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MATGEN IV Many thanks for the invitation to lecture in such a beautiful setting

MATGEN IV Many thanks for the invitation to lecture in such a beautiful setting. Introduction to Interfaces and Diffusion 1. Forward 2. Interfacial Thermodynamics 2.1 Interfacial excess properties 2.2 Gibbs adsorption equation 3. Statistical Thermodynamic Models of Segregation

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MATGEN IV Many thanks for the invitation to lecture in such a beautiful setting

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  1. MATGEN IV Many thanks for the invitation to lecture in such a beautiful setting

  2. Introduction to Interfaces and Diffusion • 1. Forward • 2. Interfacial Thermodynamics • 2.1 Interfacial excess properties • 2.2 Gibbs adsorption equation • 3. Statistical Thermodynamic Models of Segregation • 3.1 Usefulness of statistical models • 3.2 The monolayer model • 3.3 Examples of trends • 3.4 Integration of the Gibbs adsorption isotherm • 4. Anisotropy of Interfacial Properties • 5. Grain Boundaries • 6. Interfacial Equilibrium • 6.1 Equilibrium of a GB with a surface • 6.2 Equilibrium of second phases at GBs and wetting (prewetting) • 7. Diffusion in Solids – An Overview • 7.1 Introduction • 7.2 Mathematical description of (Fickian) diffusion -- apologies • 7.3 Atomic mechanisms of diffusion in alloys

  3. Forward Interfaces are regions of a microstructure that separate phases which differ in structure and/or in composition. (Exception: GBs) liquid-vapor interface (= liquid surface), solid-liquid interface, solid-solid interface (= interphase boundary, interphase interface)

  4. Interfacial Thermodynamics Follow Gibbs. Interfacial energy is denoted by  (with typical units of [mJ/m2]). It is defined as the reversible work needed to create unit area of surface, at constant temperature, volume (or pressure), and chemical potentials. Use liquid-vapor surface as an example, i.e., consider a system composed of a liquid and a vapor phase at equilibrium, separated by an interface. dividing surface, hypothetical system

  5. Interfacial excess properties Examples: interfacial excess internal energy,Es: Es = E - E' – E'' and interfacial excess no. of moles of component i: nis = ni- ni' - ni" Exception: no interfacial excess volume, V = V' + V'' Internal energies of phases ' and '' are written (as usual): and interfacial excess internal energy (no volume) is written:

  6. All other thermodynamic properties can be written from the previous relation: e.g. Gibbs free energy, G = E + PV - TS Note! The surface excess quantities are arbitrary, since they depend on precisely where the dividing surface is located within the diffuse region associated with the interface. We shall see later how this issue may be addressed.

  7. Gibbs adsorption equation and isotherm The expression for dEs, is written only in terms of extensive independent variables(dSs , dA, dnis),while the intensive variables(T, , i)are constant in the equilibrated system. This makes integration of dEs straightforward: Re-differentiating yields: Comparing with the expression for dEs, we conclude: We now define the following specific interfacial excess quantities: and where i is referred to as the adsorption of component i

  8. and obtain: Gibbs adsorption equation At constant temperature, this simplifies to the Gibbs adsorption isotherm. For a two-component system, it can be written: This may be simplified further by the use of the Gibbs-Duhem equation: It is conventional to take components 1 and 2 to be the solvent and solute, respectively, and to eliminate1from the adsorption equation: The l.h.s. of the above equation is measurable, therefore the r.h.s. cannot be arbitrary.

  9. Approximate forms of the Gibbs adsorption isotherm The chemical potential may be expressed as: 2= 2° + kTln a2, where a, is the activity and ° is the standard state chemical potential. For ideal solutions, a2 = x2,where x2is the mole or atom fraction. For dilute solutions,a2 = kox2, where ko is Henry's Law constant. In both of these cases,d2 = RTd(ln x2). Thus, in both those cases the Gibbs isotherm may be written: In particular, in dilute solutions,n'2 << n'1, so that we can write: This is the most commonly used form of the Gibbs adsorption isotherm.

  10. 3. Statistical Thermodynamic Models of Segregation The Gibbsian approach is not always easy to use •Neither , nor its variation with composition, are easy to measure in solids. •2is difficult to measure directly, because interfacial composition profiles can extend some distance from the interface, and one must determine the composition profiles of both components in the most general case. •Gibbsian thermodynamics do not provide a relationship between2 and 2. Without such information, the Gibbs adsorption isotherm cannot be integrated. These factors have led to the development of various models which can overcome some of these problems. Here we shall briefly describe one of the simplest of these models.

  11. Monolayer (surface) segregation model (regular solution approximation) xsandxare the atom fractions of the solute in the surface monolayer and the bulk, respectively, and seg is the energy of segregation, i.e. the energy change resulting from exchanging a solute atom in the bulk with a solvent atom at the surface. AandBare the surface energies of the pure components (B=solute),is the area per mole in the monolayer,  = AB – (AA + BB )/2 is the regular solution constant, ij are bond energies between i-j pairs of atoms, is the in-plane coordination and is half of the out-of plane bonds made by an atom, and Eel is the elastic strain energy of a solute atom.

  12. Typical results from this type of model Also, the relationship between xs and  is: From this relation, and the one between xs andx, it is possible to integrate the Gibbs adsorption isotherm and obtain:

  13. where kis a surface enrichment factor given by Note! The surface energy is not proportional to the adsorption, rather it is the slope, dg/dm, that is proportional to adsorption.

  14. Anisotropy of Interfacial Properties Example: energies of (100) and (111) fcc surfaces calculated by nearest neighbor bond model Each atom in (100) surface has 4 broken bonds. But breaking 4 bonds creates 2 surfaces. Number of atoms per unit area of surface is 2/a2. If energy of broken bond is  (= - eAA) then: 2(100) = 4  (2/ a2), or (100) = 4 /a2

  15. For (111) surface, each atom has 3 broken bonds. But breaking 3 bonds creates 2 surfaces. Number of atoms per unit area of surface is 4/(3 a2). Then: (111) = (3/2)  [4/(3 a2)] = 23  /a2 ~ 3.46 /a2 This approach can be used to compute the relative energies of surfaces of all possible (hkl) orientations in fcc metals (at 0 K)

  16. Data on anisotropy of surface energy can be used to determine the equilibrium crystal shape (ECS) by means of the "Wulff construction" Conversely, studies of ECS can yield information on anisotropy of gs Examples: ECS of pure Cu and of Bi-saturated Cu at ~ 900°C (with monolayer of adsorbed Bi at the surface) illustrates effects of segregation on ECS Cu Bi-saturated Cu

  17. Example:scanning electron microscope image of a Bi-saturated Cu "negative" crystal

  18. Grain Boundaries (GBs) Special type of interface in single phase materials. Play important role in properties of poly-crystalline materials. Microstructure of tetragonal TiO2 displaying EBSD contrast

  19. Schematic of GB with solute segregation Orientation space of GBs is 5-d (compared with surfaces that have 2-d orientation space). 5-d space often described by 3 Euler angles + vector perpendicular to GB plane.

  20. Alternative description of 5-d space: interface plane scheme

  21. Energies of GBs for simplified orientation space (a: symmetric tilt GB, b: symmetric twist GB e.g. energy of symmetric tilt GB (Read and Shockley): GB = B[A – ln()]

  22. Variation of GB energy in a more general orientation space (fcc metals) Note! Anisotropy of GB energy is larger than that of surface energy

  23. Example: segregation at GBs of different orientations (multilayer model)

  24. Interfacial Equilibrium GB with surface GB = (hkl)1 cos() + (hkl)2 cos() or for isotropic surface: GB = 2s cos()

  25. Example: AFM image of GB grooves at pure Cu surface

  26. Second phases at GBs and wetting For a liquid phase at a GB:GB = SL1 cos() + SL2 cos() IfSLis isotropic:GB = 2SL cos() As  -> 0 the liquid phase will spread over the whole GB. This corresponds to "complete" or "perfect" wetting of the GB Condition for complete wetting: GB = 2SL

  27. Example of GB wetting Since GB is more anisotropic than SL, there can be conditions where some high energy GBs are completely wet while low energy GBs are still dry. Wet GBs will lead to "liquid metal embrittlement"

  28. Wetting and prewetting transitions The temperature above which a given GB is wet is known as its wetting temperature (TW, shown by dashed red line on the PD below) TW adsorption T a + liq a // X X0 X Consider what happens when two-phase coexistence is approached from the single -phase domain: At T>TW, there is a jump in adsorption at GB, known as a prewetting transition. It is a precursor of the liquid film that will form when the L-phase becomes stable (the location is indicated by dotted green line) At T<TW, no adsorption transition is present

  29. c'' t = 0 t = t1 t = t2 c t2 > t1 > 0 c' 0 x Diffusion in Solids -- mathematical formalism Diffusion represents flow of material, which occurs in order to eliminate chemical potential gradients. In general, uniformity of composition leads to elimination of chemical potential gradients The fluxJ [m/A.t]at any timetand positionxalong the concentration profile is given by Fick's first law: where D [ l2/s] is the diffusion coefficient

  30. Relationship between Fickian and chemical potential formalisms From the definition of chemical potential in the case of ideal or dilute solutions,it is easily shown that: So, for these simple solutions:

  31. Back to Fick. The evolution of the profile with time and position is given by Fick's second law: For semi-infinite solids, the solutions of this PDE take the form: where A and B are constants that may be determined from the applicable initial and/or boundary conditions. For example, with: IC: c(x,0) = c' for x < 0, and c(x,0) = c'' for x > 0 A = (c'' + c')/2; B = (c'' – c')/2

  32. Atomic mechanisms

  33. Fraction of vacant sites: Rate of atom-vacancy exchange:

  34. The "self" diffusion coefficient is proportional to the product of the vacancy fraction and the rate of atom vacancy exchange This provides a hint as to the origins of the Arrhenius relation which empirically describes diffusivity

  35. Some examples of the relative magnitudes of diffusivity

  36. Summary Interfacial thermodynamics, and Gibbs' treatment of adsorption (segregation). Complements to that treatment obtained from statistical thermo, monolayer (and multilayer) models Dependence of interfacial energy on adsorption Anisotropy of interfacial energy (surfaces -- ECS, and GBs) GB structure Vector interfacial equilibrium (GB-surface, GB-second phase) and wetting Mathematical and physical descriptions of diffusion

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