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⎡ 0 0⎤ rotation T 1 ∇u +() ∇ u 0 ⎢ = B 0⎥ (No body torque) displacement 2 ⎢ ⎥ 0 ⎣⎢ 0 ⎦⎥C (x, y, z) u u=[]o v o ow u= Aξ x− xo = ξ o y− yo =η vo = Bη z− zo =ζ wo = Cζ (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 ⋅[u ∇ u =] ρ g − p ∇2 k + u ∇ ∂t Steady state No inertia Neglect gravity (xo, yo, zo) 2 ∇p = k ∇ u (Stokes equations) ∇ ⋅u = 0 (Incompressibility) ∂p ∂p ∂p u∂ ∂ v∂ w (4) =k Δ,,, u =k Δ v =k Δ w + + = 0 ξ∂ η∂ ζ∂ ∂ ξ∂ η∂ ζ u v w B.C.at the= = = 0 surface of the sphere To find solutions (1) Velocity without disturbance due to the presence of the particle the particle u= Aξ + 1 u x− xo = ξ v= Bη +1 v y− yo =η w= Cζ + 1 w z− zo =ζ (x, y, z) u1 → 0 2 2 2 v1 → 0 →ρas ∞ρ, = ξ η + ζ + w1 → 0 (xo, yo, zo) To find solutions (2) The structure of u: ∂V ∂V ∂V u = +u',', v = +v w = + w' with ∂ξ ∂η ∂ζ Harmonic functions u∂'∂ v '∂ ' w 1 2 1 ∇'u =2 0 , ∇2 v ' = 0 ,2 ∇w ' 0 = + + = − p and ∇V = p ξ∂ ∂ η∂ ζ k k Which satisfies the governing equations ∂V ∂ ∂ 1 ∂p k∇2 u =(')( k2 ∇u+ k = ∇2 V + ∇2')( u = k p)+ 0 = ∂ξ ∂ξ ∂ξ k ∂ξ ∂V ∂ ∂ 1 ∂p k∇2 v =(')( k2 ∇v+ k = ∇2 V + ∇2')( v = k p)+ 0 = 2 ∂η ∂η ∂η k ∂η ∇p = k ∇ u ∂V ∂ ∂ 1 ∂p k∇2 w =(')( k2 ∇w+ k = ∇2 V + ∇2')( w = k p)+ 0 = ∂ζ ∂ζ ∂ζ k ∂ζ u∂ v∂ ∂ w∂ ∂ V ∂ ∂V ∂ ∂V + + (')(')(')= +u + +u + + u ξ η∂ ∂ ζ∂ ξ∂ ∂ ξ∂ η∂ η∂ ∂ ζ ζ u'∂ '∂ v∂ ' w 1 1 ∇ ⋅u = 0 = ∇ 2 + +V + =p −p 0 = ξ∂ ∂ η∂ ζ k k To find solutions (3) Governing equations Harmonic function ∇ =∇⋅∇=∇⋅∇p2 p( k2 ) u2 k ( =∇ ) u 0 ∇⋅= 2 1 1 ∂ 2 2 ρ 3ξ 1 ∂ = − p ρ ∂ξ 2 ρ5 ρ 3 = 2c 2 k ∂ξ 1 ∂2 ρ 3η 2 1 2=5 − 3 2 1 ∂η ρ ρ 2 ∂ 2 2 ∂ ρ ρ a η ζ 2 1 V= c+ b +(ξ 2 − − ) ∂ 2 2 ρ 3ζ 2 1 ∂ξ ∂ξ 2 2 2 = − ∂ζ 2 ρ5 ρ 3 1 ∂ ρ u'= c − 2 , '= v 0 , ' w = 0 ∂ξ solutions for flow past a sphere 1 ∂2 ρ 3ξ 2 1 = − ∂ξ 2 ρ5 ρ 3 1 ∂2 ρ 3η 2 1 (5) = − ∂η 2 ρ5 ρ 3 1 p p='. + const ∂2 ρ 3ζ 2 1 = − ∂ζ 2 ρ5 ρ 3 55PPP355 uA=−ξ ξξ (AB22 + η + C) ζ 2 + ξξ (AB22 + η + C) ζ 2 − A ξ 22ρρρ575 55PPP355 vB=−η ηξηζ (AB22 + + C 2 ) + ηξηζ (AB22 + + C 2 ) − B η (6) 22ρρρ575 55PPP355 wC=−ζ ζξηζ (A22 + B + C 2 ) + ζξηζ (A22 + B + C 2 ) − C ζ 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 W t=[()]() n∫∫ u ⋅ ds = n ⋅ t ⋅ u ds R t= p − I2 + k E Extra P 1 ≡E[()] ∇ u +T ∇u 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 =x− υ x (xν, yν, zν) ηυ=y − υ y ζ υ=z − υ z (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== ( ) u= Ax + ν u xxx===000∑∑ ∑ ∂∂∂xxxυ ∂∂vv ∂ v v= By + v * υυ ∑ ν −B () +yyy===000 B∑∑ = ( ) B == ( ) ∂∂∂yyyυ w Az w ∂∂ww ∂ w = +∑ ν * υυ −C( + )zzz=000 C∑∑ = ( )== C == ( ) ∂∂∂zzzυ Divergence theorem Principal dilatations * ∂uυ uυ v x ∂uυ ∂uυ A= A −∫ ndV= A −ds ∫ n ( = =0 ) * ∂xυ rv ∂yυ ∂zυ AA=( 1 −φ ) ∂v v y ∂v ∂v B*= B −∫ ndVυ = B −ds ∫ υ n v ( υ = υ =0 ) BB*=( 1 −φ ) ∂yυ rv ∂xυ ∂zυ * ∂w w z ∂w ∂w CC=( 1 −φ ) C*= C −∫ ndVυ = C − ∫ds nυ v ( υ = υ =0 ) ∂xυ rv ∂xυ ∂yυ * 2=δ ABC * 2 + * 2+δ *( =2 1 2 2 φ − ) 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 k6π NP 6kπ D k * 5 5 4 =1 +φ =1 n + π P3 k 2 2 3 3 m* k NP3 = (− 1 ) n ρ 10π ρ k = N m.