A review of A new determination of molecular dimensions by Albert Einstein

Fang Xu June 11th 2007 Hatsopoulos Microfluids Laboratory Department of Mechanical Engineering Massachusetts Institute of Technology Einstein’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 Physik 19: 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", Annalen der 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 ). Albert Einstein at his • mass-energy equivalence: "Does the Inertia of a Body Depend Upon Its Energy Content?", Annalen desk in the Swiss Patent der Physik 18: 639–641, 1905 (received September office, Bern (1905) 27th ). References for viscosity of dilute suspensions

• Albert Einstein, A new determination of molecular dimensions. Annalen der Physik 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 outline

• In the presence of one particle 1)velocity profile 2)the rate of dissipation of mechanical energy • In the presence f multiple particles 1)velocity profile 2)the rate of dissipation of mechanical energy 3)viscosity • Highly concentrated suspensions: beyond Einstein’s formula • An application: Calculate Avagadro number N using viscosity and diffusion coefficients Motions of the liquid

Transport momentum Motions of the liquid (Stress)

Rigid body motions Non rigid body motion Principal dilatations Straining motion

Parallel ⎡A 00 ⎤ rotation T 1 uu )( =∇+∇ ⎢ B 00 ⎥ (No body torque) displacement 2 ⎢ ⎥ ⎣⎢ 00 C ⎦⎥ (x, y, z) = uu wv ooo ][

= Au ξ − xx o = ξ o

yy o =− η o = Bv η

zz o =− ζ o = Cw ζ (xo, yo, zo) ∇⋅u = A + B + C = 0 incompressibility Governing equations and boundary conditions for the flow past a sphere

Navier-Stokes equation: k: viscosity of (x, y, z) the fluid ∂u ρ[ ] ρ ∇+∇−=∇⋅+ 2 ukpguu ∂t Steady state No inertia Neglect gravity (xo, yo, zo) 2 ∇=∇ ukp (Stokes equations)

∇ ⋅u = 0 (Incompressibility)

∂p ∂p ∂p ∂ ∂ ∂wvu (4) Δ= uk Δ= vk Δ= wk ,,, + + = 0 ∂ ∂ ∂ ∂ ∂ ∂ζηξζηξ

B.C. = = wvu = 0 at surfacethe of spherethe To find solutions (1)

Velocity without disturbance due to the presence of the particle the particle

= ξ + uAu 1 − xx o = ξ η += vBv 1 yy o =− η ζ += wCw 1 zz o =− ζ (x, y, z)

u1 → 0 222 as v1 → 0 as ρ , ++=∞→ ζηξρ

w1 → 0 (xo, yo, zo) To find solutions (2) The structure of u: ∂V ∂V ∂V u = vu =+ ,',' wv =+ + w' with ∂ξ ∂η ∂ζ Harmonic functions ∂ ∂ ∂wvu 1''' 2 1 2 2 2wvu =∇=∇=∇ 0',0',0' + + −= p and V =∇ p ∂ ∂ ∂ζηξk k Which satisfies the governing equations ∂V ∂ ∂ 1 ∂p 2 kuk ∇=∇ 2 =+ ()'( 2 2 =∇+∇ kuVku ()' p 0) =+ ∂ξ ∂ξ ∂ξ k ∂ξ ∂V ∂ ∂ 1 ∂p 2 kvk ∇=∇ 2 =+ ()'( 2 2 =∇+∇ kvVkv ()' p 0) =+ 2 ∂η ∂η ∂η k ∂η ∇=∇ ukp ∂V ∂ ∂ 1 ∂p 2 kwk ∇=∇ 2 =+ ()'( 2 2 =∇+∇ kwVkw ()' p 0) =+ ∂ζ ∂ζ ∂ζ k ∂ζ ∂ ∂ ∂ ∂ ∂Vwvu ∂ ∂V ∂ ∂V + + = u ++ u ++ + u )'()'()'( ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ζζηηξξζηξ ∂ ∂ ∂wvu 11''' ∇ ⋅u = 0 2V +∇= + + p p =−= 0 ∂ ∂ ∂ζηξk k To find solutions (3)

Governing equations Harmonic function 2 2 2 ukukpp =⋅∇∇=∇⋅∇=∇⋅∇=∇ 0)()(

2 1 1 ∂ 2 2 ρ ξ 13 ∂ −= p ρ ∂ξ 2 ρρ35 = 2c 2 k ∂ξ 1 ∂2 ρ η 2 13 2 −= 35 2 1 ∂η ρρ 2 ∂ 22 ∂ ρ ρ a ζη 2 1 = + bcV (ξ 2 −−+ ) ∂ 2 2 ρ ζ 2 13 ∂ξ ∂ξ 2 22 −= ∂ζ 2 ρρ35 1 ∂ ρ −= 2' wvcu == 0',0', ∂ξ solutions for flow past a sphere 1 ∂2 ρ ξ 2 13 −= ∂ξ 2 ρρ35 1 ∂2 ρ η 2 13 (5) −= ∂η 2 ρρ35 1 += constpp .' ∂2 ρ ζ 2 13 −= ∂ζ 2 ρρ35

55PPP355 uA=−ξ ξξ (AB22 + η ζ ξξ + C) 2 η + ζ ξ (AB22 + + C) 2 − A 22ρρρ575 55PPP355 vB=−η ηξηζ (AB22 + + C 2 ) + ηξηζ (AB22 + + C 2 ) η− B (6) 22ρρρ575 55PPP355 wC=−ζ ζξηζ (A22 + B + C 2 ) + ζξηζ (A22 + B + C 2 ζ ) − C 22ρρρ575

u1 → 0

222 v1 →→→∞=++0 and p' 0, as ρ , ρ ξ η ζ

w1 → 0 The rate of dissipation of mechanical energy

n Force acting on the surface s s ∫∫)(])([ ⋅⋅=⋅= dsutndsuntW R +−= 2 EkIpt Extra P 1 ∇+∇≡ uuE T ])([ mechanical R>>P 2 work due to (P 46-48) the particle Neglect higher order terms

Pure fluid With the presence of multiple particles

(x,y,z)

ξυ = − xx υ (xν, yν, zν) ηυ −= yy υ

ζ υ −= zz υ

(0,0,0) d: Distance between two particles uo uν P: The radius of the particle d >> P No interaction between two particles uν:The disturbance due to one particle uo:the velocity of the pure fluid n: # of particles per volume fluid u(x,y,z) = uo + Σuν) Φ: volume of one particle the principal dilatations of the mixture

∂uu∂∂ u A* ==+=−() A (υυ ) A ( ) += uAxu ν xxx===000∑∑ ∑ ∂∂∂xxxυ ∂∂vv ∂ v += vByv * υυ ∑ ν B ==+=−()yyy===000 B∑∑ ( ) B ( ) ∂∂∂yyyυ wAzw ∂∂ww ∂ w += ∑ ν * υυ C(==+=− )zzz=000 C∑∑ ( )== C ( ) ∂∂∂zzzυ Divergence theorem Principal dilatations * ∂uυ υ xu v ∂uυ ∂uυ −= nAA −= nAdV ∫∫ ds ( = = )0 * ∂xυ rv ∂yυ ∂zυ AA −= φ)1( ∂v yv ∂v ∂v * −= nBB υ −= nBdV ∫∫ υ v ds ( υ = υ = )0 * BB −= φ)1( ∂yυ rv ∂xυ ∂zυ * ∂w zw ∂w ∂w CC −= φ)1( * −= nCC υ −= nCdV ∫∫ υ v ds ( υ = υ = )0 ∂xυ rv ∂xυ ∂yυ

δ CBA 22*2*2*2* −=++= φδ)21( Viscosity of the mixture

* : mixture

φ W2*2=+δ k(1 ) 2 Wk***=2δ 2 Volume fraction of particles φ 1+ φ 5 kk* =≈++≈++2 k(1 )(1 2φφφφ ) k (1 O (2 )) < 2 % 12− φ 2 2

Viscosity of the mixture Viscosity of the pure fluid The rheology of concentrated suspensions[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 coefficients to Avogadro number

ω: velocity; N: Avogadro number; D: diffusion coefficient; ρ: density m: molecular weight; K: drag force

Stokes’s law K RT 1 RT 1 ω = D ⋅= NP = 6πkP 6π NPk 6π Dk k * 5 5 4 1 φ 1+=+= πPn 3 k 2 2 3 3 km * NP3 = − )1( n ρ 10 ρπk = mN