Statistical Thermodynamics: Molecules to Machines

Statistical Thermodynamics: Molecules to MachinesVenkat Viswanathan May 19, 2015Module 11: Electrostatics - Debye screening and Manning condensationLearning Objectives: Analyze the role of counter-ion structuring on the electrostatic inter- actions. Discuss how counter-ions condense on highly charged objects (poly- electrolytes, proteins, and colloids).Key Concepts:Electrostatic potential, electric field, Gauss law, divergence theorem, Poissons equation, electrostatic screening, Debye- Hckel theory, Man- ning condensationElectrostatic potentialElectrostatic interactions in an aqueous environment dramatically influ- ence the thermodynamic behavior. Polyelectrolytes (charged polymers) are soluble in water, in contrast to most hydrocarbon-based polymers. Most biopolymers are polyelectrolytes, and in many instances, their bi- ological function hinges on the role of electrostatic interactions (Fig. 1).The force on a point particle with charge q in an electric field Eis given by f = qE . The potential energy can then be written as q, where is defined to be the electrostatic potential, giving E = .The electric field satisfies Gauss law, which is written as:

Figure 1: Hierarchical assembly of chromatin and the nucleosome core particle. Our meter-long genome is packaged into a micronsized nucleus with the aid of cationic proteins called histones. ES

4 ndA =V

dV(1)where is the dielectric constant of the medium (for example, =7.083 1010C2N1m2, for water at room temperature), and is thelocal charge density.We apply the divergence theorem to this to arrive at:

4 EdV =V

dV(2)VTherefore, the electrostatic potential is governed by Poissons equation: =

4(r)(3)Electrostatic potentialConsider a sphere with a radius a and total charge q. Due to symmetry, the electrostatic potential around the sphere is radial, .e. = (r). Poissons equation outside of the sphere is written as:1 d 2 dA1r2 dr r

= 0 =dr

+ A2(4)rwhere A1 and A2 are constant of integration (to be determined).Assume as r , 0; thus, A2 = 0.The electric field in the radial direction is er EGauss law, we can write:

= A1 , and, using2A12

4q

qq 1a2 4a = A1 = = r(5)This potential is valid outside the radius a (r > a). For a point particle (a 0), this is valid throughout space.The electrostatic potential provides the necessary information to find the energy. The energy of interaction between two ions with charge e (charge of an electron) separated by a distance r is:e2 1Ecoulomb =

(6) rwhere is the dielectric constant of the medium (for example, =7.083 1010C2N1m2 for water at room temperature) and the elec- tron charge e = 1.6022 1019C.Define the Bjerrum length:lB = k

e2(7)TBwhich gives the distance that two electrons are separated when their energy is equal to kBT (lB = 7 for water at room temperature).The energy of interaction for two monovalent ions in a dielectric medium is written as:Ecoulomb = kBT

lB(8)rElectrostatic potential around a charged cylinderConsider a negatively charged cylinder of radius a with a surface charge such that the average distance between the surface charges is b. The electrostatic potential in an uncharged medium ( = 0) is given by:21 = r r r r = 0 = A1 log r + A2(9)where we express the Laplacian in cylindrical coordinates and assume = (r) only.We find the boundary condition on the cylinder surface using Gauss law Using the divergence theorem, we write: ES

4 ndA =V

dV(10)This gives the boundary condition on the cylinder surface:.eE. 2ab =4e

(11)This boundary condition on the cylinder surface is rewritten as:giving A1 = 2e .

..=.r=a

2e ab

(12)For simplicity, we set 0 as r a, fixing A2 = A1 log a. Thisgives the solution for the electrostatic potential.2kBTq(r) =

log (r/a)(13)ewhere q = B /b and lB = e2/(kBT).Unlike the point charge, the electrostatic potential outside of a charged cylinder diverges logarithmically, which has dramatic consequences on the counter-ion distribution.Screened electrostatic interaction in salt (Debye-Hckel theory)When salt is added to the solution, the nature of the electrostatic in- teractions is dramatically altered. Due to the strength of electrostatic interactions, mobile counter ions have strong spatial correlations with their opposite charge. A polyanion (negative polyelectrolyte) has a flurry of positively charged counter-ions swarming around the polymer (Fig 2). The interactions between charges are dramatically reduced due to the "screening" of the electostatic interactions (Fig 3).Consider the interaction between two isolated ions in a monova- lentsalt solution with concentration ns (for example, N aCl that dis-sociates into Na+ and Cl ions) The electrostatic potential aroundthe tagged ion is governed by the Poissons equation.

Figure 2: Schematic of the salt- screening effect for a polyelectrolyte. The density of positive and nega- tive counter-ions are colored red and blue, respectively. Electro-neutrality =

4(r)(14)

requires the total charge in solution to be zero, thus the bulk concentration iswhere is the local charge density in the solution.The energy of an ion with charge q = ze (e.g. q = e for Cl) lo-cated at r in the electrostatic potential is given by E = ze(r). Assume the local concentration of counter-ions is given by a simple Boltzmann weight of the ions in the electrostatic potential . Thus,

purple (equal positive and negative)... e(r) .

. e(r) ..

Figure 3:Schematic of the salt-(r) = nse

exp

kBT

exp

+kBT

(15)

screening effect between two isolated ions.where the left term is due to local concentration of Na+, and the rightis due to Cl.Therefore, the electrostatic potential is governed by:28nse

. e . =

sinh

kBT

(16)This is the Poisson-Boltzmann equation. This nonlinear equation does not have a simple, closed-form solution. Implicit in this theory is the neglect of local concentration fluctuations, akin to a mean-field approximation.For sufficiently large lengths where 0, we can expand to lowestorder.28nse2 =

kB

(17)TWe define the Debye length (Fig 4):. kBT

lD =and assume that = (r) only.We now have:

8e2ns

(18)1 2 1r2 r2 r

=(19)rl2In the limit of large length separation r and dilute solution of counter- ions, we can write the electrostatic energy between two ions in a salt solution as:e2 EDH = 4

er/lD= kBTlBr

er/lD(20)rThis is the Debye-Hckel theory of charge screening.The Debye-Hckel theory adds a simple correction to the bare elec- trostatic potential to account for counter-ion correlations.In the absence of salt, the electrostatic interaction energy is ex- tremely long range, scaling as Ecoulomb 1/r. Salt screening leads toa short-range electrostatic interaction energy that decays exponentially with distance of separation.Charge behavior at short range (Manning condensa- tion)Neglecting the nonlinear terms in the Poisson-Boltzmann equation leads to a qualitatively incorrect behavior for distances very close to a charged object, e.g. the surface of a polyelectrolyte. For a dilute solution, theaverage distance between counter-ions is r0 n1/3. At r < r0, themean-field approximation breaks down, and we must address the inter- actions explicitly.Consider the polyelectrolyte chain to be a charged cylinder with ra- dius a (Fig. 5), which we assume to be very small (a 0 in our analysis).The charged cylinder has negative charges smeared over its surface such that the average distance between negative charges is . We want to find the behavior of an isolated ion interacting with the cylinder at distances less than the average inter-ion spacing r0.Earlier in this lecture, we found the electrostatic potential around anegatively charged cylinder in a dielectric medium to be:(r) = 2kBTq log (r/a)(21)ewhere q = lB /. The electrostatic energy between the cylinder and a positively charged monovalent ion is given by Ecyl (r) = e(r).The partition function for an isolated ion interacting with the charged cylinder within the distance r0 for small cylinder radius (a 0) scalesas:

Figure 4: Plot of the Debye length lD versus the salt concentration ns for monovalent salt in water at room tem- perature.

Figure 5: Schematic for the short-range interaction between a charged cylinder (polymer) and a surrounding solution of dilute ions. r0Q 0

drr exp

. Ecyl (r) . kBT

r00

drr exp(2q log(r)) r00

drr12q

12 2q

.r0r22q ..0

(22)The partition function diverges if q > 1.To reconcile the divergence of the free energy, the mobile ions con-dense 1 on the cylinder surface to neutralize the surface charges such1 Gerald S. Manning. Limiting Lawsand Counterion Condensation in Poly- electrolyte Solutions I. Colligative Properties. The Journal of Chemical Physics, 51(3):924, September 1969that effectively qef f = lB /ef f = 1. At short length scales, Manning condensation leads to a reduced surface charge. At long length scales, the interaction is a screened electrostatic coulomb potential.ReferencesGerald S. Manning. Limiting Laws and Counterion Condensation in Polyelectrolyte Solutions I. Colligative Properties. The Journal of Chemical Physics, 51(3):924, September 1969.

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