MESOSCALE TRANSPORT AND RHEOLOGY

Transcription

1 MESOSCALE TRANSPORT AND RHEOLOGY Jorge Viñals School of Physics and Astronomy University of Minnesota

2 MESOSCALE THEMES 1 Nonequilibrium evolution Slow ( hydrodynamic ) but unstable motion of collective variables: interfaces, topological defects, structural variables. Classical macroscopic transport models need to incorporate discontinuities (interfaces), singularities (defects), and generally solve moving boundary problems. Various mesoscale regularization schemes.

3 MESOSCALE THEMES 1 Nonequilibrium evolution Slow ( hydrodynamic ) but unstable motion of collective variables: interfaces, topological defects, structural variables. Classical macroscopic transport models need to incorporate discontinuities (interfaces), singularities (defects), and generally solve moving boundary problems. Various mesoscale regularization schemes. 2 Multiple scales and their decoupling Microscopic Laws Macroscopic Laws Mesoscopic Lawlessness (cf. R. Laughlin) Often it is not clear how to decouple time and length scales.

4 LOCAL EQUILIBRIUM Local center of mass frame Tds = du+p d(1/ρ) Laboratory frame - conservation laws Td(ρs) = d(ρe) v i dg i µdρ e = u v 2 g i = ρv i

5 LOCAL EQUILIBRIUM Local center of mass frame T ds dt = du dt +p d1/ρ dt Laboratory frame - conservation laws Td(ρs) = d(ρe) v i dg i µdρ e = u v 2 g i = ρv i

6 STRUCTURAL VARIABLES BROKEN SYMMETRIES Td(ρs) = d(ρe) v i dg i µdρ ξ i d ( i ϕ) Translational symmetry E{h(x)} = σ 2 dx h(x) 2 ξ i = σ i h(x) ϕ = h(x)

7 STRUCTURAL VARIABLES BROKEN SYMMETRIES Td(ρs) = d(ρe) v i dg i µdρ ξ i d ( i ϕ) Translational symmetry E = 1 2 dx [B( y u) 2 + K( x 2 u) 2] ξ i = Bδ iy y u Kδ ix 3 x u ϕ = u(x, y)

8 STRUCTURAL VARIABLES BROKEN SYMMETRIES Td(ρs) = d(ρe) v i dg i µdρ ξ i d ( i ϕ) Rotational symmetry E [ ˆn(x)] = 1 2 dxk ijkl ( i n j )( k n l ) h ij = Td(ρs) =... h ij d( j n i ) ( ) E ( j n i ) ρ,s,g i = K jikl k n l ϕ = ˆn(x)

9 NONEQUILIBRIUM Thermodynamic driving forces are gradients of intensive parameters, including i ξ i, j h ij, etc.

10 NONEQUILIBRIUM Thermodynamic driving forces are gradients of intensive parameters, including i ξ i, j h ij, etc. Structural stresses - Legendre transform df =... + g i dv i + ϕd( i ξ i ) Reversible stresses (low frequency/quasistatic) follow from Maxwell relation 2 f ( j ξ j ) v i = g i ( j ξ j ) 2 f v i ( i ξ i ) = ϕ. v i ġ i ϕ = ( j ξ j ) v }{{}}{{} i Reversible force Advection tϕ+v ϕ=...

11 Ginzburg Landau type F = 1 [ ] dx K ψ 2 + g(ψ) 2 tψ + v i i ψ = L 2 δf δψ Maxwell relation: ξ j = f ( j ψ) = K jψ ψ v i = i ψ tg i =... K( 2 ψ) i ψ or σ R ij = K( i ψ)( j ψ)

12 Ginzburg Landau type F = 1 [ ] dx K ψ 2 + g(ψ) 2 tψ + v i i ψ = L 2 δf δψ Maxwell relation: ξ j = f ( j ψ) = K jψ ψ v i = i ψ tg i =... K( 2 ψ) i ψ or σ R ij = K( i ψ)( j ψ) Orientational order ġ i ( j h kj ) = n k v i tn k λ kij i v j + X D k = 0, ṅ k v i = λ kmi mδ(x x ) tg i + i p l λ mli nh mn = j σ D ij

13 CONTINUUM MECHANICS s = s(e, ρ, ϕ, i ϕ) ṡ =... σ ij pδ ij i v j ξ i ( T T i ϕ) µ ϕ T ϕ ( i ϕ) = i ϕ ( i v j )( j ϕ) ṡ =... σ ( ij pδ ij ξ i j ϕ i v j + i (J ϕ µϕ i ϕv i ) T T i ξ i T ) Reversible stress when entropy/free energy depends on gradients of order parameter

14 CAHN-HILLIARD FLUID Non classical stress: (σ ij ) R = pδ ij + ξ i j ϕ so that (Ginzburg-Landau) j (σ ij ) R = i p + K( 2 ψ) i ψ High order in gradients - negligible except at interfaces. In the limit of a sharp interface K( 2 ψ) i ψ K ψ 2 κˆn σκδ(ζ)ˆn ( ) 2 dψ0 σ = K dz dz Normal stress discontinuity at the interface.

15 RHEOLOGY - UNIAXIAL FLUID { 1 [ ] 2 F = dx (q ɛ )ψ 2 2 ψ2 + g } 4 ψ4 tψ + v i i ψ = L δf δψ ψ 0(x) = ɛ 1/2 A 0 cos(q 0 x) +... Reversible stress ψ(x + u(x)) = ψ(x), σ ij = δf δ( i u j ) [ ] [ ] σ ij = i (q )ψ j ψ (q )ψ i j ψ Complex modulus σ ij (k, ω) = G ijkl (k, ω)γ kl (k, ω) G ijkl (k, ω) = G ijkl(k, ω) + ig ijkl(k, ω)

16 Order parameter relaxation causes viscoelastic response z v(z,t) = γω cos( ω t) z x G = 4ɛ 3 ( 2q 2 x q 2 z + O(γ 2 ) ) G = 32ɛγ2 q 4 3 xqz 4 ω ɛ 2 + ω 2

17 GYROID - PERIODIC MINIMAL SURFACE (Ia3d) [ 12 ψ(r) = ψ + A j e iq j r + j=1 k=1 ] 6 B k e ip k r + c.c. with 18 reciprocal lattice vectors with q 2 = 3p 2 /4 1. G = 8 j=1 [ ] ( (ω/λj ) 2 Γ j 1 + (ω/λ j ) 1 +8q φ 2 a + 2 ) 3 φ2 b, G = 8 [ ] ω/λ j Γ j 1 + (ω/λ j ) 2 j= G (ω),g (ω), G (ω) σ xz ω γ(t)

18 BLOCK COPOLYMER RHEOLOGY Perpendicular Orientation m q v y (x) Parallel Orientation z m q v x (z) x [L. Wu, T.P. Lodge, and F. Bates, J. Rheol. 49, 1231 (2005)] [ ] G ijkl (t) = G 11(t)q i q j q k q l +G 4(t)(δ ik δ jl +δ il δ jk )+G 56(t) q i (q k δ jl +q l δ kj )+q j (q k δ il +q l δ ki ) Perpendicular G 4. Parallel G 4 + G 56.

19 BLOCK COPOLYMER RHEOLOGY q Perpendicular Orientation m Two order parameters: Lamellar planes ψ(x, t) End-to-end polymer chain orientation Q ij (x, t) = m i m j 1 3 δ ij v y (x) Parallel Orientation z m q v x (z) x

20 BLOCK COPOLYMER RHEOLOGY q Perpendicular Orientation m Two order parameters: Lamellar planes ψ(x, t) End-to-end polymer chain orientation Q ij (x, t) = m i m j 1 3 δ ij v y (x) Parallel Orientation z m Free energy: Uniaxial state F S = dx { 1 2 [ (q 2 ] ɛ 2 )ψ 2 ψ2 + g } 4 ψ4 q v x (z) x Some model for polymer chains F Q (Q ij, k Q ij ) Minimal coupling i ψq ij j ψ Only simple chain diffusive relaxation studied so far (Maxwell viscoelasticity) [C. Yoo, and J. Viñals, Macromolecules 45, 4848 (2012), S. Yabunaka and T. Ohta, Soft Matter 9, 7479 (2013)].

21 RHEOCHAOS AND SHEAR BANDING Shear instability to bands of different shear rates. Non monotonic shear stress/shear rate curve due to the microstructural response of the fluid. Observed in many complex fluids: wormlike micelles, liquid crystalline polymers, colloidal suspensions, soft glasses. Elastic bursts, and rheochaos. [S. Fielding, Phys. Rev. Lett. 95, (2005)] Introduce a structure order parameter Q ij instead of phenomenological constitutive laws for the flow curve. [C. Dasgupta]

22 σ R ij [ δf = α 0 +α 1 δq ij ] δf Q ik δq kj

23 TOPOLOGICAL DEFECT MOTION AT THE MESOSCALE Hexagonal phase h Z i2 1 FS = dx (q )ψ + g (ψ) 2 ψ around defects is regular. Envelope equations on the other hand ψ(x, t) = A(X, T )e iq x = ρ(x, T )e iθ(x,t ) e iq x. I θ dl = ±2π.

24 Hexagonal phase h Z i2 1 (q )ψ + g (ψ) FS = dx 2 h i h i σij = i (q )ψ j ψ (q )ψ i j ψ

25 [A. Skaugen and L. Angheluta] [A. Yau, W. Chak, M. Kanan, G.B. Stephenson, Science 356, 739 (2017)]

26 REDUCTION FROM MESOSCALE TO MACROSCALE Order parameter expanded in slowly varying amplitudes: ψ = Ae ik 0x + Be ik 0z + c.c. where A = A(ɛ 1/2 x, ɛ 1/4 y, ɛ 1/4 z, ɛt) B = B(ɛ 1/4 x, ɛ 1/4 y, ɛ 1/2 z, ɛt)

27 ( A t = ɛa + ξ2 0 x i ) 2 y 2 A 3 A 2 A 6 B 2 A, 2q 0 ( B t = ɛb + ξ2 0 y i ) 2 x 2 B 3 B 2 B 6 A 2 B. 2q 0 Two, coupled, Ginzburg-Landau equations!

28 CAREFUL... As ɛ 0, interface width ξ = ξ 0 /ɛ 1/2 λ 0. Null terms in the solvability condition to derive GL look like (e.g.,) x+λ0 x dx A 3 e 2iq 0x = 0. What if A(X, x) is allowed to have some residual dependence on x (the O(1) scale)?

29 NONPERTURBATIVE CORRECTIONS Allowing this possibility, the lowest order envelope equations are A t B t = ɛa + ξ 2 0 x+λ0 x ( x i ) 2 y 2 A 3 A 2 A 6 B 2 A 2q 0 dx ( A 3 e 2iq 0x + A 3 e 4iq 0x ), λ 0 ( y i ) 2 x 2 B 3 B 2 B 6 A 2 B 2q 0 = ɛb + ξ0 2 x+λ0 dx ( 3 A 2 Be 2iq 0x + A 2 Be 2iq 0x ). x λ 0

30 Assume an interface, and project amplitude equations onto xa 0 and xb 0 Estimate terms of the type dxa 3 0 ( xa 0) e 2iq 0x when ξq 0 1 Continue x into complex plane, and assume that envelope on real axis results from a singularity at z = x gb + iαξ. Then dxa 3 0 ( xa 0) e 2iq 0x i( ɛ) 4 e 2q 0αξ e 2iq 0x gb Depends explicitly on x gb and is of the order of e ξ = e ξ 0/ ɛ.

31 LAW OF BOUNDARY MOTION For a grain boundary, we find, v gb = ɛ 3k0 2D(ɛ) κ2 p(ɛ) D(ɛ) cos(2k0x gb + φ) The function D(ɛ) is a friction coefficient, and p(ɛ) is a pinning force p(ɛ) ɛ 2 e α/ ɛ. p(ɛ) 0 exponentially as ɛ 0 (essential singularity). Grows quickly with ɛ.

32 v gb (t) = f D(ɛ) p(ɛ) D(ɛ) sin [2q 0x gb (t) sin(θ/2)], with (Peierls stress) p A 4 0e 2aq 0 sin(θ/2)ξ. Supercritical (second order) ξ 1/ ɛ p e 1/ ɛ 0. Subcritical (first order) ξ ξ 0 = 15λ 0 8 6πg 2 finite.

33 Relaxation of a weakly distorted boundary, BOUNDARY PINNING Boundary relaxes and moves to the right. ɛ = smooth curve. Agrees with analytic calculation. ɛ = 0.1, steps.

34 SUMMARY 1 Linear response/irreversible thermodynamics are widely used frameworks for the construction of nonequilibrium theories at the mesoscale. 2 Nontrivial response/rheology can be described from the existence of structural fields and the resulting micro stresses. 3 The mesoscale provides useful regularization schemes for the study of the dynamical evolution of moving boundaries and topological defects. 4 No general theoretical framework exists at the mesoscale. Not generally possible to decouple it from micro and macro scales. A large amount of phenomenology is required.

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