Nanoscopic approach to dynamics of liquids. Umberto Marini B. Marconi. University of Camerino and INFN Perugia, Italy. September 24, 2010. Umberto Marini ...
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
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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 of 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 n m dv dt
= vn
= F(rn ) −
X
∇rn U(|rn − rm |) − mγvn + ξ n (t)
m(6=n) j (s)i = 2γmkB T δmn δ ij δ(t − s) hξni (t)ξm
How do we contract description from phase-space (6N-DIM) → diffusion ordinary 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 ∂t f (r, v, t) + v · ∇f (r, v, t) +
Fext (r) ∂ · f (r, v, t) = Q(r, v, t) + B(r, v, t) m ∂v
Collision term Q(r, v, t) =
1 ∇v m
Z
dr0
Z
dv0 f2 (r, v, r0 , v0 , t)∇r U(|r − r0 |)
Heat bath term B (DDFT ) (r, v, t) = γ[ kBmT
∂2 ∂v2
+
∂ ∂v
· v]f (r, v, t)
Closure obtained from Decoupling (Molecular chaos) f2 (r, v, r0 , v0 , t) ≈ f (r, v, t)f (r0 , v0 , t)g2 (r, r0 , 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: i h δF − F (r)n(r, t) . ∂t n(r, t) = D∇ n(r, t)∇ δn(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. Fext (r) ∂ ∂t f (r, v, t) + v · ∇f (r, v, t) + · f (r, v, t) = m ∂v floc (r, v, t) m(v − u)2 (v − u) · C(1) (r, t) + ( − 1)C (2) (r, t) +Bbgk nkB T 3kB T Bbgk (r, v, t) ≡ −ν0 [f (r, v, t) − floc (r, v, t)] m(v−u)2 floc (r, v, t) = n(r, t)[ 2πkBmT (r,t) ]3/2 exp − 2k . B T (r,t) Umberto Marini B. Marconi (2010)
Nanoscopic approach to dynamics of liquids
September 24, 2010
11 / 26
Outline
Hydrodynamic description Continuity equation ∂t n(r, t) + ∇ · (n(r, t)u(r, t)) = 0 the momentum balance equation (K )
mn[∂t uj + ui ∂i uj ] + ∂i Pij
(1)
(1)
− Fj n − Cj (r, t) = bj (r, t)
and the kinetic energy balance equation 3 (K ) (K ) kB n[∂t + ui ∂i ]T + Pij ∂i uj + ∂i qi − C (2) (r, t) = b (2) (r, t) 2 C are are determined by interactions (self-consistent fields) and are gradients of the pressure tensor and heat flux. Z (1) (C ) Ci (r, t) = m dvQ(r, v, t)vi = −∇j Pij (r, t) (2) (C )
C (2) (r, t) = −∇i qi Umberto Marini B. Marconi (2010)
(C )
(r, t) − Pij (r, t)∇i uj (r, t)
Nanoscopic approach to dynamics of liquids
September 24, 2010
(3) 12 / 26
Outline
Interactions determine pressure and transp. coefficients Pbulk =
d i i h 1 Xh (K ) 2π 2 3 (C ) Pii + Pii nb σ g2 (σ) = kB T nb + d 3 bulk i=1
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): Z mf 2 F (r, t) = −kB T σ dkkg (r, r + σk, t)n(r + σk, t) + Gattr (r, t) (5) For slowly varying densities Fmf (r, t) = −∇µαint (r, t). Umberto Marini B. Marconi (2010)
Nanoscopic approach to dynamics of liquids
(6) September 24, 2010
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Outline
Asakura and Oosawa Entropic between hard spheres
Spheres of radius R, separated a distance 2R + D, and immersed in fluid of particles with radius r , F = −kB T ln V 0 V0 = V − F =−
8π (R + r )3 + voverlap 3
∂F NkB T ∂voverlap = = −ρkB T π(r − D/2)(2R + r + D/2) ∂D V ∂D
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 oscillations 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: Fidrag (r, t) = −γij (r)[ujimpurity (r) − uj (r)]
(7)
In the homogeneous case microscopic expression is 8 γˆij ≈ (πmkB T )1/2 σ 2 gnδij 3 (8) In the limit of small Reynolds numbers obtain mass concentration advection-diffusion equation: i KB T h ∂t c + u · ∇c = ∇ (c(1 − c)∇∆µ γ with c = ρA /ρ and ∆µ ≡
1 A µ mA
−
1 B µ , mB
D= Umberto Marini B. Marconi (2010)
(9)
kB T γ
Nanoscopic approach to dynamics of liquids
(10) September 24, 2010
16 / 26
Outline
Diffusion current ∂t ρA + ∇ · (ρA u) + ∇ · J = 0 1 n2 AB A J = −mA mB D d + DT ∇T ρ T Chemical force dA =
FA (r) FB (r) o ρA ρB n 1 1 A B , ∇µ | − ∇µ | − − T T ρnkB T mA mB mA mB D AB =
3 (kB T )1/2 . 8n (2πµAB )1/2 (σAB )2 gAB DT = αD AB
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: Z Fiviscous (r, t) = dr0 Hij (r, r0 )[uj (r0 ) − uj (r)].
(11)
60
XA=0.0 XA=0.25 XA=0.50 XA=0.75 XA=1.0
40
η/ηid 20
0
Fzviscous = Umberto Marini B. Marconi (2010)
0
0.1
0.2 0.3 Packing Fraction
0.4
0.5
∂2u ∂ 2 uz 4p z πmkB T nσ 4 g + 15 ∂x 2 ∂y 2 Nanoscopic approach to dynamics of liquids
September 24, 2010
(12) 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
8
1,2 1
6
γ
n
0,8 4
0,6 0,4
2 0,2 0
0
2
4
Umberto Marini B. Marconi (2010)
6 x
8
10
12
0
0
Nanoscopic approach to dynamics of liquids
5
x
10
September 24, 2010
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Outline
Velocity profiles
u/vT
0,0001
5e-05
0
0
2
4
6
x/σΑΑ
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
0,20
0,02
0,10
nσΑΑ
3
0,04
0
0 1e-02
u/vT
2e-02 5e-03
1e-02 0
0
Umberto Marini B. Marconi (2010)
2
4 6 x/σAA
0
2
4 6 x/σΑΑ
Nanoscopic approach to dynamics of liquids
0 8 September 24, 2010
22 / 26
Outline
Lattice Boltzmann in a nutshell Lattice Boltzmann strategy is applied as a numerical solver. Based on 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: ∂t fi (r, t) + vi · ∂r fi (r, t)] '
fi (r + vi δt, t + δt) − fi (r, t) δt
Collisional stage fi (r + ci , t + 1) − fi (r, t) = wi
Umberto Marini B. Marconi (2010)
K X fi loc (r, t) − fi ( 1 (l) (l) C (r, t)h (c ) + i α τ0 v 2l l! α l=0 T
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
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