On the hydrodynamic diffusion of rigid particles

On the hydrodynamic diffusion of rigid particles O. Gonzalez

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.

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

Spherical bodies

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.

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

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

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.

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).

Arbitrary bodies

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.

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

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

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.

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.

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

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.

Asymptotic analysis

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

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

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.

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

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

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

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

Application

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 ].

Results for straight model: D c 5 vs sequence length 4.5 dimensionless Dt x n 4 3.5 3 2.5 2 1.5 1 0 20 40 60 80 100 120 140 160 basepairs n Curves: numerics w/r = 10, 11,..., 15Å (top to bottom). Symbols: experiments (ultracentrifuge, light scattering, electrophoresis). Estimated radius: r = 10 15Å.

Results for curved model: D c 5 vs sequence length 4.5 dimensionless Dt x n 4 3.5 3 2.5 2 1.5 1 0 20 40 60 80 100 120 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 0.5 0 0 20 40 60 80 100 120 140 160 basepairs n Curves: numerics w/r = 12, 11,..., 18Å (top to bottom). Symbols: experiments (birefringence, light scattering). Estimated radius: r = 13 17Å.

Results for curved model: D n 4 vs sequence length 3.5 dimensionless Dr x n^3 3 2.5 2 1.5 1 0.5 0 0 20 40 60 80 100 120 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.

Numerical method

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

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).

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

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

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.

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

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.

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.

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

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

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

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.

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.

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.

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.

Conditioning: singular values σ vs parameters λ, φ 13 σ min 1 7 1 σ max 0 0 0.2 0.4 0.6 0.8 1 λ Results for method with p = 0 and l = 1. log 10 (σ max /σ min ) 3 2.5 2 1.5 1 0.5 0.1 0.3 0.5 0.7 0.9 λ φ/φ Γ = 1 8 (dots), 2 8 (crosses), 3 8 (pluses),..., 7 8 (triangles). Condition number σmax σ min 10 1.5 for (λ, φ/φ Γ ) near ( 1 2, 1 2 ).

Accuracy: computed load f hyd vs mesh size h 28.6 28.55 F 28.5 28.45 4 8 12 16 20 1/h log 10 F 1 2 3 4 5 6 7 0.8 1 1.2 1.4 log 10 1/h Results for method with p = 0 and various l, λ, φ. Convergence is visible; limited by iterative solver.