Physical and Interfacial Electrochemistry 2013 Lecture 7. Material Transport in Electrochemical Systems.

Transcription

1 Physial and Interfaial Eletrohemistry 3 Leture 7. Material Transport in Eletrohemial Systems. Material Transport Proesses 3 main mehanisms: iffusion Mass transport in onentration gradient Migration (ondution) Mass transport in potential gradient Convetion Mass transport via hydrodynami flow iffusion and onvetion most important in eletrohemistry sine eletromigration is usually suppressed in experiments. Material transport in a ondensed phase may be desribed from either a marosopi (in terms of partial differential equations, irreversible thermodynamis) or a mirosopi (essentially statistial or stohasti) viewpoint. Material flux. Rate of material transport is defined in terms of a material flux f (also denoted as J ) with units mol m - s -. This is defined as the quantity of material flowing through a referene plane of area A per unit time. d The flux of a speies through an element of area d is a vetor quantity given by f Molar onentration (mol m -3 ) v v Fluid veloity vetor ms -

2 iffusive Material Transport iffusive material transport driven by presene of onentration gradient. Mathematially desribed by the Fi iffusion equations (material flow expressions diretly analogous to Fourier heat flow (energy transfer in a temperature gradient) expressions. Marosopially, mathematial expressions previously developed for heat flow an be used with suitable modifiation to orresponding material transport problem. Steady State Fiian iffusion. High solute onentration Net diffusive flux Low solute onentration We onsider two ases: steady state (time independent) diffusion and transient (time dependent) diffusion. In the first ase = (x) only whereas in the seond ase = (x,t). Assume driving fore for diffusion is proportional to hemial potential gradient. We introdue a diffusive pseudo-driving fore F whih is related to the gradient in hemial potential via: F F d dx A.E Fi We now assume that diffusive pseudo fore F is assoiated with a diffusive Flux f whih serves as a measure of the diffusion rate. We use irreversible thermodynamis to relate flux and fore: f F F... Assuming F is small orresponding to small departure from equilibrium (linear irreversible thermodynamis) we neglet terms in F and higher. Also Sine if F = then f = (no effet exists without the orresponding ause) then =. This leaves us with the assignment that we have a linear relationship between Flux (the effet) and driving fore (the ause): Introduing the definition of hemial potential: RT ln a a d dln RT d RT dx dx dx d f F dx RT d d RT dx dx Ideal solution approximation Ativity oeffiient =. d f F dx We introdue the phenomenologial diffusion oeffiient (unit: m s - ) RT We arrive at the Fi equation of steady state diffusion: f f - d dx 3-

3 More generally if the solution is not ideal the ativity oeffiient differs from unity. f d d d dx d d ln d ln dx Note that typially the diffusion oeffiient is independent of onentration. iffusion oeffiients vary over many orders of magnitude. Typial values are: Ions in aqueous solution: -5-6 m s -. Ions in polymeri matries: -8-3 m s -. Solid state diffusion: m s -. High hemial potential Negative hemial potential gradient iffusive flux iffusive riving fore Low hemial potential Time ependent (T) Fiian iffusion. Inward flux f x Unit area We onsider a diffusing speies passing through a retangular volume element. Outward flux f x dx f f x x (x) Inward flux x x x dx Outward flux x dx x x x dx dx x dx f f dx x f x dx f dx x f x x x dx ( x dx) dx x x x 3- - t t Equation of ontinuity f x t t Fi T diffusion Equation. 3

4 General analysis of ion transport in eletrolyte solution. We onsider the fores ating on an ion in a visous medium suh as an eletrolyte solution (assumed strutureless). Charged speies move under the influene of a gradient in eletrohemial potential with a driving fore F. Movement of the ion in the visous solvent is opposed by a visous fritional fore F v proportional to the veloity v of the ion. Total fore F F F m Initial ondition t v v v dv dt v N dv v dt m N A exp t N A m A LE in veloity v t v ss F v N Steady state (SS) veloity A F F v N A v Frition oeffiient Units:gs - Material flux of ion in solution (taes iffusion and migration into aount). f v ss A F u N u N A Eletrohemial mobility Units: mol g - s - f Steady State material flux expression for speies j in fluid may be derived. j z jf j j j j jv RT f f j t j, f j, M f j j, C iffusion Migration Convetion Equation of ontinuity (fluid mehanis) relates time rate of hange of speies onentration with the divergene of the material flux. Hene relates steady state and time varying material transport equations. j z F j j j t RT Time dependent material transport equation for flow of speies j in fluid medium an then be derived. v iffusion Migration Convetion j j In many situations in eletrohemistry we wor under onditions where an exess of supporting eletrolyte is used and so the migration omponent an be negleted. This results in the onvetive - diffusion equation. This expression relevant to desribe transport in flowing streams or to rotating eletrodes. j j j jv t In a stationary (unstirred) solution and if the eletrode does not rotate then we an neglet fored onvetion and we have the Fi time dependent diffusion equation. j j j t This equation is solved subjet to the stipulation of speified initial and boundary onditions via standard analytial or numerial tehniques to obtain an expression for the onentration profile of reatant as i funtion of distane and time (r,t). The urrent response may be subsequently derived via differentiation with respet to distane. 4

5 5 Planar iffusion Retangular Cartesian Co-ordinates (x,y,z). t z y x,,, x y z Cylinderial iffusion Cylinderial Polar Co-ordinates (r,,z) r P(r,,z) z y z x ),,, ( t z r Spherial iffusion Spherial Polar Co-ordinates (r,,) P(r,,) r z y x ),,, ( t r P(x,y,z) Choie of o-ordinate system depends on type of eletrode geometry. Co-ordinate System Fi iffusion Equation Cartesian Cylinderial Polar Spherial Polar z y x t z r r r r r t sin sin sin r r r r r r t t Time dependent Fi diffusion equation (TFE) in various Co-ordinate systems. Primary objetive is to solve the TFE in a form appropriate for a partiular eletrode geometry, subjet to various initial and boundary onditions to obtain expressions for both the onentration profile and the onentration gradient.

6 iffusion in a medium of infinite extent: a prototype diffusion problem. t= t=t t=t t x= x= x= x= We onsider an infinite volume of pure solvent at equilibrium. We let a sharply defined spie of solute be introdued into the entral region of the solvent at time t =. This defines the origin at x =. The solute diffuses through the solvent until there is a homogeneous dispersion of solute moleules throughout the solvent. The problem is to quantify the diffusive spread of solute through the medium in either diretion from the origin as a funtion of distane and time. t x x, x, t x exp 4t Initial ondition 4t The solution to the initial value problem is a Gaussian funtion. t x x, x x, t exp 4t 4t Short times Initial injetion t = -3 t =.5 x - t = - t =. t =. / 3 Long times Limx, t x x Solute spreads out -4-4 x 6

7 Passive diffusion in a finite membrane. We examine the diffusion of a moleule through a membrane of finite thiness L from a donor ompartment to an aeptor ompartment. This is a basi model for the transdermal delivery of drugs (niotine path). t x L x,, t L, t onor ompartment x t u L L u u membrane u, u, u, fl u Aeptor ompartment Normalised variables Normalised Initial & boundary onditions iffusion Flux (rate) = partition oeffiient = onentration in membrane We sale the boundary value problem and express in dimensionless variables. n exp sin u, n n iffusive flux Total amount transferred through Membrane. Membrane onentration profile n u N N L n d n exp n n exp n n n L 6 Potential step hronoamperometry : reversible proess. We desribe the transient eletrohemial tehnique alled Potential Step Chronoamperometry (PSCA) in whih a potential step of magnitude E is applied to an eletrode and a Nernstian eletrohemial redox transformation ours in response to the potential step perturbation ausing a urrent to flow. We solve the Fi diffusion equation for the problem to derive an expression for the transient Faradai urrent response as a funtion of time. Normalised reatant onentration profile (x,t)/. Transient urrent response i(t) E Pulse width ( s) Expt. duration Transient oulometri response q(t) C R = mm =. x -5 m s - Potential Profile E =.8 V 7

8 t x x t t,,, x t Red(aq) Ox(aq) ne d d d d x,, x,, erf Aerf Berf erf erf erf erf B A A erf erf x x R x, t erf erf t t x x O ( x, t) erf erf t t Error funtion erf erf z z exp y dy z erf z exp y z Error funtion omplement dy Reatant R ompletely oxidized at eletrode surfae iffusive depletion (diffusion layer) C R = mm =. x -5 m s - 8

9 f f d erf dz i nfa z exp z x x erf exp t t. t t iffusion layer thiness Typial hronoamperometri urrent Transient response profile to applied Potential step. Charateristi diffusion time i( t) nfaf nfa t Cottrell Equation di nfa S C d( t ) i iffusion ontrolled onditions. t Cottrell Plot Potential step hronoamperometry. Apply potential step of amplitude E to an eletrode. Redox reation ours as a result of the potential step perturbation. Measure resulting urrent variation with time. Analyse i vs t response urve to obtain transport and ineti information. If ET proess is Nernstian (faile ET) then the transient urrent response is governed by the Cottrell equation. The diffusion oeffiient may be obtained readily. If both and are both nown then eletrode area A may be evaluated via PSCA analysis. nfa i t / S t C Potential E Current i / t E Time t Time t E E Cottrell Plot n F iffusion oeffiient i SC A S C t -/ 9

10 Nernst diffusion layer As potential step experiment proeeds the depletion of reatant speies near the eletrode surfae inreases. This is termed the diffusion layer of the eletrode. Eletroative material must diffuse aross this zone in order to reat at the eletrode. Hene as time progresses the diffusion layer beomes thier and the rate of diffusive transport falls and so does the observed diffusion limited urrent. iffusion layer thiness iffusion limited flux Charateristi diffusion time iffusive mass Transport oeffiient f x nfa i nfaf At long times however, rather than the urrent falling off to zero as predited by the Cottrell equation, the experimentally measured urrent tends to an approximately steady value. This is onsistent with a model in whih The bul solution beyond a ritial distane d from the eletrode is well Mixed due to the onset of natural onvetion (movement of the solution brought about by density differenes), so that the onentration of the eletroative speies is maintained at a onstant bul value. In the Nernst model the onentration hanges linearly in the diffusion layer and the diffusive flux and diffusion limited urrent is readily determined from Fis st law. ouble layer region Interfaial ET Eletrode Transport via Moleular diffusion only iffusion layer (stagnant) A ET B A Transport and inetis in eletrohemial systems: Nernst diffusion layer approximation. Simple diffusion layer model neglets onvetion effets and also simplifies analysis of Fi diffusion equations. Important to note that NL model is approximate. In reality the zone of mixing via natural onvetion and the stagnant zone of diffusion merge into one another. Bul solution (well stirred) Material flux f Hydrodynami layer

11 istane sales in eletrohemistry Eletrode. nm nm nm. m m m Inner Helmholtz Plane Outer Helmholtz Plane iffuse ouble Layer Aqueous solution iffusion Layer. mm mm Hydrodynami Boundary Layer distane Potential Step Chronooulometry. The harge q(t) passed during the appliation of a potential step an be readily evaluated via integration of the Faradai urrent transient response urve. t t nfa / qt () itdt () t dt nfa t / Analysis indiates that the harge inreases with the square root of eletrolysis time. The extent of eletrolysis expressed in terms of the amount of reatant speies onsumed an also be readily evaluated and is shown to depend on the eletrode area, the reatant onentration and on the square root of time and reatant diffusion oeffiient. q( t) N( t) A nf Anson equation t q( t) ql qads nfa t / q(t) The total hronooulometri response ontains ontributions from the Faradai ET reation, double layer harging and possibly, reatant adsorption. A plot of q(t) versus t / is linear and is termed an Anson Plot. I S A A nfa q L nfa t Anson Plot

12 Charging and Faradai urrents. Voltammetri measurements rely on the examination of ET proesses at solid/liquid interfaes. Analytial uses of voltammetry rely on measuring urrent flow i i F i C as a funtion of analyte onentration. However applying a potential programme to an eletrode neessitates the harging of the solid/liquid interfae up to the new applied potential. This auses a urrent to flow whih is independent of the onentration of analyte. Hene the observed urrent is the sum of two ontributions, the harging urrent and the Faradai urrent. The Faradai urrent i F is of primary analytial interest sine this quantity is diretly proportional to the bul onentration of the analyte speies of interest. i Eletrode i C i F C L R CT R S Solution The objetive is to maximise i F and minimise i C. Note that i C is always present and has a onstant residual value. As the onentration of analyte dereases i F dereases and will approah the value of the residual value of i C. This plaes a lower limit on analyte detetion and hene on the use of voltammetry as an analytial appliation. Potential Step Chronoamperometry: ouble Layer Charging. At the beginning of a potential step experiment the urrent flowing is mainly attributed to double layer harging. The urrent flowing at short times (tens of milliseonds or less) is due to supporting eletrolyte ion movement near the eletrode whih auses the interfaial double layer to form. We model the interfae region in the absene of Faradai ativity as an equivalent iruit onsisting of a resistor R S and a apaitor C L in series. We alulate the harge flowing when a potential E is applied aross this RS series assembly. We introdue the relationship between urrent and harge to obtain a differential equation whih may be readily integrated subjet to a suitable initial ondition using standard methods (e.g. Laplae Transforms). E ir S q C L dq i dt dq q dt R C S t q L E R S q t EC L dq i ( t) dt t exp RSC We predit that a plot of ln i versus time t should be linear with a slope S = /R S C L and interept I = ln(e/r S ). Hene the solution resistane R S and the double layer apaitane C L may be evaluated from this semi-logarithmi plot. L EC E t exp RS RSC L L t exp E ln i ln RS RC time onstant R S C L R C The urrent harging the double layer apaitane drops to /e or 37% of its initial value at a time t = and to 5% of its initial value at t = 3. Hene if R S = Ω and C L = F then t = s and L harging is 95 % omplete at t = 6 s. S L t

13 Charging urrent and Faradai urrent ontribution an be omputed for various transient eletrohemial tehniques E F i E E f E i i F (t) Charging urrent deays muh more rapidly than Faradai urrent. i F i i i F C E i i C E t exp RS RSC L i C (t) nfa t / i F / / t Tae reading when i C is small. Potential step tehnique. A similar quantitative analysis an be done for other EC methods suh as yli and linear potential sweep voltammetry. = f /f Reversible ET Eletrode A B Heterogeneous ET Solution Situation pertains when ET inetis are fast and Nernstian. A = F(E-E )/RT f a Mass transport a Normalised steady State diffusion equations Boundary onditions f f exp exp u v u u v v u v a b x u v a a F RT v u v ln u a b u v a a exp E E Normalised urrent Nernst equation 3

14 = f /f Eletrode Heterogeneous Mass transport ET A B Solution A F(E-E )/RT Irreversible ET Normalised Current / F RT when f f / ln exp E E / Situation pertains when ET inetis are very sluggish Quasi reversible ET = f /f Rapid ET.5 Slow ET Most general situation: Intermediate between = F(E-E )/RT reversible and irreversible limits. f exp Normalised urrent f exp exp 4

15 General shape adopted by urrent/voltage urve for a simple outer sphere redox ouple A/B. Region of mixed Rate ontrol. Both ET and MT Important..5 iffusion : material transport due to existene of a onentration gradient Material transport via diffusion rate limiting Eletrode Heterogeneous ET Interfaial ET Kinetis rate limiting Normalised urrent A B Solution A B Mass transport = F(E-E )/RT Normalised potential ox A B e B red e A Net urrent zero. Interfaial ET balaned. Nernst eqn. pertains. Normalised flux f a exp exp exp exp b a F RT E E General urrent/ potential response for outer sphere ET proess, both forms of redox ouple present in solution. Heterogeneous ET A B small limit ET ontrol Redution large limit MT ontrol Oxidation A = F(E-E )/RT B Mass transport 5

16 Random wal model of diffusion We now examine diffusive matter transport from a mirosopi point of view. The most appropriate desription of moleular movement on a mirosopi sale is that of a random or stohasti proess. Moleular displaements are best analyzed using statistial methods and probability theory. The trajetory or path of a diffusing moleule is desribed in terms of a random wal. The diffusing partiles move in a series of small unorrelated random steps and gradually move away from their original positions. We assume that the partiles an jump through a distane L and do so in a time. Hene the distane overed by a partile in a time t is given by (t/)l. origin Step length L We onsider the following problem. The partile starts off from x = and an move right or left: eah hoie is equally probable hene the probability = ½ for eah ase. In a random proess eah suessive step is independent of the previous one (analogous to the meandering of a drunen undergraduate). The diretion whih the partile will tae is not influened by preeding steps. The problem is to determine the probability that the partile will be found at a distane x from the origin at a time t after taing a ertain number n of steps. This orresponds to a speifi wal (experiment). We then repeat the proess many times in order to obtain good statistial averaging (one has many wals) distane uring this time interval the diffusing partile will have taen n steps where the net displaement s=t/. The total number of steps is given by n = n R + n L. Also the net distane travelled, x= n R L n L L, where L is the step length. Hene n R n L = x/l. The probability of the partile being loated at a distane x after n steps of length L is simply given by the probability that of the n steps, n R ourred to the right and n L ourred to the left. Sine eah step an our in one of two diretions (left or right) eah with a probability of ½, then after n steps, the probability of a right step or a left step is (½) n. We now must determine the number of ways of taing n R steps in a total of n steps. This problem is the same as determining the number of ways of hoosing n R objets from a box ontaining n objets irrespetive of the order. This an be readily alulated from simple probability theory as the binomial oeffiient: n! /n R!.n L! = n! /n R!(n-n R!). Hene the probability P n (x) of being at position x after taing (in a total of n steps) n R steps to the right and n L = n-n R steps to the left in any order is simply obtained by multiplying the probability of this sequene, (½) n, by the number of possible sequenes of suh steps (the Binomial oeffiient). This yields the Binomial istribution Funtion. n n! Pn ( x) n! n n! R s x R L x n nr nl, nr nl L x nr n n s L x n nr n n s L P n s Binomial istribution Funtion n! n ns ns!! 6

17 Random Wal Binomial istribution Funtion. The Binomial distribution funtion has been alulated for n =,, and 5 and presented in histogram form. We note that as the number of steps n inreases, then the distribution funtion P n (s) beomes more defined and approahes the symmetri bell shape harateristi of the Gaussian distribution. We an alulate the average and root mean square displaement using standard methods. Hene the width of the probability distribution varies as n /. n The mean square distane travelled by a diffusion partile after s spn s n steps, eah step being of length L an also be readily s evaluated. This is alled the mean free path. s s rms n s s P s n s n n s x rms rms x x L s rms rms nl L nl When n is very large one an show (the maths are tedious) that the Binomial distribution funtion hanges to a Gaussian funtion. s P n s exp n n Now only even values of s are allowed if n is even And only odd values are allowed if n is odd. Hene this expression is valid for n and s values of Similar parity (even/odd harateristi). P n (s) will be zero if there is a differene in parity between n and s. Hene we must develop an expression for the probability as a funtion of the distane parameter x. 7

18 We therefore alulate the probability that a partile is loated in an interval between x and x + dx at some time t. P n n s n n x dx nl L x nl xdx P sds exp ds exp exp dx n nl We now assume that the diffusing moleule taes z steps per unit time and we note that n = zt = t/. Hene the probability expression redues to the following form. x We reall the solution to the Fi diffusion Px, tdx exp dx equation for infinite diffusion from a point L zt zl t soure. Hene the probability that a partile whih was initially x x, t exp loated at x = at t = will be found at a position x at 4t 4t a subsequent time t an be readily defined. The expression provides an expression for the diffusive Hene both the marosopi solution to spread as a funtion of time. the diffusion equation and the mirosopi solution derived via x appliation of probability theory to many Px, t exp Lzt zl t small steps of a randomly diffusing moleule, yield the same result The fundamental step frequeny z = / (where denotes the fundamental jump time) and the mean free path L are now ombined together in a new onstant fator alled the diffusion oeffiient. This results in a modified form of the probability for diffusive spread. x Px, t exp L zl 4t 4t We reall that <x > = nl and also n = zt. Hene the mean square distane travelled in a time t is <x > = zl t. Hene the rms distane travelled is related to the diffusion oeffiient as follows. x rms x ztl zl t t Einstein-Smoluhowsi Equation iffusion regarded as an ativated rate proess (Eyring). L We an adopt a more detailed view of the fundamental random hopping event. We - have introdued two fundamental parameters, the mean jump distane L and the jump frequeny z = /. The diffusion oeffiient is related to these two quantities L via well defined expressions whih depend on whether the diffusive proess is or We now examine the struture of a liquid medium in whih diffusion is taing plae. The struture of a liquid is loal in extent and transitory in time and mobile in spae. uring a random wal the diffusing speies jumps out of one site in the liquid struture and enters into another adjaent site. The mean jump distane L represents the average distane between sites and will therefore depend on the detail of the liquid struture. There is a formal analogy here between ion jumping and hemial reation AB + C A + BC, sine formally we have a jump of an atom B from a site in A to a site in C. Both are ativated rate proesses. We disuss the approah adopted by Eyring to diffusion when it is viewed as an ativated rate proess. Oupied site ion Vaant site Before ion jump Vaant site ion After ion jump Oupied site The fundamental idea behind the Eyring approah is that the potential energy (or the Gibbs energy) of a system of partiles involved in a rate proess varies as the partiles moves to aomplish the proess. If the Gibbs energy of the system is plotted versus the position of the moving partile then the energy of the system has to attain a ritial value termed the ativation free energy G * for the proess to be aomplished. 8

19 Pre-jump site Jump distane L Jumping speies Post-jump site Gibbs energy * G Ativated state System has to ross an ativation energy barrier for rate proess to our. Pre-jump site Post-jump site Position The number of times per seond, z, that the rate proess ours (the jump frequeny) for diffusive hopping from site to site an be evaluated via the Eyring equation. * BT G BT S z exp exp h RT h R * H exp RT * We develop the Eyring expression for the diffusion oeffiient as follows. L L BT G exp h RT * L BT S exp h R * H exp RT * Numerial fator defining imension of diffusion proess. = ½ (-), = /6 (3-) Ativation energy for diffusion We an define an Arrhenius equation for diffusion. The diffusion oeffiient exhibits an Arrhenius type behaviour with hanges in temperature T. Typially the ativation energy for diffusion is a. 5 J/mol. A E A, exp RT BT S A L e exp h R * ln E A, d ln R d(/ T ) Pre-exponential fator Base of natural logarithm /T 9

20 Linear sweep and yli voltammetry. Linear sweep voltammetry (LPSV) and yli voltammetry (CV) are the most well nown and popular eletrohemial tehnique in use today. In both tehniques a defined time varying potential ramp is applied to a woring eletrode ausing an interfaial ET reation to our. The urrent flowing from this proess is monitored and the resulting plot of urrent versus applied potential (the voltammogram) is reorded. Again the quantitative basis of the method is the solution of the time dependent Fi diffusion equation. This initial and boundary value problem annot be solved totally analytially but the integral equation obtained whih desribes the voltammetri urrent response has to be solved numerially. Normalised Current i nfa nf RT a Linear Potential Sweep Voltammetry. Reversible ET Niholson-Shain Current funtion NS Integral equation for normalised urrent ` d exp Normalised variables a b u v a a nf t RT nf E t E RT E t E i t i nf x RT Integral defining normalised urrent E P E. 3V 4 seh d P.446 iffusive boundary value problem u u v v u u v v u v u v exp Randles Sevi equation for pea urrent i P nfa.446 nfa nf RT a P nf RT a

21 Analytial Approximation to LPS Voltammogram for reversible ET.5 NS () NS () nf(e-e )/RT Numerial Solution Not good for large Analytial Approximation nf(e-e )/RT a.9933 b. 3 NS ` NS a a b b b a.944 b.5 b.3349 i nfa nf RT a 3 d exp NS exp Prasad & Sangaranarayanan JEC, 3 LPS Voltammetry. Reversible ET = -5 m s - ; T = 98K ; n = ; =. M Sweep rate =. Vs - igisim Programme

22 Cyli voltammetry (CV) is a form of eletrohemial spetrosopy. In CV the potential applied to the eletrode is swept in a linear manner from an initial value E i to a given intermediate value E l, and then the diretion of the potential san is reversed to a final value E f. Usually the initial and final potentials are the same. The output obtained is alled a voltammogram and is a plot of urrent versus applied potential. A series of peas appears at definite potentials orresponding to the ourrene of oxidation and redution reations at the eletrode surfae. A major experimental variable is the sweep rate de/dt =. This defines the experimental time sale. Typially the shape of the voltammogram as a funtion of sweep rate is examined. Pea potentials E P, pea urrents i P, pea separations E P and pea widths at half pea heights d are measured as a funtion of sweep rate. Cyli Voltammetry (CV) Potential E i E f E E i time E E f Cyli Voltammetry. Reversible ET. CV profile obtained via igital Simulation (using ommerial software, CH Instruments) of Time ependent Fi iffusion equations for A and B. Impliit Finite ifferene Method Used. Analytial solution of diffusion equations even for the simple ase of reversible ET, results in the generation of an integral equation whih an only be solved numerially. Hene digital simulation is a powerful tehnique when more ompliated situations (irreversible ET, oupled ET and homogeneous hemial reations) are onsidered. CV profile for a reversible ET proess will exhibit sharp and well defined peas with a defined pea separation. Redution A Oxidation e B E =.3 V

23 CV : iagnosti Features If ET proess is rapid (Nernstian E rev proess) then we expet that: Pea urrent varies linearly with both the bul onentration of the reatant and the square root of the sweep rate aording to the Randles-Sevi equation. The voltammetri pea potentials are invariant with respet to hanging sweep rate. The oxidation/redution pea urrent ratio is approximately unity at all sweep rates. The standard redox potential is the average of the anodi and athodi pea potentials. i.446nfa i P i P, ox P, red E E P P, ox E.446nFA nf RT P, red nf RT / / S RS.58 V n S / RS ( T 98K) i P E P E E Randles-Sevi Plot S RS, ox P, red Cyli Voltammetry. Reversible ET = -5 m s - ; T = 98K ; n = ; =. M Sweep rate =. Vs - 3

24 LPS Voltammetry: Irreversible ET. E P E / E = -4 m/s = -5 m /s b =.5 E =.3 V E / E Eletrode inetis are finite: ause distortion of voltammogram. Cyli Voltammetry : Effet of slow ET inetis = -7 m s - = m s - = -3 m s - = -5 m s - Sweep rate =. V/s = -5 m /s C = mm Pea shapes beome less sharp as ET inetis beome more sluggish. 4

25 Shape of CV response depends on ineti faility of interfaial ET. As eletrohemial rate onstant dereases, the anodi/athodi pea separation E P inreases and the pea urrent dereases. ET proesses are termed Reversible (E rev ) if >. m/s Quasireversible (E qr ) if ( -5 < <. ) m/s Irreversible (E irrev ) if < -5 m/s. CV profiles beome more drawn out if interfaial ET inetis are sluggish. It an sometimes be diffiult to distinguish between the effets of slow ET inetis and finite solution resistane effets when analysing the shape of the CV profile sine both effets an result in CV profile distortion. We an evaluate by noting E P at some given san rate n. Using the theoretial woring urve shown aross we an then alulate the orresponding value of Y N and hene evaluate. etermination of ineti parameters using CV. ne P /mv N N / OX RE nf OX RT R.S. Niholson, Anal. Chem. 965, 37, Cyli Voltammetry Quasi Reversible ET Kinetis 5

26 Effet of apaitative urrent flow. Net urrent onsists of both Faradai (i F ) and Charging (i C omponents. These vary with sweep rate in different ways. The harging urrent omponent beomes important when the onentration of redox speies is low or the san rate is large. This is a major limitation of CV as a tehnique. This an be addressed using ultramiroeletrodes. i.446nfa F nf / RT Ei ic CL C RS time onstant R C S L L t exp Limitations of yli voltammetry Charging urrent ontribution Important when sweep rate is large and when onentration of eletroative speies is low. Unompensated solution resistane Important espeially in non-aqueous solutions and at fast sweep rates. Both effets an be minimized if the size of the woring eletrode is made very small. Hene eletrohemial studies using ultramiroeletrodes (UME) are now very ommon. UME systems minimize both harging urrent and unompensated solution resistane effets in CV studies and also result in greatly enhaned rates of material transport to the eletrode surfae. This failitates measurement of fast ET proesses. 6

27 Ultramiroeletrodes Ultramiroeletrodes (UME s) are of small size, typially between.6 mm radius. Most ommon onstrution involves sealing a wire into soft glass and exposing the end to form an inlaid dis eletrode. Have a wide range of miroeletrode geometries inluding dis, dis array, miroring, interdigitated array, hemisphere et. iffusion to a miroeletrode is not planar but is onvergent and an be approximated as spherial. UME systems minimize both harging urrent and unompensated solution resistane effets in CV studies. Planar iffusion Spherial iffusion In CV the harging urrent has a limiting value given by the produt n C L. Sine the double layer apaitane is proportional to the eletrode area then it will vary as the square of the eletrode radius. The Faradai urrent under steady state onditions varies as the eletrode radius. Hene the ratio of the Faradai to the harging urrent is improved as the eletrode radius dereases. This advantage more than offsets any disadvantage derived from the small urrent flow that must be measured at miroeletrodes. iffusion to a miroeletrode is very effiient and results in greatly enhaned rates of material transport to the eletrode surfae. This failitates measurement of fast ET proesses. iffusive rate onstant a eletrode radius Typial range of eletrode size. Small size = new hemistry and physis! Better spae, time and energy resolution. Standard eletrode size Muh urrent researh Lots of new Chemistry here! 7

28 Literature referenes K. Stuli et al. Miroeletrodes :definitions, haraterization and appliations. Pure Appl. Chem., 7() R.J. Forster. Miroeletrodes: new dimensions in eletrohemistry. Chem. So. Rev. 994, pp C.G. Zosi. Ultramiroeletrodes: design, fabriation and haraterization. Eletroanalysis 4 () 4-5. J. Heinze. Ultramiroeletrodes in eletrohemistry. Angew. Chem. Int. Ed. Engl., 3 (993) M.V. Mirin et al. Nanometer-sized eletrohemial sensors. Anal. Chem., 69 (997)

29 Miroeletrode Gallery Sigmoidal shape: harateristi of radial diffusion. Pea Shape : harateristi of Planar diffusion. 9

30 UME s Radial vs. Planar iffusion Radial iffusion Redox wave: sigmoidal shape I ss = 4nFr o C o * I ss san rate independent o C o * Planar iffusion Redox wave: normal shape I p / o / C Planar and non planar diffusion. Potential sweep voltammetry performed on maroeletrodes results in harateristi pea shaped urrent / voltage urves whereas sigmoidal shaped responses are obtained for miroeletrodes. The rms distane travelled by a diffusion speies d=(t) /. When d a, the eletrode radius non planar onvergent diffusion will predominate, and the voltammetri response will be sigmoidal rather than pea shaped. 3

31 Spherial/Hemispherial eletrode Miroeletrode geometries: Some quantitative results. t r r r i i 4 nf nf a a Sphere Hemisphere Inlaid is eletrode t r r r z i a SS urrent proportional to eletrode radius a. i i nf d ds a a a 4nF S Sphere Hemisphere Inlaid is Steady state diffusion Limited urrent a Surfae diameter Typial geometries are spherial and inlaid dis. Spherial easy to analyze mathematially. Inlaid dis easy to fabriate, but diffiult to treat mathematially. Miroeletrode experiments produe sigmoidal voltammograms harateristi of steady state onditions Experiments using ultramiroeletrodes very good for performing aurate ineti measurements. isadvantages inlude low urrents and diffiult theory! 3

32 We mae an approximate alulation of the i F /i C ratio for an inlaid mirodis eletrode of radius a under steady state onditions. C i C L CL X A X X H H H a a Helmholtz layer thiness z i F i i F C 4nF a 4nF a K X H a a K 4nFX H Inlaid mirodis a The ratio of Faradai urrent to harging or apaitive urrent inreases with : Smaller eletrode radii Slower sweep rates Larger redox speies onentrations. r 3

33 Nano-sized arbon fibre eletrodes. Chen & Kuerna, Eletrohem. Commun. 4 () 8-85 Chen & Kuerna, J. Phys. Chem.B. 6 () a. 9nm Eletrophoreti paint a 38nm Nanoeletrode 33

34 UME s: advantages,disadvantages and appliations. Advantages Redued apaitane. Low IR drop. isadvantages Noise. Eletrohemial fouling Analyte Method of eletrode preparation. Stray apaitane. Maintenane. Cost Fragile onstrution. UME appliations Fast eletrode inetis (high sweep rate CV). Biologial systems. Study reations at low temperatures (frozen glasses in solvents of low to moderate permittivity without eletrolyte in solvents of high resistane in solid state in gas phase. evelopment of analytial tools to probe eletrode/solution interfae (SECM, Bard et al.). Sanning eletrohemial mirosopy (SECM) iameter (typially nm) of UME tip determines lateral resolution, hene Tip geometry important. Hemispherial diffusion to tip. Aquire urrent vs tip position (x,y). Analyze variation of urrent vs normalised tip/substrate distane L. Modes of operation: Feedba (positive or negative) Colletion/Generation. 34

35 SECM Operation Tip mounted on a three axis positioning stage driven by piezoeletri atuators ubder software ontrol. As tip is sanned in raster pattern aross sample surfae (substrate) the software produes a map of the eletrohemial response of the tip. Generally tip urrent is displayed although tip voltage an also be aquired. If substrate is ondutive then substrate potential and urrent an be displayed. Any onventional EC measurement (CV, PSCA et) an be arried out with SECM. Great flexibility in operational mode employed: Feedba Mode Generation/Colletion Mode, Potentiometri SECM Penetration Experiments Equilibrium Perturbation Mode SECM Imaging Mirometer size metal tips used in form of inlaid diss. Nanometer sale tips fabriated via EC ething methods. Theory well developed using approximate analytial methods espeially Finite ifferene and Finite Element numerial methods. 35

36 36

37 Feedba Modes In the feedba mode the tip potential is set to a value at whih a partiular speies in the solution suh as a redox mediator is onsumed. Feedba is the term used to indiate that the measured tip urrent is influened by the rate at whih the mediator is regenerated at the substrate. The substrate may also be subjeted to an independent bias and serve as a seond woring eletrode, although it is not neessary that the substrate be another eletrode or even be ondutive. The feedba effet is sensitive to the tip-substrate separation d and this distane an be expressed in units of tip radius as L = d/a. At large values of normalised distane L the tip diffusion layer is not effeted by the substrate and is independent of L. An inlaid mirodis is often used sine this tip surfae is parallel to the substrate surfae and maximizes the feedba effet. Conial eletrodes of the type used in STM produe smaller feedba effets than the dis. For an oxidizable redox mediator A the proess ouring at the tip during a feedba experiment is A ne - B. Now if E T is poised suffiiently positive then the rate of reation is diffusion ontrolled. For large L and if an inlaid dis is used then i T = 4nF a. When L is smaller the produt speies B an diffuse to the substrate where it an be redued ba to A via B + ne - A. This reation produes an additional flux of A to the tip and hene an inrease in tip urrent and so i T > i T. When this reation is also diffusion limited the tip urrent reahes a maximum value. This phenomenon is termed positive feedba. As tip-substrate distane d dereases, i T inreases without limit. Negative feedba is the terminology used for situations where the produt B does not reat at the substrate surfae sine it may be insulating or the A/B reation may be inetially irreversible. 37

38 For negative feedba at small L values i T < i T beause the substrate now hinders diffusion of speies A from the bul to the tip surfae. The smaller L gets the smaller will be the ratio i T /i T. For both positive and negative feedba, variations of i T an be related to hanges in L, and used to image the substrate topography by sanning the tip over the substrate surfae. Sine the mass transport rate in positive feedba experiments at small L is typially /d ~. m/s, onvetive effets arising from tip motion (typially m/s) during imaging an be negleted. Sine the magnitude of the feedba is sensitive to the rate of mediator regeneration at the substrate surfae, the measured feedba/distane behaviour (the approah urve) provides information on the inetis of the proess at the substrate. uring imaging the z position of the tip is usually fixed and therefore an independent measurement of the tip-substrate distane is useful to de-onvolute substrate inetis from the effets of topography. Hene feedba mode ineti studies are often made at fixed x-y lateral positions but with the tip sanning over a range of z values to generate an approah urve. Negative Feedba e - A B e - A e - B A B A A A A A A B e - i T, 4nFa Insulating substrate i T, Condutive substrate i T i T, i T i T Sanning eletrohemial Mirosopy (SECM) : Positive and negative feedba., i T i T L L Positive Feedba i T, 38

39 Generation/Colletion Mode Generation/olletion (G/C) mode refers to experiments in whih speies are generated at the substrate independent of the tip reation and are subsequently deteted at the tip, or vie versa. These two different G/C modes are referred to as substrate generation/tip olletion SG/TC and tip generation/substrate olletion TG/SC. The ratio of fluxes at the tip and the substrate defines the olletion effiieny. Owing to the relative sizes of tip and substrate the olletion effiieny approahes % in the TG/SC mode. eviations of the olletion from % an be used to investigate homogeneous hemial inetis of unstable speies in the tip/substrate gap. The rates of hemial reations following an initial eletron transfer may be determined by measuring the distane dependene of the olletion effiieny: at small L the reverse reation at the substrate ompetes with the homogeneous hemial reation. ual eletrode generator/olletor experiments. Generator Generator Quinn et al. Eletrohem. Commun. 4() First attempt at ring/dis Generator olletor devie. is Colletor A B A A B A Colletor Ring Negative feedba Insulating substrate Positive feedba Conduting substrate Behaviour of reation intermediates probed. 39