INTERFACE DYNAMICS OF MODEL SYSTEMS:FROM LOCAL PROPERTIES TO BOUNDARY CONDITIONS

Guillaume Galliero, LFCR, Pau University, France

Collaborators :H. Hoang, H. Si Hadj Mohand, LFCR, Pau Univ.

D. Legendre, IMFT, Toulouse Univ.

16th May 2018Workshop on Modeling and Simulation of Interface DynamicsIMS - Singapore

MOTIVATIONS

2OIL AND GAS ISSUES

Quantifying accurately Multiphasic Fluid Dynamics in porous medium is crucial for the Petroleum Engineering

Various tools are available for that purpose but most of them are limited when employed to deal with “new” systems:

Complex resources : Sour/Acid Gases, Shales gas/oil … Complex conditions : Extreme HP/HT, Tight reservoirs …

Complex recovery methods: Smart Water, Polymer Flooding …

Data and more physically based models are required

But experiments are hard to achieve (HSE, HP …)

METHODOLOGY

4FROM NANO TO GIGA-SCALE: DIFFERENT TOOLS

ms

ms

ns

ps

fs

Å nm mm mm

Ab initioQuantum

Atomistic

Monte-CarloMolecular dynamics

Mesoscale

Lattice Gas, LBDPD, DSMC, KMC …

Time

Length

Continuum

Finite Difference/VolumeFinite ElementDiscrete Element …

Coarse graining

The choice is guided by time/length scalesDirect or Indirect Coupling between scales/tools is possible

Reservoir Simulations …

Pore/Core-scale modeling

NanoPore/interfaces

Molecular screening

Conservation Eqs.

Boltzmann Eq.

Newton Eqs.

Schrödinger Eq.

Adapted from Gubbins & Moore, IECR (2010)

5OUR GROUP STRATEGY

Develop/use molecular simulations codes to study/quantifyfinely fluid behavior at interfaces at the nanoscale

Thermophysical propertiesTension, Diffusion …

Physical mechanismsTransport, Osmosis …

Direct/indirect upscalingBoundary conditions …

g

qRqA

INTERFACE DYNAMICS:ISSUES IN A SIMPLE SYSTEM

7INTERFACE DYNAMICS AT THE NANOSCALE

g

E.g.: A nanodroplet of a simple fluid moved by body forces on a perfectly flat and rigid surface

Viscous stress (Slip ?)

Fluid Inhomogeneities? (Local density/viscosity ?)

Uncompensated Young Tension (Dynamic microscopic angle?)

Viscous stress (Boundary Conditions)

5 nm

FROM LOCAL PROPERTIES …

g

Fluid Inhomogeneities? (Local density/viscosity ?)

LOCAL FLUID TRANSPORT PROPERTIES ININHOMOGENOUS FLUIDS

H. HOANG, G. PIJAUDIER-CABOT (LFCR) AND F. MONTEL (TOTAL) Hoang and Galliero, PRE (2012), JCP (2013), JPCM (2014), PRE (2015)

10FLUID INHOMOGENEITIES ISSUES

L*

20 30 40 50 60

h*

2

4

6

8

Liquid Gas

Solid

Density of a “oil” drop in a nanopore

Low High

3 n

m

At solid interfaces, fluids are strongly inhomogeneous

(SFA, DFT …) Spatialy variable viscosity ?

Couette like simulations of a simple fluid (LJ) in a nanoslit

x*

-4 -2 0 2 40

0.2

0.4

0.6

0.8

1

bulk

=0.291

bulk

=0.408

bulk

=0.519

bulk

=0.625

p* = 0.5p* = 0.75p* = 1.25p* = 2

Density profiles

5 nm

11A VARIABLE VISCOSITY ?

The local viscosity computed by MD cannot be deduced from the local thermodynamic variables alone (T and )

Local Viscosity profile Local Viscosity profile

p = 20 MPa p = 80 MPa

-- ----

-- -- ---- -- -- --

---- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --

x*

m

-4 -3 -2 -1 00

1

2

3

4

5

There are non local effects

MD simulationvdW model--

--

--

--

----

-- -- -- ----

-- -- --

--

-- -- -

- -- -- -- --

-- -- -- -- -- -

--- -- --

---- -

---

x*

m

-4 -3 -2 -1 00

0.2

0.4

0.6

0.8

P*=0.5 P*=2

The fluid viscosity is variable in space !

Hoang & Galliero, PRE (2012)

12INTRODUCING NON LOCAL EFFECTS

Using the microscopic formulation of the momentum flux, shear viscosity is decomposed in two parts: t c

Translation Collisions

To introduce non-local effects, c is assumed to be a function of the density averaged over a typical length

2

2

1

x x s ds

0

c

bulk

xx x with e.g.

This method can be more accurate using more evolved weight functions and a perturbation scheme

Similar to Bitsanis et al. J. Chem. Phys. 1987

Hoang & Galliero, JCP (2013), JPCM (2014)

13NON LOCAL VISCOSITY ?

Effective viscosity can multiplied by an order of magnitude relatively to the bulk

Non local viscosity is well modeled by the simplest DFT

Can be combined with DGT/DFT to introduce fluid inhomogeneities …

Local Viscosity profiles Local Viscosity profiles

p = 20 MPa p = 80 MPa

----

--

--

----

-- -- -- ----

-- -- --

--

-- -- -

- -- -- -- --

-- -- -- -- -- -

--- -- --

---- -

---

x*

m

-4 -3 -2 -1 00

0.2

0.4

0.6

0.8

-- ----

-- -- ---- -- -- --

---- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --

x*

m

-4 -3 -2 -1 00

1

2

3

4

5

MD simulationvdW modelNon-local model

P*=0.5 P*=2

Hoang & Galliero, PRE (2012)

TO SHEAR AT FLUID-FLUID INTERFACES …

g

Viscous stress (Slip ?)

VELOCITY SLIP AT FLUID-FLUID INTERFACES

Galliero, PRE (2010), Bugel et al. Micro&Nanofluid. (2011)

M. BUGEL, J.P. CALTAGIRONE (I2M, BORDEAUX, FRANCE)

16FLUID-FLUID SHEAR STRESS ISSUES

The partial slip (Navier B.C.) at fluid-solid interfaces is well known, but what about Fluid-Fluid interfaces ?

Hybrid and MD simulations of Fluid-fluid interfaces under shear

Slip s known to occur at polymer-polymer interfaces

Liquid-Gas Liquid-Liquid

17HYBRID SIMULATIONS OF A DIPHASIC COUETTE FLOW

Bugel et al., M&N (2011)

Hybrid MD-CFD simulations

Coupling on primary variables (v, T)Schwartz alternating method

Liquid Gas

At the liquid-gas interface a partial slip seems to occur !

Discrete

Continuous

18VELOCITY JUMP AT A LIQUID-LIQUID INTERFACE

x* position

0 10 20 30 40 50

i*

0.0

0.2

0.4

0.6

0.8

vz*

-0.4

-0.2

0.0

0.2

0.4

The velocity profile exhibits a “jump” at the interface

Density

Velocity

Lx 30 nm

Phase I & II have the same properties

but are weakly miscible (e12=kije, with kij<1)

Galliero, PRE (2010)

19INTERFACIAL VISCOSITY IN SIMPLE FLUIDS ?

x* position

6 8 10 12 14 16 18 20

vz*

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

kij = 0.125

kij = 0.375

kij = 0.625

vz*

Velocity jump for various miscibility

0.0 0.5 1.0 1.5

(b

ulk-

int)/

bu

lk

0.0

0.1

0.2

0.3

0.4

0.5

Interfacial viscosity

Phase I

Phase II

The interfacial viscosity is lower than the one deduced from profile (DGT?)

In a water wet nanopore (10 nm) the theoretical oil relative permeability is roughly two times higher when the partial slip is taken into account

The partial slip increases with the interfacial tension

Interfacial self-diffusion is probably inversely proportionally increased …

YOUNG TENSION AT THE CONTACT LINE …

g

Uncompensated Young tension (Dynamic microscopic angle?)

CONTACT LINE DYNAMICS:(NANO?)DROPLETS SPECIFICITIES

H. HOANG (LFCR), S. DELAGE-SANTACREU (LMAP, PAU, FRANCE)

R. LEDESMA, D. LEGENDRE (IMFT, TOULOUSE, FRANCE) Hoang et al., in preparation

22YOUNG STRESS ISSUES

Young stress occurs at the Contact Line but is it valid at the nanoscale?

MD simulations of nanodroplets on a perfectly flat and rigid surface

Equilibrium Dynamic (external force)

g

Lennard-Jones 2D droplets on LJ CFC solid surface for 0.4p <q < 0.85p

10 nm

23

LV

SVSL q

cos ( )z q

x

zx

-20 -10 0 10 20-0.2

-0.1

0

0.1

0.2

Local shear stress (zx)

z=0

z=5

Liquid GasGas

Young tension ?

( ) ( , )dRInt

LInt

x

zxInt xz x z x

EQUILIBRIUM: YOUNG TENSION (1/2)

At the contact line the (integrated) shear stress is macroscopicallyequal to the Young tension i.e.

At equilibrium, shear stress is localized at the Liquid Vapor interface and decreases when z increases (consistent with Young tension)

Irving-Kirwood/Method of Plane

24

z

[In

t-c

os(

q)]

0 2 4 6 8

0

0.2

0.4

0.6

z

[In

t-c

os(

q)]

0 2 4 6 8

0

0.1

0.2

0.3

0.4

z

0 2 4 6 80

0.5

1

1.5

2C=0.55

C=0.25

Young stress =

shear stress

Shear tension at the contact line differs from cos(q)!

But Young stress = shear stress at liq/vap interface above zY some molecular sizes (zy scales with q-1)

Shear minus Young tensions (q 0.85p)

EQUILIBRIUM: YOUNG TENSION (2/2)

How shear tension compares to Young tension ?

Shear minus Young tensions (q 0.4p)

Young stress =

shear stress

( ) ( , )dRInt

LInt

x

zxInt xz x z x

25DYNAMICS: SHEAR/YOUNG TENSION (qEq = p/2)

Even out of equilibrium Shear tension=Young tension

above z 5 !

z

In

t

0 2 4 6 8-0.2

0

0.2

0.4

0.6

Equil. State

Non-Equil. State (Ad)

Non-Equil. State (Re)

Shear minus Young tension (Ca=0.2)

cos(q)

In

t

0.1 0.2 0.3 0.4

0.1

0.2

0.3

0.4

Ca

Ca

Shear tension at z=5 vs Ca

CL dynamics (at z>5) driven by [cos (qNE)-cos(qEq)] +viscous stress (Ca)

z

[In

t-c

os(

q)]

0 2 4 6 8-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Young tension is compared with shear tension at different z and different Ca …

The influence of Ca on shear tension is not negligible

qR qA

Consistent with Qian et al. PRE 2003, Ren and E, PF 2007

26A SPECIFIC MICROSCOPIC PICTURE ?

The problem becomes more complex when an adsorbed layer (macromolecules) occurs …

In not too wetting cases, classical laws are really robust !

Classical (homogeneous) continuum breaks down (q-1)

Uncompensated Young Stress [cos (qNE)-cos(qEq)]

Viscous stress(Navier BC)

How upscaling MD results when dealing with wetting case (thin films …) ?

TO BOUNDARY CONDITIONS

LIQUID-LIQUID NANOCOUETTE FLOW:WHICH BOUNDARY CONDITIONS ?

H. HOANG H. SI HADJ MOHAND (LFCR)

D. LEGENDRE (IMFT, TOULOUSE, FRANCE)

29BOUNDARY CONDITIONS ISSUES

There exists many CL boundary conditions options but whichone is the most appropriate when all scales are solved ?

Comparison between CFD and MD simulation on a liq-liq Couette flow

Seminal works of Robbins, Koplik, Qian, Ren ...

Phase I & II (Lennard-Jones fluids) have the same properties but are non miscible

30CFD MODELS (JADIM)

Tested dynamic angle models :

Static : qd = qY

Generalized Navier BC (GNBC*) : cosqY

- cosqd=

Ucl

gB

cl

cosqY

- cosqd=

Ucl

g mv

l3exp

s(1+ cosqY)

nkBT

æ

èç

ö

ø÷Molecular Kinetic Theory (MKT**) :

f=14.3 (em)1/2/3

Navier-Stokes (NS) Equations +

Volume Of Fluids (VOF, interface)

Reference case : MD of Qian et al., Phys. Fluids, 2003

Imposed Slip length + Dynamic angle

Bcl=3.02 (em)1/2/3

Used slip length model (Navier): UW

- V = l¶U

¶nW

Mesh size < Slip length

where l=1.63

qd

Equilibrium angle: qY = p/2

+V

-V

*Qian et al. PRE (2003), Ren and E, PF (2007), **Blake and co-workers

NS Simulation – GNBC model

with normalized friction Bcl=3.02

Unstable case : the MKT “friction” is too large

NS Simulation – MKT model

with normalized friction f=14.3

31CONTACT ANGLE FRICTION CONTROLS THE STABILITY

NS - Static model NS – GNBC model

Too much slip Slip is modified by friction

-V

V

(friction Bcl=3.02)

32VOF-NS VERSUS MD: PREDICTIONS

No Contact Line model seems to be fully predictive when combined with VOF-NS

GNBC: Bcl=3.01 => Bcl=1.7

MKT: f=14.3 => f=1.7

Static: l=1.63 => l=1.2

Velocity profiles

33VOF-NS VERSUS MD: ADJUSTED MODELS

If adjusted, all CL models yield reasonable results !

NS+VOF describes well this nanoflow when all scales are solved

OUTCOMES

35OUTCOMES

Shear viscosity can be variable in space at the Contact Lineo Non locals effects on viscosity occur close to a solid surfaceo Interfacial region may lead to apparent fluid-fluid slip

And for future works …

Include non local density/viscosity …. introduce upscaling …

Macroscopic laws are very robust in liquids if all scales are solvedo Uncompensated Young tension dominant at the CL (but at z>0)o For Liq-Liq Couette flow, CL models works well when fitted

Sheared Liq-Gas

THANK YOU FOR YOUR ATTENTION !

AcknowledgmentsPau University

CNRSAnd TOTAL

All colleagues collaborating

EXTRAS

Boundary conditionsConfinement effectsWettability …

Contact anglesDiffusion coefficientsSlip length ...

38

MOLECULAR SIMULATIONS: WHAT FOR ?

Molecular ModelForce Fields

Molecular SimulationsMC,MD

“Exact” emerging properties/fields

Test/Development of Theories

Pseudo-experimental Data

, q, Pc, k,, D, V …

39

MOLECULAR SIMULATIONS: HOW IT WORKS ?

No assumption concerning the physical phenomena that may emerge!

MOLECULAR MODEL

PHYSICAL PROPERTIES (X)

Molecular Dynamics (Temporal evolution)

Monte Carlo (Statistical evolution)

n ( 106-108)configurations

X a physical property 105-107 timesteps (t 1 ns)

1

MD

X X t dtt

1

1

n

iMCi

X Xn

A set of N interacting

particles

Average over configurations

Average over time

Ergodic Theorem MC MD

X X

40

MOLECULAR DYNAMICS: HOW IT WORKS ?

At a time step t each centre of forces (atom or molecule) is characterised by its position r(t), its velocity v(t) and its acceleration a(t)

Newton’s Equation + Effective interaction potentials (V)

dt

vdmF i

i

j

ij

ij

ij

ij rr

VF ˆ

i

iiii

m

tFttvttrttr

)(

2

1)()()(

2

New position at t + t (various integrator)

i

Fij

vi

i

(t)

Fijvi

(t + t)

j

Explicit Scheme, “easily” parallelised (domain decomposition)Computation time N2 (can be reduced to N logN)

Quasi-experimental process (data with error bars ...)

41

FORCE FIELDS: THE CORE OF THE PROBLEM

From hard-sphere representation to a full atom (ab initio) description, but no ideal model

12 6

4LJUr r

e

Dispersive/repulsive interactions: usually a Lennard-Jones pot.but Mie/Exp pot. are used

e

ULJ

r

Polar interactions: isotropic or point charges

Internal degrees of freedom: Bonding, Flexion …

Mixtures: Combining rules (Lorentz-Berthelot …)

The choice is guided by the goal (pseudo-experimental vs understanding)

Non-Bonded

42

HOW COMPUTING MACROSCOPIC QUANTITIES ?

N

i

N

ij

ijijc

c

EV

P1

23

1rF

2

1

x

x

TN dxxPxP

Field properties

Sound velocity, …

Dynamic properties

A flux is imposed (heat, momentum, …) and the response is measured (Non-Equilibrium)

All physical properties are “mesurable” in the ad hoc ensemble

The property is deduced from the fluctuations (Equilibrium)

Momentum/Energy

Exchange

Equilibrium/Thermodynamic properties

The strain, velocity, temperature, pressure, concentration … fields are deduced from averages (over time or ensemble)

43METHODOLOGY

Extensive Molecular Dynamics simulations on well controlled Vapor/Liquid/Solid systems

Variable wetting properties C=0.25 (q 0.85p )

Lennard-Jones interactions

Static and migration under external force configurations

12 6

4Ur r

C

e

Fluid-Fluid: C=1; Fluid-Solid: C =0.25-0.55

q

C=0.55 (q 0.4 p )

Sessile Lennard-Jones droplets on a CFC flat rigid LJ solid

Static nanodroplet Moving nanodroplet

Vapor

Liquid

44EQUILIBRIUM : CONTACT ANGLE

The Contact angle is determined by a circle fit (using Gibbs dividing surface and extrapolated at z=0.5)

C

qp

0.2 0.3 0.4 0.5 0.6

0.4

0.5

0.6

0.7

0.8

0.9

ef-s=Cef-f

Contact Angle at z=0.5

The contact angle is proportionalto fluid-solid interactions

Consistent with literature

The contact angle is weakly size dependent (here R 5nm)

x

z

-15 -10 -5 0 5 10 150

5

10

15

20

x

z

-20 -10 0 10 200

10

20

30

C=0.25 (q 0.4p ) C=0.55 (q 0.85p )

45EQUILIBRIUM: YOUNG-LAPLACE EQUATION (1/2)

As well known Pz= cst everywhere but not Px (cf. = (Pz-Px)dz)

Vapor pressure profile (q p/2)

PzPx

When a curved fluid-fluid interface is present a pressure (capillary)

difference between the fluids exist

PLiqPVap

LV

Liq VapP PR

Liquid pressure profile (q p/2)

zL

oca

lP

ress

ure

0 2 4 6 8 10

-0.5

0

0.5

1

1.5

z

0 2 4 6 80

0.5

1

1.5

PzPx

Density

But what occurs at the nanometer scale ?

Young-Laplace Eq.

z

Loca

lP

ress

ure

0 2 4 6 8 10-0.05

0

0.05

0.1

PzPx

46EQUILIBRIUM: YOUNG-LAPLACE EQUATION (2/2)

, , 0.6 z Liq z VapR P P

0.6LV

Young-Laplace equation, LV = RP, is well respected for such nano-droplets (using normal pressure)

The curvature correction to LV (Tolman’s length) is negligible !

From liquid/vapor interfaces simulations one can compute surface tension Liquid Vapor

and from the droplet normal pressure

In 3D droplets, line tension is negligible as well … (works of Bresme …)

47EQUILIBRIUM: YOUNG-DUPRÉ LAW (1/2)

Contact angles vs C

LV

SVSL q

1cos SV SLY

LV

q

From Young –Dupré law

From circle fit

From the computation of all three , the Young angle can be estimated

At the contact line the three (Sol/Vap, Sol/Liq and Liq/Vap)

tensions compensate each otherscosLV Y SV SL q

Young-Dupré law

Young angle, qY, is not equal to the measured angle at z=0.5 !

C

qp

0.2 0.3 0.4 0.5 0.6

0.3

0.4

0.5

0.6

0.7

0.8

48EQUILIBRIUM: YOUNG-DUPRÉ LAW (2/2)

Why so: Px (and so ) is affected by the surface

over long-range (cf. vdW forces)

Finite size effects on the result when q 0 (zY > h)Macroscopically described by disjoining pressure (when h < zY)

Distance at which qY = q, noted zY,

Scales 1/qY

zYq Is there a distance from the solid surface at which qYD=q,

i.e. macro angle = micro angle?

Distance at which qY=q

1/qY

z Y

0.45 0.6 0.75 0.9 1.050

1

2

3

4

5

6

7

h

49DYNAMICS: DROPLET (qEq = p/2) UNDER AN EXTERNAL FIELD

Droplet under a « gravity » field Flow field (Ca=0.2)

C = 0.45qEq p/2

FExt

VC

entr

oid

0.001 0.002 0.003 0.004 0.005 0.006

0.02

0.04

0.06

0.08

Droplet velocity vs external force

A friction coefficient can bededuced (here x=F/V 0.07)

linear regime

By varying Fext, the capillary numberis changed

Ca=V/ 0.05-0.35

Slip length at Liq/Sol 1

Non linear response occurs for smaller q and high V

50DYNAMICS: ADVANCING/RECEDING CONTACT ANGLES (qEq = p/2)

x

z

-15 -10 -5 0 5 10

6

8

10

12

14

16

qAqR

Dynamic contact angles (at z=5) qA,R (at z=5) vs Ca

Ca

Abs(q

q

Eq)

0 0.1 0.2 0.3 0.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

CaA

bs(

q

q3 E

q)

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

A hydrodynamic behavior holds for both angles at z = 5 !

qR

qA

Even with nano-droplets (but not too low q), hydrodynamics seem to be well respected above zYs!

At both contact lines advancing and receding angles can be measured

Circle fit is stil valid

Hydrodynamics model predict that q3 Ca