Quantum Mechanicsin
Nanotechnology
Thomas PrevenslikQED Radiations
Discovery Bay, Hong Kong
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
1
Classical physics assumes the atom always has heat capacity, but QM requires the heat capacity to vanish at the nanoscale
QM = quantum mechanics
Unphysical results with Classical Physics
Nanofluids violate mixing rules
Thermal conductivity of thin films depends on thickness
Nanostructures do not charge
The Universe is expanding
Nanoparticles do not damage DNA
Molecular Dynamics is valid for nanostructures
And on and on
Background
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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QM Consequences
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Without heat capacity, the atom cannot conserve EM energy by the usual increase in temperature.
Conservation proceeds by the creation of QED induced non-thermal EM radiation that charges the nanostructure
or is lost to the surroundings
QED = quantum electrodynamicsEM = electromagnetic.
Fourier’s law that depends on temperature changes is not applicable at the nanoscale
3
Advantages of QM
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Unphysical interpretations of the nanoscale are avoided
Nanofluids obey mixing rules
Thermal conductivity of thin films remains at bulk
Nanostructures create charge or emit EM radiation
The Universe is not expanding
Nanoparticles damage DNA Molecular Dynamics is valid for nanostructures
Nanocomposites cross-link by EUV radiation
And on and on4
QM at the Macroscale
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Applying a nano coating on macrostructures avoids natural convection and conserves heat by emission of
QED radiation instead of temperature increases
Suggesting:
QED is the FOURTH mode of Heat Transfer?( 3 modes known: Conduction, Radiation, Convection)
Turbine blade coolingCooling of Conventional Electronics
Moore’s law and 13.5 nm LithographyDrinking water Purification
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4th Mode of Heat Transfer
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
QED radiation
NanoCoating avoids natural convection and conserves Joule heat by QED radiation instead of
temperature increase
Joule heat
ConventionalElectronics
Coating
Natural convection
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Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Theory
Heat Capacity of the Atom
TIR Confinement
QED Heat Transfer
QED Emission Spectrum
7
Heat Capacity of the Atom
1 10 100 10000.00001
0.0001
0.001
0.01
0.1
TIR Confinement Wavelength - l - microns
Pla
nck
Ene
rgy
- E
- e
V
1
kT
hcexp
hc
E
NEMS
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
In MEMS, atoms have heat capacity, but not in NEMS
MEMS kT 0.0258 eV
Classical Physics
QM
8
Since the RI of coating > electronics, the QED radiation is confined by TIR
Circuit elements ( films, wires, etc) have high surface to volume ratio, but why important?
The EM energy absorbed in the surface of circuit elements provides the TIR confinement of QED radiation.
QED radiation is spontaneously created from Joule heat dissipated in nanoelectronics.
f = (c/n) / and E = hf
TIR Confinement
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
For thin film of thickness d, = 2d
For NPs of diameter D, = D9
QED Heat Transfer
Excitons
Excitons = Hole and Electron Pairs → Photons
QED Excitons = EM radiation + Charge
Conservation by QED Excitons is very rapidQabs is conserved by photons before thermalization only after which phonons respond
No thermal conduction 0Fourier solutions are meaningless
Conductivity remains at bulk
Q|¿|¿
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Phonons
Qcond
Charge
QED Radiation
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QED Emission Spectrum
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
1 10 100 10000.001
0.01
0.1
1
10
Coating Thickness - d - nm
QE
D R
adia
tion
Wav
elen
gth
- -
mic
rons
Zinc Oxide
Silicon
IR
VIS
UV
EUV
QED radiation emission in VIS and UV radiation
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Applications
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Thin FilmsQED Heat Transfer
Electronics Circuit DesignNanocompositesEUV Lithography
Validity of Molecular DynamicsNanochannels
Expanding Universe
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Thin Films
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Thermal Conductivity
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
The reduced thermal conductivity of thin films has been known for over 50 years.
Today, the BTE derives the steady state thickness dependent conductivity of thin films.
BTE = Boltzmann transport equation.
But the BTE solutions show reduced conductivity only because QED radiation loss is not included in heat balance.
If the QED loss is included, no reduction in conductivity The conductivity remains at bulk.
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QED Heat Transfer
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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QED v. Natural Convection
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Classical convective heat transfer dissipates heat Q by,
H is the heat transfer coefficient, and A the surface area.
By QM , the temperatures of the coating and surroundings are the same, T = To
QED heat transfer is significant, hQED >> H
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Electronics Design
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Electronics Design
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
0.001 0.01 0.1 1 10 100 10000.0001
0.001
0.01
0.1
1
10
100
1000
Characteristic Size - d = / 2 - microns
TIR
Pla
nck
Ene
rgy
E =
hc
/ 2nd
- e
V
n = 3
n = 1.5Zinc Oxide
Optimum Design 0.05 < d < 20 microns
Fourier equation and BTE invalid Use QED heat transfer
Optimum
No 1/f NoiseNo Hot Spots
1/f Noise
No Hot Spots
NEMS Silicon
E > 3 eVCharged atoms
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Optimum NEMS/MEMS electronics circuit element occurs with 0.05 to 20 micron thick printed circuits.
• No hot spots or 1/f noise
• Design electronic circuits using QED
QED supersedes natural convection, but requires nanoscale coatings on heat transfer surfaces
Optimum
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Nanocomposites
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Mechanical Properties
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Nano Composites comprising NPs in a polymer are observed to display significantly enhanced mechanical properties.
The NPs are thought to enhance the polymer properties by forming an interphase adjacent the NP.
But the mechanism is not well understood. 21
Interphase Dilemma
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Rationally, the design of nanocomposites cannot proceed without knowing the interphase properties
Stress-strain curves are required, but tensile tests are not possible because the interphase is nanoscopic.
Currently, MD has been proposed to derive the properties of the interphase.
But MD simulations based on Lennard-Jones or even ab-initio potentials can never be shown to duplicate the stress-strain curve of the interphase, if unknown
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Design of Nano Technology?
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
The interphase dilemma is similar to the difficulty in the rational design throughout nanotechnology
Solution
Experimental characterization . (Build and test, forget computer simulations)
Hand wave classical physics to obtain unphysical explanations
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RVE Characterization?
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
In nanocomposite design, assume a stress-strain curve for the interphase and use the RVE procedure in 3D
FEA with ANSYS and COMSOL.
RVE stands for representative volume element.
The FEA should simulate the experimental test of the nano-composite design application.
Iterate on the assumed stress-strain curve until the true stress-strain curve is found upon convergence.
But the RVE approach is meaningless, as the experiment already verifies if the nanocomposite
design is acceptable.
Need experimental stress-strain curve24
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Radical polymerization may be dismissed as enhancements are observed without photo initiators.
UV induced cross-linking may be dismissed as nanocomposite properties are enhanced even if the polymer
is known not to exhibit UV cross-linking.
Only if EUV radiation is used do ALL polymers cross-link. EUV stands for extreme ultraviolet.
Enhanced properties of nanocomposites are therefore caused by the EUV cross-linking of the polymer.
What is the source of EUV?
QED Induced Radiation
Cross-linking Mechanism
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Characterization
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Prepare polymer tensile specimens, say < 1 mm diameter wires or 3 micron thick flat geometries from
the natural polymer.
Determine the wavelength of the EUV emission expected from the NPs based on their diameter and RI
. 1 10 100
1
10
100
1000
NP diameter - d - nm
QE
D W
ave
len
gth
-
- n
m
Silicon
Zinc Oxide
EUVUV + VIS
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EUV Source
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Table-top EUV sources have recently been developed similar to that used in EUV lithography.
But QED induced EUV provides a far simpler way of irradiating the tensile specimens
TensileSpecimen
EUV
Coating
Vacuum chamberTensile
specimen
Coating
EUV Source
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Electrical current is passed through the housing by applying voltage in short pulses. Joule heat is produced, but the
temperature in the coating does not increase because of QM.
Instead, QED creates EUV to irradiate the tensile specimen. The wavelength of the EUV is given by
= 2 nd.
For zinc oxide having n = 2 and taking d = 10 nm, QED creates 40 nm EUV.
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EUV Fluence from NPs
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
The NPs emit a EUV fluence F,
F = 1.5 NkT / A
where N is the number of atoms in the NP, d is the atom diameter; and A is the NP surface
N = (D/d)³ and A = D².
At 300 K, the carbon atom d = 0.134 nm gives the steady EUV fluence F = 0.82 mJ/cm². During thermal processing at
temperatures T ~ 500 K, F exceeds 2 mJ/cm².
EUV Lithography 1-10 mJ/cm².
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EUV Lithography
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Moore’s law
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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EUV lithography with light at 13.5 nm is planned in the next generation of computer chips.
However, difficulty in producing the EUV light source is questioning whether extending Moore's law � is possible
The difficulty in extending Moore’s law may be traced back to classical physics that requires EUV light to be created upon
the ionization of atoms in high temperature plasmas.
Nevertheless, LPP have evolved as the primary source of EUV light in 13.5 nm lithography.
LPP = laser produced plasmas
LPP Lithography
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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LPP systems for 13.5 nm computer chips are very expensive costing as much as USD 120 million.
The LPP plasma requires high energy 20 kW CO2 lasers to vaporize tin and lithium targets.
Collector mirrors require a multilayer coating to reflect the largest amount of 13.5 nm EUV light.
Periodic heating of mirrors at 400 C is required to evaporate tin and lithium debris in order to maintain the reflectivity and
enable long lifetimes.
LPP Light Sources
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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The LPP light sources use high power CO2 lasers to heat solid tin and gaseous helium targets, the plasmas of which
produce the EUV light by atomic emission.
EUV light is collected and focused by an elliptical mirror that delivers the focused EUV light to the silicon wafer
EUV by QED
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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A heater is provided on the back surface, the heat flowing through the lens thickness into the coating is converted by
QED into EUV light that is focused on the wafer.
For zinc oxide n ~ 2, and d < 5 nm, the EUV < 20 nm
BackSurface Heater
Nano Coating Focal
Point
Spherical Lens
EUV
The EUV by QED comprises a glass lens provided on the front surface with a nanoscale zinc oxide coating
QED Lithography
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Unlike the LPP requirement of high mirror reflectivity, QED lithography only requires a zinc oxide nanoscale coating.
Instead of high energy CO2 lasers, QED lithography is far more efficient as pulsed < 5 W power.
QED lithography avoids the need for debris control.
LPP requires large 320 mm diameter collector mirror. But QED lithography uses small < 100 mm spherical glass lenses.
Nano-structuring of materials using desktop LPP lithography may be performed with a hand-held EUV Source.
Validity of Molecular Dynamics
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Molecular Dynamics MD is commonly used to simulate heat transfer at the nanoscale in the belief:
Atomistic response using L-J potentials (ab initio) is more accurate than macroscopic finite element FE programs, e.g.,
ANSYS, COMSOL, etc.
In the following, it is shown:
FE gives equivalent heat transfer to MD, but both are invalid at the nanoscale by QM
And present:
Invalid and valid MD solutions by QM
Valid and Invalid MD
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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MD and FE Restrictions
MD and FE are restricted by statistical mechanics SM to atoms having thermal heat capacity
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Validity
Historically, MD simulations of the bulk performed under PBC assume atoms have heat capacity
PBC = periodic boundary conditions
In the macroscopic bulk being simulated, all atoms do indeed have heat capacity
MD is therefore valid for bulk PBC simulations
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Today, MD is not used for bulk simulations, but rather for the atomistic response of discrete nanostructures
Problem is MD programs based on SM assume the atom has heat capacity that is the cause of the
unphysical results, e.g.,
Conductivity in Thin films depends on thickness
Nanofluids violate mixing rules, etc
Why is this so?
MD Problem
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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MD - Discrete and PBC
Akimov, et al. “Molecular Dynamics of Surface-Moving Thermally Driven Nanocars,”
J. Chem. Theory Comput. 4, 652 (2008). Sarkar et al., “Molecular dynamics simulation of effective thermal conductivity and study of enhance thermal transport in nanofluids,”
J. Appl. Phys, 102, 074302 (2007).
MD for Discrete kT = 0, But MD assumes kT > 0
Car distorts but does not move
Macroscopic analogy, FE = MD
Classical Physics does not work
QM differs No increase in car temperature
Charge is produced by excitons Cars move by electrostatic interaction
MD for kT > 0 is valid for PBC because atoms in macroscopic nanofluid
have kT > 0
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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MD - NW in Tensile Test
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
Lw
w
F F
T. Prevenslik, “Nanowire Stiffening by Quantum Mechanics , MNHTM2013-220025, Hong Kong, Dec. 11-14, 2013
Silver 38 nm NWs x 1,5 micron long were modeled in a smaller size comprising 550 atoms in the FCC configuration with at an atomic spacing of 4.09 Ȧ. The
NW sides w = 8.18 Ȧ and length L = 87.9 Ȧ.
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MD - NW in Tensile Test
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
To obtain valid MD solutions, the Coulomb force Fij between atoms is modified by the ratio of thermal energy UkT of the atom to the
electrostatic energy UES from the QED induced charge by the excitons.
𝑈𝑘𝑇=32𝑘𝑇 𝑔𝑟𝑖𝑝
𝑈 𝐸𝑆=3𝑒2
20 𝑜𝑅𝑎𝑡𝑜𝑚
= 0.0065
𝐹 𝑖𝑗=e2
4 𝑜𝑅𝑖𝑗2
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MD - NW in Uniaxial Tension
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100008.74E-09
8.76E-09
8.78E-09
8.80E-09
8.82E-09
8.84E-09
8.86E-09D
isp
lam
en
t L
oa
din
g -
- m
Solution Time Step
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
-2E+05
-1E+05
0E+00
1E+05
Str
ess
- x
, y
, z
-
psi x and y
z
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000E+00
1E+07
2E+07
3E+07
4E+07
Yo
un
g's
M
od
ulu
s -
Y -
p
si
Solution Time Step
= 0.5 Ȧ
= 0.15 Ȧ
= 0.25 Ȧ
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MD – NW in Triaxial Tension
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
-50000
0
50000
100000
150000
200000
250000
300000
Solution Time Step
Str
ess
- ps
i
x and y
z
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000E+00
1E+07
2E+07
3E+07
4E+07
5E+07
6E+07
Solution Time Step
You
ng's
Mod
ulus
- Y
- p
si
= 0.001 Solution 15% of kT
= 0.002
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Solution Time Step
Poi
sson
's R
atio
-
=0.001
= 0.002
IncompressibleLimit
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Nanochannels
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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High Fluid Flow
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Water flow through nanochannels is observed to be 2-5 orders of magnitude higher than predicted by the Hagen-
Poiseuille equation of continuum mechanics
Slip at the channel wall cannot explain the high flow because the calculated slip-lengths exceed the slip on non-wetting
surfaces by 2 to 3 orders of magnitude.
High flow is more likely caused by the size effect of QM that causes the viscosity of the fluid to vanish in nanochannels allowing the Hagen-Poiseuille equation to remain valid as
the Bernoulli equation for frictionless flow.
QM Restrictions and QED
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Vanishing viscosity is the consequence of QM denying the atom the heat capacity to conserve viscous heating by an
increase in temperature.
Instead, viscous heat is conserved by QED inducing atoms in fluid molecules to create EM radiation
The EM radiation ionizes the fluid molecules, the Coulomb repulsion of atoms avoiding atomic contact to reduce viscosity.
Charged Atom Flow
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Neuron Synapse
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Lennard-Jones Potential
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Radius - R
Ato
m a
nd C
harg
e P
oten
tial
s
Atom
Charge0
Atom + Charge
U=4 [(𝑟 )
12
−( 𝑟 )
6] - Repulsion - Attractive
Simulate vanishing viscosity by taking the attractive potential 0
Valid MD Simulations
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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MD valid by QM require the viscous heat is conserved by charging the atoms – not by an increase in temperature. MD
solutions are therefore made near absolute zero temperature,
Conserve viscous heat by creating charge repulsion between atoms usually conserved by temperature
Hence, a discrete 2D model comprising 100 atoms in a BCC configuration of liquid argon under a constant shear stress
was selected
2D Distorted MD Model
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Bernoulli Equation
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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0 20000 40000 60000 80000 100000 120000 1400001E-08
1E-07
1E-06
1E-05
1E-04
1E-03
Iteration
Vis
cosi
ty -
-
Pa
- s
= 120 k
= 1.2 k
Viscosity ( reduced 144 X )
QED induced charged flow in nanochannels converges to frictionless flow given by the Bernoulli equation.
Expanding Universe
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Background
Prior to 1910, the Universe was thought static and infinite
In 1916, Einstein‘s theory of relativity required an expanding or contracting
Universe
In 1929, Hubble measured the redshift of galaxy light that by the Doppler Effect showed the Universe was expanding.
But you probably do not know
Cosmic dust of submicron NPs permeate space and redshift galaxy light without Universe expansion
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Dusty Galaxies
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
NGC 3314
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Redshift Z > 0 without Universe expansion
NP
Surface AbsorptionQED under TIR
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
QED Redshift
Vc=
(Z+1 )2 −1
( Z+1 )2+1 0.966 !!!
NP Velocity
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Redshift Photon
lo = (1+Z)
Z=𝑜−
In ISM, D < 500 nm.Take D = 300 nm, n = 1.5 o = 900 nm
Z = 6.4
o
Single galaxy photonLyman Alpha
= 121.6 nm
QED Redshift
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
0 0.05 0.1 0.15 0.2 0.250
2
4
6
8
10
12
0
0.2
0.4
0.6
0.8
1
1.2
Cosmic Dust NP radius - D/2 - microns
QE
D R
edsh
ift -
Z
Gal
axy
velo
city
rat
io -
V/c
V/c
Z
Z
Ly- = 0.1217 micron
Amorphous Silicate: n = 1.5
H- = 0.656 micron
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Redshift v. Wavelength?
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Hubble’s redshift by the Doppler effect requires the same Z for ALL wavelengths
QED induced Z is not the same for ALL wavelengths
Available data supports Doppler shift at low Z < .05(Astrophys J 123, 373-6, 1956)
To obtain Hubble Z, redshift measurements Zmeas are corrected with measured Z for Ly- and H- lines,
Z = Zmeas – ( ZLy- - ZH-)
Water Purification
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QED Induced UV
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Theory
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Disinfection occurs as the body heat from the hands of the person holding the drinking bowl is transferred to the coating.
Because of QM, the body heat cannot increase the coating temperature as the heat capacity vanishes under TIR.
Instead, conservation proceeds by QED inducing the heat to be converted to UV radiation. The TIR wavelength ,
= 2 n d
n and d are the refractive index and thickness of the coating.
Optimum UV wavelength to destroy bacteria is 250 - 270 nm
Zinc oxide coating having n = 2 requires d = 65 nm.
UV Intensity
Isfahan University of Technology - Quantum Mechanics in Nanotechnology - October 8-9, 2014
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Guidelines for the UV intensity suggest the minimum dose at all points in the water 16 to 38 mW / cm2. For a 20 cm
drinking bowl, the required heat is about 5 to 10 W.
The 5 to 10 W is consistent with the sudden application of body temperature TH = 37 C to the coating at TC = 20 C
where, is the density, C the heat capacity, and A the area of the coating. H is the heat transfer coefficient between hand and bowl. QM requires C to vanish instantaneous UV.
Questions & Papers
Email: [email protected]
http://www.nanoqed.org
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