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Computer Simulation of Colloids Jürgen Horbach Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft und Raumfahrt, Linder Höhe, Köln

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lecture 1 introduction lecture 2 introduction to the Molecular Dynamics (MD) simulation method lecture 3 case study: a glassforming Yukawa mixture under shear lecture 4 introduction to the Monte Carlo (MC) method lecture 5 case study: phase behavior of colloid-polymer mixtures studied by grandcanonical MC lecture 6 modelling of hydrodynamic interactions using a hybrid MD-Lattice Boltzmann method

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**Computer Simulations of Colloids Introduction**

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**(1) colloids as model systems: different interaction potentials and**

hydrodynamic interactions (2) correlation functions: describe the structure and dynamics of colloidal fluids (3) outline of the forthcoming lectures

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**(1) colloids as model systems: different interaction potentials and**

hydrodynamic interactions (2) correlation functions: describe the structure and dynamics of colloidal fluids (3) outline of the forthcoming lectures

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**colloids as model systems**

size: 10nm – 1µm, dispersed in solvent of viscosity large, slow, diffusive → optical access can be described classically tunable interactions, analytically tractable → theoretical guidance soft → shear melting phenomena studied via colloidal systems: glass transition, crystallisation in 2d and 3d, systems under shear, sedimentation, phase transitions and dynamics in confinement, etc.

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**hard sphere colloids confirmed theoretically predicted crystallization**

(of pure entropic origin) study of nucleation and crystal growth test of the mode coupling theory of the glass transition

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**tunable interactions I**

hard spheres soft spheres volume fraction volume fraction, hair length and density

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**tunable interactions II**

charged spheres super-paramagnetic spheres number density, salt concentration, charge magnetic field

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**tunable interactions III**

entropic attraction in colloid-polymer mixtures volume fraction of colloidal particles, number of polymers, colloid/polymer size ratio colloid-polymer mixtures may exhibit a demixing transition which is similar to liquid-gas transition in atomistic fluids transition is of purely entropic origin

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**hydrodynamic interactions**

interaction between colloids due to the momentum transport through the solvent (with viscosity η) Dij: hydrodynamic diffusion tensor decays to leading order ~ r -1 (long-range interactions!)

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**simulation of a colloidal system**

choose some effective interaction between the colloidal particles but what do we do with the solvent? simulate the solvent explicitly describe the solvent on a coarse-grained level as a hydrodynamic medium Brownian dynamics (ignores hydrodynamic interactions) ignore the solvent: not important for dynamic properties (glass transition in hard spheres), phase behavior only depends on configurational degrees of freedom

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**(1) colloids as model systems: different interaction potentials and**

hydrodynamic interactions (2) correlation functions: describe the structure and dynamics of colloidal fluids (3) outline of the forthcoming lectures

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**pair correlation function**

np(r) = number of pairs (i,j) with ideal gas: Δr small i r fluid with structural correlations:

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**pair correlation function: hard spheres vs. soft spheres**

soft sphere fluid: hard sphere fluid:

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**static structure factor**

microscopic density variable: homogeneous fluid: static structure factor: low q limit: large q:

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**typical structure factor for a liquid**

1st peak at ~2π/rp where rp measures the periodicity in g(r) dense liquid → low compressibility and therefore small value of S(q→0) in general S(q) appropriate quantity to extract information about intermediate and large length scales S(q) can be measured in scattering experiments

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**van Hove correlation function**

generalization of the pair correlation function to a time-dependent correlation function proportional to the probability that a particle k at time t is separated by a distance |rk(t) - rl(0)| from a particle l at time 0 self part:

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**intermediate scattering functions**

coherent intermediate scattering function: incoherent intermediate scattering function: can one relate these quantities to transport processes?

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**F(q,t) for a glassforming colloidal system**

Eckert, Bartsch, Faraday Discuss. 123, 51 (2003)

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**↓ ↓ diffusion constants self-diffusion interdiffusion**

tagged atoms in I concentration differences in I and II ↓ ↓ diffusion process homogeneous distribution of tagged particles in I+II homogeneous distribution of red and blue atoms in I+II

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**calculation of the self-diffusion constant**

Einstein relation: Green-Kubo relation: vtag(t): velocity of tagged particle at time t the two relations are strictly equivalent!

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**Computer Simulations of Colloids Molecular Dynamics I: How does it work?**

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**Outline MD – how does it work?**

numerical integration of the equations of motion periodic boundary conditions, neighbor lists MD in NVT ensemble: thermostatting of the system MD simulation of a 2d Lennard-Jones fluid: structure and dynamics

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**Newton‘s equations of motion**

classical system of N particles at positions Upot : potential function, describes interactions between the particles simplest case: pairwise additive interaction between point particles solution of equations of motion yield trajectories of the particles, i.e. positions and velocities of all the particles as a function of time

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**microcanonical ensemble**

consider closed system with periodic boundary conditions, no coupling to external degrees of freedom (e.g. a heat bath, shear, electric fields) particle number N and volume V fixed → microcanonical ensemble momentum conserved, set initial conditions such that total momentum is zero total energy conserved:

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**simple observables and ergodicity hypothesis**

potential energy: kinetic energy: pressure: here < ... > is the ensemble average; we assume the ergodicity hypothesis: time average = ensemble average

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**a simple model potential for soft spheres**

WCA potential: σ=1, ε=1 cut off at rcut=21/6σ, corresponds to minimum of the Lennard-Jones potential

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**a simple problem N = 144 particles in 2 dimensions:**

starting configuration: particles sit on square lattice velocities from Maxwell-Boltzmann distribution (T = 1.0) potential energy U = 0 due to d > rcut d>rcut L=14.0 how do we integrate the equations of motion for this system?

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the Euler algorithm Taylor expansion with respect to discrete time step δt bad algorithm: does not recover time reversibility property of Newton‘s equations of motion → unstable due to strong energy drift, very small time step required

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**the Verlet algorithm consider the following Taylor expansions:**

addition of Eqs. (1) and (2) yields: equations (1) and (2): velocity form of the Verlet algorithm symplectic algorithm: time reversible and conserves phase space volume

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**implementation of velocity Verlet algorithm**

do i=1,3*N ! update positions displa=hstep*vel(i)+hstep**2*acc(i)/2 pos(i)=pos(i)+displa fold(i)=acc(i) enddo do i=1,3*N ! apply periodic boundary conditions if(pos(i).lt.0) then pos(i)=pos(i)+lbox else if(pos(i).gt.lbox) pos(i)=pos(i)-lbox endif call force(pos,acc) ! compute forces on the particles do i=1,3*N ! update velocities vel(i)=vel(i)+hstep*(fold(i)+acc(i))/2

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neighbor lists force calculation is the most time-consuming part in a MD simulation → save CPU time by using neighbor lists Verlet list: cell list: update when a particle has made a displacement > (rskin-rcut)/2 most efficient is a combination of Verlet and cell list

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**simulation results I initial configuration with u=0:**

after 105 time steps:

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**energies as a function of time**

→ determine histogramm P(u) for the potential energy for different N

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pressure

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**fluctuations of the potential energy**

dotted lines: fits with Gaussians width of peaks is directly related to specific heat cV per particle at constant volume V compare to canonical ensemble

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**simulations at constant temperature: a simple thermostat**

idea: with a frequency ν assign new velocities to randomly selected particles according to a Maxwell-Boltzmann (MB) distribution with the desired temperature simple version: assign periodically new velocities to all the particles (typically every 150 time steps) algorithm: take new velocities from distribution total momentum should be zero scale velocities to desired temperature

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**energies for the simulation at constant temperature**

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**pair correlation function**

very similar to 3d no finite size effects visible

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**static structure factor**

S(q) also similar to 3d no finite size effects low compressibility indicated by small value of S(q) for q→0

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**mean squared displacement**

diffusion constant increases with increasing system size !?

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**long time tails in 3d Green-Kubo relation for diffusion constant:**

low density high density cage effect in dense liquid: Levesque, Verlet, PRA 2, 2514 (1970) Alder, Wainwright, PRA 1, 18 (1970) → power law decay of velocity autocorrelation function (VACF) at long times

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**Alder‘s argument Stokes equation describes momentum diffusion:**

backflow pattern around particle: consider momentum in volume element δV at t=0 volume at time t: amount of momentum in original volume element at time t → d=2: refined theoretical prediction

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**effective diffusion constants**

→ larger system sizes required to check theoretical prediction !

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**literature 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: From Algorithms to Applications (Academic Press, San Diego, 1996) K. Binder, G. Ciccotti (eds.), Monte Carlo and Molecular Dynamics of Condensed Matter Systems (Societa Italiana di Fisica, Bologna, 1996)

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**Computer Simulations of Colloids Molecular Dynamics II: Application to Systems Under Shear**

with Jochen Zausch (Universität Mainz)

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**shear viscosity of glassforming liquids**

1013 Poise → 10-3 Poise → dramatic slowing down of dynamics in a relatively small temperature range

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**F(q,t) for a glassforming colloidal system**

Eckert, Bartsch, Faraday Discuss. 123, 51 (2003)

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**glassforming fluids under shear**

hard sphere colloid glass: confocal microscopy apply constant shear rate drastic acceleration of dynamics Weissenberg number: τ(T): relaxation time in equilibrium Besseling et al., PRL (2007) central question: transient dynamics towards steady state for W >>1 problem amenable to experiment, mode coupling theory and simulation

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outline simulation details: interaction model, thermostat, boundary conditions dynamics from equilibrium to steady state? dynamics from steady state back to equilibrium?

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**technical 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?

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**model 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

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**equations of motion and thermostat**

use of thermostat necessary + colloid dynamics → solve Langevin equation equations of motion: dissipative particle dynamics (DPD): → Galilean invariant, local momentum conservation, no bias on shear profile low ξ (ξ=12): Newtonian dynamics, high ξ (ξ=1200): Brownian dynamics

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**Lees-Edwards boundary conditions**

no walls, instead modified periodic boundary conditions shear flow in x direction, velocity gradient in y direction, “free“ z direction shear rate: linear shear profile

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**further simulation details**

generalized velocity Verlet algorithm, time step system size: equimolar AB mixture of 1600 particles (NA=NB=800) L= , density ρ= , volume fraction Φ≈48% at each temperature at least 30 independent runs temperature range 1.0 ≥ T ≥ 0.14 (40 million time steps for production runs at T=0.14)

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**self diffusion B particles slower than A particles weak temperature**

dependence of D for sheared systems if along T = 0.14: W ~

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**shear stress: equilibrium to steady state**

averaging over 250 independent runs definition: maximum around marks transition to plastic flow regime steady state reached on time scale time scale on which linear profile evolves?

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shear profile velocity profile becomes almost linear in the elastic regime maximum in stress not related to evolution of linear profile

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**structural changes: pair correlation function**

pair correlation function not very sensitive to structural changes → consider projections of g(r) onto spherical harmonics

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**structure and stresses**

how is the stress overshoot reflected in dynamic correlations?

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**mean squared displacement**

occurrence of superdiffusive regime in the transient plastic flow regime superdiffusion less pronounced in hard sphere experiment (→ Joe Brader)

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**EQ to SS: superdiffusive transient regime**

y: gradient direction z: free direction measure effective exponent:

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**EQ to SS: Incoherent intermediate scattering function**

transients can be described by compressed exponential decay: tw=0: β ≈ 1.8 → different for Brownian dynamics?

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**EQ to SS: Newtonian vs. overdamped dynamics**

change friction coefficient in DPD forces: → same compressed exponential decay also for overdamped dynamics

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**shear stress: from steady state back to equilibrium**

decay can be well described by stretched exponential with β≈0.7 stressed have completely decayed at

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**SS to EQ: mean squared displacement**

stresses relax on time scale of the order of 300 (“crossover“ in tw=0 curve) then slow aging dynamics toward equilibrium

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conclusions phenomenology of transient dynamics in glassforming liquids under shear: binary Yukawa mixture: model for system of charged colloids, using a DPD thermostat and Lees-Edwards boundary conditions EQ→SS: superdiffusion on time scales between the occurrence of the maximum in the stress and the steady state SS→EQ: stresses relax on time scale 1/ , followed by slow aging dynamics

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**Computer Simulations of Colloids Monte Carlo Simulation I: How does it work?**

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**Outline MC – how does it work?**

calculation of π via simple sampling and Markov chain sampling Metropolis algorithm for the canonical ensemble MC at constant pressure MC in the grandcanonical ensemble

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**→ Markov chain sampling**

calculation of π first method: integrate over unit circle second method: use uniform random numbers → direct sampling → Markov chain sampling Werner Krauth, Statistical Mechanics: Algorithms and Computations (Oxford University Press, Oxford, 2006)

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**calculation of π : direct sampling**

estimate π from algorithm: Nhits=0 do i=1,N x=ran(-1,1) y=ran(-1,1) if(x2+y2<1) Nhits=Nhits+1 enddo estimate of π from ratio Nhits/N

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**calculation of π : Markov chain sampling**

algorithm: Nhits=0; x=0.8; y=0.9 do i=1,N Δx=ran(-δ,δ) Δy=ran(-δ,δ) if(|x+Δx|<1.and.|y+Δy|<1) then x=x+Δx; y=y+Δy endif if(x2+y2<1) Nhits=Nhits+1 enddo optimal choice for δ range: not too small and not too large here good choice δmax=0.3

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**calculation of π with simple and Markov chain sampling**

(N = 4000, δmax = 0.3) simple sampling (N = 4000): run Nhits estimate of π run Nhits estimate of π

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**simple sampling vs. Markov chain sampling**

probability density: observable: simple sampling: A sampled directly through π Markov sampling: acceptance probability p(old→new) given by

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**more on Markov chain sampling**

Markov chain: probability of generating configuration i+1 depends only on the preceding configuration i important: loose memory of initial condition consequence of algorithm: points pile up at the boundaries shaded region: piles of sampling points δmax

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**detailed balance consider discrete system: configurations**

a, b, c should be generated with equal probability detailed balance condition holds

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**detailed balance in the calculation of π**

acceptance probability for a trial move one can easiliy show that detailed balance: analog to the time-reversibility property of Newton‘s equations of motion

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**a priori probabilities**

moves Δx and Δy in square of linear size δ defines a priori probability pap(old→new) different choices possible generalized acceptance criterion:

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**canonical ensemble problem: compute expectation value**

direct evaluation of the integrals: does not work! simple sampling: sample random configurations samples mostly states in the tails of the Boltzmann distribution

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**idea of importance sampling**

sample configurations according to probability choose , then this can be realized by Markov chain sampling: random walk through regions in phase space where the Boltzmann factor is large

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**importance sampling choose transition probability such that**

to achieve this use detailed balance condition possible choice yields no information about the partition sum

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**Metropolis algorithm displacement move acceptance probability**

N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, JCP 21, 1087 (1953) old configuration displacement move trial configuration ΔE = E(trial) - E(old) acceptance probability yes no ΔE ≤ 0 ? new config. = trial config. r < exp[-ΔE/(kBT)] ? r uniform random number 0 < r < 1 no new config. = old config.

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**implementation of the Metropolis algorithm**

do icycl=1,ncycl old=int[ran(0,1)*N] ! select a particle at random call energy(x(old),en_old) ! energy of old conf. xn=x(old)+(ran(0,1)-0.5)*disp_x ! random displacement call 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 averages enddo

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**dynamic interpretation of the Metropolis precedure**

probability that at time t a configuration r (N) occurs during the Monte Carlo simulation rate equation (or master equation): equilibrium: detailed balance condition one possible solution:

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**remarks Monte Carlo measurements:**

first equilibration of the system (until Peq is reached), before measurement of physical properties random numbers: real random numbers are generated by a deterministic algorithm and thus they are never really random (be aware of correlation effects) a priori probability: can be used to get efficient Monte Carlo moves!

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**Monte Carlo at constant pressure**

system of volume V in contact with an ideal gas reservoir via a piston V0 - V acceptance probability for volume moves: V trial move:

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**grand-canonical Monte Carlo**

system of volume V in contact with an ideal gas reservoir with fugacity insertion: V0 - V V removal:

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**textbooks on the Monte Carlo simulation method**

D. P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical 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: From Algorithms to Applications (Academic Press, San Diego, 1996) K. Binder, G. Ciccotti (eds.), Monte Carlo and Molecular Dynamics of Condensed Matter Systems (Societa Italiana di Fisica, Bologna, 1996)

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**Computer Simulations of Colloids Monte Carlo II: Phase Behaviour of Colloid-Polymer Mixtures**

collaborations: Richard Vink, Andres De Virgiliis, Kurt Binder

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**depletion interactions in colloid-polymer mixtures**

polymers cannot move into depletion zones of colloids with their center of mass gain of free volume if there is overlap of the depletion zones of two colloids effective attraction between colloids (of entropic origin) → demixing transition possible into colloid-rich phase (liquid) and colloid-poor phase (gas)

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**Asakura-Oosawa (AO) model**

in the following AO model for Rp/Rc = 0.8: → exhibits demixing transition into a colloid-poor phase (gas) and a colloid-rich phase (liquid) → temperature not relevant; what is the corresponding variable here?

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**Outline phase diagram: AO model for RP/RC = 0.8: binodal**

critical point (FSS) interfacial tension interfacial width capillary waves

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**grandcanonical MC for the AO model**

volume V, polymer fugacity zP and colloid fugacity zC fixed → removal and insertion of particles problem 1: high free energy barrier between coexisting phases in the two-phase region (far away from the critical point) → use successive umbrella sampling problem 2: low acceptance rate for trial insertion of colloidal particle, high probability that it overlaps with a polymer particles → use cluster move

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**successive umbrella sampling**

problem: high free energy barrier between coexisting phases

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cluster move problem: low acceptance rate for trial insertion of a colloid → cluster move sphere of radius δ and volume Vδ around randomly selected point try to replace nr ≤ nP polymers (A) by a colloid (B): reverse move (C+D): R.L.C. Vink, J. H., JCP 121, 3253 (2004)

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**phase diagram critical point from finite size scaling**

→ 3D Ising universality class → estimate interfacial tension by γ = ΔF/(2A) (Binder 1982)

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finite size scaling cumulant ratio → critical value ηrP,cr≈0.766

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interfacial tension γ*≡(2RC)2γ as a function of the difference in the colloid packing fraction of the liquid (L) and the vapor (V) phase at coexistence → DFT (M. Schmidt et al.) yields accurate result

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**interfacial 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)

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capillary wave theory free energy cost of spatial interfacial fluctuations (long wavelength limit) result for mean square amplitude of the interfacial thickness combination with mean-field result by convolution approximation → determination of intrinsic width w0 by simulation not possible

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**is CWT valid? logarithmic divergence?**

lateral dimension Lx = Ly large enough?

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**finite size scaling II map on universal 3D Ising distribution**

account for field mixing → AO model belongs to 3D Ising universality class

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**conclusions discrepancies to mean-field results**

AO model belongs to 3D Ising universality class interfacial broadening by capillary waves much more on the AO model in R.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).

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**Computer Simulations of Colloids Modelling of hydrodynamic interactions**

collaboration: Apratim Chatterji (FZ Jülich)

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**simulation of colloids**

size 10 nm – 1 μm mesoscopic time scales solvent: atomistic time (~ 1 ps) and length scales (~ Å) simulation difficult due to different time and length scales → coarse graining of solvent‘s degrees of freedom

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**Langevin equation many collisions with solvent particles**

on typical time scale of colloid systematic friction force on colloid Brownian particles of mass M : fr,i uncorrelated random force with zero mean: fluctuation-dissipation theorem:

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**decay 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 momentum conservation

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**generalized Langevin equation**

idea: couple velocity of colloid to the local fluid velocity field via frictional force point particle: problems: determination of hydrodynamic velocity field from solution of Navier-Stokes equation rotational degrees of freedom → colloid-fluid coupling?

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**lattice Boltzmann method I**

: number of particles at a lattice node , at a time t, with a velocity discrete analogue of kinetic equation: Δi: collision operator velocity space from projection of 4D FCHC lattice onto 3D (isotropy) moments of ni :

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**lattice Boltzmann method II**

linearized collision operator: equilibrium distr.: recover linearized Navier-Stokes equations: thermal fluctuations: add noise terms to stress tensor

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**lattice Boltzmann algorithm III**

collision step: compute propagation step: compute

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**colloid fluid momentum exchange**

sphere represented by (66) uniformly distributed points on its surface each point exchanges momentum with surrounding fluid nodes

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**velocity correlations at short times**

give particle a kick and determine decay of its velocity blue curves: red curves: fluid velocity at the surface of the sphere

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Cv(t) and Cω(t) I: moment of inertia independent of R and ξ0

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**thermal fluctuations ok?**

linear response theory: velocity relaxation of kicked particle in a fluid at rest = velocity autocorrelation function of particle in a thermal bath

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charged colloids

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**colloids in electric field**

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