1 classicalscalingeffects 131ahrenkiel.sdsmt.edu/courses/fall2019/nano702/lectures/1...introduction...
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IntroductionMany of the forces familiar from the study of classical physics have varied scaling with size, and as a consequence, forces which dominate at macroscopic length scales may become relatively less (or more) important as the size of objects are scaled down. This notion, and the fundamental limits on miniaturization of man‐made machines, are the major themes explored in Feynman's lecture, “There's plenty of room at the bottom".
"You must not fool yourself – and you are theeasiest person to fool." Richard Feynman
pdf available at http://calteches.library.caltech.edu/1976/
Classical scaling examples
• Scaling of Mechanical SystemsScaling of a simple harmonic oscillatorScaling of the oscillations of a Beam
• Scaling of electrical circuitsScaling of an RLC CircuitNano‐rlectro‐mechanical machines (NEMS)
• Scaling of Thermal PropertiesHeating/cooling rate of a nano‐sized object
• Scaling of Fluid MechanicsScaling of a nano‐particle undergoing Brownian motionMotor proteins
• S.
w w w
h h
Scale all dimensions bya factor α (α<1)
h
w
Scaling of a simple, harmonic oscillatorConsider a simple harmonic oscillator, which could represent any object that exhibits a vibrational resonance:
Newton’s force law, no damping
F ma kx mx
Solution: 0sinx t A t
20 0sin 0m k A t
0mx kx
0 k m →
Natural frequency:
m V Mass:
F LYA L
Elasticity: YAF L kx
L
2A A k k
Natural frequency generally increases as size decreases
Young’s modulus
3m V m 3V V
L L YAF L k xL
Scale all dimensions by a factor α
00 k m
Scaled:
Scaled:
Example: Aluminum
80 1.6 10 rad sk m 0 1599 rad sk m
690 N mYAkL
21 mmA
1 cmL
31000 cmV
2.7 kgm V
32.7 g cm 69 GPaY
00 254 Hz
2f
510
0.1 mL
2100 nmA
31 mV
69 N mYAkL
2.7 fgm V
00 25.4 MHz
2f
L
A
V
Scaling of the oscillations of a beamSuppose a sound wave traveling at speed v is generated by the free oscillation of a beam:
The wavelength of any sustained resonance will be 2LN
where N is a positive integer.
The speed, wavelength, and frequency are related by v f
The frequency scales with the length of the beam as ff
The lowest (fundamental) frequency has N=1. The first harmonic has N=2, etc.
In other words, the pitch of the resonating beam varies inversely with length.
L
2NvfL
Scale only the beam length: L L
Scaling of an LC circuitNow let us examine an LC circuit. How does the resonance frequency of such a circuit scale?
Inductance:
The resonance frequency is given by
0I I C LThe current I satisfies the differential equation:
The solution is: 0sinI t A t
20 01 sin 0C A t L
0
1C
L
0 AC
t
L L
2A A t t
Scaled:
00
0 AC
t
C C Capacitance:
20
ANL
L
0
1C
L
Scaled: 20
ANL
L2A A
L L
Nano‐electro‐mechanical systems (NEMS)Nano‐scale circuits can be expected to operate at higher frequencies. Nano‐electro‐mechanical systems (NEMs) can have resonant frequencies in the ~100 MHz to the GHz range.
Note: The characteristic time for charging/dischargingthe capacitor in an RLC circuit is given by:
RC
2A A L L
Resistance scales as:LRA
Resistance: LRA
R R
R C Thus, charging times are invariant with length scale.
C C Capacitance scales as:
Heating and cooling of a nanoparticleConsider a spherical object of radius r and specific heat Cv in a thermal reservoir with temperature difference ΔT
Rate of heat transfer through nanoparticle surface: T
dQ k A Tdt
Total amount of heat exchanged to reach equilibrium: VQ C m T
V
dQ d TC mdt dt
1T
V
k Ad TR T Tdt C m
T
V
k AR T
C m
2A A 3m m
1R R
Cooling rate:
Scaled:
Faster cooling for smaller‐sized particles:
Terminal velocity in a fluid
Net gravitational force:
RgF
dragF
drag 6F Rv
Consider the terminal velocity of a sphere in a viscous fluid falling earth’s gravity.
fluidgF m m g m g
For large objects, the drag force is approximately dragF Av
3fluid
43
m V R
24A R
tAv m g
drag gF FNet force is zero at terminal velocity
3t
m g gv RA
t tv v Scaled: R R
For smaller objects in a fluid with viscosity η, the drag force with laminar flow is given by Stoke’s lawAt terminal velocity:
226 9t
m g gv RR
6 tRv m g
2t tv v Scaled: R R
Example: Au spheres in waterAu: 319.3 g cm 31.0 g cm
51.8 10 Pa s
1 Pa s 10 P (poise) Note:
water:
226 9t
m g gv RR
29.8 m sg
1 mmR
510
10 nmR
Thermal and diffusive motion2
B
1 12 2
m v k T1 22 B
th
k Tv v
m
m V Mass: 3m V m 3V V Scaled:
3 2th thv v Thermal velocity (speed) increases as particle size decreases.
Equipartition theorem (1‐D):
For a system of many particles, we need to consider diffusion by thermal motion.
C Jt x
CConcentration:
DDiffusion coefficient:
JFlux:Continuity equation:
CJ Dx
Fick’s Law:
2
2
C CDt x
Diffusion equation:Combine
Diffusion‐I
2, e ,ikx
xC x t dx C x t
Consider the concentration as the probability density for the position of one particle in a large ensemble:
,0C x x
,C x t
The particle starts from a known position: Take the Fourier transform:
2
22
, 4 ,C x t k C x tx
,0 1C x x
, ,C x t C x tt t
2
2
, ,C x t C x tDt x
2, 4 ,C x t k D C x tt
2, exp 4C x t A k Dt
2, exp 4C x t k Dt
21 2 2 4, exp 4 e eikx k Dt
kC x t k Dt dk
2 21, exp
42xC x t
DtDt
Inverse Fourier transform:
Fourier transform of 2nd derivative:We need to solve: Partial differentiation allows:
Combine: Integrate over time:
Initial condition:
Diffusion‐II
We need the diffusion coefficient to estimate this.
For a collection of particles in a viscous medium:
drift diffusionJ J J
drift tJ v C diffusion
CJ Dx
drag exttv
F F
At terminal velocity: exttv F
16 R
If drag 6 tF Rv Total flux:
mobility
Zero net force:
,C x t
x
0t
w tWidth of the probability distribution:
4w t Dt
Diffusion‐III
0 e E x kTC x C
BD k T
Assume a thermal distribution:
extE x F x With a uniform force applied (e.g., gravity):
drift ext ext 0 e E x kTJ F C x F C
extdiffusion 0 e E x kTD FC xJ D C
x kT
drift diffusion 0J J J
B24
3k Tt
w t DtR
Einstein relation
We can relate diffusion coefficient to mobility by assuming near‐equilibrium conditions.
B
6k T
DR
In this case:
higher energy
lower energy
gF
Diffusion‐IVCompare drift vs. diffusion:
B23k Tt
w tR
229t
gd t v t R t
When does distance drifted exceed distance diffused?:
d t w t
t
w ttv
2 2 5
272t
w kTv g R
5
Smaller particles remain suspended longer
//drift
//diffusion
B3
3k Tw
g R
...and diffuse farther
Example: Au spheres in water
Bth
k Tv
m1) Thermal velocity:
Au spheres: 319.3 g cm
1 mmR
0.0259 eVBk T Thermal energy (300 K):
7.2 nm sthv
510
10 nmR
0.23 m sthv
2 2 5
272
kTg R
2) Diffusion time:
2
3H O 1.0 g cm
211.0 10 s 41.0 10 s 2.7 h
B3
3k Tw
g R
3) Diffusion length: 0.02 fmw 2.2 mmw
Motor proteins
One such nano‐machine in living cells is a motor protein, driven by chemical energy and functioning in the aqueous environment of a cell. An example would be the motor protein kinesin, which can move a lipid raft many times its own mass along a microtubule, apparently oblivious to the forces of gravity.
kinesin
lipid raft
microtubule
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