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This research was supported by the Australian Research Council Irina V. Belova 1, Graeme E. Murch 1, Nagraj S. Kulkarni 2 and Yongho Sohn 3 1 The University of Newcastle, Callaghan, AUSTRALIA 2 Oak Ridge National Lab, USA 3 University of Central Florida, Fl, USA This paper is dedicated to the memory of the late John Manning Simultaneous Measurement of Tracer and Interdiffusion Coefficients in a Diffusion Couple

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Talk outline Talk outline: 1.Introduction. 2.Theory development for metallic/ionic systems and multicomponent alloys. 3.Model testing of the theory. 4.Application to interpreting data from Manning’s (1959) thin (3 nm) sandwich radioisotope interdiffusion experiment. 5. Application to addressing a thick (30-100 nm) sandwich stable isotope interdiffusion experiment (SIMS-based). 6. Conclusions.

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There are two basic ways of measuring diffusion coefficients from composition profiles: The tracer diffusion experiment. Used to obtain tracer diffusion coefficients: bulk and grain boundary, but only at a single alloy composition. Tracer diffusion coefficients are most appropriate as input to mobility databases e.g. as associated with the Materials Genome Initiative. The interdiffusion experiment. Used to obtain interdiffusion coefficients as a function of composition. Intrinsic diffusion coefficients can also be measured (at the composition of the inert marker plane). In a multi-marker interdiffusion experiment a set of intrinsic diffusion coefficients at different compositions are, in principle, obtainable. If the diffusion zone contains a dilute region then the impurity diffusion coefficient and the ‘enhancement factor’ (for interdiffusion!) are, in principle, obtainable. Introduction

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As we will see, the development of an isotopic diffusion flux formalism allows for the measurement of tracer diffusion coefficients (as functions of composition) in an interdiffusion experiment. This would make it possible to ‘fast-track’ the obtaining of the composition dependence of tracer diffusion coefficients. And we can now process Manning’s data that are over 50 years old ! Introduction cont. Can tracer diffusion and interdiffusion experiments be combined, perhaps for the purpose of obtaining the (very useful) composition dependence of the tracer diffusion coefficients? Yes. It’s been done twice to our knowledge: (Manning 1959 for Ag-Cd) and Meyer and Slifkin (1966 for Ag-Au) But there was no theoretical formalism available to process the data !!

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Introduction cont. Our Strategy: The Onsager flux equations of non-equilibrium thermodynamics are a very convenient starting point for this problem. When the Onsager flux equations for isotopes are combined with linear response theory the form of the flux equations simplifies dramatically. Linear response theory is not based on any specific kinetics or diffusion mechanism. Therefore the isotopic flux equations are applicable to diffusion in solids or liquids and to any diffusion mechanism. Linear response theory: A.R. Allnatt and A.B. Lidiard, Atomic Transport in Solids, Cambridge University Press, 1993, pp 227-233.

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Interdiffusion in metals/alloys: - Usually assumed to proceed via the bulk and with the vacancy mechanism and with vacancies at equilibrium (vacancy sources and sinks are sufficiently active). Then interdiffusion is controlled by the faster moving species, and the interdiffusion coefficient is given by the Darken-Manning relation. - If vacancies are not in equilibrium then the driving force on the vacancies (X V ) 0 and the diffusion equations are more complicated. - There is evidence that in some systems (like VSi) grain boundary diffusion plays a significant role in interdiffusion, therefore the interdiffusion process must then, in effect, be decoupled. Introduction cont.

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Interdiffusion in ionic compounds: - If atoms on (say) the anion sublattice i are much more mobile than the atoms on the cation sublattice then the description of interdiffusion of the cations is similar to interdiffusion in metals (i.e. the Darken-Manning Equation is applicable). - If the atoms on the anion sublattice are much less mobile than the atoms on the cation sublattice then the interdiffusion of the cation atoms is controlled by the slower moving atoms and the interdiffusion coefficient is given by the Nernst-Planck Equation. - In general, both forms, Darken-Manning and Nernst-Planck, may operate. Introduction cont.

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General Theory of Isotopic Fluxes Onsager flux expressions: (the vacancy diffusion mechanism is assumed for simplicity but any diffusion mechanism or combination is possible): Belova and Murch, Phil Mag Letters, 2014 (For convenience we will assume that the vacancies are at equilibrium X V =0 but we are not restricted to this.) Where i,j = A,B,C,… are the atomic components in the system.

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Assume now that there is an identifiable/measurable isotope A1 of the A atoms. The remaining A atoms will be called (collectively) isotopes A2. (This could be one or a combination of more than one different real isotopes of A.) Then we have for the forces X A1, X A2 and X A : General Theory of Isotopic Fluxes cont.

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For the applicability of the isotopic fluxes approach we need to have This conditions can be guaranteed only if: c A1 1 /c A2 1 c A1 2 /c A2 2 where c A1 1, c A1 2 and c A2 1, c A2 2 are the compositions for, say, the left (1) and right (2) hand ends of the diffusion couple respectively. General Theory of Isotopic Fluxes cont.

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where N is the number of lattice sites per unit volume. We have that: F A1A2 = F A1 = F A2 = F A, F A1B = F A2B = F AB, etc. (F i is a function describing correlation between movements of two different atoms of the i species and is the same for all isotopes of i, F ij, i ≠ j, is a function describing cross-correlation between movements of i and j species and, again, is the same for all i species isotopes.) Linear response theory relations: General Theory of Isotopic Fluxes cont.

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Then we have simply that: It is convenient now to focus only on the fluxes J A1 and J A. General Theory of Isotopic Fluxes cont. And, it is true that (Murch and Dyre 1990, Lidiard and Allnatt 1991), These are the general expressions for the isotopic fluxes relative to the lattice frame.

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Note: 1. There is no explicit thermodynamic factor here. 2. The only requirement for these equations to be useful is for the two parts 1, 2 of a diffusion couple to satisfy: After fairly standard manipulations, our final exact Diffusion Equations (Fick’s 2 nd Law) are (in the laboratory frame): or General Theory of Isotopic Fluxes cont. c A1 1 /c A2 1 c A1 2 /c A2 2

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Application to isotopic interdiffusion (targeting say A) in a multicomponent metallic system ABC….. in the usual couple geometry A xA B xB C xC … xA+xB+xC+…=1.0 A=A1+A2+… Standard interdiffusion experiment A yA B yB C yC … yA+yB+yC+…=1.0 A=A1+A2+… Assuming that c A1 1 /c A2 1 c A1 2 /c A2 2 Isotope A1 is a chosen isotope (the most convenient); Isotope A2 = all remaining isotopes collectively then the new interdiffusion analysis can be applied directly. (In practice, this geometry requires an expensive bulk isotope enhancement. Natural variations of isotopic ratios could be used in a few cases.) General Theory of Isotopic Fluxes Identifiable isotopes A1, B1, C1,… of all atomic components of the system with the remaining A, B, C, … atoms being called (collectively) isotopes A2, B2, C2,…. Further generalization of the formalism is possible

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We can now test our formalism by: 1.Setting up a model diffusion couple with some input tracer diffusion and interdiffusion coefficients. 2. Determine numerically the composition profiles. 3. Process the composition profiles as is done experimentally. 4. Compare the resulting diffusion coefficients with the input diffusion coefficients.

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Test Example. The diffusion couple is set up as: A1 0.6 A2 0.2 B 0.2 – A1 0.2 A2 0.3 B 0.5 With input D A * = (1.67 - 7.78 c B + 11.11c B 2 )D 0, where D 0 = 10 -15 m 2 /s and D B * = (3.22 - 15.56 c B + 22.22 c B 2 ) D 0,a diffusion time = 10 4 s and a thermodynamic factor = -3.44 + 31.11c B - 44.45c B 2. For 0.2 c B 0.5, 1 and has a maximum at c B = 0.35. We also provide an (input) interdiffusion coefficient as: For the above set of diffusion and composition parameters the resulting composition profiles are as follows: Numerical testing

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Numerically simulated composition profiles of the initial diffusion couple A1 0.6 A2 0.2 B 0.2 – A1 0.2 A2 0.3 B 0.5 with a parabolic dependence of D A * and D B * on composition after a diffusion anneal of 10 4 s. Numerical testing cont. A1 0.6 A2 0.2 B 0.2 – A1 0.2 A2 0.3 B 0.5

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To analyse these profiles we used our Diffusion Equations and the Sauer and Freise modification of the Boltzmann-Matano method. The following two equations will have to be applied to the B component and A1 isotope profiles: Numerical testing cont.

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Symbols – results of our analysis, solid lines – exact values for the ratios. Comparison of output and input D’s Numerical testing cont. The range of compositional access to the tracer diffusion coefficient is seen to be a bit narrower than that of the interdiffusion coefficient. Why ? At the ends of the diffusion zone, the c A1 and c A2 profiles have the same gradients and therefore cancel in the analysis (we used a relative tolerance of 10 -5 ).

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Ag AgCd (15.7-27.3 at%Cd) Tracer source Ag 110 or Cd 109 3 nm thick J.R. Manning, Phys. Rev. A 116 (1959) p. 69 and PhD thesis, 1958. Manning’s experimental tracer profile data has never been interpreted in terms of obtaining tracer diffusion coefficients. Manning’s 1959 radio-isotope sandwich layer interdiffusion experiment.

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J.R. Manning, Phys. Rev. A 116 (1959) p. 69 T=1053K T=900K Application of our analysis to analysing Manning’s ‘sandwich’ interdiffusion couple (cont.) Ag and Cd tracer concentration profiles (arb. units)

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J.R. Manning, PhD thesis (1958), Pp 52, 54, 55 T=1053K T=1000K Application of our analysis to Manning’s ‘sandwich’ interdiffusion couple (cont.) Ag and Cd tracer concentration profiles (arb. units)

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J.R. Manning, Phys. Rev. A 116 (1959) p. 69 T=1053K T=1000K Application of our analysis to Manning’s ‘sandwich’ interdiffusion couple (cont.)

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We have: binary alloy AB where the A atom population consists of two isotopes: A1 and A2 In the laboratory frame we have that: These are general expressions that are independent of kinetic theory. Note again that the A1 flux has the tracer diffusion coefficient D A* and the interdiffusion coefficient combined. Application of our analysis to Manning’s ‘sandwich’ interdiffusion couple (cont.)

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We have: 1. Interdiffusion profile (c A, c B ), invariant wrt Boltzmann transform x/t 1/2 i.e. its shape does not change with time; 2. Isotope profile c A1 (<< 1.0) that has a shape that changes with time, but according to the classical solution this change is simply proportional to 1/t 1/2 : We introduce the Boltzmann variable: Application of our analysis to Manning’s ‘sandwich’ interdiffusion couple (cont.)

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Time Simple Solution: Isotope profile in the absence of the concentration gradient Application of our analysis to Manning’s ‘sandwich’ interdiffusion couple (cont.)

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and: Then for the isotope profile we have then: I.V. Belova, N.S. Kulkarni, Y.H. Sohn & G.E. Murch, Phil Mag, 2014, http://dx.doi.org/10.1080/14786435.2014.965234 Application of our analysis to Manning’s ‘sandwich’ interdiffusion couple (cont.)

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Applying the usual integrating step and allowing for a constant shift a ′ in, solving for D A * : or This is equivalent to the Matano plane condition Application of our analysis to Manning’s ‘sandwich’ interdiffusion couple (cont.) I.V. Belova, N.S. Kulkarni, Y.H. Sohn & G.E. Murch, Phil Mag, 2014, http://dx.doi.org/10.1080/14786435.2014.965234

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Interdiffusion Coefficient Ag tracer diffusion coefficient J.R. Manning, Phys. Rev. A 116 (1959) p. 69 Results of application of our analysis of Manning’s ‘sandwich’ interdiffusion couple. Solid points: Independent tracer diffusion coefficient D Ag * (Schoen PhD thesis, University of Illinois 1958) Open symbols from our isotopic interdiffusion analysis T=1053K I.V. Belova, N.S. Kulkarni, Y.H. Sohn & G.E. Murch, Phil Mag, 2014, http://dx.doi.org/10.1080/14786435.2014.965234

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Interdiffusion Coefficient J.R. Manning, Phys. Rev. A 116 (1959) p. 69 T=1000K Solid points: Independent tracer diffusion coefficient D Cd *, D Ag * (Schoen PhD thesis, University of Illinois 1958) Cd tracer diffusion coefficient Ag tracer diffusion coefficient Open symbols from our isotopic interdiffusion analysis Results of application of our analysis of Manning’s ‘sandwich’ interdiffusion couple, cont. I.V. Belova, N.S. Kulkarni, Y.H. Sohn & G.E. Murch, Phil Mag, 2014, http://dx.doi.org/10.1080/14786435.2014.965234

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Interdiffusion Coefficient J.R. Manning, Phys. Rev. A 116 (1959) p. 69 T=900K Solid points: Independent tracer diffusion coefficient D Cd *, (Schoen PhD thesis U of Illinois 1958) Cd tracer diffusion coefficient Open symbols from our isotopic interdiffusion analysis Results of application of our analysis of Manning’s ‘sandwich’ interdiffusion couple, cont. I.V. Belova, N.S. Kulkarni, Y.H. Sohn & G.E. Murch, Phil Mag, 2014, http://dx.doi.org/10.1080/14786435.2014.965234

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A y B 1-y A x B 1-x Modification of Manning’s 1959 experiment for modern SIMS analysis using a thick layer of stable isotopes Application of our formalism to address a ‘thick sandwich’ interdiffusion couple, where the A1 layer is now of ‘finite’ thickness. Isotope A1 layer of about 30-100 nm - -10-30 times thicker than in Manning’s radiotracer experiments but typical of isotopically enriched foils in SIMS experiments.

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Initial profiles: c A1 c A = c A1 + c A2 A1 ‘sandwich’ layer of finite thickness Application to the ‘sandwich’ interdiffusion couple of A-B, where the A1 layer is of finite thickness, cont. xx c A2 A1A2

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Our basic Diffusion Equations remain the same: Now we have that and Application to the ‘sandwich’ interdiffusion couple A-B, where the A1 layer is of finite thickness, cont. Comes from the composition spike of A 1

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This can be done using the known ‘natural’ ratio of c A1 0 /c A2 = r : We need to have information on Application to the ‘sandwich’ interdiffusion couple A-B, where the A1 layer is of finite thickness, cont.

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Applying the Boltzmann transformwe have for A1 that and for A: Application to the ‘sandwich’ interdiffusion couple A-B, where the A1 layer is of finite thickness, cont.

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Application to the ‘sandwich’ interdiffusion couple A-B, where the A1 layer is of finite thickness, cont. This pair of nonlinear diffusion equations (they are more complicated in this case) can again be solved using standard numerical methods. A constant shift a ′ in can again be identified using the criteria as before:

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Final (after diffusion anneal) profile: c A1 c A2 c A = c A1 + c A2 A1 thick “sandwich” layer Application to the ‘sandwich’ interdiffusion couple A-B, where the A1 layer is of finite thickness, cont. xx A2A1

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Summary: ● The Onsager formalism was combined with exact relations from linear response theory to address isotopic fluxes in interdiffusion. ● The resulting exact expressions for the isotopic fluxes and the Diffusion Equations are surprisingly simple. ● When applied to the sandwich interdiffusion experiment, the new formalism makes it possible to obtain tracer diffusion coefficients (and their composition dependence) from a determination of the interdiffusion composition profile and the isotopic composition profiles. ● The formalism was applied to analyse, for the first time (!), Manning’s (1959) radio-isotope sandwich interdiffusion experiment (6 couples in total) to give the tracer diffusion coefficients and their composition dependences. There was excellent agreement with independent measurements. ● The formalism was further developed to address the thick sandwich interdiffusion experiment (for SIMS-based experiments).

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Thank you very much!

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