background image
Limitations
Limitations
:
:
particle
particle
-
-
exchange of rigid molecules, dense phases
exchange of rigid molecules, dense phases
Possible applications: phases equilibrium in complex situations
Review of thermodynamical ensembles
Gibbs ensemble simulation method
Possible applications:
Phase coexistence, adsorption,
swelling, partitioning
Applicable for dense phases and
large molecules
Grand Canonical Monte Carlo simulations of
adsorption of LJ nitrogen
(colored molecules) at
77 K in a model porous glass of 47% porosity and
3.23 nm mean pore size. (figure taken from Dr.
Lev Gelb's website at Washington
University in St. Louis:
http://www.chemistry.wustl.edu/~gelb/cpgvis.html)
Reference: L. D. Gelb and K. E. Gubbins,
Langmuir 14, 2097 (1998)
Snapshot of GCMC simulations of LJ nitrogen at 77
K in a model activated carbon (CS1000) at a bulk
pressure of P/P0=1. The rods and the spheres
represent C-C bonds and nitrogen molecules,
respectively. The model activated carbon was
obtained from constrained reverse Monte Carlo
simulations (figure taken from J. Pikunic et al.,
Langmuir
19, 8565 (2003))
T=290 K
T=290 K
Freezing of CCl
4
in a multi-walled carbon
nanotube with D = 5 nm
Snapshots of typical configurations of the adsorbed
phase. The carbon nanotube walls are not shown for
clarity
(taken from F. R. Hung et al., Mol. Phys., in press
(2004))
Applications and examples:
Applications and examples:
There are three kinds of ensembles:
Microcanonical
Microcanonical
Ensemble
Ensemble
No interaction
with reservoir (i.e. large
simulated system).
Canonical Ensemble
Canonical Ensemble
Can exchange heat and work with
reservoir
(i.e. all particles are confined
within a finite volume with periodic
boundary conditions).
Grand Canonical Ensemble
Grand Canonical Ensemble
Can exchange heat, work, and
particles with reservoir
(particles are
exchanged between the reservoir and
the simulated volume) : T, V and µ are
specified
"Computer Simulation of Phase Equilibria", Nigel Wilding (2001)
Grand Canonical Monte Carlo Method
A simulation based on generating a
sequence of equilibrium configurations
from probability distributions
equilibrium phase diagram of a given
model system
Statistical methods require the analysis of a large number of configurations that
are generally not available due to inherent the simulation limitations in size and
length. In order to ensure micro-reversibility and ergodicity assumptions (e.g.
time-average = spatial-average), averages are taken over predefined
thermodynamical ensembles.
Isolated system:
microcanonical ensemble
(N, V, E are common)
System in contact with
an energy and particle
reservoir: grand
canonical
ensemble
(µ, V, T are common)
System in contact with
an energy reservoir:
canonical
ensemble
(N, V, T are common)
Thermodynamical
Thermodynamical
ensembles
ensembles
Applications and examples:
Applications and examples:
Université de Reims
CHAMPAGNE ARDENNE
INRA UMR FARE
Fractionnement des Agro Ressources
et Emballages
CPCB Moulin de la Housse, BP 1039,
F 51687 REIMS cedex 2
Developed by Panagiotopoulos (
1987
), the basic idea in the Gibbs ensemble
method is to simulate phase coexistence properties by following the evolution
in phase space of a system composed of two distinct regions. The two regions
in the simulation system represent the two coexistence phases at a phase
transition. However, there are
no physical interfaces between the two regions
.
In general,
the two regions have different densities and compositions
, and
are
at thermodynamic equilibrium with each other (thermodynamic contact without
physical contact)
.
For Gibbs ensemble simulation, an NVT
condition refers to:
the combined volume of the two
regions and total number of molecules
are constant
the combined system is completely
surrounded by an infinite medium of
constant temperature.
For multiple phase coexistence,
necessary and sufficient conditions
are that all
the phases be at
thermal, mechanical
, and
chemical equilibrium
with each other.
Chemical equilibration:
Particle exchange
equilibrates chemical
potential.
Mechanical equilibration:
Volume exchange
equilibrates pressure.
Thermal equilibration allow
random displacement of
particles within each regions.
1
1
2
2
, ,
, ,
N V T
N V T
1
1
2
2
1, ,
1, ,
N
V T
N
V T
-
+
1
1
2
2
,
,
,
,
N V
V T
N V
V T
+
-
Gibbs ensemble - Algorithm
The equilibrium between A & B
is simulated via 2 boxes that
exchange energy and matter
without physical contact
A
B
Box 1
Box 2
Gibbs ensemble method
Gibbs ensemble method
Mesoscopic modelling
(II/II)
Particle exchange equilibrates
chemical potential
Volume exchange
equilibrates pressure
Incidentally, the coupled moves enforce mass and volume balance
Description without
A-B interface
Description with
A-B interface
, ,
, ,
, ,
b a
b a
b a
a
a
a
a
T P N
T V N
V E N
G
A
S
T
N
N
N
µ
=
=
= -
ln exp(
)
ex
kT
U
µ
= -
-
exp(
)
ln
ex
V
U
kT
V
µ
-
= -
NVT ensemble
NPT ensemble
Critical Properties of
Critical Properties of
Alkanes
Alkanes
(
(
Siepmann
Siepmann
et al.
et al.
, 1993)
, 1993)
L
Lennard
J
Jones fluid adsorbed on a random porous
solid composed of a matrix of non-overlapping
spheres (see inset)
Results were obtained by thermodynamic integration
of GCMC simulation results by Monson and
coworkers
et cetera
Concerted moves to detangle chains
Concerted moves to detangle chains
orientation trials
orientation trials
chain growth
chain growth
lattice approx.
lattice approx.
Conductor
Conductor
-
-
like Screening Model for Real Solvents (
like Screening Model for Real Solvents (
Klamt
Klamt
et al.
et al.
, 1995)
, 1995)
Gas
Gas
-
-
Liquid Transition in Pores:
Liquid Transition in Pores:
Capillary
Capillary
Condensation
Condensation
Calculations of
Calculations of
chemical potential
chemical potential
Panagiotopoulos, A. Z., "Direct Determination
of Phase Coexistence Properties of Fluids by
Monte Carlo Simulation in a New Ensemble ",
Mol. Phys., 61, 813 (1987)
Sorption isotherms
Sorption isotherms