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Page 1: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the
Page 2: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Date: 11/08/11

Presented by : Dr Mark Bankhead, NNL

Mesoscopic simulationsof molten silicates usingthe LAMMPS codeLAMMPS Users' Workshop, Albuquerque, New Mexico,

9-11 August 2011

Page 3: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 3

Modelling at the National Nuclear Laboratory

• The National Nuclear Laboratory was formally establishedon the 1st April 2008.

• It was formed predominately from the R&T division ofBNFL.

• NNL operates from 5 UK sites.

Modelling forms the cornerstone of the NNL’s scientific and technicalcapability, underpins signature research programmes in wastemanagement, decommissioning and new build,

• Modelling accounts for approximately 15% of direct revenue

• About 100 scientists and engineers across multiple disciplines

Page 4: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 4

Chemical and Materials Modelling

Plant support

Safety cases

R&D

University links

Business areasCapabilities

Page 5: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 5

Chemical and Materials modelling

• Fundamental methods (such as MD) makes an importantcontribution across all the business areas.

• Customer drivers from difficult challenges and need to innovate.

• LAMMPS has replaced commercial software for moleculardynamics simulation.

'Synroc ' Zirconolitefor Pu/ immobilisation

Fission gas diffusionin MOX fuel

Borosilicate glassphysical properties

Parameters for IXplant kinetic models

Page 6: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 6

Challenging problems: Glass melt viscosity

Chemistry and fluid flow with Meso-scale methods

The operation of many nuclear plants relieson knowledge of the transport properties ofchemically complex fluids and solids:

• Example problem is the convection anddischarge of vitrification melter at Sellafield.

• Continuum scale modelling depends on theviscosity (depends on feed composition)

MD approaches are challenging, canmesoscale methods help meet the challenge.

LAMMPS is an ideal tool for prototyping newmethods.

Page 7: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 7

An introduction to Dissipative Particle Dynamics

• Dissipative Particle Dynamics - off-lattice simulationapproach for fluids derived from LGCA,

• Mesoscale approach to simulation of matter (analogous toLattice Boltzmann methods in fluid mechanics),

• Mesoscale means it should work as well at micro andcontinuum scales,

DPD has been applied to model a diverse range of systems :

• Fluid flow (pipes and porous media)

• Phase equilibria (polymer melts)

• Complex fluids (Emulsions and colloidal suspensions)

• Particle packing (concrete aggregates, liquid crystals)

Page 8: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 8

DPD Methodology

Groot-Warren method (LAMMPS):

Forces acting on beads:

Fi = Fij

C + Fij

D + Fij

R( )j≠ i∑

• Describes condensed matter

• Internal energy isrepresented, but not conserved

• Momentum is conserved,preserving hydrodynamics

Page 9: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 9

Forces in the GW DPD model

• Conservative force definesthe chemical interactionswithin a material

• Weight constant defines asoft interaction potential

FijC = aωC r̂ij

−=

c

ijC

rr

• 'a' determines strength ofsoft repulsion

Page 10: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 10

Forces in the GW DPD model

• Weight constants :

ωD = ωR( )2

• Fluctuations arise from the random and dissipativeforce terms in the model:

• controls the strength of the dissipative force.

ζσ TkB2=

Dissipative force (frictional drag)

Random (stochastic) force (Brownian motion)Th

erm

ost

at

ijR

ijR

ij rF ˆξωσ=

ijijijD

ijDij rrvF )) )( ⋅−= ωζ

Page 11: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 11

Boundary conditions and unit systems

ρ = ρrc3

p = prc3 / kBT

E = E / kBT

a = arc / kBT

Particle density

Energy

Scalar Pressure

Conservative force

• Boundary conditions are can be imposed forperiodicity, additional forces and solid objects:

• rc defines the length scale in DPD, kBT is aconvenient unit of energy (reduced units):

Page 12: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 12

How can we apply DPD?

Simulations With DPD

1. A choice for the inter-particle forces, so we can modelmaterials of interest;

2. Applying appropriate boundary conditions, so we cansimulate problems of interest;

3. Finding a means of analyzing the simulations, to investigatethe effect of change;

DPD could be a useful technique for the nuclear industry if we canaddress the following (Hoover 2006):

Page 13: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 13

Coarse Graining Matter in DPD

Properties: momentum& ‘chemical potential’

Reduce degrees of freedom:Molecular Structure to ‘Blobs’

rc =ρmin ⋅VM

i

Av

3

i

ci V

r3

Page 14: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 14

Regular Solution Theory

Chemical potential (driving force) and activity (degree) of mixingcan be derived from the Gibb’s free energy of mixing .

The excess heat of mixing of a regular solution (SE (entropy ofmixing) & VE (volume of mixing)=0).

• Energy of mixing for 1 mol of solution:

• Change of cohesive pressure as a result of mixing:

A12 = c1 + c2 − 2c12

∆mUV = x1V1 + x2V2( )A12φ1φ2 ≡ GRSTm

2112 ccc =

Cohesive energy density forms the basis of the Hildebrand(solubility) parameter:

δ = c1/2 = (−U / V )1/2 ( ) Tm

Tvap V

zRTH −∆=2δ

Page 15: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 15

Regular solution theory for non-regular mixtures

• Non-regular mixtures

• The method introduced by Hansen defines the parameters for:

• Dispersion d

• Polar interactions p

• Hydrogen bonds h

• Note that as vm is the volume of the mixture, which may not be asum of the mole-fraction of the molar volumes of its components(i.e. there may be a volume change on mixing).

GRSTM = vmφ 1−φ( ) δ1

d −δ2d( )2

+ δ1p −δ2

p( )2+ δ1

h −δ2h( )2( )

Page 16: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 16

DPD and solubility parameters

• Correspondence between RST and DPD (Travis 2007)

Equate the expression for the free energy of mixing from DPD tothat of Regular Solution Theory:

• The expression for the a11 and a22 terms can be obtained byindependently from the relationship between the internal pressureand the Virial part of the DPD free energy of mixing:

(δ1 − δ 2 )2 = −rc4 N A

2αa11

v12 − 2

a12

v1v2

+a22

v22

42

21

11ci r

aαρδ

= ai =δi

2

αρi2

∆ = −α ρ12a11 + ρ2

2a22 − 2ρ1ρ2a12[ ]RST DPD (reduced units)

DPD (reduced units)

Page 17: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 17

Inorganic materials: e.g. SiO2

Conservative force terms for SiO2

ρ with temperature for SiO2

∆H vap0

TB

= s0( )g− s0( )l

)( riT

iT

i TTdTdVVV r −+=

∆H vapT = ∆Hvap

0 + T T − T BP( )CpT −Cp0( )

δ 2 = ∆Hvap − RT( ) zVm

Density:

Conservative force:

i

ci V

r3

δ i =rc

3

kBTδ i

Page 18: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 18

Dissipative and random forces

• Transport Properties of materials with DPD

DPD is very interesting as a tool to model fluid flow problems.

• Applied to generic systems, a broad spectrum of fluid behaviour isobserved.

Dissipation refers to the energy loss in a material over time. Theconstant ij (zeta) controls the strength of the dissipative force.

• A number of empirical methods have been developed towards thederivation of these forces. (Shown later)

A universal method for modelling transport properties remainsbeyond reach.

Page 19: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 19

Using DPD with LAMMPS

LAMMPS is very flexible:

• LAMMPS scripting can be used toembed DPD parameters usingpolynomial functions as a function ofsystem temperature,

• NNL have recently modifiedpair_style dpd to work withfix_adapt,

• Hybrid potentials possible,

• Performance is very good:

(36 steps/ sec on 24 cores for 100Kparticles at NNL)

Page 20: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 20

Poiseuille flow simulations SiO2 viscosity

• Bead types determined viaRST approach.

• 3D model of a fluid betweenparallel infinite plates

(LAMMPS ~100K particles 4hon 24 cores)

• Non-equilibrium boundaryconditions applied. Flowinduced due to gravity

(Duong-Hong, Karniadakis).

Page 21: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 21

Boundary condition validation

Instantaneous temperature profileperpendicular to walls with short

time-step (0.005 DPD units)

DPD SiO2 Fluid Densityperpendicular to walls

Page 22: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 22

Viscosity and dissipative forces

DPD SiO2 Viscosity with ijDPD SiO2 Viscosity with ij

( ij determines the time-scale)

Page 23: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 23

Navier Stokes behaviour of DPD fluids

Stress vs strain Velocity Profiles

Page 24: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 24

Benchmarking the method

Modelling SiO2 rheology

• Method works for regularmixtures.

• Qualitative trends inviscosity.

• Non-regular mixturescould be modelled by usingan extension of RST.

Predicted viscosity

vs. Temperature

650K to 2200K

Page 25: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 25

Conclusions on the method

• Where do we stand with DPD?

Research has delivered a robust & universal method to describe theconservative forces in a DPD simulation.

Numerical experiments can be used to establish the scope of thedissipative (ζ) constant.

• Applying this approach means that DPD can be used to modelqualitative trends in fluid behaviour, this is potentially industriallyuseful, if generally unsatisfactory.

• Academic challenge remains to provide a robust method tocalculate these parameters so real fluids can be modelled withcertainty.

Page 26: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 26

Future applications of LAMMPS

• Complex fluids and complexgeometries,

• Hybrid models mixingmesoscale methods (e.g. DPD+ LJ),

• Structural models

• Long-term goal toimplement SPAM/SPH modelsin LAMMPS,

PD model in LAMMPS of fracture of 'grout' due to theexpansion of an internal body.

Microstructure ofSandstone usedfor DPDpermeabilitysimulations

Page 27: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 27

Acknowledgements

• Thanks to sponsors and collaborators

This work would not have been possible without the help andsupport of the following

• National Nuclear Laboratory (ex. BNFL, NSTS & Nexia Solutions),

• Sellafield Ltd.,

• Nuclear Decommissioning Authority,

• Dr S. Owens (NNL),

• Dr K. Travis (Sheffield University),

Page 28: Presented by : Dr Mark Bankhead, NNL...Slide 10 Forces in the GW DPD model • Weight constants : D =(R) 2 • Fluctuations arise from the random and dissipative force terms in the

Slide 28

References

• K P Travis et al, J. Chem. Phys., vol 127, (2007)

• A F M Barton, Handbook of Solubility Parameters, 2nd ed(2000)

• K Mecke & C H Arns, Fluids in porous media, J Phys. Cond.Matt., 17, (2005)

• Duong-Hong et al, Comput. Mech., 35: 24–29, (2004)

• Pivkin & Karniadakis, PRL, 96, 206001 (2006)

• Hoover, SPAM, Adv, Nonlinear Dynamics, 25 (2006)