On the hydrodynamic diffusion of rigid particles

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Gerard Golden
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1 On the hydrodynamic diffusion of rigid particles O. Gonzalez

2 Introduction Basic problem. Characterize how the diffusion and sedimentation properties of particles depend on their shape. Diffusion: Sedimentation: Applications. Molecular separation techniques, structure determination; particle transport, mixing in microfluidic devices.

3 Outline Introduction Spherical bodies Arbitrary bodies Asymptotic analysis Application to DNA (Numerical method, if time permits)

4 Spherical bodies

5 Classic model for spherical bodies Setup. Consider a dilute solution of identical spheres in a fluid subject to external loads. Physical domain Configuration space ext f x E R 3 E R 3 ρ(x, t) # spheres per unit volume of E. f ext (x, t) external body force. µ, T fluid viscosity, temperature.

6 Modeling assumptions Consider locally time-averaged forces and motion for each particle and assume: 1. Net force balance. f f hydro ext f ext + f hydro + f osmotic = 0. f osmotic 2. Hydrodynamic force model. f hydro v f hydro = 6πγµv, γ radius. γ

7 3. Osmotic force model. Modeling assumptions f f osmotic = ψ, ψ = kt ln ρ. 4. Conservation of mass. B ρ dv + ρv n da = 0, B E. t B B

8 Resulting model on E Equations. Combining 1-3 and localizing 4 we get f ext 6πγµv kt ρ Eliminating v gives ρ = 0, ρ t + [ρv] = 0. ρ t = [D ρ Cρf ext ] D = kt 6πµγ, C = 1 6πµγ. Remark. Various experiments can measure D or C and hence γ.

9 Example: centrifuge experiment rotor air solute + solvent spin ω cell air solvent solute r a r b elapsed time ρ t = 0 ρ t > 0 ρ t >> 0 r a r b r a r f (t) r b r a r b D and/or C can be determined from speed of moving front r f (t).

10 Arbitrary bodies

11 Model for arbitrary bodies Setup. Consider a dilute solution of identical bodies in a fluid subject to external loads. Physical domain Configuration space Local coord space ext f τ ext (x,r) (q,η) E R 3 E x SO 3 R 12 E xa R 6 ρ(q, η, t) # bodies per unit volume of E SO 3. (f ext, τ ext )(q, η, t), external body force, torque. µ, T fluid viscosity, temperature.

12 Modeling assumptions Consider locally time-averaged loads and motion for each particle and assume: 1. Net force and torque balance. [ ] ext f + τ ext f τ ext hydro f c τ hydro f osmotic osmotic τ or [ ] hydro [ ] osmotic f f + = τ τ F ext + F hydro + F osmotic = 0 R 6 [ ] 0 0 where [ ] f F = Λ T local basis components. τ

13 Modeling assumptions 2. Hydrodynamic force model. [ ] hydro [ ] [ ] f L1 L = 3 v τ L 2 L 4 ω f hydro hydro τ c Γ v ω or [ ] v ω or [ ] [ M1 M = 3 f M 2 M 4 τ V = MF hydro R 6 where ] hydro L = L(Γ, c), M = M(Γ, c) [ R ] 6 6 v M = Λ 1 MΛ T, V = Λ 1. ω

14 3. Osmotic force model. Modeling assumptions F F osmotic = ψ ψ = kt ln ρ, = ( q, η ). 4. Conservation of mass. B t B ρg dv + B ρgv N da = 0 B E A.

15 Resulting model on E SO 3 Equations. Combining 1-3 and localizing 4 we get F ext M 1 V kt ρ Eliminating V gives ρ = 0, (ρg) t + [ρgv] = 0. ρ t = g 1 [gd ρ gρcf ext ] D = kt M(Γ, c), C = M(Γ, c). Remark. Model is fully coupled b/w translations and rotations.

16 Detail on hydrodynamic model f hydro hydro τ v ω [ f τ ] hydro [ ] [ ] L1 L = 3 v L 2 L 4 ω

17 Detail on hydrodynamic model f hydro hydro τ v ω u(x), p(x) c x Ω Γ = Ω µ u = p in R 3 \Ω u = 0 in R 3 \Ω u = v + ω (x c) on Γ u, p 0 as x [ f τ ] hydro [ ] [ ] L1 L = 3 v L 2 L 4 ω M(Γ, c) = L(Γ, c) 1 R 6 6, where L(Γ, c) is a Dirichlet-to-Neumann map.

18 Asymptotic analysis

19 Basic question Question. What does the coupled model imply about various observable densities of interest? ρ c t =? where ρ c is # of ref points c per unit volume of E. c E R 3 E R 3

20 Basic question Question. What does the coupled model imply about various observable densities of interest? ρ n t =? where ρ n is # of ref points n per unit area of S 2. n E R 3 S R 3 2

21 Scale separation Result. For particles of arbitrary shape, there is a natural scale separation for dynamics on E and SO 3. E particle E x SO3 L 1 Translations: t E = time to diffuse across E Rotations: t S = time to diffuse across SO 3 L t S t E ( ) l 2 L The two-scale structure is ideal setting for asymptotics; the small param is ε = l/l << 1.

22 Limiting model on E Result. For particles of arbitrary shape Γ and mobility tensor M(Γ, c), the leading-order equation on E on the scale t E is ρ c t D c = kt 3 tr[m 1(Γ, c)], = [D c ρ c ρ c h ext ] h ext = avg ext load ρ c = # of ref points c per unit volume of E. c E R 3

23 Property of model on E Result. The diffusivity D c depends on body shape Γ and ref point c. For each Γ, there is a unique c R 3 such that D c = min c R 3 D c. c c * Γ

24 Limiting model on S 2 Result. For particles whose shape Γ and mobility tensor M(Γ, c) satisfy an elongation condition wrt the body axis n, the leading-order equation on S 2 on the scale t S is ρ n t D n = kt 2 tr[p nm 4 (Γ, c)p n ], = D n ρ n P n = proj orthog to n ρ n = # of ref points n per unit area of S 2. n S 2 R 3

25 Property of model on S 2 Result. The diffusivity D n depends on body shape Γ and ref vector n, but not ref point c. For each Γ, there is at least one n S 2 such that D n = min n S 2 D n. n * c n Γ

26 Application

27 Estimation of hydrated radius Problem. Given experimental measurements of D c and D n for various sequences, we seek to fit the radius parameter r in a geometric model. Γ(S, r), S = DNA sequence. r =? D c = kt 3 tr[m 1(Γ(S, r), c)], D n = kt 2 tr[p nm 4 (Γ(S, r), c)p n ].

28 Results for straight model: D c 5 vs sequence length 4.5 dimensionless Dt x n basepairs n Curves: numerics w/r = 10, 11,..., 15Å (top to bottom). Symbols: experiments (ultracentrifuge, light scattering, electrophoresis). Estimated radius: r = 10 15Å.

29 Results for curved model: D c 5 vs sequence length 4.5 dimensionless Dt x n basepairs n Curves: numerics on straight model (same as before). Open symbols: experimental data (same as before). Crosses, pluses: numerics on curved model w/r = 10, r = 15Å. Estimated radius: r = 12 17Å.

Results for straight model: D n 3 vs sequence length 2.5 dimensionless Dr x n^3 2 1.5 1 0.

30 Results for straight model: D n 3 vs sequence length 2.5 dimensionless Dr x n^ basepairs n Curves: numerics w/r = 12, 11,..., 18Å (top to bottom). Symbols: experiments (birefringence, light scattering). Estimated radius: r = 13 17Å.

31 Results for curved model: D n 4 vs sequence length 3.5 dimensionless Dr x n^ basepairs n Curves: numerics on straight model (same as before). Open symbols: experimental data (same as before). Crosses, pluses: numerics on curved model w/r = 12, r = 18Å. Estimated radius: r = 10 12Å.

32 Numerical method

33 Numerical method for M(Γ, c) f hydro hydro τ v ω [ f τ ] hydro [ ] [ ] L1 L = 3 v L 2 L 4 ω

34 Numerical method for M(Γ, c) f hydro hydro τ v ω u(x), p(x) c x Ω Γ = Ω µ u = p in R 3 \Ω u = 0 in R 3 \Ω u = U[v, ω] on Γ u, p 0 as x [ f τ ] hydro [ ] [ ] L1 L = 3 v L 2 L 4 ω The computation of M(Γ, c) = L 1 (Γ, c) R 6 6 requires six solutions of the exterior Stokes equations with data U[v, ω](x) = v + ω (x c).

35 Boundary integral formulation Stokes kernels (singular solns): G(x, y) single-layer, H(x, y) double-layer.

36 Boundary integral formulation Stokes kernels (singular solns): G(x, y) single-layer, H(x, y) double-layer. Actual, parallel surfaces: Γ actual, γ parallel, 0 < φ < φ Γ offset param. n(y) y ξ φ γ Γ= Ω

37 Boundary integral formulation Stokes kernels (singular solns): G(x, y) single-layer, H(x, y) double-layer. Actual, parallel surfaces: Γ actual, γ parallel, 0 < φ < φ Γ offset param. n(y) y ξ φ γ Γ= Ω Mixed representation: u(x) = λ γ G(x, ξ)ψ(y(ξ)) da ξ + (1 λ) Γ H(x, y)ψ(y) da y 0 < λ < 1 interpolation param, ψ potential density.

38 Boundary integral formulation Stokes kernels (singular solns): G(x, y) single-layer, H(x, y) double-layer. Actual, parallel surfaces: Γ actual, γ parallel, 0 < φ < φ Γ offset param. n(y) y ξ φ γ Γ= Ω Mixed representation: u(x) = λ γ G(x, ξ)ψ(y(ξ)) da ξ + (1 λ) Γ H(x, y)ψ(y) da y 0 < λ < 1 interpolation param, ψ potential density. Integral equation: Given U find ψ s.t. lim x o x u(x o ) = U(x) for all x Γ. x o R 3 \Ω

39 Integral operators: Properties of formulation A G ψ + A H ψ + cψ = U (A G ψ)(x) = Γ G λ,φ (x, y)ψ(y) da y regular (A H ψ)(x) = Γ Hλ (x, y)ψ(y) da y weakly singular.

40 Integral operators: Properties of formulation A G ψ + A H ψ + cψ = U (A G ψ)(x) = Γ G λ,φ (x, y)ψ(y) da y regular (A H ψ)(x) = Γ Hλ (x, y)ψ(y) da y weakly singular. Solvability theorem: Under mild assumptions, there exists a unique ψ C 0 for any Γ C 1,1, φ (0, φ Γ ), λ (0, 1) and U C 0.

41 Integral operators: Properties of formulation A G ψ + A H ψ + cψ = U (A G ψ)(x) = Γ G λ,φ (x, y)ψ(y) da y regular (A H ψ)(x) = Γ Hλ (x, y)ψ(y) da y weakly singular. Solvability theorem: Under mild assumptions, there exists a unique ψ C 0 for any Γ C 1,1, φ (0, φ Γ ), λ (0, 1) and U C 0. Mobility tensor: Solutions for six independent sets of data are required to determine M. (v, ω) U ψ (f hyd, τ hyd ) L M. }{{} 6 times

42 Locally-corrected Nystrom discretization Arbitrary quadrature rule: y b nodes, W b weights, h > 0 mesh size, l 1 order. Γ= Ω

43 Locally-corrected Nystrom discretization Arbitrary quadrature rule: y b nodes, W b weights, h > 0 mesh size, l 1 order. Partition of unity functions: ζ b (x) y b, ζb (x) y b, ζ b + ζ b = 1. Γ= Ω

44 Locally-corrected Nystrom discretization Arbitrary quadrature rule: y b nodes, W b weights, h > 0 mesh size, l 1 order. Partition of unity functions: ζ b (x) y b, ζb (x) y b, ζ b + ζ b = 1. Discretized operators: (A G Γ= Ω h ψ)(x) = b G λ,φ (x, y b )ψ(y b )W b (A H h ψ)(x) = b ζ b(x)h λ (x, y b )ψ(y b )W b + ζ b (x)r x (y b )ψ(y b ) R x local poly correction at x, p 0 degree of correction.

45 Locally-corrected Nystrom discretization Arbitrary quadrature rule: y b nodes, W b weights, h > 0 mesh size, l 1 order. Partition of unity functions: ζ b (x) y b, ζb (x) y b, ζ b + ζ b = 1. Discretized operators: (A G Γ= Ω h ψ)(x) = b G λ,φ (x, y b )ψ(y b )W b (A H h ψ)(x) = b ζ b(x)h λ (x, y b )ψ(y b )W b + ζ b (x)r x (y b )ψ(y b ) R x local poly correction at x, p 0 degree of correction. Moment conditions: R x chosen s.t. A H h g = AH g for all local polys g at x up to degree p.

46 Properties of discretization A G ψ + A H ψ + cψ = U A G h ψ h + A H h ψ h + cψ h = U Solvability theorem: Under mild assumptions, there exists a unique ψ h C 0 for any Γ C 1,1, φ (0, φ Γ ), λ (0, 1) and U C 0.

47 Properties of discretization A G ψ + A H ψ + cψ = U A G h ψ h + A H h ψ h + cψ h = U Solvability theorem: Under mild assumptions, there exists a unique ψ h C 0 for any Γ C 1,1, φ (0, φ Γ ), λ (0, 1) and U C 0. Convergence theorem: Under mild assumptions, if Γ C m+1,1 and ψ C m,1, then as h 0 ψ ψ h 0 l 1, p 0, m 0 ψ ψ h Ch l 1, p = 0, m 1 ψ ψ h Ch min(l,p,m) l 1, p 1, m 1.

48 Conditioning: singular values σ vs parameters λ, φ 13 σ min σ max λ Results for method with p = 0 and l = 1. log 10 (σ max /σ min ) λ φ/φ Γ = 1 8 (dots), 2 8 (crosses), 3 8 (pluses),..., 7 8 (triangles). Condition number σmax σ min for (λ, φ/φ Γ ) near ( 1 2, 1 2 ).

49 Accuracy: computed load f hyd vs mesh size h F /h log 10 F log 10 1/h Results for method with p = 0 and various l, λ, φ. Convergence is visible; limited by iterative solver.

Modeling of a centrifuge experiment: the Lamm equations

Modeling of a centrifuge experiment: the Lamm equations O. Gonzalez and J. Li UT-Austin 29 October 2008 Problem statement Setup Consider an experiment in which a 2-component solution (solute + solvent)