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 u αγ ( r ) = q α q γ [1 − exp( − κ αγ r )] r ◮ Charged hard spheres r < σ αγ and u αγ ( r ) = q α q γ u αγ ( r ) = ∞ for for r ≥ σ αγ r Symmetric versions q α = ± q ; κ αγ = 1 / d ; σ αγ = 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 c ext ( r ) = δ q ext exp( i k · r ) ; δ q ext → 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 q α ρ α � = 0 in the bulk ? • Apply the external electrostatic potential ϕ B ( r ) created by the external uniform charge distribution c B = − � α q α ρ α → � H B = H + q α i ϕ B ( r i ) i [ Jancovici ,1981] (Exact solution for the 2D OCP)

NON-NEUTRAL TCP/Auxiliary System S ∗ : TCP immersed in a background with density c B • c 2 � | r i − r | + 1 c B � H ∗ = H + � B Λ 2 d r d r ′ q α i d r | r ′ − r | 2 Λ i (i) Homogeneous and neutral : � α q α ρ α + c B = 0 (ii) lim TL β F ∗ / Λ = f ∗ ( β, ρ 1 , ρ 2 ) with ρ 1 and ρ 2 independent • S B : TCP submitted to ϕ B (i) Same densities distribution functions as S ∗ (ii) F B differs from F ∗ by electrostatic energy of � α q α ρ α F B = F ∗ + 1 q α q γ � Λ 2 d r d r ′ � ρ α | r ′ − r | ρ γ 2 αγ S = S ∗ for � • α q α ρ α = 0 → f ( β, ρ ) = f ∗ ( β, ρ 1 , ρ 2 ) with ρ = ρ 1 + ρ 2

DFT/Framework • Grand-canonical description of S with inhomogeneous fugacities z α ( r ) = exp( βµ α − β V ext α ( r )) induced by external potentials → Inhomogeneous densities ρ α ( r ) = z α ( r ) δ ln Ξ δ z α ( r ) → Free energy functional � d r ( βµ α − β V ext β F ( ρ 1 ( · ) , ρ 2 ( · )) = α ( r )) ρ α ( r ) − ln Ξ • Fundamental equation of DFT : δ F δρ α ( r ) = µ α − β V 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 ρ α ( r ) where the reference ingredients depend on the homogeneous system S ∗ β F ( ρ 1 ( · ) , ρ 2 ( · )) = 1 ρ α ( r ) q α q γ � d r d r ′ � | r ′ − r | ρ γ ( r ′ ) 2 α,γ � d r f ∗ ( ρ 1 ( r ) , ρ 2 ( r )) + + 1 � � M ∗ d r αγ ( ρ 1 ( r ) , ρ 2 ( r )) ∇ ρ α ( r ) · ∇ ρ γ ( r ) + ... 12 α,γ M ∗ αγ : 2nd moments of c ∗ αγ ( r ) + β q α q γ / 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 ϕ ext ( r ) created by c ext ( r ) = δ q ext exp( i k · r ) → V ext α ( r ) = q α ϕ ext ( r ) ◮ Take the Laplacian of DFT equation ◮ Take δ q ext → 0 and linearize with respect to δρ α ( r ) = C α ( k ) exp( i k · r ) ◮ Take k → 0 and keep lowest-order terms • Linear coupled equations determine C α ( k ) up to order k 2 included ( χ − 1 αγ = ∂ 2 f ∗ /∂ρ α ∂ρ γ ): (4 πβ q 2 1 + χ − 1 11 k 2 ) C 1 ( k )+(4 πβ q 1 q 2 + χ − 1 12 k 2 ) C 2 ( k ) = − 4 πβ q 1 δ q ext (4 πβ q 1 q 2 + χ − 1 21 k 2 ) C 1 ( k )+(4 πβ q 2 2 + χ − 1 22 k 2 ) C 2 ( k ) = − 4 πβ q 2 δ q ext

FOURTH MOMENT/Thermodynamic Expression • Compare induced charge � α q α δρ α ( r ) obtained by DFT to linear response expression q α δρ α ( r ) = − 4 πβ k 2 ˜ � S ( k ) δ q ext exp( i k · r ) α S ( k ) = I 2 k 2 + I 4 k 4 + ... → with ˜ ρ 2 (4 π ( q 1 − q 2 )) 2 β 3 ( χ − 1 11 χ − 1 22 − χ − 2 I 4 = − 12 ) χ T with isothermal compressibility χ T T = ρ 2 ∂ 2 f ρ 2 ∂ 2 f ∗ ∂ 2 f ∗ ∂ 2 f ∗ χ − 1 β ( q 1 − q 2 ) 2 ( q 2 + q 2 ∂ρ 2 = − 2 q 1 q 2 ) 2 1 ∂ρ 2 ∂ρ 2 β ∂ρ 1 ∂ρ 2 1 2

FOURTH MOMENT/About the derivation • Assume analytic properties of f ∗ in the vicinity of the neutral point ( ρ 1 , ρ 2 ) associated with S • The One-Component Plasma expression is recovered 1 I OCP = − [ Vieillefosse , 1977] 4 (4 π q ρ ) 2 βχ OCP T • 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 z α ( r ) = exp( βµ α − β q α V B ( r )) principle of topological reduction : inhomogeneous weights z α ( r ) = exp( βµ α − β q α V B ( r )) → homogeneous weights ρ α (suppresion of articulation points) • Chain resummations [ Abe-Meeron ] → Mayer-like diagrams [weights ρ α , bonds built with the screened Debye potential exp( − κ r ) / r and the short-range part of interactions] • Low-density expansion of I 4 : 4 πβκ 2 + πβ 2 1 α ρ α ) 2 + I (0) � q 3 + O ( ρ 1 / 2 ) I 4 = − κ 5 ( 4 α,γ 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 = ρ c and T → T + c ◮ I 4 diverges similarly to χ T ◮ Violation of the [ Stillinger-Lovett ] sum rule at the critical point I 2 � =(4 πβ ) − 1 Remarks : (i) Possible violation of SL for I 2 at the critical point observed before [ Caillol,1995 ] (ii) Violation of SL for I 2 contradicts theoretical predictions/models (iii) For the asymmetric mean-spherical model : violation of SL for I 2 and divergency of I 4 [ Aqua-Fisher ,2004]

CRITICAL BEHAVIOURS/Plausible Scenario • Sum rule for I 4 → possible divergency because of χ T ...but what about ( χ − 1 11 χ − 1 22 − χ − 2 12 ) ?! • For the symmetric RPM, I 4 is proportional to ( χ − 1 11 − χ − 1 12 ) : The divergency of I 4 implies divergencies in the χ − 1 αγ THE PICTURE AT THE CRITICAL POINT (i) The system becomes dielectric with I 2 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 I 4 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 I 4 • 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)