Thermal fluctuations in multiphase lattice Boltzmann methods Fathollah Varnik, Markus Gross ICAMS, Ruhr-University Bochum, Germany ERC-Workshop “Multiphase physics at the micro- and nano-scales” University of Rome Tor Vergata, 16-17 May 2013

Introduction Newtonian mechanics Lattice Gas averaging Thermal fluctuations inherently present averaging Boltzmann eq. discretization Lattice Boltzmann Thermal fluctuations fully eliminated Navier-Stokes equations

Need for fluctuations in LB deterministic B.eq.: distribution function f = avg. # of particles in a phase space cell (r…r+dr , c…c+dc) but particles are permanently scattered in and out of each phase space cell  “instantaneous” distribution function fluctuates  Fluctuating Boltzmann equation: noise Kadomtsev (Sov. Phys. JETP 1957) , Bixon, Zwanzig (Phys. Rev. 1969) Fox, Uhlenbeck (Phys. Fluids 1970)

Examples Some examples for the importance of thermal fluctuations in LB: Brownian particles in a solvent [Ladd, PRL 1993] Hybrid MD-LB approaches Polymer dynamics in a solvent [Ahlrich, Dünweg,J Chem Phys 1999] Nano-confined polymers [Usta, Butler, Ladd, PRL 2007] DNA translocation [Farahpour, Maleknejad, FV, Ejtehadi, Soft Matter 2013]

(small mean-free path, sub-sonic flows) Ladd’s approach stochastic Boltzmann: stochastic Navier-Stokes: Chapman- Enskog noise xi fluct. stress R (small mean-free path, sub-sonic flows) [Ladd, PRL 1993] Stress fluctuations uncorrelated in space and time This ensures equipartition on hydrodynamic (large) length scale Further improvement by thermalizing the kinetic (ghost) modes  Equipartition at all length scales [Adhikari, Stratford, Cates, Wagner, EPL2004]

Test: Decay of velocity autocorrelation [Ladd, PRL 1993] Symbols: Fluctuating LB Solid lines: Theory

Why fluctuations in non-ideal fluid LB? Capillary fluctuations vapor interfacial fluctuations induced by thermal motion in the bulk liquid Critical dynamics T liquid-vapor coexistence metastable region spinodal r

Need for fluctuations… Colloids in a drop competition between Brownian motion capillary flow colloidal crystallization at the substrate multitude of residual colloid patterns observed on the substrate (after evaporation)

Non-ideal fluid LB force-based model [Lee/Fischer (PRE 2006), He/Shan/Doolen (PRE 1998)]: ideal gas equilibrium distribution interaction force: ( LB force Fi) modified-equilibrium model [Swift/Osborn/Yeomans (PRL 1995)]: pressure tensor constraint:

What is needed? Covariance matrix of fluctuations: Previous studies: Ideal gas  [Ladd, JFM 1994, Adhikari et al., EPL (2005)] [Dünweg, Schiller, Ladd, PRE (2007)] [Gross, Adhikari, Cates, FV, PRE (2010)] [Ollila, Denniston, Karttunen, Ala-Nissila, PRE (2011)] Non-ideal gas: Within forced-based non-ideal LB, we encountered difficulties in handling non-linear advection term and force term Route via Discrete Boltzmann Equation

Procedure: Start with Fluctuating Discrete Boltzmann Equation (FDBE) Bring the equations to a form similar to generalized Langevin equation Derive expression for the covariance matrix of fluctuations Translate it to the Lattice Boltzmann Framework

Generalized Langevin equation A set of coupled Langevin equations Onsager, Machlup (1953) / Fox, Uhlenbeck (1970) Apply to the one dimensional case friction random force equipartition

Fluctuating DBE Thermal noise in Discrete Boltzmann Equation: Linearize DBE, express it in terms of moments and Fourier transform: This is a set of coupled Langevin equations, thus: Onsager, Machlup (1953) / Fox, Uhlenbeck (1970) Gross, Cates, Varnik, Adhikari (J. Stat. Mech. 2011) time-evolution operator: equilibrium correlation matrix:

Elements of the G-matrix How to choose the equilibrium correlations of the distribution function? ideal gas [Kadomtsev (Sov. Phys. JETP 1957) / Bixon, Zwanzig (Phys. Rev. 1969) / Fox, Uhlenbeck (Phys. Fluids 1970) / Adhikari et al (EPL 2005)] non-ideal gas [Klimontovich (1973) / Gross, Adhikari, Cates, Varnik (Phys. Rev. E 2010)] Note: this guarantees

Noise covariance matrix Force based LB: Using linearized force, one obtains the same result as for an ideal gas Non-ideal fluid interactions are reversible and do not contribute to dissipation The same non-correlated noise also for non-linear forces are expected (and found!) on a lattice, however, microscopic reversibility is often violated Modified equilibrium LB: Spatially correlated noise! [Gross, Cates, FV, Adhikari (J. Stat. Mech. 2011)] [Gross, Adhikari, Cates, FV (Phys. Rev. E, 2010)]

Hydrodynamic limit Fluctuation-Dissipation Relation for a continuum square-gradient, two-phase fluid: FDT identical to the single-phase fluid case [Kim/Mazenko (J. Stat. Phys. 1991)] This is violated by the modified equilibrium non-ideal LB! bulk viscosity R: random stress tensor shear viscosity

Hydrodynamics of the LB fluid Fluctuation-Dissipation Relation (LB fluid) modified-equilibrium (Swift/Yeomans) LBM: modified bulk viscosity: random stress tensor spatially correlated force-based (Lee/Fischer) LBM: standard (‘ideal gas’) expressions for the viscosities (indep. of k) random stress tensor same as for a single-phase fluid (uncorrelated) [Kim/Mazenko (J. Stat. Phys. 1991)] shear viscosity bulk viscosity

Benchmark Momentum correlations: Density correlations: Sufficient to consider

Benchmark Capillary fluctuations: vap. liq. h(r) vap. deviation from straight line due to Fourier-transform of discrete Laplacian [Gross, Cates, FV, Adhikari (J. Stat. Mech. 2011)]

Stochastic lubrication equation [Davidovitch, Moro, Stone (PRL 2005)] Spreading of nanodroplets Stochastic lubrication equation [Davidovitch, Moro, Stone (PRL 2005)] h(t) r(t) Noise dominated spreading r  t1/6 log r perfectly wetting substrate r  t1/10 Tanner’s law (no noise) prediction: thermal fluctuations enhance late-time spreading rate log t

Spreading of nanodroplets  cross-section through a 3D simulation

Thank you for listening Summary Analysis of fluctuating Discrete Boltzmann Equation: Ideal gas noise within force-based non-ideal fluid LB Spatially correlated noise for modified equilibrium LB Use of ideal gas noise works well for a certain parameter range [Gross, Cates, FV, Adhikari (J. Stat. Mech. 2011)] [Gross, Adhikari, Cates, FV (Phys. Rev. E, 2010)] [Gross, Adhikari, Cates, FV (Phil. Trans. R. Soc. A 2011)] Thank you for listening