Mesoscopic simulation of the dynamics of confined complex ...Mesoscopic simulation of the dynamics...

Flowing Complex Fluids Research Group Department of Chemical and Biological EngineeringUniversity of Wisconsin-Madison

Mesoscopic simulation of the dynamics ofconfined complex fluids

M. D. GrahamDept. of Chemical and Biological Engineering

Univ. of Wisconsin-Madison


• Microfluidics and DNA• Richard Jendrejack• Hongbo Ma• Juan Hernandez• Yeng-Long Chen•Eileen Dimalanta• Kyubong Jo•Chris Stoltz• Prof. Juan de Pablo• Prof. David C. Schwartz• NSF: UW-NSEC

• Cellular fluid mechanics• Sam Anekal• Patrick Underhill• Pratik Pranay• Juan Hernandez• NSF: CBET

Group/Collaborators/support


Microfluidics and complex fluids

genomic DNA (Schwartz)

biological fluids

particle-based assays(Edelstein et al.)

droplet reactors (Ismagilov et al.)

dispersions (Anna)

Surface-based biochemicalassays (Jensen et al.)


Confined polymers: fundamental issues

• Diffusion• Relaxation• Polymer conformations in flow• Effect of flow on adsorption• Mechanisms of center of mass migration and apparent slip in polymer solutions

Goal: simulations and theory to shed light on these issues

2Rg2h


Optical mapping for genomics(Schwartz lab; OpGen Inc.)

DNA is stretched by flow and electrostatically adsorbed to a surfacewithin a microfluidic device.

The optical map can be usedas a scaffold to orient short, detailedsequences (contigs) and thus accel-erate construction of a full genome.

Restriction enzymes cleave the adsorbed DNA molecules at well-defined positions, providing landmarks along the sequence, an “optical map” of the genome.


Adsorption depends on flow protocol

Reservoir of DNA solution

Syringe pump

Advancing meniscus

Dimalanta 2004

Negligible adsorption

Significant adsorption

Cationic capture surface



Coarse-grained models of polymers

atomistic (~ Å)

worm-like chain (~ 100 nm)(bending potential)

bead-spring chain (~µm)

Representation Resolution


Force balance for a chain in solution

dri = v i + Mij ⋅ f j + kT∂∂rj

⋅M ji

⎛

⎝⎜⎞

⎠⎟j=1

Nb

∑⎡

⎣⎢⎢

⎤

⎦⎥⎥dt + 2 Bij ⋅dw j

j=1

Nb

∑

imposed flow

spring forceexcluded volume

wall forcethermal fluctuations

Mij =1ζ(Iδ ij +ζΩij )

mobility tensor

BijBkjj=1

Nb

∑ = kTMik

fluctuation-dissipationtheorem

′ v = Ωiji, j=1

Nb

∑ ⋅ fj

hydrodynamic interaction tensor

→ velocity field generated by motion of macromolecule→ Solution to a Stokes problem


Coupling polymer dynamics to fluid dynamics

• Each “bead” is a regularized point force• Beads experience Stokes drag and Brownian force • Noise and drag are coupled!• Two-way coupling between fluid motion and polymer motion

• Approaches to this coupling:• Continuum solvent

• Green’s function methods (us, John Brady & co.)• Stochastic Navier-Stokes (Paul Atzberger)• Lattice Boltzmann (Tony Ladd)

•Particulate solvent• Dissipative particle dynamics (Bruce Caswell)• Stochastic rotation dynamics (Ron Larson)


Coupling polymer dynamics to fluid dynamics:Green’s function methods

Computation time for a naïve algorithm in an unboundedgeometry!!: N3 -- not good!

But…


Coupling polymer dynamics to fluid dynamics:Green’s function methods

Fast Stokes solver for (regularized) point forces in yourfavorite geometry => V = M . F

+Chebychev polynomial approximation to M1/2 (Fixman 1986)

+Derivative-free stochastic integration scheme (Fixman 1978, Hinch etal.1995)

Matrix-free linear algebra routines (GMRES etc.)+

=Brownian dynamics with hydrodynamic interactions in O(N) (or so)time in your favorite geometry.

Jendrejack et al., 2000, 2003, Hernandez-Ortiz et al. 2007, 2008


GGEM: fast method for point particles inStokes flow

• Near-field details– easy to regularize– Exponentially decaying

(short range)– Easy to calculate, i.e.

analytical expression– O(Nparticles)

• Long-range interactions– Correct boundary

conditions at walls,– Solution through O(Nmesh)

CFD method– Generalizes to

Stokesian dynamics,immersed boundarymethod

Jendrejack et al ‘03, Hernandez et al ‘07, Anekal et al. ‘08


GGEM

Force density: ρ(x) = f(xv ) δ (x − xv ) − g(x − xv )[ ] + g(x − xv ) v

N

∑“local” “global”

• Conventional Ewald sum approachto electrostatics:

• Local: exponentially decaying fcns.

• Global: smooth solution in Fourierspace

• This idea is not restricted to periodicdomains.

ρ = ρl + ρg ⇒ v = v l + vg


GGEM: local

Analytical FREE-SPACE solns.(i.e. no Boundary Conditions)

Neighbor list with size given by the smearing function

v l (x) = Gl (x − xv ) ⋅ f(xv )v

N

∑O(N) Green’s function-like calculation:

Gl (x) =18πη

δ +xxr2

⎡⎣⎢

⎤⎦⎥erfc(αr)

r−18πη

δ −xxr2

⎡⎣⎢

⎤⎦⎥2απ 1/2

e(−α2r2 )

⎥⎦⎤

⎢⎣⎡ −= − 22)(

2/3

3

25)(

22

rerg r απα αSmearing function:


GGEM: global

Find the solution on a mesh with αΔx << 1

Sparse/iterative Methods: O(Mesh points)

−∇pg +η∇2vg + ρg = 0∇ ⋅vg = 0

Solution of Stokes Equations:

with no-slip boundary Conditions: vg = −v l

After the O(Mesh) calculation a O(N) interpolation is performed:

GGEM = O(N)


Point force regularizationSmearing function g is aregularized delta function:Use it to replace the point force,with

GlR (x) = 1

8πηδ +

xxr2

⎡⎣⎢

⎤⎦⎥erf(ξr)r

−erf(ξr)r

⎡⎣⎢

⎤⎦⎥+

18πη

δ −xxr2

⎡⎣⎢

⎤⎦⎥2επ 1/2

e(−ξ2r2 ) −

2απ 1/2

e(−α2r2 )⎡

⎣⎢⎤⎦⎥

Regularized:

Beadhydrodynamic

radius: a

α−1

ξ−1

ξ−1 =3aπ


Jendrejack et al., JCP (2000,2002)

L = 21 µm (λ-phage)Ns = 10 springsNk,s = 19.8 Kuhns/spring

v = 0.0019 µm3/Kuhna = 0.077 µm (h* = 0.15)bk = 0.106 µm (Nk = 198Kuhns)

equilibrium stretch (1.5 µm)relaxation time (4.1 s)diffusivity (0.0115 µm2/s)

Smith and Chu, 1998Smith, Perkins and Chu, 1996

match

model parameters experimental data

• Model parameters determined by direct comparison to equilibrium experimental data for 21 µm DNA.

• Quantitative agreement with experiment for both transient and steady flows over a wide range of Weissenberg numbers.

• Molecular weight dependence of properties is “built in” because HI and EV are included in model

• Expect good performance in microfluidic simulations

Parametrization with free solution data


Expt

Channel Heights = 0.8 microns – 10 micronsλ-phange DNA: chain Lengths Ns = 10 – 120 (21 microns – 234 microns)

Square channels

Slit channels

• Static confinement transition at Rg,bulk/H ≈0.4

• Power-law scaling in highly confined regime as predicted by good solvent blob models(de Gennes, Douad, Brochard)

• Hydrodynamic screening in highly confinedchannels -> Rouse dynamics

• Good agreement with experimentalmeasurements of Doyle group for λ-DNA

Equilibrium results for confined chains

Stretch Diffusivity

Jendrejack et al. JCP, 2003, Chen et al. 2004

2/3

1/4

-2/3


• Polymer migrates from wall: hydrodynamic depletion layer forms• Confinement (wall effect) is crucial• Molecules move faster than average fluid velocity

Dilute solution: pressure-driven flow

0We =

We = 1.3, ˙ γ = 3.98s−1

Center of mass distribution42 µm DNA, H = constant = 9.3 Sbulk

Distance along channel vs. time

10.5 µm21 µm42 µm84 µm

We = 10, ˙ γ = 30.8s−1

We = 100, ˙ γ = 308s−1

Jendrejack et al 2003 ,2004


DNA migration in oscillatory flow

40 µm

Before Oscillation 8 min Later

Experiment (T2 DNA, We=50, 0.25 Hz, square channel,Jo & Schwartz 2005)


Experimental resultsSimulation results

Before Oscillation 40 seconds 6 minutes

Chen et al 2005

Migration in oscillatory flow:experiment vs. simulation


velocityfluid.maxvelocityDNAavg.=fR )2//(max Hv=γ

Sugarman & Prud’homme (1988), Chen et al 2005

25 µm

Detection points at 25 cm and 200cm

detector

λ-DNA in microcapillary flow

Simulations: square channelExperiments: cylindrical channel

Comparison to capillary flow experiments

No adjustable parameters


Hydrodynamic migration mechanism

The flow induced by each endof a horizontal relaxing chainpushes the other end awayfrom the wall. (Jendrejack et al.2004)y

A horizontally aligned stressletabove a wall induces a wall-normal flow (cf. Leal 1980, Smart& Leighton 1991, Hudson 2003)

dumbbell

equivalent stresslet

induced velocity

Point force above a wallred: vy>0, blue: vy<0.

Near-field

Far-field


Kinetic theory for a single chain near awall: dumbbell model

number density(probability)

Migration tensor (3rd order)

Polymer stressKirkwood diffusivity(conformation-dependent)

Flux expression in low Re, point-dipole limit

Thick depletion layers in uniformshear flow

Migration and diffusionbalance at steady state:

Depletion layer thickness

can be much larger than Rgat high Weissenberg number

Ma et al. (2005, 2006)


Spatial development of depletion layer

Depletion layer thickness:

Uniform shear flow in semi-infinite domain

These balance in entrance region We=10,FENE-P model

Weak singularity (smoothed by diffusion)

Migration is fastest near the wall => molecules “pile up” to form a moving, spreading shock-like concentration profile

Ld

Lx

Similarity solution


Spatial development of depletion layerPlane Poiseuille flow, full numerical solution

We = 10

B/Ld ≈ 20

FENE-P Dumbbell

• From theory:

• Entrance length forFENE-P model: Lx ~ We3

• Fully developed profileonly when δh ≈ Ld:

“pileup”residence time

>>diffusion time across Ld

y


Concentration and geometry effects

Hernandez-Ortiz et al. (2008)

Equilibrium Wi = 20, φ = 10-4, HI Wi = 20, φ = 0.12, HI

Complex geometry >104 beads FEM for ud, Fixman methods for

random term

Migration away from wallsweakens with concentration Dilute: Significant depletion

from cavity Concentrated: Very weak

depletion from cavity Consistent with experiments


Adsorption depends on flow protocol

Reservoir of DNA solution

Syringe pump

Migration suppressesadsorption

“Fountain flow”at meniscus promotesadsorption



Consider reference frame moving with meniscus:


Blood flow in the microcirculation Red blood cells flow in center of arterioles,

white blood cells “marginate” What do drug delivery particles do? Many diseases/injurues disrupt normal blood

flow (heart disease, diabetes, hemorrhage) Minute amounts of “drag reducing” polymer

added to the bloodstream can improve tissueoxygenation

Dynamics of blood flow in small vessels iscomplex and poorly understood

Distribution of red and white blood cells in microcirculation


Polymer additives decrease cell-free layer thickness in invitro experiments

RBCs in buffer: fewcells near bottomwall

buffer+50 ppm PEO:many more cells nearbottom wall

Proposed particles for drug delivery: (a) liposomes, (b)wormlike micelles, (c-d) nonspherical polymeric particles


GGEM/IBM GGEM can be used as the basis for a variant of the immersed boundary method, valid

for Stokes flow. The local interactions are not put on the mesh but are included in thelocal (analytical) solution.

This can also be viewed as a low budget boundary integral method. Is there a convergence proof for this variant of IBM?


-8 -6 -4 -2 0 2 4 6 80

0.5

1

1.5

2

Δx/a

Δy/a

SimulationLac et. al.(2007)

Pair collision of elasticcapsules in shear

Δy

Δx


Red blood cells in capillaries - preliminary!



Summary• Particle level simulation and theory for flexible polymers (DNA) in microchannels, including hydrodynamic effects of confinement

• Fast simulation methods allow exploration of concentration, geometry, flow

• DNA dynamics in microchannel flow- Thick depletion layers in flow: dominated by wall-modified hydrodynamic interactions- nontrivial concentration dependence in complex flows -- not understood

• Flux expression for solution of dumbbells near a wall- depletion layer thickness normal stress- slow evolution of depletion layer

• Similar framework holds for other systems, such as blood.

Mesoscopic simulation of the dynamics of confined complex ...Mesoscopic simulation of the dynamics...

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