So, what we're going to do is revisit where we left off last time and continue to talk about the structure of the double layer. Again, we're working out of just following exactly what was in Bard and Faulkner. And if you remember last time, we got to this point where by making the assumptions from our model that we had thermal equilibrium between the charge distribution in any lamella at a position x away from the interface and the potential that was dropped up to that point that holds those charges toward that interface that we could build a self-consistent analytically treatable model for a one-to-one electrolyte of what the charge distribution ought to be at room temperature as a function of that potential drop. And you could see before that one-to-one electrolyte that analytical solution shows that it goes like these hyperbolic tangents which for small potential drops can be linearized so that the potential relative to its full potential drop is screened with distance exponentially and the attenuation kappa is the so-called Dubai screening length which for one-molar electrolytes is very short because most of the potential is rapidly screened by having a high electrolyte concentration able to approach that electrode surface and screen that potential drop from the residual ions in solution and is in the order of three angstroms when you calculate all of those known quantities and plug into that expression at one molar ionic strength in aqueous solution and goes like the square this goes like the square root of the concentration and so it's on the order of ten angstroms at a millimolar thirty angstroms something like that and the actual numbers that result from this at different concentrations are in a table and Barton Faulkner so three angstroms is the minimum value of the screening length that's about one molar and then it's thirty angstroms a hundred angstroms something like that at millimolar concentrations and those are good numbers to know about of course in a metal the Dubai screening length is essentially zero because all the charge on the surface of the metal is where the charge is and there's no electric field nor potential drop in the bulk of any metal phase so that's the limiting case of having the ultra high ionic strength if you want to think about it that way that a liquid can't come close to and the most you can do is collapse it down to something like three angstroms because of that potential drop you see that the potential profiles here exactly plotted should if you dropping a thousand millivolts say a volt it'll take on the order of twenty angstroms for it to go to one over e of its value at ten to the minus two molar as I said because there's a factor of ten less the square root of this is factored a hundred less but the screening length goes like the square root of that and yeah this is down to one over e but yeah but there's another problem with that so here's the exponential limiting form right but one over cap is thirty four angstroms right so there's something else in there that I'll talk about later on this is the limiting case right here right alright so why is that we need to go a little further so to go a little further we now know analytically for that model what the potential drop ought to be as a function of distance that's not what we measure we don't measure the potential drop as a function of distance we measure the charge density on that electrode because if you remember from the electro capillary equation we get the surface tension the first derivative of the surface tension with respect to potential is the charge density and the second derivative the first derivative of the surface charge density with respect to potential is the differential capacitance so at a solid electrode we access the differential capacitance from which we can integrate to get the charge density or at mercury electrode we can differentiate the surface tension to get the charge density but in neither of neither of those two do we directly measure the potential drop profile in the electrolyte so to access what we can measure experimentally we have to take this model that is predicted what the ion and concentrations are and what the potential drop that results from that is in the electrolyte and convert that into the two things we can measure the surface charge density and the differential capacitance that results from both of those distributions of charges spatially so we need to do that to do that the most convenient way is to draw a Gaussian box so draw a Gaussian box of surface area A and of course if you draw a Gaussian box from Gauss's law the total charge is the integral of the electric field over the whole surface area of that box but the electric field strength is zero at all points on the surface except the interface where the magnitude of the field is the gradient of the potential evaluated at x equals zero because that is the field there and so in fact this integral over all the surface of the box reduces to an integral just over that area of the first derivative of the potential with respect to position so the field gradient right there we can solve for that from what we knew because we know again just relating that this is the Gauss's law the charge density is related to the area times the dielectric constant times that that's just a rearrangement of what we had before what we knew from our model from that charge distribution model what the charge density on the surface is because we knew what the potential gradient had to be because if you know the potential profile with respect to distance you can take its derivative since we have a Cauch function as a function of distance we can take the derivative of the Cauch we get a cinch function back out so d phi dx looks like cinch of the argument of that with the constants in front some of which were in here grouped all together and so this must be the charge density because it is the derivative of the potential with respect to distance in that Gaussian box over the only place where the integration in the box is non-zero so this allows us to relate that analytical treatment of the potential gradient to something we can measure the charge density and to the extent that that treatment was valid it predicts that the charge density should go like the hyperbolic sign of these things where this of course is related monotonically then to the charge on the electrode now you can say but we don't know the potential drop because we can't measure that but we can measure the differential capacitance right so we can predict back and use that then to infer what the potential drop is and therefore infer what the charge density is right so here's what the differential capacitance should do because we need to take the derivative of this with respect to potential and so now we're differentiating not with respect to distance but with respect to the variable that we're imposing experimentally and fortunately the derivative of sinh returns back Akash except that now it's Akash of an argument with an independent variable that's potential instead of distance but it's fine and more stuff comes out here because of that derivative that we took so that this is now the pre-factor in front again just a bunch of constants and for dilute aqueous solutions then these are the values of those constants and you can see that that predicts the order of magnitude of the differential capacitance should be on something like a few hundred microfarads per centimeter squared and depend in that form on the ionic strength furthermore because it goes like a hyperbolic cosine that's a symmetric function around zero and it's roughly exponential increasing because the hyperbolic cosine is e to the x plus e to the minus x over two and so it predicts the functional form of this which is in qualitative agreement with if you remember the actual measured differential capacitance of solid electrode was not a constant value which is why we started this in the first place but increased in roughly this kind of curvature form as a function of potential away from the point of zero charge and you also remember that it was going up as the ionic strength went up and that the dependence that we measured experimentally actually tended to flatten out as the ionic strength went up so those things are in qualitative agreement with this but not quantitative agreement this model is called the GUI Chapman model because the only thing that went into the model of the double layer molecularly was that we had atomic charges that were point charges whose density in the solution was given by the interplay between kt and the potential drop that resulted as a function of equilibration of those charges in various different layers with the potential they're trying to screen so there's a reason why we can immediately see why this will fail it fails around the point of zero charge to give us the right differential capacitance and gives us this fall off which doesn't mimic that which one measures experimentally quantitatively for at least one good reason let's see if we can guess what that one good reason might be because I was very careful to tell you if you listened very carefully just a minute ago to what was actually used as the model of this charge density for these ions in solution I said they were point charges but real molecules are not point charges and that's the next thing we have to put in that modification is called Stern's modification what happened and the reason that that capacitance could arbitrarily get lower because there's no bound on this right this capacitance can get lower or higher with different ionic strengths and in fact that charge density distribution that we saw if you went back and looked at it could have gotten arbitrarily thin and closer and closer to the electrolyte as needed to screen those charges relative to kt as I applied more and more potential the distance as you pointed out over which that potential was screened could be shorter than the size of one molecule doing that screening and that's what this predicts but that's not in fact physically reasonable so we have to impose another condition and the physical condition is Stern's modification oh no so what Stern said was I want to conceptually make these ions come no closer than one certain distance and that distance I will call the outer Helmholtz plane right and that's an important concept there's going to be an inner Helmholtz plane and outer Helmholtz plane if I prevent ions physically from screening charge at an arbitrarily small distance away from my metal electrode but say that a say a chloride ion which is solvated by water could maybe strip part of the waters around it but can never lose them all and even if it did itself has a finite size and so if you drew the center point of charge density it's never arbitrarily close to the electrode surface it's somewhere on that chloride ion physically away from that metal then you say that that very first layer must have a capacitance and a certain dielectric constant and with that can never be decreased below a certain value that's going to therefore allow you to bound the total potential that could drop to that first plane and from then on you can draw planes that are more like GUI Chapman Stern GUI Chapman planes and the reason for that is if you saw what happened here in that graph that I just showed notice that at low potential drops at low potential drops the limiting form isn't too different depending on which potential drop it is but at high potential drops when there's a lot of potential to be dropped this model the GUI Chapman model predicts a very effective screening because these point charges can come arbitrarily close to the interface and so you can see that if you can find the ability to screen that charge and its screen says only some fraction of the potential there that the rest of it would have a lower amount of potential that needed to be screened and that would be in the more diffuse layer so called that extends in his thermal equilibration lamellae away from the remainder of the electrode surface and this compact layer where these ions are squeezed up to the surface but can't be squeezed any further than a certain distance acts more like a true capacitor because the charge density on that capacitor is proportional to the applied voltage and the thickness and dielectric properties of that capacitor are basically independent of the charge density that are put on either of those two plates so we need to introduce because of the finite ion size a separate capacitance in series with this capacitance which is the rest of the structure of the double layer and that's what Stearns modification is so what you do analytically is you say that the total potential drop I'm going to partition into two separate potential drops one potential drop that drops across this inner Helmholtz layer to this plane and that is essentially treated like a pure capacitor and then the remainder of the potential drop whatever is left behind that I will treat within the GUI Chapman model of ions diffuse and of a much larger screening length than can be achieved in the compact layer when you work through that you get all the same equations in the electrolyte except that only a fraction of the potential from phi 2 to the total potential drop is done across the diffuse layer and so the potential drop might look something like that you would have because this is a fixed capacitor over that first finite distance a linear drop in the potential from one plate to the other because at a constant dielectric constant you'll get a linear potential drop from x equals 0 to that first plane and then the remainder of this potential will be dropped across the diffuse layer within the GUI Chapman model you could in fact add a more sophisticated model which allows for the finite ion size of these ions as well and never lets them pack closer than a certain lamellar distance and in fact going even further forces cations to be partially screened by anions and then cations and not just makes all the negatives onto a positively charged electric people do this but that's can't be solved analytically so the simple GUI Chapman model just makes the solution a bunch of point negative charges for its point charges but prevents them within Stearns modification from coming arbitrarily close to the electric surface the effect of doing this and adding Stearns modification is simple the total capacitance of the double layer is the capacitance of that Helmholtz layer that inner Helmholtz plane capacitor in series with the diffuse layer for two capacitors in series then if you remember the way they add they the reciprocal of their total capacitance is the sum of the reciprocals of the individual capacitances and therefore the smallest one wins in the total capacitance right that's just simple circuit element analysis and that's what happens analytically here too if you go through it so that there is a position of these inner Helmholtz layer and the outer Helmholtz plane that is on the order of a few angstroms three or four angstroms and so you would put into that value something like three or four angstroms to deduce what the capacitance and therefore what the potential is dropped across that first plane is and then the remainder will be dropped across the diffuse layer within the GUI Chapman model you can see two things first thing is at least that this is all analytically known it's the GUI Chapman model here's the hyperbolic Cosh that gave us this parabolas this is in series with the other capacitance the Helmholtz capacitance the second thing is that in principle the Helmholtz capacitance is known analytically if we estimate this to be on the order of a molecular spacing three or four angstroms therefore we should be able to bound what the double layer total capacitance is because this can't be arbitrarily small because this is a finite size and this is the dielectric constant of the solvent in which we're in so we should be able to bound that and if you remember what we saw is indeed we saw that was a bound and so at low electrolyte concentration in fact mostly what you see is that hyperbolic cosine like dependence but at high electrolyte concentration you see something much different than that and that makes sense because at high electrolyte concentration most of the ions are as close to the electric surface as they can get and you're actually as you're applying potential just adding more charge density to a fixed distance and there's a very little amount of potential dropped across the diffuse layer but most of the potentials dropped across the compact layer and so the capacitance should be constant with applied potential that's just a pure double sided capacitor as you go to lower and lower ionic strength then you can't drop much of the potential across the compact layer because most of the ions aren't sucked up right to there and they're mostly distributed in their Boltzmann equilibrium in the diffuse layer and in that case you'll see the hyperbolic cosine dependence of the capacitance on potential and this is just a property of when Helmholtz capacitance dominates and CSC is at high ionic strength when CSC becomes big compared to CH then you just see CH and it's a constant capacitance at low ionic strength the diffuse capacitance is the small one and it dominates because they're adding reciprocally and remember that one is explicitly going like the square of the concentration the diffuse layer capacitance and so at some concentration it's going to be either smaller or bigger than the Helmholtz capacitance which to first order isn't going like the concentration of ions in solution that's just defined by the molecular distance that those ions connect closest approach come to that electrode surface and so you can see immediately that for these two different physical capacitors adding in series at low concentrations the diffuse capacitance will dominate the total and you'll see the hyperbolic cosh and at high concentrations you should lose that because the Helmholtz layer capacitance dominates the total and it's to first order independent of potential. Now if you remember this dip is what we saw experimentally near the point of zero charge and we saw it for things like sodium fluoride at low ionic strength predominantly and we also saw at high ionic strength the capacitance tending to become independent of potential so the Gouy-Chatman-Stern model semi-quantitatively captures what we observe experimentally there are however a few important differences and those can be used to further tell us about double layer structure one important difference is if you compare the magnitude of the capacitance that is measured to the magnitude that will be calculated in a solvent like water of course we know the dielectric constant of water we can call this double layer something like three or four angstroms and so if this is 78 and this is three or four because this is going like the inverse that predicts what the capacitance that double layer should be and you can measure what it is and disagreement with that by a significant amount quantitative disagreement so let's see the way this goes this is this inverse is proportional to the capacitance this dielectric constant therefore establishes a value for what the Helmholtz capacitance ought to be but what we measure is a number much lower than that so at high ionic strengths what you measure in that limiting capacitance is much lower than what you would predict if you use the bulk dielectric constant for that compact double layer and to the extent that there are no other capacitors in series because they've all been swamped out by going to the high concentration this tells you that the effective dielectric constant of the Helmholtz layer must be much smaller or this thickness is much bigger than a molecular thickness to the extent that we now make an assumption that the molecular thickness is about right for what this compact layer is and you can get that by looking at the transition in concentration between when the diffuse layer dominates and when the Helmholtz layer dominates you deduce that it's the dielectric constant of that compact layer that is very different than that of water and in fact you would predict the differential capacitance as we saw on the order of about a hundred microfarads per centimeter squared but what you actually measure is a differential capacitance on the order of 10 microfarads per centimeter squared 10 to 20 so what this means is it means that the dielectric constant of this compact layer is much lower than that of bulk water and is on the order of 4 in fact is a pretty good number 4 to 10 the interpretation of that is that the structure of the material in that first layer is not that of pure liquid water but is much closer to that of highly structured water like ice that makes sense because you could envision let's say a chloride ion trying to get to closest approach to this electrode surface will not strip all its waters as it does that but maybe a few and the rest will be tightly held because it's lost the easy ones that are unbound it doesn't want to lose the others and they form a hydrogen bonded network that's essentially ordered and oriented as these anions are trying to pack very closely to screen the positive charge on this electrode surface or negative charge depending on what you applied you can go deeper by building molecular models and orient the dipoles and calculate from quantum mechanics from given molecular model structures what dielectric constant of that should be and people have tried to do that as well but that's extremely model dependent depending on exactly how many waters that you produce you think you've been lost from a given ion exactly where you put them and how you orient them so we won't want to go that deeply but the key important points are that it's that you need at least at least these two parallel plate capacitors in series in order to correctly model double layer measured capacitance if you do that you get it semi quantitatively right you get the cost dip around the point of zero charge at low ionic strength you get a higher but limiting capacitance not as high as it should be if the dielectric constant were that of the bulk solvent by about a factor of 10 and more independent of potential but not completely independent if you remember the sodium chloride measured number looked something if I went back to it we could find it see if we can find it was way back there look like that so here's the hyperbolic cost region here's the high ionic strength region where the potential dependence of the capacitance is flattening out on the order of 10 or 20 microfarads per centimeter squared is promised less than that and therefore limited by this number right because this is a capacitor in series with something and therefore the sum is never going to exceed this because the only other option is that the diffuse capacitance is bigger than this since they're adding in series the smallest one will always win and therefore even if there were regions of bulk water those capacitances must be higher than this region and this is the limiting capacitance region of all that's in series in that solution right but you see it's not completely independent of potential and the interpretation of that molecularly is that as you go away from the point of zero charge you start to be able to strip more of the waters off of the chlorides because as you apply more potential to them instead of just a chloride being solvated by waters trying to screen a point charge you're demanding more and more chloride screen that they're trying to snuggle up as close as they can and you're going to perturb their packing and arrangement as you apply more and more electric field strength to that compact layer the exact nature of this people are trying to probe by second harmonic generation and surface enhanced ramen and look at the dipoles and see their structure and interplay them with pretty heavy duty quantum mechanics calculations to try to model that whole dipole layer and their whole fields of people you know who've been studying this for a long time on electric surfaces so I won't go into more details there but did want to give you that flavor of what's happening yeah this asymmetry in part but another part in other part yeah in part it's thought that that's the case you're charging a positively in one case negatively in the other remember also that these are the surface excesses right and so you're you're naturally gonna not necessarily go to zero because that doesn't mean you've eliminated say all fluoride adsorption when it's here it's competitive fluoride with water and sodium with water right so you're changing something continuously all the way along okay so let me go ahead there are two other reasons why we need aha oops okay why we need to know about double layer structure and I think neither of them are in these slides but that's okay I can tell you about them the first is let me go back to that graph right there perfect when an ion comes up to an electrode a redox active ion and we apply a potential to it depending on where it is in the solution it will experience different amounts of driving force in its potential that is dropped that it sees right and you can see that when you involve this compact layer to first order you can say that the redox active ions like ferrocene or ferro cyanide ferrocene is a bad example because it's charged and how well it could penetrate would depend on the ionic strength of supporting electrolyte and whatever else is around it remember this double layer collapses fast and then the redox stuff happens is the model which this occurs that if you have something neutral say like ferrocene that it can come to the outer Helmholtz plane but never further in than that and so you can see the fraction of the applied potential that it experiences is a function of both the ionic strength concentration in the double layer in the electrolyte and the potential dropped across the diffuse layer that it doesn't see right it only really sees the potential dropped across the compact layer because if it can exchange electrons and it's in constant it's in solution in fact there's a distribution in solution of rate constants they're all distance dependent rate constants they're going to be roughly exponentially falling off with distance so the ones closest are going to dominate the overall measured rate and so the overall measured rate is going to be dominated by the ones that are close and experiencing only a fraction of the total potential dropped applied across the electrode so they don't to first order experience the potential dropped in the diffuse layer but do experience the potential dropped across the compact layer so you need to correct in fact the applied potential the over potential and everything you put in the Toffel equation a rigorous electro analysis of current versus over potential curves would not use the applied potential as the over potential but would use the applied potential corrected for at a given ionic strength the amount that is dropped across the diffuse layer and only use the amount that's dropped across the double layer as the potential that actually is being applied to that iron and this modification is in fact a double layer modification of the rate constant that is conventionally used in careful electro analysis so up until now I think everyone always told you well the things you put into a Butler-Volmer equation and the things you put into the Toffel slopes were the simple over potential relative to equilibrium that you applied to that electrode that's reasonably true at high ionic strengths in very conductive media because in that case the compact layer is where most of the potential is dropped and a relatively small fraction is dropped in the diffuse layer because this is a smaller capacitance that's big and this is limiting but in very diffuse media with the very low electrolyte concentration then that won't be the case and so you have to correct your over potential for the potential that the ion that you're measuring the electrochemistry of is really experiencing and you could make that correction analytically if you knew the capacitance that you measured for that supporting electrolyte under those conditions and you put in Stern's modification and you want to subtract off this stuff and only get that as to what the redox active ion can experience and if you look in fact on another page in Barton Faulkner after this they'll show you to do exactly that the second reason why this is important is that there's another effect of driving ions into this compact layer and I said I used ferro sinit as a bad example but in fact for this now it's a good example. Let's suppose that I was foolish enough to use as my in my gedonkin experiment as my redox active ion a part of my supporting electrolyte like instead of trying to reduce one millimolar ferricyanide in KCL I just decided that potassium ferricyanide is a good salt and water I'm gonna throw in one molar potassium ferricyanide and reduce it. What would happen? Due to the double-layer buildup I would have essentially a fixed charge at a given applied potential at a fixed distance away from the surface of my redox active species and it would essentially not be diffusing it's adsorbed on the electrode but at a fixed distance away from the electrode although I can't tell the difference when I'm making a measurement so I would see something that is essentially weakly like the square root or independent to first order of concentration of what I had the solution that if I did a potential step I would see a lot of the charge go into those adsorbed species and not go like T to the one half that's in the initial pulse and then later on I see the diffusive species take over as they continue to move toward the electroactive part that can do more the interface so what you would see in that kind of experiment would be you would take and do a potential step in the presence of now let's do it right let's do it with one molar sodium chloride and a little bit of ferroferri cyanide if there were an adsorption of ferroferri cyanide or if it were part of the double layer formation that what you would see would be in that initial potential step you would see a charge that was greater than the charge needed to just charge the capacitance of the double layer in inert electrolyte and that additional charge would be the ferrodeic charge to a fixed concentration of non diffusive species that was essentially part of forming the ions in this compact layer and the way you tell the difference is to do exactly what I just said as you start without that species there and you just make a potential step and measure the total amount of initial charge and there should be no T to the one half charge because there is no ferrodeic species in solution then you do it again with your electroactive species and there's going to be an apparent increase in the charge at T equals zero and that apparent increase isn't because you've got a different double layer structure per se because it's still one more sodium chloride but because there's some of that electroactive species that can't get any closer but at equilibrium stuck right there in that compact at the edge of that compact layer so it appears in the time intercept at T equals zero of an Anson plot in chrono-kilometre because you extrapolate the T to the one half amount of the charge down to zero that total amount is the non diffusive part and the non diffusive part is the sum of the capacitive part plus anything that looks like its capacitive this extra charge this non diffusive because the ions were stuck right there that were electroactive now it need not be a freely diffusing ion it could actually be a physically adsorbed ion for instance iodide on platinum physis orbs and actually in some cases chemis orbs into platinum makes chemical bonds and you can oxidize or reduce that and release it into the solution again if you do a potential step you'll see that as a non diffusive component in the intercept of chrono-kilometre and the extra charge you measure on top of what you inferred should have been there only in a purely double layer case is that adsorbed charge and so you use this Stearns modification to give you the model of what that double layer charge is supposed to be so as to do two things one is to see if there's any extra adsorbed charge and two is for non adsorbed species to correct the potential they actually experience relative to the applied potential by what's left across that inner Helmholtz plane to that compact layer and not correct for what's in the diffuse layer okay yeah the equilibrium potential no the equilibrium potential is not because that's an electrochemical potential the kinetics will be different but if you measure the the electric chemical potential you don't get something for free all right their concentrations are changing and there are over so that balances itself out but if you made us a kinetic measurement which is what a current over potential relationship is then they are different at different ionic strengths because of this phenomenon the difference is limiting and becomes less important and that's that's in part is so when some when Andy I'm sure told you all electrochemists work at really high ionic strengths of supporting electrolytes that's exactly why they do it is to minimize having to make this correction right that's still what all it provided there's no electron hopping from one to the next all right because you no longer have an inert layer of something like chlorides there that can't be electroactive so in fact electrons can hop from one ferry cyanide to the next ferro cyanide and if you brought them in because you now you have a non-electro inert thing it's still confounding right you've got electron hopping yeah but but yeah so I've told you what I think you need to know about double layer and if you want to know more details you can just read the rest of the chapter and Barton Faulkner on that but I think most of the key points I already covered the rest of what I want to do probably in the next 15 or 20 minutes today is to switch topics and talk about one other electro analytical method that Andy didn't talk about in chem 117 that we've agreed is incredibly useful and you should know about and that's microelectroids so there's no good definition of what a microelectrode is exactly because electrochemist started out with mercury drops of a certain size and then went to mercury pools they went to solid electrodes but a reasonable definition at least of a spherical microelectrode is one that without convection reaches steady state on an experimental time scale that you care about now why can we so typically it might be a micron or a few microns or 10 microns is the critical diameter it's not 10 millimeters because we already know that from a given diffusion coefficient that times tau goes like l squared over d and if you're looking at things that are kind of a millimeter away from a plane and we know diffusion coefficients are 10 to the minus 5 semi or squared per second roughly of ions and water then if you put in 10 to the minus 1 of of l and square it 10 to the minus 2 and divide by d 10 to the minus 5 it takes kind of a thousand seconds before something could reach steady state and it won't even do it then because after a thousand seconds convection comes in even from thermal gradients in the room and acoustic noises and things so you can really never last in a t to the one-half and a potential step on a planar electrode out to a thousand seconds it deviates from linearity for these other experimental artifact reasons right if you had an incredibly quiet non-thermal gradient room there's no reason the cotrel equation equation shouldn't hold forever but it never does past a few seconds in the real world on the other hand for a few seconds on the real world at small enough critical dimensions the current will reach steady state and will not die like t to the one-half and for electrodes whose currents reach steady state within a second or two or less on that order they're called micro electrodes other people have different definitions of them based on physical size but that's a good operating definition if they reach steady state or something that looks like it in a potential step experiment within an experimental timescale that electrochemists care about on the order of a second or something like that they're called the micro electrode now why would that be the case let's suppose we have a spherical drop remember that there are two terms in the in the solution of a current potential relationship in the diffusion limited regime for that and I understand that Andy did do that the first one is the t to the minus one-half term that leads to the cotrel equation when you solve that from the Laplace transform and did the potential step and the second term is the one he told you to ignore because the first term is the one that led to the cotrel part the NFAD to the one-half c square root of pi in the denominator t to the minus one-half the second term is NFAD c over the radius and has no time dependence in it now for a fairly large spherical drop even at relatively short times this term is big compared to that and what that's saying is that let's make a very large spherical drop the earth looks flat on a short enough time horizon locally and so does a mercury drop a mercury drop looks flat and looks like a planar electrode on a short enough time horizon and what's that time horizon that time horizon just has to be such that this term is big compared to that term and that happens for any reasonable time before you lose the t to the one-half dependence to these other artifacts I mean at long enough time this term will get small compared to that term because your diffusion layer grows bigger and bigger and eventually when it grows big enough you realize that your landscape is telling you the earth is curved and is not flat anymore because you're expanding your view of the horizon right and so you're expanding your view of the diffusion horizon think about the diffusion boundary that you apply to a spherical drop as it starts to grow at time it's growing like t to the one-half and eventually if you wait long enough that time horizon that that diffusion gradient won't be just planar and linearly related to the curvature of the mercury drop but will start to curve itself and you'll notice that it curves from the rest of the transport in the solution to that curved surface and you won't think it's a plane anymore the problem is that for any reasonable size mercury drop if this is a millimeter the time needed to realize that it's curved is sufficiently long that these other things start to interfere with your ability to measure the Cotrell current dependence anyway and you can't ever reach that regime on the other hand if this is a very small electrode then even at short times you lose the Cotrell dependence quickly and you see this term which is independent of time the reason you can see why radial dependence should be independent of time I don't remember I don't know if Andy did that or not he actually showed you the term right yeah probably when you solve the diffusion equation but said forget about it because we're going to look at at the other term so the way to think about is why does this go down like t to the one half it goes down like t to the one half because all the flux goes down because the in a in a planar system with time the projected area linear area away from the electrodes constant you lose more and more concentration because you have to reach further out with time like the square root of time so your current your flux goes down like the square root of time but now let's think about what happens to a curved surface with time around that hemisphere you're reaching further and further out in the diffusion layer right you're going further and further away but the area over which you can suck ions in is increasing like r squared and that increased area exactly cancels the flux over the increased area exactly cancels the decreased flux that you get from longer time so that this sphere expanding with time gives you a constant flux to a spherical point to a spherical sink unlike a linear sink that gives you a flux that goes like t to the minus one half it's because the area in that annual in that ring then that sphere is expanding with time and it's going like a squared with time the flux is going down like the square root the square root cancels the square to be a constant for linear you only see the square root because the area is constant okay so that's good so well known spherical diffusion is independent of time so clearly it's steady state for microelectrode when this is small enough to have that dominate that the cartrell equation for a potential step at even short times becomes NFADC over r zero now we know independently how to measure D because we know the concentration and we can measure the diffusion coefficient of that a macro electrode from a cartrell potential step now there are no adjustable parameters and so therefore we can use this to tell us what the effective radius of that microelectrode is just like if we knew the diffusion coefficient we could have used the measured current to tell us what the area of our electrode was but the important point is that these read steady state very quickly because of course this gets bigger than this at shorter times like the square root of time linearly with the radius changing and people have made electrodes the way they make STM tips of poking metal through pieces of glass that have radii that are micron or less when the critical length was a millimeter then that defined when the square root of time took over and it didn't take over for a second but if it goes down by 10 to the fourth then this can take over in much much faster times than that in milliseconds or less you read steady state so of course in a potential step if you could time resolve it fast enough and a microelectrode you would see a cartrell like behavior initially because it looks planar at very short times until the diffusion layer grows to be comparable to the radius of the object and then you know it's curved and then it becomes steady state but there's a lower bound on the time with which electrochemist can make measurements and the bound isn't electronic what's the bound the double-layer capacitance and resistance right it's the RC time constant to charge the double layer which is a physical bound because you've got to move those ions from the bulk of the solution through they're not through wires they're real ions with real masses in a real electrolyte until you have to move with a certain resistance to charge up a certain capacitance and there's a certain RC time constant that's associated with doing that and you can't accept there's exceptions but in general measure anything of significance on a faster time than the double-layer charging time right now fortunately as we'll see microelectrode's help us get around some of that too spherical microelectrode's are the ideal geometry for microelectrode's because of course it's radial diffusion reaches steady state but it's very hard to make objects that are very small that are perfect spheres it's easier to make them that are discs right you took a wire and embedded it through a piece of glass and then broke it off and polished it and made smaller and smaller wires and the projection then is a disc so a disc is the most common microelectrode but a disc isn't really purely spherical diffusion in fact it's more complicated than that it's kind of planar diffusion that way and radial diffusion out that way so what's clear is that small enough discs at long enough times look like point sources of radial diffusion and we'll ask him toad to a purely radial diffusion case but the details are important and you can solve for that and they've been solved right you've got to write the concentrations as a function of the radius away from the disc in the x and y direction since the disc will be independent of whether it's x or y and then the z direction is a unique direction and then this time and you write your standard del squared expand it out and then the boundary conditions just like normal and solve for it right and then we have a fourth condition because usually these discs are sealed in the things that are electrically inactive and so there's no flux there at all so the fourth boundary condition is that there's no flux in that plane for r greater than r0 greater than the diameter of your disc so now that we know all those boundary conditions we can plug in and in fact see that there's a function that function at long times asymptotes to one this is as a function of the radius of the disc it has to because at long times that disc just looks like a point source of which radial diffusion is occurring and is independent of t and this is exactly the same thing as you would get of it there's an area 4 pi times the area of the disc divided by its radius and at long times it's just the disc effective area looks like a hemisphere of that radius it's not a sphere there's a factor of 2 and a pi in there because it it's a little point disc instead of a curved sphere as the effective area so this is just a correction relative what we saw before by the area of a disc of radius r relative to the area of a hemisphere or a sphere of radius r and the rest of it is just reaching steady state because there's no time dependence right but at earlier times the terms do have the t to the one-half in them and in fact at very short times this is just going to be a cattrel equation because now it just looks like a plane or electrode on that time scale that a fusion layer just sees the disc as a plane and it'll exactly reduce to the cattrel equation so you knew this had to be the first term in the expansion and you knew the last term had to go to one and in between terms you don't know until someone like John Newman solves them on a computer for you okay so a disc reaches steady state and in fact at a disc ultra micro electrode they're the short times here's an intermediate time and then the long time regime looks something like that reach a steady state right here's a cattrelian like short time asymptoting down to the steady state long time radial diffusion limit okay pizza wheel of death so what about a cylinder so cylinders are wires sticking up just take a wire right a micron wire a fine gauge wire and don't get in a solution can't call it a plane or electrode now it's cylindrical diffusion and you solve it again here's del squared now it's going to be radial diffusion away from that cylinder but there's a end to the cylinder which is z and is going to be planar diffusion there the longer the wire the more it looks like radial diffusion the less there's a planar component you need the ratio of the planar area to the total amount stuck in the solution to figure out exactly what it is but clearly at short at long enough times when the planar diffusion part has gone away oops then it's going to be radial diffusion what's happening here right the other important point is that however unlike okay so let's think about what happens to a cylinder relative to what happens to a sphere what happens to a sphere is that the area through which the concentration flux is going is increasing like r squared right so it's when you have a cylinder of essentially an infinite length the area through which things are increasing is increasing like 2 pi r right so that immediately tells you that if one was exactly compensating the other in the spherical case to get the steady state that a cylinder won't get to steady state because the area is increasing fast enough to make up for the control induced flux decrease it's only going like r not like r squared and in fact what happens is that it looks like planar diffusion like like spherical diffusion except that an asymptotes in denominator like log of time so it's weakly dependent on time it doesn't have t to the one half but it doesn't reach a rigorous steady state right so cylinders are pseudo steady state geometries because the current will go up very slowly with time like 1 over log of time but will never not go up anymore at all now this is actually unfortunate because it's easier to make measurements on top of something you know that's a steady state interpreted then something goes like 1 over log time but if you know it analytically you can deal with it now the reason I went to cylinder is because the other easy geometry to make physically is a band ultra microelectro the way you would make a band is either lithographically right down a metal line on an insulating substrate like gold on glass that you would mask and make a micron or less and make it very long so you don't have edge effects to worry about or another way is you evaporate say gold on to a piece of glass and then evaporate something else insulating on top of that and the way Henry White has done this at Utah is then you just break it in half and you put the edge in a solution and now the solution sees that line of metal sandwiched by two insulating things his only problem is the way he did that when he broke it is that then he had to eliminate the variations in the glass that was cracked so he polished it with 0.05 micron alumina which probably many people believe might well have roughened that 10 nanometer band of gold to be much more than 10 nanometers because of the alumina grit that was used to do the polishing that's a debatable point but you see the idea so the band also has a critical dimension and it reaches a quasi steady state just like the cylinder because again there's one dimension the z dimension in which we'll have planar diffusion and then there's the radial dimension but it's not really truly radial anymore because it's not a cylinder it's a band and so you see the time dependence of that's yet different still goes like the log of time but there's a correction for the width of the band because the wider the band the more it looks like a planar electrode and you can see this must be the case because it's our standard dimensionless kind of thing here a dt over a length squared right because at small times the diffusion layer hasn't grown and this just looks like a planar macro electrode of a certain width and a certain length and a long enough times for the diffusion layer to understand that this finite in this width direction then it starts to be radial diffusion out there but it can't increase like the full a squared that you would do in a hemisphere but it's not like the cylinder that's completely uniformly going out because there's a plane of obstructed diffusion across the band that you don't have when you had the cylinder so it turns out that it goes like the log of the time with the w squared correction in it and we couldn't know this exactly but we could have guessed it would have if I told you the cylinder went like log time that this would go like log time with some correction there with the w and it turns out there's a correction of w in front too because it accounts for the fact that this is not fully having a diffusion layer go like w squared so you're getting less flux than you ought to have been if it was purely cylindrical okay so why do people use these they use these because they reach steady state these were the purely diffusive expressions right in each case there's a steady state or a pseudo steady state current that is related to NFA times the concentration of species solution times the mass transport coefficient that at a rotating disk you saw it came from the leverage equation that got you to steady state but the here you don't have to rotate the solution at all the solution generates its own steady state flux because of this hemispherical or pseudo hemispherical diffusion it's always good to reach steady state because you don't have to plot anything as a function of time the solution is that the mass and stirs itself furthermore at very small electrodes because these things are going like one over the critical dimension if the dimension of the electrode is on the order of a micron or less the mass transfer coefficients that can be obtained here are higher than those you can obtain what did Andy say with a before the leverage equation broke down with a Volkswagen engine on it okay because you're going like square root of the rotation velocity right and you have to maintain laminar flow and the Reynolds number has to be below a hundred to maintain laminar flow so it doesn't get turbulent so there's a limit on the mass transfer coefficient that you can get by rotating an electrode and remember what we're trying to do in kinetics is we always have mass transport as a as a bound to fight right because if the rate constant for electron transfer is greater than mass transport then the current at a given potential is limited by the rate of mass transport and you end up in the cut trail limit or the leverage limit and you're not sensitive to the rate constant you're trying to measure if you could continue to stir that leverage faster and faster and faster arbitrarily quickly there will be no rate constant that would in principle be experimentally not measurable because you could supply the material arbitrarily fast and therefore in a potential measurement at any potential you could always apply it faster than it could be reduced or oxidized and the current would eventually limit not at the leverage part but you deviate from leverage because you'd be limited by kinetics but in practice you can't reach that because you start to get the turbulence and leverage breaks down when you go into turbulence so you and it going like the square root of that rotation velocity so you can only reach mass transfer velocities of on the order of a tenth of a centimeter per second which means that if the rate constants are faster than a tenth of a centimeter per second or so that you're not sensitive to them and all you're measuring is the flux of those ions to the surface not the rate constant at which they're reduced or oxidized that turns out to be about the same time as what you can measure in the deviation from the cut trail equation and potential step which if you remember when you solved in a potential step the cut trail equation was the diffusion limited current and for rate constants that were slower than a certain value you would measure less current than that because you would be kinetically limited and when you actually look at the rate constants that you can measure for times in the cut trail equation of the planar electrode where you can still maintain t to the one-half dependence on the order of a second it again turns out to be about a tenth of a centimeter per second rate constant and so things faster than that at macroelectrists can't be measured but at microelectrodes you can get much faster mass transport coefficients because in fact they go like one over r and so in principle you could get very fast mass transport coefficients as if you were stirring the solution in the Leavitch equation with your Volkswagen engine by going down to a very small microelectrode radius that would reach steady state extremely quickly and have that diffusional area just expand with time to counteract any change in flux okay so that's the advantage but there's one more advantage let's see if we have it and the one more advantage is yes if I'm going to get to it so here's what a psychedelic tamagram would look like on a microelectrode if you go slow enough we're going to go from something here that's reduced okay we got it okay we're going to reduce something right if we go slow enough in scan then we reach the limiting current because of course the rate constant is getting more is getting higher as we get to higher over potential relative to the form potential eventually when the rate constant gets faster where diffusion limited if this is a radial microelectrode that current reaches steady state with time remember in a CV you're convoluting potential and time on that axis which means if it doesn't change with time it doesn't dip like a normal CV anymore it just plateaus out so this asymptote is the Cotrell equation analog it tells you the radius of the microelectrode and the concentration of the iron in solution furthermore there are only two possible functional forms of this one is the case when the rate constant is so fast that you never deviate from the Nernst equation at the as a boundary condition of the concentration of oxidized reduced at the electrode surface and in that case this just measures out an equation that just looks like a no-toffle it's purely the thermodynamic I minus I lim over I as if you remember you saw probably in the first week there was an I minus I lim over I dependence in the absence of kinetics that just comes from the Nernst equation at a boundary condition at the electrode surface relative to species in the bulk okay so this is analytically knowable all you need to know is the concentration and I minus I lim over I that's in the case of pure thermodynamic limit but in the case of kinetic limit then this will be stretched out relative to an I minus I lim over I because at potentials here that aren't a high enough over potential to be mass transport limited you'll be kinetic limited and so the current will reach steady state at that potential but its value will be lower than if it were Nernst boundary condition limited and from the functional form of this this maps out essentially the toffle behavior in that region and so it maps out the rate constant versus potential this would be exactly analogous to doing this experiment at a Levych stirred micro at Levych stirred disk at a high enough stirring velocity that mass transport wasn't limiting your current at that potential but kinetics was contributing or limiting to it and if you remember the Kuteki Levych plots you had one over I lim was one over the rotation velocity plus one over the rate constant because they were adding in series and the smallest one and so the limiting case would be the rate constant wins at high enough rotation velocities this is exactly the analog to that and you can see that you're more and more sensitive to those rate constants as the electric diameter gets smaller and smaller because you're reaching higher and higher mass transport fluxes and therefore taking less values of the rate constant in order to compete with that and be limited by the kinetic values there so you can make a steady state current voltage measurement without storing the electrode and use it to map out the rate constant as a function of electrode potential in mass transport regimes and therefore rate constant regimes that you can't access with normal electrodes now if you scan more quickly that's not what you see because after all if I scan this quickly enough let's suppose this electrode is small but I really go fast then the diffusion gradient never is time to grow and it never knows it's a spherical electrode it thinks it's a plane and so at very fast times you should recover a CV just like a normal diffusion limited cyclical tamagra will tamagram and in between is an in between behavior right so slow scan rates reach steady state fast scan rates go to planar diffusion and you need Nicholson and Shane to analyze those and in between or in between the nice thing about this is that the peak and because this is Nicholson and Shane the fast time ones tell you the area of the electrode in the diffusion coefficient and the slow time ones can therefore use those numbers self consistently to tell you the radius the electrode and therefore there are no adjustable parameters and it's all internally measured within one set of data at high mass transport flexes and there's one other more important point here which is why couldn't we do this measurement at any electrode you know why couldn't we take a planar electrode that's a millimeter long and go to a hundred thousand volts a second and beat out any rate constant why why did I tell you you couldn't go faster than a tenth of a volt a second than a tenth of a centimeter per second rate constant in a normal electrochemical experiment you believe me you should say but just go a hundred volts a second you'll be at a tenth of a centimeter per second rate constant so why can't we do that why can't we go a hundred volts a second at a conventional electrode at a millimeter-sized electrode yeah what about the double layer yeah it's actually more than that it's not only that you don't give it time to charge is just that there's a resistance to and the resistance of the solution scales with the electrode area right because you've got two plates the counter electrode and the working electrode and you've got to move ions through that medium between the working electrode and the counter electrode and the current that you're demanding even from a given concentration solution scales like the area right so the absolute magnitude of the current density would be constant the magnitude of I is linearly with area the time with which you can do any potential measurement is given by the Rc time constant of the system and the resistance is proportional to the electrode area which means that macroscopic areas inherently demand because they're demanding more total current more resistance which means the RC time constant for a macroscopic cell is on the order of like a millisecond you could do a potential step faster than that but the cell itself can't charge because it's in its internal resistance times the double layer capacitance charging is on the order of a millisecond time so you can't make a measurement on that system faster than a millisecond by even if you have fast electronics the system itself has an RC time constant of about a millisecond for a macro electrode but of course since that are scales like the current and the current scales like the area if you go down to a microelectrode you win like R squared in the absolute magnitude of the current now it turns out the capacitance actually goes up so but the net effect of that is still linear with the electrode area so the time constant of the cell goes down on the order of linearly with the critical dimension of your electrode so it could be a long band electrode and the critical dimension is in its length or a long cylinder it doesn't matter what its length is it matters what its critical dimension is what its radius is of the cylinder of the band as that gets smaller your RC time constant gets smaller approximately linearly with that critical dimension and that opens up shorter time windows for you to do potential step experiments or CVs I think we can see that yep so that's what a psychic voltamogram looks like at different sweep rates if you do it a macroelectrode you'd start somewhere like there and you see your peak if you go faster it starts to get washed out and if you go really fast the peak is really small and the reason for that is you're sweeping this potential like crazy through the electronics but the system doesn't even know what you're doing because it's internal RC time constant is so long that your it thinks you're barely moving the potential that it can respond to and then you're going back the other way in your psychic voltamogram basically you swept it back and forth before it even figured out you moved it at all and that's just a function of the RC time constant basically the electric chemical cell in the electrolyte is a filter with a given RC time constant on what you do to your applied potential it's a low pass filter right above some time you could be stepping it like crazy and it just can't respond and the time constant with which is a low pass filter depends essentially linearly on the critical dimension and so you can go by going to microelectrode you can do psychic voltametry at 100,000 volts a second and people have done that now it's still better many people believe to not do psychic voltametry but to do the steady state analysis provided that what you're measuring is a pure electron transfer rate limited reaction just oxidize go to reduced and back sure you can get that out of a CV at 100,000 volts a second you're going to apply Nicholson and Shane and your forward versus reverse splitting and your peak height are going to fit Nicholson and Shane and you get the rate constant out of that but you could also get it out of just a steady state current voltage deviation from the Nernst equation from the island minus I over island because you've got very effective mass transport and you make a steady state measurement the advantage of making the steady state measurement is no IR correction you have to worry about you have to worry about what you have to worry about IR correction but you have to worry about it in a much different way then you have to worry about it affecting the CV because you're doing a steady state measurement ok on the other hand CV is still valuable for square schemes and for other things when the product isn't stable like suppose you have a transient intermediate and you want to know what its potential is or you want to know what the first potential is to oxidize the first thing and as a follow-up the first thing goes to the second thing if you do a steady state measurement you might depending on the rate determining step have a convolution of all that in your current but if you can go to 100,000 volts a second the only thing you see first is the first thing and then you can watch it just like people like psychedelic treatment for other complex chemical mechanisms do other stuff right so each has its role but the point is that microelectros allow you to access very fast times and or depending on how you choose to do it very high mass transport coefficients so you can measure rate constants in principle that are even on the order of I think if the electrode diameter is on the order of a few nanometers then in principle that's an equivalent mass transport flux of about a thousand centimeters per second to give you a feeling what a thousand centimeters per second is I don't know if Andy told you what the limiting ion movement is to a plane it's on the order of 10 to the fourth centimeters per second for ions we know from electric chemistry from semiconductor electric chemistry that for electrons the thermal velocity makes them tend to the seven centimeters per second to a plane and ions are a thousand times heavier on average than an electron because it's just the ratio of the proton mass of a few protons and neutrons relative to a few electrons about a factor of a thousand and so the thermal velocity at room temperature of ions in solution is easily shown from stat mech to be on the order about 10 to the fourth centimeters per second it's just kt over h it's just the partition theorem is one half mv squared right and the m is a lot is a thousand times less so plug that in figure at the velocity you'll find that ions in solution bang into a surface at a collision limit of velocity of about 10 to the fourth centimeters per second now for an electrochemical reaction not every collision is going to lead to an electron exchange because a lot are not fruitful especially if they're activation barriers to that and so you could see that if you can measure rate constants on the order of a thousand centimeters per second they're very close to the collision limited ones and very low reorganization energies would be needed to even get you close to that so there would be very few if any electrochemical rate constants that ought not to be accessible a steady state by measurement at these high mass flux nanometer sized electrodes now there's actually debate as to whether or not anyone has actually ever successfully made a nanometer sized well-defined electrode to do these measurements at but there's not much debate that if you could do that that you could measure essentially all electrochemical rate constants no matter how fast even the so-called reversible ones won't be reversible when you start to approach the ability to measure every collision as to whether or not it's going to react so we won't talk about the literature examples of that because that's beyond what we want to do but I'll tell you it's debatable as to whether or not people have done that but it is an interesting extension of microelectrodes into the nanometer regime and the question is whether you would call them if microelectrodes were millimeters ultra microelectrodes were microns whether they would be like super duper ultra microelectrodes or nanodes or something else is another story so with that I covered the two topics I want to get to decide whether or not there is more to actually talk about that we care about next week so I'll let you know whether or not we're actually gonna have more to say on Tuesday or Thursday we're done