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

2. 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

3. Computer Simulations of Colloids Introduction

4. (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

5. (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

6. 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.

7. 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

8. tunable interactions I
hard spheres
soft spheres
volume fraction 
volume fraction, hair length and density

9. tunable interactions II
charged spheres
super-paramagnetic spheres
number density, salt concentration, charge
magnetic field

10. 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

11. 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!)

12. 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

13. (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

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

15. pair correlation function: hard spheres vs. soft spheres
soft sphere fluid:
hard sphere fluid:

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

17. 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

18. 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:

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

20. F(q,t) for a glassforming colloidal system
Eckert, Bartsch, Faraday Discuss. 123, 51 (2003)

21. ↓ ↓ 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

22. 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!

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

24. 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

25. 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

26. 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:

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

28. 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

29. 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?

30. 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

31. 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

32. 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

33. 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

34. simulation results I initial configuration with u=0:
after 105 time steps:

35. energies as a function of time
→ determine histogramm P(u) for the potential energy for different N

36. pressure

37. 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

38. 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

39. energies for the simulation at constant temperature

40. pair correlation function
very similar to 3d
no finite size effects
visible

41. 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

42. mean squared displacement
diffusion constant increases with increasing system size !?

43. 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

44. 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

45. effective diffusion constants
→ larger system sizes required to check theoretical prediction !

46. 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)

47. Computer Simulations of Colloids Molecular Dynamics II: Application to Systems Under Shear
with Jochen Zausch (Universität Mainz)

48. shear viscosity of glassforming liquids
1013 Poise →
10-3 Poise →
dramatic slowing down of dynamics in a relatively small temperature range

49. F(q,t) for a glassforming colloidal system
Eckert, Bartsch, Faraday Discuss. 123, 51 (2003)

50. 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

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

52. 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?

53. 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

54. 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

55. 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

56. 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)

57. self diffusion B particles slower than A particles weak temperature
dependence of D for
sheared systems if
along T = 0.14:
W ~

58. 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?

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

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

61. structure and stresses
how is the stress overshoot reflected in dynamic correlations?

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

63. EQ to SS: superdiffusive transient regime
y: gradient direction
z: free direction
measure effective exponent:

64. EQ to SS: Incoherent intermediate scattering function
transients can be described by
compressed exponential decay:
tw=0: β ≈ 1.8 → different for
Brownian dynamics?

65. EQ to SS: Newtonian vs. overdamped dynamics
change friction coefficient
in DPD forces:
→ same compressed exponential decay also for overdamped dynamics

66. 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

67. 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

68. 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

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

70. 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

71. → 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)

72. 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

73. 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

74. 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

75. 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

76. 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

77. detailed balance consider discrete system: configurations
a, b, c should be generated with equal
probability
detailed balance condition holds

78. 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

79. 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:

80. 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

81. 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

82. importance sampling choose transition probability such that
to achieve this use detailed balance condition
possible choice
yields no information about the partition sum

83. 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.

84. 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

85. 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:

86. 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!

87. 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:

88. grand-canonical Monte Carlo
system of volume V in contact with an ideal gas reservoir with
fugacity
insertion:
V0 - V V
removal:

89. 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)

90. Computer Simulations of Colloids Monte Carlo II: Phase Behaviour of Colloid-Polymer Mixtures
collaborations: Richard Vink, Andres De Virgiliis, Kurt Binder

91. 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)

92. 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?

93. Outline phase diagram: AO model for RP/RC = 0.8: binodal
critical point (FSS)
interfacial tension
interfacial width
capillary waves

94. 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

95. successive umbrella sampling
problem: high free energy barrier between coexisting phases

96. 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)

97. phase diagram critical point from finite size scaling
→ 3D Ising universality class
→ estimate interfacial tension by γ = ΔF/(2A) (Binder 1982)

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

99. 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

100. 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)

101. 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

102. is CWT valid? logarithmic divergence?
lateral dimension Lx = Ly large enough?

103. finite size scaling II map on universal 3D Ising distribution
account for field mixing
→ AO model belongs to 3D Ising
universality class

104. 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).

105. Computer Simulations of Colloids Modelling of hydrodynamic interactions
collaboration: Apratim Chatterji (FZ Jülich)

106. 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

107. 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:

108. 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

109. 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?

110. 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 :

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

112. lattice Boltzmann algorithm III
collision step: compute
propagation step: compute

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

114. 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

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

116. 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

117. charged colloids

118. colloids in electric field