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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
Diffusion equation (Fourier):


 2
t
x
2
Fundamental, or...
only numerology?
Diffusion equations:
Heat:

T  Θ T  ...
2

t
x
2
Θ = α, m s
2 -1
Diffusion equations:
Diffusion of mass (1855):

 Θ ...
2

t
x
2
Θ = D, m s
2 -1
Diffusion equations:
Hydrodynamics (noncompressible fluid):

υ  Θ υ  ...
2

t

x
2
Θ = ν, m s
2 -1
Quantum mechanics:
φ ii 

i
2 i
t 22mm  x
2 2
2φ
i  1
2
Quantum mechanics:


i

2
t
2m  x
2
2
free particle…
Question:
Why
i

?
2m
Answer… P-K-C hypothesis
Economy…
… diffusion equation
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
But…. money is not
Economy & diffusion…
conserved!!!
Merton & Sholes Nobel
price „helped in”…
…grand failure in 2008!!!


 2
t
x
2
< 10-35 - ???
Planck scale- 10-35
nucleus
- 10-16
atoms
- 10-10
biology
- 10-4
mechanics - 1,
Earth - 107
cosmology – 1027
> 1030 ???


  2  ...
t
x
2
Fundamental !!!
Challenges
everywhere…
Mechano-chemistry:
Darken & stress,
Uniqueness…
Electro-chemistry:
Nernst-PlanckPoisson + drift
Applied:
Reactive interdiffusion…
Real geometry…
Nernst-Planck-Poisson Problem
ci
 div  ciid 
t
id  Bi  j Fj
ch
el
F


grad



 i i
j j
Walther Hermann NERNST
1864-1941
F
div E 
Siméon Denis POISSON
1781 -1840
Max PLANCK
1858-1947
zc


i
i i
Nernst-Planck-Poisson Problem
Walther Hermann NERNST
1864-1941
Max PLANCK
1858-1947
Siméon Denis POISSON
1781 -1840
i  1( Nernst
), 2( Planck
~ 1900),
Unsolved:
uniqueness,
quasi-stationary
Unsolved:
NPP ionic
+ drift…
Problems,
multi-component
systems…
W. Kucza (2009): converge…
3(~ 1960), 4?...
Nernst-Planck:
  Bi  j Fj
d
i
d
Ji
: 
d
ci i
Flux is not limited to diffusion…
Ji 
d
Ji
 ci
drift
Bi-velocity: Wagner (1933), Darken (1948),
Danielewski & Holly (Cracow >1994)...
c  i ci (t , x)  const.
Show…
c  i ci (t, x)  const.
??
R1….
No stress…
Ωi = Ω = const.
Material reference frame (Darken: 1948);
Lagrange, substantial, material etc…derivative
d
;
dt
D
;
Dt
D
; Dt
Dt 
Internal reference frame (Darken 1948):
Lagrange, substantial, material… derivative
D

    grad
Dt   t
 or
 m
material velocity = 
or
?

D

M
    grad
Dt  M t
local centre of composition:

M
ci
:  i i
c

D
m
    grad
Dt  m t
local centre of mass:
i
 : i i

m
D

    grad
Dt  t
None of
them!!!
local volume velocity:
i ci
 : 
i
i 1 c
r
If not:
c  i ci (t, x)  const.
Then?
J
i
d
i
(t , x )  0
Vegard
law
?
EOS
?
 J i (t , x )  0
i
 z FJ (t, x)  0
  J (t , x )  c    0
i i
i
i
d
i
drift
i
We need different approach…
Darken!!!
Bi-velocity…
Lattice sites not
conserved!
„Zig-zag Road”…
to the target
Euler ~250 ago: only...
m
19th century: Cauchy, Navier, Lamé…
Stephenson (1988): drift & m
Cracow (1994):
vd & drift
up to 2007: only m
Öttinger (2005): „something is missing”
Brenner (2006): Fluid Mechanics Revisited…
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
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)
150 years of diffusion:
Number of laws decreases…
Complexity increases…
Do we „stay with”: m,
ρυ, q, U only ?
o
Λ
Λ

A(x)
Dynamics &
diffusion?
B
100%
A
100%
λ 1(0)
xm
t = 0+
x
xm  f  1 ,   t , x  

Does
x
depend
on
time,
i.e.,
x
(t)
or
x
=
const?
m
m
m
I law: F = 0
x = const.
m
Λ
Dynamics &
diffusion?
Λ

A(x)
o
a)
B
100%
A
100%
λ1(0)
xm
Central problem!!!
t= 0
x
xm : const 
xm  f  1 ,   t , x  
f  1  t  ,   t , x  
E.g., diffusion and
t 
deformation, stress,
Yes
!
reactions...
x
x
λ (t)

A = const.
c)
1
λ (0)
m
+
… fundamentals
Hopeless? only!
Euler’s theorem:
f (x1, , xr;…) is called homogeneous of the m-th degree in the
variables x1,…, xr if:
f  k x1 ,..., k xr ;...  k f  x1 ,..., xr ;...
m
several identities follow, e.g.:
 f  k x1 ,..., k xr ;...
m 1
 mk f  x1 ,..., xr ;...
i xi
 k xi

  p, T , N1 ,..., N r    i
Ni
N i
  p, T , N1,..., N r   i i N i
Volume densities:
whole mixture:
  c  1
V
i-th component:   i ci
V
i
From Euler theorem:
1
d  t 
d
d
c

d
x

c

d
x


i
i


i
dt t 
dt t 
dt
The molar volume is the nonconserved property
But… is transported by components velocity field.
Fundamentals II
The Liouville transport theorem:
d
 fi

f
d
x


div
f

d
x


i
i
i




dt   t 
t

 t  
fi is a sufficiently smooth function (e.g., have first
derivative, C1) and υi is defined on fi
Liouville:
Conservation of component (fi = ci)
d
 ci

ci d x   
 div  cii   d x  0

d t  t 
t

 t  
ci
 div  cii   0
t
The Liouville theorem & the Volume Continuity,
fi(t,x) = „volume density” = ci(t,x) Ωi(t,x)
1
1
  i ci i
d
c  dx   
 div


i i i

d t  t 
t
 t  

const
d
 div
c


d
x
div
c



0


i
i
i
i
i
i
i
dt t  i



i ciii  d x


The volume density conservation law or…
equation of volume continuity at constant
volume:
div
 c     0
i
i
i i
Overall drift velocity:
      
drift
D
drift 
dT
tr
Finally due to Liouville
the bi-velocty method :
d
 div  i ci ii
dt
i
 div  ii   0 for i  1, 2,..., r
t
Di
ext


Div



g
r
ad
V
 i i Dt
i

Volume continuity:
Conservation of mass:
Momentum conservation:
Energy conservation:

i
D ui
i
Dt

i


i

i i
m
1
3

pI : grad  i  div J
Dsi
J q  p
Entropy production: i i
 div
 As
Dt
T
i
i  
drift

d
i
q
Dsi
J q  p
Entropy production: i i
 div
 As
Dt
T
i
m

p
1
1

As  i i grad i i  i ui iid grad  J q grad  i iid grad i  0
T
T
T
T
Brenner 2009, Danielewski & Wierzba 2010:
 L11
 q 
 M    L
J 
 21
L12   grad ln T 
L22   grad  
L-matrix is both symmetric and non - negative as required by LIT!
• Entropy production term is always positive
• Bi-velecity method is consistent with LIT
Conservation laws in material reference frame
Bi-velocity vs. LIT
Planck-Kleinert Crystal
Diffusion, stress, reactions
& more
Reactions & Interdiffusion: Multiples
Experiment:


Cu-Sn-Ag
Cu-Ag-Sn-Ni
Fabrication and vacuum
annealing:
0.20cm
3.00cm
35.0°
200oC
T= 150, 180,
t = 3h, 30h, 100h, 7days, 14days…
Cu
Ag
Ni
Sn
M. Pawełkiewicz, EMPA & AGH
Fabrication
 Sectioning
International PhD School Switzerland – Poland
Cu


Sn
after heat treatment: t=4h and T=180 C
International PhD School Switzerland – Poland
Interdiffusion & stress
mechano-chemistry
Interdiffusion
Diffusion couple
Co
Ni-Cu
52
wt% Ni
Fe
51Ni-Fe
wt% Ni
t=0
Model of Interdiffusion
Diffusion couple
c(t,x)
t>0
B. Wierzba 2008
D
e
p
ext

 Div       gradV
Dt 
Future...
   Bi       
d
i
ch
i
el
i
m
i

electro-mechano-chemistry
Bi-velocity method…
at the Planck scale
Oliver Heaviside (1850-1925)
• Impedance
• Complex numbers
• Heaviside function
• Maxwell reformulated
"Mathematics
is anmy
experimental
"I
do not refuse
dinner simply
science, and
do not come
because
I dodefinitions
not understand
the
first, but later on."
process
of digestion."
Planck-Kleinert Crystal
M. Danielewski, “The Planck-Kleinert Crystal”, Z. Naturforsch. 62a, 564-568 (2007).
Zeilinger: soccer balls diffract…
http://www.quantum.univie.ac.at/research/c60/
Soccer Balls Diffract
http://www.quantum.univie.ac.at/research/c60/
Professor Anton Zeilinger:
Experiment & theory for C60 and C70,
C60F48: world record (108 atoms) in matter interferometry.
J. Clerk Maxwell, Phil. Trans. R. Soc. Lond.,
155 (1865) 459-512
Already on…
„Planck-Kleinert Crystal”
„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.”
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
Vacuum… No! [5]→ “There is no information without
representation”
[1]
[2]
[3]
[4]
F. Wilczek, Phys. Today 52, 11 (1999).
J. L. Lebowitz, Rev. of Modern Phys. 71, S347-S357 (1999).
M. Bojowald, Nature 436, 920-921 (2005).
H. Kleinert, Ann. Phys. 44, 117 (1987).
[5] W. Żurek, Nature, 453, (2008), 23.
Volume continuity:
d
 div
div
c
c 0 


i i i i
dt
cic
Conservation of mass: div
 div
 0i 1,..., r
 ciic0 for
t t
DDm i
ext


Div



grad
V
Momentum conservation: 
i i
 Div   gradV
DtDt m 
i
D i
I law: i i
Dt
i
D 
1

i  i 
i :pGrad
I : grad
 i div
Jq J q
  div
Dt  3

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 !!!
P-KC: single crystal, „super ideal”, enormous
intrinsic energy, bounded (???), etc…
Volume
continuity
div  c   0

 div   
t

     
Mass
conservation:
m
Navier-Lame
+ diffusion:
D m

 Div 
Dt 
d
m
Div  2 graddiv uσ   divgrad uσ
Energy
conservation:
De

 Div   
Dt 
Included: the entropy production as a result
of defect formation and diffusion…
L. D. Landau, E. M. Lifshits, “Fluid Mechanics”,
2nd ed., (Butterworth-Heinemann, Oxford 1987).
Processes:
1. Transverse wave
2. Longitudinal wave
3. lattice deformation
(Kleinert 2003)
4. Pi diffusion (mass)
5. Heat transfer…
+ stationary, traveling
&
their combinations!
The physical constants (ideal regular fcc lattice)
Physical Quantity
Lattice parameter
Unit
Planck length
Symbol for unit
lP
Poisson ratio in ideal fcc crystal
Mass of particle
Planck mass
Frequency of the internal process
Inverse of the
Planck time
Lamé constant
Energy density
Number of particles in unit cell
Value in SI units
SI unit
Reference
1.61624(12)·10-35
m
NIST
0.25
-
Cauchy &
Poisson
mP
2.17645(16)·10-8
kg
NIST
fP = 1/tP
1.85486(98)·1043
s-1
NIST
kgm-1s-2
This work
1.853237194·10114
4
National Institute of Standards and Technology, Reference on Constants, Units
and Uncertainty, http://physics.nist.gov (2006).
This work
The energy of volume deformation field:
The energy of the torsion field:
    i
m
.
m
Re


m
Re
m
Im
m
Im
Gravity:
Mass
conservation:

 div   
t
Energy
conservation:

Volume continuity:
De
 Div   
Dt 
div     d   0
[defects] ≈ const at: T= const
in Planck-Kleinert crystal:
 div  P
m
d
P
  e
already Newton…
Siméon-Denis Poisson:
div grad
where:
m
Re
 4 G  M
G  l P c mP
2
G  6.674189 10
NIST data: G
= 6.6742(10)
11
-11
• 10
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
4. The “dark energy” → energy of the DIPP’s.
Remark:
1. Planck length = Schwarzschild radius
2. mP  mWIMP  mgraviton  m  Higg ' s boson 
Electromagnetism:
Zero diffusion .
So far !
Only transverse wave:
Navier-Lame
& no diffusion:

D 
Dt

  L
graddiv 
σ
mP f P

1)   0    
d
2) ρ = ρ0  const.
3) div  0

5/5/2017
L
mP f P

divgrad
σ
Transverse waves in P-KC:
2 x
t
2
3
= 8lP
  L
mP
3
divgrad x + 8lP
L
grad div x
mP
Equivalent form:
 x
2
t
2
=
c

x
L
2
2
+ c rot rot x
Equation of the transverse wave:
:
1  x
=
rotr
ot
x
2
2
c t
2
c  8l
.
3
P
    L  m  299 792 km s
1
P
299 792. 5 (NIST)
m


t
=     L  rotrot x
B   m 
  L
B

t
1
0

  L
B
0

t
2
fP
rot

2
f P rot x
E
f P2
rotE
2
f P rot x

Full set Maxwell eqs. in vacuum:
analogous simple transformations...
Quantum mechanics:
Mass
conservation:
Energy
Conservation:
and…

 div   
t
De

 Div   
Dt 
    i
m
m
Re
m
Im
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

m
Im
The energy flux:
J
d
M
   M BM grad 
m
Im
   eBM grad 
m
Im
  e
 div   e 

 t
 J d    eB grad  m
 M
M
Im
 e
m
 BM div   e grad Im 
t
  e
m

B
div

e
grad


M
Im 
 t

 0
m
m
e  2 exp 2  Re  i Im  0

 BM   mP M  BP


m
m
 Re
 i Im

exp  2
2
t
c


m
m


 mP BP
 Re

 i  Im
m
div  exp  2

 grad Im 
2
M
c





m
m
m
m



 Re
 mP BP
 Re

 i Im
 iIm
m
div  exp  2
 exp  2

 grad Im 
2
2
c
M
c






 t

m
m
2


exp


i

c


Re
Im



mP BP

2
m

div  grad Im 
t
M
2
mP BP c

i

div grad   E
t
2M
2

h
i

div grad   E
t
4 M
h  2 mP BP c
2
The physical constants at the Planck scale and...
four time scales!
Physical Quantity
Value in SI units
SI unit
Reference
Volume of PC cell
4.222 002 828·10-105
m3
This work &
NIST
Planck density
2.062 008 662·1097
kg m-3
This work &
NIST
Young modulus
4.633 092 986·10114
kg m-1s-2
This work
Planck mass mobility
5.391 213 982·10-44
s
This work
Defects self-diffusion coefficient
2.422 685 816·10-27
m2s-1
This work
Planck constant
6.626 069 311·10-34
6.626 069 3(11)·10-34
kg m2 s-1
This work
NIST
Gravitational constant
6.6742(10)·10-11
6.674176·10-11
m3 kg-1 s-2
NIST
This work
Speed of longitudinal wave
519 255 240
m s-1
This work
Speed of transverse wave
299 792 153
299 792 458
m s-1
This work
NIST
Physical reality at the Planck scale:
• Faster than light velocity of longitudinal wave…
Kleinert: "fine-tuning" to make all “sound speeds” equal or…
consider:
  3c
L
• 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.
• Transverse wave ≡ electromagnetic wave
• DIPP’s → Dark Matter → Dark Energy
• Waves involving temperature ≡ “the
second sound” described by Landau and
Lifschitz, etc…
Conclusions
Fluxes → Nernst-Planck formulae
Collective behavior → standing wave (”particle”)
Conclusions
Multi-phase and
multi-component
Today in R1.
Tomorrow…. R3
Future:
• new experimental methods
•new processes to predict
• all methods developed in math and physics
will be usefull...
Forthcoming:
   Bi       
d
i
ch
i
el
i
m
i

electro-mechano-chemistry
Remark:
Planck-Kleinert Crystal: straightforward!
vs.
Complexity of diffusion processes
in multicomponent… systems
END
Fundamentals I
Euler - the volume & molar volume:
  kn1 ,..., knr ; T , p   k   n1 ,..., nr ; T , p 
*
*
… homogeneous of the 1st degree in the variables n1,…, nr
The volume and molar volume:
  kn1 ,..., knr ; T , p   k   n1 ,..., nr ; T , p 
*
*
  N1 ,..., N r ; T , p   k   n1 ,..., nr ; T , p 
*
… homogeneous of the 1st degree in the variables N1,…, Nr
Ni = ci/c is molar ratio