noc20-me28-lec15

[Music] [Music] welcome to the fourth week of this open course on diffusion in multi-component solids this is the fifteenth lecture in this series and in this lecture I will discuss how diffusion in multi-component system can be described based on unsub's formalism of ficks law we will also see how gives to him equation along with use of an appropriate frame of reference helps to reduce the number of inter diffusion coefficients from n square to n minus one square in an N component system so we have seen the expression for ficks law for diffusion of a component J equal to minus D DC by DX now in this expression D is constant which is called diffusivity and but it is valid when the concentration gradient is very small because only then D is constant because we have better this expression for D so to one-sixth gamma alpha square where gamma is the successful jump frequency and alpha is the jump distance if the composition changes both gamma and alpha would change right so D is a function of composition so only when we have a dilute solution or small concentration gradient you have B constant and you can use this expression but if the concentration gradient is larger then we know that D has to vary with composition because the in the concentration gradient C varies as a function of X so obviously D will also vary with X in fact under the concentration gradient or if the concentration gradient is larger and if there are interactions between the atoms the thermodynamic interactions are stronger then in fact the jump in forward direction is not same as in the backward direction so we need to take that into account and expressions for diffusivity will become more complicated okay so D is a function of composition but it does not depend upon the gradient of composition just like in furriers law thermal conductivity does not depend upon the temperature gradient and in row homes law electrical conductivity does not depend upon gradient of electric potential similarly diffusivity does not depend upon gradient of concentration but it does depend upon composition okay so that keep that in mind in fact whenever there is a diffusion occurring at any given plane already at again given X the composition also varies with T right so we should more appropriately write this as minus de dou C by dou X partial of Z with respect to X this sign for partial I'll call it dope because C is a function of X and T so how we take into account parameter T we'll talk in the next class but right now keep this in mind okay in fact this is valid only for binary systems diffusion of component one into the other if we have multi-component system then we need to modify this expression right so let's look at ficks law is applied to multi-component diffusion the multi-component system means it has typically three or more components so if we consider n component systems let's say component one - III - - - and then we use on such as formalism and extend the ficks law using one size formalism so using one size formalism we express the flux as a linear combination of all the concentration gradient the flux of a component I so if we write the flux of component say one its denote this by J 1 tilde so Tilda denotes that we are considering inter diffusion or diffusion due to the strong concentration gradients so J 1 tilde as a linear combination of all concentration gradients since we have n components there are n concentration variables and n concentration gradients right so we need to express this as linear concentration gradients linear combination of n concentration gradients so which we write as dou C J by dou X where J is equal to 1 to N and the corresponding coefficients we denote as d the effect of 1 on to flux of 1 we didn't touch d 1 1 tilde and obviously we gives negative sign similarly d 1 2 tilde and so on for component n it is d 1 n tilde similarly the flux of to we express as d 2 1 tilde to C 1 by 2 X minus D 2 2 tilde those C 2 by dou X by dou X and we have n such flux equations J n tilde so how many coefficients we have for each flux we have 1 2 3 n terms and there are n such fluxes so n times n the total n square these are called intra diffusion coefficients okay but all of them are not independent why because there are constraints on both the concentration gradients and the fluxes which means not all concentration gradients are independent similarly not all fluxes are independent provided we use the right frame of reference right I have introduced you to the different frames of reference that we use so it is worth to mention here that most of the time the volume fixed reference frame is used for diffusion measurements okay so that is most popularly used frame of reference so in basically volume fixed frame which we consider then all n fluxes are not again independent so we can reduce some of these coefficients so let's see how do we reduce the number of it efficients using the constraints on gradients and fluxes so first consider the constraint on concentration gradients and its origin lies in the gibbs-duhem equation like we have seen the gibbs-duhem equation during the refreshers on thermodynamics right so if we apply gibbs duhem equation to partial molar volumes what do I get X 1 DV 1 bar plus X 2 GV 2 bar X and DV and bar equal to 0 so if we multiply both sides by VM which is the molar volume divided divided by GM which is the molar volume so V 1 bar V 2 bar here are the partial molar volumes of components 1 2 up to component N and X 1 X to denote the mole fractions of components 1 2 so we can write X 1 by VM [Music] now what is X I over VM is what CI right there C is the concentration in number of moles per volume unit better Q per centimeter cube right so we can write C 1 DV 1 bar plus C 2 D V 2 bar C n DV and bar equal to 0 which means Sigma C I V VI bar equal to 0 but you know what is civi bar is the civi bar is the volume fraction remember we have derived this some time back so civi bar is Phi I which is nothing but the volume fraction of I and so Sigma C I VI bar should be 1 the summation of all volume fractions should be equal to 1 so if we take the differential you know Sigma C I D VI bar plus Sigma V I bar DC I should be equal to 0 and this we know from here is already 0 so we have Sigma e I bar DC I equal to 0 so if you differentiate with respect to x coordinate we can write more specifically for this as V 1 bar those even by dou X plus V 2 bar dou C 2 by dou X 1 V and bar Lucienne by dou X equal to zero so this is the constraint that we have to use so we can write the constraint we can write here dou C n by dou X is equal to n minus 1 component V n minus 1 bar over V n bar dou C n minus 1 by dou X so if we substitute for dou C n by dou X suppose if we substitute for J 1 tilde now if we substitute in the equation for J 1 tilde substitute for dou C n by dou X which replace this term by this term on the right hand side so this negative negative becomes positive and we have D tilde 1 n times V 1 bar over V n bar times 2 C 1 by D X and so on so we will get and then we rearrange to take out the gradient terms individually then we get minus D 1 1 tilde minus D 1 and tilde over V and bar minus d went to tilde 2 V 2 bar over he and bar those C 2 by dou X minus 1 to the N minus 1 term even D 1 n minus 1 tilde tilde to be n minus 1 bar over V n bar to C n minus 1 by dou X so each of this term we can substitute with one term so this one we can substitute with one term which we denote as d1 when tilde superscript 3 superscript n sorry so each of this term we denote as d1 1 tilde superscript and in this particular case similarly this one will tilde n minus 2 X so in general we can write though I J tilde n is equal to di J tilde minus D are n tilde times V J bar over V n bar which means not all of these n coefficients are independent but we have only n minus 1 independent coefficients for a given flux term now how many such fluxes are there there is also a constraint on fluxes and what is that constraint and if we use volume fixed frame the constraint is Sigma GI tilde VI bar should be equal to zero remember our fluxes here are expressed as expressed in the unit of moles per centimeter square or meter square per second so we can write J 1 tilde plus J 0 which means J and tilde is given as minus even bar over V n bar n minus 1 bar over V and bar J so again not all n fluxes are independent so N 1 is dependent only n minus 1 independent fluxes so we have each flux expressed by n minus 1 independent terms and there are n minus 1 such independent fluxes so total n minus 1 square inter diffusion coefficients so in general we can express if you write equations for all n minus 1 fluxes here then we express the flux of any components ji tilde equal to minus summation of D I J tilde n dose CJ by dou X J here where is from 1 to n minus 1 and also I is from 1 to n minus 1 and so there are n minus 1 square coefficient here so this is how we can express in to diffusion in a n component system we can describe into diffusion in n component system by using ficks law extended to the multi-component system by all sergel's formalism so we need n minus 1 square inter diffusion coefficient to completely describe inter diffusion in an N component system moreover the set of these n minus 1 square inter diffusion coefficients are functions of composition which means with varying composition this set will vary or for a given composition we have a fixed set of n minus 1 square into diffusion coefficients if I describe this for a ternary system in order to understand this so consider a ternary system of copper nickel zinc so which component to treat as dependent component is our choice we can treat either copper or nickel or zinc so if I treat zinc as dependent then I have two independent fluxes one for copper one for nickel so I can write J tilde copper as minus D tilde copper copper superscript zinc the superscript refers to the dependent component times the gradient of copper and tilde copper nickel J nickel tilde should be minus D zinc superscript C copper by dou X minus D so I have for inter diffusion coefficients in a ternary system right so n minus 1 square 3 minus 1 is 2 2 square is 4 because there are two independent fluxes and two independent concentration gradients so in this case d copper copper d nickel nickel or di J tilde n where I equal to J in this case they are d copper copper zinc and nickel nickel zinc these are called main entry diffusion coefficients right so d-iii tilde ends are called mein inter diffusion coefficients and they quantify the effect of concentration gradient of I on its own inter diffusion flux right so it is basically the contribution of concentration gradient of I on its own inter diffusion flux di J tilde n where I not equal to J are referred to as cross or interactive inter diffusion coefficients so in this case D copper nickel and D nickel copper are the two cross inter diffusion coefficients and they are basically the contribution of concentration gradient of J on the flux of I so these are basically the quantification of what we refer to as diffusional interactions or the cross effects so these cross coefficients quantify the cross effects meaning if we consider D copper nickel zinc term when we use zinc as dependent it means how the gradient of nickel is affecting the flux of copper similarly D nickel copper indicates how the gradient of copper is affecting the flux of nickel so D copper nickel quantify the interaction of ECAL with copper and nickel copper quantify is the interaction of copper with nickel during the diffusion clear so now when we write the cross terms and the main terms right for example in a ternary system if I write J 1 tilde equal to minus B 1 1 tilde the main term most of the time it will be positive although it is not necessary because day one when tilde 3 is d 1 1 tilde minus D 1 3 tilde but d13 the cross effect when 3 would be much smaller compared to the main term D 1 1 so most of the time this term will be positive the cross term D 1 2 tilde 3 may be negative or positive what does it mean basically if T 1 2 tilde is negative or positive if D 1 2 tilde is positive which means the inter diffusion flux of 1 is enhanced down the gradient of 2 and reduced up the gradient of 2 similarly if D 1 2 tilde is negative it means Center diffusion flux of 1 is enhanced up the gradient of 2 and reduced down the gradient of 2 so let's try to illustrate this so let us call this as term 1 which is the main term remember we need to take into account both gradient and coefficient right so let us call this as term 1 this is term 2 including this negative sign so let's suppose this is always positive then the first term should be depending upon the sign of the gradient remember it again the entire term may be positive or negative so what is the convention typically the flux if it moves from left to right it is given a positive sign if the flux is from right to left it is given a negative sign so let us consider two cases so I am allowing the diffusion between two different compositions of Ally 1 2 3 so let's say first one is 70/30 zero and second terminal Ally is hundred percent of three write this as 1 3 the second case so in this case one component one component two are moving in the same direction in other words the both the gradients have same sign in this case what is the sign of gradient of one and two negative gradients okay so both those even by dou X is less than zero and dou C 2 by dou X is less than zero the second case I am considering when the two gradients are opposite so let's assemble 70 is zero 30 with zero 100 zero so the second alloy contains 100% second terminal 100% of component two to start with this is again 70 this is component 1 this is component 3 so in this case now the two gradients are opposite so dou C 1 by dou X is negative but dou C 2 by dou X is positive yet as I'm increasing X the concentration of 2 is increasing that's why the gradient is positive so if B 1 2 tilde 3 is greater than 0 what does it mean if d 1 2 tilde is greater than 0 first term is positive we know because this gradient is negative so the term is positive if d 1 2 is also positive and this term is negative so this term will be positive so both the terms are are positive which means the flux of 1 is enhanced by the nones down the gradient of 2 right because D 1 2 is positive if it was negative what should happen term 1 is positive but D 1 2 is negative so this overall term becomes negative because negative negative is positive and there is one more negative term here so the first term is positive second term is negative so the cross term is having a subtractive effect that's why we say the flux of one is reduced so here a term 1 is greater than 0 but term 2 is less than 0 right so this is an additive effect when d12 is positive well d12 is negative subtractive effective so we say that the diffusion of one is reduced down the gradient of two if d12 is negative now in this case we'll have the reverse scenario right if d1 to tilde is greater than 0 then obviously this is positive and this is positive here the gradient also positive so this term is negative and so we will have subtractive effect so we say if d1 to tilde is positive the flux of component 1 is reduced up the gradient of 2 and if d1 2 tilde is less than 0 we will similarly have additive effect so now here we have expressed the fluxes of copper and nickel with zinc as dependent component now I can very well write these equations with copper as dependent component right so if we write let's say copper as dependent component then I will get equations for two independent fluxes will be zinc and nickel by D zinc zinc Tilda copper dou C zinc by dou X minus D zinc nickel Tilda those helical by dou X the nickel flux will be D tilde so I get a different set of inter diffusion coefficients now these two sets are different like d nickel nickel with zinc as dependent is not the same as d nickel nickel sorry this should be copper so d nickel nickel with copper as dependent is not same as d nickel nickel with zinc as dependent ok but they are interconvertible that you can show then also have the equations with which we can convert from when dependent component to the set with another dependent component for any for NN component system that you do as exercise basically there are only four independent coefficients for ternary we don't say four plus four eight because this you once you have this set you can obtain the next set so you can use them depending upon which your component you want to use as dependent component okay now let's try to apply on suggest formalism of ficks law to binary system binary means two components right so we have only one independent division clerks one independent concentration gradient so if we treat two as dependent right jaywen tilde equal to minus D 1 1 tilde to dou C 1 by dou X and if we treat one as dependent you can write flux 2 is independent you can write it as J 2 tilde is equal to minus D 2 2 tilde 1 go see through by dou X in the original on Hajus formalism we will write DJ 1 tilde is equal to minus D 1 when u tilde dou C 1 by dou X minus D 1 2 tilde those C 2 by dou X J 2 tilde to minus and if we substitute for Lucy to by dou X here or for dou C 1 by dou X here we will get this to one of these two equations right so that's how we are reducing four coefficients to square for 2n minus 1 square 2 minus 1 is only 1 but again these two they seem to be different but they are not let's see how because we can write V 1 bar those even by the x + V 2 bar those c2 by dou X equal to 0 which means dou c2 by dou X is equal to minus V 1 bar over V 2 bar those even by dou X so if we substitute in the equation for J 2 tilde we should get negative negative positive D - 2 tilde 1 times V 1 bar over V 2 bar times dou C by dou X let's call this equation 1 but we know because we are using volume fixed frame even bar J 1 tilde plus V 2 bar J 2 tilde equal to 0 which tells me J 2 tilde equal to minus V 1 bar over V 2 bar J 1 tilde and if we substitute for J 1 tilde from here we get V 1 bar over V 2 bar times DS 1 1 tilde to go C 1 by dou X call this equation 2 so if you compare the right hand sides of 1 & 2 it tells me D 2 2 tilde 1 is should be equal to D 1 1 tilde - so both the equation coefficient should be same so in binary in effect we have only 1 coefficients we can express both the fluxes using 1 coefficients J 1 tilde we call it as D tilde minus G tilde those even by dou X and J 2 tilde minus La Russa - by dou X so the binary inter-diffusion simplifies to just knowing one enter diffusion coefficients here so wearing partial molar volumes is a complicated case okay so typically during the experimental measurements it is commonly assumed that the molar volumes remain constant if the molar volume is constant what does it imply it implies that all the partial molar volumes are also constant and the same okay you can show this so this implies that that means even bar equal to V 2 bar equal to V 3 bar in component system this makes the life simpler because it not only simplifies the constraint for example Sigma V I bar J I tilde should be 0 and since all VI bars are same this just implies to that Sigma J tilde equal to 0 it also means that UV velocity of volume fixed frame is zero because there is no change in volume there is no net flow of volume even in lab fixed frame and since because of that J tilde is equal to j 0 the volume fixed frame coincides with the last extreme je0 refers to the Fluxus determined in love fixed frame okay and that's why for experimental analysis most of the time molar volumes are of junk constant in fact we will show it later even when the partial molar volumes are constant but changing which means V 1 bar cannot equal to V 2 bar or equal to V 3 bar but VI bar themselves are constant with composition that even in that case UV equal to zero and the LA fixed frame coincides with volume fixed frame clear we will stop here for today [Music] [Music] [Music]