Why can like charges attract ?Why can like charges attract ?Strong-coupling approaches in soft matterStrong-coupling approaches in soft matter

Emmanuel TrizacUniversité Paris-Sud

Varenna / July 12 2012

In collaboration with Ladislav Samaj (Bratislava)

The question for todayThe question for today

When coulombic interactions prevail...

Effective interactions between like-charged colloids ?

Overall neutrality / 2 colloids fixed free energy of fluid of small ions...ATTRACTION or REPULSION ?

large + small → “dressed” large

Like charge attraction: Like charge attraction: an old controversy...an old controversy...

...but in essence mean-field ...but in essence mean-field ∇2 r = 2sinh [ r ]

Bottomline : Bottomline : mean-field mean-field repulsiverepulsive

Neu, Sader, Chan, Raimbault, E.T (1998-2000)

∇2 r =

2sinh [ r ]Prototype :

Idea :

Result : Hessian matrix of f is positive definite repulsion

Poisson-Boltzmann

AND YET... Like-charge attraction clearly possible at short distances: numerical simulations (Monte Carlo, integral equations...) + experiments

Leaving aside “long range anomalies” reported in supposedly mean-field regime

clumping of strands of actin proteins

IllustrationsIllustrations

DNA bundles

colloidal aggregates

But also : cohesion of cement, etc

T

J. Shin et al

Lobaskin and Linse

Holm / Sayar

Possibility of like-charge attraction at ''short'' distances: known since 1980s (Guldbrand, Jonson, Wennerstrom, Linse, Kjellander, Marcelja...)

Driven by correlation effects, missed at mean-field (Poisson-Boltzmann) level

Question: which analytical theory for strong coupling ?

No reliable approach on the market…

Goal is to fill this gap

… and bottomline : Monte Carlo simulations

experiments

The modelThe model simple dielectric medium uniformly charged plates point-like ions (or small hard spheres) Coulomb interactions (plate-ion ; ion-ion)

Units and parametersUnits and parameters

(Gouy)

(Bjerrum)

coupling constant

→

When does mean-field apply ?When does mean-field apply ?

Mean-field limit

(Netz 2001)

1

charged plate,

counter-ion, charge -q e

Here : study STRONG COUPLING1

Goal : interplate pressureGoal : interplate pressure under strong coupling = large under strong coupling = large ΞΞ

in general

Clear on the 1 interface problemAlso the case for 2 plates problem (interactions)Rouzina, Bloomfield, Shklovskii, Levin, Netz, Naji, Podgornik, Lau, Pincus...

In strongly coupled regime...In strongly coupled regime...Key point Key point : :

formation of 2D Wigner crystals for strongly formation of 2D Wigner crystals for strongly correlated electrons at charged interfacescorrelated electrons at charged interfaces

WignerWigner 1934Grimes/AdamsGrimes/Adams 1979 (→ e at He surface)

what is going on ? → simple physical picture → beyond the simple picture ?

Like-charge attraction: the simple pictureLike-charge attraction: the simple picture2 charged planes 2 charged planes <0, no electrolyte (counter-ions <0, no electrolyte (counter-ions ⊕ ⊕ only)only)

P = Pcin Pél

Pcin = k B T ϱcontact , repulsivePél = −2

2/ , attractive

a⊥e µ

µ ∝ 1/

= e2 / ( kT)

contact theorem, essential (exact)

Start with the 2 plates at a large distanceStart with the 2 plates at a large distance

Now bring the two plates very closeNow bring the two plates very close

Like-charge attraction: the simple pictureLike-charge attraction: the simple picture2 charged planes 2 charged planes <0, no electrolyte (counter-ions <0, no electrolyte (counter-ions ⊕ ⊕ only)only)

When d < a⊥, profile (r) uniform contact ≈ mean ≈ 2d

P ² (-1+2µ/d) attraction if d >2 µ

Argument requires µ < a⊥, min, > 1

and d < dcrit = ≈ 7 Å in water

Expansion from (known) ground state exact result !

Explains failure of mean-field !

P = Pcin

Pél

Pcin

= kB

T ϱcontact

, repulsive

Pél= −2 2/ , attractive

a⊥e µ

µ ∝ 1/

0 2µ a⊥

drepulsion attraction ?

= e2 / ( kT)

contact theorem, essential (exact)

UNBINDING !UNBINDING !

Excellent agreement with Monte Carlo, under large Symbols : MC (Moreira and Netz, EPJE, 2001)

Does it work ?Does it work ?

Take home message ITake home message I

Leading Strong Coupling order :Leading Strong Coupling order :

follows, follows, herehere, from, from

SINGLE PARTICLE PICTURESINGLE PARTICLE PICTURE

→ very essence of Netz et al Virial Strong Couling approach (the dominant one on the market)

Take home message II : forget it !Take home message II : forget it !

SINGLE PARTICLE PICTURESINGLE PARTICLE PICTURE

may hold beyond Strong Couplingmay hold beyond Strong Coupling

andand

does does notnot always apply to SC regime always apply to SC regime

worse : in general does notnot apply

Rest of the talk : ● Discuss a case where single particle picture provides correct leading behaviour : 1 plate or 2 plates without dielectric images

● Illustrate the “forget it” message : including dielectric mismatch

The single particle picture for a single plate + ionsThe single particle picture for a single plate + ions

Exact resultExact result

Beyond the simple picture :Beyond the simple picture :strong coupling regime for 1 plate (I)strong coupling regime for 1 plate (I)

Original expansion (Netz/Moreira, 2001) : field theory : virial/fugacity expansion in inverse powers of coupling π z3 (l

B )² σ

+ renormalization of infrared divergences through neutrality MEANING : from 1 particle to 2 particles correction

But correction has wrong prefactor (compared to Monte Carlo...) !Correct theory ??Reconsider expansion... possible from knowledge of the ground state Wigner crystal formed at vanishing T ; hexagonal structure Idea : consider relevant first excitations...

z = z / Gouy = z / µ~

µ = b = 1/(2 π lB σ)

Same idea as A. Lau & P. Pincus (2002)...

… but assumed counter-ions stick to plates

… excitations considered not relevant at short d.

Wigner crystal formed at vanishing T Idea : consider relevant first excitations...

Beyond the simple picture :Beyond the simple picture :the Wigner SC approachthe Wigner SC approach

One ion shifted from its lattice position

with

so that S = 11.034...Generalized Riemann function

Beyond the simple picture :Beyond the simple picture :strong coupling regime for 1 plate (II)strong coupling regime for 1 plate (II)

But above is not sufficient ; “excite” all particles

Transverse d.o.f. (also harmonic, not coupled to z) not relevant ;Single particle density follows from Boltzmann weight

Beyond the simple picture :Beyond the simple picture :strong coupling regime for 1 plate (III)strong coupling regime for 1 plate (III)

… compare to simulations : both and density profile

1/ in original works (that are thus incorrect)“cured” by introduction of a scaling parameter

We get :

OK !

Any fugacity/virial-like expansion doomed to fail

Beyond the simple picture :Beyond the simple picture :strong coupling regime for 1 plate (IV)strong coupling regime for 1 plate (IV)

Comparison to Comparison to Monte Carlo simulationsMonte Carlo simulations

Counter-ion profile close to a single plate

Field theoryFugacity exp

Back to like charge attraction... Back to like charge attraction... what is the ground state ?what is the ground state ?

Still controversial... (Falko, Goldoni, Peeters, Messina, Lowen, Netz, Lobaskin)

Solve this auxiliary problem first ! Electronic bilayer

side view

““Auxiliary” study :Auxiliary” study :Ground state of bilayer Wigner crystalsGround state of bilayer Wigner crystals

Phases identified but thresholds ? (controversial) critical behaviour ?

Looking forLooking fornew series representation of energies of all phasesnew series representation of energies of all phases

Compute energy : start by “intra”-layer term , embed within a disc

Subtract background, diverging disc radius making use of gamma identity

Heavy use of Poisson summation formula

The resulting series :

Interest : 3-pronged Discuss analytically critical behaviour Compute analytically all stability windows Exceptional convergence properties (4 terms ↔ 15 digits !)

SummarySummary

2nd order 1st order

(distance)

Like-charge attraction forLike-charge attraction forthe two plates problemthe two plates problem

Strong coupling → relevant ground state is structure I

Plate 1Plate 2

As for 1 plate case : energy cost of a given excitation

Like charge attraction forLike charge attraction forthe two plates problemthe two plates problem

Repeat 1 plate analysis...Single particle density follows from Boltzmann weight

Is correction interesting ?YES : new results in like-charge attraction regimeUsual trick : contact theorem (z=0)

Almost insensitiveto ground state

→ Exact result ; virial approach breaks down

→ Exact result... → … but but how large should Ξ be ?→ virial approach breaks down (2 body vs N body correction)→collective effects show up

WSC equation of state versus MCWSC equation of state versus MC

Monte Carlo

Lines : WSC predictionSymbols : MC (Moreira and Netz, EPJE, 2001)

Like-charge attraction phase diagramLike-charge attraction phase diagram

Monte CarloAsymp exact prediction 1/4

Exp. requires : large , finite d / does not cover full attract range! ~

Good nose

??ChenWeeks

More results :More results :The asymmetric caseThe asymmetric case

Generalize the previous approach...… ground state ? Complicated in generalbut working at fixed d and taking large → simple hexagonal structure

Field theoretic treatment (VSC) was not worked out

~

Density profile obtained from a cumulant expansion technique

Leading : Kanduc et al, 2008

close to 11.034...

Comparison to Monte Carlo IComparison to Monte Carlo I(from Kanduc, Trulson, Naji, Burak, Forsman, Podgornik 2008)(from Kanduc, Trulson, Naji, Burak, Forsman, Podgornik 2008)

Order 0

Order 1

= 86 = 0.5

Monte Carlo

Comparison to Monte Carlo IIComparison to Monte Carlo II(from Kanduc, Trulson, Naji, Burak, Forsman, Podgornik 2008)(from Kanduc, Trulson, Naji, Burak, Forsman, Podgornik 2008)

Order 0

Order 1

= 86 = 0

Monte Carlo

The pressure follows (contact theorem)The pressure follows (contact theorem)

(technical limitation)

Order 0 Order 1

Phase diagramPhase diagram

Good agreement with Monte Carlo (Kanduc et al)Good agreement with Monte Carlo (Kanduc et al)

The “forget it” take home message :The “forget it” take home message :VSC / single particle picture breaks in generalVSC / single particle picture breaks in general

Consider planar charged surface + dielectric mismatch

Take > ' → repulsive images VSC : 1 particle + image + plate potential

Cannot holdCannot hold (length scale analysis)Ions no longer “close to the surface”(was key point validating single part picture)

Monte Carlo : Courtesy of Y. Levin / A. dos Santos

>> '

Bonus feature : Bonus feature : still works at “low” couplingsstill works at “low” couplings

MC data from Moreira / Netz, EPL 2002

>> '

PerspectivesPerspectives

Some technical improvements to be achieved... long distance behaviour

Curved colloids ? in progress (G. Tellez, J. Mallarino)

Electrolytic systems ?

MessageMessage

Strong coupling : refers to large Ξ regime

“Single particle picture” should not be confused with strong coupling

Even when single particle picture holds (eg for leading order), corrections do not follow from virial route.

In general : Virial SC approach incorrect Virial breaks down due to collective effects

Thank youThank you

References :Phys. Rev. Lett. 106, 078301 (2011) for strong couplingPhys. Rev. E 84, 041401 (2011)Eur. Phys. Journal E. 34, 20 (2011) for 2D resultsPhys. Rev B 85, 205131 (2012) for bilayer ground state