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RMC-5 Budapest, Sept 20 - 22, 2012
Some statistical mechanics of simplenon-equilibrium two-phase systems
Holger Maier∗, Manfred J. HampeFachgebiet Thermische VerfahrenstechnikTechnische Universitat Darmstadt
Philippe A. Bopp∗∗Department of ChemistryUniversite Bordeaux [email protected]
∗ now at Wacker Chemie, Burghausen / Cleveland TN∗∗ 2009/2010 on sabbatical leave atMax Planck-Institut fur Mathematik in den Naturwissenschaften, Leipzig,and Materials Chemistry, Angstrom Laboratory, Uppsala University,funded by the Wenner-Gren Stiftelserna Stockholm and the Max Planck-Institut
This file (pdf) can be downloaded: http://loriot.ism.u-bordeaux1.fr/or3.2/pabsem.html
RMC-5 Budapest, Sept 20 - 22, 2012
UPPSALA
DARMSTADT
BORDEAUX
LEIPZIG
RMC-5 Budapest, Sept 20 - 22, 2012
Starting point of the investigation
If an interface is subjected to a gradient, e.g. of temperature, motions will
arise. They are believed to be an important factor in the mass transport across
such interfaces, e.g. in liquid-liquid extraction in chemical engineering.
0 0.2 0.4 0.6 0.8 1
tem
pera
ture
mole fraction of B
A - B mixture
’Simple’ phase diagram
for an A-B mixture
CP : Critical Points
UCST : Upper Critical Solution TemperatureUCEP : Upper Critical End Point
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−rich phase −rich phase
flows
concentrationgradient
interfacial region WARM
effectSoret
COLD
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Marangoni Convection
– Any variation in temperature or composition results in local variationsof surface tension .... and thus induces convective motion.This effect is called the Marangoni convection.
– ... temperature induced convection called thermocapillary convection
– In terrestrial environments, Marangoni convection is usuallyovershadowed by buoyancy-driven flows,in reduced gravity ... (it) could become very important.
– ... Marangoni effects may be an important factor ... in thin fluid layers.
– Marangoni flows arise in applications like coating, electronic cooling,welding, wafer drying ......
from Denis Melnikov:”Development of numerical codes for the study of Marangoni convection”Thesis, Universite Libre de Bruxelles, 2004
RMC-5 Budapest, Sept 20 - 22, 2012
Aim of this work
To study the Marangoni convections not in terms of
macroscopic variables (surface tension, viscosity, ...)
but in terms of microscopic ones (i.e. intermolecular interactions)
Tool: Molecular Dynamics (MD) computer simulations
- a type of molecular simulations
which allows to study structural and dynamical properties
in and out of equilibrium
RMC-5 Budapest, Sept 20 - 22, 2012
The Essentials of MD Simulations
- Input: Interatomic and/or intermolecular interactions
Molecular Dynamics (MD) (numerical) Simulation
- Output: A sample of the classical trajectory of an ensemble ofN (= 1,000 - 100,000) such atoms or molecules in a periodic arrangementunder proper external conditions (e.g. pressure, temperature,temperature gradient, ...)(Depending on the conditions (equilibrium, non-equilibrium, ...), this could be a sample
of the micro-canonical, canonical, .... ensemble, or more complicated things)
- Analyze the trajectory by means of statistical mechanics→ e.g. internal energy ∆U , (self-)diffusion Ds, viscosity η,
X-ray or neutron scattering functions S(Q), S(Q,ω), ....
RMC-5 Budapest, Sept 20 - 22, 2012
Interactions: Spherical particles, Argon-like (i.e. extremely simplified)
Lennard-Jones (for computational ease)
Uij(rij) = 4εij ·(
(σijrij
)12 − (σijrij
)6)
for the A-A, B-B, and A-B interactions.
In most casesUA−A = UB−B 6= UA−B
to make the system symmetric, UA−B controls the miscibility.
(Rule of thumb:UA−B strong, deep compared to UA−A and UB−B → very miscibleUA−B weak, shallow compared to UA−A and UB−B → little or not miscible)
Masses: mA = mB = mAr = 40 amu
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- Equilibrium MD (EMD) (usual)
- Non-Equilibrium MD (NEMD):Non-stationaryStationary
Non-stationary: Perturb the system at some time t0 and watch relaxation(usually many times, and take averages)
Stationary: Perturb the system continuously (e.g. heat and cool) so thatits global state is independent of time:∂∂t = 0
We will use here stationary non-equilibrium MD simulations
RMC-5 Budapest, Sept 20 - 22, 2012
First: An MD Demixing ’Experiment’t = 0 ns t = 0.16 ns t = 0.36 ns
t = 0.6 ns t = 1.0 ns t = 1.36 ns
t = 2.4 ns t = 2.8 ns t = 3.0 ns
t = 5.0 ns ↑ interface ↑ ↑ interface↑
RMC-5 Budapest, Sept 20 - 22, 2012
loca
l den
sity
compare to molecular dimension
Schematic density profiles across liquid-liquid interface
coldwarm
Gibbs dividing surface
interfacial width?
A−rich phase B−rich phase
RMC-5 Budapest, Sept 20 - 22, 2012
Model liquid-liquid interfaces between partially miscible Lennard-Jonesparticles with Argon masses have been studied previously inMD simulations at equilibrium [1,2].
We focus here on such systems subject to atemperature gradient parallel to the interfacesand try to find (microscopic) Marangoni convection [3].( A flow near an interface from the warm to the cold region )
[1] A Molecular Dynamics Study of a Liquid-Liquid-Interface: Structure and Dynamics,J. B. Buhn, Ph. A. Bopp, and M. J. Hampe, Fluid Phase Equilibria, 224, 221 (2004)
[2] Structural and dynamical properties of liquid-liquid interfaces: A systematic moleculardynamics study, J. B. Buhn, Ph. A. Bopp, and M. J. Hampe, J. Mol. Liq. 125, 187 (2006)
[3] Uber die Ausbreitung eines Tropfens einer Flussigkeit auf der Oberflache einer anderen,
C. Marangoni, Annalen der Physik und Chemie 143, (1871)
RMC-5 Budapest, Sept 20 - 22, 2012
Previous: Model liquid-liquid interface at constant temperature
0
0.005
0.010
0.015
0.020100 K
116 K132 K
108 K126 K
138 K
-60 -40 -20 0 20 40 60
ρ [ A
-3 ]
z [ A ]
Density profilesof A- and B-type particlesat 6 different temperatures(only one half of the boxis depicted)
0
5
10
15
20
100 110 120 130 140
W [
A ]
T [ K ]
W = w0 (1 - T/Tc)
−ν Analytical expression fitted toW-values from two different types of fitsto the Figure aboveTop: Particles of equal sizeBottom: Particle size ration 1.2
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Schematically:System with a temperature gradient parallel to the interfaces:
T
"HEATING"
.."COOLING"
Q
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Simulations
We perform equilibrium (NV E) or (NV T ) (for reference)and stationary non-equilibrium (NV T1T2)Molecular Dynamics (MD) computer simulationson systems like the one sketchedand on one-phase systems of similar compositions.
Systems are usually simulated for some 75 nanoseconds, which needs about70 CPU-hours per run for the smallest system and up to 1000 for the largest.
The (many!) analyses of the trajectories may also be quite CPU-time con-suming (0.1 - 0.5 the time for the simulation)
RMC-5 Budapest, Sept 20 - 22, 2012 Model system with 3-D Periodic Boundary Conditions
periodicboundaryconditions
cold hot
yellow rich phase
green rich phase
interface
usually ≈ 5000 - 10000 particles (and up to 10 times more)(NV T1T2)-ensemble with
< T >≈ 0.8 - 0.9 Tc , ∆T = T2 − T1 = 10− 40 K tsimulation ≈ 75 ns
red: Thermostat < T > +0.5 ·∆T blue: Thermostat < T > −0.5 ·∆TParticles here yellow/green instead of red/blue above.
RMC-5 Budapest, Sept 20 - 22, 2012
Computing the average particle velocities in small volume elementsresults in a field∗ typically like this
-40
-30
-20
-10
0
10
20
30
40
-50 -40 -30 -20 -10 0 10 20 30 40 50
y [1
0-10 m
]
z [10-10m]
N-Ar1Ar2-120-10
warm
cold
warm
∗Projections of locally averaged particle velocities onto the laboratory y − z plane
RMC-5 Budapest, Sept 20 - 22, 2012
The analysis is often difficult because of poor statistics(even with very long simulation runs)
locally averaged particle velocity = < v >x,y,z
= <1
Ni(x, y, z)
Ni(x,y,z)∑
j=1
~vj(x, y, z) >ti,ti+δt
(x, y, z) = a small volume around x, y, z
in general, however, < v >x,y,z ≯ vi(t) (for a particle i)
RMC-5 Budapest, Sept 20 - 22, 2012
Things that can be studied:
– Technical parameters:
Influence of the ’thermostats’, the cut-off radii of interactions, simulation
box size, ...
– Physical parameters:Influence of the temperature gradient (∆T/(Ly/2)): Linear response?Influence of the interactions (ε and σ parameters): MiscibilityInfluence of the particle masses (mA,mB)Influence of the external conditions (e.g. pressure)etc. etc. etc.
RMC-5 Budapest, Sept 20 - 22, 2012
We discuss here only:
(i) Influence of the temperature gradient
(ii) Influence of the LJ cross parameters, i.e. (cum grano salis) miscibility
(iii) Influence of the box size
RMC-5 Budapest, Sept 20 - 22, 2012
(i) Influence of the temperature gradient
100 - 140 K (120 ± 20) 120 - 160 K (140 ± 20)
0.0
2.0
4.0
-5.0 -2.5 0.0 2.5 5.0
|y| [
nm]
z [nm]
N-Ar5Ar6-0.6-1.0-3346-3524-4.74x8.00x9.40-100-140 sim544 sim566 sim595
MeansLocYZComVelYZ.eps
1 m/s
0.0
2.0
4.0
-5.0 -2.5 0.0 2.5 5.0
|y| [
nm]
z [nm]
N-Ar5Ar6-0.6-1.0-3346-3524-4.74x8.00x9.40-120-160 sim806 sim820 sim835
MeansLocYZComVelYZ.eps
1 m/s
110 - 130 K (120 ± 10) 115 - 125 K (120 ± 5)
0.0
2.0
4.0
-5.0 -2.5 0.0 2.5 5.0
|y| [
nm]
z [nm]
N-Ar5Ar6-0.6-1.0-3346-3524-4.74x8.00x9.40-110-130 sim545 sim567 sim596
MeansLocYZComVelYZ.eps
1 m/s
0.0
2.0
4.0
-5.0 -2.5 0.0 2.5 5.0
|y| [
nm]
z [nm]
N-Ar5Ar6-0.6-1.0-3346-3524-4.74x8.00x9.40-115-125 sim546 sim568 sim597
MeansLocYZComVelYZ.eps
1 m/s
Compare with top half of previous Figure
hot
cold
hot
cold
RMC-5 Budapest, Sept 20 - 22, 2012
Conclusions (i):
→ Direction of average velocities always hot to cold inthe vicinity of the interface
→ Velocities increase with increasing temperature gradient
→ Average velocities decrease with increasing average temperature(i.e. decrease with decreasing distance from the critical linesin our cases Tc ≈ 160− 180 K)
=⇒ This is as expected for a macroscopic Marangoni pattern(under ’moderate’ conditions)
RMC-5 Budapest, Sept 20 - 22, 2012
(ii) Influence of the cross interaction (→ miscibility, interfacial width)
100 - 140 K (120 ± 20)
0.0
2.0
4.0
-5.0 -2.5 0.0 2.5 5.0
|y| [
nm]
z [nm]
N-Ar5Ar6-0.6-1.0-3346-3524-4.74x8.00x9.40-100-140 sim544 sim566 sim595
MeansLocYZComVelYZ.eps
1 m/s
0.0
2.0
4.0
-5.0 -2.5 0.0 2.5 5.0
|y| [
nm]
z [nm]
N-Ar5Ar6-0.5-1.0-3346-3524-4.74x8.00x9.40-100-140 sim1212 sim1218 sim1238
MeansLocYZComVelYZ.eps
1 m/s
Same system as above
Less miscible: Thinner interfaces
hot
cold
hot
cold
RMC-5 Budapest, Sept 20 - 22, 2012
Conclusions (ii):
→ At a given temperature thinner interfaces (less miscible liquids)favor the convection vortices(This was tested only in a few cases)
RMC-5 Budapest, Sept 20 - 22, 2012
(iii) Influence of the box sizeWhen the size of the simulation box is increased, are the vortices becominglocalized near the interfaces (schematically case A)or do they move out into the bulk (schematically case B)?
or
Interface
hot
cold
A
B
RMC-5 Budapest, Sept 20 - 22, 2012
100 - 140 K (120 ± 20)
reference system
as above
0.0
2.0
4.0
-5.0 -2.5 0.0 2.5 5.0
|y| [
nm]
z [nm]
N-Ar5Ar6-0.6-1.0-3346-3524-4.74x8.00x9.40-100-140 sim544 sim566 sim595
MeansLocYZComVelYZ.eps
1 m/s
0.0
2.0
4.0
-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0
|y| [
nm]
z [nm]
N-Ar5Ar6-0.6-1.0-13384-14096-4.74x8.00x36.70-100-140 sim679 sim732 sim834
MeansLocYZComVelYZ.eps
1 m/s
RMC-5 Budapest, Sept 20 - 22, 2012
0.0
2.0
4.0
6.0
8.0
-20.0 -15.0 -10.0 -5.0 0.0 5.0 10.0 15.0 20.0
|y| [
nm]
z [nm]
N-Ar5Ar6-0.6-1.0-26768-28192-4.74x16.00x36.70-100-140 sim746 sim846 sim1179 sim1241 sim1259 sim 1302 sim1402 sim1572
MeansLocYZComVelYZ.eps
1 m/s
Same scale as previous graphsBiggest system simulated, 54960 particles
RMC-5 Budapest, Sept 20 - 22, 2012
Conclusions (iii):
→ The centers of the vortices remain in the vicinity of the interfaces and donot move out into the bulk when the box size is increased (case A).Since the vortices remain associated with the interfaces it is really an interfacialphenomenon, not a bulk phenomenon(which we also know from the fact that we never saw anything like thatin one-phase simulations under similar conditions (Soret systems))
RMC-5 Budapest, Sept 20 - 22, 2012
Further Analyses (I):Stress-tensor S (according to H.Heinz, W.Paul, K.Binder, Phys.Rev. E72,066704, 2005)
normal stresses Szz
hot
cold-5 -4 -3 -2 -1 0 1 2 3 4 5
z [nm]
0
1
2
3
4
y [n
m]
-48
-46
-44
-42
-40
-38
[MP
a]
normal stresses Syy
hot
cold-5 -4 -3 -2 -1 0 1 2 3 4 5
z [nm]
0
1
2
3
4
-48
-46
-44
-42
-40
-38
[MP
a]
Tensor components Sαβ = Iαβ +Kαβ ↑ interfaces ↑(K: kinetic, I: interactions)
-1
-0.5
0
0.5
1
-4 -2 0 2 4
Syz
[MP
a]
z [nm]
Shear stresses Syz [10nm - 30nm]
IyzKyzSyz
RMC-5 Budapest, Sept 20 - 22, 2012
Further Analyses (II):Relation to Soret system(concentration gradient resulting from a temperature gradient in a mixture)
-2.0
-1.0
0.0
1.0
2.0
-5.0 -2.5 0.0 2.5 5.0
∂(µ(
ρ))/∂
y [n
m-4
]
z [nm]
density y-gradients (arb. units) and definition of domains
grad ρAr6z=10z=11z=12z=13z=14z=15
grad ρAr5
ρAr5+Ar6 (different scale)
RMC-5 Budapest, Sept 20 - 22, 2012
0
0.5
1
1.5
0.0 2.0 4.0
ρ Min
[nm
-3]
y [nm]
Local density of MINORITY component in the mixture
cold hot
centerz=10z=11z=12z=13z=14z=15
Soret systemEquil. mixturesat local temperature
RMC-5 Budapest, Sept 20 - 22, 2012
16
17
18
19
20
21
0.0 2.0 4.0
ρ Maj
[nm
-3]
y [nm]
Local density of MAJORITY component in the mixture
cold hot
centerz=10z=11z=12z=13z=14z=15
Soret systemEquil. mixturesat local temperature
RMC-5 Budapest, Sept 20 - 22, 2012
-0.06
-0.04
-0.02
0.00
0.0 10.0 20.0 30.0 40.0
ST
[-]
Lz [nm]
Soret coefficient away from interfaces for different system z-dimensions
NE Int. Sys.corr. Soret Sys.
RMC-5 Budapest, Sept 20 - 22, 2012
Take Home Message
- The convection is always directed from hot to cold in thevicinity of the interfaces
- The convective patterns are localized close to the interfaces
- Even with very high temperature gradients, the systemsrespond linearly
- Molecular transport mechanisms dominate convective ones inall non-equilibrium simulations
- Normal thermal diffusion (Soret effect) is observed away fromthe interfaces
- Local ’phase equilibria’ are seen along the interfaces
=⇒ Thermocapillary effect demonstrated at the microscopic scaleMarangoni Convection at the molecular level
RMC-5 Budapest, Sept 20 - 22, 2012
Acknowledgments
A large part of this work was carried out during PAB’s sabbatical leave fromUniversite Bordeaux 1 in the academic year 2009/2010.
He thanks Dr. Maxim Fedorov and Prof. Kersti Hermansson and their cowork-ers for their hospitality in Leipzig and Uppsala. Financial support by theMax Planck-Institut fur Mathematik in den Naturwissenschaften (MPI-MISLeipzig) and the Wenner Gren Foundation (Stockholm) is also gratefully ac-knowledged.