050120 Derivations

Derivation of the Sterically Modified Poisson-Boltzmann Equation Vincent Chu∗, Sebastian Doniach†‡ April 29, 2005

Abstract In a 1997 paper, Borukhov et al. derived a modification to the PoissonBoltzmann equation (PBE) that incorporated steric repulsion due to finite ion sizes in the case of a solution with ion and counterion species of equal atomic volume.[1] In many systems of interest, the behavior if ions and counterions in solution is essential in understanding a variety of physical and biological phenomena. However, in these systems, the ion and counterion species typically have different sizes and so a further modification of the PBE must be completed. This write-up will present a derivation of the PBE to a general case with an ion and counterion species of different sizes and charges.

1

Background

This paper describes a generalization of the Poisson-Boltzmann equation (PBE) first published by Borukhov.[1] In his paper, Borukhov modified the PBE to take into account the finite size of two equal-volume ion species in solution. This paper builds upon that work by relaxing the requirement that both ion species be composed of ions of equal size. The modified PBE was obtained through a variational method by minimizing the free energy of a lattice gas with respect to the the ion concentrations and the electric potential ψ. Borukhov’s modification to the PBE for an ion solution composed of a positive ionic species with charge +e and negative ion species of −ze is: ∇2 ψ =

4πzec− ezβeψ − e−βeψ b ² 1 − φ0 + φ0 (ezβeψ + ze−βeψ )/(z + 1)

(1)

3 where φ0 = (z + 1)a3 c− b , a is the volume of a single ion (assumed to be the same for both positive and negative ions), and c− b is the bulk ∗ Department

of Applied Physics, Stanford University of Physics, Stanford University ‡ Correspondence should be sent to: McCullough Building, 476 Lomita Mall, Stanford, CA 94305-4045. Email: [email protected] † Department

1

concentration of negative ions. Note that the original PBE is recovered in the limit of very small ion concentrations (φ0 → 0): h i 4ψzec− b ezβeψ − e−βeψ (2) ² In many physical and biological systems of interest where the interaction of ionic solutions is an important consideration, the PBE equation has long been employed to model the behavior of the ions in solution, particularly around charged objects like proteins. However, the PBE equation is of limited use in these situation because it neglects the ion size as noted before. Borukhov’s solution is a step forward but suffers from the fact that in most ion solutions, the two ionic species are typically not of equal size.

∇2 ψ =

2

Bulk Ionic Solutions as Lattice Gases

Consider a ionic solution composed of two species: a positive ion species carrying charge +e and a negatively charged counterion species of charge −ze. Let the volume of the negatively charged ions be a3 and the volume of the positively charged ions ka3 .1 We also impose a charge neutrality condition: Nb+ = zNb−

(3)

Nb±

where give the total number of positive/negative ions in the bulk solution. We wish to model this system as a cubical lattice gas with a lattice spacing a3 . In our model, the size of the largest ion sets the scale of the lattice. For a lattice gas with N total sites, the volume of the gas is simply V = N a3 . For the lattice gas as a whole, the partition function is given by: ("

Z=

k   X n

n=0

k

#

x

n

)N

+y

h

= (1 + x)k + y

iN

(4)

where x ≡ eβµ+ , y ≡ eβµ− and µ± is the chemical potential of the positive and negative ion species. Note that the local potential ψ does not enter into the partition function because the solution as a whole is neutral. The equilibrium number of each ion species is simply given by:



Nb+



 Nb−



=

1 ∂[log Z] kx(1 + x)k−1 =N β ∂µ+ (1 + x)k + y

=

y 1 ∂[log Z] =N β ∂µ− (1 + x)k + y







(5)

± We can define the following bulk concentrations n± b and cb : 1 In general, positively charged ions tend to have smaller radii than negatively charged ions; hence we choose the smaller ion species to be the positively charged one.

2



3 c+ b a

≡

n+ b ≡

≡

n− b



3 c− b a









Nb+ kx(1 + x)k−1 = N (1 + x)k + y

Nb− y ≡ = N (1 + x)k + y





(6)

± c± b gives the bulk number density per volume while nb gives the bulk number density per lattice site of the positive and negative ion species. Note that the charge neutrality condition implies that:



Nb+

Nb−

  =

n+ b =z n− b

(7)

We would like to obtain a relationship between the chemical potentials µ± and the bulk concentrations of ions. Eqs. 6 define this relationship; solving these equations for x and y yields:

x ≡ eβµ+

y ≡ eβµ−

=

(n+ b /k) − 1 − nb − n+ b /k

=

(zn− b /k) 1 − (k + z)n− b /k

=

n− b 1 − n− b

=

n− b 1 − n− b

 

1 − n− b + 1 − n− b − nb /k

k

1 − n− b 1 − (k + z)n− b /k

k

(8)

where we have used the charge neutrality condition to eliminate n+ b (Eq. 7). Eqs. 8 sets the relationship between the chemical potentials µ± and the bulk concentration of ions, an externally controlled parameter.

3

Statistics at Individual Lattice Sites

We now examine the concentrations of ions at individual lattice sites; for single lattice sites, the charge neutrality condition (Eq. 7) no longer applies and now the effect of the electric potential ψ must be taken into account, treated in the context of this theory as a mean-field. Howver, the partition function for each individual site is very similar to the partition function of the bulk solution (Eq. 4) and we may make immediate use of the results in the prior section. The partition function of a single lattice site is: Z = (1 + x)k + y

(9)

where we now define x ≡ eβ(µ+ −eψ) and y ≡ eβ(µ− +zeψ) . We can then immediately write down the relations between the concentrations at individual lattice points and the Boltzmann factors x, y:

3



c + a3 c a







n+ ≡

≡

N− y n ≡ = N (1 + x)k + y



− 3



N+ kx(1 + x)k−1 = N (1 + x)k + y

≡





−

(10)

We can now then express the Boltzmann factors x, y as functions of the concentrations in the same way as the previous section (Eqs. 8).

x ≡ eβ(µ+ −eψ)

=

(n+ /k) 1 − n− − n+ /k

y ≡ eβ(µ− +zeψ)

=

n− 1 − n−



1 − n− 1 − n− − n+ /k

k

(11)

Note that in this case, the concentrations are spatially varying functions of position from lattice site to lattice site. The free energy is simply F = −τ log Z where τ = 1/β. The entropy can then be determined by taking the derivative with respect to τ : S = −n+ log (n+ /k) − n− log (n− ) − k(1 − n− − n+ /k) log (1 − n− − n+ /k) +(1 − n− ) log (1 − n− )k−1 (12)

4 Modification of the Poisson-Boltzmann Equation Poisson’s equations allows us to solve for the electric potential ψ given a particular charge distribution ρ: 4π ρ (13) ² where ² is the dielectric constant of the solvent medium. For our lattice gas the net charge density at each lattice site is simply: ∇2 ψ = −

ρ = (en+ − zen− )/a3 = ec+ − zec−

(14)

Eq. 13 and Eq. 14 can be combined to yield the modified PoissonBoltzman Equation: 4πe (zn− − n+ ) (15) ²a3 Substituting Eqs. 10 for the concentrations n± and Eqs. 8 for eβµ± yields the final modified Poisson-Boltzmann equation: ∇2 ψ =



k−1

3 k−1 βzeψ 3 −βeψ − 3 (1 − c− e − e−βeψ 1 − c− )cb a /k 4πzec− b a ) b a − z(1 − e b ∇ ψ=   k − 3 − 3 − 3 − 3 k−1 βzeψ −βeψ ² 1 − cb a − z(1 − e )cb a /k + cb a (1 − cb a ) e (16) 2

4

3 If we define the constant φ0 ≡ (k + z)c− b a /k to be the bulk fraction of a lattice cell volume occupied by ions, we can re-write the PoissonBoltzmann equation in the following form:



k−1

3 −βeψ 3 k−1 βzeψ (1 − c− e − e−βeψ 1 − φ0 + (zc− 4πzec− b a /k)e b a ) b ∇ ψ=   3 −βeψ k + c− a3 (1 − c− a3 )k−1 eβzeψ ² 1 − φ0 + (zc− b a /k)e b b (17) 2

5 Obtaining the PBE Through Variational Methods It should be mentioned that the modified PBE (Eq. 17) can be obtained through a variational principle by writing down the free energy F − T S of the lattice gas and minimizing it with respect to the electric potential ψ and the ion concentrations. For a lattice gas, the total free energy is obtained by intergrating the free energy density over all lattice sites:

U

=

−T S

=

Z h

−

kb T a3

i ² |∇ψ|2 + ec+ ψ − zec− ψ − µ+ c+ − µ− c− dV 8π

Z



c+ a3 log (c+ a3 /k) + c− a3 log (c− a3 )

+k(1 − c− a3 − c+ a3 /k) log (1 − c− a3 − c+ a3 /k) i

− (1 − c− a3 ) log (1 − c− a3 )k−1 dV

(18)

Minimizing Eq. 18 with respect to the potential ψ yields Poisson’s equation (Eq. 15). Variation with respect to c± yields the relations Eqs. 10; together these three sets of equations yields the modified PBE (Eq. 17).

References [1] I. Borukhov, D. Andelman, and H. Orland, Physical Review Letters 79 (3), 435-438 (1997).

5