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Metody opisu dyfuzji wielu składników, unifikacja metody dyfuzji wzajemnej i termodynamiki procesów nieodwracalnych Marek Danielewski Interdisciplinary.

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Presentation on theme: "Metody opisu dyfuzji wielu składników, unifikacja metody dyfuzji wzajemnej i termodynamiki procesów nieodwracalnych Marek Danielewski Interdisciplinary."— Presentation transcript:

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

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6 Quantum mechanics: φ φ

7 free particle…

8 Question: Why Answer… P-K-C hypothesis

9 Economy… … diffusion equation

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

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

12 < ??? Planck scale nucleus atoms biology mechanics - 1, Earth cosmology – > ???

13 Fundamental !!!

14 Challenges everywhere… Mechano-chemistry: Darken & stress, Uniqueness… Electro-chemistry: Nernst-Planck- Poisson + drift Applied: Reactive inter- diffusion… Real geometry…

15 Nernst-Planck-Poisson Problem Siméon Denis POISSON Walther Hermann NERNST Max PLANCK

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

17 Nernst-Planck:

18 Flux is not limited to diffusion…

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

20 No stress… Ω i = Ω = const. R1…. ??

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

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

23 local centre of composition:

24 local centre of mass:

25 local volume velocity: None of them!!!

26 If not: Then?

27 EOS ? Vegard law ?

28 We need different approach… Darken!!!

29 Bi-velocity…

30 Lattice sites not conserved!

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34 „Zig-zag Road”… to the target

35 Öttinger (2005): „something is missing” 19 th century : Cauchy, Navier, Lam é … Stephenson (1988): drift &  m up to 2007: only  m Brenner (2006): Fluid Mechanics Revisited… Cracow (1994): v d & drift

36 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

37 150 years of diffusion equation: Defects „everywhere & always”… (1918 Frenkel) Nonstoichiometry is a rule… (1933 Schottky & Wagner) Lattice sites are not conserved (1948 Kirkendall & Darken) Diffusion velocity… (~1900 Nernst & Planck) Darken problem has a unique solution (2008 Holly, Danielewski & Krzyżański) Darken problem is self-consistent with LIT (ActaMat 2010, Danielewski & Wierzba)

38 150 years of diffusion: Number of laws decreases… Complexity increases… Do we „stay with”: m, ρυ, q, U only ?

39 Dynamics & diffusion? Does x m depend on time, i.e., x m (t) or x m = const?

40 Dynamics & diffusion?

41 Hopeless?… fundamentals only!

42 Euler’s theorem: f (x 1,, x r ;…) is called homogeneous of the m-th degree in the variables x 1,…, x r if: several identities follow, e.g.:

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44 Volume densities:

45 From Euler theorem:

46 The molar volume is the nonconserved property But… is transported by components velocity field.

47 Fundamentals II The Liouville transport theorem: f i is a sufficiently smooth function (e.g., have first derivative, C 1 ) and υ i is defined on f i

48 Liouville: Conservation of component (f i = c i )

49 The Liouville theorem & the Volume Continuity, f i (t,x) = „volume density” = c i (t,x) Ω i (t,x)

50 The volume density conservation law or… equation of volume continuity at constant volume:

51 Overall drift velocity:

52 Finally due to Liouville the bi-velocty method :

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55 Brenner 2009, Danielewski & Wierzba 2010: Entropy production term is always positive Bi-velecity method is consistent with LIT

56 . Fulfill the fundamental conditions: acceleration does not depend on internal energy, centre of mass is not affected by diffusion, drift velocity is the unique frame of reference diffusion fluxes generate Darken’s drift.

57 Diffusion fluxes, the Nernst-Planck formula:

58 Conservation laws in material reference frame Bi-velocity vs. LIT Diffusion, stress, reactions & more Planck-Kleinert Crystal

59 Reactions & Interdiffusion: Multiples T= 150, 180, 200 o C t = 3h, 30h, 100h, 7days, 14days… Experiment: Cu-Sn-Ag Cu-Ag-Sn-Ni Fabrication and vacuum annealing: Cu Ag Ni Sn M. Pawełkiewicz, EMPA & AGH

60 International PhD School Switzerland – Poland Fabrication Sectioning

61 International PhD School Switzerland – Poland after heat treatment: t=4h and T=180 C Sn Cu  

62 Interdiffusion & stress mechano-chemistry

63 Interdiffusion Ni-Cu Ni-Fe

64 Model of Interdiffusion

65 B. Wierzba 2008

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67 Future...

68 Bi-velocity method… at the Planck scale

69 Oliver Heaviside ( ) Impedance Complex numbers Heaviside function Maxwell reformulated "Mathematics is an experimental science, and definitions do not come first, but later on." "I do not refuse my dinner simply because I do not understand the process of digestion."

70 Planck-Kleinert Crystal M. Danielewski, “The Planck-Kleinert Crystal”, Z. Naturforsch. 62a, (2007).

71 Zeilinger: soccer balls diffract…

72 Soccer Balls Diffract

73 Professor Anton Zeilinger: Experiment & theory for C60 and C70, C 60 F 48 : world record (108 atoms) in matter interferometry.

74 J. Clerk Maxwell, Phil. Trans. R. Soc. Lond., 155 (1865) Already on… „Planck-Kleinert Crystal”

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76 „ The assumption, therefore, that gravitation arises from the action of the surrounding medium in the way pointed out, leads to the conclusion that every part of this medium possesses, when undisturbed, an enormous intrinsic energy, and that the presence of dense bodies influences the medium so as to diminish this energy wherever there is a resultant attraction. As I am unable to understand in what way a medium can possess such properties, I cannot go any further in this direction in searching for the cause of gravitation.”

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79 Physics Today [1] → “The persistence of ether” Statistical mechanics [2] → dimensions become large  quantum properties emerge. Quantum space [3] → analogous to crystal... Kleinert [4] → Einstein gravity from a defect model [1] F. Wilczek, Phys. Today 52, 11 (1999). [2] J. L. Lebowitz, Rev. of Modern Phys. 71, S347-S357 (1999). [3] M. Bojowald, Nature 436, (2005). [4] H. Kleinert, Ann. Phys. 44, 117 (1987). Vacuum… No! [5]→ “There is no information without representation” [5] W. Żurek, Nature, 453, (2008), 23.

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81 Planck crystal Planck foam Planck Scale Physics World Crystal World Condensate Scopus & Google: Planck-Kleinert Crystal

82 The Planck-Kleinert Crystal → World Crystal (three-dimensional quasi-continuum): - Frenkel disorder - defects form solid solution - defects diffuse - „classical” conservation laws - volume continuity & material reference frame! - double valued deformation field !!!

83 Volume continuity Mass conservation: Navier-Lame + diffusion: Energy conservation: P-KC: single crystal, „super ideal”, enormous intrinsic energy, bounded (???), etc…

84 Included: the entropy production as a result of defect formation and diffusion… L. D. Landau, E. M. Lifshits, “Fluid Mechanics”, 2 nd ed., (Butterworth-Heinemann, Oxford 1987).

85 4/29/2015

86 + stationary, traveling & their combinations! Processes: 1.Transverse wave 2.Longitudinal wave 3. lattice deformation (Kleinert 2003) 4. P i diffusion (mass) 5. Heat transfer…

87 Physical QuantityUnitSymbol for unitValue in SI unitsSI unitReference Lattice parameterPlanck lengthlPlP (12)· mNIST Poisson ratio in ideal fcc crystal0.25- Cauchy & Poisson Mass of particlePlanck massmPmP (16)·10 -8 kgNIST Frequency of the internal process Inverse of the Planck time f P = 1/t P (98)·10 43 s -1 NIST Lamé constantEnergy density · kgm -1 s -2 This work Number of particles in unit cell4This work National Institute of Standards and Technology, Reference on Constants, Units and Uncertainty, (2006).http://physics.nist.gov The physical constants (ideal regular fcc lattice)

88 The energy of volume deformation field: The energy of the torsion field:.

89 Mass conservation: Energy conservation: Volume continuity: Gravity:

90 , In P-KC: or in equivalent form:

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92 [defects] ≈ const at: T= const

93 in Planck-Kleinert crystal:

94 already Newton…

95 Siméon-Denis Poisson:

96 NIST data: G = (10)

97 4. The “dark energy” → energy of the DIPP’s. 1.The Diffusing Interstitial Planck Particles (DIPP’s) = WIMP’s 2. DIPP’s create the gravitational interaction between matter. 3. The “dark matter” → DIPP’s

98 Remark: 1. Planck length = Schwarzschild radius

99 Electromagnetism: Zero diffusion. So far !

100 4/29/2015 Navier-Lame & no diffusion: Only transverse wave: 2) ρ = ρ 0  const.

101 Transverse waves in P-KC: Equivalent form:

102 . Equation of the transverse wave: :

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104 Full set Maxwell eqs. in vacuum: analogous simple transformations...

105 Other Maxwell  “empty lattice” and low deformation. From noncompressible flow follows:. The charged particles are not considered  differentiating equation for E : The identity. The Maxwell system in vacuum.

106 Mass conservation: Energy Conservation: and… Quantum mechanics:

107 1.The deformation, its velocity and diffusion of defects are now assumed to be negligible

108 The energy flux: 1.The process that governs de Broglie waves is the fast internal process. 2.We analyze the case when the driving force of the transport (the collective Planck mass movement, i.e., the movement of a complex of particles showing an energy E and mass M = E c -2 ), is controlled by the imaginary part of mechanical potential

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113 The physical constants at the Planck scale and... four time scales! Physical QuantityValue in SI unitsSI unitReference Volume of PC cell · m3m3 This work & NIST Planck density ·10 97 kg m -3 This work & NIST Young modulus · kg m -1 s -2 This work Planck mass mobility · sThis work Defects self-diffusion coefficient · m 2 s -1 This work Planck constant · (11)· kg m 2 s -1 This work NIST Gravitational constant (10)· · m 3 kg -1 s -2 NIST This work Speed of longitudinal wave m s -1 This work Speed of transverse wave m s -1 This work NIST

114 Physical reality at the Planck scale: Faster than light velocity of longitudinal wave… Kleinert: "fine-tuning" to make all “sound speeds” equal or… consider: The different velocities are related to specific force field → a real quantities that mark different time-scales. The Diffusing Interstitial Planck Particles (DIPP’s) → gravity The collective behavior of the Planck particles → the particle: Schrödinger equation follows.

115 Transverse wave ≡ electromagnetic wave DIPP’s → Dark Matter → Dark Energy Waves involving temperature ≡ “the second sound” described by Landau and Lifschitz, etc…

116 Conclusions Fluxes → Nernst-Planck formulae Collective behavior → standing wave (”particle”)

117 Conclusions Multi-phase and multi-component Today in R1. Tomorrow…. R3

118 Future: new experimental methods new processes to predict all methods developed in math and physics will be usefull...

119 Forthcoming:

120 Planck-Kleinert Crystal: straightforward! vs. Remark: Complexity of diffusion processes in multicomponent… systems

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122 Physical QuantityUnitSymbol for unitValue in SI unitsSI unitReference Lattice parameterPlanck lengthlPlP (12)· mNIST Poisson ratio in PKC0.25-Poisson Mass of particlePlanck massmPmP (16)·10 -8 kgNIST Frequency of the internal process Inverse of the Planck time f P = 1/t P (98)·10 43 s -1 NIST Lamé constant Energy density · kgm -1 s -2 This work Number of particles in unit cell4This work National Institute of Standards and Technology, Reference on Constants, Units and Uncertainty, (2006).http://physics.nist.gov The physical constants in Planck-Kleinert Crystal

123 The collective movement (“the particle like” behavior): - in fluids (e.g., vortex, soliton, etc.), - in solids (e.g., complex defects and standing waves*. The particle  wave → E = M c 2 The particle is a complex → by analogy its mobility, B M : *J. Giannoulis and A. Mielke, Nonlinearity 17, (2004).

124 Fundamentals I Euler - the volume & molar volume: … homogeneous of the 1 st degree in the variables n 1,…, n r

125 The volume and molar volume: N i = c i /c is molar ratio … homogeneous of the 1 st degree in the variables N 1,…, N r

126 Maxwell equations in vacuum → assume: - the rotation of the deformation vector = -E - the mass flux due to deformation = B. Multiply both sides by:


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