Presentation on theme: "Computer Simulation of Colloids Jürgen Horbach Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft-"— Presentation transcript:
1Computer Simulation of Colloids Jürgen Horbach Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft und Raumfahrt, Linder Höhe, Köln
2lecture 1introductionlecture 2introduction to the Molecular Dynamics (MD) simulation methodlecture 3case study: a glassforming Yukawa mixture under shearlecture 4introduction to the Monte Carlo (MC) methodlecture 5case study: phase behavior of colloid-polymer mixtures studied bygrandcanonical MClecture 6modelling of hydrodynamic interactions using a hybrid MD-LatticeBoltzmann method
4(1) colloids as model systems: different interaction potentials and hydrodynamic interactions(2) correlation functions: describe the structure and dynamics ofcolloidal fluids(3) outline of the forthcoming lectures
5(1) colloids as model systems: different interaction potentials and hydrodynamic interactions(2) correlation functions: describe the structure and dynamics ofcolloidal fluids(3) outline of the forthcoming lectures
6colloids as model systems size: 10nm – 1µm, dispersed in solvent of viscosity large, slow, diffusive → optical accesscan be described classicallytunable interactions, analytically tractable→ theoretical guidancesoft → shear meltingphenomena studied via colloidal systems:glass transition, crystallisation in 2d and 3d, systems under shear,sedimentation, phase transitions and dynamics in confinement, etc.
7hard sphere colloids confirmed theoretically predicted crystallization (of pure entropic origin)study of nucleation andcrystal growthtest of the mode couplingtheory of the glasstransition
8tunable interactions I hard spheressoft spheresvolume fraction volume fraction, hair length and density
9tunable interactions II charged spheressuper-paramagnetic spheresnumber density, salt concentration, chargemagnetic field
10tunable interactions III entropic attraction in colloid-polymer mixturesvolume fraction of colloidal particles,number of polymers, colloid/polymersize ratiocolloid-polymer mixtures may exhibit a demixing transition which issimilar to liquid-gas transition in atomistic fluidstransition is of purely entropic origin
11hydrodynamic interactions interaction between colloids due to the momentum transport throughthe solvent (with viscosity η)Dij: hydrodynamic diffusion tensordecays to leading order ~ r -1 (long-rangeinteractions!)
12simulation of a colloidal system choose some effective interaction between the colloidal particlesbut what do we do with the solvent?simulate the solvent explicitlydescribe the solvent on a coarse-grained levelas a hydrodynamic mediumBrownian dynamics (ignores hydrodynamicinteractions)ignore the solvent: not important for dynamicproperties (glass transition in hard spheres),phase behavior only depends on configurationaldegrees of freedom
13(1) colloids as model systems: different interaction potentials and hydrodynamic interactions(2) correlation functions: describe the structure and dynamics ofcolloidal fluids(3) outline of the forthcoming lectures
14pair correlation function np(r) = number of pairs (i,j) withideal gas:Δr smallirfluid with structural correlations:
15pair correlation function: hard spheres vs. soft spheres soft sphere fluid:hard sphere fluid:
17typical structure factor for a liquid 1st peak at ~2π/rp where rpmeasures the periodicityin g(r)dense liquid → lowcompressibility and thereforesmall value of S(q→0)in general S(q) appropriatequantity to extract informationabout intermediate and largelength scalesS(q) can be measured inscattering experiments
18van Hove correlation function generalization of the pair correlation function to a time-dependentcorrelation functionproportional to the probability that a particle k at time t is separated bya distance |rk(t) - rl(0)| from a particle l at time 0self part:
19intermediate scattering functions coherent intermediate scattering function:incoherent intermediate scattering function:can one relate these quantities to transport processes?
20F(q,t) for a glassforming colloidal system Eckert, Bartsch, Faraday Discuss. 123, 51 (2003)
21↓ ↓ diffusion constants self-diffusion interdiffusion taggedatoms in Iconcentrationdifferencesin I and II↓↓diffusion processhomogeneous distribution oftagged particles in I+IIhomogeneous distribution ofred and blue atoms in I+II
22calculation of the self-diffusion constant Einstein relation:Green-Kubo relation:vtag(t): velocity of tagged particle at time tthe two relations are strictly equivalent!
23Computer Simulations of Colloids Molecular Dynamics I: How does it work?
24Outline MD – how does it work? numerical integration of the equations of motionperiodic boundary conditions, neighbor listsMD in NVT ensemble: thermostatting of the systemMD simulation of a 2d Lennard-Jones fluid:structure and dynamics
25Newton‘s equations of motion classical system of N particles at positionsUpot : potential function, describes interactionsbetween the particlessimplest case: pairwise additive interaction between point particlessolution of equations of motion yield trajectories of the particles, i.e.positions and velocities of all the particles as a function of time
26microcanonical ensemble consider closed system with periodic boundary conditions, no couplingto external degrees of freedom (e.g. a heat bath, shear, electric fields)particle number N and volume V fixed → microcanonical ensemblemomentum conserved, set initial conditions such that total momentumis zerototal energy conserved:
27simple observables and ergodicity hypothesis potential energy:kinetic energy:pressure:here < ... > is the ensemble average; we assume the ergodicity hypothesis:time average = ensemble average
28a simple model potential for soft spheres WCA potential:σ=1, ε=1cut off at rcut=21/6σ,corresponds to minimumof the Lennard-Jonespotential
29a simple problem N = 144 particles in 2 dimensions: starting configuration: particlessit on square latticevelocities from Maxwell-Boltzmanndistribution (T = 1.0)potential energy U = 0 due tod > rcutd>rcutL=14.0how do we integrate the equations of motion for this system?
30the Euler algorithmTaylor expansion with respect to discrete time step δtbad algorithm: does not recover time reversibility property of Newton‘sequations of motion→ unstable due to strong energy drift, very small time step required
31the Verlet algorithm consider the following Taylor expansions: addition of Eqs. (1) and (2) yields:equations (1) and (2): velocity form of the Verlet algorithmsymplectic algorithm: time reversible and conserves phase space volume
32implementation of velocity Verlet algorithm do i=1,3*N ! update positionsdispla=hstep*vel(i)+hstep**2*acc(i)/2pos(i)=pos(i)+displafold(i)=acc(i)enddodo i=1,3*N ! apply periodic boundary conditionsif(pos(i).lt.0) thenpos(i)=pos(i)+lboxelseif(pos(i).gt.lbox) pos(i)=pos(i)-lboxendifcall force(pos,acc) ! compute forces on the particlesdo i=1,3*N ! update velocitiesvel(i)=vel(i)+hstep*(fold(i)+acc(i))/2
33neighbor listsforce calculation is the most time-consuming part in a MD simulation→ save CPU time by using neighbor listsVerlet list:cell list:update when a particle hasmade a displacement> (rskin-rcut)/2most efficient is a combinationof Verlet and cell list
34simulation results I initial configuration with u=0: after 105 time steps:
35energies as a function of time → determine histogramm P(u) for the potential energy for different N
37fluctuations of the potential energy dotted lines: fits withGaussianswidth of peaks is directlyrelated to specific heat cVper particle at constantvolume Vcompare to canonicalensemble
38simulations at constant temperature: a simple thermostat idea: with a frequency ν assign new velocities to randomly selectedparticles according to a Maxwell-Boltzmann (MB) distributionwith the desired temperaturesimple version: assign periodically new velocities to all the particles(typically every 150 time steps)algorithm:take new velocities from distributiontotal momentum should be zeroscale velocities to desired temperature
39energies for the simulation at constant temperature
40pair correlation function very similar to 3dno finite size effectsvisible
41static structure factor S(q) also similar to 3dno finite size effectslow compressibilityindicated by smallvalue of S(q) for q→0
42mean squared displacement diffusion constant increases with increasing system size !?
43long time tails in 3d Green-Kubo relation for diffusion constant: low densityhigh densitycage effect indense liquid:Levesque, Verlet,PRA 2, 2514 (1970)Alder, Wainwright,PRA 1, 18 (1970)→ power law decay of velocity autocorrelation function (VACF) at long times
44Alder‘s argument Stokes equation describes momentum diffusion: backflow pattern aroundparticle:consider momentum in volume element δV at t=0volume at time t:amount of momentum in original volumeelement at time t →d=2:refined theoretical prediction
45effective diffusion constants → larger system sizes required to check theoretical prediction !
46literature M. Allen, D. J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1987)D. Rapaport, The Art of Molecular Dynamics(Cambridge University Press, Cambridge, 1995)D. Frenkel, B. Smit, Understanding Molecular Simulation: FromAlgorithms to Applications(Academic Press, San Diego, 1996)K. Binder, G. Ciccotti (eds.), Monte Carlo and Molecular Dynamicsof Condensed Matter Systems(Societa Italiana di Fisica, Bologna, 1996)
47Computer Simulations of Colloids Molecular Dynamics II: Application to Systems Under Shear with Jochen Zausch (Universität Mainz)
48shear viscosity of glassforming liquids 1013 Poise →10-3 Poise →dramatic slowing down of dynamics in a relatively small temperature range
49F(q,t) for a glassforming colloidal system Eckert, Bartsch, Faraday Discuss. 123, 51 (2003)
50glassforming fluids under shear hard sphere colloid glass: confocal microscopyapply constant shear ratedrastic acceleration of dynamicsWeissenberg number:τ(T): relaxation time in equilibriumBesseling et al., PRL (2007)central question: transient dynamics towards steady state for W >>1problem amenable to experiment, mode coupling theory and simulation
51outlinesimulation details: interaction model, thermostat, boundary conditionsdynamics from equilibrium to steady state?dynamics from steady state back to equilibrium?
52technical requirements (1) model potential(2) external shear field produces heat: thermostat necessary(3) can we model the colloid dynamics realistically?(4) how can we shear the system without using walls?
53model potential: a binary Yukawa mixture requirements: → binary mixture (no crystallization or phase separation)→ colloidal system (more convenient for experiments)→ possible coupling to solvent: density not too high
54equations of motion and thermostat use of thermostat necessary + colloid dynamics → solve Langevin equationequations of motion:dissipative particle dynamics (DPD):→ Galilean invariant, local momentum conservation, no bias on shear profilelow ξ (ξ=12): Newtonian dynamics, high ξ (ξ=1200): Brownian dynamics
55Lees-Edwards boundary conditions no walls, instead modified periodicboundary conditionsshear flow in x direction,velocity gradient in y direction,“free“ z directionshear rate:linear shear profile
56further simulation details generalized velocity Verlet algorithm, time stepsystem size: equimolar AB mixture of 1600 particles (NA=NB=800)L= , density ρ= , volume fraction Φ≈48%at each temperature at least 30 independent runstemperature range 1.0 ≥ T ≥ 0.14 (40 million time steps for productionruns at T=0.14)
57self diffusion B particles slower than A particles weak temperature dependence of D forsheared systems ifalong T = 0.14:W ~
58shear stress: equilibrium to steady state averaging over 250independent runsdefinition:maximum around markstransition to plastic flow regimesteady state reached ontime scaletime scale on which linearprofile evolves?
59shear profilevelocity profile becomes almost linear in the elastic regimemaximum in stress not related to evolution of linear profile
60structural changes: pair correlation function pair correlation function notvery sensitive to structuralchanges→ consider projections of g(r)onto spherical harmonics
61structure and stresses how is the stress overshoot reflected in dynamic correlations?
62mean squared displacement occurrence of superdiffusive regime in the transient plastic flow regimesuperdiffusion less pronounced in hard sphere experiment (→ Joe Brader)
64EQ to SS: Incoherent intermediate scattering function transients can be described bycompressed exponential decay:tw=0: β ≈ 1.8 → different forBrownian dynamics?
65EQ to SS: Newtonian vs. overdamped dynamics change friction coefficientin DPD forces:→ same compressed exponential decay also for overdamped dynamics
66shear stress: from steady state back to equilibrium decay can be well described by stretched exponential with β≈0.7stressed have completely decayed at
67SS to EQ: mean squared displacement stresses relax on time scale ofthe order of 300 (“crossover“ intw=0 curve)then slow aging dynamics towardequilibrium
68conclusionsphenomenology of transient dynamics in glassforming liquids under shear:binary Yukawa mixture:model for system of charged colloids, using a DPD thermostat andLees-Edwards boundary conditionsEQ→SS: superdiffusion on time scales between the occurrence of themaximum in the stress and the steady stateSS→EQ: stresses relax on time scale 1/ , followed by slow agingdynamics
69Computer Simulations of Colloids Monte Carlo Simulation I: How does it work?
70Outline MC – how does it work? calculation of π via simple sampling and Markov chain samplingMetropolis algorithm for the canonical ensembleMC at constant pressureMC in the grandcanonical ensemble
71→ Markov chain sampling calculation of πfirst method: integrate over unit circlesecond method: use uniform random numbers→ direct sampling→ Markov chain samplingWerner Krauth, Statistical Mechanics: Algorithms and Computations(Oxford University Press, Oxford, 2006)
72calculation of π : direct sampling estimate π fromalgorithm:Nhits=0do i=1,Nx=ran(-1,1)y=ran(-1,1)if(x2+y2<1) Nhits=Nhits+1enddoestimate of π from ratio Nhits/N
73calculation of π : Markov chain sampling algorithm:Nhits=0; x=0.8; y=0.9do i=1,NΔx=ran(-δ,δ)Δy=ran(-δ,δ)if(|x+Δx|<1.and.|y+Δy|<1) thenx=x+Δx; y=y+Δyendifif(x2+y2<1) Nhits=Nhits+1enddooptimal choice for δ range:not too small and not too largehere good choice δmax=0.3
74calculation of π with simple and Markov chain sampling (N = 4000, δmax = 0.3)simple sampling (N = 4000):run Nhits estimate of πrun Nhits estimate of π
75simple sampling vs. Markov chain sampling probabilitydensity:observable:simple sampling: A sampled directly through πMarkov sampling: acceptance probabilityp(old→new) given by
76more on Markov chain sampling Markov chain: probability of generating configuration i+1 dependsonly on the preceding configuration iimportant: loose memory of initial conditionconsequence of algorithm: points pile up at the boundariesshaded region:piles of sampling pointsδmax
77detailed balance consider discrete system: configurations a, b, c should be generated with equalprobabilitydetailed balance condition holds
78detailed balance in the calculation of π acceptance probability for a trial moveone can easiliy show thatdetailed balance: analog to the time-reversibility property of Newton‘sequations of motion
79a priori probabilities moves Δx and Δy in square of linear size δ defines a prioriprobability pap(old→new)different choices possiblegeneralized acceptance criterion:
80canonical ensemble problem: compute expectation value direct evaluation of the integrals: does not work!simple sampling: sample random configurationssamples mostly states in the tails of the Boltzmann distribution
81idea of importance sampling sample configurations according to probabilitychoose, thenthis can be realized by Markov chain sampling: random walk throughregions in phase space where the Boltzmann factor is large
82importance sampling choose transition probability such that to achieve this use detailed balance conditionpossible choiceyields no information about the partition sum
83Metropolis algorithm displacement move acceptance probability N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, JCP 21, 1087 (1953)old configurationdisplacement movetrial configurationΔE = E(trial) - E(old)acceptance probabilityyesnoΔE ≤ 0 ?new config. =trial config.r < exp[-ΔE/(kBT)] ?r uniform randomnumber 0 < r < 1nonew config. = old config.
84implementation of the Metropolis algorithm do icycl=1,ncyclold=int[ran(0,1)*N] ! select a particle at randomcall energy(x(old),en_old) ! energy of old conf.xn=x(old)+(ran(0,1)-0.5)*disp_x ! random displacementcall energy(x(new),en_new) ! energy of new conf.if(ran(0,1).lt.exp[ -beta*(en_new-en_old) ] ) x(old)=x(new)if( mod(icycl,nsamp).eq.0) call sample ! sample averagesenddo
85dynamic interpretation of the Metropolis precedure probability that at time t a configuration r (N) occurs duringthe Monte Carlo simulationrate equation (or master equation):equilibrium:detailed balance condition one possible solution:
86remarks Monte Carlo measurements: first equilibration of the system (until Peq is reached), beforemeasurement of physical propertiesrandom numbers:real random numbers are generated by a deterministic algorithm andthus they are never really random (be aware of correlation effects)a priori probability:can be used to get efficient Monte Carlo moves!
87Monte Carlo at constant pressure system of volume V in contact with an ideal gas reservoir via a pistonV0 - Vacceptance probability for volumemoves:Vtrial move:
88grand-canonical Monte Carlo system of volume V in contact with an ideal gas reservoir withfugacityinsertion:V0 - VVremoval:
89textbooks on the Monte Carlo simulation method D. P. Landau, K. Binder, A Guide to Monte Carlo Simulations inStatistical Physics(Cambridge University Press, Cambridge, 2000)W. Krauth, Statistical Mechanics: Algorithms and Computations(Oxford University Press, Oxford, 2006)M. Allen, D. J. Tildesley, Computer Simulation of Liquids(Clarendon Press, Oxford, 1987)D. Frenkel, B. Smit, Understanding Molecular Simulation: FromAlgorithms to Applications(Academic Press, San Diego, 1996)K. Binder, G. Ciccotti (eds.), Monte Carlo and Molecular Dynamicsof Condensed Matter Systems(Societa Italiana di Fisica, Bologna, 1996)
90Computer Simulations of Colloids Monte Carlo II: Phase Behaviour of Colloid-Polymer Mixtures collaborations: Richard Vink, Andres De Virgiliis, Kurt Binder
91depletion interactions in colloid-polymer mixtures polymers cannot move intodepletion zones of colloidswith their center of massgain of free volume if there isoverlap of the depletion zonesof two colloidseffective attraction betweencolloids (of entropic origin)→ demixing transition possible into colloid-rich phase (liquid) andcolloid-poor phase (gas)
92Asakura-Oosawa (AO) model in the following AO model for Rp/Rc = 0.8:→ exhibits demixing transition into a colloid-poor phase (gas) and acolloid-rich phase (liquid)→ temperature not relevant; what is the corresponding variable here?
93Outline phase diagram: AO model for RP/RC = 0.8: binodal critical point (FSS)interfacial tensioninterfacial widthcapillary waves
94grandcanonical MC for the AO model volume V, polymer fugacity zP and colloid fugacity zC fixed → removaland insertion of particlesproblem 1: high free energy barrier between coexisting phases in thetwo-phase region (far away from the critical point)→ use successive umbrella samplingproblem 2: low acceptance rate for trial insertion of colloidal particle, highprobability that it overlaps with a polymer particles→ use cluster move
95successive umbrella sampling problem: high free energy barrier between coexisting phases
96cluster moveproblem: low acceptance rate for trial insertion of a colloid → cluster movesphere of radius δ and volume Vδ around randomly selected pointtry to replace nr ≤ nP polymers (A)by a colloid (B):reverse move (C+D):R.L.C. Vink, J. H., JCP 121, 3253 (2004)
97phase diagram critical point from finite size scaling → 3D Ising universality class→ estimate interfacial tension by γ = ΔF/(2A) (Binder 1982)
99interfacial tensionγ*≡(2RC)2γ as a function of the difference in the colloid packingfraction of the liquid (L) and the vapor (V) phase at coexistence→ DFT (M. Schmidt et al.)yields accurate result
100interfacial profile mean field result: → strong finite size effects , Lx,y=31.3, Lx,y=23.1→ strong finite size effects(not due to critical fluctuations)
101capillary wave theoryfree energy cost of spatial interfacial fluctuations (longwavelength limit)result for mean square amplitude of the interfacial thicknesscombination with mean-field result by convolution approximation→ determination of intrinsic width w0 by simulation not possible
103finite size scaling II map on universal 3D Ising distribution account for field mixing→ AO model belongs to 3D Isinguniversality class
104conclusions discrepancies to mean-field results AO model belongs to 3D Ising universality classinterfacial broadening by capillary wavesmuch more on the AO model inR.L.C. Vink, J.H., JCP 121, 3253 (2004);R.L.C. Vink, J.H., JPCM 16, S3807 (2004);R.L.C. Vink, J.H., K. Binder, JCP 122, (2005);R.L.C. Vink, J.H., K. Binder, PRE 71, (2005);R.L.C. Vink, M. Schmidt, PRE 71, (2005);R.L.C. Vink, K. Binder, J.H., Phys. Rev. E 73, (2006);R.L.C. Vink, K. Binder, H. Löwen, PRL 97, (2006);R.L.C. Vink, A. De Virgiliis, J.H. K. Binder, PRE 74, (2006);A. De Virgiliis, R.L.C. Vink, J.H., K. Binder, Europhys. Lett. 77, (2007);K. Binder, J.H., R.L.C. Vink, A. De Virgiliis, Softmatter (2008).
105Computer Simulations of Colloids Modelling of hydrodynamic interactions collaboration: Apratim Chatterji (FZ Jülich)
106simulation of colloids size 10 nm – 1 μmmesoscopic time scalessolvent:atomistic time (~ 1 ps) andlength scales (~ Å)simulation difficult due to different time and length scales→ coarse graining of solvent‘s degrees of freedom
107Langevin equation many collisions with solvent particles on typical time scale of colloidsystematic friction force on colloidBrownian particles of mass M :fr,i uncorrelated random force with zero mean:fluctuation-dissipation theorem:
108decay of velocity correlations consider single Brownian particle of mass M :→ exponential decay of velocity autocorrelation function (VACF)correct behavior: power law decay of VACF due to local momentumconservation
109generalized Langevin equation idea: couple velocity of colloid to thelocal fluid velocity field via frictional forcepoint particle:problems:determination of hydrodynamic velocity field from solutionof Navier-Stokes equationrotational degrees of freedom → colloid-fluid coupling?
110lattice Boltzmann method I : number of particles at a lattice node , at a time t, witha velocitydiscrete analogue of kinetic equation:Δi: collision operatorvelocity space from projection of 4DFCHC lattice onto 3D (isotropy)moments of ni :
111lattice Boltzmann method II linearized collision operator:equilibrium distr.:recover linearized Navier-Stokes equations:thermal fluctuations: add noise terms to stress tensor
112lattice Boltzmann algorithm III collision step: computepropagation step: compute
113colloid fluid momentum exchange sphere represented by (66)uniformly distributed pointson its surfaceeach point exchangesmomentum with surroundingfluid nodes
114velocity correlations at short times give particle a kick and determine decay of its velocityblue curves:red curves:fluid velocity atthe surface of thesphere
115Cv(t) and Cω(t)I: moment of inertiaindependent of R and ξ0
116thermal fluctuations ok? linear response theory:velocity relaxation of kicked particle in a fluid at rest =velocity autocorrelation function of particle in a thermal bath