tension by molecular simulation

Lehrstuhl für Thermodynamik
Prof. Dr.
-
Ing. H. Hasse
A
new
route
to
evaluate
the
curvature
dependence
of
the
surface
tension
by
molecular
simulation
St. Petersburg, 25
th
June 11
M. T. Horsch,
1
,
2
,
3
S. K. Miroshnichenko,
2
J. Vrabec,
2
A. K. Shchekin,
4
E. A. Müller,
3
G. Jackson,
3
and
H. Hasse
1
1
TU Kaiserslautern
2
Universität Paderborn
3
Imperial College
4
St. Petersburg State Univ.
Lehrstuhl für Thermodynamik
Prof. Dr.
-
Ing. H. Hasse
The formal
dividing
surface
25
th
June 11
2
Martin
Horsch
The
thermodynamic
laws
derived
by
J. W. Gibbs
are
valid in
general
.
Concrete
values
for
interfacial
properties
,
however
,
depend
on
the
way
the
internal
structure
of
a
phase
boundary
is
projected
to
two
dimensions
, i.e. on
the
choice
of
a
dividing
surface
.
„
take
some
point
[
…
]
and
imagine
a
geometrical
surface
to
pass
through
this
point
and
all
other
points
which
are
similarly
situated
[
…
]
called
the
dividing
surface
[
…
]
all
the
surfaces
which
can
be
formed
in
the
described
manner
are
evidently
parallel“
[J
.
W
.
Gibbs,
On
the
equilibrium
of
heterogeneous
substances
(
1876
/
77
),
S
.
380
]
.
Thermodynamic
properties
of
an
interface
are
determined
by
its
internal
3
-
dimensional
structure
Relations
of
axiomatic
thermodynamics
can
be
applied
to
a
formal
2
-
dimensional
surface
.
Lehrstuhl für Thermodynamik
Prof. Dr.
-
Ing. H. Hasse
The Laplace radius
R
L
of the surface of tension couples
dV’
and
dA
.
Thermodynamic
equilibrium
condition
3
Martin
Horsch
The Laplace equation:
Surface tension =
differential
free energy per surface area




dF dA dF dF
dA p dV S dT dN

 
 
  
 
   
Relation between
dV’
and
dA
:
L
2
R dA dV


Mechanical equilibrium:
L

2
p dV dA
p
R



 
 
25
th
June 11
Lehrstuhl für Thermodynamik
Prof. Dr.
-
Ing. H. Hasse
Virial
route
to
the
surface
tension
4
Martin
Horsch
10
2
10
3
10
4
surface tension in units of

-2
0,0
0,2
0,4
0,6
10
2
10
3
10
4
droplet size in particles
0.8

0.7

0.95


0
LJTS fluid

0

0
Vrabec et al.
2006, 2008
The normal pressure
p
N
(
z
) de
-
cays near the Laplace radius
R
L
.
Very significant decrease of
γ
.
Mechanical approach:
Bakker
-
Buff equation


out
2 2
L N T
in
( ) ( )
R dz z p z p z


 



out
3
2 3
N
in
2 ( )
p dp z z

 

Irving
-
Kirkwood pressure tensor




 

N
3
{,} ( )
( ) ( )
4
ij ij
i j z
ij
f
p z kT z
z r
S
s r
25
th
June 11
Lehrstuhl für Thermodynamik
Prof. Dr.
-
Ing. H. Hasse
Variational
route
to
the
surface
tension
5
Martin
Horsch
Non
-
linear
terms
represent
the
contribution
due
to
fluctuations
.
Small
influence
of
curvature
on
γ
.
Test area method:
(
Source
: Sampayo et al., 2010)
equimolar radius
R
e
in units of
ζ
deviation of
Δ
U
from mean value
probability
density
surface
tension
in
units
of
εζ
-
2
LJSTS fluid at
T
= 0.8
ε
Canonical partition function yields

 
   
 
 
ln exp
U
F T
T




     
2 3 4
,,
f U U U U
For infinitesimal deformations,
γ
=
Δ
F
/
Δ
A
with
25
th
June 11



 
2
e e
4.
A R R
Lehrstuhl für Thermodynamik
Prof. Dr.
-
Ing. H. Hasse
Analysis
of
spherical
density
profiles
6
Martin
Horsch
distance from the centre of mass in units of

0
2
4
6
8
density x

3
0.01
0.1
1
T
= 0.75

equimolar radius
R
e
capillarity radius
R
0
= 2

0
/

p
LJTS fluid
The approach of R. C. Tolman (1949) is based on the quantities:
•
Equimolar radius
R
e
(from the density profile)
•
Laplace radius
R
L
= 2
γ
/
Δ
p
of the surface of tension (requires
γ
)
•
Surface tension
γ
as a function of
1/
R
L
(which requires
γ
…)
Without previous knowledge of
γ
(
R
L
)
,
this set of variables is inconvenient.
Novel approach: Use
Δ
p
instead of
1/
R
L
, use
R
0
= 2
γ
0
/
Δ
p
instead of
R
L
.
25
th
June 11
Lehrstuhl für Thermodynamik
Prof. Dr.
-
Ing. H. Hasse
VLE at curved interfaces: Limiting cases
7
Martin
Horsch
pressure in units of
p
c
-4
-2
0
2
4
6
8
10
chemical potential in units of
T
c
3
4
5
6
7
8
0.6
T
c
0.8
T
c
RKS (

= 0)
spinodal
spinodal
planar
•
Droplet + metastable vapour
•
Bubble + metastable liquid

 
L
2
p
R
Planar limit: The curvature changes
its sign and the radius
R
L
diverges.
Spinodal limit: For the external
phase, metastability breaks down.
25
th
June 11
Lehrstuhl für Thermodynamik
Prof. Dr.
-
Ing. H. Hasse
Droplet
size
in
equilibrium
8
Martin
Horsch
Simulation results support the capillarity approximation for
n
> 1 000.
p''
in units of 10
-3



-3
40
50
1000
10000
100000
0.9

0.95

capillarity,
i.e.

=

0
simulation
droplet size
n
in particles
p''
in units of 10
-3



-3
4
8
16
20
100
1000
10000
0.7

0.65

0.8

ten Wolde-
Frenkel
criterion
LJTS
25
th
June 11
Lehrstuhl für Thermodynamik
Prof. Dr.
-
Ing. H. Hasse
The
planar
limit
9
Martin
Horsch
Transformation: “
R
0
rather than
R
L
”
Compare the radii
R
0
and
R
e
The Tolman length
δ
=
R
e
–
R
L
characterizes the dependence of
the surface tension on curvature.
… for the Tolman length:
capillarity radius
R
0
in units of

6
8
10
12
14
16
equimolar radius
R
e
in units of

6
8
10
12
14
16
T
= 0.85

T
= 0.75

T
= 0.65

LJTS

p
from

profiles

p

from IK
p
tensor
(Vrabec
et al.
)

= 1


= -1

25
th
June 11






p
Δ
d
γ
d
R
d
γ
γ
d
δ
p
Δ
p
Δ
0
L
0
0
0
lim
1
lim
2
1









e
R
L
R
R
R
R
R
δ








0
e
0
e
e
lim
lim
“excess equimolar radius”
η
=
R
e
–
R
0
Lehrstuhl für Thermodynamik
Prof. Dr.
-
Ing. H. Hasse
0
2
4
6
0
2
4
6
R
0
/

radius in unis of

R
e
R
L
0.0
0.2
0.4
0.6
-1
0
1

p
in units of


-3
deviation in units of


=
R
e

-

R
L

=
R
e

-

R
0
0.0
0.4
0.8
0.0
0.2
0.4
0.6
0.8

p
in units of


-3
surface tension in units of

/

2
T
= 0.7

LJTS
Tolman
scenario


= 0.51

(

=
1.2


p
)
capillarity
spinodal
limit
The
spinodal
limit
10
Martin
Horsch
25
th
June 11
Lehrstuhl für Thermodynamik
Prof. Dr.
-
Ing. H. Hasse
Conclusion
11
Martin
Horsch
•
The virial (Irving
-
Kirkwood) and variational (test area) approaches
lead to contradictory results for the curvature dependence of
γ
.
•
Without knowledge of the surface tension, it is impossible to
determine the Laplace radius
R
L
of the surface of tension.
•
In terms of the capillarity radius
R
0
(instead of
R
L
) and the pressure
difference
Δ
p
(instead of 1/
R
L
), Tolman’s approach can still be applied.
•
For the LJTS fluid, the planar limits of the Tolman length
δ
and the
excess equimolar radius
η
are smaller in magnitude than
ζ
.
•
This result is consistent with “Tolman’s scenario” (
R
→ 0 and
γ
→ 0)
for the spinodal limit, if
δ
does not depend on higher powers of
Δ
p
.
25
th
June 11