Some statistical mechanics of simple non-equilibrium two ...nphys/rmc5/talks/Bopp_RMC.pdfSome...

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


UPPSALA

DARMSTADT

BORDEAUX

LEIPZIG


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


−rich phase −rich phase

flows

concentrationgradient

interfacial region WARM

effectSoret

COLD


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


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


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), ....


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


- 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


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↑


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


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)


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


Schematically:System with a temperature gradient parallel to the interfaces:

T

"HEATING"

.."COOLING"

Q


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.


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


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)


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.


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


(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]



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]



1 m/s

0.0

2.0

4.0

-5.0 -2.5 0.0 2.5 5.0

|y| [

nm]

z [nm]



1 m/s

Compare with top half of previous Figure

hot

cold

hot

cold


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)


(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]



1 m/s

0.0

2.0

4.0

-5.0 -2.5 0.0 2.5 5.0

|y| [

nm]

z [nm]



1 m/s

Same system as above

Less miscible: Thinner interfaces

hot

cold

hot

cold


Conclusions (ii):

→ At a given temperature thinner interfaces (less miscible liquids)favor the convection vortices(This was tested only in a few cases)


(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


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]



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]



1 m/s


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


1 m/s

Same scale as previous graphsBiggest system simulated, 54960 particles


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))


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


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)


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


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


-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.


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


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.

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