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Molecular Dynamics Study of Charge Inversion of a Rod-Shaped Macroion by Polyelectrolyte Counterions

Motohiko Tanaka (NIFS, Japan)A.Yu.Grosberg (U.Minnesota, USA)

http: //dphysique.nifs.ac.jp/E-mail: mtanaka @nifs.ac.jp

Outline:* Electrophoresis by molecular dynamics simulations* A cylindrical macroion with polyelectrolyte counterions

Gel Symposium 2003 - Nov.17-21, 2003 (U. Tokyo, Japan)

■ Definition:A macroion attracts many counterions (coions also follow)so that they are more than sufficient to neutralize the macroion, leading to reversal of charge sign of net charge.

■ Phenomenon in:room-temperature electrolyte solution.

■ Occurrence conditions:1. Electrostatic correlations are strong:

(electrostatic energy) > (thermal energy)

2. Asymmetry exists between counterions /coionsCounterions are multivalent , or small in size

● Strong Coulombic correlations of surface counterions(not of charge cloud) are essential.

Charge inversion (over-charging /-screening)

+ -

-

Applications of charge inversion

● Stable colloidal dispersion is formed by electrostatic mutual repulsion.

● Gene therapy – gene delivery to living cellsvia electrostatics + hydrophobicity

ExperimentsSukhishvili et al. (1993): single DNAWalker and Grant (1996): latex+DNAGelbart, Pincus et al. (1998): polyelectrolyteHidalgo-Alvarez et al.: latex particles

Theory and simulations

Gonzalez-Tovar et al. (1985-2003): hypernetted chain theorySjostrom et al.(1996); Greberg and Kjellander (1998): MCNguyen and Shklovskii (2000-2001); Levin (2002): analytical theoryMessina, Holm and Kremer (2000-2003): MDTanaka and Grosberg (2001-2003): MD

+

+

++

++

++-

--

+

cell-

Surface double layers at charge inversion

Perez et al. Mol.Phys. (2002)

Hypernetchain theory (Gonzales-Tovar, Lozada-Cassou et al., 1985-2003)

Poisson-Boltzmann ion-ion correlations

Radial distribution functions become non-monotonic for counterions and coions

>> An electrostatic double layer is formedat the macroion surface

Poisson-Boltzmann theory is not valid:No ion-ion correlations includedNo charge inversion predicted

g(x)

counterions

coions

Poisson-BoltzmannHypernetchain

x/a

Hypernetchain

ion radius

Poisson-Boltzmann

Series of our molecular dynamics simulations:

1. Static model (immobile macroion):

observable: RDF J.Chem.Phys. (2001)

2. Electrophoresis model (mobile macroion):

observable: mobility Euro.Phys.J. E (2002)

3. Asymmetric saltA rod macroion with polyelectrolyte

observable: mobility and RDF Phys.Rev.E (2003)

>> Is a single DNA charge inverted? Cond-mat/0311009

Molecular dynamics of charge inversion: Static macroion

Small ions moving in the Langevin (fictitious) fluid A fixed macroion

Ions in the vicinity of a macroionRadial distribution functions

counterions

coions

r/a

integratedcharge

Tanaka and Grosberg (J.Chem.Phys. 2001)

Q / Q ~ 160%by a static observable

peak 0

counterions

coions

Q (r

) /Q

oρ

(r)

charge inverted

Why do we apply an electric field?

1. Static profile (radial dist. function) is not alwaysa good index of “charge inversion”, because only bound ions can drift with a macroion.

>> Electrostatically bound only if Φ(ion) > kT

2. Mobility is the direct proof of charge inversion, and is based on the net charge.

-- the drift direction reveals the sign of the net charge of the complex

3. Electrophoretic mobility is important in applications:

material separation gene delivery

Other conditions:

(1) Charge neutrality is kept

(2) Use excess counterions

(3) Strong electrostatic correlations

MD simulation with particle solvent

Boundary condition: periodicHeat bath – drain Joule heat

macroion

coioncounterion

neutral

A macroion (Rod)CounterionsCoionsMany neutral particles (solvent)

Players:

Electrophoresis study by molecular dynamics

● Particles: A macroion, counterions, coions, saltNeutral particles as solvent – non Langevin fluid

● Newton equations of motion

● Periodic boundary conditions – Ewald sum (PPPM method)

● Thermal bath only on neutral particles to drain Joule heat

Electrophoresis study of charge inversion

Macroion and its vicinity

All ionsin

tegr

ated

char

ge(a

t pe

ak)

drift

spe

ed

time (τ ~ 1 ps)

charge inverted

trivalent counterionsmonovalent coions

λ = 5a σ ~ 0.26e/aB2

non-inverted

Tanaka and Grosberg (EPJ, 2002)

Time history of charge and drift

Not much drift corresponding to Q

0

Drift speed against the applied electric fieldLinear / nonlinear regimes of electrophoresis

drift

spe

ed o

f the

mac

roio

n co

mpl

ex

linearregime

nonlinearregime

disruption of the complex

E= 10 V/cm !!6

electric field

Real numbers of charge inversion phenomenon

Phenomenon in solution (water) ε ~ 80 Requirements: strong electrostatic correlations

multivalent counterions

Physical scales

electric field E << E with E ~ 10 V/cm (nonlinear)

E d / εk T ~ 0.1 at room temp.time > 10 ps

drift speed of a macroion v ~ Q E/n ~ 0.01 cm/s

for E = 100 V/cm

c c

c

6

B

*d

w

Estimate the net charge of the complex

Measure the friction by velocity decay

time

V ~ exp( -t /τ )

Bare macroion(no small ions)

Macroion complex

By momentum balance

for µ = 0.5 µ , one has Q = 4e (Q = -30e)About 15% of the bare charge

doubled for a fatmacroion

0*

0

V

Hydrodynamic interactions?

In MD simulations, a thermal bath is adopted to drain the Joule heat due to the applied electric field.

Observations:● Solvent flow was Fourier analyzed, but no flow pattern

was identified.● Runs with and without the thermal bath almost agreed.

Counterflow around the moving macroion is screenedat short distances (comparable with the Debye length),

because electrostatically interacting ion environment absorbs the momentum associated with the macroion.

This effect was proposed previously.. (Long, Ajdari et al. 1996, Viovy 2000)

A spherical macroion with monovalent salt

(a)

(b)

mob

ility

salt ionic strength

σ ~ 0.26 e/a ■, 0.08 e/a ●2

w/ excess Z-ions

w/o excess Z-ions

counterions coionstrivalent monovalent salt0.0017 /a = 1 mol/l

0.08 e/a = 0.65 C/mµ ~ 21 (µm/s)/(V/cm)

2 22

0

w/

w/o

2□

σ /σ ~ (n Z)Nguyen & Shklovskii (2000)

* 1/2

Tanaka (PRE, 2003)

sI

A rod-shaped macroion with monovalent salt

trivalent counterionsmonovalentsalt

mob

ility

σ = 0.08, 0.06 and 0.04 e/a ~ 2σ2

salt ionic strength

for n ~ 0.006/a (3.6mol/l)3sI

µ ~ 21 (µm/s)/(V/cm)0

● ○ ■

µ=0 (spherical)

DNA

2σDNA

Charge inversion of the DNA

● There is a threshold of surface charge density for a rod-shaped macroion to get charge inverted:

σ ~ 0.19 C/m < σ

What if counterions are polyelectrolytes,as are usual in the DNA experiments?

DNA

2c, rod

A rod-shaped macroion with polyelectrolyte

with3-3-3 PE

with1-1-1 PE

mob

ility

σ ~ σ

σ = 4σ

For σ of the DNA

ionic strength of Z-ions

sphere

µ = 0.01µ ~ 3(µm/s)/(V/cm)0

n = 0.005/a ~ 0.3 mol/ l3zI

With 3-3-3 PE

µ > 0

µ < 0

DNA

DNA

rod

rod

Tanaka (cond-mat/0311009)

Inclusion of short-range attraction potential

● Mobility reversal occurs in MD simulation if a small attraction Lennard-Jones potential is included

ε = 1 k T µ > 0

for, however, rather a long polyelectrolyte chain of ten monomers.

cf. Small mobility reversal was derived by theory for surfactantwith strong hydrophobic tails χ= - 6k T (Silva et al. 2001)

LJ B

B

Reversed mobility enhancement

Entanglement of long polyelectrolyte counterions

Dragging by long polyelectrolyte counterions 1-1-1-1-

Rotating rods of finite length

3e-3e-3e polyelectrolyte e-e-e polyelectrolyte

Integrated peak charge

Angle betweenrod and x-axis

Drift speed ofthe rod

E

E

counterion

coion surface charge

Q０

<V > > 0x

Charge inversion of the DNA

There is a certain threshold of surface charge density for the single DNA to get charge inverted

σ ~ 0.19 C/m < σ

We have found that1. Electrostatic effect alone is not sufficient.2. Polyelectrolyte counterions are effective.3. Short-range attraction (hydrophobicity) and

electrostatics may be collaborating in actual situations.

4. Unpredicted: Entanglement of polyelectrolyte.

DNA2

c, rod

Conclusion

● Charge inversion was confirmed in terms of reversedelectrophoretic mobility. The net inverted charge was estimated to be 15% (at most) of the bare macroion charge.

● There is a threshold of surface charge density, at which the correlation energy of surface counterions is

（Ze） / 2εR ~ 5k T (R : Wigner-Seitz cell radius)

● In terms of reversed mobility, a rod macroion as is closer to the plane than a sphere is more persistent to added salt.Polyelectrolyte counterions are also favorable for chargeinversion.

For the single DNA – polyelectrolyte as counterions + hydrophobicity …..

WS B2

WS

Acknowledgments

Special thanks toColleagues, especially

Professor Toyoichi Tanaka （MIT, deceased)Professor Alexander (Shura) Grosberg (UMN)

Financial support:Ministry of Education, Science and Culture of Japan

Computing facilities:Minnesota Supercomputing Institute (USA)Institute for Space and Astronautical Science (Japan)Boewulf PC cluster (in-house)

References

1. M.Tanaka and A.Yu. Grosberg, Giant charge inversion of a macroion due tomultivalent counterions and monovalent coions: Molecular dynamics study, J.Chem.Phys., 115, 567-574 (2001).

2. M.Tanaka and A.Yu. Grosberg, Electrophoresis of charge inverted macroioncomplex : Molecular dynamics study, Euro.Phys.J., E7, 371-379 (2002).

3. M.Tanaka, The effects of asymmetric salt and a cylindrical macroion on chargeinversion: Electrophoresis by molecular dynamics simulations, Phys.Rev.E68,in press (2003).

4. M.Tanaka, Electrophoresis of a rod macroion under polyelectrolyte salt:Is DNA charge inverted?, cond-mat/0311009 (2003).

5. Charge inversion of a macroion in electrolyte solvent: A rotating rodwith polyelectrolyte counterions, Slow Dynamics in Complex Fluids (AIP Conference Series, 2004).

1. Ionic soft condensed matters (Polymers, Charge inversion), 2. First principle molecular dynamics (Quantum mechanics), 3. High-temperature plasmas (Magnetic reconnection, Mesoscale particle code, Planetary shocks), 4. Method of molecular dynamics and Boewulf PC cluster, 5. Published papers and books (Cover pictures)＊Video movies of molecular dynamics simulations

Ionic Soft Condensed Matters

First Principle (ab initio) Molecular Dynamics Method and Tools of

Molecular Dynamics

Cover PicturesPublications

High Temperature Plasmas

First proof of Collisionless Magnetic ReconnectionDevelopment of Mesoscale Particle CodePlanetary Shocks

Charge inversionGraphen destruction

Boewulf PC cluster

Planetary shockScalapack on PGI & Red Hat Linux 7.3

Pentium 4 and its performance

http://dphysique.nifs.ac.jp/

Documents

Molecular Dynamics Study of Charge Inversion of...Applications of charge inversion Stable colloidal dispersionis formed by electrostatic mutual repulsion. Gene therapy – gene delivery