MIT10_626S11_lec28 - V Electrostatics Lecture 28...

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V. Electrostatics Lecture 28: Electrostatic Correlations MIT Student (and MZB) 1. Mean-Field Theory Continuum models like the Poisson-Nernst-Planck equations are “mean-field approximations” which describe how discrete ions are affected by the mean concentrations c i and potential φ . Each ion migrates in the mean electric field, which is produced by the mean charge density, not by the discrete, fluctuating charges in the molecular system. The self-consistent system of PNP equations we have derived thus far is µ i = k B T ln( γ i c i ) + z i e F i = M i c i i c i t + ∇⋅ F i = 0 −∇⋅ ( ε ) = ρ = z i ec i i However, discrete ion-ion interactions are a significant component of the excess chemical potential for a charged species in a bulk electrolytic solution. To accurately model such systems, it is important to account for these discrete interactions. 2. Bjerrum length What is the length scale below which electrostatic correlations are important? In very dense charged systems, it is the ion size, as in solvent free ionic liquids (see below). In typical electrolytes, however, the relevant scale is the Bjerrum length, where the bare Coulomb energy between two elementary charges is balanced by the thermal fluctuation energy: e 2 e 2 4 πε l B = k B T l B = 4 k B T At larger length scales, we may expect that thermal fluctuations are strong enough to justify replacing discrete ion-ion Coulomb forces with a continuum mean-field theory. In water at room temperature, the Bjerrum length is 0.7nm, which is only a few molecular lengths, so it makes sense to try to use mean-field theories based on the continuum PNP equations (such as Gouy-
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Lecture 28: Electrostatic correlations 10.626 (2011) Bazant Chapman) to describe the diffuse part of the double layer, at least at low salt concentrations, when the Debye length greatly exceeds the Bjerrum length. Note that these two length scales are related as follows: λ D = 1 4 π l B z i 2 c 0 i = 1 8 l B I where I = 1 z i 2 c 0 is the molar ionic strength, which arises in Debye-Huckel theory, based on 2 i linearization of the PNP mean-field theory above for small voltages. (See also below.) The condition D l B , which is needed to justify a mean-field theory of the diffuse part of the double layer, thus corresponds to I 6 1 4 3 l B 3 which says that the mean volume per ion must be at least six times larger than a sphere whose radius is the Bjerrum length. Put another way, the “correlation volume” within one Bjerrum length of an ion should contain fewer than ~6 neighboring ions for the Debye-Huckel mean-field theory to hold.
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This note was uploaded on 11/27/2011 for the course CHEMICAL E 20.410j taught by Professor Rogerd.kamm during the Spring '03 term at MIT.

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MIT10_626S11_lec28 - V Electrostatics Lecture 28...

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