Tracer diffusion in polymer solutions - ESPResSoespressomd.org/html/ess2012/Day5/pkosovan.pdft.de Tracer diffusion in polymer solutions Peter Košovan1 Olaf Lenz1 Christian Holm1 Apostolos

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Tracer diffusion in polymersolutions

Peter Košovan1 Olaf Lenz1 Christian Holm1

Apostolos Vagias2 Kaloian Koynov2 George Fytas2

1. Institute for Computational Physics, University of Stuttgart2. MPI for Polymer Research, Mainz

ESPRESSO Summer School, Stuttgart, 12.10.2012

P. Košovan et. al. Tracer diffusion 1/27

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Diffusion – how and why?

• Diffusant in an environment

• Drug release, chemical reaction kinetics

• Biology

Methods to study diffusion in solutions

• Particle tracking

• Scattering of light, neutrons, X-rays . . .

• PFG NMR

• Fluorescence recovery after photobleaching (FRAP)

• . . .

• Fluorescence Correlation Spectroscopy


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Fluorescence correlation spectroscopy (FCS)

• Fluorophores(fluorescent tracers)

• Selectivity

• Single-molecule sensistivity

• Sub-micron sized focal spot

• Tracer diffusion in itsenvironment

• Popular in biosciences• Works also in vivo

Focusedvolume

Excitation beam

Flu

ore

sce

nce Objective

optics

Laser source

Collector optics

Dichroic mirror

Detector

Schematic representation ofexperimental setup


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Typical FCS tracers

W. Chiuman, Y. Li, Nucl. Acids Res. (2007) 35 (2): 401-405 doi: 10.1093/nar/gkl1056


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Analysis of FCS experiments

• Autocorrelation of fluorescence intensity fluctuations

G(t) =〈δI(t0)δI(t0 + t)〉〈δI(t)〉2

• Inverse problem – solved by curve fitting

• Single-component diffusion

G(t) =1N

(1 +

4Dtw2)−1(

1 +4Dt

s2w2)−0.5

• Multi-component diffusion + photophysical relaxation

G(t) =Q(t)

N

n∑i=1

Fi

[(1 +

4Di tw2

)−1(1 +

4Di ts2w2

)−0.5]


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A glimpse on experimental FCS data

Two different tracers in dilute polymer solution

0.00

0.20

0.40

0.60

0.80

1.00

10-6

10-5

10-4

10-3

10-2

10-1

N G

(t)

/ Q

(t)

t / s

(a) Experiment

A488 2.3e-3 g/ml

Rh6G 2.6e-3 g/ml

-0.05

0.00

0.05

10-6

10-4

10-2

Re

sid

ua

lst / s


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Crowding-induced slowdown

10-3

10-2

10-1

100

10-6

10-5

10-4

10-3

10-2

10-1

100

D / D

0

c [g/ml]

c = c*

A488

A647

Polymer


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Concentration regimes in polymer solutions

Dilute Semidilute Concentrated

• Overlap concentration:

c∗ =N

43πR

3g∼ N

N−3ν∼ N−2ν ≈ N−1.2


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Diffusion of polymers in solution

Rouse: no hydrodynamics

• Real concentrated solutions• Langevin thrermostat• Rouse time

τR ∼ τ0N(1+2ν)

• Long time scales τ � τR

DR =kBTζR

=kBTNζ

Zimm: hydrodynamics

• Real dilute solutions• LB-fluid, DPD• Zimm time

τZ ∼ τ0N3ν

• Long time scales τ � τZ

DZ =kBTζZ∼∼ kBT

ηsbNν

• Zimm is faster than Rouse for the same N


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One more tracer

10-3

10-2

10-1

100

10-6

10-5

10-4

10-3

10-2

10-1

100

D / D

0

c [g/ml]

c = c*

A488

A647

Rh6G

Polymer


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. . . and one more

10-3

10-2

10-1

100

10-6

10-5

10-4

10-3

10-2

10-1

100

D / D

0

c [g/ml]

c = c*

A488

A647

Rh6G

QD

Polymer


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Clue: polymer-tracer interactions

10-2

10-1

100

101

102

103

0 10 20 30 40 50

Inte

nsity [A

.U.]

Distance from surface [µm]

glass gel solution

Rh6G

QD

A488

A647


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Model system setup

• 20 athermal polymers(WCA potential)

• Chain length M = 50

• 10 athermal tracers(WCA potential)

• 5 attractive tracers(LJ potential)

• Variation of �LJ• Concentrations:

c/c∗ ∈ [10−3 : 102]

• σLJ = 1 nm, D0 = D(Rh6G)

• Langevin thermostatP. Košovan et. al. Tracer diffusion 13/27

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Relevant observables

• G(t) from simulation trajectory

G(t) =〈

exp(−∆x

2(t)w2

− ∆y2(t)

w2− ∆z

2(t)s2w2

)〉,

F. Höfling et. al., Soft Matter, 7, 1358–1363 (2011)

• Realization in ESPRESSO:

set fcs [correlation new obs1 $tracer_positions \

corr_operation fcs_acf $wx $wy $wz dt 1.0 \tau_max $tot_time tau_lin 16];

correlation $fcs autoupdate start...# integration loop...correlation $fcs finalizecorrelation $fcs write_to_file "Gt.dat"


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Fits to G(t)

0.00

0.20

0.40

0.60

0.80

1.00

10-8

10-7

10-6

10-5

10-4

10-3

N G

(t)

/ Q

(t)

t / s

(b) Simulation

c = 3.2e-04 g/ml

Attractive

Repulsive

-0.05

0.00

0.05

10-8

10-6

10-4

Re

sid

ua

lst / s


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Athermal tracers: crowding-induced slowdown

10-3

10-2

10-1

100

10-6

10-5

10-4

10-3

10-2

10-1

100

D / D

0

c [g/ml]

c = c*

A488

A647

Athermal (sim)

Polymer

Polymer (sim)


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Attractive (�LJ = 2.5): two-component diffusion

10-3

10-2

10-1

100

10-6

10-5

10-4

10-3

10-2

10-1

100

D / D

0

c [g/ml]

c = c*

PNiPAAm

Polymer (sim)

Rh6G 280k

Attractive (sim)


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Attractive (�LJ = 2.0): gradual slowdown

10-3

10-2

10-1

100

10-6

10-5

10-4

10-3

10-2

10-1

100

D / D

0

c [g/ml]

c = c*

A488

A647

Athermal (sim)

QD

Weak Attr. (sim)

Polymer (sim)

PNiPAAm


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Amplitudes = fraction of bound tracers

K =[T][P][TP]

0.00

0.20

0.40

0.60

0.80

1.00

10-5

10-4

10-3

10-2

10-1

Fslo

w

or

f

c [g/ml]

Fslow (ε=2.5)


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K =[T][P][TP]

0.00

0.20

0.40

0.60

0.80

1.00

10-5

10-4

10-3

10-2

10-1

Fslo

w

or

f

c [g/ml]

Fslow (ε=2.5)

f (ε=2.5)


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K =[T][P][TP]

0.00

0.20

0.40

0.60

0.80

1.00

10-5

10-4

10-3

10-2

10-1

Fslo

w

or

f

c [g/ml]

f (ε=2.0)

f (ε=2.5)

Fslow (ε=2.5)


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K =[T][P][TP]

0.00

0.20

0.40

0.60

0.80

1.00

10-5

10-4

10-3

10-2

10-1

Fslo

w

or

f

c [g/ml]

f (ε=2.0)

f (ε=2.5)

Fslow (ε=2.5)

Fsow Rh6G

Fslow A488


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Comparing different length scales

Tracer trajectory (�LJ = 2.5) compared to different focal spot sizes

w ≈ 30 nm w ≈ 100 nmJ. Enderlein, PRL 108, 108101 (2012)


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History-dependent observable: binding lifetime

• Definition: time interval between two events(binding or unbinding)

• Event = change of a state• Need to know the state in the past• Realization in ESPRESSO:

set bound_lft [observable new interaction_lifetimes \type $type_tr_att type $type_mon $cut 1];

...for {set i 0} {$i < $maxi} {incr i} {

integrates $nsteps;observable $bound_lft update

}...set lifetimes observable $bound_lft print;


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Measuring the bound lifetimes

10-3

10-2

10-1

100

0 2 4 6 8 10 12 14 16 18

Pro

babili

ty

Lifetime, τ [µs]

(a) Bound state lifetimes

ε=2.0

ε=2.5

10-3

10-2

10-1

100

0 2 4 6 8 10 12 14 16 18

Pro

babili

ty

Lifetime, τ [µs]

(b) Free state lifetimes

ε=2.0

ε=2.5

• Survival probabilityP(t) = exp(−t/τ)


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Quantitative comparison of length scales

L2bound = 6Dboundtbound

L2free = 6Dfreetfree

�LJ tbound[µs]

Lbound[nm]

tfree[µs]

Lfree[nm]

2.0 0.45 11 4.63 1012.5 2.17 54 2.53 75.0

�LJ = 2.0 : Lbound � w = 30 nm

�LJ = 2.5 : Lbound ≈ w


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Effective diffusion coefficient (�LJ = 2.0)

Deff = fDbound + (1− f )Dfree

10-3

10-2

10-1

100

10-6

10-5

10-4

10-3

10-2

10-1

100

D / D

0

c [g/ml]

c = c*

DeffQD

Weak Attr. (sim)

Polymer (sim)

PNiPAAm


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Resolved issues

• Diffusion slowdown critically depends on polymer-diffusantinteractions

• Universal behaviour for athermal diffusants

• Slowdown deep in the dilute solution for interacting diffusants

• FCS can resolve the two processes when attraction is strongenough

• It yields Deff when attraction is weaker

• Clear comparison of length scales from simulations


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Open questions

• Deviatoin of Rh6G slowdown from simple binding.

• Saturation of Fslow ≈ 0.5

• Origin of slow A488 slow component at high c


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Thanks and Acknowlegement

• Experimental collaborators:A. Vagias, K. Koynov, G. Fytas

$ DFG SPP 1259 Intelligente Hydrogele

Thank you for your attention!


MotivationExperimentsSimulation modelSummary and conclusions

Documents

Tracer diffusion in polymer solutions - ESPResSoespressomd.org/html/ess2012/Day5/pkosovan.pdft.de Tracer diffusion in polymer solutions Peter Košovan1 Olaf Lenz1 Christian Holm1 Apostolos