noc20-me28-lec39

[Music] [Music] welcome to the 39th lecture in this open course on diffusion in multi-component solids in this lecture I shall discuss the derivation of expression for the diffusion flux of a component in terms of chemical potential gradients we will see that for the substitutional diffusion by vacancy mechanism the flux of a component depends not only on its own chemical potential gradient but also on the chemical potential gradients of other diffusing components it is said that gradient in chemical potentials is the fundamental driving force for diffusion today we will see why it is so we will try to derive the expression for diffusion flux under the chemical potential gradients we will use the simple kinetic theory for this derivation this type of equation has been derived long back by seeds Fisher Holloman terrible and by body I will try to explain this theory in today's conventional terms we have seen the expression for diffusivity be equal to 1/6 gamma lambda square in which gamma is the successful jump frequency of an average atom and lambda is the jump distance jump frequency term gamma involves two types of contributions one is the number of times an atom has energy equal to or greater than the activation energy for migration and second the probability that an adjacent site is vacant so that belongs to the activation barrier for vacancy formation in the absence of any driving force we have seen at their regular atomic positions the atoms are lying in a potential energy well so if this is an atom e and it jumps onto the next site it's called this V there is no net change in the Gibbs free energy cause of this jumper but in order for this jump to occur the atom has to cross this energy barrier which is known as activation energy for migration Delta G M when there is a driving force present the situation is little bit different whenever there is a driving force the jumps are biased in a particular direction because if the atom jumps in that particular direction it causes decrease in Gibbs free energy so the first one this diagram we have shown for the condition in absence of any driving force will see in presence of driving force this potential energy well configuration is little bit different okay so this potential energy wells will look something like this so there is a driving force in the x-direction let's say so when the item jumps from this position into the next site here we'll see that it is associated with decrease in Gibbs free energy and the decrease in Gibbs free energy is referred to as the driving force obviously we can see for the forward jump for our class here let's call the jumps from left to right as forward jump and the jumps from right to left as reverse jump of the atom so for the forward jump we can see the atom has to cross a lesser energy barrier than a reversion so for example if the atom in this position wants to jump back it has to cross a bigger activation barrier than for the forward jump and because of the symmetric condition we can show that the activation barrier for the forward jump can be written as Delta G M minus F by 2 and activation barrier for the reverse jump would be Delta G M plus F by 2 now if we denote the number of times an average atom has an energy equal to or greater than the activation energy for Migration per second as gamma a let's call it the jump frequency of Adam this is an absence of driving force and this is only the migration part so as you see the actual jump frequency gamma will have the migration part as well as the vacancy formation part so in my notation here gamma a is the jump frequency which includes only the migration part and gamma a is in the absence of any driving force so obviously when there is a driving force present like here the forward jump frequency let's call it gamma plus will be different than the reverse jump frequency let's call it Yama - so in the absence of driving force gamma a will be given as gamma a equal to MU 0 exponential minus Delta G M by K T where nu 0 is the Debye frequency the number of time atom vibrates per second so obviously gamma plus would be given by nu 0 exponential minus Delta G M - if by - by K T and gamma - would be nu 0 exponential minus Delta G M plus F by 2 over K T so we can write this exponential term as exponential minus Delta G M by K T times exponential F over to Katie so obviously gamma plus would be equal to gamma a exponential F / - Katie and gamma - would be gamma a exponential minus F / - Katie so gamma a here again represents the jump frequency of atom a in absence of any driving force so the jump frequencies in the forward and reverse direction will be modified by this driving force F and the driving force is usually a positive quantity and so F will be given as minus Delta G so Delta G is the difference between the Gibbs free energy of this side and this side and it will include only the non-ideal part so we'll come back to it later so with this we can use an approximation exponential x is approximately 1 plus X if X is much less than unity now in these expressions for gamma plus and gamma minus F is usually much less than KT and so we can use this approximation and we can write gamma plus is equal to gamma a times 1 plus F over 2 K T and gamma minus is equal to gamma a times 1 minus F over 2 K T and the difference between the forward and reverse jump frequencies would be gamma plus minus gamma minus should be equal to gamma a times F over KT so it will be clear why we got the expression for the difference between gamma plus and gamma minus as we proceed so we have now expressed the forward and reverse jump frequencies in terms of the driving force and the jump frequency in absence of the driving force now let's try to evaluate the diffusion fluxes in terms of this jump frequencies so if we consider two atomic planes they'll plane 1 and plane 2 and the distance between the two planes is lambda so each plane can be thought of surrounded by an element of volume lambda if we consider unit cross sectional area let's say the number of atoms of a on plane 1 are in a and the concentration of the volume element around plane 1 is C a and so we know in a is equal to volume is lambda times C a and if we consider that there is a concentration gradient from plane 1 to plane 2 then the number of atoms on plane two would be na plus lambda times dou a by dou X that is the gradient of number of atoms with respect to X and so in terms of C a it can be written as lambda times C a plus lambda square dou C a by dou X now we will be treating here the diffusion that occurs by vacancy mechanism so it is important to know the vacancy concentrations on the two planes so if we express the vacancy concentration of plane 1 as X 1 V and there is also a gradient in vacancy concentration if we allow for the gradient in vacancy concentration the vacancy concentration on plane 2 would be X when V Plus lambda times the gradient in X 1 V with respect to X with this if we want to evaluate the inter diffusion flux across a plane that lies in the middle of plane 1 and plane - we need to consider how many atoms are jumping from plane one to plane two per unit times and how many atoms are jumping from plane to to plane one per unit time and take the difference of the two so the difference will give me how many atoms of a are crossing this plane at the middle per unit time per unit cross sectional area so that's the diffusion flux across this plane in the middle of plane 1 and plane - so if I denote the number of atoms jumping from plane went to plane two per unit time as J a12 - that would be equal to the jump frequency of atoms a and this I am considering again including only the migration part times once the atom has the required energy to jump the next side also has to be reckoned so this will also depend upon the probability that a vacancy exists on that Jason site it depends upon the vacancy concentration on claim to here so x1 v this is at plane two times the number of atoms of a on plane one so this would be equal to gamma a plus times the vacancy concentration on plane two is given by this term this would be X when V + lambda 2 X when V by dou X times the number of atoms of a on plane one is lambda C a so this is J a from plane went to plane two similarly J a from playing to to plane one would be given by the jump frequency of atoms a and this will be the reverse from frequency which is gamma a minus times the vacancy concentration on plane one which is X when V times the number of atoms on plane two which is lambda C a plus lambda square times dou C a by dou X and so the inter diffusion flux J a is given by the difference between 1 and 2 so if we take Jas J a 1/2 2 minus J a 2 2 1 if we arrange the terms containing lambda square together we get lambda square times gamma a plus C a2 x1 v by dou X minus gamma a minus X 1 V times dou C by dou X plus gamma a plus minus gamma a minus x mi plus X 1 V lambda CA X 1 V times C a times lambda so now it would be clear why we evaluated the difference between gamma a plus and gamma a minus let's denote this as equation three so if we now substitute for gamma a plus gamma a - and for the difference between gamma a plus and gamma a - from these equations here you will get J equal to gamma a lambda square into 1 plus F over 2 KT times C a dou X 1 V by dou X - 1 minus F over 2 K T times X when we do see a by dou X plus gamma a times F over KT times X when we times C a times lambda it's called this equation 4 so we have got the equation for diffusion flux of a in terms of the driving force F and the jump frequency gamma a in the absence of driving force and the gradient of vacancy concentration as well as the vacancy concentration and lambda the jump length so this will be true for any type of driving force and the specific expression will try to derive for a driving force which is the chemical potential gradient so let's try to see what form the driving force F tags when there is a chemical potential gradient for this let's go back to this diagram of plane 1 and plane to hear a when an atom a jumps from plane 1 to plane - what is happening is the of atoms of a on plane to have increased by one whereas the number of vacancies on plane to have decreased by one and the reverse has occurred in plane one that is and the number of atoms of a on plane one has decreased by one and the number of vacancies on plane one has increased by one and we know the increase in Gibbs free energy when an atom is added to a homogenous alloy is the chemical potential of the atom we can similarly define the chemical potential of a vacancy it is the change in Gibbs free energy when an atom is removed from the interior and placed onto the surface in which process when extra lattice site has been created so if there is a chemical potential gradient suppose let us say chemical potential of a is decreasing from plane one to plane - obviously there will be a net decrease in Gibbs free energy and this will act as a driving force for diffusion from plane went to plane - so if we consider a binary alloy of a and B and since we are considering the diffusion by vacancy mechanism let's also consider vacancy as one more species which is diffusing this is because by creation or destruction of vacancies there is a change in number of lattice sites and so we need to consider vacancy as one more species here so we can write the total number on any plane of lattice sites as n equal to an A plus NB plus and we and the gifts free energy of this plane can be expressed as G equal to MU a na plus mu P and B plus mu V and V now G can also be expressed as for any homogeneous alloy as G 0 plus G X s minus T Delta is some ideal so in this expression G 0 is the Gibbs free energy before mixing this part represents the change in Gibbs free energy if the mixing was ideal that is this part is related to the change in configurational entropy because of mixing and G X s is the excess Gibbs free energy which is basically the change in Gibbs free energy because of all the other reasons than this ideal configurational entropy change now when Adam is jumping from 1 to 2 the driving force associated with this jump will be related to the change in Gibbs free energy only because of the non-ideal part and so we can write the change in Gibbs free energy as the atom of a jumps from 1 to 2 as Delta G is equal to and dou G X s by Doh and a minus dou G X s by Doh n V on plane at X plus lambda minus dou G X s by Doh and a minus dou G X s by Doh n V at the plane X now this term dou G X s by doin a at denotes the change in excess Gibbs free energy because of addition of one atom of a whereas this term no GX s by Doh Envy denotes the change in Gibbs free energy because of addition of one vacancy and since the vacancy is jumping away from the plane at X plus lambda that is from plane two here there is a negative sign attached to it and this total expression we can write as lambda times dou by dou X of this difference between dou G X s by Doh na and dou G X s by doing V and since F is simply minus Delta G we get the expression for F as f is equal to minus lambda dou by dou X of dou G X s by Doh in a minus 2 G X s by dou n V now if we substitute for F from this equation let us call this as equation 5 into equation 4 here it will be a long expression but we can approximately neglect any term in lambda which is greater than lambda square now since the term F involves a term lambda so we can see here and there is a term lambda square outside the bracket here so lambda square times F will involve a term in lambda cube which we will neglect and so this expression can be written as j a equal to gamma a lambda square a dou x 1 v by dou x minus x 1 b dou C by dou x if here can be written as minus gamma a X when we see a lambda square into 1 by K T times dou by dou X of dou G X s but I do in a minus dou G X s by Doh NV let's call this equation 6 now we can write this dou X 1 V by dou X term as X 1 V times dou Ln of X 1 V by dou X and similarly this term second term we can write as C a times dou lnca by dou X so we have the coefficient of C a time X 1 V here and see a times X 1 B again here and so this whole term we can take out we can denote it as gamma a times X 1 V lambda square C a times dou Ln X 1 V by dou X minus dou Ln C a by 2x this one we can draw it as minus 1 by K T dou by dou X dou G X s by 2 and a minus dou G X s I do envy this equation 7 now let's try to find out the nature of dou G X s by do in a and dou G X s by du and V from this equation we can write G as G 0 plus G X s minus Delta s em ID can be written as T 2 K Ln Omega and we know for the mixing of a B and vacancies we can write Ln Omega as Ln of total number of atoms n factorial divided by n factorial times in B factorial times in V factorial this we have derived when we went over the statistical model of mixing during our thermodynamic refresher so this can be written simply as n ln n minus and a ln n a minus NP l n mb + v if we take the derivative of G with respect to n a which we know by definition is the chemical potential of new a which should be on the right-hand side here G 0 is constant so dou G 0 by dou n a is 0 so we have the term dou G X s by dou n a minus KT times dou Ln Omega by dou n a so if I now look into this if I take the derivative with respect to a similarly dou anybody n is 1 and n B and in V are independent of n a so that the derivative of n B and in V with respect to n a would be 0 so we can write this as Ln n - Ln a if you take the minus sign out and we know n a by is the mole fraction of a or the atom fraction of a which is entered as X a so this will be equal to minus Ln X a and so we get the expression for MU a as dou G X s by Doh in a plus KT Ln X a X is the mole fraction of a with a similar logic we can write mu V equal to dou G X s by Doh + v + kt ln of x when v x 1 v is the side fraction of vacancies and from here if we substitute for dou G X s by Doh na and dou G X s by Doh in V in two equations seven here we will get J a equal to gamma a X 1 V lambda square C a times dou ln x 1 V by dou X and dou Ln C a by 2 X - 1 over KT times dou by dou X of dou G X s by dou n a is mu a minus kt ln x a and dou G X s by Doh NV is mu v minus kt ln x 1 v so this would be minus mu v plus kt ln x one week now this katie and katie will get cancelled and so dou ln x 1 V by dou X get cancel with this term and so we get the expression for J a as j a equal to if we take the mu a minus mu v term together will get - gamma a x1v lambda square seed a times 1 by K T dou by dou X of mu a minus mu V + gamma a X 1 V lambda square C a times dou Ln X a by dou X minus dou Ln C a by dou X we know the mole fraction X a is given as the concentration C a times the molar volume V M and so if we substitute here Doyle and XA dou lnca by dou X terms will get cancelled and so we are left with the gradient of L and VM with respect to X and so finally we get the expression for J a under chemical potential gradient as a is equal to minus gamma a X 1 V lambda square C a times 1 by K T dou mu a minus mu V dou X plus gamma a X 1 V lambda square C a tow Ln V M by dou X so this is the general expression for inter diffusion flux under chemical potential gradient you can see there are three contributions here one is from the chemical potential gradient of a also the chemical potential gradient of vacancy and also from the change in molar volume with X and the molar volume will change with X if the molar volume is a function of composition because the composite chan is changing with X so the first two terms here are basically the Fijian flux the flux because of the chemical potential gradients and the third term here is the drift term associated with the change in molar volume obviously if the molar volume will change with composition there will be a volume flow associated with the diffusion process and it will give rise to a drift and so there will be a contribution from drift to the diffusion flux of a now if we assume that the vacancy equilibrium is quickly attained at every plane that means if there is no excess vacancy flow then the chemical potential of vacancy is zero and then the only contribution will be from the chemical potential gradient of a and the molar volume change in addition if VM is assumed to be independent of composition and the only contribution will be from chemical potential gradient of a so we can write J a equal to minus gamma a X 1 V lambda square C a times 1 by K T dou mu a by 2 X and we can write this as minus L a dou mu a by dou X so we have derived the equation for diffusion flux of a in terms of the chemical potential gradient of a and it is related to the chemical potential gradient of a through this term L a this is similar phenomenological type of equation that we used for flux as a function of concentration gradient which were related through the inter diffusivity term so more fundamentally the diffusion is driven by the chemical potential gradient now this term L a which is called kinetic coefficient you can see frequent these two sides la is equal to gamma a x1v lambda square si a times 1 over KT and if we look at this gamma a x1v lambda square this is nothing but the gamma lambda square and if we add the geometric factor corresponding to the specific type of lattice so for cubic this geometric factor is usually 1/6 so 1/6 gamma lambda square and we know this is the answer diffusivity or what you call as pressure diffusivity so we can write this as L a is equal to da star times C a over KT where da star is the tracer diffusivity and the stressor diffusivity we know is essentially under negligible concentration gradient so when there is no concentration gradient or negligible concentration gradient we are talking about the self or impurity diffusivity which is generated by the a star it is referred to as tracer diffusivity so we can express the kinetic coefficient in terms of the tracer diffusivity of the particular atom now let's look into more general case when there is a net vacancy flow which is existing which is bound to happen if da star is not equal to D B star B star is the pressure diffusivity of B in that case we have to consider dou mu V by dou X term also so we know by gibbs-duhem equation x-a domu a by dou X plus XP dou mu P by dou X plus X 1 V dou mu V by dou X should be equal to 0 and so we can express the chemical potential gradient of vacancy in terms of the chemical potential gradients of a and B so that will be dou mu V by dou X should be equal to minus X a over X 1 V dou mu a by dou X minus X B over X 1 V dou mu B by dou X and if we substitute for dou mu V by dou X from here into this expression for J a and neglecting the dependence of molar volume on composition we will get the equation of the sword G a is equal to minus L a times dou mu a by dou X plus X a over X 1 B dou mu a / 2x + 6 P over X V dou mu P by dou X so G a can be expressed as a function of chemical potential gradients of both the components a and B will be minus L a times 1 plus X over X when V dou mu a by dou X minus L a times X B over X when V dou mu P by 2 X so we can in general express this as J equal to minus L a a dou mu a by dou X minus L a B dou mu B by dou X so the diffuse clerks of a will be guided by chemical potential gradients of both a as well as B the relation with chemical potential gradient of a would be expressed through this main kinetic coefficient la a and the contribution from the chemical potential gradient of B will be expressed through this cross kinetic coefficient l a B and this will be true whenever the vacancy diffusion is involved or whenever we are considering the substitutional diffusion by vacancy mechanism because to more or less extent there will always be a net vacancy flow as long as da star is not equal to D B star only the relative contributions will keep changing depending upon the difference between da star and DB star now we did this for binary but this is in general true for an N component system so in general we can express the diffusion of a component I J I as minus summation of L IJ dou mu J by dou X and J here varies from 1 to N so we have obtained the Unser's relation for diffusion under chemical potential gradient so it is important to note here as long as the vacancy mechanism is involved in order to define diffusion flux we need both the main kinetic coefficient similar to la here as well as the cross kinetic coefficient L IJ when I is not equal to J similar to L a be here one more point that I would like you to note here is that the dependence of molar volume on composition will also give rise to a contribution to the flux of a and it comes through this term doel and vm by though x so only when we assume that the molar volume is independent of composition we can ignore the drift associated with the diffusion and that's why we define the volume fixed frame of reference alright so we derived the relation that expresses the diffusion flux of a component in terms of its own chemical potential gradient and when there is a net vacancy flow that is when there is a difference in the tracer diffusivities of different diffusing species then the contribution will also come from the chemical potential gradients of other components and in general the flux can be expressed by this own seizure type of relation thank you [Music] [Music] [Music]