A review of A new determination of molecular dimensions by … · 2016-01-28 · • G. K....
Transcript of A review of A new determination of molecular dimensions by … · 2016-01-28 · • G. K....
A review of A review of A new determination of molecular A new determination of molecular dimensions dimensions byby Albert EinsteinAlbert Einstein
Fang Xu
June 11th 2007Hatsopoulos Microfluids Laboratory
Department of Mechanical EngineeringMassachusetts Institute of Technology
Einstein’s Miraculous year: 1905Einstein’s Miraculous year: 1905
• the photoelectric effect: "On a Heuristic Viewpoint Concerning the Production and Transformation of Light", Annalen der Physik 17: 132–148,1905 (received on March 18th).
• A new determination of molecular dimensions. This PhD thesis was completed on April 30th and submitted on July 20th, 1905. (Annalen der Physik19: 289-306, 1906; corrections, 34: 591-592, 1911)
• Brownian motion: "On the Motion—Required by the Molecular Kinetic Theory of Heat—of Small Particles Suspended in a Stationary Liquid", Annalender Physik 17: 549–560,1905 (received on May 11th).
• special theory of relativity: "On the Electrodynamics of Moving Bodies", Annalen der Physik 17: 891–921, 1905 (received on June 30th ).
• mass-energy equivalence: "Does the Inertia of a Body Depend Upon Its Energy Content?", Annalender Physik 18: 639–641, 1905 (received September 27th ).
Albert Einstein at his desk in the Swiss Patent office, Bern (1905)
References for viscosity of dilute suspensionsReferences for viscosity of dilute suspensions
• Albert Einstein, A new determination of molecular dimensions. Annalen derPhysik 19: 289-306, 1906; corrections, 34: 591-592, 1911 (English translation: Albert Einstein, Investigations on the theory of the Brownian movement, Edited with notes by R. FÜrth, translated by A.D. Cowper, Dover, New York, 1956)
• G. K. Batchelor, An introduction to fluid dynamics, p246 Cambridge University press, Cambridge, 1993
• Gary Leal, Laminar Flow and Convective Transport Processes Scaling Principles and Asymptotic Analysis, p175, Butterworth-Heinemann, Newton, 1992
• William M. Deen, Analysis of transport phenomena, p313, Oxford university press, 1998
outlineoutline
• In the presence of one particle1)velocity profile2)the rate of dissipation of mechanical energy
• In the presence f multiple particles1)velocity profile2)the rate of dissipation of mechanical energy3)viscosity
• Highly concentrated suspensions: beyond Einstein’s formula
• An application: Calculate Avagadro number N using viscosity and diffusion coefficients
Motions of the liquidMotions of the liquid
Rigid body motions
Motions of the liquid
Non rigid body motionStraining motion
Parallel displacement
rotation⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=∇+∇
C B A
uu T
000000
21)(
][ wv uu ooo=
ζηξ
CwBvAu
o
o
o
===
ζηξ
=−=−=−
o
o
o
zzyyxx
(x, y, z)
(xo, yo, zo)
Transport momentum(Stress)
0=++=⋅∇ CBAu
(No body torque)
Principal dilatations
incompressibility
Governing equations and boundary conditions Governing equations and boundary conditions for the flow past a spherefor the flow past a sphere
(x, y, z)
(xo, yo, zo)
ukpguutu 2][ ∇+∇−=∇⋅+∂∂ ρρ
ukp 2∇=∇
0=⋅∇ u
0,,, =∂∂
+∂∂
+∂∂
Δ=∂∂
Δ=∂∂
Δ=∂∂
ζηξζηξwvu wkp vkp ukp
Navier-Stokes equation:
Steady state No inertia Neglect gravity
(4)
(Incompressibility)
B.C. spherethe of surfacethe at wvu 0===
k: viscosity of the fluid
(Stokes equations)
To find solutions (1)To find solutions (1)
1
1
1
wCwvBvuAu
+=+=+=
ζηξ
disturbance due to the particle
Velocity without the presence of the particle
(x, y, z)
(xo, yo, zo)
ζηξ
=−=−=−
o
o
o
zzyyxx
00
0
1
1
1
→
++=∞→→
→
wζηξρ ,ρ as v
u222
To find solutions (2)To find solutions (2)
The structure of u:
pk
V 12 =∇
',',' wVwvVvuVu +∂∂
=+∂∂
=+∂∂
=ζηξ
ζζζζ
ηηηη
ξξξξ
∂∂
=+∂∂
=∇+∇∂∂
=+∂∂
∇=∇
∂∂
=+∂∂
=∇+∇∂∂
=+∂∂
∇=∇
∂∂
=+∂∂
=∇+∇∂∂
=+∂∂
∇=∇
ppk
kwVkwVkwk
ppk
kvVkvVkvk
ppk
kuVkuVkuk
0)1()'()'(
0)1()'()'(
0)1()'()'(
2222
2222
2222
011'''
)'()'()'(
2 =−=∂∂
+∂∂
+∂∂
+∇=
+∂∂
∂∂
++∂∂
∂∂
++∂∂
∂∂
=∂∂
+∂∂
+∂∂
pk
pk
wvuV
uVuVuVwvu
ζηξ
ζζηηξξζηξ
with
and
ukp 2∇=∇
0=⋅∇ u
Which satisfies the governing equations
0',0',0' 222 =∇=∇=∇ wvu pk
wvu 1'''−=
∂∂
+∂∂
+∂∂
ζηξ
Harmonic functions
To find solutions (3)To find solutions (3)
0)()( 222 =⋅∇∇=∇⋅∇=∇⋅∇=∇ ukukpp
2
2 1
2ξρ
∂
∂= c
kp
Governing equations
35
2
2
2
35
2
2
2
35
2
2
2
131
131
131
ρρζ
ζρ
ρρη
ηρ
ρρξ
ξρ
−=∂
∂
−=∂
∂
−=∂
∂
)22
(2
122
22
2
2
2 ζηξξρ
ξρ
−−+∂
∂+
∂∂
=abcV
0',0',
1
2' ==∂
∂−= wvcu
ξρ
Harmonic function
solutions for flow past a spheresolutions for flow past a sphere
(5)
3 5 52 2 2 2 2 2
5 7 5
3 5 52 2 2 2 2 2
5 7 5
3 5 52 2 2 2 2 2
5 7 5
5 52 25 52 25 52 2
P P Pu A ( A B C ) ( A B C ) A
P P Pv B ( A B C ) ( A B C ) B
P P Pw C ( A B C ) ( A B C ) C
ξ ξ ξ η ζ ξ ξ η ζ ξρ ρ ρ
η η ξ η ζ η ξ η ζ ηρ ρ ρ
ζ ζ ξ η ζ ζ ξ η ζ ζρ ρ ρ
= − + + + + + −
= − + + + + + −
= − + + + + + −
(6)
35
2
2
2
35
2
2
2
35
2
2
2
131
131
131
ρρζ
ζρ
ρρη
ηρ
ρρξ
ξρ
−=∂
∂
−=∂
∂
−=∂
∂
.' constpp +=
1
2 2 21
1
0
0 and p' 0, as ρ , ρ ξ η ζ0
u
vw
→
→ → →∞ = + +
→
The rate of dissipation of mechanical energyThe rate of dissipation of mechanical energy
])([21
2
)(])([
TuuE
EkIpt
dsutndsuntW
∇+∇≡
+−=
⋅⋅=⋅= ∫ ∫
P
R
R>>P
s
n
(P 46-48)Neglect higher order terms
Force acting on the surface s
Extra mechanical work due to the particle
Pure fluid
With the presence of multiple particlesWith the presence of multiple particles
d: Distance between two particles P: The radius of the particle d >> PNo interaction between two particlesuν:The disturbance due to one particleuo:the velocity of the pure fluidu(x,y,z) = uo + Σuν)
(0,0,0)
(xν, yν, zν)
(x,y,z)
υυ
υυ
υυ
ζηξ
zzyyxx
−=−=−=
uo uν
n: # of particles per volume fluidΦ: volume of one particle
the principal dilatations of the mixturethe principal dilatations of the mixture
∑∑∑
+=
+=
+=
ν
ν
ν
wAzw
vByv
uAxu 0 0 0
0 0 0
0 0 0
*x x x
*y y y
*z z z
u u uA ( ) A ( ) A ( )x x xv v vB ( ) B ( ) B ( )y y yw w wC ( ) C ( ) C ( )z z z
υ υ
υ
υ υ
υ
υ υ
υ
= = =
= = =
= = =
∂ ∂ ∂= = + = −
∂ ∂ ∂∂ ∂ ∂
= = + = −∂ ∂ ∂∂ ∂ ∂
= = + = −∂ ∂ ∂
∑ ∑
∑ ∑
∑ ∑
)0(
)0(
)0(
*
*
*
=∂∂
=∂∂
−=∂∂
−=
=∂∂
=∂∂
−=∂∂
−=
=∂∂
=∂∂
−=∂∂
−=
∫∫
∫∫
∫∫
υ
υ
υ
υυ
υ
υ
υ
υ
υ
υυ
υ
υ
υ
υ
υ
υυ
υ
υ
yw
xw ds
rzwnCdV
xwnCC
zv
xv ds
ryvnBdV
yvnBB
zu
yu ds
rxunAdV
xunAA
v
v
v
v
v
v
)1()1()1(
*
*
*
φ
φ
φ
−=
−=
−=
CCBBAA
)21(22*2*2*2* φδδ −=++= CBA
Divergence theorem Principal dilatations
Viscosity of the mixtureViscosity of the mixture
Viscosity of the pure fluidViscosity of the mixture
* : mixture
Volume fraction of particles
* 2
2
2
W 2δ k(1 )2
2δ
1 52 (1 )(1 2 ) (1 ( )) 2 1 2 2 2
* * *
*
W k
k k k k O %
φ
φφ φ φ φ φ
φ
= +
=
+= ≈ + + ≈ + + <
−
The The rheologyrheology of concentrated suspensionsof concentrated suspensions[1][1]
• Viscosity does not obey Einstein’s formula• Normal stress• Particles migrate• Stress does not reach steady state instantaneously… …
[1] Andreas Acrivos, The rheology of concentrated suspensions: latest variations on a theme by Albert Einstein, Indo-US Joint Conference’04, Mumbai, Indian, 2004
An application: from viscosity and diffusion An application: from viscosity and diffusion coefficients to Avogadro numbercoefficients to Avogadro number
NPkRTD 16
⋅=πkP
Kπ
ω6
=
3*
34
251
251 Pn
kk πφ +=+=
mNn ρ=
)1(10
3 *3 −=
kkmNP
ρπ
DkRTNP 16π
=
ω: velocity; N: Avogadro number;D: diffusion coefficient; ρ: densitym: molecular weight; K: drag force
Stokes’s law