Determination of the interfacial tension and the
pressure tensor for curved surfaces and
inhomogeneous non-spherical particles
Cátedra Manuel Sandoval Vallarta 2015
12 April 2016
Departamento de Física, CBI
Universidad Autónoma Metropolitana (UAM), Iztapalapa, México
George Jackson
Acknowledgements
·
Collaborators:
José G. Sampayo (Imperial)
Enrique de Miguel (Huelva)
Alexandr Malijevský (Prague)
Erich A. Müller (Imperial)
Paul Brumby (Tokyo)
Enrique de Miguel (Huelva)
Andrew J. Haslam (Imperial)
Rik Wensink (Orsay)
·
Funding:
Outline
•
Brief review of thermodynamics of interfaces
•
Molecular simulation of the interfacial tension
Mechanical approach - Pressure tensor (Kirkwood & Buff, 1949)
Thermodynamic approach - Free energy difference (Bennett, 1976)
Finite-size scaling approaches - Landau free energy (Binder, 1982)
Test area (TA) method - Free energy perturbation (Gloor et al., 2005)
•
Issues with the computation of the interfacial tension of liquid
drops and curved interfaces
Test area (TA) method (Sampayo et al., 2010, Lau et al., 2015)
•
The pressure tensor and interfacial properties of anisotropic
systems (rod-like particles near substrates)
Mechanical Definition of Pressure
Clausius (1822-1888)
•
Pressure Virial - Phil. Mag. 40, 122-127 (1870)
dr
r
r
g
r
r
u
r
kT
f
r
V
kT
P
i
j
i
ij
ij
2
2
(
)
(
)
3
2
3
1
Planar fluid interfaces
•
Mechanical (pressure tensor) route to the tension
Surface area
Pressure tensor
Tangential & normal components
L
L
L
z
L
/2
+
L
/2
z
y
x
L
L
L
z
L
/2
+
L
/2
z
y
x
z
y
x
2
L
A
r
r
r
r
P
zz
yy
xx
P
P
P
0
0
0
0
0
0
Planar fluid interfaces
•
Work of deforming the interface
Infinitesimal deformation of area
work done due to deformation in
x
direction
tangential contribution
normal contribution
z
y
x
z
y
x
work done due to deformation in
z
direction
3
0
1
1
,
1
,
0
1
V
V
L
L
A
L
L
L
A
L
L
A
A
A
x
z
Planar fluid interfaces
•
Mechanical definition of interfacial tension
Total work done
Interfacial tension – mechanical expression
water 25°C
P
z
P
z
z
A
P
P
z
z
A
z
z
P
A
z
z
P
A
W
W
W
T
T
N
L
L
N
L
L
T
N
T
d
)
(
d
)
(
)
(
d
)
(
d
)
(
2
/
2
/
2
/
2
/
P
P
z
z
A
W
T
V
d
)
(
bar
750
~
)
(
bar
03
.
0
~
bar
750
m
N
10
5
.
7
~
)
(
nm
1
~
m
mN
75
~
1
7
2
•
Mechanical expression of Kirkwood & Buff (1949)
Interfacial tension in terms of pressure virials
Kirwood-Buff relation is natural choice in MD.
Not straight forward for discontinuous potentials.
Planar fluid interfaces
i
j
i
ij
ij
ij
ij
ij
i
j
i
ij
ij
ij
ij
ij
ij
ij
i
j
i
ij
Molecular Simulation of Interfacial Tension
•
Mechanical route of Kirkwood & Buff (1949)
Interfacial tension
•
Kirwood-Buff relation is natural choice in MD simulation.
•
Accuracy can be poor (large values of pressure tensor).
•
Not straight forward for discontinous potentials.
Lee & Barker (1974), Lui (1974)
Chapela, Saville & Rowlinson (1975)
Rao & Levesque (1976)
i
j
i
ij
ij
ij
ij
ij
i
j
i
ij
Molecular Simulation of Interfacial Tension
•
Mechanical route of Kirkwood & Buff (1949)
Molecular Simulation of Interfacial Tension
•
Mechanical route of Kirkwood & Buff (1949)
•
Mechanical expression
Interfacial tension in terms of pressure virials
Buff (1955)
Thomson
et al.
(1984)
Drops of liquid
Schofield and Henderson (1982)
Non-unique nature of pressure tensor, tension
and surface of tension for drops
r
r
r
r
P
P
P
N
g
l
d
)
(
8
1
3
0
3
r
r
P
r
r
P
r
P
N
N
T
2
0
Molecular Simulation of Pressure Tensor
•
Mechanical route of Kirkwood & Buff (1949)
Average component of pressure (homogeneous fluid)
Virial equation for the pressure
Components of pressure tensor (inhomogeneous fluid)
i j i ij
ij
ij ij ij i j i
ij ij
dr
r
du
r
r
r
V
kT
f
r
V
kT
P
1
1
(
)
i j i ij ij ij
ij i j i
ij zz yy xx
dr
r
du
r
V
kT
V
kT
P
P
P
P
1
1
(
)
3
r
f
i j i
j i N ij ij ij ij N
i j i
j i T ij ij ij ij ij T
i j i
j i ij ij
z
z
z
dr
r
du
r
z
A
kT
z
z
P
z
z
z
dr
r
du
r
y
x
A
kT
z
z
P
f
r
V
kT
P
)
,
,
(
)
(
1
)
,
,
(
)
(
2
1
,
,
1
2 2 2
Fluid Interfaces
•
Thermodynamic definition of interfacial tension
Change in Helmholtz free energy
Total Helmholtz free energy
Surface free energy
SdT
PdV
dN
d
dF
i
i
i
VT
N
idA
dF
A
N
PV
F
i
i
i
i
s
i
s
N
A
F
when
0
A
N
F
F
F
F
i
s
i
i
Molecular Simulation of Interfacial Tension
•
Thermodynamic route - free energy difference
Helmholtz free energy from partition function
Free energy difference of two systems 0 and 1 (Bennett, 1976)
Interfacial tension
Miyazaki, Barker, Pound (1976)
Salomons & Mareschal(1991)
Moody & Attard (2004)
•
Requires simulation of at least two systems.
•
Care has to be taken with
N
s
contribution.
N
N
N
N
Z
kT
U
d
N
Q
Q
kT
F
3
3
!
/
exp
!
1
ln
r
1
1
0
1
0
1
0
1
0
1
/
exp(
/
exp(
ln
ln
ln
kT
U
W
kT
U
W
kT
Z
Z
kT
Q
Q
kT
F
F
F
A
F
A
F
F
F
s
Molecular Simulation of Interfacial Tension
•
Finite size scaling (Binder, 1982)
Grand partition function (Allen & Hansen 2002)
Restricted partition function for systems with N
*
Landau free energy (restricted grand potential)
Potoff & Panagiotopoulos (2004)
0
0
3
)
(
/
exp
/
exp
/
exp
!
1
N
N
N
N
N
kT
d
U
kT
N
kT
Q
N
N
r
)
(
)
/
exp(
)
(
)
(
)
/
exp(
)
(
*
*
*
0
*
N
kT
Q
N
N
N
N
kT
Q
N
N
N
(
)
ln
(
)
ln
(
)
ln
where
(
)
(
)
*
*
*
*
*
kT
N
kT
N
kT
N
N
Molecular Simulation of Interfacial Tension
•
Finite size scaling (Binder, 1982)
Finite size effects (Binder, 1986)
Interfacial tension
Hunter & Reinhardt (1995), Errington (2003)
Singh, Kofke, Errington (2003)
•
Useful for critical region
•
Progressively difficult at lower temperatures
exp(
/
)
exp(
/
)
lim
)
(
)
(
lim
max
min
kT
L
kT
N
N
x
s
L
L
L
2
2
2
2
ln
L
kTC
L
L
kTx
A
A
L
L
s
Molecular Simulation of Interfacial Tension
•
Thermodynamic route - free energy perturbation
Free energy difference of two systems 0 and 1
System 1 is a perturbation of 0
0
1
0
1
0
1
ln
ln
Z
Z
kT
Q
Q
kT
F
F
F
0
1
0
0
1
0
0
0
1
0
1
/
exp
/
exp
/
exp
/
exp
/
exp
/
exp
U
U
U
kT
U
Z
Z
kT
U
d
kT
U
kT
U
d
kT
U
d
kT
U
d
Z
Z
N
N
N
N
r
r
r
Molecular Simulation of Interfacial Tension
•
Thermodynamic route - free energy perturbation
Free energy difference (Zwanzig, 1954)
Longuet-Higgins (1951) – mixtures of conformal fluids
Barker (1951), Pople(1952) – polar fluids
0
0
1
ln
exp
/
ln
kT
U
kT
Z
Z
kT
F
Molecular simulation of free energy derivatives
•
Other free energy perturbation methods
Chemical potential – Test particle method
Widom (1963)
Pressure – Test volume method (isotropic)
Eppenga & Frenkel (1984)
Harismiadis
et al.
(1996); Vörtler & Smith (2000)
Pressure tensor – Test volume method (anisotropic)
de Miguel & Jackson (2006)
Brumby
et al.
(2010)
NN N N VT
kT
U
kT
V
N
kT
Q
Q
kT
N
F
/
exp
ln
/
1
ln
ln
lim
1
3
0 00
ln
lim
ln
1
/
exp
/
lim
0
1
kT
V
V
U
kT
Molecular Simulation of Interfacial Tension
•
Thermodynamic route - free energy perturbation
Test Area (TA) perturbation approach (Gloor et al., 2005)
Area of system 0
Area of system 1
A
A
A
1
0
1
0
A
0
,
0
,
0
,
z
y
x
L
L
L
A
L
L
A
L
L
L
z
z
x
y
x
1
1
0
,
1
,
0
,
1
,
1
,
1
0
V
Molecular Simulation of Interfacial Tension
•
Thermodynamic route - free energy perturbation
Test area (TA) free energy perturbation
Interfacial tension
forward or central difference
Gloor, Jackson, Blas, de Miguel, JCP, 123, 134703 (2005)
•
as accurate than mechanical route.
•
only requires simulation of a single system.
•
useful for complex atomistic potentials.
•
easy to extend to other types of interface/mixtures.
•
can be used for discontinous potentials (with care).
/
0
exp
ln
U
kT
kT
F
Molecular Simulation of Interfacial Tension
•
Vapour-liquid interface of LJ system
MD Mechanical
Alejandre 1999
TAMC
Gloor 2005
*
2
kT
Molecular Simulation of Interfacial Tension
•
Vapour-liquid interface of LJ system
MD mechanical
Alejandre 1999
TAMC
Gloor 2005
MC thermo
Mareschal 1991
MC FSS
Hunter 1995
MC FSS
Potoff 2000
*
2
kT
T
*
Molecular Simulation of Interfacial Tension
•
Vapour-liquid interface of SW system
TAMC
Gloor 2005
MC FSS
Singh 2003
*
2
kT
Molecular Simulation of Interfacial Tension
•
Vapour-liquid interface of SW system
TAMC
Gloor 2005
MC FSS
Singh 2003
*
MD mechanical
Alejandre 1999
*
2
kT
Molecular Simulation of Interfacial Tension
•
Vapour-liquid interface of SW system
TAMC
Gloor 2005
*
2
kT
Molecular Simulation of Interfacial Tension
•
Vapour-liquid/vapour-nematic interface of GB
*
2
kT
Molecular Simulation of Interfacial Tension
•
Vapour-liquid interface of 2CLJ
TAMC
MD mech.
kT
T
*
A drop of liquid
•
Rowlinson,
J. Phys.: Condens. Matter
,
6,
A1 (1994):
“What could apparently be simpler than a drop of liquid?
… [the] system throws up a set of mechanical,
thermodynamic and statistical mechanical problems that
are still matters of acute controversy”.
“mechanical problems”
What do we know about liquid drops?
·
Classic Text:
J. S. Rowlinson and B. Widom
Molecular Theory of Capillarity
(Clarendon Press, Oxford, 1982)
·
Reviews:
J. R. Henderson
“Statistical mechanics of spherical interfaces”
in
Fluid Interfacial Phenomena
edited by C. A. Croxton (Wiley, New York, 1986)
A. Malijevsky and G. Jackson
A perspective on the interfacial properties of
nanoscopic liquid drops
What do we know about liquid drops?
What do we know about liquid drops?
Young (1805)
Laplace (1806)
P
gP
lR
•
Mechanical expression for pressure difference
Laplace, Traité Mécanique Céleste, 10. Sur L´Action Capillaire (1806)
What do we know about liquid drops?
Gibbs (1876-1878)
•
Thermodynamic perspective
l
V
g
V
R
A
0
)
(
C
R
dA
dF
dR
C
dA
dN
dV
P
dV
P
dT
S
dF
S
S
NVT
l
l
g
g
What do we know about liquid drops?
§
What do we know about liquid drops?
§
§
died September 5, 1948
What do we know about liquid drops?
•
Tolman curvature correction
§
§
died September 5, 1948
Tolman (1949)
tension
of
surface
)
(
)
(
2
length
Tolman
interface
planar
of
tension
surface
1
2
1
)
(
*
S
S
S
S
S
g
R
R
R
P
P
P
R
R
l
What do we know about liquid drops?
What do we know about liquid drops?
Reliable estimates of the Tolman length
•
Penetrable sphere model – Hemingway et al. (1981)
•
Ising model/curvature expansion – Wortis and Fisher (1984)
•
Hard spheres against spherical wall/FMT – Bryk et al. (2003)
mean-field theory exactly
solvable at zero temperature
symmetrical density profile
mean-field theory appropriate
near critical point
always negative for physical
values of packing fractions
2
0
0
.
02
2
2
1
R
R
)
1
2
1
(
Conflicting findings for LJ liquid drops
R
R
)
1
2
1
(
Conflicting findings for LJ liquid drops
•
Square Gradient Theory (SGT)
Rayleigh (1892), van der Waals (1893), Cahn and Hilliard (1958)
R
R
)
1
2
1
(
r
s
r
r
r
r
;
6
2
2
2
homo.
s
c
s
d
kT
C
C
f
Conflicting findings for LJ liquid drops
•
Square Gradient Theory (SGT)
Falls et al. (1981)
Guermeur et al. (1985)
Blokhuis and Kuipers (2006)
SGT
0
0
0
R
R
)
1
2
1
(
Conflicting findings for LJ liquid drops
•
Square Gradient Theory (SGT)
•
Classical Density Functional Theory (DFT)
Evans (1979)
Falls et al. (1981)
Guermeur et al. (1985)
Blokhuis and Kuipers (2006)
SGT
0
0
0
R
R
)
1
2
1
Conflicting findings for LJ liquid drops
•
Square Gradient Theory (SGT)
•
Density Functional Theory (DFT)
Falls et al. (1981)
Guermeur et al. (1985)
Blokhuis and Kuipers (2006)
bubbles
drops
SGT
0
0
0
R
R
)
1
2
1
(
0
0
0
Conflicting findings for LJ liquid drops
•
Molecular Simulation
Thomson et al. (1984)
Nijmeijer et al. (1992)
?
0
0
?
0
VT
FSS
R
R
)
1
2
1
(
Conflicting findings for LJ liquid drops
Conflicting findings for LJ liquid drops
Molecular Simulation of Planar Fluid Interfaces
•
Thermodynamic route - free energy perturbation
Test Area (TA) perturbation approach (virtual change)
Area of system 0
Area of system 1
0
1
1
A
A
0
A
0
,
0
,
0
,
z
y
x
L
L
L
1
1
0
,
1
,
0
,
1
,
1
,
z
z
x
y
x
L
L
L
L
L
1
0
V
•
Thermodynamic route - TA method
Test area (TA) free energy perturbation
Interfacial tension
forward/backward or central difference
Molecular Simulation of Planar Fluid Interfaces
Gloor, Jackson, Blas, de Miguel,
JCP
, 123, 134703 (2005)
•
only requires simulation (MD or MC) of single system.
•
useful for complex atomistic potentials.
•
easy to extend to other types of interface/mixtures.
•
useful for discontinuous potentials (with care).
/
0
exp
ln
U
kT
kT
F
A
kT
U
kT
A
F
A
F
A
A
NVT
0
0
0
/
exp
ln
lim
MD mechanical
Alejandre 1999
TAMC
Gloor 2005
MC Bennett
Mareschal 1991
MC FSS
Hunter 1995
MC FSS
Potoff 2000
•
Thermodynamic route - TA method
Molecular Simulation of Planar Fluid Interfaces
Planar vapour-liquid interface of Lennard-Jones fluid
LJ-STS
*
*
T
Molecular Simulation of Drops
•
Thermodynamic route - free energy perturbation
Test Area (TA) perturbation
Reference
system 0
Perturbed
system 1
Perturbed
system 1’
z
y
x
z
y
x
0
A
0
A
1 1 1 1 Prolate e z e y x z y x R R R R R L L L L L
1 1 1 1 Oblate e z e y x z y x R R R R R L L L L L 2 04
R
eMolecular Simulation of Drops
•
Thermodynamic route - free energy perturbation
Test Area (TA) perturbation
Reference
system 0
Perturbed
system 1
Perturbed
system 1’
z
y
x
z
y
x
0
A
0
A
1 1 1 1 Prolate e z e y x z y x R R R R R L L L L L
1 1 1 1 Oblate e z e y x z y x R R R R R L L L L L 2 04
R
eMolecular Simulation of Drops
•
Thermodynamic route - free energy perturbation
Test Area (TA) perturbation
Reference
system 0
Perturbed
system 1
Perturbed
system 1’
z
y
x
z
y
x
0
A
0
A
1 1 1 1 Prolate e z e y x z y x R R R R R L L L L L
1 1 1 1 Oblate e z e y x z y x R R R R R L L L L L 2 04
R
e•
Thermodynamic route - TA method
Molecular simulation of drops
Liquid drops of LJ particles
TA
Sampayo et al. 2010
Thermo
Thomson et al. 1984
Thermo
El Bardouni et al. 2000
Mechanical
Vrabec et al. 2006
LJ-STS
VT
Schrader et al. 2009
FMT-MF
Malijevsky et al. 2010
•
Thermodynamic route - TA method
Analyse different terms of free energy perturbation:
Molecular simulation of interfacial tension
expand exponential
We now expand the logarithm
...
6
1
2
1
1
ln
1
...
!
3
1
!
2
1
1
ln
1
exp
ln
1
0
3
3
0
2
2
0
0
3
2
0
U
U
U
U
U
U
U
F
0
!
i
i
x
i
x
e
...
3
1
2
1
1
•
Thermodynamic route - TA method
Molecular simulation of interfacial tension
Collecting the terms we obtain the so-called
high-temperature expansion:
leading-order contribution
...
...
6
1
3
1
...
6
1
2
1
...
6
1
1
3 0 3 3 0 2 2 0 2 0 3 3 0 2 2 0 0 3 3 0 2 2 0
U
U
U
U
U
U
U
U
U
F
3
2
...
6
1
2
1
3 3 0 0 0 2 0 3 2 2 2 0 0 2 1 0
F
U
U
U
U
F
U
U
F
U
F
•
Thermodynamic route - TA method
Molecular simulation of interfacial tension
For infinitesimal deformations:
To leading order:
F
F
1
U
0
•
Thermodynamic route - TA method
Molecular simulation of interfacial tension
For pair-wise additive potentials:
The interfacial tension to leading order:
Lekner and Henderson (1977)
At leading order the thermodynamic route is equivalent
to the mechanical route of Kirkwood and Buff.
i
A
A
NVT
z
U
z
y
U
y
x
U
x
A
N
A
U
A
F
dA
dF
2
1
2
1
0
0
0
lim
lim
i
j
i
ij
ij
ij
ij
ij
ij
dr
r
du
r
z
y
x
A
dr
r
du
r
z
y
x
A
N
N
)
(
1
)
(
1
)
(
1
2
)
1
(
2
2
2
2
1
12
12
12
12
2
12
2
12
2
1
2
12
2
1
N
i
j
i
•
Thermodynamic route - TA method
Molecular simulation of interfacial tension
Planar vapour-liquid interface of LJ particles
LJ-STS
2
0
0
2
2
0
1
2
1
U
U
F
U
F
•
Thermodynamic route - TA method
Molecular simulation of interfacial tension
Planar vapour-liquid interface of LJ particles
Only leading order “mechanical” term contributes to
the interfacial tension. No appreciable fluctuations.
Thermodynamic and mechanical routes equivalent.
LJ-STS
2
0
0
2
2
0
1
2
1
U
U
F
U
F
•
Thermodynamic route - TA method
Molecular Simulation of Interfacial Tension
Liquid drops of LJ particles
LJ-STS
2
0
0
2
2
0
1
2
1
U
U
F
U
F
•
Thermodynamic route - TA method
Molecular Simulation of Interfacial Tension
Liquid drops of LJ particles
Second order “fluctuation” term now contributes
markedly to the interfacial tension.
Thermodynamic and mechanical routes NOT equivalent!
LJ-STS
2
0
0
2
2
0
1
2
1
U
U
F
U
F
•
Thermodynamic route - TA method
Molecular Simulation of Interfacial Tension
Liquid drops of TIP4P/2005 Water
•
Thermodynamic route - TA method
Molecular Simulation of Interfacial Tension
Liquid drops of TIP4P/2005 Water
Inconsistent findings for tension of drops of water
TA TIP4P/2005
Lau et al. (2014)
Mitosis TIP4P/2005
Joswiak et al. (2014)
Thermo Excision LJ-Dipolar
Samsonov et al. (2003)
TA DPD
Ghoufi & Malfreyt (2011)
TA TIP4P/2005
Conclusions
•
First-order mechanical (virial) contribution characterises the
planar interfacial tension.
•
Large energetic fluctuations (second order) contribute to the
interfacial tension of nanoscale drops in addition to
mechanical contribution.
•
Additional entropic contribution to the free energy of drops.
•
The first-order mechanical (virial) route to the tension of drops
is inappropriate.
•
The correct “virial” expression must contain second-order
terms of the type.
•
Implications for classical nucleation theory (CNT)
dx
dU
dx
dU
x
x
dx
dU
x
dx
dU
Simulation of confined non-spherical particles
Lekkerkerker and co-workers (1997, 2006)
Allen and co-workers (1999-2002)
Dijkstra, van Roij & Evans (2000)
L
Z
= 42
Confined hard spherocylinders, NVT, L/D = 10
L
Z
= 24
Confined hard spherocylinders, NVT, L/D = 10
L
Z
= 11
Confined hard spherocylinders, NVT, L/D = 10
•
L
Z
= 6
Confined hard spherocylinders, NVT, L/D = 10
Confined hard spherocylinders, NVT, L/D = 10
Confined hard spherocylinders, NVT, L/D = 10
Confined HSC system
Paul Brumby, PhD thesis, Imperial College London,2010
P
NNormal pressure
•
HSC
L/D
= 10
•
Hard-wall confinement
•
Surface wetting
•
Nematic ordering
•
Surface Adsorption
Tangential pressure
P
TP
Tinterfacial tension
γ
y
z
x
P
N
b
0
d
Lz
z
z
n
T
z
L P
P
Calculation of the components of the
pressure tensor and surface tension
Volume perturbations
Isotropic
For non-spherical particles both the
positive (expansive) and negative
(compressive) volume changes have
to be considered in evaluating the
pressure tensor
x
