Poisson-Boltzmann theory with Duality

Transcription

1 Poisson-Boltzmann theory with Duality A.C. Maggs CNRS+ESPCI, Paris June 016

2 Introduction to Poisson-Boltzmann theory introduction and historic formulation in terms of densities transformation to potential Dual formulation fluctuations weakness

3 Introduction Approximate theory of charged ions very different from theory of simple fluids (short ranged interactions) Gouy + Chapman: theory of charged surfaces, capacitance of charged surfaces. Debye-Hückel: chemical potential and free energy calculations

4 Introduction Mean field Gives an approximation to the free energy for monovalent ions Boltzmann distribution c i = ci 0 e βe i φ Poisson equation φ = ρ/ɛ = (ρ e + i c i e i )/ɛ for definitness mostly work with symmetric electrolyte fluctuations are neglected

5 Free energy formulation Imagine coarse-graining space, boxes side a. Number of particles in box n i = a 3 c i. Total number of states Ω = N! i n i! Partition function Z = n i Ωe βuc where U c is the electrostatic energy: U c = 1 1 ρ(r) 4πɛ r r ρ(r ) dr dr

6 Free energy ln Ω = ln N! i (n i ln(n i ) n i ) Z = n i e βf With (grand) free energy F = k B T ( i (n i ln n i ) n i ) + U c i n i µ index i is a box and species index. Minimize: k B T ln (n i /a 3 c 0 i ) = e i φ i

7 Potential as variable Can write the free energy as a function of potential ) F = ( λ cosh(βe 0 φ) ɛ( φ) + ρ e φ dr However concave... Write and U c = 1 φ 1 φ = + 1 ( φ) k B T (n i ln n i n i µn i ) = ( k B Tn i e i n i φ) (( φ) =ρ e φ + λ cosh(βe 0 φ) ) standard form of PB functional, c 0 = e βµ /a 3, λ = k B Tc 0

8 Rescaling action f = ɛ ( φ) λ cosh(βeφ) + ρ e φ ( ) f = 1 ( φ) + cosh( Γ φ) ρ e φ where φ = e 0 βφ Lengths Debye length scaling single parameter family l D = (ɛk B T /e 0c 0 ) Γ 1 = l 3 D c 0

9 Complex action for electrostatics field theory integral for φ U c = Take symmetric electrolyte ɛ ( φ) + iρφ Can perform sums over n ± ρ = ρ e + (n + n )e 0 S n = (eβµ e iβe 0φ ) n n! e eβµ+ie 0 βφ f (φ) = λ cos(φe 0 β) + ɛ ( φ) Z = Dφ e βf + iφρ e

10 Mean field theory again Use integration in the complex plane to find the real form of the action again, using cos(ix) = cosh(x). However we now have systematic ways of improving the result. Analytics possible in the limit of Γ small weak coupling However more difficult when Γ large, or intermediate

11 Loop expansion: Expand in Γ Quadratic expansion about mean-field solution F loop = k BT log e 0 β λ cosh(βe 0 φ) ɛ Matrix determinant, hard to evaluate in general geometries, can evalute in bulk F loop = k BT d 3 q log(λe 0β + ɛq ) Strongly divergent, compare to empty box: F loop = k BT d 3 q log ( (λe 0β /ɛq ) + 1 ) ne 0 aɛ + O(k BT κ 3 )

12 Excluded volume Introduce model with no more than single atom on each site. Sum becomes: Giving S (1 + βa 3 λe iβe 0φ + βa 3 λe iβe 0φ ) f k B T log(1 + βλa 3 cos(e 0 βφ)) Mean field equations then have f = ρ e φ ( φ) / k B T log(1 + βλa 3 cosh(e 0 βφ)) Applications ionic fluids weakness symmetric volumes

13 Ionic fluids Important for battery technology

14 Solving the PB equations: Pressure Analogy in one dimension to Lagrangian. Immediate first integral The Hamiltonian ɛ ( ) φ λ cosh(βe 0 φ) = P z Has interpretation as pressure (perfect gas, plus Maxwell) Inverted potential analogy Lagrangian and Hamiltonian analogy boundary conditions as speed

15 Relaxational Dynamics: Orland define flux: continuity: µ = F c = ln c i/c 0 + e i φ J i = D µ c i t = div (J i) Thus generalising to finite temperatures c i t = D ln c i e i ρ + ξ(t)

16 Fluctuations beyond PB Poisson-Boltzmann does not correctly describe heterogeneous dielectric materials, One loop correct includes however important extra contributions image charges, Born energies

17 where are we going Why is the free energy concave? it would be useful for numerical work Many complaints in the literature on the sign/concavity in Poisson-Boltzmann functions Can we evaluate the loop energies in general geometries? Aquasol (Pasteur, Delarue) Full exact electrostatics too

18 Duality in electrostatics Concave, local electrostatics U φ = { ɛ ( φ) } + ρ f φ Introduce new variable E } U = { ɛ E + ρ f φ + D ( φ + E) Integrate by parts } U = { ɛ E + D E φ(div D ρ f ) Legendre transformation { } D U = ɛ φ(div D ρ f )

19 PB functionals There is a well known variational principle in terms of φ: Symmetric electrolyte G = ρ f φ ɛ ( φ) ktc 0 cosh(qφβ) Non-convex Poisson-Boltzmann functional { } D G = + φ(ρ f div D) ktc 0 cosh(qφβ) Variations wrt φ give simply a Legendre transform G = D ɛ + L(cosh)[ρ f div D]

20 Transformed form The reciprocal free energy density that we require is g(ξ) = k BTξ q sinh 1 (ξ/qc 0 ) k B T 4c0 + ξ /q, ξ = (ρ f div D) 8 7 cosh(φ) φ 3 L(g)[ ξ ] ξ Minimum strictly equivalent to that of the conventional functional but locally minimizable

21 PB-Langevin Add mobile dipoles to Poisson-Boltzmann f = ρ f φ ( φ) Dual variations λ ion cosh (βqφ) λ dip sinh(βp 0 φ ) βp 0 φ f = ρ f φ E g(φ) h(e) + D ( φ + E) Legendre transform h f = φ(ρ f div D) E g(φ) + h(p) + E (D P) Two more transforms, P polarization: f = (D P) + h(p) + g(ρ f div D)

22 Conclusions Convexity easy minimization when coupling to other physics (biomolecules)