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