheng Yang an Gou-Jen Wang equivaen circui of an infiniesima uni wih a isance x from he ine of his nanochanne is schemaicay iusrae in Figure 5. i i -i V i i i V i -V r x x+x Figure 5. The equivaen circui of an infiniesima uni The ime epenen poenia graien an iffusion curren graien aong he axia irecion a posiion x can be escribe as, Vi () rii () x (6) ii() Vi() C (7) Equaions (6) an (7) can be rearrange as, Vi() Vi() rc (8) ii() ii() rc (9) When he foowing parameer subsiuion is appie, he above Equaions become he orinary ifferenia equaions as Equaions () an (). crx y 4 () Vi() Vi() cr x / y 4 () Vi() Vi() cr y () Equaions () an () can be inegrae o obain a n orer ifferenia equaion as, Vi() Vi() + y y y () Hence, Vi () A exp( y ) y (4) an y y Vi () A e y+ B Aerf ( y) + B (5) y y erf ( y) e y where π enoes he error funcion. Equaion (5) inicaes ha V i () ony epens on y. When an x, V i () V () is he iniia voage in vesse. The consans A an B in Equaion (5) are, i A B V () (6) The ion iffusion inuce poenia an curren insie a nanochanne can be sove from Equaion (5) respecivey as, Vi ( ) V ()[ erf( )] τ (7) an V () x ii ( ) exp( ) rx πτ τ (8) 4 τ crx where is he ime consan.. Impeence of Nanochanne To fuy uiize Equaion 8 o esimae he iffusion curren hrough a nanochanne, he uni resisance r an capaciance efine in Figure 5 have o be eermine in avance. In genera, i is ifficu o experimenay measure hem. For a nanochanne as shown in Figure wih hickness, he overa iffusion impeance across i can be moee by [9], D cs () Zs () RW ψ () s (9) where R W enoes he resisance across he channe. R W can be furhere cacuae accoring o, V RW ( ) qad c () The overa capaciance across he nanochanne can aso be esimae in erms of R W as, CW RW D () The uni resisance r an capaciance efine in Figure 5 are hen approximae by Equaion an, respecivey. R W r C C W 4. COMPUTER SIMULATIONS () Compuer simuaions of he iffusion curren hrough an anoic auminum oxie (AAO) hin fim wih nanochannes by using Equaion 8 are conuce. The overa resisance R W an capaciance C W of an AAO hin fim are 8.9 Ω an.8 µf, respecivey []. Accoringy, he uni resisance r an capaciance can be cacuae. TIMA Eiions/ENS6 - - ISBN

4 Ming-Chun Chien, Feng-Cheng Yang an Gou-Jen Wang Figure 6 iusraions he profies of he posiion epenen iffusion curren hrough an AAO hin fim for an eecrochemisry ce as shown in Figure wih iniia ion concenraion raio c ()/c ()6. x -5 iffusion curren(a) x x ( m) Figure 6. Profies of he posiion epenen iffusion curren hrough an AAO hin fim Figure 7 epics he rajecories of he iffusion currens a x for ion concenraion raios 6 an, respecivey. Diffusion curren (A).5.5 x -5 iffusion raio 6 iffusion raio Figure 7. Trajecories of he iffusion currens a x 5. EXPERIMENTAL VERIFICATION To furher verify he vaiiy of Equaion 8, an eecrochemisry bah ha conains wo vesses separae by an AAO hin fim wih pore iameer aroun 5 nm as shown in Figure 8 is bui. The CH67A eecrochemisry anayzer by CH Insrumens is use o measure he ion currens. The eecroysis use is poassium chorie (KC). In genera, pores wih iameer of 5 nm shou aow boh he caion (K +,.7nm) an anion (C -,.8 nm) o penerae simuaneousy. Therefore, he KC eecroysis is no suiabe o verify Equaion 8 ha is vai for iffusion curren inuce by ions wih same poariies. However, here are 4. K + ions an /5 C - ion on average fowing in he channe ue o he eecric osmosis insie a nonochanne []. I is reasonabe o assume ha he caions conribue mos of he iffusion curren. 4 Figure 8. AAO hin fim o separae vesses wih ifferen ion concenraions Figure 9 compares he experimena an simuaion iffusion currens for he coniion of c ()/c ()6. There is ony a sma eviaion beween he heoreica an experimen aa. The vaiiy of he propose moe of iffusion curren hrough nanochannes can be verifie. Diffusion curren (A) x Simuaion Experimen Figure 9. Comparison of he heoreica an experimena iffusion curren 6. CONCLUSIONS Diffusion propery aows ions o rave from higher concenraion area o ower concenraion area. When a hin fim wih porous nanochannes is impemene o separae souions wih ion concenraion graien, iffusion inuce curren can be eece. Diffusion ions raveing aong a nanochanne are infuence by boh he iffusion fux an he eecric osmosis fux ha resus from he eecrica oube ayer. In his arice, a nove moe ha consiers boh he iffusion an eecric osmosis effecs o escribe he propery of he iffusion curren hrough a nanochanne is propose. Equivaen impeance of a nanochanne is moee as a series connecion of finie RC circuis. The impeance of he eecrica oube ayer ha causes he eecric osmosis fux is represene by a capacior an he buffer souion is moee by a resiser. Compuer simuaions using a nanoporous anoic auminum oxie (AAO) hin fim o separae souions wih ion concenraion graien are conuce. An eecrochemisry bah ha conains wo vesses separae TIMA Eiions/ENS6-4 - ISBN

5 Ming-Chun Chien, Feng-Cheng Yang an Gou-Jen Wang by an AAO hin fim is bui o carry ou he experimens. A high egree coincience beween he heoreica an experimena aa is observe. ACKNOWLEDGMENTS The auhors wou ike o aress heir grauae o he Naiona Science Counci of Taiwan for financia suppor uner gran NSC-94--E-5-. The work was conuce in he Cener of Nanoscience an Nanoechnoogy a he Naiona Chung-Hsing Universiy, Taiwan. [9] J. Bisquer, A. Compe., Journa of Eecroanayica Chemisry 449, -,. [] H. J. Oh, K. W. Jang, C. S. Chi, Bu. Korean Chem. Soc. (), 4, 999. [] C. L. Garner, W. Nonner, R. S. Eisenberg, Journa of Compuaiona Eecronics, 5-, 4. REFERENCES [] G. T. A. Kovacs, Micromachine ransucers sourcebook, McGraw-Hi: New York, 998. []S. K. W. Deringer, T. E. McKnigh, an N. L. Jeon, Anayica Chemisry 7, 4-46,. []A. D. Srock, S. K. W. Deringer, A. Ajari, I. Mezic, H. A. Sone, G. M. Whiesies, Science 95, ,. [4]T. J. Johnson, D. Ross, L. E. Locascio, Anayica Chemisry 74, 45-5,. [5]W. E. Aey an B. J. Aer, Physica Review Leers 4(), , 979. [6]M. H. Lee, Physica Review Leers 85(), 4-45,. [7]B. Ph. van Miigen, P. D. Bons, B. A. Carreras, R. Sánchez, European Journa of Physics 6, 9-95, 5. [8] K. A. Snyer, Concree Science an Engineering, (), 6-4,. [9] G. L. Fain, Moecuar an ceuar physioogy of Neurons, Harvar Universiy Press, 999. [] M. H. Lee, Physica Review Leers 85, 4-45,. []J. F. Smaey, M. D. Newon, S. W. Feberg, Eecrochem. Commun., 8-88,. [] F. F. Saniago, J. Bisquer, G. G. Bemone, G. Boschoo, A. Hagfe, Soar Energy Maerias & Soar ces 87, 7, 5. []J. Bisquer, V. S. Vikhrenko, Eecrochem. Aca. 47, ,. [4] W. Preis, W. Sie, Soi Sae Ionics, 86-88, , 996. [5]G. M. Whiesies, J. P. Mahias, C. T. Seo, Science, -9, 99. [6] M. Park, C. Harrison, P. M. Chaikin, R. A. Regiser, D. H. Aamson, Science, 4-44, 997. [7]C. A. Schier, W. Srunz, Eecrochimica. Aca. 46, 69 65,. [8]C. Hiz, A. Lasia, Journa of Eecroanayica Chemisry 5,,. TIMA Eiions/ENS6-5 - ISBN