Electrochemistry 2013

Transcription

1 Physial and Interfaial Eletrohemistry 13 Leture 5 Eletrode Exess negative harge density Solution Potential Helmholtz Layer Eletrode/solution Interfae + + s Exess positive harge density x H a..6 nm istane C L

2 The struture of the eletrified interfae The interfae between two dissimilar i il interfaes is eletrified. Almost all surfaes arry an exess eletri harge. Hene when two dissimilar phases ome into ontat, harge separation ours in the interfaial region whih results in the generation of an interfaial potential differene or eletri field. The term eletrified interfae (or double layer region) is used to desribe the arrangement of harges and oriented solvent dipoles at the interfae between an eletrode and an eletrolyte. A simplisti piture of this region (termed the Helmholtz model) is presented in the figure outlined aross. As we have seen this region is very thin. The struture of the double layer region an be pitured as a parallel plate apaitor with a plate separation of moleular dimension. E field generated Bottom line : an eletrial ouble layer is set up at M/S interfae. Exess positive harge density Exess negative harge density

3 Eletroneutrality is valid in bulk solution. Consider a metal eletrode in ontat with an aqueous solution ontaining salt (e.g KCl (aq)). The solution ontains solvated harged ions and solvent dipoles. Fores experiened by ions and solvent moleules in bulk of solution are isotropi : spherial symmetry operates. Ions and water moleules (on a time average) experiene fores whih are position and diretion independent. There is no net alignment of solvent dipoles, and positive and negative ions are equally distributed throughout any volume element of the solution. Eletroneutrality operates in bulk solution region very far from eletrode surfae.

4 Eletroneutrality breaks down in surfae region. What about the solution region next to the eletrode surfae? In this region fores experiened by ions and solvent dipoles are no longer isotropi and homogeneous. The fores are anisotropi beause of the presene of the eletrode phase. New solvent struture, different from that of the bulk, develops beause of the phase boundary. Eletroneutrality t lit breaks down on the solution side of the interfae. There will be a net orientation of solvent dipoles and a net exess harge in any volume element of the solution adjaent to the eletrode surfae. The solution side of the interfae beomes eletrified.

5 Interfaial harge separation generates high interfaial Efield. One the solution side of the interfae beomes eletrified (aquires a net or exess harge), an eletri field will operate aross the phase boundary. Sine the metalli phase ontains harged partiles, the latter will respond to this E field. The free eletrons will move away from or move towards the interfae depending on the diretion of the E field. Thus a net harge will be indued on the metal, whih will be equal in magnitude and opposite in sign to that on the solution side of the phase boundary. Thus harge separation ours aross the M/S interfae, and this gives rise to an interfaial potential differene. Typially the potential differene is a. 1. V. However the spatial dimensions of the interfae region are very small, typially 1 nm thik. Thus the eletri field strength present at the M/S interfae will be typially 1 7 Vm 1 whih is very large.

6 Eletrified interfae. Formal definition. iti Term eletrified interfae used to desribe the arrangement of harges and oriented solvent dipoles at the interfae between an eletrode and an eletrolyte solution. We now present some simple approximate models to desribe the properties of the eletrified interfae. We examine three simple models : Helmholtz ompat layer model GouyChapman diffuse layer model Stern model. A key idea whih we develop involves the representation of interfaial struture in terms of an eletrial equivalent iruit element (speifially a single apaitor or series of apaitors). We initially adopt a simple pitorial desription. This will be followed owed by a more quantitative tat mathematial analysis. s.

7 Eletrode Exess positive harge density Exess negative harge density Simple representation of eletrode/solution interphase region : Helmholtz ompat layer model. Eletrode Exess negative harge density + + Solution + Solution Eletri field present at interfae r CL CH x H Exess positive harge density + + Struture of thin double layer region modelled as a parallel plate apaitor with a plate separation of moleular dimension. C L Potential x H a..6 nm Helmholtz Layer s istane C L 6 F m

8 Numerial alulations Using the Helmholtz Model. We need the following Relationships from basi Physis. C H Surfae harge density on metal (Cm ) C H E x H Eletri field Strength (Vm 1 ) Helmholtz Capaitane (Fm ) Interfaial Potential differene (V) = permittivity of vauum = x 1 1 Fm 1 r r = dieletri onstant of solution. xh r ( bulk) 78 ( Helmholtz) 5 6 istane between plates of apaitor r Eletrode Exess positive harge density Exess negative harge density Solution C L Potential x H a..6 nm Helmholtz Layer s istane A fundamental problem is assigning a value for the dieletri onstant of the solvent in the thin Helmholtz region. Solvent struture in this region differs onsiderably from that of the bulk solution. Have onsiderable dieletri i saturation effets and so dieletri onstant will be muh lower than that assoiated with the bulk solution. The dieletri onstant may also vary rapidly with distane in interfae region.

9 Potential istribution in Helmholtz Compat Layer. Potential distribution obtained using the PoissonBoltzmann equation whih relates harge density and eletrostati potential. d dx r (x) Radius of solvated ion = x H Ions treated as point harges. Hene an assume that exess harge density between eletrode surfae and OHP is zero, hene =. x M M S x H x M d dx d dx x x H Linear potential profile in ompat t layer. S x x M x x H M M x H S S

10 This simple Helmholtz piture is not omplete sine it predits that the double layer apaitane is a onstant independent of onentration and the potential applied to the eletrode. Experiment has shown that the double layer apaitane varies with the latter quantities in a somewhat omplex manner and so this simple piture of the double layer region must be modified df d to take this observation into aount. Hene we onlude that a more elaborate model is required. However the simple Helmholtz model will hold quite well for eletrolyte onentrations greater than.5 M.

11 The diffuse double layer. How does a 3 distribution ib ti of harge arise? We have negleted the disordering effet of the thermal motion of the ions in the solution. This opposes the ordering tendeny due to operation of eletrostati fores in the interfae region. Thermal and eletrostati fores results in an equilibrium. The exess harge density s S ounterbalaning the exess harge density s M on the metal, is at a maximum lose to the eletrode surfae. It diminishes i i in an approximate exponential manner with inreasing distane from the eletrode surfae, giving rise to a diffuse spae harge layer adjaent to the eletrode. etailed analysis indiates that the thikness of the diffuse layer region will depend both on the potential applied t the eletrode, and on the onentration of ions present in the eletrolyte solution.

12 GouyChapman model of diffuse double layer. A more sophistiated piture (termed the GouyChapman model) assumes that the exess harge density on the solution side of the interfae an be represented in terms of a three dimensional spae harge region. This is presented shematially in the figure presented below. This harge distribution arises as follows. It is reasonable to suppose that the disordering fores arising from the thermal energy of the ions should oppose the ordering tendeny indued in the interphase region by eletrostati fores. Consequently instead of onsidering a simple ompat layer (termed the Helmholtz layer) these thermal and eletrostati fores are assumed to result in an equilibrium, in whih the exess harge density S ounterbalaning the harge density M on the metal, is at a maximum lose to the eletrode surfae, and would diminish in an exponential manner with inreasing distane from the eletrode surfae, thereby giving rise to a diffuse layer adjaent to the eletrode. etailed analysis indiates that the thikness of the latter region will depend on the potential applied to the eletrode and on the onentration of ions in the eletrolyte. Eletrode Exess positive harge density Exess negative harge density Solution OHP L a. 11 nm In the GouyChapman model of the interfae region, it is assumed that the exess harge density on the solution side of the interfae an be represented in terms of a three dimensional spae harge. s x iffuse layer thikness

13 GouyChapman model of diffuse double layer. Eletrode M Exess negative harge density C L r ze C osh L kbt r osh r L C osh osh C, + r C At potential of, L Zero harge: + + r + C, L + s x Exess positive harge density d Solution 4 ze L sinh Charge density in diffuse layer L a. 11 nm Valid when is small. osh1 L = ebye Length. Measures diffuse layer thikness.

14 iffuse layer thikness. The diffuse layer thikness is alled the ebye Length and is given the symbol L. In many books this is denoted as 1/. For a z,z eletrolyte the ebye length is given by the expression aross. Evaluation of the onstants gives a useful expression for omputation. L 1/ k B T r RT z e z F L 1/ r T 1e8 z Note that the ebye Length varies inversely with the square root of the eletrolyte onentration. Hene as the solution beomes more onentrated the thikness of the diffuse layer dereases. For instane for a simple 11 eletrolyte suh as NaCl at 98 K, L = 3.5 nm when = 1 4 mol dm 3. However when is inreased to.1 mol dm 3 the ebye length dereases to.96 nm. We reall this onept when we disussed the H theory of Ion/ion interations. L / m Note that the ebye Length inreases as the ioni onentration dereases. The diffuse layer thikness will be greatest for the most dilute solutions. mol m 3 or mm L /m 1e9 1e /mol m 3 (1,1) eletrolyte, water r = 78, T = 98K

15 The PoissonBoltzmann equation (I). From lassial eletrostatis we use the Poisson equation whih relates the dieletri displaement vetor and the loal volume density of harge (the number of harges per unit volume). E permittivity permittivity of vauum r E dieletri onstant 8.854x1 1 Fm 1 ivergene operator eletri field vetor Eletri field vetor E an be related to the eletrostati potential using basi physis. E Gradient operator This is the form of the Poisson equation whih relates the harge density and the eletrostati potential.

16 The PoissonBoltzmann equation (II). We now need to evaluate the harge density. The volume density of harge is obtained by adding together the produt of the harge q j and onentration j of eah ioni speies j in the solution next to the eletrode surfae. Ion valene q j j j j z j e j fundamental harge We use the Boltzmann equation of statistial mehanis to obtain a relationship between the loal ounterion onentration j and the bulk onentration j. To do this we need to evaluate the work w j done in bringing the ion from a referene point at infinity, up to a point distane r from the eletrode surfae. We assume that this work is purely eletrial in nature. w j j j r z e( ) ( r) q r j j w j exp kbt j j exp z j e( r) k B T

17 The PoissonBoltzmann equation (III). L 1 1 kbt ze We are now in a position to write down the PB equation. ebye Length, z,z eletrolyte This is a fairly ompliated equation to solve from first ( r ) 1 priniples. The exat form of z je j ( r ) j the differential equation depends on the geometry. 1 z je r We shall assume a z,z eletrolyte l t z je j ep exp suh as KCl or NaCl. j kbt The geometry determines the form that the operator takes. z z z A planar geometry is used for z,z valent maroeletrodes, whereas a spherial eletrolyte geometry is adopted for ultramiroeletrodes. d x ze ze ze Planar geometry exp exp dx kbt kbt d ze ze sinh k B T dx 1 r d d r dr dr Spherial geometry r d r dr 1 ze d ze dr ze sinh kbt ze ze exp exp kb T kb T The PB equation is solved for.

18 etailed analysis utilising eletrostati theory and statistial mehanis (Albery 1975) shows that the normalised potential distribution in the diffuse layer is given by : 1 tanh tanh exp p In short one uses the Poisson equation of eletrostatis whih relates the eletrostati potential to the harge distribution generating g it and the Boltzmann distribution law of lassial statistial mehanis whih desribes the way that ions are distributed throughout the diffuse layer to develop a hybrid equation alled the Poisson Boltzmann (PB) equation whih is then solved using standard mathematial tehniques to obtain the expression outlined above. This proedure is well desribed in many of the books dealing with fundamental physial eletrohemistry. In the full PB expression above denotes the eletrostati potential at any distane in the diffuse layer, and is the eletrostati potential at the surfae of the eletrode. For small values of the normalised potential we write that tanh and the potential distribution within the diffuse layer deays aording to the following simple exponential relationship : exp ze k T x B L ze k T B The variation of normalised potential with normalised distane for various values of omputed using the full PB expression is presented in the next slide. ld Below = 1 we note that deays exponentially with distane. For > 5 and a limiting urve is obtained. We note also that deays to zero when is greater than 3. This means that the strong eletri field of the eletrode is ontained within a distane of a. 3L from the eletrode.

19 Poisson/Boltzmann equation : planar surfae. d 1 sinh d ze kbt ze k T Normalised potential B Neglet ompat layer x x L Variation of eletrostati potential with distane in the diffuse layer region. The potential is effetively exponentially deaying with distane from solid surfae. Normalised distane exp Small surfae potential ebyehukel approximation Large surfae potential tanh 1 exp 1 tanh tanh Full solution of PB equation Thikness of iffuse layer exp

20 The Poisson Boltzmann equation (IV). The PB equation fully desribes the pertinent eletrial properties of the diffuse layer. However it an only be solved analytially for a few speial situations. For the most general ases a numerial solution has to be adopted. The PB equation for flat planar surfaes an be rigorously solved for z,z eletrolytes. This annot be done rigorously for spherial surfaes. Approximate solutions of varying degrees of auray have been produed. A reasonable approah valid both for planar and spherial interfaes involves the ebyehukel approximation, whih results in the transformation of the PB equation into a linear form as indiated below. This approximation will be valid provided that the potential at the surfae of the eletrode is not too large. Approximate form H approximation Of potential distribution x Planar d ze k exp x exp BT geometry L dx

21 Charge density in diffuse layer region. x dx 1 d If the potential applied to the metal surfae is small we an assume a linear approximation and For a z,z eletrolyte the loal harge density is given by ze ze exp ze ze exp sinh sinh sinh sinh d d ebye Length 4ze 4ze d sinh d d d sinh osh sinh sinh z e k T 4ze 4ze ze k T B Note that t the sign of is opposite to that of. Hene a positive implies that negative ions are attrated to interfae and vie versa. B

22 We define the differential iffuse Layer apaitane C We define the differential apaitane of the diffuse layer as sinh B d d T k e z C d d T k ze d d d d C B M osh k B T e z ih 4 ze Charge density within diffuse layer is osh B B T k ze T k e z sinh W ll th t b L th (diff osh osh B C T k ze T k 1 1 We reall that ebye Length (diffuse layer thikness is e z os C C depends on potential. It is not onstant T k ze L B 1 1 T k e z B It is not onstant.

23 Variation of diffuse layer apaitane with potential.

24 How good is the diffuse layer Theory in pratie? iffuse layer model also applies for olloidal l partile/solution l interfae. ouble layer modelling still being Performed at researh level l to various degrees of sophistiation.

25 We an also ompute the distribution of ions in the diffuse layer (Albery 1975). Now ions within the diffuse layer are not all of the same sign. The harge density on the eletrode is balaned by an aumulation of harges of opposite sign (termed ounter ions) and by a defiit it of harges of the same sign (termed oions) ompared with their onentrations in the bulk solution region. Now the ion distribution in the diffuse layer is given by the Boltzmann distribution ib ti law : exp ze exp k B T (1) It is possible to show that (Albery 1975) the oion (labeled +) and the ounterion (labeled ) distributions are given by the following expression : 1 tanh f 1 oth f () In the former expressions the funtion f( ) will depend on the sign of the potential applied to the eletrode aording to : f lntanh f lntanh These expressions, although omplex, arry an important message. In simple terms they state that almost all the harge on the metal is balaned by the aumulation of ounter ions and relatively little by the expulsion of oions. The ounterions therefore enjoy a greater signifiane than the oions. This breakdown in the eletroneutrality is outlined in the figure presented in the next slide for various values of. Here eqn.1 and eqn.. have been used. It is lear that even for quite small values of the ounterion and oion onentrations in the diffuse layer will exhibit signifiant deviations from their bulk solution values where eletroneutrality is maintained. We note that for > 5 eletroneutrtality is restored. For instane for =.5 the onentrations are altered by a fator of 3. For larger values of the onentration of the repelled oions approah zero in the interphase region near the eletrode surfae whereas the onentration of the attrated ounterion is many times its value in the bulk solution.

26 Eletroneutrality breakdown in diffuse layer region : Planar surfae. Counter ion exess 1 tanh f oth 1 f lntanh f f ln tanh Counterion onentration inreases lose to harged solid surfae and oion Coion depletion onentration dereases lose to harged surfae.

27 Typial variation of C L with applied potential. Hg/aqueous KCl interfae. Capaitane maximum Modern models inorporating speifi adsorption of ions in the inner ompat layer, allied with a model for the water struture in the inner layer explain the apaitane maximum Constant apaity Region. Explained by Helmholtz model Capaitane minimum Explained by GouyChapman model

28 Stern model of the interfae region. Neither the Helmholtz ompat layer model nor the Gouy Chapman diffuse layer model is totally satisfatory. In the GC model the solvated ions are modelled as point harges. This neglet of ion size is unrealisti. In reality the solvated ion an only approah the eletrode surfae to a distane equal to its solvated radius a. Hene a more logial approah is to ombine the features of the Helmholtz l and Gouy Chapman models. This was done by Stern. The Stern model is as follows. Next to the eletrode we have a region of high eletri field and low dieletri onstant n ( r value a. 6) with a row of firmly held ounter ions. Beyond that there is an ioni atmosphere (the diffuse layer) where there is a balane between the ordering eletrostati fore and disordering thermal motions. The dieletri onstant inreases rapidly with distane in this region. The eletrial potential varies linearly with distane (a. hydrated ion radius) within the inner ompat layer and dereases in an approximate exponential manner with distane within the diffuse layer, deaying to zero in the bulk solution.

29 Stern model of solid/solution interphase region. The GouyChapman model of the double layer region assumes that ions behave like point harges. This neglet of ioni i size is not very realisti. Ions are hydrated in aqueous solutions and so the latter speies an only approah the eletrode to a distane equal to the radius of a hydrated ion. Consequently, a more logial approah is to ombine the features of the Helmholtz and GouyChapman models. This was done by Stern. Hene a better piture for the double layer is as follows.next to the eletrode we have a region of high eletri field and low dieletri onstant with a row of firmly held ounterions. Beyond that there is an ioni atmosphere (the diffuse layer) where there is a balane between the eletrostati t ti fores and random thermal motions. This idea is presented in figure.6. The potential variation is also shown shematially. The potential varies linearly with distane within the inner ompat layer and dereases approximately exponentially with distane within the outer diffuse region. The line of demaration between the ompat and diffuse regions is alled the Outer Helmholtz Plane (OHP). Consequently the interfae region may now be represented in terms of a simple iruit model onsisting of two apaitanes labeled C H and C in series. The total double layer apaitane C L is therefore given by the following reiproal expression. C L x H r CL CH C C r zeh osh L kbt L r osh H

30 Stern model of solid/solution interphase region. 1 C L 1 C H 1 C C L C H C Series arrangement of apaitors. The smaller of the two apaitanes will determine the overall apaitane. If C H and C are of very different size then the term ontaining the larger one may be negleted. The diffuse layer apaitane will predominate when the solution onentration is low.

31 A reasonable model of the eletrode/solution interfae. Linear potential Variation with distane The eletrode/solution interfae is modelled d as a series arrangement of two apaitors. This is an equivalent iruit representation of the interfae. Exponential Variation of Potential with distane x H From basi physis : L Compat layer C L C H C Series arrangement of apaitors. 1 C L 1 C H 1 C Total apaitane iffuse layer

32 Further elaborations on interfaial struture The model for the double layer presented here is not at all sophistiated. Muh work is urrently underway to develop better models for the interfae region. For instane the inner layer apaity CH is independent of eletrolyte onentration but it depends strongly on the harge density of the eletrode and somewhat more weakly on temperature. Negative ions may also desolvate and speifially adsorb on an eletrode surfae. This influenes the potential distribution at the interfae. There also will be an oriented layer of water moleules attahed to the metal surfae. This will give rise to a surfae potential ti. The metal surfae is also more omplex than we have assumed. Beause of the small eletroni mass, eletroni density an extend for a ertain distane out from the eletrode surfae. Typially the eletroni density dereases exponentially with a deay length of a..5 nm. Sine the eletroni density of metals is high, this spill over will give rise to an appreiable negative exess harge outside the metal, whih for an unharged surfae, must be balaned by an equal and opposite positive exess harge within the metal. This gives rise to a signifiant surfae potential. The eletri field in the double layer will distort the eletroni distribution ib ti and hange the surfae potential ti. This will in turn effet the magnitude of the Helmholtz apaity C H. Hene in reality the situation at an eletrode/solution interfae is quite omplex. We will not disuss these matters here. Instead we will present a piture of the urrently aepted model of the interphase region in the figure presented in the next slide. This is termed the BM model (Bokris and Reddy 197). Note that it represents the situation on the solution side of the interfae in terms of a triple layer. Many of the ompliating at features mentioned in the previous paragraph are illustrated. An Inner Helmholtz Plane (IHP) is defined and is regarded as the lous of the eletrial entres of speifially adsorbed ions. A primary and seondary water layer is introdued with onsiderably differing dieletri properties. The jury is still out on whether the BM represents the true piture of the eletrified interfae region. The reader is referred to Bokris and Khan (1993) and Shmikler (1996(a), 1996(b)) for further details on the urrent state of play.

33 The BM triple layer Model of the metal/solution Interfae. One urrently aepted model is the BM model. It represents the situation on the solution side of the interfae in terms of a triple layer. an inner helmholtz plane (IHP) is introdued and is regarded as the lous of the eletrial enters of speifially adsorbed ions. These anions are strongly adsorbed onto the eletrode surfae and are partially desolvated. This fat auses a reversal of eletrostati potential in the region between the IHP and the OHP. x H A primary and seondary water layer is introdued with differing dieletri properties. x H The primary water layer ( r = 5) is loated immediately adjaent to the metal eletrode surfae. The seondary water layer (e r = 36) is loated as a hydration sphere around a solvated ation and anion.

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35 Polarizable and nonpolarizable interfaes. C L Interfaial struture Eletrode/solution interfae No leakage of Charge aross M/S interfae Ideally Polarizable Interfae : R CT Eletrial equivalent iruit R CT Ideally nonpolarizable Interfae: R CT Measures ET Aross interfae Charge transfer ours aross M/S interfae

36 Simple equivalent iruit representation of eletrode/solution interfae region. L harging urrent i i C C L R S Resistane of solution Faradai urrent i F i i i C F R CT Solution Eletrode Evaluation of C L (and hene i ) always neessary when making kineti measurements at short timesales. ouble layer harging urrent always present in addition to Faradai urrent in eletrohemial measurements.

37 Experimental interrogation of eletrode/solution interfaes. Conventional eletrohemial tehniques. Based on measurement of urrent, potential and harge. CV, RV, RRV, PSCA, CIS et. Applied both to marosized and miroeletrodes. Theory, instrumentation, and pratie well developed. d No diret information on mirosopi struture of eletrode/solution interfae. Spetrosopi tehniques. Provides useful hemial information anout speies at interfaes. FTIR, Raman, UV/VIS, XPS, EXAFS, Ellipsometry, EC/NMR (new tehnique, very speialised, limited appliation at present). Sanning probe mirosopy. High resolution topographial imaging of eletrode surfaes with atomi resolution. Surfae reativity also probed with high spatial resolution. STM, AFM, SECM. R f t PA Ch i t A H tt T h i d M h i Refer to: P.A. Christensen, A. Hamnett, Tehniques and Mehanisms in Eletrohemistry, Chapman and Hall, UK, 1994 for details onerning Spetrosopi tehniques.