Mean-Field Description of Ionic Size Effects

Transcription

1 Mean-Field Description of Ionic Size Effects Bo Li Department of Mathematics and NSF Center for Theoretical Biological Physics University of California, San Diego Work Supported by NSF, NIH, CSC, CTBP ICMSEC, Chinese Academy of Sciences, Beijing June 17, 2011

2 Biomolecular Interactions Charge effect. Left: no charges. Right: with charges.

3 Some existing works on special cases of size effects. V. Kralj-Igliç and A. Igliç. A simple statistical mechanical approach to the free energy of the electric double layer including the excluded volume effect. J. Phys. II (France), 6: , I. Borukhov, D. Andelman, and H. Orland. Steric effects in electrolytes: A modified Poisson-Boltzmann equation. Phys. Rev. Lett., 79: , V. B. Chu, Y. Bai, J. Lipfert, D. Herschlag, and S. Doniach. Evaluation of ion binding to DNA duplexes using a size-modified PoissonBoltzmann theory. Biophys. J, 93: , 2007.

4 Some existing works on special cases of size effects (cont d) G. Tresset. Generalized PoissonFermi formalism for investigating size correlation effects with multiple ions. Phys. Rev. E, 78:061506, X. Shi and P. Koehl, The geometry behind numerical solvers of the Poisson Boltzmann equation, Commun. Comput. Phys., 3, , A. R. J. Silalahi, A. H. Boschitsch, R. C. Harris, and M. O. Fenley, Comparing the predictions of the nonlinear Poisson Boltzmann equation and the ion size-modified Poisson Boltzmann equation for a low-dielectric charged spherical cavity in an aqueous salt solution, J. Chem. Theory Comput. 6, , 2010.

5 Sizes of ions are different! Sodium: 3.34 Å, Chloride: 2.32 Å.

6 Main references Bo Li, Minimization of electrostatic free energy and the Poisson Boltzmann equation for molecular solvation with implicit solvent, SIAM J. Math. Anal., 40, , Bo Li, Continuum electrostatics for ionic solutions with nonuniform ionic sizes, Nonlinearity, 22, , Shenggao Zhou, Zhongming Wang, and Bo Li, Mean-field description of ionic size effects with non-uniform ionic sizes: A numerical approach, Phys. Rev. E, 2011 (in press). Bo Li, Xiaoliang Cheng, and Zhengfang Zhang, Dielectric boundary force in molecular solvation with the Poisson Boltzmann free energy: A shape derivative approach, 2011 (submitted to SIAM J. Applied Math.).

7 Outline 1. The Classical Poisson Boltzmann Theory 2. Mean-Field Models with Ionic Size Effects 3. Non-Uniform Sizes: Constrained Optimization 4. Numerical Results 5. Conclusions

8 1. The Classical Poisson Boltzmann Theory

9 Consider an ionic solution occupying a region Ω. ρ f : Ω R: given, fixed charge density c j : Ω R: concentration of jth ionic species cj : bulk concentration of jth ionic species q j = z j e : charges of an ion of jth species z j : valence of ions of jth species e : elementary charge β: inverse thermal energy Poisson s equation: Charge density: Boltzmann distributions: ε(x)ε 0 ψ(x) = ρ(x) ρ(x) = ρ f (x) + M j=1 q jc j (x) c j (x) = c j e βq jψ(x) The Poisson Boltzmann Equation (PBE) εε 0 ψ + M q j cj e βqjψ = ρ f j=1

10 M PBE εε 0 ψ + q j cj e βqjψ = ρ f j=1 The Debye Hückel approximation (linearized PBE) εε 0 ψ εε 0 κ 2 ψ = ρ f Here κ > 0 is the ionic strength or the inverse Debye Hückel screening length, defined by κ 2 = β M j=1 q2 j c j εε 0 The sinh PBE (q 2 = q 1, c 2 = c 1 ) εε 0 ψ 2qc 1 sinh(βqψ) = ρ f

11 Electrostatic free-energy functional 1 M [ G[c] = ρψ + β 1 c j ln(λ 3 c j ) 1 ] M µ j c j 2 dv Ω j=1 ρ(x) = ρ f (x) + M j=1 q jc j (x) εε 0 ψ = ρ j=1 (+ B.C., e.g., ψ = 0 on Ω) Λ : the thermal de Broglie wavelength µ j : chemical potential for the jth ionic species Equilibrium conditions (δg[c]) j = q j ψ + β 1 ln(λ 3 c j ) µ j = 0 c j (x) = c j e βq jψ(x) Minimum electrostatic free-energy (note the sign!) G min = εε M ( 1) 0 2 ψ 2 + ρ f ψ β 1 cj e βqjψ dv Ω j=1

12 Theorem (B.L. 2009). The functional G has a unique minimizer c = (c 1,...,c M ). There exist constants θ 1 > 0 and θ 2 > 0 such that θ 1 c j (x) θ 2 x Ω j = 1,...,M. All c j are given by the Boltzmann distributions. The corresponding potential is the unique solution to the PBE. Remark. Bounds are not physical! A drawback of the PB theory. Proof. By the direct method in the calculus of variations, using: Convexity. G[λu + (1 λ)v] λg[u] + (1 λ)g[v] (0 < λ < 1); Lower bound. Let α R. Then the function s s(ln s + α) is bounded below on (0, ) and superlinear at ; A lemma (cf. next slide). Q.E.D.

13 G[c] = 1 M [ ρψ + β 1 c j ln(λ 3 c j ) 1 ] M µ j c j 2 dv Ω j=1 j=1 Lemma (B.L. 2009). Given c = (c 1,...,c M ). There exists ĉ = (ĉ 1,...,ĉ M ) that satisfies the following: ĉ is close to c; G[ĉ] G[c]; there exist constants θ 1 > 0 and θ 2 > 0 such that θ 1 ĉ j (x) θ 2 x Ω j = 1,...,M. Proof. By construction using the fact that the entropic change is very large for c j 0 and c j 1. Q.E.D. slns O s

14 PBE: εε 0 ψ B (ψ) = ρ f [ εε0 ] Define: I[φ] = Ω 2 φ 2 ρ f φ + B(φ) dv Notation B(ψ) = β 1 M j=1 c j ( ) e βqjψ 1 H 1 g(ω) = {φ H 1 (Ω) : φ = g on Ω} Theorem. The functional I : H 1 g(ω) R has a unique minimizer ψ. The minimizer is the unique solution to the PBE. B o ψ

15 PBE: εε 0 ψ B (ψ) = ρ f [ εε0 ] I[φ] = 2 φ 2 ρ f φ + B(φ) dv Ω Proof. Step 1. Existence and uniqueness by the direct method. Step 2. Key: The L -bound. Let λ > 0 and define λ if ψ 0 (x) < λ, ψ λ (x) = ψ 0 (x) if ψ 0 (x) λ, λ if ψ 0 (x) > λ. I[ψ] I[ψ λ ], ψ λ ψ, the properties of B, and the uniqueness of maximizer = ψ = ψ λ for large λ. Step 3. Routine calculations. Q.E.D.

16 2. Mean-Field Models with Ionic Size Effects

17 Electrostatic free-energy functional 1 M [ G[c] = ρψ + β 1 c j ln(a 3 2 j c j ) 1 ] M µ j c j dv Remarks. Ω j=0 ρ(x) = ρ f (x) + M j=1 q jc j (x) εε 0 ψ = ρ c 0 (x) = a 3 0 j=1 (+ B.C., e.g., ψ = 0 on Ω) ] [ 1 M i=1 a3 i c i(x) a j (1 j M): linear size of ions of jth species a 0 : linear size of a solvent molecule c 0 : local concentration of solvent Derivation from a lattice-gas model only for the case of a uniform size: a 0 = a 1 = = a M. G[c] is convex in c = (c 1,...,c M ).

18 Theorem (B.L. 2009). The functional G has a unique minimizer (c 1,...,c M ), characterized by the following two conditions: Bounds. There exist θ 1, θ 2 (0, 1) such that θ 1 a 3 j c j (x) θ 2 x Ω j = 0, 1,...,M; Equilibrium conditions (i.e.,(δg[c]) j = 0 for j = 1,...,M) ( aj a 0 ) 3 log ( a 3 0c 0 ) log ( a 3 j c j ) = β (qj ψ µ j ) j = 1,...,M. Proof. Similar to the case without the size effect. Q.E.D. Remark. The bounds are non-physical microscopically!

19 Lemma (B.L. 2009). Given c = (c 1,...,c M ). There exists ĉ = (ĉ 1,...,ĉ M ) that satisfies the following: ĉ is close to c; G[ĉ] G[c]; there exist θ 1 and θ 2 with 0 < θ 1 < θ 2 < 1 such that θ 1 a 3 j ĉ j (x) θ 2 x Ω j = 0, 1,...,M. Proof. By construction in two steps. First, take care of c 0. Then, take care of c j (j = 1,...,M). Q.E.D.

20 ( aj a 0 ) 3 log ( a 3 0c 0 ) log ( a 3 j c j ) = β (qj ψ µ j ) j = 1,...,M. The case of a uniform size: a 0 = a 1 = = a M = a. The generalized Boltzmann distributions c j = The generalized PBE cj e βq jψ 1 + a 3, j = 1,...,M. M i=1 c i e βq iψ εε 0 ψ + M j=1 q jc j e βq jψ 1 + a 3 M j=1 c j e βq jψ = ρ f A variational principle: ψ minimizes the convex functional I[φ] = εε M 0 2 φ 2 ρ f φ + β 1 a 3 log 1 + a 3 cj e βq jφ dv Ω j=1

21 ( aj a 0 ) 3 log ( a 3 0c 0 ) log ( a 3 j c j ) = β (qj ψ µ j ) j = 1,...,M. The general case: Implicit Boltzmann distributions Set D M = {u ( = (u 1,...,u M ) R M : u j > 0, j = 0, 1,...,M} u 0 = a0 3 1 ) M j=1 a3 j u j f j (u) = a 3 j a 3 0 log ( a 3 0 u 0 ) log ( a 3 j u j ), j = 1,...,M. Lemma (B.L. 2009). The map f : D M R M is C and bijective. Proof. It is clear that f is C. f is injective. det f 0, use det(i + v w) = 1 + v w. f is surjective. Note: f j (u) = r j j z = z/ u j = 0, where z(u) = M [ ( ) ] M u j log a 3 j u j 1 + r j u j. j=0 Construction: min DM z < min DM z. So all j z = 0. Q.E.D. j=1

22 Set g = (g 1,...,g M ) = f 1 : R M D M B j (φ) = g j (β(q 1 φ µ 1 ),...,β(q M φ µ M )) [ B 0 (φ) = a0 3 1 ] M j=1 a3 j B j(φ) M φ Define B(φ) = q j B j (ξ)dξ φ R Assume j=1 M j=1 q jb j (0) = 0 0 (electrostatic neutrality) Lemma (B.L. 2009). The function B is strictly convex. Moreover, M > 0 if φ > 0, B B (φ) = q j B j (φ) = 0 if φ = 0, j=1 < 0 if φ < 0, and B(φ) > B(0) = 0 for all φ 0. Proof. Direct calculations using the Cauchy Schwarz inequality to show B > 0. Also, use the neutrality. Q.E.D. o ψ

23 G[c] = 1 M [ ρψ + β 1 c j ln(a 3 2 j c j ) 1 ] M µ j c j dv Ω Theorem (B.L. 2009). j=0 j=1 The equilibrium concentrations (c 1,...,c M ) and corresponding potential ψ are related by the implicit Boltzmann distributions c j (x) = B j (ψ(x)) x Ω, j = 1,...,M. The potential ψ is the unique solution of the boundary-value problem of the implicit Poisson Boltzmann equation εε 0 ψ B (ψ) = ρ f. This is the Euler Lagrange equation of the convex functional [ εε0 ] J[φ] = 2 φ 2 ρ f φ + B (φ) dv. Q.E.D. Ω

24 3. Non-Uniform Sizes: Constrained Optimization

25 Electrostatic free-energy functional in (ψ, c)-formulation [ εε0 ] Minimize F[ψ, c] = Ω 2 ψ 2 + β 1 Q(c) dv { M j=1 Q(c) = c [ ( j log Λ 3 ) ] c j 1 without size effect M j=0 c [ ( ) ] j log a 3 j c j 1 with size effect with the constraints Conservation of mass: c j dv = N j, j = 1,...,M Ω Charge neutrality: M j=1 N jq j + f dv + σ ds = 0 Ω Γ ( Poisson s equation: εε 0 ψ = f + ) M j=1 q jc j in Ω Boundary condition: εε 0 n ψ = σ on Ω Notations N j : total number of ions of the jth species f : volume charge density σ : surface charge density

26 Electrostatic free-energy functional in (E, c)-formulation [ εε0 ] Minimize F[ψ, c] = Ω 2 E 2 + k B TQ(c) dv { M j=1 Q(c) = c [ ( j log Λ 3 ) ] c j 1 without size effect M j=0 c [ ( ) ] j log a 3 j c j 1 with size effect with the constraints Conservation of mass: Charge neutrality: c j dv = N j, j = 1,...,M Ω M j=1 N jq j + f dv + σ ds = 0 Gauss s Law: εε 0 E = f + M j=1 q jc j in Ω Boundary condition: εε 0 E n = σ on Γ Compatibility: E = 0 in Ω Ω Γ

27 Nondimensionalization The Bjerrum length: l B = βe2 4πεε 0 c j = 4πl B c j Λ = (4πl B ) 1/3 Λ f = 4πl Bf e ψ = βeψ N j = 4πl B N j a j = (4πl B ) 1/3 a j σ = 4πl Bσ e E = βee

28 Nondimensionalized (ψ, c)-formulation [ ] 1 Minimize F [ψ, c ] = Ω 2 ψ 2 + Q (c ) dv { M ( Q (c j=1 ) = c j log c j 1 ) without size effect ( log c j 1 ) with size effect M j=0 c j with the constraints c j dv = N j, j = 1,...,M Ω M j=1 N jz j + f dv + σ ds = 0 Ω Γ ( ψ = f + ) M j=1 z jc j in Ω n ψ = σ on Γ

29 Nondimensionalized (E, c)-formulation [ ] 1 Minimize F [E, c ] = Ω 2 E 2 + Q (c ) dv { M ( Q (c j=1 ) = c j log c j 1 ) without size effect ( log c j 1 ) with size effect M j=0 c j with the constraints c j dv = N j, j = 1,...,M Ω M j=1 N jz j + f dv + σ ds = 0 E = f + M i=1 z jc j in Ω E n = σ on Γ E = 0 in Ω Ω Γ

30 A Lagrange multiplier method for the case without size effect min max (E,c) (ψ,λ) L(E, c, ψ, λ) L(E, c, ψ, λ) = F(E, c) Ω ψk(e, c)dv + M j=1 λ jh j (c j ) K(E, c) = E M j=1 z jc j f H j (c j ) = z j c j dv z j N j, j = 1,...,M Conditions for a saddle point Ω E L = E + ψ = 0 ψ L = K(E, c) = 0 ci L = log c j + z j ψ + λ j z j = 0, j = 1,...,M λj L = H j (c j ) = 0, j = 1,...,M

31 Finally, solve numerically c j = N je z jψ Ω ezjψ dv Algorithm ψ = M j=1 + Boundary Conditions in Ω, j = 1,...,M z j N j e z jψ Ω e z jψ dv + f 0. Initialize ψ 0, k = 0, ω (0, 1), and tol > Find ψ that solves the boundary-value problem of M z j N j e z jψ k ψ = Ω e z jψ k dv + f. j=1 2. If ψ ψ k < tol, then stop. Otherwise, set ψ k+1 = ωψ k + (1 ω)ψ and k k + 1, and go to Step 1.

32 An augmented Lagrange multiplier method for the case with the size effect min max ˆL(E, c, ψ, λ, r) (E,c) (ψ,λ) ˆL(E, c, ψ, λ, r) = F(E, c) j=1 EˆL = E + ψ = 0 E = ψ Ω ψk(e, c)dv M M r j + λ j H j (c j ) + 2 [H j(c j )] 2 ψˆl = K(E, c) = 0 ψ = M j=1 z jc j + f a 3 ( j cjˆl = a0 3 log 1 ) M i=1 a3 i c i + log c j + (λ j + ψ)z j + r j z j H j (c j ) = 0, j = 1,...,M λjˆl = H j (c j ) = 0, j = 1,...,M j=1

33 Algorithm 0. Initialize c (0), ψ (0), λ (0), and r (0). Fix β > 1. Set k = Solve the boundary-value problem of Poisson s equation with c (k) i to get ψ (k+1). 2. Use Newton s method to solve for c (k+1) with ψ (k+1), λ (k), and r (k). 3. Update the Lagrange multipliers and the penalty parameters λ (k+1) j r (k+1) j = λ (k) j + r (k) j H j (c (k+1) j ), j = 1,...,M, = βr (k) j, j = 1,...,M. 4. Test convergence. If not, set k k + 1 and go to Step 1.

34 Newton s method for solving a3 ( j a0 3 log 1 ) M i=1 a3 i c i + log c j + (λ j + ψ)z j ( ) + r j zj 2 c j dv N j = 0 in Ω, j = 1,...,M. Ω Uniform discretization with N grids and cell volume v θj m a3 ( j a0 3 log 1 M i=1 a3 i ci m + r j z 2 j ) + log c m j + (λ j + ψ m )z j ( v N i=1 ci j N j ), m = 1,...,N, j = 1,...,M. Iteration for k = 1, 2,... for j = 1,...,M Newton s method for θ m j = 0 (m = 1,...,N) end for end for

35 Fix j. Denote Θ = (θ 1 j, θ2 j,, θn j ) T and c = (c 1 j, c2 j,, cn j ) T. The system of equations: θj m = 0 (m = 1,...,N) Θ(c) = 0. ( ) Θ 1 c = diag 1 ξ 1,..., ξ N + r j zj 2 ve e ( ) ξ m 1 a 6 1 j = cj m + a0 3 M a3 0 i=1 a3 i cm i N m=1 ξm det Θ c = 1 + r jzj 2 v N > 0 m=1 ( ) ξm Θ 1 = diag( ξ 1,...,ξ N) r j zj 2 v c 1 + r j zj 2 v N ξ ξ m=1 ξm Newton s iteration ( cj m cj m γξ m θj m r jzj 2 v ) N p=1 θp j ξp 1 + r j zj 2 v, m = 1,...,N. N p=1 ξp

36 4. Numerical Results

37 Computational setting The Bjerrum length l B = 7 Å Ω = (0, L) (0, L) (0, L) Ball B c of radius R centered in Ω Fixed surface charges σ = Ze on B c Example 1. M = 2, z 1 = 1, z 2 = +1, a 1 = 3.34 Å, a 2 = 2.32 Å, a 0 = 2.75 Å, N 1 = 120, N 2 = 60, Ze = 60e, R = 8Å, L = 80Å, grid points.

38 Ionic concentrations in the mid-plane z = 40Å.

39 Total particle number of counterion (a) Iteration steps Total particle number of coion (b) Iteration steps Convergence of total numbers of counterions (left) and coions (right) in iteration.

40 x 10 3 Total charge of the system (c) Iteration steps Convergence of total charges in iteration.

41 CPU time (s) AugLagMulti SMPBmove O(Nlog(N)) Number of grid nodes: N Log-log plot of the CPU time vs. the number of grid points. The O(N log N) complexity results from FFT.

42 Example 2. M = 2, z 1 = 1, z 2 = +1, N 1 = 2Z, N 2 = Z, R = 14Å, L = 160Å, grid points. Concentration of counterion (M) Classical PB theory a 0 =10Å, a 1 =10Å, a 2 =10Å a 0 =10Å, a 1 =10Å, a 2 =2Å a 0 =8Å, a 1 =10Å, a 2 =2Å a 0 =8Å, a 1 =8Å, a 2 =2Å Distance to a charged surface Concentration of counterions with linear size a 1, and Z = 60. Note that 1/a 3 1 = M when a 1 = 10Å and 1/a 3 1 = M when a 1 = 8Å. Observation: Maximal packing!

43 Concentration of counterion (M) σ= e/å 2 σ= e/å 2 σ= e/å 2 σ= e/å Distance to a charged surface Variation of surface charges with a 0 = 8Å, a 1 = 10Å, a 2 = 2Å. Observation: (1) Threshold. (2) Higher charge, wider saturation.

44 Example 3. M = 3, z 1 = +3, z 2 = +2, z 3 = +1, Z = 200, Concentration of counterion (M) (a) z 1 N 1 = z 2 N 2 = z 3 N 3 = Z/3, R = 10Å, L = 80Å, grid points Distance to a charged surface Concentration of counterion (M) Distance to a charged surface (a) The classical PB solution: no size effect. (b) The size effect with a 0 = a 1 = a 2 = a 3 = 5Å. Observation: non-monotonocity and stratification! (b)

45 Concentration of counterion (M) (c) Distance to a charged surface Concentration of counterion (M) (b) Distance to a charged surface (c) a 0 = 4Å and a +1 = a +2 = a +3 = 5Å. (d) a 0 = 2Å and a +1 = a +2 = a +3 = 5Å. Observation: Smaller solvent molecules, larger discrepancy.

46 Concentration of counterion (M) (a) Distance to a charged surface Concentration of counterion (M) (b) Distance to a charged surface Denote α i = z i /a 3 i (i = 1,...,M), the valence-to-volume ratios. (a) a 0 = 2Å, a +3 = 7Å, a +2 = 6Å, a +1 = 5Å. α +2 : α +3 : α +1 = : : 1. (b) a 0 = 2Å, a +3 = 7Å, a +2 = 5Å, a +1 = 6Å. α +2 : α +3 : α +1 = : : 1. Observation: The valence-to-volume ratios are key parameters!

47 Concentration of counterion (M) (c) Distance to a charged surface Concentration of counterion (M) (d) Distance to a charged surface (c) a 0 = 2Å, a +3 = 7Å, a +2 = 6Å, a +1 = 4Å. α +1 : α +2 : α +3 = : : 1. (d) a 0 = 2Å, a +3 = 8Å, a +2 = 6Å, a +1 = 4Å. α +1 : α +2 : α +3 = : : 1. Observation: The valence-to-volume ratios are key parameters!

48 5. Conclusions

49 Summary Minimization of electrostatic free-energy functional: unique set of equilibrium concentrations and electrostatic potential. Uniform size: generalized PBE. Non-uniform sizes: implicit PBE. Constrained optimization methods for the case of non-uniform ionic sizes. Predictions and discoveries: Counterion saturation near the charged surface; Counterion stratification near the charged surface; The ionic valence-to-volume ratios α j = z j a 3 j (j = 1,...,M) not just the valences z j (j = 1,...,M), are the key parameters in the stratification.

50 Discussions Not included and studied: optimal packing; the Stern layer; and charge inversion. Analytical studies of the differences between a uniform size and non-uniform sizes. Any consequences of the discovery of the importance of the valence-to-volume ratio? Mean-filed models still can not predict the ion-ion correlations. New, consistent, and efficient models? Applications to variational implicit solvation.

51 Thank you!