V. Electrostatics. Lecture 25: Diffuse double layer structure

Transcription

1 V. Electrostatcs Lecture 5: Dffuse double layer structure MIT Student Last tme we showed that whenever λ D L the electrolyte has a quas-neutral bulk (or outer ) regon at the geometrcal scale L, where there s very lttle mean charge densty ρ = z ec compared to the total charge densty c 0, or more precsely ρ = O(ε ) where ρ = ρ/ec 0, ε = λ D /L 1. In order to satsfy electrostatc boundary condtons, however, dffuse charge exsts n thn quas-equlbrum double layers (whch are mathematcal boundary layers ). The on profles are approxmately n thermal equlbrum (µ constant), even when there s a non-zero current of flud flow, due to the small scale λ D L. [Note: the double layer can go out of equlbrum f c 0 0 at a lmtng current, or a very fast transent can occur, e.g. hgh frequency mpedance wth ω D/λ D.] 1 Posson-Boltzmann Equaton We start wth an assumpton of quas-equlbrum, so that the chemcal potental µ =constant. We separate the electrc potental φ nto two parts: φ = φ + ψ, where φ s the (approxmately) constant bulk potental and ψ s the part due to dffuse charge. From the quas-equlbrum assumpton, the concentratons and charge densty are n equlbrum wth the spatally varyng part ψ, so c = c eq (ψ), ρ = ρ eq (ψ). The generalzed Posson-Boltzmann equaton s derved by usng ths charge densty n Posson s equaton (standard electro-statcs) to get (ε ψ) = ρ eq (ψ). (1) 1

2 Lecture 5: Dffuse double layer structure For a dlute soluton, the concentratons wll follow a Boltzmann dstrbuton, ( ) eq z eψ c = ν c 0 exp, () k B T where ν s the stochometrc coeffcent defned as c 0, /c 0, relatng the bulk concentraton of an ndvdual speces to the bulk salt concentraton c 0 where ψ = 0. Usng ths term for the concentratons, the defnton of the charge densty, and assumng a constant permttvty the generalzed Posson-Boltzmann equaton smplfes to ( ε zeψ ) ψ = z eν c 0 exp. (3) k BT For small potentals ( ψ k B T/e), we can expand the exponental to get ] z ε eψ ψ z eνc 0 [1 + O(ψ ) k B T z eνc 0 ψ (z e) ν c 0 kb T ψ ρ bulk kb T (z e) ν c 0. (4) We know that ρ bulk 0, and we can recognze the coeffcent of ψ as 1 /λ D, so λ ψ = ψ, λ εkbt D D =. (5) (z e) ν c 0 Ths small voltage lnearsaton s known as the Debye-Huckel Equaton, and λ D s called the Debye screenng length. In one dmenson, the Debye- Huc kel equaton can be solved easly f the potental at the surface s known (ψ(0) = ψ D ), x/λ D λ d ψ = ψ ψ [ ψ D ( e D dx = ) z ψ(0) = ψ D c (x) = ν c eψ D e kb T We can sketch ths soluton for the case of a bnary electrolyte: x/λ D ].

3 Lecture 5: Dffuse double layer structure The concentraton profles relax to the bulk values at large x (whch could be slowly varyng, but are depcted as constant here). The ntegrated area of both curves tends to the same value, so that there s no net adsorpton of charge near the surface n the lmt of low voltage. 1.1 Capactance of Double Layer (n the Debye-Huc kel lmt) The capactance s of the double layer s calculated by related the surface charge to the surface potental. The net charge brought to the surface can easly be calculated by ntegrated over the charge densty: (z e) ν c ε ε q = ρ( ψ( x)) dx = 0 k B T 0 ψ(x)dx = ( λ D ψ D ) = ψ D 0 λ D λd We expect that the net charge n the double layer should balance the surface charge, so ths calculated value s also the surface charge. The capactance s then calculated usng ts defnton dq s dq C D =. dψ D dψd Thus, the capactance n the Debye-Huc kel lmt s C DH ε D =. (6) λ D Interestngly, the double layer behaves lke a parallel plate capactor of wdth λ D. 3

4 Lecture 5: Dffuse double layer structure Double Layer at Hgh Voltage At large voltages, we cannot lnearse the Posson-Boltzmann equaton as above. Instead, we have to solve the full nonlnear problem, whch s straghtforward for the case of a symmetrc (z : z) dlute electrolyte n one dmenson, whch s referred to as the Guoy-Chapman Model. The Posson- Boltzmann equaton n ths case s d ψ ( ) ε = ρ eq = c e zeψ/k B T 0ze e z eψ/k B T. dx We frst non-dmensonalze usng the thermal voltage scale ( ψ = zeψ/k B T ) and the Debye length ( x = x/λ D ) yeldng d ψ dx = ψ = snh ψ = ρ. (7) Ths dfferental equaton can be ntegrated by multplyng both sdes by ψ and ntegratng to get ψ = snh ψ ψ ψ = snh ψ ψ 1 (ψ ) = (cosh ψ 1) (snce ψ 0, ψ 0 as x ) 1 (ψ ) = snh ( ψ/) ψ = snh( ψ/). We could ntegrate further to get the potental, but we are most nterested n the effect on capactance, whch we can fnd from ths result. The total charge can be calculated wth d ψ dψ q = ρdx = ε dx = ε. dx dx 0 0 x=0 Pluggng n the value of ψ from above, we get q = snh( ψ D/) c D = dq s dψ D = cosh( ψ D/). 4

5 Lecture 5: Dffuse double layer structure Fnally, we brng back the dmensonal quanttes to get ( ) GC ε zeψ D C D = cosh k B T λ D (8) As expected, for small ψ D ths result smplfes to the Debye-Huc kel lmt from above. We can plot these equatons to get to get an dea of how these varous lmts compare: At large potentals ons begn to concentrate at the strongly charged surface and molecular crowdng effects take over, ndcated as the dashed lne. 5

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