Kinetic Theory for Rigid Dumbbells

Transcription

1 Kinetic Theory for Rigid Dumbbells Hector D. Ceniceros University of Santa Barbara Mathematics Department University of California, Santa Barbara Universidade de São Paulo, November H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

2 Outline 1 Discussion: Elastic Dumbbell 2 Rigid Dumbells 3 Concentrated Solutions H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

3 Dilute Solution of Elastic Dumbbells!" #" #"!" $!" $ #" $" ψ t ( = Q Q ψ ) H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

4 Dilute Solution of Elastic Dumbbells!" #" #"!" $!" $ #" ψ t $" ( = Q Q ψ ) Substituting Q we get the Smoluchowski or Fokker-Planck Equation ψ t = Q [{ v Q 2ζ Fc (Q) + 1ζ } ] (Fe2 Fe1 ) ψ + 2kT ζ 2 Q ψ H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

5 Dilute Solution of Elastic Dumbbells!" #" #"!" $!" $ #" ψ t $" ( = Q Q ψ ) Substituting Q we get the Smoluchowski or Fokker-Planck Equation ψ t = Q [{ v Q 2ζ Fc (Q) + 1ζ } ] (Fe2 Fe1 ) ψ + 2kT ζ 2 Q ψ When the flow is no longer uniform ψ t has to be substitute for Dψ Dt H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

6 Dilute Solution of Elastic Dumbbells!" #" #"!" $!" $ #" ψ t $" ( = Q Q ψ ) Substituting Q we get the Smoluchowski or Fokker-Planck Equation Dψ Dt = Q [{ v Q 2ζ Fc (Q) + 1ζ } ] (Fe2 Fe1 ) ψ + 2kT ζ 2 Q ψ When the flow is no longer uniform ψ t has to be substitute for Dψ Dt H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

7 Flow-Structure, Coupled System The Flow pi + µ 2 v + τ p = 0 v = 0 H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

8 Flow-Structure, Coupled System The Flow The microstructure Dψ Dt = Q pi + µ 2 v + τ p = 0 v = 0 [{ v Q 2ζ Fc (Q) + 1ζ (Fe2 Fe1 ) } ψ τ p = n QF c n Q(F e 1 Fe 2 ) nkt I ] + 2kT ζ 2 Q ψ H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

9 Flow-Structure, Coupled System The Flow The microstructure Dψ Dt = Q pi + µ 2 v + τ p = 0 v = 0 [{ v Q 2ζ Fc (Q) + 1ζ (Fe2 Fe1 ) } ψ τ p = n QF c n Q(F e 1 Fe 2 ) nkt I ] + 2kT ζ 2 Q ψ Recall QF c = QF c ψ(q, t)dq, etc H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

10 Oldroyd B: Hookean Springs Hookean springs, F c = HQ (and no external forces) τ p = nh A nkt I, A = QQ H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

11 Oldroyd B: Hookean Springs Hookean springs, F c = HQ (and no external forces) Note that because τ p = nh A nkt I, A = 4kT ζ I 4 ζ HA A = QQ we have a closed system and we don t need ψ to evaluate A and τ p. H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

12 Oldroyd B: Hookean Springs Hookean springs, F c = HQ (and no external forces) Note that because τ p = nh A nkt I, A = 4kT ζ I 4 ζ HA A = QQ we have a closed system and we don t need ψ to evaluate A and τ p. Oldroyd B Model pi + µ 2 v + τ p = 0 v = 0 τ p + λ H τ p = 2nkT λ H D H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

13 Oldroyd B: Hookean Springs Hookean springs, F c = HQ (and no external forces) Note that because τ p = nh A nkt I, A = 4kT ζ I 4 ζ HA A = QQ we have a closed system and we don t need ψ to evaluate A and τ p. Oldroyd B Model pi + µ 2 v + τ p = 0 v = 0 τ p + λ H τ p = 2nkT λ H D Note τ p = Dτ p Dt v τ p τ p v T H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

14 Non-Linear Dumbbells and a Closure Approximation The (Warner) FENE spring: F c = HQ 1 Q 2 /Q 2 0 Q Q 0 QF c f (A) So we cannot longer obtain a closed system. H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

15 Non-Linear Dumbbells and a Closure Approximation The (Warner) FENE spring: F c = HQ 1 Q 2 /Q 2 0 Q Q 0 QF c f (A) So we cannot longer obtain a closed system. Peterlin Approximation F c = HQ 1 Q 2 /Q 2 0 and QF c = H 1 tr(a)/q 2 0 A = f (tra)ha H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

16 Non-Linear Dumbbells and a Closure Approximation The (Warner) FENE spring: F c = HQ 1 Q 2 /Q 2 0 Q Q 0 QF c f (A) So we cannot longer obtain a closed system. Peterlin Approximation F c = HQ 1 Q 2 /Q 2 0 and QF c = H 1 tr(a)/q 2 0 A = f (tra)ha FENE-P Model pi + µ 2 v + τ p = 0; v = 0 τ p = nf (tra)ha nkt I A = 4kT ζ I 4 ζ f (tra)ha H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

17 Rigid Dumbbells %!"! #! "# $#!"# "# $!"# $ "# &# H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

18 Rigid Dumbbells %!"! #! "# $#!"# "# $!"# $ "# &# This is a system with a constraint. We don t know the force of the constraint acting on each bead. H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

19 Rigid Dumbbells %!"! #! "# $#!"# "# $!"# $ "# &# This is a system with a constraint. We don t know the force of the constraint acting on each bead. But at r c 1 ( ) F h Fh ( ) F b Fn ( F e F e ) 1 = 0 (1) H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

20 Equations of Motion Also the forces along the connector Lu = r 1 r 1 do not generate any motion perpendicular to it: [( ) ( ) (I uu) F h 1 Fh 1 + F b 1 Fb 1 + ( F e 1 1) ] Fe = 0 (2) H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

21 Equations of Motion Also the forces along the connector Lu = r 1 r 1 do not generate any motion perpendicular to it: [( ) ( ) (I uu) F h 1 Fh 1 + F b 1 Fb 1 + ( F e 1 1) ] Fe = 0 (2) We need expressions for the forces: F h j = ζ ( ṙ j v j ) = ζ ( ṙ c jl u v 0 v r c 1 2 jl v u ) F b j = j kt R ln ψ L where R = u u is the gradient on the surface of the sphere: R = e θ θ + e φ 1 sin θ φ H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

22 Equations of Motion II Substituting these forces in (1) and (2) we get: ṙ c = v 0 + v r c + 1 ( F e 2ζ 1 + F e 1) u = v u v : uuu 1 6λ R ln ψ + 1 [ ( (I uu) F e ζl 1 F e 1)] where λ = ζl 2 /12kT is a time constant. H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

23 Equations of Motion II Substituting these forces in (1) and (2) we get: ṙ c = v 0 + v r c + 1 ( F e 2ζ 1 + F e 1) u = v u v : uuu 1 6λ R ln ψ + 1 [ ( (I uu) F e ζl 1 F e 1)] where λ = ζl 2 /12kT is a time constant. We write the last term as (2/L)RΦ, Φ=potential independent of r c. Also observing that v u v : uuu = u v u and setting D r = 1/(6ζ) we get ( u = u v u D r R ln ψ + 1 ) kt RΦ H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

24 Equations of Motion II Substituting these forces in (1) and (2) we get: ṙ c = v 0 + v r c + 1 ( F e 2ζ 1 + F e 1) u = v u v : uuu 1 6λ R ln ψ + 1 [ ( (I uu) F e ζl 1 F e 1)] where λ = ζl 2 /12kT is a time constant. We write the last term as (2/L)RΦ, Φ=potential independent of r c. Also observing that v u v : uuu = u v u and setting D r = 1/(6ζ) we get ( u = u v u D r R ln ψ + 1 ) kt RΦ We can now substitute into the continuity equation ψ t = R ( u ψ) H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

25 The Smoluchowski Equation Smoluchowski Equation for Rigid-rod like Molecules Dψ Dt ( = D r R Rψ + ψ ) kt RΦ R (u v u ψ) (3) H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

26 The Smoluchowski Equation Smoluchowski Equation for Rigid-rod like Molecules Dψ Dt ( = D r R Rψ + ψ ) kt RΦ R (u v u ψ) (3) We can obtain an equation for the second moment A = uu from (3) : A = 2D r (I 3A) 2 v : Q D r kt where Q = uuuu. (RΦ)u + u(rφ) H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

27 The Smoluchowski Equation Smoluchowski Equation for Rigid-rod like Molecules Dψ Dt ( = D r R Rψ + ψ ) kt RΦ R (u v u ψ) (3) We can obtain an equation for the second moment A = uu from (3) : A = 2D r (I 3A) 2 v : Q D r kt (RΦ)u + u(rφ) where Q = uuuu. Clearly, we don t have a close system; an equation for Q would involve the sixth moment and so on... H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

28 The Smoluchowski Equation Smoluchowski Equation for Rigid-rod like Molecules Dψ Dt ( = D r R Rψ + ψ ) kt RΦ R (u v u ψ) (3) We can obtain an equation for the second moment A = uu from (3) : A = 2D r (I 3A) 2 v : Q D r kt (RΦ)u + u(rφ) where Q = uuuu. Clearly, we don t have a close system; an equation for Q would involve the sixth moment and so on... Mathematical Temptation: Closure Approximation, Q f (A). H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

29 The Polymeric Stress Due to the constraint, we can t apply the procedure we used for an elastic dumbbell. But it turns out that the Kramers-Kirkwood formula is still valid: τ p = n 1 j= 1 R j F h j H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

30 The Polymeric Stress Due to the constraint, we can t apply the procedure we used for an elastic dumbbell. But it turns out that the Kramers-Kirkwood formula is still valid: τ p = n 1 j= 1 R j F h j Substituting R j = 1 2 jlu and Fh j = ζ( ṙ j v 0 v r j ) we get τ p = 1 2 nl2 ζ u( u v u) H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

31 The Polymeric Stress Due to the constraint, we can t apply the procedure we used for an elastic dumbbell. But it turns out that the Kramers-Kirkwood formula is still valid: τ p = n 1 j= 1 R j F h j Substituting R j = 1 2 jlu and Fh j = ζ( ṙ j v 0 v r j ) we get τ p = 1 2 nl2 ζ u( u v u) and substituting the expression for u we get (Kramer s) Stress τ p = n kt D r v : Q + 3nkT A + n urφ nkt I H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

32 Doi s Theory for a Concentrated Solution Φ is replaced by a mean field potential describing intermolecular interactions Maier-Saupe Φ = 3 ktua : uu 2 Models excluded volume effects: finite volume of macromolecules. H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

33 Doi s Theory for a Concentrated Solution Φ is replaced by a mean field potential describing intermolecular interactions Maier-Saupe Φ = 3 ktua : uu 2 Models excluded volume effects: finite volume of macromolecules. Marrucci-Greco (adds spatial elasticity) Φ = 3 ( ) 2 ktu A + l A : uu H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12

34 Doi s Theory for a Concentrated Solution Φ is replaced by a mean field potential describing intermolecular interactions Maier-Saupe Φ = 3 ktua : uu 2 Models excluded volume effects: finite volume of macromolecules. Marrucci-Greco (adds spatial elasticity) Φ = 3 ( ) 2 ktu A + l A : uu Doi uses a pre-averaged rotational diffusivity D r = D r (1 S 2 ) 2 S 2 = (S : S)/2 and S = A 1 3 I, D r constant. Translation diffusivity neglected. H. D. Ceniceros (UCSB) Complex Fluids 10/ / 12