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Metody opisu dyfuzji wielu składników,

Metody opisu dyfuzji wielu składników, unifikacja metody dyfuzji wzajemnej i termodynamiki procesów nieodwracalnych Marek Danielewski. Interdisciplinary Centre for Materials Modeling AGH Univ. o f Sci. & Technolog y , Cracow, Poland Będlewo, Czerwiec 2013. φ. φ. Quantum mechanics:.

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Metody opisu dyfuzji wielu składników,

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  1. Metody opisu dyfuzji wielu składników, unifikacja metody dyfuzji wzajemnej i termodynamiki procesów nieodwracalnych Marek Danielewski Interdisciplinary Centre for Materials Modeling AGH Univ. of Sci. & Technology, Cracow, Poland Będlewo, Czerwiec 2013

  2. φ φ Quantum mechanics:

  3. Quantum mechanics: free particle…

  4. Question: Why Answer… P-K-C hypothesis

  5. Economy… … diffusion equation

  6. The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1997: "for a new method to determine… the value of derivatives" Myron S. Scholes Long Term Capital Management Greenwich, CT, USA Robert C. Merton Harvard University

  7. But…. money is not conserved!!! Merton & Sholes Nobel price „helped in”… …grand failure in 2008!!! Economy & diffusion…

  8. < 10-35 - ??? Planck scale- 10-35 nucleus - 10-16 atoms - 10-10 biology - 10-4 mechanics - 1, Earth - 107 cosmology – 1027 > 1030 ???

  9. Fundamental !!!

  10. Walther Hermann NERNST 1864-1941 Max PLANCK 1858-1947 Siméon Denis POISSON 1781 -1840 Nernst-Planck-Poisson Problem

  11. Siméon Denis POISSON 1781 -1840 Walther Hermann NERNST 1864-1941 Max PLANCK 1858-1947 Nernst-Planck-Poisson Problem Unsolved: uniqueness, quasi-stationary Problems, multi-component ionic systems… Unsolved: NPP + drift… W. Kucza (2009): converge…

  12. Nernst-Planck:

  13. Fluxis not limited to diffusion…

  14. Bi-velocity: Wagner (1933),Darken (1948), Danielewski & Holly (Cracow >1994)... Show…

  15. ?? R1…. No stress… Ωi =Ω = const.

  16. Material reference frame (Darken: 1948); Lagrange, substantial, material etc…derivative

  17. Internal reference frame (Darken 1948): Lagrange, substantial, material… derivative

  18. local centre of composition:

  19. local centre ofmass:

  20. local volume velocity: None of them!!!

  21. If not: Then?

  22. Vegard law ? EOS ?

  23. We need different approach… Darken!!!

  24. Bi-velocity…

  25. Lattice sites not conserved!

  26. „Zig-zag Road”… to the target

  27. 19th century: Cauchy, Navier, Lamé… Cracow (1994): vd & drift Stephenson (1988):drift &m up to 2007: only m Öttinger (2005): „something is missing” Brenner (2006): Fluid Mechanics Revisited…

  28. Brenner in „Fluid Mechanics Revisited” (Physica A, 2006) 1. Complemented: volume fixed RF 2. Was polite to not notice: conflict between RF’s … in our papers

  29. 150 years of diffusion equation: Diffusion velocity… (~1900 Nernst & Planck) Defects „everywhere & always”… (1918 Frenkel) Nonstoichiometry is a rule… (1933 Schottky & Wagner) Lattice sites are not conserved (1948 Kirkendall & Darken) Darken problem has a unique solution (2008 Holly, Danielewski & Krzyżański) Darken problem is self-consistent with LIT (ActaMat 2010, Danielewski & Wierzba)

  30. 150 years of diffusion: Number of laws decreases… Complexity increases… Do we „stay with”: m, ρυ, q, U only ?

  31. Dynamics & diffusion? Does xmdepend on time, i.e.,xm(t) or xm= const?

  32. Dynamics & diffusion?

  33. … fundamentals only! Hopeless?

  34. Euler’s theorem: f(x1, , xr;…) is called homogeneous of the m-th degree in the variables x1,…, xr if: several identities follow, e.g.:

  35. Volume densities:

  36. From Euler theorem:

  37. The molar volume is the nonconserved property But… is transported bycomponents velocity field.

  38. Fundamentals II The Liouville transport theorem: fi is a sufficiently smooth function(e.g., have first derivative, C1) andυi is defined on fi

  39. Liouville: Conservation of component (fi = ci)

  40. The Liouville theorem & the Volume Continuity, fi(t,x) =„volume density” = ci(t,x) Ωi(t,x)

  41. The volume density conservation law or… equation ofvolume continuity at constant volume:

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