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 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: d r n = v n dt m d v n � − m γ v n + ξ n ( t ) = F ( r n ) − ∇ r n U ( | r n − r m | ) dt m ( � = n ) � ξ i n ( t ) ξ j m ( s ) � = 2 γ mk B T δ mn δ ij δ ( t − s ) 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 ) + F ext ( r ) · ∂ ∂ v f ( r , v , t ) = Q ( r , v , t ) + B ( r , v , t ) m Collision term Q ( r , v , t ) = 1 � � d r ′ d v ′ f 2 ( r , v , r ′ , v ′ , t ) ∇ r U ( | r − r ′ | ) m ∇ v ∂ 2 Heat bath term B ( DDFT ) ( r , v , t ) = γ [ k B T ∂ v 2 + ∂ ∂ v · v ] f ( r , v , t ) m Closure obtained from Decoupling (Molecular chaos) f 2 ( r , v , r ′ , v ′ , t ) ≈ f ( r , v , t ) f ( r ′ , v ′ , t ) g 2 ( 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: δ F � � ∂ t n ( r , t ) = D ∇ n ( r , t ) ∇ δ n ( 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. ∂ t f ( r , v , t ) + v · ∇ f ( r , v , t ) + F ext ( r ) · ∂ ∂ v f ( r , v , t ) = m ( v − u ) · C (1) ( r , t ) + ( m ( v − u ) 2 f loc ( r , v , t ) � � − 1) C (2) ( r , t ) + B bgk 3 k B T nk B T B bgk ( r , v , t ) ≡ − ν 0 [ f ( r , v , t ) − f loc ( r , v , t )] 2 π k B T ( r , t ) ] 3 / 2 exp � − m ( v − u ) 2 � m f loc ( r , v , t ) = n ( r , t )[ . 2 k 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 mn [ ∂ t u j + u i ∂ i u j ] + ∂ i P ( K ) − F j n − C (1) ( r , t ) = b (1) ( r , t ) ij j j and the kinetic energy balance equation 3 2 k B n [ ∂ t + u i ∂ i ] T + P ( K ) ∂ i u j + ∂ i q ( K ) − C (2) ( r , t ) = b (2) ( r , t ) ij i C are are determined by interactions (self-consistent fields) and are gradients of the pressure tensor and heat flux. � C (1) d v Q ( r , v , t ) v i = −∇ j P ( C ) ( r , t ) = m ( r , t ) (2) i ij C (2) ( r , t ) = −∇ i q ( C ) ( r , t ) − P ( C ) ( r , t ) ∇ i u j ( r , t ) (3) i ij Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 12 / 26
Outline Interactions determine pressure and transp. coefficients d P bulk = 1 n b + 2 π � � � � P ( K ) + P ( C ) � 3 n 2 b σ 3 g 2 ( σ ) bulk = k B T ii ii d i =1 Rewrite interaction as a sum of specific forces: � F mf ( r , t ) + F drag ( r , t ) + F viscous ( r , t ) � C (1) ( r , t ) = n ( r , t ) . (4) We identify the force, F mf , acting on a particle at r with the gradient of the so-called potential of mean force (attractive+repulsive): � F mf ( r , t ) = − k B T σ 2 dkkg ( r , r + σ k , t ) n ( r + σ k , t ) + G attr ( r , t ) (5) For slowly varying densities F mf ( r , t ) = −∇ µ α 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 of particles with radius r , F = − k B T ln V ′ V ′ = V − 8 π 3 ( R + r ) 3 + v overlap F = − ∂ F ∂ D = Nk B T ∂ v overlap = − ρ k B T π ( r − D / 2)(2 R + r + D / 2) 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: F drag ( r , t ) = − γ ij ( r )[ u impurity ( r ) − u j ( r )] (7) i j In the homogeneous case microscopic expression is γ ij ≈ 8 3( π mk B T ) 1 / 2 σ 2 gn δ ij ˆ (8) In the limit of small Reynolds numbers obtain mass concentration advection-diffusion equation: ∂ t c + u · ∇ c = K B T � � ∇ ( c (1 − c ) ∇ ∆ µ (9) γ m A µ A − 1 1 with c = ρ A /ρ and ∆ µ ≡ m B µ B , D = k B T (10) γ Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 16 / 26
Outline Diffusion current ∂ t ρ A + ∇ · ( ρ A u ) + ∇ · J = 0 J = − m A m B n 2 1 � D AB d A + D T � T ∇ T ρ Chemical force � 1 d A = ρ A ρ B � F A ( r ) − F B ( r ) m A ∇ µ A | T − 1 �� m B ∇ µ B | T − , m A m B ρ nk B T ( k B T ) 1 / 2 D AB = 3 . (2 πµ AB ) 1 / 2 ( σ AB ) 2 g AB 8 n D T = α D AB Umberto Marini B. Marconi (2010) Nanoscopic approach to dynamics of liquids September 24, 2010 17 / 26
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