the lattice electrokinetics...

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Simulating Soft Matter 2017:The Lattice Electrokinetics Algorithm

2017-10-10 1 / 27Institute for Computational PhysicsUniversity of Stuttgart

The Lattice Electrokinetics AlgorithmMichael Kuron, Georg RempferOctober 10th, 2017

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Subproject C.5 –

Macromolecular translocation through nanopores

C. Holm

G. Rempfer, F. Weik

14.09.2016Institute for Computational Physics

University of Stuttgart 16 / 20

positions / velocitiesionic densites flow field

P3Mestatics

MDintegrator

FVintegrator

LBintegrator

forces

FFTestatics

FFTelectrostatics

ionic fluxes fluid forces

fluidcoupling

shortrange

Electrokinetics Molecular Dynamics Lattice-Boltzmann

Current developments in Espresso

LB-EK point particle coupling

C.5 Makromolekularer Transport durch nanoskalige Poren

Prof. Dr. Christian Holm

Georg RempferSimon Schöll

Diffusion

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IntroductionApplication¢ Charge transport in solution¢ Electrophoresis, Electro-osmotic flow, dynamics of colloids

and polymers in solution, sedimentation, …

Method¢ Separation of time scales à treat solute ions as continuum¢ More efficient than explicit ions because these are slow at

sufficiently sampling the volume outside the Debye layer¢ Time-dependent Poisson-Boltzmann solver¢ Exploit lattice-Boltzmann’s efficiency for hydrodynamics¢ Performance independent of salt concentration

Diffusion

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The Electrokinetic Equations

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The Stokes Equation¢ Derived from mass and momentum conservation¢ Incompressible Newtonian flow¢ Low Reynolds number

𝛻 ⋅ 𝜌𝑢 = 0𝜂𝛻(𝑢 − 𝛻𝑝 + 𝑓./0 = 0

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The Nernst-Planck EquationFick’s law of diffusion:

Add migration in electric field:

Add advective transport due to fluid flow:

Continuity equation:

𝚥2 = −𝐷2𝛻𝜌2

𝚥2 = −𝐷2𝛻𝜌2 −𝐷2𝑘6𝑇

𝑧2𝑒𝜌2𝛻Φ

𝚥2 = −𝐷2𝛻𝜌2 −𝐷2𝑘6𝑇

𝑧2𝑒𝜌2𝛻Φ + 𝜌2𝑢

𝛻 ⋅ 𝚥2 = −𝜕𝜌2𝜕𝑡

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The Poisson Equation

𝜌=>?@A. =B𝑧2𝑒𝜌2

�

2

𝛻(Φ = −4𝜋𝑙G𝑘G𝑇𝑒 B𝑧2𝜌2

�

2

𝑙H =IJ

KLMNMO2PQ≈ 0.7 nm

¢ Homogeneous dielectric coefficient

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The Fluid CouplingMotion of solutes exerts a force on the fluid via frictional coupling

𝚥2UVWW = −𝐷2𝛻𝜌2 −𝐷2𝑘6𝑇

𝑧2𝑒𝜌2𝛻Φ

f./0 =B𝑘G𝑇𝐷2

𝚥2UVWW�

2

G. Rempfer et al., doi:10.1063/1.4958950, 2016

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The Lattice Electrokinetics Algorithm

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The AlgorithmDiscrete solution of the electrokinetic equations

Fabrizio Capuania)FOM Institute for Atomic and Molecular Physics (AMOLF), Kruislaan 407, 1098 SJ Amsterdam,The Netherlands

Ignacio Pagonabarragab)Departament de Fısica Fonamental, C. Martı i Franques 1, 08028 Barcelona, Spain

Daan FrenkelFOM Institute for Atomic and Molecular Physics (AMOLF), Kruislaan 407, 1098 SJ Amsterdam,The Netherlands

!Received 2 March 2004; accepted 21 April 2004"

We present a robust scheme for solving the electrokinetic equations. This goal is achieved bycombining the lattice-Boltzmann method with a discrete solution of the convection-diffusionequation for the different charged and neutral species that compose the fluid. The method is basedon identifying the elementary fluxes between nodes, which ensures the absence of spurious fluxesin equilibrium. We show how the model is suitable to study electro-osmotic flows. As an illustration,we show that, by introducing appropriate dynamic rules in the presence of solid interfaces, we cancompute the sedimentation velocity !and hence the sedimentation potential" of a charged sphere.Our approach does not assume linearization of the Poisson–Boltzmann equation and allows us fora wide variation of the Peclet number. © 2004 American Institute of Physics.#DOI: 10.1063/1.1760739$

I. INTRODUCTION

The study of the dynamics of suspensions of chargedparticles is interesting both because of the subtle physicsunderlying many electrokinetic phenomena and because ofthe practical relevance of such phenomena for the behaviorof many synthetic and biological complex fluids.1,2 In par-ticular, electrokinetic effects can be used to control the trans-port of charged and uncharged molecules and colloids, usingelectrophoresis, electro-osmosis, and related phenomena.3 Asmicro-fluidic devices become ever more prevalent, there arean increasing number of applications of electro-viscous phe-nomena that can be exploited to selectively transport mate-rial in devices with mesoscopic dimensions.4

In virtually all cases of practical interest, electroviscousphenomena occur in confined systems of a rather complexgeometry. This makes it virtually hopeless to apply purelyanalytical modeling techniques. But also from a molecular-simulation point of view electroviscous effects present a for-midable challenge. First of all, the systems under consider-ation always contain at least three components; namely asolvent plus two !oppositely charged" species. Then, there isthe problem that the physical properties of the systems ofinterest are determined by a number of potentially differentlength scales !the ionic radius, the Bjerrum length, theDebye–Huckel screening length and the characteristic size ofthe channels in which transport takes place". As a result,fully atomistic modeling techniques become prohibitivelyexpensive for all but the simplest problems. Conversely,standard discretizations of the macroscopic transport equa-

tions are ill-suited to deal with the statistical mechanics ofcharge distributions in ionic liquids, even apart from the factthat such techniques are often ill-equipped to deal with com-plex boundary conditions.

In this context, the application of mesoscopic !‘‘coarse-grained’’" models to the study of electrokinetic phenomenain complex fluids seem to offer a powerful alternative ap-proach. Such models can be formulated either by introducingeffective forces with dissipative and random components, asin the case of dissipative particle dynamics !DPD",5 or bystarting from simplified kinetic equations, as is the case withthe lattice-Boltzmann method !LB".

The problem with the DPD approach is that it necessar-ily introduces an additional length scale !the effective size ofthe charged particles". This size should be much smaller thanthe Debye screening length, because otherwise real charge-ordering effects are obscured by spurious structural correla-tions; hence, a proper separation of length scales may bedifficult to achieve. A lattice-Boltzmann model for electro-viscous effect was proposed by Warren.6 In this model, thedensities of the !charged" solutes are treated as passive scalarfields. Forces on the fluid element are mediated by thesescalar fields. A different approach was followed in Ref. 7,where solvent and solutes are treated on the same footing!namely as separate species". This method was then extendedto couple the dynamics of charged colloids to that of theelectrolyte solution. As we shall discuss below, both ap-proaches have practical drawbacks that relate to the mixingof discrete and continuum descriptions.

The LB model that we introduce below appears at firstsight rather similar to the model proposed by Warren. How-ever, the underlying philosophy is rather different. We pro-

a"Electronic mail: [email protected], [email protected]"Electronic mail: [email protected]

JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 2 8 JULY 2004

9730021-9606/2004/121(2)/973/14/$22.00 © 2004 American Institute of Physics Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 129.69.120.241 On: Mon, 10 Oct

2016 12:08:15

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Solving the Navier-Stokes EquationsLattice-Boltzmann: see Ulf’s talk tomorrow

7

Uniform Grids

• Domain Decomposition:• regular decomposition into blocks containing uniform grids

• Parallelization:• data exchange on borders between blocks via ghost layers

waLBerla, an ultra-scalable multi-physics simulation framework for piecewise regular gridsChristian Godenschwager - FAU Erlangen-Nürnberg - September 3, 2015

receiverprocess

senderprocess

𝑓Y 𝑟 + 𝑐YΔ𝑡, 𝑡 + Δ𝑡 − 𝑓Y 𝑟, 𝑡= ^

_(𝑓Y

.a 𝑟, 𝑡 − 𝑓Y 𝑟, 𝑡 )

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Solving the Diffusion-Migration ProblemStart with the continuum problem:

Discretize using 2-point finite differences:

In 3D: calculate fluxes across links to 18 face and edge neighbors

Need to only actually calculate half of them

𝑗 𝑥, 𝑡 = −𝐷𝜕𝜌𝜕𝑥 𝑥, 𝑡 −

𝐷𝑧𝑒𝑘6𝑇

𝜌 𝑥, 𝑡 𝜕Φ𝜕𝑥 (𝑥, 𝑡)

𝑗 𝑥 +Δ𝑥2 , 𝑡 = −𝐷

𝜌 𝑥, 𝑡 − 𝜌 𝑥 − Δ𝑥, 𝑡Δ𝑥

− fgI2hQ

i j,k li jmnj,k(

o j,k mi jmnj,knj

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Solving the Advection ProblemLB fluid velocities are known on lattice nodesFluxes need to be calculated on lattice links

To avoid interpolation of fluid velocity, use volume-of-fluid method

�v

vxdt

vydt

(1� vx)dt

(1� vy)dt

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Propagating the FluxesStart with the continuum continuity equation:

Integrate it and apply Gauß’s divergence theorem:

Consider this expression for a single lattice cell:

Determine A0 = 1+2√2 by requiring <x2>=6Dt for pure diffusion

𝜕𝜌𝜕𝑡 𝑟, 𝑡 = −𝛻 ⋅ 𝚥(𝑟, 𝑡)

𝜕𝜕𝑡 p 𝜌(𝑟, 𝑡)d𝑉 = −p 𝚥(𝑟, 𝑡) ⋅ 𝑛td𝐴

�

vN

�

wN

𝑉x𝜌 𝑟, 𝑡 + Δ𝑡 − 𝑉x𝜌 𝑟, 𝑡 = −𝐴xΔ𝑡B𝑗Y(𝑟, 𝑡)�

Y

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Solving the Electrostatics ProblemStart with the continuum equation:

Discretize the Laplace operator using a 3-point stencil (7-point in 3D):

Write as matrix:

Solve using Gauß-Seidel, SOR, Conjugate Gradient, GMRES, Multigrid, …

𝜕(

𝜕𝑥( Φ(𝑥, 𝑡) = −𝜌(𝑥, 𝑡)

Φ 𝑥 + Δ𝑥, 𝑡 − 2Φ 𝑥, 𝑡 + Φ(𝑥 − Δ𝑥, 𝑡)Δ𝑥( = −𝜌(𝑥, 𝑡)

−2 1 0 ⋯ 0 11 −2 1 ⋯ 0 00 1 −2 ⋯ 0 0⋮ ⋮ ⋮ ⋱ ⋮ ⋮1 0 0 ⋯ 1 −2

Φ(0)Φ(Δ𝑥)Φ(2Δ𝑥)

⋮Φ(𝑁Δ𝑥)

= −

𝜌(0)𝜌(Δ𝑥)𝜌(2Δ𝑥)

⋮𝜌(𝑁Δ𝑥)

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Solving the Electrostatics Problem via FFTTake the discrete equation:

Write each term using a discrete Fourier transformation:

Φ 𝑥 + Δ𝑥, 𝑡 − 2Φ 𝑥, 𝑡 + Φ(𝑥 − Δ𝑥, 𝑡)Δ𝑥( = −𝜌(𝑥, 𝑡)

1Δ𝑥( 𝑁�

B 𝑒(LY2~ jlnj Φ�(𝑘)

~m^

2�x

−2

Δ𝑥( 𝑁�B 𝑒

(LY2~ jΦ� 𝑘

~m^

2�x

+1

Δ𝑥( 𝑁�B 𝑒

(LY2~ jmnj Φ�(𝑘)

~m^

2�x

= −1𝑁�B 𝑒

(LY2~ j𝜌t(𝑘)

~m^

2�x

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Solving the Electrostatics Problem via FFT (cont’d)Simplify:

This has to be true for each summand because DFT is an orthonormal basis.Solve for the potential:

1Δ𝑥( B 𝑒

(LY2~ j 𝑒

(LY2~ nj + 𝑒m

(LY2~ nj − 2 Φ�(𝑘)

~m^

2�x

= −B 𝑒(LY2~ j𝜌t(𝑘)

~m^

2�x

Φ� 𝑘 = −Δ𝑥(

2 cos 2𝜋Δ𝑥𝑁 𝑘 − 2

𝜌t(𝑘)

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Boundary ConditionsFixed objects: single colloid / colloid crystal, polymer in fixed conformation, walls, …¢ Impermeable to solute: set flux to zero¢ Fluid sticks to them: no-slip boundaryMoving objects: multiple (moving) colloids or polymers¢ Object much lager lattice spacing: moving boundary¢ Object size similar to lattice spacing: force coupling Reactive surfaces: catalysts¢ Surface cells convert solute species into another¢ Non-zero flux boundary

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ImplementationsESPResSo¢ Soft matter simulation tool from the University of Stuttgart¢ EK code developed by Georg¢ Easiest to use, GPGPU implementation¢ Some new innovative features (grid refinement, particle coupling, thermalization)¢ Tutorial in the afternoonwaLBerla¢ Lattice-Boltzmann HPC framework from FAU Erlangen¢ EK code developed by me¢ Other new innovative features (grid refinement, moving boundaries, reactions)

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Some Applications

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Colloidal ElectrophoresisAs simulated in Raspberry tutorial (tomorrow)

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Electro-osmotic Flow¢ Channels with charged walls

(as simulated in LB tutorial)¢ Nanopores¢ Porous media¢ Microfluidic devices

G. Rempfer

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Some Extensions and More Applications

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Chemical Reactions / Self-ElectrophoresisBulk Reactions¢ Convert ions of reactant species into ions of product species¢ Input parameters: stoichiometric ratio (reaction equation) and reaction rate

Boundary Reactions¢ Take place at a colloid’s surface, replace the no-flux boundary condition¢ Typically used for catalytic reactions

M. Kuron

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Coupling to Moving Particles¢ Similar to Ladd boundaries in LB¢ What to do with solute ions in cells that

get claimed/vacated by a particle?

6

T = 0

~v

T = 0.25

~v

T = 0.5

~v

T = 0.75

~v

T = 1

~v

T = 1.25

~v

Figure 1. Illustration of the mass conservation modificationto the Ladd boundary scheme to make it usable for EK. Cellswhose center is inside the particle are considered to be bound-ary nodes. The arrows indicate how solute is drawn intovacated cells (panes 2, 3, and 6) and expelled from newly-overlapped cells (panes 4 and 5).

during the time steps after a cell has been claimed orvacated. To reduce these effects, we propose a partialvolume scheme, which is illustrated in Fig. 2.

In the following, (r, t) is a field describing the volumefraction of the cell at r that is overlapped by a particle,with = 1 meaning that the cell is completely insidethe particle and = 0 completely outside. In the cal-culation of the diffusive fluxes (18), the concentrationsare replaced with ones that take into account that all so-lute resides in the non-overlapped part of the cells. Toprevent the resulting diffusive fluxes from diverging as ! 1, we renormalize them by scaling them with thevolume. This leads to the following modified expressionfor the flux:

jdiff

ki

(r ! r+ c

i

, t)

=

D

k

agrid

✓⇢k

(r, t)

1� (r, t) �⇢k

(r+ c

i

, t)

1� (r+ c

i

, t)

◆

� Dk

zk

e

2kB

Tagrid

(1 + 2p2)

✓⇢k

(r, t)

1� (r, t)

+⇢k

(r+ c

i

, t)

1� (r+ c

i

, t)

◆⇥ (�(r, t)� �(r+ c

i

, t))

�

⇥ (1� (r, t)) (1� (r+ c

i

, t)) . (23)

With this change, refilling vacated cells as per Eqs. (21)and (22) is no longer necessary. They can be set to zeroconcentration and will be filled up by the diffusive fluxagain as increases. We determine numerically bysub-dividing each cell into 8 equally-sized cells and de-termining how many of them are completely inside andcompletely outside the particle. For those cells that areneither, the subdivision is recursively repeated up to amaximum depth of 4. Expelling solute from a cell thatis claimed by a particle is, however, still necessary —even with the modified expression for the flux — as the

T = 0

~v

T = 0.1

~v

T = 0.2

~v

T = 0.3

~v

T = 0.4

~v

T = 0.5

~v

Overlapped volume : 0 1

Figure 2. Illustration of the partial volume scheme for mov-ing boundaries in EK. The shading of the cells inside theparticle corresponds to the overlapped volume to indicatehow the particle’s charge is distributed across the cell layerat its surface. In the calculation of the diffusive flux, the con-centrations are scaled with 1 � to determine the effectiveconcentrations.

cell is not necessarily completely empty by the time itis claimed. The expelled amount of solute with Eq. (23)is much smaller than with Eq. (18) and thus the effectof this sudden change on the simulation is reduced toacceptable levels.

One further source of sudden variations in solute fluxesis the change in electrostatic potential when the vol-ume across which a particle’s charge is distributed variesdue to the fluctuation in the number of boundary cells.Therefore, when calculating the electrostatic potential,each particle’s total charge Q = Ze is distributed amongall cells that are at least partially overlapped by thatparticle:

⇢b

(r, t) = Ze (r, t)

Vp

, (24)

with Vp

the particle’s (non-discrete) volume. Inhomo-geneous charge distributions are also possible as long asthe charge in a cell varies smoothly as the cell is slowlyclaimed or vacated by the colloidal particle.

IV. VALIDATION

We implement our new algorithm using the waLBerlaframework68. It supports several lattice Boltzmann mod-els, including the one introduced in Section III A, andcorrectly handles the moving LB boundaries describedin Section III B. We already added an implementationof the EK model described in Section III C. waLBerlaprovides excellent scaling on high-performance computeclusters and contains advanced features, such as grid

6

T = 0

~v

T = 0.25

~v

T = 0.5

~v

T = 0.75

~v

T = 1

~v

T = 1.25

~v

Figure 1. Illustration of the mass conservation modificationto the Ladd boundary scheme to make it usable for EK. Cellswhose center is inside the particle are considered to be bound-ary nodes. The arrows indicate how solute is drawn intovacated cells (panes 2, 3, and 6) and expelled from newly-overlapped cells (panes 4 and 5).

during the time steps after a cell has been claimed orvacated. To reduce these effects, we propose a partialvolume scheme, which is illustrated in Fig. 2.

In the following, (r, t) is a field describing the volumefraction of the cell at r that is overlapped by a particle,with = 1 meaning that the cell is completely insidethe particle and = 0 completely outside. In the cal-culation of the diffusive fluxes (18), the concentrationsare replaced with ones that take into account that all so-lute resides in the non-overlapped part of the cells. Toprevent the resulting diffusive fluxes from diverging as ! 1, we renormalize them by scaling them with thevolume. This leads to the following modified expressionfor the flux:

jdiff

ki

(r ! r+ c

i

, t)

=

D

k

agrid

✓⇢k

(r, t)

1� (r, t) �⇢k

(r+ c

i

, t)

1� (r+ c

i

, t)

◆

� Dk

zk

e

2kB

Tagrid

(1 + 2p2)

✓⇢k

(r, t)

1� (r, t)

+⇢k

(r+ c

i

, t)

1� (r+ c

i

, t)

◆⇥ (�(r, t)� �(r+ c

i

, t))

�

⇥ (1� (r, t)) (1� (r+ c

i

, t)) . (23)

With this change, refilling vacated cells as per Eqs. (21)and (22) is no longer necessary. They can be set to zeroconcentration and will be filled up by the diffusive fluxagain as increases. We determine numerically bysub-dividing each cell into 8 equally-sized cells and de-termining how many of them are completely inside andcompletely outside the particle. For those cells that areneither, the subdivision is recursively repeated up to amaximum depth of 4. Expelling solute from a cell thatis claimed by a particle is, however, still necessary —even with the modified expression for the flux — as the

T = 0

~v

T = 0.1

~v

T = 0.2

~v

T = 0.3

~v

T = 0.4

~v

T = 0.5

~v

Overlapped volume : 0 1

Figure 2. Illustration of the partial volume scheme for mov-ing boundaries in EK. The shading of the cells inside theparticle corresponds to the overlapped volume to indicatehow the particle’s charge is distributed across the cell layerat its surface. In the calculation of the diffusive flux, the con-centrations are scaled with 1 � to determine the effectiveconcentrations.

cell is not necessarily completely empty by the time itis claimed. The expelled amount of solute with Eq. (23)is much smaller than with Eq. (18) and thus the effectof this sudden change on the simulation is reduced toacceptable levels.

One further source of sudden variations in solute fluxesis the change in electrostatic potential when the vol-ume across which a particle’s charge is distributed variesdue to the fluctuation in the number of boundary cells.Therefore, when calculating the electrostatic potential,each particle’s total charge Q = Ze is distributed amongall cells that are at least partially overlapped by thatparticle:

⇢b

(r, t) = Ze (r, t)

Vp

, (24)

with Vp

the particle’s (non-discrete) volume. Inhomo-geneous charge distributions are also possible as long asthe charge in a cell varies smoothly as the cell is slowlyclaimed or vacated by the colloidal particle.

IV. VALIDATION

We implement our new algorithm using the waLBerlaframework68. It supports several lattice Boltzmann mod-els, including the one introduced in Section III A, andcorrectly handles the moving LB boundaries describedin Section III B. We already added an implementationof the EK model described in Section III C. waLBerlaprovides excellent scaling on high-performance computeclusters and contains advanced features, such as grid

M. Kuron, doi:10.1063/1.4968596

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Self-Electrophoresis + Moving Particles

M. Kuron

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Grid refinement¢ All the interesting things happen in the Debye layer¢ We don’t want to waste computational effort on the bulk fluid¢ Use a lower resolution further away from boundaries or dynamically refine where the gradients

are steep

I. Tischler, M. Lahnert

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Force-coupled ParticlesSimilar to Dünweg point-coupling in LB

S. Schöll, R. Kaufmann

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Subproject C.5 –

Macromolecular translocation through nanopores

C. Holm

G. Rempfer, F. Weik

14.09.2016Institute for Computational Physics

University of Stuttgart 16 / 20

positions / velocitiesionic densites flow field

P3Mestatics

MDintegrator

FVintegrator

LBintegrator

forces

FFTestatics

FFTelectrostatics

ionic fluxes fluid forces

fluidcoupling

shortrange

Electrokinetics Molecular Dynamics Lattice-Boltzmann

Current developments in Espresso

LB-EK point particle coupling

C.5 Makromolekularer Transport durch nanoskalige Poren

Prof. Dr. Christian Holm

Georg RempferSimon Schöll

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Thermal fluctuations¢ Significant for nano-scale problems¢ More complex problem than for LB¢ Needed for correct polymer conformations

R. Kaufmann

the lattice electrokinetics...

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