Lecture: Debye – Huckel Theory

Dr. Ronald M. Levy

ronlevy@temple.edu

Statistical Thermodynamics

1

What is Debye – Huckel Model(approximation)?

1. Cations and anions embedded in dielectric continuum

2. Vijelec = qiqj/ϵrij

3. Vijnon−elec, hard spheres with radius σ/2

4. We assume the electrolyte solution is dilute

+

-

-

+

ϵ

With the approximation above, following properties will be calculated in this chapter 1. ΔA ≡ A − A0, excess free energy to charge the ions 2. Wij r , The Potential of mean force between an ion pair

2

A. Potential Due to a set of + and – charges

𝜙 𝑟 = 𝑞𝑗+

𝜖|𝑟 − 𝑟 𝑗|𝑗

+ 𝑞𝑘−

𝜖|𝑟 − 𝑟 𝑘|𝑘

= 𝜌+(𝑟 ′)

𝜖|𝑟 − 𝑟 ′|ⅆ𝑟 ′ +

𝜌−(𝑟 ′)

𝜖|𝑟 − 𝑟 ′|ⅆ𝑟 ′

= 𝜌 (𝑟 ′)

𝜖|𝑟 − 𝑟 ′|ⅆ𝑟 ′

Discrete

Continuous

B. Poisson Equation Relates potential 𝜙 to the charge density

𝛻2𝜓 𝑟 = −4𝜋

𝜖𝜌(𝑟)

C. Gauss’ Law

𝛻𝜓ⅆ𝑠 =4𝜋

𝜖[charge enclosed by the surface 𝑄𝑡𝑜𝑡]

3

Close surface

4

Statistical mechanics of weak electrolyte solutions

𝐴 − 𝐴0 ≡ Electrostatic Helmholtz free energy of the primitive model for N cations and N anions in volume V

𝐴 − 𝐴0 = −𝑘𝑇 ln𝑍𝑁

𝑍𝑁°

𝑍𝑁 = ⋯ 𝑒−𝛽𝑈𝑁 ⅆ𝑟 1⋯ⅆ 𝑟 𝑁

𝑍𝑁° = ⋯ 𝑒−𝛽𝑈𝑁

°ⅆ𝑟 1⋯ⅆ 𝑟 𝑁

𝑈𝑁 =1

2 𝑈

(𝑟 𝑖𝑗)(𝑠)

𝑖≠𝑗

+1

2 𝑞𝑖𝑞𝑗

𝜖|𝑟 𝑖 − 𝑟 𝑗|𝑖≠𝑗

Short range Electrostatic

𝑈𝑁° =1

2 𝑈

(𝑟 𝑖𝑗)(𝑠)

𝑖≠𝑗

Hard sphere

5

We need to determine the average potential acting on an ion

𝜓𝑗 𝑟𝑗 = 𝑞𝑖

𝜖|𝑟 𝑖 − 𝑟 𝑗|𝑖≠𝑗

Potential acting on ion j at 𝑟 𝑗 due to all the other ions

Then

𝑈𝑁,𝑒𝑙𝑒𝑐 =1

2 𝑞𝑗𝜓𝑗(𝑟 𝑗)

Total electrostatic energy for an instantaneous configuration of the ions

We need to determine the canonically average 𝜓𝑗 and 𝑈𝑁,𝑒𝑙𝑒𝑐. The canonical

average of 𝜓𝑗 𝑟𝑗 is given by:

𝜓𝑗 = ⋯ 𝜓𝑗(𝑟 𝑗)𝑒

−𝛽𝑈𝑁 ⅆ𝑟 1⋯ⅆ 𝑟 𝑁

⋯ 𝑒−𝛽𝑈𝑁 ⅆ𝑟 1⋯ⅆ 𝑟 𝑁 Notice that 𝜓𝑗 is independent of 𝑟 𝑗

The electrostatic contribution to the helmholtz free energy of the electrolyte solution can now be expressed as a charging integral:

A − 𝐴0 = 𝜓𝑗(𝜆) 𝑞𝑗ⅆ𝜆𝑗

1

0𝑗

6

To derive the Debye-huckel Results, we derive a Poisson Equation for the electrostatic potential of a system in which one of the ions is fixed at the origin and the positions of the remaining charges are canonical averaged.

𝜓(𝑟 )1 = ⋯ 𝜓𝑗(𝑟 )𝑒

−𝛽𝑈𝑁 ⅆ𝑟 1⋯ⅆ 𝑟 𝑁

⋯ 𝑒−𝛽𝑈𝑁 ⅆ𝑟 1⋯ⅆ 𝑟 𝑁

𝜓(𝑟 )1 is the average electrostatic potential at 𝑟 when ion 1 is fixed at the origin and positions of all the others is averaged

Now, take the lapacian of both sides

𝛻2 𝜓(𝑟 )1 = ⋯ 𝛻2𝜓𝑗(𝑟 𝑗)𝑒

−𝛽𝑈𝑁 ⅆ𝑟 1⋯ⅆ 𝑟 𝑁

⋯ 𝑒−𝛽𝑈𝑁 ⅆ𝑟 1⋯ⅆ 𝑟 𝑁= −4𝜋

𝜖𝜌(𝑟 )1

Now, 𝜌(𝑟 )1 is the average charge density at 𝑟 due to all the ions when ion number 1 is fixed at the origin. The charge density at 𝑟 can be expressed in terms of the pairs correlation function, which in turn can be expressed in terms of the potential of mean force, which can be expressed in terms of the averaged electrostatic potential. We will proceed as follows:

𝜌(𝑟 )1 → 𝑔(𝑟 ) → 𝑤(𝑟 ) → 𝜓(𝑟 )1

7

𝜌(𝑟 )1 = 𝐶𝑠𝑞𝑠𝑔𝑠(𝑟 )2

𝑠=1

+ and - charge

Concentration of ion s in the bulk

Charges on ion s

Substituting this into the Poisson equation, the average electrostatic potential gives:

𝛻2 𝜓(𝑟 )1 = −4𝜋

𝜖𝜌 𝑟 1 = −

4𝜋

𝜖 𝐶𝑠𝑞𝑠𝑔𝑠 𝑟 2

𝑠=1

= −4𝜋

𝜖 𝐶𝑠𝑞𝑠𝑒

−𝛽𝑊𝑠 𝑟 2

𝑠=1

Where we have used the relationship between 𝑔𝑠 𝑟 and the PMF W

This equation is exact up to his point. The first Debye-Huckel approximation is to approximate the PMF between ions by the average electrostatic potential.

𝑊𝑠 𝑟 ≈ 𝑞𝑠 𝜓 𝑟 1 ≡ 𝑞𝑠𝜙1(𝑟)

Debye-Huckel approximation, which neglects the effects of non-electrostatics short range interactions

To simplify, 𝜙1(𝑟) is canonically averaged electrostatic potential at r with an ion fixed at the origin

8

Substituting this into the differential equation in previous page yields:

𝛻2𝜙1 𝑟 =4𝜋

𝜖 𝐶𝑠𝑞𝑠𝑒

−𝛽𝑞𝑠𝜙1(𝑟 )2

𝑠=1

This is Non-linear Poisson-Boltzmann equation for the average electrostatic potential. It combines electrostatics with statistical mechanics (the factor 𝛽 is an indicator of statistical mechanics averaging). This equation can be solved on computer and is used for biophysical modeling, for example, to study the ionic atmosphere around DNA, and the effects of salt on the electrostatic potentials of proteins. Non-linear differential equation like Poisson-Boltzmann equation are more difficult to study than linear differential equation. The second Debye-Huckel approximation is to linearize the Poisson-Boltzmann equation

𝑐𝑠𝑞𝑠𝑒−𝛽𝑞𝑠𝜙1(𝑟 )

𝑠

≅ 𝑐𝑠𝑞𝑠𝑠

− 𝛽 𝑐𝑠𝑞𝑠2𝜙1(𝑟 )

For small 𝜙1(𝑟 ) or for low concentration

9

𝛻2𝜙1 𝑟 = (4𝜋𝛽

𝜖 𝑞𝑠

2𝑐𝑠𝑠

)𝜙1 𝑟

Substituting this into the PB equation yields:

𝑟 > 𝑎

𝐾2

𝐾2 =8𝜋

𝜖𝑘𝑇𝐼 𝐼 =

1

2 𝑞𝑠

2𝑐𝑠𝑠

Debye-Huckel screening parameter

Ion radius = 𝑎/2

So is proportional to the ionic strength, and the debye huckel equation can be expressed:

𝛻2𝜙1 𝑟 = 0 0 < 𝑟 ≤ 𝑎 𝛻2𝜙1 𝑟 = 𝐾

2𝜙1 𝑟 𝑟 > 𝑎

Boundary conditions

1. 𝜙1 𝑟 → 0 as 𝑟 → ∞

2. 𝜙1 𝑟 and 𝜖𝜙1 𝑟

𝑑𝑟 continuous at 𝑟 = 𝑎

Remember 𝜙1 𝑟 = 𝜓 𝑟 1

10

Because 𝜙1 𝑟 is spherically symmetric, we use spherical coordinate to solve ∆𝐻.

1

𝑟2ⅆ

ⅆ𝑟𝑟2ⅆ𝜙1ⅆ𝑟= 𝑘2𝜙1(𝑟 )

General solution to this 2nd order differential equation is: (check by substitution)

𝜙1 𝑟 =𝐴1𝑒−𝐾𝑟

𝑟+𝐵1𝑒+𝐾𝑟

𝑟 𝐵1 = 0 because if boundary conditions at ∞

𝑟 > 𝑎

1

𝑟2ⅆ

ⅆ𝑟𝑟2ⅆ𝜙1ⅆ𝑟= 0 0 < 𝑟 ≤ 𝑎

Solution is:

𝜙1 𝑟 =𝑞1𝜖𝑟+ 𝐵2

11

We find 𝐴1 and 𝐵2 by applying the boundary conditions

𝜙1 𝑟 →𝑞1𝜖𝑎+ 𝐵2 𝑟 → 𝑎 𝑟 < 𝑎

𝜙1 𝑟 →𝐴1𝑎𝑒−𝐾𝑎 𝑟 → 𝑎 𝑟 > 𝑎

gives 1 equation for 𝐴1 and 𝐵2

𝜖ⅆ𝜙1ⅆ𝑟→ −𝑞1𝑎2

𝜖ⅆ𝜙1ⅆ𝑟→ −𝜖𝐴1𝑎2𝑒−𝐾𝑎 − 𝜖

𝐾𝐴1𝑎𝑒−𝐾𝑎

𝑟 → 𝑎 𝑟 < 𝑎

𝑟 → 𝑎 𝑟 > 𝑎

gives 2 equation for 𝐴1 and 𝐵2

Solving for 𝐴1 and 𝐵2, we get:

𝐴1=𝑞1𝑒𝐾𝑎

𝜖(1 + 𝐾𝑎) 𝐵2= −

𝑞1𝐾

𝜖(1 + 𝐾𝑎)

𝑟 > 𝑎

12

The Debye-huckel Electrostatic Potential is then:

𝜙1 𝑟 =𝑞1𝜖𝑟−𝑞1𝐾

𝜖(1 + 𝐾𝑎) 0 < 𝑟 ≤ 𝑎

𝜙1 𝑟 =𝑞1𝑒−𝐾(𝑟−𝑎)

𝜖𝑟(1 + 𝐾𝑎)≈𝑞1𝑒−𝐾𝑟

𝑟 𝑟 > 𝑎

For 𝐾𝑎 ≪ 1, the screened interaction between 𝑞1 at the origin and 𝑞2 at r is given by:

𝑊12 = 𝑞2 𝑟 𝜙1 𝑟 =𝑞1𝑞2𝑒

−𝐾𝑟𝑒𝐾𝑎

𝜖𝑟(1 + 𝐾𝑎)

≈𝑞1𝑞2𝑒

−𝐾𝑟

𝜖𝑟

For dilute solution

For dilute solutions, as 𝐼 → 0, it can be shown that the PMF between 2 ions is:

𝑊12 =𝑞1𝑞2𝑒

−𝐾𝑟

𝜖𝑟

Exact Stat Mech As 𝐾 → 0

So, Debye-Huckel theory is an exact limiting law for dilute solutions

Where 𝐾2 ∝ 𝐼 𝐼 =1

2 𝑞𝑠2𝑐𝑠𝑠 ionic strength

Debye-Huckel results

“Screened Coulumb Potential”

13

Let us examine the charge distribution and the total net charge surrounding the charge 𝑞1 at the orgin:

𝑄𝑇𝑜𝑡 = 𝜌(𝑟) 4𝜋𝑟2ⅆ𝑟1

∞

𝑎

Total charge between r and r+dr

𝜌(𝑟)1 = −𝜖

4𝜋∇2𝜙1 𝑟 = −

𝜖

4𝜋𝐾2𝜙 𝑟 = −

𝜖𝐾2

4𝜋

𝑞1𝑒−𝐾(𝑟−𝑎)

𝜖𝑟(1 + 𝐾𝑎)

Poisson Equation Debye-Huckel approximation

𝑄𝑇𝑜𝑡 = −𝑞1𝐾2

(1 + 𝐾𝑎)𝑒−𝐾(𝑟−𝑎)𝑟

∞

𝑎

ⅆ𝑟 = −𝑞1

𝑝(𝑟)

Charge neutrality Total net charge surrounding 𝑞1 is equal and opposite

𝑝 𝑟 ⅆ𝑟 = −𝑞1𝐾2

(1 + 𝐾𝑎)𝑒−𝐾(𝑟−𝑎)𝑟ⅆ𝑟 𝑟 > 𝑎

Solution to Debye-Huckel equation for r>a

14

𝑝(𝑟)

𝐾𝑟 1 2

𝑃𝑚𝑎𝑥 at 𝑟 = 1/𝐾

The ion distribution has a probability maximum at 𝑟 = 1/𝐾

𝐾−1~3𝑐1/2 = 30Å at 25℃ where 𝑐 = 0.01𝑀

So for a 0.01M salt solution, ions are ~30Å apart on average and can ignore short range effects