��������� ��� �� ���� �� ������ ������ �� �� � ��������������� �� � ������������� ������� ���
�������� ����� �� � �� ��������
�������� ������� �� ���� �������� ���� ��������� ���������� ��� �� ��������� ����������� �������� ��
!"���# ��$���%����% ��&# ��$���%��'�% (�)���# )������*���+��+,�
������ ������� �� ��� ������� � ����� ��� �� ������ �� �������� ������ ������������ ��� ��� ��� �� ��������� �� �� �������� ��� ������ !� ������ ���� ������������������ ����������� �� ��� ������� ���� ����� ����� ��� �� ����� � ������� ���������� ��� ��� ��� ���� � ����� �� ��� ������ ��� ������� � ��� �� �������"���� �� ���"�������� ����#�� $%"&' (���������� �� �� �� � �������� ���� � �������� ������ �� ���������� ��� ������� �� ���� ������� ���������� ������� ���� �����$)*' ! ��� �������� ������� �� ������ ������� $+&' �� ������� ��� ������ ������������� � ���� ������� �������� � ��������� � ����� ���������� �� ������ ���� ������� ������� ������# ������ ���� ��� �#����� �������� ,��� � ��� �������� ���������� ��� ������� ������ $&' ��� �� ������ �� ��� ������� ������# ��� ��������� � �������� ������ �� � �� ����� � �� ��� ��������������� ������� �� � �� ��� ������ �������
! ��� ������ ��������� �� ���� ��� � ���� �� � ���"������ ������� ��� �������� ����������� ����� �� ������ ��������� ��� � ���� �� �������� � ������� ������ ��"������ �� - ����� � ����� . / ��� ��������� ����� �� � �������� ����� ��# 0������ ��� 1���� �2������ �� ����� ��� ���� �������� ���� ��� ������������� ������������� 3����"4��� �������� ������ ���� � ������ ������� �������� ,��� � �� � /� ������ ����� �� ������ ��������� ��� ���� �� ����� ��� ������� ������ �� ���� ������������� �� ����� ��� ���������� ���� ��� ,���"��5� ������� ��������� ����� �� ������ ��� ����������� �� � . ���6� �� � ��� 78����� ����� �� . ����� . 6 ��� ��"���� �������� ���������� ���������� ������ �� ��� 4���� ���� �������� � ��� �#������������� ,��� � ��� �� ������ ����� � � ���� ���� ��� ������ ��������� �� ��� � ���� "���� ���������� � ��� �������� �� ������ ��� ������� �� ����� �������� ���������� ����� 9�� � ����� �� $&%/%%'
��� ������� ������ ����� �� ��� ��-���� �.�������� � :�� % ��� ���� . �������;������� /�/)�� � �,���� �������� /�/+�� � ���� �������� �� /�/<�� � �/ == �>�� ,���� �2������ ��� ������� ������ ����� �� ��� 91� �� / %* �>�� ��� ����� �� ?"��� �� ��8�����
:����� %@ ��� ��������������� ������� �� ��� ���"������ ������� �� ���� ������ ����������� �� �������� ���� ��� ��� ������ ������� �,���� �������A / ++ �>��� ������� ������� ���� �������� �� ����� ������� ��2����A / == �>��� ��������
:����� ;@ ��� ���"������ �������� ���� ��� �������� �� ��������� �� ����� ���� $����'�2��� �� $�����' ������ ��� ������ ����� �� �������� ��� �������� �� ����� ��� ����������� ���� �� ���� ��� �������� �� ������ ��� �������A ����� ������� ��� ������ ��������
�� ��� � ��������� - =/ ��� ������5���� �� ��� ������� �� �� � ;%����������B���� ��� ���� ��������� �� . ��������� $%;' ��� ����� ���� �� ������ = * �� & ) ���������� ��������� �� �#������ ����� ������� ���� �� �C��� ��� ����� �� ��� �����"���������� 0� ��� ���� ��� ������� ��� ��� � �������� ������� �� �������� �� ������ � ��� ������� �� ����� ����� �� �������� ���� �������� �� ��� ������ ������� ���������������� $%=' !� �� ���� ���� ��� ���"������ ������� �� ���� ��������� �� ������������ ��� ��� ��������� ������� �� ��� ���� ������ �� ������� ������ �����
:����� ; ����� ��� ������ ������� ��� ���� ��� ��������� �� ����� ��� ��� ����"���� ��� :�� ��������� ��� ����� ������ �� ���� ������� ��� ����� �� �������� ��� "�������� �� ����� �� =/ ����� ��� ��� ������� ��� ������� ��� � ����� ��� �������� �� ��� ��� ����� ����� ��� ��� %/ ��� �������� �� ����� �� ���� ��� ������� ���� ��������� � �� ��� ;/ ����� �������� �� ��� ��� ������� �� ��� ������ ������� ������� ! ��� ���� ��� � ����� ��� ��� ������ �������� �� ��� �������� �� ���� ��� �������� ������ ��� ���� ������ �������� �� ����� � � /�/;��� 0�� ��� ��� ������ ����� �� ��� ������� �������� �� �� ������� �� � ���� � ��� ����,��� ������� ������� �� ������� � � =�%���
���� ���� ��� ��������� � ��� ���� � ��� 1� %6/=6;%) �;//=� ���� ��� ������ �� D������� ������� �� E����� �� 4��� ��� ���������� ��� ��������� ���� ����������������� �� ��� F������� �� ������� E������������ !�������
����������
%� D ��5����"����� � 3�5���"������ 9 4 G������ �+ /"�)+!"�+ %0 =+% �%*)6� ;� G 0 0����� �� E 7 ���� /�����. � ���� 11�� ;;* �%**+� =� � � 1�� � � �� �������� �� 7 ! E��������� !"�+ 2��+ 3���+ %� %6+) �;///� <� H ������ � G��� �� I I����� !"�+2��+3���+ %� )&; �;///� 6� � 3�5���"������ �� D ��5����"����� �+/�����. ���� �+���+ �0� ;)6 �;//%� +� � ����� �� � �� �������� �+/"�)+!"�+11� 6+& �;//%� &� � ����� � �� �������� ( �+!"�+�+�($� =&% �;//;� )� � B I����� B � I����� 4�����,+/"�)+�'� & �%**6� *� I D���� � H E�, � �� �� �+��.+/"�)+� ��� 6/;= �;//;� %/� 9 3�� 4 "3 B��� �� � �8���� !"�+2��+3���+� $' =)6) �%**+� %%� 4 "3 B��� 2��+��.+!"�+� $� )%= �;///� %;� � 4 ���� �+/"�)+!"�+ �1� *;< �%*+*� %=� � ����� 3� ��)� ���.�)��5�0�0�1� �;//=�
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/