SPH for microfluidic suspensions: Surface tension, wetting ... for... · © Fraunhofer-Institut für Werkstoffmechanik IWM 2 T. Breinlinger 30.06.2014 SPH FOR MICROFLUIDIC SUSPENSIONS:

© Fraunhofer-Institut für Werkstoffmechanik IWM

SPH FOR MICROFLUIDIC SUSPENSIONS: SURFACE TENSION, WETTING AND SOLID PARTICLES

T. Breinlinger

30.06.2014


2

T. Breinlinger

30.06.2014

SPH FOR MICROFLUIDIC SUSPENSIONS: SURFACE TENSION, WETTING AND SOLID PARTICLES

„ANTZ“ Universal Pictures

Microfluidic suspensions:

• Drops

• Particles

• Wetting/dewetting

• Drying


3

Group Dr. Torsten Kraft Powders and fluidic systems

Simulation of powdertechnological processes and process chains

Granulation

Powder transfer

Powder compaction

Tape casting, screen printing

Drying, debinding

Sintering

Simulation of manufacturing processes

Modeling of complex suspensions

Microfluidic system design

Determination of material data

Analysis of various materials

Process optimization by simulation

Abrasive processing with bonded grain

Filling, compaction and sintering of a ceramic

seal disc

Wetting of a structured surface

Granulation via spray drying

Sintering warpage of a printed LTCC


4

AGENDA

Smoothed Particle Hydrodynamics (weakly compressible)

Surface tension in SPH

Different approaches

Wetting in SPH

How to incorporate wetting in surface tension models

Applications

Modeling suspensions with SPH

Different approaches

Applications


5

Weakly compressible SPH Basic equations

Density summation

Momentum equation

Pressure term

Viscous term

Surface tension force

𝜌𝑖 = 𝑚𝑖 𝑊𝑖𝑗

𝑗

𝜕𝒗𝑖𝜕𝑡

=1

𝜌𝑖−𝛁𝑝𝒊 + 𝜂𝛁2𝒗𝑖 + 𝑭 𝑠

𝑖 + 𝒈

𝛁𝑝𝑖 = 𝑝𝑖 + 𝑝𝑗𝑚𝑗

𝜌𝑗𝑗𝛁𝑊𝑗𝑖

𝜂𝑖𝛁

2𝒗𝑖 = 𝜉4𝜂𝑖𝜂𝑗

𝜂𝑖 +𝜂𝑗

𝒗𝑖 − 𝒗𝑗 ⋅ 𝒙𝑖 − 𝒙𝑗

𝒙𝑖 − 𝒙𝑗2+ 𝛽ℎ2𝑗

𝑚𝑗

𝜌𝑗𝛁𝑊𝑖𝑗

𝑭𝑖𝑠 =…

J.J. Monaghan, Rep. Prog. Phys. 68 (2005)

A. Colagrossi et al., J. Comput. Phys. 191 (2003)

P.W. Cleary, Appl. Math. Model. 22 (1998)

S. Adami et al., J. Comput. Phys. 229 (2010)


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Surface Tension in SPH


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Surface tension Introduction

Caused by:

Cohesive forces among liquid molecules.

Effects:

Contraction of liquid surface.

Pressure jump across surface.

Δp = −𝜎 div 𝒏

𝜎: surface tension

𝒏: surface normal


8

Surface tension Modeling techniques in SPH

Based on cause

Pairwise forces

Based of effect

CSF model (Continuum surface force)

S. Nugent and H.A. Posch, Phys. Rev. E 62 (2000)

A. Tartakovsky and P. Meakin, Phys. Rev. E 72 (2005)

J.U. Brackbill et al., J. Comput. Phys. 100 (1992)

J.P. Morris, Int. J. Numer. Meth. Fl. 33 (2000)



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Surface tension Implementation of pairwise forces in SPH

Nugent & Posch:

Large neighborhood required.

Tartakovsky & Meakin:

Attractive at long range, repulsive at short range.

Both models require case calibration.

Just as high EOS stiffness, these forces can add molecular viscosity.



𝑭𝑖𝑠 = 𝑠𝑖𝑗𝑐𝑜𝑠

1.5𝜋

3ℎ𝑟𝑗 − 𝑟𝑖 𝑟𝑖𝑗 𝑟𝑖𝑗 ≤ ℎ

𝑝 =𝜌𝑘𝑇

1 − 𝜌𝑏− 𝑎𝜌2 van der Waals EOS


10

Surface tension Implementation of CSF in SPH

Surface tension force

Color function

Gradient of color function

Surface normal

Surface curvature

𝑭𝑖𝑠 = 𝜎𝜅𝒏𝛿

𝒏 = 𝛁𝑐 𝛁𝑐

𝑐𝑖𝑗 = 1,0,

𝑖𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑜𝑓 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑝ℎ𝑎𝑠𝑒𝑠 𝑖𝑓 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑝ℎ𝑎𝑠𝑒.

𝛁𝑐𝑖 = 𝒏𝛿 = 𝑉𝑖2 + 𝑉𝑗

2 𝜌𝑖𝜌𝑖 + 𝜌𝑗

𝛁𝑊𝑖𝑗𝑗𝑑𝑖𝑓𝑓. 𝑝ℎ𝑎𝑠𝑒

𝜅 = 𝛁𝒏𝑖 = 𝑑 𝒏𝑖 − 𝒏𝑗 𝑉𝑗𝛁𝑊𝑖𝑗𝑗

𝒙𝑖 − 𝒙𝑗 𝑉𝑗 𝛁𝑊𝑖𝑗𝑗



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Surface tension Modeling techniques in SPH - Summary

Based on cause

Pairwise forces

+ Free surface possible

+ Intrinsic wetting

- Calibration required

- „Raspberry“ clustering

- Artificial viscosity

Based of effect

CSF model

+ Contact angle and surface tension as input parameter

- More complex to implement

- Two phases required




J.P. Morris, Int. J. Numer. Meth. Fl. 33 (2000)



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Wetting in SPH (for CSF-based models)


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Wetting Contact line treatment

3 phase contact line governed by Young equation

𝜎 cos 𝜃𝑒𝑞 + 𝛾𝑠𝑙 − 𝛾𝑠𝑣 = 0

2 phase problem governed by Young-Laplace equation

Δp = −𝜎 div 𝒏

Δp


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Normal correction method

Prescribed surface normal near walls

𝒏𝑡𝑙 = 𝒏𝑡 sin 𝜃𝑒𝑞 −𝒏𝑤 cos 𝜃𝑒𝑞

No treatment Prescribed normal

𝜃𝑐𝑢𝑟𝑟𝑒𝑛𝑡 = 𝜃𝑒𝑞 = 60°



15

Normal correction method

Prescribed surface normal near walls

What happens without additional treatment:

Spurious currents

Especially at triple line

Even induced there?



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The normal correction should be smoothed

Sharp correction Smoothed correction

𝒏∗𝑖 =𝑓𝑖𝒏+ 1−𝑓𝑖 𝒏𝑡𝑙

𝑓𝑖𝒏+ 1−𝑓𝑖 𝒏𝑡𝑙 𝑖 with

𝑓𝑖(𝑑𝑤,𝑖) = 𝑑𝑤,𝑖 𝑑𝑚𝑎𝑥

𝑑𝑤,𝑖 = 𝑚𝑖𝑛 𝒙𝑖𝑗 ⋅ 𝒏𝑤,𝑖 − 𝛿

𝜃𝑐𝑢𝑟𝑟𝑒𝑛𝑡 = 60°, 𝜃𝑒𝑞 = 90°

T. Breinlinger et al., J. Comput. Phys. 243 (2013)


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Wetting Spurious currents

The smoothed normal correction scheme helps to reduce spurious currents


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Wetting Solution of Young-Laplace equation

3D drop on solid substrate (color code: pressure)

𝜃=60° 𝜃=150°

𝜌𝑙 𝜌𝑔 = 1

𝜂𝑙 𝜂𝑔 = 1

𝑔 = 0


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Wetting Equilibrium shape of drops

2D drop on solid substrate

Sharp normal correction Smoothed normal correction

𝜌𝑙 𝜌𝑔 = 1


𝑔 = 0


20

Wetting Pinning effects

2D drop crossing an edge

𝜌𝑙 𝜌𝑔 = 1000


𝑔 = 0.007…0.014


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Suspensions in SPH


22

Suspensions Different approaches

A suspension is a complex fluid consisting of a liquid phase and particles.

Suitable approach for modeling depends on

Level of detail

System size

Homogeneous Macroscopic Mesoscopic Microscopic

Level of detail System size


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Homogeneous approach:

Describes suspension as homogeneous fluid with complex rheological model.

Macroscopic approach:

Uses two (or more) intersecting and interacting continuous fluids to describe both phases.

Mesoscopic approach:

Uses discrete element (DEM) model of solid particles and local averaging for coupling with SPH.

Microscopic approach:

Uses discrete solid particles and fully resolves the flow around them (in CFD known as „immersed boundaries“).

Suspensions Different approaches


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Suspensions Homogeneous approach

Uses „regular“ SPH.

Complex rheology:

Shear thickening/thinning

Thixotropy

Can be implemented relatively easily in SPH as 𝜂 = 𝑓(𝛾 ) or 𝜂 = 𝑓 𝛾 in case of thixotropy.

Shear rate 𝛾 Time 𝑡

Vis

cosi

ty 𝜂

Vis

cosi

ty 𝜂

Shear thinning

Shear thickening

Shear load Relaxing

A. Wonisch et al., J. Am. Ceram. Soc. 94 (2011)


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Suspensions Homogeneous approach

Example: Tape casting

SPH continuum simulation of slurry

-> shearrate 𝛾 in system is known.

Jeffery equation

gives particle orientation for non-spherical particles.

Including the shear history, this gives very good argeement with the microstructure found in experiments.

Macroscopic shear rate

Microscopic orientation

Φ 𝑡 = 𝑎𝑟𝑐𝑡𝑎𝑛 𝑟𝑒tan 𝛾 𝑡

𝑟𝑒 + 𝑟𝑒−1


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Suspensions Mesoscopic approach

Discrete element method DEM solves Newton‘s equations for individual solid particles.

SPH solves Navier-Stokes equations for fluid phase.

Coupled by local averaging.

calculate coupling forces on DEM

integrate them on SPH

normalize by 1 = 𝑚𝜌 𝑊 based on DEM particles to ensure

conservation of momentum.

The size ratio of DEM vs. SPH particles should not be too far from 1.

DEM too small -> large summations -> slow.

DEM too large -> averaging no longer justified.

vi vj

Rj

ri

rj

wi

wj

T. Breinlinger et al., Proc. 6th SPHERIC (2011)

M. Robinson et al., Proc. 6th SPHERIC (2011)

D. Gao & A. Herbst, Int. J. Comp. Fluid Dyn. 23 (2009)


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Main loop


Coupling SPH and DEM

Coding perspective

Determine Neighbors

Calculate density, stress tensors etc.

Calculate interaction

forces Integration

Determine volume

fractions

Calculate coupling forces

Determine Neighbors

Calculate interaction

forces Integration Calculate

Overlaps

SPH

D

EM


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Example 2: Spray drying process - Drying of individual droplets

Solid primary particles with DEM

Cohesion depending on moisture

Fluid solvent with SPH

Newtonian fluid

Interaction of DEM/SPH via coupling forces

Drag, capillary

Drying model in SPH

Isothermal

Discrete phase transition

Solvent

(SPH)

Solid particles

(DEM)

Droplet


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Example 2: Spray drying process - Drying of individual droplets

Results

Binary phase change works for pure SPH but induces buckling and bulging in coupled DEM-SPH simulations.

Pure drop

(SPH)

Suspension drop

(SPH+DEM)


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Mesh based CFD as an alternative for spray drying process.

Binary drying model in SPH causes instabilities.

Mesh based Volume of Fluid (VOF) Method allows for continuous phase change.

VOF is implemented in OpenFOAM („interFOAM“).

Implemented a customised coupled VOF-DEM solver in OpenFOAM.

Momentum conservation

DEM substepping in time

Coupling forces for capillary force

Cohesion depending on moisture


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Mesh based CFD as an alternative for spray drying process.

Results

Drying and granulation can be simulated using VOF+DEM.

Granule morphology depends on the relation of cohesive vs. capillary forces.

strong medium weak

cohesive vs. capillary forces:


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Direct numerical simulation of suspensions with SPH

Fluid via “regular” SPH: appropriate properties (density, viscosity etc.)

Solid particles via rigid bodies (rigid clusters of SPH particles)

Modes of interaction

Fluid-Fluid -> regular SPH

Fluid-Solid -> regular SPH

Solid-Solid -> hard spheres

Suspensions Microscopic approach

Solvent

Solid particles


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About the rigid clusters

A rigid body k consists of a cluster of rigidly linked SPH-particles 𝑖 ∈ 𝑠𝑘 (sub-particles)

Total cluster force is the sum of all sub-particles forces:

Quaternion 𝑞𝑘 = 𝜉𝑘 , 𝜂𝑘 , 𝜁𝑘 , 𝜒𝑘 gives the cluster orientation

Time integration of 𝑞𝑘 through rigid body solver:

𝑓𝑘 = 𝑓𝑖𝑖∈𝑆𝑘

force on rigid body: torque on rigid body: 𝑡𝑘 = 𝑏𝑖 × 𝑓𝑖𝑖∈𝑆𝑘

bi: position of the sub-particles relative to the cluster center of mass

𝑞𝑘 𝑡 + Δ𝑡 = 𝑞𝑘 𝑡 + 𝑞 𝑘 𝑡 Δ𝑡 +1

2𝑞 𝑘(𝑡)Δ𝑡

2 − 𝜆𝑘(𝑡)𝑞 𝑘(𝑡)∆𝑡2

with 𝜆𝑘(𝑡) being the Lagrangian multiplier satisfying 𝑞𝑘 2 𝑡 + Δ𝑡 = 1

I.P. Omelyan, Phys. Rev. E 58 (1998)


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Example: platelet orientation

2d parallelization using 12 CPUs

3d representative volume cell with Lees-Edwards-boundary-conditions

Edge length: 250 µm

Initial SPH-particle spacing Dx=5 µm ->125000 SPH particles

h/Dx=1.05, qubic spline kernel

Random cluster orientation

Shear rate: 100 s-1

Shear duration: 1 s

Fluid viscosity: 0.42 Pa s

x

z


35


x

z


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Conclusions

Surface tension & wetting

Surface tension can be accurately modeled using SPH.

Pairwise forces

Intrinsic wetting

Molecular viscosity

Free surfaces

CSF

Technical parameters

Wetting must be modelled

Smoothed normal correction should be used

Suspensions

Different approaches available.

Level of detail vs. system size.

Complex rheology models can be very useful for large scale simulations.

Drying of a particle suspension with SPH can be difficult.

On the microscale, cluster-based SPH modeling can be very powerful.

Documents

SPH for microfluidic suspensions: Surface tension, wetting ... for... · © Fraunhofer-Institut für Werkstoffmechanik IWM 2 T. Breinlinger 30.06.2014 SPH FOR MICROFLUIDIC SUSPENSIONS: