SUM RULES FOR IONIC FLUIDS AND CRITICAL BEHAVIOURS Angel Alastuey - - PowerPoint PPT Presentation

SUM RULES FOR IONIC FLUIDS AND CRITICAL BEHAVIOURS

Angel Alastuey

ENS Lyon and CNRS, FRANCE

November 6, 2015 Collaborator : Riccardo Fantoni (Trieste, ITALY)

INTRODUCTION/Sum Rules

• Sum Rules relate

Microscospic correlations → Macroscopic or universal quantities

• Sum rules express screening properties :

◮ Conceptual

interest

◮ Constraints

for approximate theories

• Valid for the bulk or interfaces, static or dynamical properties

Many contributions from JANCO [Jancovici-Samaj,2010]

INTRODUCTION/Two-Component Plasmas

• S : Classical two-component plasmas with

Coulomb interactions regularized at short distances :

◮ Charged soft spheres

uαγ(r) = qαqγ r [1 − exp(−καγr)]

◮ Charged hard spheres

uαγ(r) = ∞ for r < σαγ and uαγ(r) = qαqγ r for r ≥ σαγ Symmetric versions qα = ±q ; καγ = 1/d ; σαγ = σ → RPM

• Liquid-gas transition

[Caillol-Leveque,2014] [Das-Kim-Fisher,2012] Question: Why the FOURTH MOMENT OF CHARGE CORRELATIONS diverges when approaching the critical point ? [Das-Kim-Fisher,2011] (Refined finite-size scaling)

INTRODUCTION/Strategy and Outline

• Strategy :

◮ Apply cext(r) = δqext exp(ik · r) ; δqext → 0 (i) ; k → 0 (ii) ◮ Local approach → Non-neutral homogeneous TCP ◮ Suitable framework → Density Functional Theory ◮ Compare with Linear Response using ˜

S(k) FOURTH MOMENT of S(r) in terms of THERMODYNAMIC FUNCTIONS

• Outline :
• 1. Non-neutral TCP
• 2. Density Functional Theory
• 3. Fourth moment
• 4. Critical behaviours

NON-NEUTRAL TCP/External Potential

• No external potential →

Always neutrality in the bulk [Lieb-Lebowitz,1972] (Proof for quantum Coulomb matter) How to induce a net charge distribution

α qαρα = 0 in the bulk ?

• Apply the external electrostatic potential ϕB(r) created by

the external uniform charge distribution cB = −

α qαρα →

HB = H +

• i

qαiϕB(ri) [Jancovici,1981] (Exact solution for the 2D OCP)

NON-NEUTRAL TCP/Auxiliary System

• S∗ : TCP immersed in a background with density cB

H∗ = H +

• i

qαi

• Λ

dr cB |ri − r| + 1 2

• Λ2 drdr′

c2

B

|r′ − r| (i) Homogeneous and neutral :

α qαρα + cB = 0

(ii) limTL βF ∗/Λ = f ∗(β, ρ1, ρ2) with ρ1 and ρ2 independent

• SB : TCP submitted to ϕB

(i) Same densities distribution functions as S∗ (ii) FB differs from F ∗ by electrostatic energy of

α qαρα

FB = F ∗ + 1 2

• Λ2 drdr′

αγ

ρα qαqγ |r′ − r|ργ

• S = S∗ for

α qαρα = 0 → f (β, ρ) = f ∗(β, ρ1, ρ2) with

ρ = ρ1 + ρ2

DFT/Framework

• Grand-canonical description of S with inhomogeneous fugacities

zα(r) = exp(βµα − βV ext

α (r)) induced by external potentials

→ Inhomogeneous densities ρα(r) = zα(r) δ ln Ξ δzα(r) → Free energy functional βF(ρ1(·), ρ2(·)) =

• dr(βµα − βV ext

α (r))ρα(r) − ln Ξ

• Fundamental equation of DFT :

δF δρα(r) = µα − βV ext

α (r)

Remark : Functional F is not exactly known in general

DFT/Gradient Expansion

• Slow spatial variations → Expansion of F in powers of the

spatial derivatives of ρα(r) where the reference ingredients depend

• n the homogeneous system S∗

βF(ρ1(·), ρ2(·)) = 1 2

• drdr′

α,γ

ρα(r) qαqγ |r′ − r|ργ(r′) +

• drf ∗(ρ1(r), ρ2(r))

+ 1 12

• dr
• α,γ

M∗

αγ(ρ1(r), ρ2(r))∇ρα(r) · ∇ργ(r) + ...

M∗

αγ : 2nd moments of c∗ αγ(r) + βqαqγ/r; direct correlations c∗ αγ

[Hohenberg-Kohn,1964] (Functional of the electron gas) [Yang-Fleming-Gibbs,1976] (Square-gradient terms)

DFT/Induced Densities

• Consider S submitted to ϕext(r) created by

cext(r) = δqext exp(ik · r) → V ext

α (r) = qαϕext(r) ◮ Take the Laplacian of DFT equation ◮ Take δqext → 0 and linearize with respect to

δρα(r) = Cα(k) exp(ik · r)

◮ Take k → 0 and keep lowest-order terms

• Linear coupled equations determine Cα(k) up to order k2

included (χ−1

αγ = ∂2f ∗/∂ρα∂ργ):

(4πβq2

1 +χ−1 11 k2)C1(k)+(4πβq1q2 +χ−1 12 k2)C2(k) = −4πβq1δqext

(4πβq1q2 +χ−1

21 k2)C1(k)+(4πβq2 2 +χ−1 22 k2)C2(k) = −4πβq2δqext

FOURTH MOMENT/Thermodynamic Expression

• Compare induced charge

α qαδρα(r) obtained by DFT to linear

response expression

• α

qαδρα(r) = −4πβ k2 ˜ S(k)δqext exp(ik · r) with ˜ S(k) = I2k2 + I4k4 + ... → I4 = −

ρ2 (4π(q1−q2))2β3 (χ−1 11 χ−1 22 − χ−2 12 )χT

with isothermal compressibility χT χ−1

T = ρ2

β ∂2f ∂ρ2 = ρ2 β(q1 − q2)2 (q2

2

∂2f ∗ ∂ρ2

1

+ q2

1

∂2f ∗ ∂ρ2

2

− 2q1q2 ∂2f ∗ ∂ρ1∂ρ2 )

FOURTH MOMENT/About the derivation

• Assume analytic properties of f ∗ in the vicinity of the neutral

point (ρ1, ρ2) associated with S

• The One-Component Plasma expression is recovered

I OCP

4

= − 1 (4πqρ)2βχOCP

T

[Vieillefosse, 1977]

• The Binary Ionic Mixture (point charges with the same signs

immersed in a background) expression is recovered [Suttorp,2008] (Uses the BGY hierarchy)

FOURTH MOMENT/Check at low densities

• Mayer diagrams for S∗ built with zα(r) = exp(βµα − βqαVB(r))

principle of topological reduction : inhomogeneous weights zα(r) = exp(βµα − βqαVB(r)) → homogeneous weights ρα (suppresion of articulation points)

• Chain resummations [Abe-Meeron] → Mayer-like diagrams

[weights ρα, bonds built with the screened Debye potential exp(−κr)/r and the short-range part of interactions]

• Low-density expansion of I4 :

I4 = − 1 4πβκ2 + πβ2 κ5 (

• α,γ

q3

αρα)2 + I (0) 4

+ O(ρ1/2) Sum Rule WORKS!

CRITICAL BEHAVIOURS/Observed Divergencies

• Grand-canonical Monte Carlo simulations for the symmetric

RPM [Das-Kim-Fisher,2012] Approach of the critical point with ρ = ρc and T → T +

c ◮ I4 diverges similarly to χT ◮ Violation of the [Stillinger-Lovett] sum rule at the

critical point I2=(4πβ)−1 Remarks : (i) Possible violation of SL for I2 at the critical point observed before [Caillol,1995] (ii) Violation of SL for I2 contradicts theoretical predictions/models (iii) For the asymmetric mean-spherical model : violation of SL for I2 and divergency of I4 [Aqua-Fisher,2004]

CRITICAL BEHAVIOURS/Plausible Scenario

• Sum rule for I4 → possible divergency because of χT...but

what about (χ−1

11 χ−1 22 − χ−2 12 ) ?!

• For the symmetric RPM, I4 is proportional to (χ−1

11 − χ−1 12 ) :

The divergency of I4 implies divergencies in the χ−1

αγ

THE PICTURE AT THE CRITICAL POINT (i) The system becomes dielectric with I2 finite (the fraction of free charges vanishes) (ii) Algebraic tails appear in the large-distance behaviour of S(r) [Martin-Gruber,1983] (iii) The divergency of I4 is consistent with a slow algebraic decay of S(r) and the vanishing of free-charges density

CONCLUSION AND PERSPECTIVES

• Usefulness of the sum rule for I4
• Further investigation of the interplay between critical

correlations and the Coulomb interaction (use of the BGY hierarchy)

• Consequences for the universality class ?
• Application to the two-dimensional case (Cross-over from

Kosterlitz-Thouless transition to a liquid-gas transition)