Monte-Carlo simulations of the structure of complex liquids with various interaction potentials Aljaž Godec Advisers: prof. dr. Janko Jamnik and doc. dr. Franci Merzel National Institute of Chemistry
Dec 26, 2015
Monte-Carlo simulations of the structure of
complex liquids with various interaction
potentialsAljaž
Godec
Advisers: prof. dr. Janko Jamnik and doc. dr. Franci Merzel
National Institute of Chemistry
Contents
1. Introduction
2. Statistical mechanics of complex liquids
3. Spherical multipole expansion of the electrostatic interaction energy
4. Monte-Carlo simulations of ensembles of anisotropic particles
5. How to present the results of MC simulations?
6. Conclusions and considerations for future work
Introduction
What are complex liquids?
simple liquid
anisotropic particles, COMPLEX POTENTIALS
hard spheres, Lennard-Jones
particles-
SIMPLE ISOTROPIC
POTENTIALS
Importance of complex liquids?
≈ρvapour
ρbulk
complex liquid
ΔF= ΔU-TΔS
ΔU = 0 ΔU > 0Hydrophobic interactions
Introduction
≈ρvapour
ρbulk
S
hard sphere in LJ fluid
S
Angular correlations completely ignored!!
Statistical mechanics of complex liquids
Assumption: separable Hamiltonian
(intermolecular interactions have no effect on the quantum states)H=Hclass+Hquant
two sets of independent quantum states (i.e. eigenstates can be taken as a product)
H n E n
cl qun n n with energy
En=Encl+En
qu
The partition function factorisesQ = Q cl Q qu
qu qun i
i
E and
Q ( exp( / ))qu Nqu i
i
kT individual molecular energy
Consequence of the above assumption: the contributions of quantum coordinates to physical properties are independent
of density
classical: centre of mass and the external rotational degrees of freedom
quantum mechanical vibrational and internal rotational degrees of freedom
Statistical mechanics of complex liquids
Probability density for the classical states
The classical Hamiltonian can be split into kinetic and potential energy
( ) exp( ( )) /N N N N N N N NP p H p Z r p r p
d d d d exp( ( ))N N N N N N N NZ p H p r p r p
H=Kt+Kr+U(rNωN)
2
1
/ 2N
t ii
K p m
2
1 , ,
/ 2N
r ii x y z
K J I
0 0
0 0
0 0
xx
yy
zz
I
I
I
I
Iα
( ) ( ) ( ) ( )N N N N N N N NP p P P p P r p p r
In Monte-Carlo calculations we need only the configurational probability density, but
we introduce a new distribution P'(rNpNωNJN)
( ) ( ) NN N N N N N N NP P p Jac r p J r p
1 2
( )...
( )
NN
NN
pJac Jac Jac Jac
J
( )f pJ
Statistical mechanics of complex liquids
new probability density
1
( )sin
( )x y z
p p pJac Jac
J J J
d d d' d ( ) 1N N N N N N N NP r p J r p J
( ) ( ) NN N N N N N N NP P p Jac r p J r p
1( ) ( )(sin ...sin )N N N N N N N NNP P p r p J r p
11( ) '( )(sin ...sin )N N N N N N N N
NP P r p J r p J
it is convinient to introduce a new distribution P(rNpNωNJN)
d d d d ( ) 1N N N N N N N Np P r p r p J
( ) exp( ( )) /N N N N N N N NP H Z r p J r p J
d d d d exp( ( ))N N N N N N N NZ H r p J r p J
We can write
Statistical mechanics of complex liquids
We can now directly factorize( ) ( ) ( ) ( )N N N N N N N NP P P P r p J p J r
t r cZ Z Z Z2
1( ) exp( / 2 ) /
NN
i ti
P p m Z
p
2
1 , ,( ) exp( / 2 ) /
NN
i ri x y z
P J I Z
J
( ) exp( ( )) /N N N NcP U Z r r
2
1
d exp( / 2 )N
Nt i
i
Z p m
p2
1 , ,
d exp( / 2 )N
Nr i
i x y z
Z J I
J
d d exp( ( ))N N N NcZ U r r
1 2( ) ( ) ( )... ( )NNP P P Pp p p p
1 2( ) ( ) ( )... ( )NNP P P PJ J J J
the ps and Js of different molecules are uncorrelated
furthermore
1 1 1 1( ) ( ) ( )... ( )x y zP P p P p P pp
1 1 1 1( ) ( ) ( )... ( )x y zP P J P J P JJ
N N3 3
1Q d d exp( ( ))
! !N N C
N N N N N Nt r t r
ZU
N N
r r
thus we can directly integrateΛt=h/(2πmkT)1/2
Λr=(h/(8π2IxkT)1/2)×
(h/(8π2IykT)1/2)(h/(8π2IzkT)1/2)
Ω=8π2 (4π)
Spherical multipole expansion of the electrostatic interaction energy
electrostatic interaction
1 2
1 2 1 1 2 2
1 2 1 2
1 2 1 212
1( ) ( ) ( ) ( )*m m m
l l l l m l m lml l l m m m
A r r r Y Y Yr
12 6
1 2 12 1212 12
1 2 12 12 12
4q q
Ur r r
a molceule= a distribution of charges (placed in the atomic centres);Atoms have finite sizes and also interact with polarization interactions
electrostatic interaction between two molceules= interaction between two
charge distributionsspherical harmonic expansion of r12
-1=|r+r2-r1|-1
Pair potential energy:
dispersion polarisationexchange repulsion(finite size of atom)
z
y
x
r·
·q1
q2
r1
r2
14( / ) ( ) ( )*
2 1l li
i i lm i lmi i l mi
qq r r Y Y
l
r r
potential of a charge distribution:
Spherical multipole expansion of the electrostatic interaction energy
mth component of the spherical multipole moment of order l :
Q ( )llm i i lm i
i
q r Y
1 2 1 1 2 2
1 2 1 2
112 1 2 1 2/ ( ; )Q Q ( )*l
l l l m l m lml l m m m
u A r C l l l m m m Y
interaction between two charge distributions= ∑(interactions of components of multipole moments of charge
distributions)
z
y
x
r
Introduction of body-fixed coordinate frame:
x’
z’ y’
z
y
x
x’
z’ y’
x’
z’
y’
Ω
' '( ') ( ) ( )lmlm mm lmY D Y
z’’
y’’
x’’
x’
z’
Spherical multipole expansion of the electrostatic interaction energy
z
y
x
y’r·
·q1
q2
r1
r2
xyz: space-fixedx’y’z’ and x’’y’’z’’: body-fixed
Q ( )*Qllm mk lk
n
D
1 2
1 2 1 1 2 2 1 1 2 2
1 2 1 2 1 2
112 1 2 1 2 1 2Q Q / ( ; ) ( )* ( )* ( )*l ll
l l l k l k m k m k lml l k k m m m
u A r C l l l m m m D D Y
calculated only once
What is gained?example: molecule consisting of four charges
17 terms /pair
5 (10) terms /pair
Spherical multipole expansion
TIP5P water model
Relation between multipole moments in the space-fixed and body-fixed
coordinate frames:
1 2 1 1 2 2
1 2 1 2
112 1 2 1 2/ ( ; )Q Q ( )*l
l l l m l m lml l m m m
u A r C l l l m m m Y
12 6
12 12 1 212 12
12 12
4i j ij
q qU
r r r
Monte-Carlo simulations of ensembles of anisotropic particles
What do we do in a MC calculation?2
1
( )dx
x
F f x x2
1
( ( ) / ( )) ( )dx
x
F f x P x P x xP(x)... probability density
function
( )
( ) trials
fF
P
Monte-Carlo: perform a number of trials τ: in each trial choose a random number ζ from P(x) in the interval
(x1,x2)
How to choose P in a way, which allows the function evaluation to be concentrated in the region of space that makes importatnt
contributions to the integral?
Construction of P(x) by Metropolis algorithmgenerates a Markov chain of
states1. outcome of each trial depends only upon the preceding trial
2. each trial belongs to a finite set of possible outcomes
exp( ) / cP U Z
Monte-Carlo simulations of ensembles of anisotropic particles
a state of the system m is characterized by positions and orientations of all moleculesprobability of moving from m to n = πmn
N possible states πmn constitute a N×N matrix, π
each row of π sums to 1
probability that the system is in a particular state is given by the probability vector ρ=(ρ1, ρ2, ρ3,..., ρm, ρn,..., ρN)
(2) (1)ρ ρ π
(3) (1)ρ ρ ππ
probability of the initial state = ρ(1)
lim lim (1) N
N ρ ρ π
equilibrium distribution
Microscopic reversibility (detailed balance):
mn m nm n
if ( )
( / ) if ( )mn n m n m
mnmn n m n m n m
U U
U U
Metropolis: 1mm mnm n
exp( ( ))nn m
m
U U
Monte-Carlo simulations of ensembles of anisotropic particles
How to accept trial moves?Metropolis:
- allways accept if Unew ≤ Uold
- if Unew>Uold choose a
random number ζ from the interval [0,1]
0
1
exp(-βΔU)
Unew-UoldΔUnm
ζ 1
ζ 2×
×allways accept
accept
reject
How to generate trial moves?
1 max
1 max
1 max
0,5
0,5
0,5
i
i
i
x x r
y y r
z z r
translation
rotation max' ( 0,5)
maxcos ' cos ( 0,5) cos
max' ( 0,5)
How many particles should be moved?sampling
efficiency:
2 / ir CPU time
~kT reasonable acceptance 2
2 4
1...
2
0 ( ) ( )
a a bi i ia a b
i i i
ab i
U UU r r r
r r r
f U r
O
U
2 / ( )i abr kT f U
1. N particles, one at a time:
CPU time ~ nN
2 /( ) /i abr CPU kT nf U
2. N particles in one move: 2 /( ) /i abr CPU kT Nnf U
CPU time ~ nN
2 / ( )i abr NkT f U
2 / ( )i abr kT f U sampling efficiency down
by a factor 1/N
Monte-Carlo simulations of ensembles of anisotropic particles
How to represent results (especially angular correlations)?
( ) ! ( ) !exp( ( )) /N N N N N Ncf N P N U Z r r r
we introduce a generic distribution function:
d d ( ) !N N N Nf N r r
! 1( ) d d exp( ( )) / d d ( )
( )! ( )!h h N h N h N N N h N h N N
c
Nf U Z f
N h N h
r r r r r
we further introduce a reduced generic distribution function:
ideal gas:
1 1 2 2( ) ( ) ( )... ( )h hh hf f f f r r r r
1 1 1 1 1 1 1 1 1 1d d ( ) ( ) d d ( )f f V N r r r r r
homogenous isotropic fluid:
1 1( ) /f r
How to present the results of MC simulations?
generally:
pair correlation function:
spherical harmonic expansion of the pair correlation function in a space fixed frame:
( ) ( ) /h h h h h hf g r r
2
12 1 2 3 32
2
1 2 1 22
( 1)( ) d ...d d ...d exp( ( )) /
( ) ( ) ( ) ( )
N NN N c
i j i ji j
N Ng U Z
r r r r
r r r rδ(ω)=δ(φ) δ(cosθ) δ(χ)
1 2
1 1 2 2
1 2 1 2 1 2
1 2 1 2 1 2 1 2 1 2 1 2( ) ( ; ; ) ( ; ) ( )* ( )* ( )*l lm n m n lm
l l l m m m n n
g g l l l n n r C l l l m m m D D Y r
1 1 2 2
1 2 1 2
1 2 1 2 1 2 1 2 1 2( ) ( ; ) ( ; ) ( ) ( ) ( )*l m l m lml l l m m m
g g l l l r C l l l m m m Y Y Y r
linear molecules:
intermolecular frame ω=0φ :
1 1 2 2( ) ( ) ( )... ( ) ( )h h h hh hf f f f g r r r r rangular correlation function,
g(rhωh) :
1 2
1 2
1 2 1 2 1 2( ; ) l m l ml l m
g g l l m r Y Y r
1/ 2
1 2 1 2 1 2
2 1( ; ) ( ; 0) ( ; )
4l
lg l l m r C l l l mm g l l l r
1 2
1 2
1 2 1 2 1 2( ; ) l m l ml l m
g g l l m r Y Y r
removing the m dependence:
reconstruction
EXAMPLE: dipoles in LJ spheres
r
φθ2
How to present the results of MC simulations?
Conclusions and considerations for the future
- we have briefly reviewed the statistical mechanics of complex liquids
- in order to reduce the number of interaction terms that have to be evaluated in each simulation step a spherical multipole expansion of the electrostatic interaction energy was made
- the basics of the Monte-Carlo method for simulation of ensembles of anisotropic particles were provided along with useful methods for representing the results of such simulations.
- finally results of MC simulations of dipoles embedded in Lennard-Jones spheres were briefly presented.
- employ such simulations to study biophysical processes, such as the hydrophobic effect
- possibility of including polarization effects basis for developing a polarizable water model for biomolecular simulations