Nanoscopic approach to dynamics of liquids Umberto Marini B. Marconi - - PowerPoint PPT Presentation

Nanoscopic approach to dynamics of liquids

Umberto Marini B. Marconi

University of Camerino and INFN Perugia, Italy

September 24, 2010

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 1 / 26

Motivations

Motivations

New areas of physics, materials science and chemistry come in at the

• nanoscale. At nanoscale dimensions different physical phenomena

start to dominate. A central question in nanofluidics concerns the extent to which the hydrodynamic equations hold at the nanoscale. New techniques available: electrowetting, drop/bubble microfluidics, soft-substrate actuation, electro-osmotic pumps, electrophoresis, static mixing, flow focusing, etc. Nanofluidic computing where basic computing elements such as logic gates may be incorporated into very small scale devices. Enable nanofluidic technology by directly incorporating computing functions.

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 2 / 26

Motivations

Transport in a nanochannel

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 3 / 26

Motivations

A fluid in a pipe

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 4 / 26

Motivations

Properties at the nanoscale

When structures approach the size regime corresponding to molecular scaling lengths, new physical constraints are placed on the behavior of the fluid. Fluids exhibit new properties not observed in bulk, e.g. vastly increased viscosity near the pore wall; they may effect changes in thermodynamic properties and may also alter the chemical reactivity

• f species at the fluid-solid interface.

Large demand for studying transport in nanofluidic devices, multiphase dynamics , interfacial phenomena At small scales Navier-Stokes equation breaks down Consider the discrete nature of fluids and hydrodynamics in a workable scheme Represent non ideal gas behavior via a bottom-up approach or coarse graining procedure instead of fine graining methods.

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 5 / 26

Outline

OUTLINE

Kinetic approach: evolution equation for the 1-particle phase space distribution. Balance equations for conserved quantities. Hydrodynamics Transport coefficients. Lattice Boltzmann Equation implementation. Numerical test: Poiseuille flow of hard spheres in a narrow pore Conclusions.

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 6 / 26

Outline

Microscopic description of inhomogeneous fluids

Phenomenological Langevin equation:

drn dt

= vn m dvn

dt

=  F(rn) −

• m(=n)

∇rnU(|rn − rm|)   − mγvn + ξn(t) ξi

n(t)ξj m(s) = 2γmkBTδmnδijδ(t − s)

How do we contract description from phase-space (6N-DIM) → diffusion

• rdinary 3d space?

Answ: At equilibrium via integral eqs. method or DFT. Non-equilibrium...

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 7 / 26

Outline

Evolution eq. 1-particle phase-space distribution

Kinetic equation

∂tf (r, v, t) + v · ∇f (r, v, t) + Fext(r) m · ∂ ∂vf (r, v, t) = Q(r, v, t) + B(r, v, t)

Collision term Q(r, v, t) = 1 m∇v

• dr′
• dv′f2(r, v, r′, v′, t)∇rU(|r − r′|)

Heat bath term B(DDFT)(r, v, t) = γ[kBT

m ∂2 ∂v2 + ∂ ∂v · v]f (r, v, t)

Closure obtained from Decoupling (Molecular chaos) f2(r, v, r′, v′, t) ≈ f (r, v, t)f (r′, v′, t)g2(r, r′, t|n)

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 8 / 26

Outline

Approaches: DDFT and Kinetic equation

When friction γ is large: ∂tn(r, t) = D∇

• n(r, t)∇

δF δn(r, t) − F(r)n(r, t)

• .

(1) F free energy functional of density. Method works when colloidal particles due to the strong interaction with the solvent reach rapidly a local equilibrium. Velocity distrib. function is ≈ Maxwellian. Density evolves diffusively towards the equilibrium solution. Smoluchovski description appropriate. The Solvent acts as an HEAT BATH . Noise and friction are intimately connected through Fluctuation-dissipation.

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 9 / 26

Outline

Dynamics of molecular liquids vs. colloidal suspensions

Colloidal dynamics is overdamped. Relaxation occurs via diffusion. (One conserved mode) No Galilei invariance. Molecular liquids have inertial dynamics, 5 conserved modes First 5 (hydrodynamic) moments of f (r, v, t) privileged status. Hard modes (short lived) absorb energy from the soft modes and restore global equilibrium.

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 10 / 26

Outline

How to combine microscopic and hydrodynamic description?

• Eq. of state requires better description of structure. Revised

Enskog theory. Simplifly transport equation by exactly treating contributions to hydrodynamic modes while approximating non hydrodynamic terms via an exponential relaxation ansatz. ∂tf (r, v, t) + v · ∇f (r, v, t) + Fext(r) m · ∂ ∂vf (r, v, t) = floc(r, v, t) nkBT

• (v − u) · C(1)(r, t) + (m(v − u)2

3kBT − 1)C (2)(r, t)

• +Bbgk

Bbgk(r, v, t) ≡ −ν0[f (r, v, t) − floc(r, v, t)] floc(r, v, t) = n(r, t)[

m 2πkBT(r,t)]3/2 exp

• − m(v−u)2

2kBT(r,t)

• .

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 11 / 26

Outline

Hydrodynamic description

Continuity equation ∂tn(r, t) + ∇ · (n(r, t)u(r, t)) = 0 the momentum balance equation mn[∂tuj + ui∂iuj] + ∂iP(K)

ij

− Fjn − C (1)

j

(r, t) = b(1)

j

(r, t) and the kinetic energy balance equation 3 2kBn[∂t + ui∂i]T + P(K)

ij

∂iuj + ∂iq(K)

i

− C (2)(r, t) = b(2)(r, t) C are are determined by interactions (self-consistent fields) and are gradients of the pressure tensor and heat flux. C (1)

i

(r, t) = m

• dvQ(r, v, t)vi = −∇jP(C)

ij

(r, t) (2) C (2)(r, t) = −∇iq(C)

i

(r, t) − P(C)

ij

(r, t)∇iuj(r, t) (3)

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 12 / 26

Outline

Interactions determine pressure and transp. coefficients

Pbulk = 1 d

d

• i=1
• P(K)

ii

+ P(C)

ii

• bulk = kBT
• nb + 2π

3 n2

bσ3g2(σ)

• Rewrite interaction as a sum of specific forces:

C(1)(r, t) = n(r, t)

• Fmf (r, t) + Fdrag(r, t) + Fviscous(r, t)
• .

(4) We identify the force, Fmf , acting on a particle at r with the gradient of the so-called potential of mean force (attractive+repulsive): Fmf (r, t) = −kBTσ2

• dkkg(r, r + σk, t)n(r + σk, t) + Gattr(r, t)

(5) For slowly varying densities Fmf (r, t) = −∇µα

int(r, t).

(6)

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 13 / 26

Outline

Asakura and Oosawa Entropic between hard spheres

Spheres of radius R, separated a distance 2R + D, and immersed in fluid

• f particles with radius r, F = −kBT ln V ′

V ′ = V − 8π 3 (R + r)3 + voverlap F = −∂F ∂D = NkBT V ∂voverlap ∂D = −ρkBTπ(r − D/2)(2R + r + D/2)

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 14 / 26

Outline

Fluids at substrates

Near a repulsive wall a dense fluid of hard spheres displays pronounced

• scillations on a nanoscale.

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 15 / 26

Outline

Non equilibrium forces

The drag force is proportional to the velocity difference between impurity and fluid: F drag

i

(r, t) = −γij(r)[uimpurity

j

(r) − uj(r)] (7) In the homogeneous case microscopic expression is ˆ γij ≈ 8 3(πmkBT)1/2σ2gnδij (8) In the limit of small Reynolds numbers obtain mass concentration advection-diffusion equation: ∂tc + u · ∇c = KBT γ ∇

• (c(1 − c)∇∆µ
• (9)

with c = ρA/ρ and ∆µ ≡

1 mA µA − 1 mB µB,

D = kBT γ (10)

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 16 / 26

Outline

Diffusion current

∂tρA + ∇ · (ρAu) + ∇ · J = 0 J = −mAmB n2 ρ

• DABdA + DT

1 T ∇T

• Chemical force

dA = ρAρB ρnkBT 1 mA ∇µA|T − 1 mB ∇µB|T − FA(r) mA − FB(r) mB

• ,

DAB = 3 8n (kBT)1/2 (2πµAB)1/2(σAB)2gAB . DT = αDAB

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 17 / 26

Outline

Viscous force force and shear viscosity

Viscous force of non local character: F viscous

i

(r, t) =

• dr′Hij(r, r′)[uj(r′) − uj(r)].

(11)

0.1 0.2 0.3 0.4 0.5 Packing Fraction 20 40 60 η/ηid

XA=0.0 XA=0.25 XA=0.50 XA=0.75 XA=1.0

F viscous

z

= 4 15

• πmkBTnσ4g

∂2uz ∂x2 + ∂2uz ∂y2

• (12)

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 18 / 26

Outline

Inhomogeneous Diffusion in a slit

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 19 / 26

Outline

Microscopic profiles and mobility tensor

Diffusion is normal, but non isotropic, parallel and normal mobility are different

2 4 6 8 10 12 x 0,2 0,4 0,6 0,8 1 1,2 n 5 10 x 2 4 6 8 γ

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 20 / 26

Outline

Velocity profiles

2 4 6 x/σΑΑ 5e-05 0,0001 u/vT

Figure: Velocity profiles of the two species for a channel of width H = 6σAA and load F = 0.001. according to the toy model.

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 21 / 26

Outline

Self-consistent Numerical solution of transport equation

0,02 0,04 nσΑΑ

3

0,10 0,20 2 4 6 x/σAA 1e-02 2e-02 u/vT 2 4 6 8 x/σΑΑ 5e-03 1e-02

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 22 / 26

Outline

Lattice Boltzmann in a nutshell

Lattice Boltzmann strategy is applied as a numerical solver. Based

• n discretization of velocities on a lattice. No hydrodynamic equations

need to be solved. The distribution function is replaced by an array of 19 populations, f (r, v, t) → fi(r, t). Minimal velocity set employed (D3Q19, D3Q27) The propagation of the populations achieved via a time discretization to first order and a forward Euler update: ∂tfi(r, t) + vi · ∂rfi(r, t)] ≃ fi(r + viδt, t + δt) − fi(r, t) δt Collisional stage fi(r + ci, t + 1) − fi(r, t) = wi

K

• l=0

1 v2l

T l!C (l) α (r, t)h(l) α (ci) + f loc i

(r, t) − fi( τ0

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 23 / 26

Outline

For a practical scheme we need to

Evaluate integrals by Gauss quadratures in r-space Have a good representation of the radial distribution function Fischer-Methfessel (1980) Boundary conditions simple (eg no-slip via bounce-back) Disentanglement of spatial/velocity discretization (i.e. u∇u term in NS eq.). No need to solve Poisson eqn for pressure. Navier-Stokes is recovered for small gradients .

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 24 / 26

Outline

Conclusions

Starting from a microscopic level we have obtained a governing eq. for f (r, v, t) describing both equilibrium structural properties and transport properties. Hydrodynamic vs. non-hydrod. modes splitting proves a convenient route. Attractive potential tails: contribute to the energy of the fluid, but less important for collisional dissipation. Future work must include more general interactions and geometries.

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 25 / 26

Outline

Bibliography

UMBM and S. Melchionna, Lattice Boltzmann method for inhomogeneous fluids

Europhysics Letters, 81, 34001 (2008)

UMBM and S. Melchionna

Kinetic Theory of correlated fluids: From dynamic density functional to Lattice Boltzmann methods Journal of Chemical Physics, 131, 014105 (2009).

Phase-space approach to dynamical density functional theory

• J. Chem. Phys. 126, 184109 (2007)

UMBM and P. Tarazona

Nonequilibrium inertial dynamics of colloidal systems J. Chem. Phys. 124, 164901 (2006).

Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 26 / 26