Semi-flexible dense polymer brushes in flow -simulation & theory

Frank Romer and Dmitry A. Fedosov

Forschungszentrum Julich GmbHTheoretical Soft Matter and Biophysics (ICS-2/IAS-2)

Institute of Complex Systems & Institute for Advanced Simulation

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Introduction

theoretical & computational methods

from meso scale to continuum

hydrodynamic interactions

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Motivation

Semi-flexible polymers at high density stiff anchored onto a surface:

blood vessels lung mucus layerinner ear

glycocalyx brush structure on the endothelial surface layer1 (EGL)

periciliary layer of the lung airway2

vestibular3 & auditory4 sensory epithelium

1S. Weinbaum et al., Annu. Rev. Biomed. Eng. 9, 121–167 (2007).2B. Button et al., Science 337, 937–941 (2012).3K. Legendre et al., J. Cell Sci. 121, 3347–3356 (2008).4J. Tolomeo and M. Holley, Biophys. J. 73, 2241–2247 (1997).

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Motivation

Modeling blood flow in vessels with explicit EGL:

scaling problem!→ implicit modeling or coarse graining

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Motivation

Can we predict the properties (e.g. height) of a dense semi-flexiblepolymer brush in shear flow for a given set of parameters:

flexural rigidity EI ,

grafting density σ and

shear rate γ on top?

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Already done?

There is a lot experimental, simulation & theoretical work done onfull-flexible polymer brushes (PB), but only a few on semi-flexible PBs.

Semi-flexible polymer brushes in equilibrium:

D. V. Kuznetsov and Z. Y. Chen, J. Chem. Phys. 109, 7017–7027(1998): scaling theory

G. G. Kim and K. Char, Bull. Korean Chem. Soc. 20, 1026–1030(1999): continuous chain model

C.-M. Chen and Y.-A. Fwu, Phys. Rev. E 63, 011506 (2000): MCsimulation

Semi-flexible polymer brushes in flow:

Y. W. Kim et al., Macromolecules 42, 3650–3655 (2009): theory &simulation

F. Romer, D. A. Fedosov polymer brushes in flow 6 / 34

Already done?

EI d2θds2 = −3πη cos θ(s)

∫ Ls u(s) ds

with: s2udy2 = 3πσ u(y)

cos θ(y)

no volume-exclusion interaction

grafting density only influence velocity

max. grafting density studied σ = 0.03

model only works for small deformations

No, it is not done!

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Simulations

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Dissipative Particle Dynamics (DPD)simulation method

Dissipative Particle Dynamics (DPD) was introduced by Hoogerbrugge andKoelman in 19925.

Particles in DPD represent clusters ofmolecules and interact through simplepair-wise forces: Fi =

∑j 6=i (FC

ij + FRij + FD

ij ).

DPD system is thermally equilibratedthrough a thermostat defined by forces: FR

ij

and FDij .

The DPD scheme consists of the calculation of the position and velocitiesof interacting particles over time. The time evolution of positions andvelocities are given by: dri = vi dt and dvi = Fi dt.

5P. Hoogerbrugge and J. Koelman, Europhys. Lett. 19, 155 (1992).F. Romer, D. A. Fedosov polymer brushes in flow 9 / 34

Smoothed Dissipative Particle Dynamics (SDPD)simulation method

Espanol and Revenga combined in Smoothed Dissipative ParticleDynamics (SDPD)6 best features of the Smoothed ParticleHydrodynamics (SPH)7 and DPD methods.

SDPD allows to introduce an arbitrary equation of state:- e.g. control compressibility.

SDPD particles has a well defined size and volume:- consitent scaling of thermal fluctuations of the fluid,- while dynamics of immersed objects is scale-free8

In SDPD transport coefficients (viscosity, thermal conductivity) are adirect input.

6P. Espanol and M. Revenga, Phys. Rev. E 67, 026705 (2003).7R. Gingold and J. Monaghan, Mon. Not. R. Astron. Soc. 181, 375 (1977).8A. Vzquez-Quesada et al., J. Chem. Phys. 130, 034901 (2009).

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Systemsimulation model

The SDPD ensemble:slit like geometry (2D PBC)

wall & fluid: SDPD particle

reflection planes at walls

Poiseuille flow in x direction:fx ,i = (dP/dx)/ρ

polymers grafted on a tetragonal lattice with alat

friction between fluid & polymer beads

We characterize the system by:

grafting density: σ = (alat/d)−2

flexibility: lp/L = EI/(kBTL)

rate: ˜γ = L3ηγ/kBT

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Polymer Beamsimulation model

The Polymer beam is represented by:

N = 10 + 1 tangential bonded beads of diameter d

1st two beads fixed

extensible worm-like chain model like

friction interaction with fluid

repulsive interaction between beads (WCAa)→ volume exclusion

aJ. D. Weeks et al., J. Chem. Phys. 54, 5237–5247 (1971).

Discretized elastic potential9 for isotropic elastic cylinder of diameter d :

U ({rk}) =∑N−1

i=0

[kb2d [ri ,i+1 − d ]2 + EI

d [1− cos ∆θ]]

with: EI/kb = d2/169X. Schlagberger and R. R. Netz, Europhys. Lett. 70, 129 (2005).

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Results

We have simulated systems with:

grafting densities σ from 0.01 to 1,

and beam elasticities lp/L of 10 and 100

at shear rates ˜γ between 101 and 106.

The Reynolds number, referring to the maximum velocity the polymerbrush is exposed at the tip and beam elasticity, varies betweenRe = ρud

η = impuls convectiondiffusion = 10−6 − 10−1 → Stokes regime10.

Height of brush is calculated from the first moment of the polymermonomer density profile11:

h = 2∫

yρ(y)dy∫ρ(y)dy

10E. Guyon et al., Physical hydrodynamics, (Oxford University Press, 2001).11M. Deng et al., J. Fluid Mech. 711, 192–211 (2012).

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Resultscomparison of methods

Comparison with LB and BD simulations from Y. W. Kim et al.,Macromolecules 42, 3650–3655 (2009):

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Resultsfrom SDPD simulations

Polymer brush height as a function of shear rate on top of the brush:

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Theory

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Theoretical Model

The polymer beam is treated as a cantilever.

Uniform behavior

Quasi 2D: no perturbation in z

Internal coordinates alongcontour line: s, θ(s)

Euler-Bernoulli beam theory:

EI dr(s)ds ×

d3r(s)ds3 = F(s)× dr(s)

ds

EI d2θ(s)ds2 = Fy (s) sin θ(s)− Fx (s) cos θ(s)

Hydrodynamic drag force: Fd(s)

Volume exclusion force: Fv(s)Boundary conditions:θ(s)|s=0 = 0 at the grafting surface, and dθ/ds|s=L = 0 at the free end.

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Shear force on the beamtheoretical model

The effective shear force acting on the beam atposition s:

Fd(s) =∫ L

s fd(s ′)ds,

with the local force density due to hydrodynamicinteraction for the x-component

f dx (s) = f d

⊥(s) cos θ(s) + f d‖ (s) sin θ(s),

and the y -component

f dy (s) = f d

⊥(s) sin θ(s)− f d‖ (s) cos θ(s)

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Shear force on the beamtheoretical model

Shear force components:f d⊥(s) = ζ⊥(s)ηux (s) cos θ(s) and f d

‖ (s) = ζ‖(s)ηux (s) sin θ(s)

Friction coefficient from slender body theorya,piecewise approximated as a cylinderb:

ζ⊥ = 8πln(L/d)− 1

2+ln(2)

,

ζ‖ = 4πln(L/d)− 3

2+ln(2)

.

aR. G. Cox, J. Fluid Mech. 44, 791–810 (1970).bC. H. Wiggins et al., Biophys. J. 74, 1043–1060 (1998).

Local velocity ux (s) depend on the hydrodynamicpenetrationc.

cS. T. Milner, Macromolecules 24, 3704–3705 (1991).

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Velocity profiletheoretical model

Local velocity u(s) depend on the hydrodynamic penetration into thebrush12:

Like Kim et al.13 we introduce an equation similar to the Brinkmanequation14 for flow in porous media but depending on local density:

d2udy2 = ζσ u(y)

cos θ(y) →d2ux (s)

ds2 = ζ(s)ux (s) σd2 cos θ(s)− dux (s)

dsdθ(s)

ds tan θ(s),

with boundary conditions: u(y)|y=0 = u(s)|s=0 = 0, and du/dy |y=L = γL

resp. du/ds|s=L = γL sin θ(L).

12S. T. Milner, Macromolecules 24, 3704–3705 (1991).13Y. W. Kim et al., Macromolecules 42, 3650–3655 (2009).14H. C. Brinkman, Appl. Sci. Res. A1 A1, 27 (1947).

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Volume exclusiontheoretical model

Discretise the beam: N = bL/dc spheres

Positions are given as a function of s:

the fx(s) =

∫ s0 sin θ(s ′)ds ′ ∧ y(s) =

∫ s0 cos θ(s ′)ds ′

Assume a repulsive interaction:

gij =

{rij 6 d : εg

kB Td [(d/rij )

α − 1]rij

rij

rij > d : 0

}with εg & 50 and α ≥ 3.

Force density fvn on a sphere n ∈ [0...N − 1] ina discretized beam:

fvn =∑

jgnj

d

Fv(s) = fvbs/dc (d ds/de − s/d) +∑N−1

n=ds/de fvn d

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Theoretical Solution

Brinkman-like equation

d2ux (s)ds2 = ζ(s)ux (s) σ

d2 cos θ(s)− dux (s)ds

dθ(s)ds tan θ(s)

Beam equation

EI d2θ(s)ds2 = Fy (s) sin θ(s)− Fx (s) cos θ(s)

Solver scheme:

0 guess a configuration: θ0(s) for s ∈ [0, L]

1 calculate volume exclusion force → Fv(s)

2 solve Brinkman-like equation → u(s)⇒ Fd(s)

3 Beam equation & DuFort-Frankel scheme15 → θn(s)

4 if∫ L

0 |θn − θn−1| ds > tol : goto 1© else : found final configuration ;)

15E. C. Du Fort and S. P. Frankel, Math. Comp. 7, 135–152 (1953).F. Romer, D. A. Fedosov polymer brushes in flow 22 / 34

ResultsSDPS vs theory

Comparison between SDPD simulation and our theoretical model:brush height

Figure : Relative brush height vs shear rate normalized by polymer elasticity

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ResultsSDPS vs theory

Comparison between SDPD simulation and our theoretical model:brush height

overestimation of volume-exclusion term Fv ← neglected perturbation in z

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Resultserror analysis

Figure : Relative deviation of brush height vs shear rate.

N 106RMS 14.6 %AAD 9.5 %Bias 8.3 %

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Resultstheoretical model

Prediction of the response to shear flow of dense polymer brushes.

EGL, σ ≈ 1

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ResultsSDPS vs theory

Comparison between SDPD simulation and our theoretical model:velocity profile

Figure : normalized velocity profiles, ∂P/∂x = const.

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Resultstheoretical model

Prediction of flow rates in channels or tubes.

micro-vascular flow

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ResultsSDPS vs theory

Comparison between SDPD simulation and our theoretical model:Relative apparent viscosity

Figure : Relative apparent viscosity vs shear rate scaled by elasticity.

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Resultstheoretical model

Prediction of app. viscosity as input for continuous methods.

microfluidic devices

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Conclusion

Simulations of polymer brushes with

densities varying between 0.01 (no interaction between polymers)and 1 (maximum density of SC packing),with elasticities in range of 2 orders of magnitude andunder top shear loads in range of 6 orders of magnitude.

Simulations h(σ,EI , γ) in good agreement with previous studies.

We propose the first theoretical model describing dense semi-flexiblepolymer brushes in a wide parameter range:

Good agreement with simulations: (∆h)max ≈ d .Model reproduces all features shown in simulation.Systematic overestimation of height due to reduced dimensionality.

Now we can predict the interaction of a densesemi-flexible polymer brush with shear flow!

F. Romer and D. A. Fedosov, Europhysics Letters 109, 68001 (2015)

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Outlook

Direct mechanical stress or deformation of the brush.

→ simulation of brush compression:preliminary results in goodagreement with theoretical model

⇒ add viscoelasticity to vessel walls:model mechanical transduction offorces due to e.g. EGL–RBCinteractions

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Acknowledgements

D. A. Fedosovco-author

Julich Supercomputing Centre (JSC)

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