1. Mean-Field Theory. 2. Bjerrum length

Transcription

1 1. Mean-Feld Theory Contnuum models lke the Posson-Nernst-Planck equatons are mean-feld approxmatons whch descrbe how dscrete ons are affected by the mean concentratons c and potental φ. Each on mgrates n the mean electrc feld, whch s produced by the mean charge densty, not by the dscrete, fluctuatng charges n the molecular system. The self-consstent system of PNP equatons we have derved thus far s µ k B T ln(γ c ) + z eφ F M c µ c t + F (ε φ) ρ z ec However, dscrete on-on nteractons are a sgnfcant component of the excess chemcal potental for a charged speces n a bulk electrolytc soluton. To accurately model such systems, t s mportant to account for these dscrete nteractons.. Berrum length What s the length scale below whch electrostatc correlatons are mportant? In very dense charged systems, t s the on sze, as n solvent free onc lquds (see below). In typcal electrolytes, however, the relevant scale s the Berrum length, where the bare Coulomb energy between two elementary charges s balanced by the thermal fluctuaton energy: e e 4πεl B k B T l B 4πεkB T At larger length scales, we may expect that thermal fluctuatons are strong enough to ustfy replacng dscrete on-on Coulomb forces wth a contnuum mean-feld theory. In water at room temperature, the Berrum length s.7nm, whch s only a few molecular lengths, so t makes sense to try to use mean-feld theores based on the contnuum PNP equatons (such as Gouy-

2 Lecture 8: Electrostatc correlatons 1.66 (11) Bazant Chapman) to descrbe the dffuse part of the double layer, at least at low salt concentratons, when the Debye length greatly exceeds the Berrum length. Note that these two length scales are related as follows: 1 1 4πl B z c 8πl B I where I 1 z c s the molar onc strength, whch arses n Debye-Huckel theory, based on lnearzaton of the PNP mean-feld theory above for small voltages. (See also below.) The condton l B, whch s needed to ustfy a mean-feld theory of the dffuse part of the double layer, thus corresponds to 1 3 I 6 4π 3 l B whch says that the mean volume per on must be at least sx tmes larger than a sphere whose radus s the Berrum length. Put another way, the correlaton volume wthn one Berrum length of an on should contan fewer than 6 neghborng ons for the Debye-Huckel mean-feld theory to hold. 3. Correlaton Functons How can we go beyond mean feld theory, f we know the nteractons between dscrete partcles (e.g. Coulomb)? We smply need a statstcal descrpton of the lqud that gves us the probabltes of fndng dfferent local onc confguratons, whose energes we could n prncple calculate. From experments (e.g. neutron scatterng) or smulatons (e.g. molecular dynamcs, Monte Carlo, etc.), t s possble to measure statstcal correlatons between dscrete partcles, related to ther nteractons. For a gven system of ons of speces and speces, the number of pars of sad on, separated by a dstance r to r+dr s gven by n (r) 4πr g (r)c c where g (r) s the par correlaton functon (g s unty n a unform deal gas). In an electrolyte, we have g (r) for dfferent types of on pars. g +- (r) s the counter-on par correlaton functon, and g -- (r) s the co-on par correlaton.

3 Lecture 8: Electrostatc correlatons 1.66 (11) Bazant 3.5 g (r) r 1 r r 3 r Image by MIT OpenCourseWare. FIG. 1 Typcal g(r) for a lqud. The frst neghbor dstance s r 1, the second s r, etc. h(r) g +- (r) g (r) g++(r) o r (A) -.1 Image by MIT OpenCourseWare. FIG. 3 Total correlaton functons for a monovalent bnary electrolyte wth dameter of the on 5Å. g ++ (r) s the par correlaton functon for a central atom and a neghborng co-on and shows repulson. g +- (r) s the counter-on par correlaton functon and shows attracton. The sold lnes result from asymptotc analyss of the double layer and the dashed lnes result from settng the mean force potental equal to the sum of the core and electrostatc asymptotes.

4 Lecture 8: Electrostatc correlatons 1.66 (11) Bazant For par nteractons, the excess chemcal potental of speces s µ k B T lnγ ex 1 K (r)4πr g (r)c dr + many-body terms where K (r) par nteracton energy. The factor of 1 evaluatng all of the parwse nteractons. prevents double-countng when 4. Electrostatc Correlatons n a Dlute Electrolyte (Debye-Hückel Theory) The bare coulomb nteracton n a delectrc solvent for pont charges s gven by (z e )(z e) K 4πεr To calculate on profles n the screenng cloud, the regon of excess dffuse charge or dffuse countercharge, of a sphercal central on of speces I, we use the Debye-Hückel approxmaton c (r) g ( r)c c e z eψ k B z eψ T c 1 k B T where ψ s the perturbaton of electrostatc potental n the screenng cloud of the central on and s ψ φ φ. The fluctuatons have energy on the order of k B T whch s of a small enough magntude that the lnearzaton n the above equaton s vald. The lnear response of a screened central on s gven by the Debye-Hückel equaton for a general dlute electrolyte ψ ψ Over the length scales consdered, the potental vares only wth dstance from the central on. The potental s thus a functon of r only and the above equaton smplfes to d dψ r ψ r dr dr Because the fluctuatons n the potental decrease wth ncreasng dstance from the central on ( φ φ as r ), ψ( ). Applyng ths boundary condton, the soluton to the above equaton s a modfed sphercal Bessel functon e r λ D ψ(r) A r

5 Lecture 8: Electrostatc correlatons 1.66 (11) Bazant The constant A can be evaluated usng Gauss s law for a pont charge. The electrc feld generated about a pont charge s ndstngushable from that at the surface of a sphercally symmetrc charge dstrbuton of the same total charge. dψ z e ε dr r a 4πa z e dψ e a A 1 λ D 4πa a ε dr r a a e a A z e 4πε(a + ) Incorporatng these boundary condtons gves the screened Coulomb potental e a z e ψ(r) 4πεr(1 + a ) The par correlaton functon then becomes ( a r ) λ (z D g (r) 1 e)(z e)e 4πεk B Tr(1 + a ) Usng ths defnton for the par correlaton functon and gnorng many-body terms, the excess chemcal potental for speces s 1 (z e)(z e) (z e)(z e)e ( a r ) ex µ 1 4πr c dr a 4πεr 4πεk B Tr(1 + a ) (z e) c ( a r ) (z e ) ( z e) e z ec dr 8πεr a 8πε εk (1 + a BT ) The frst term n the above summaton s the product of the bare Coulomb potental and the bulk charge densty. Under bulk neutralty condtons, ths term s zero but would otherwse dverge. The excess chemcal potental smplfes to (z e) 1 µ ex k T ln(γ B ) 8 πε a + where s the Debye screenng length. The screenng length can also be wrtten n terms of the molar onc strength I

6 Lecture 8: Electrostatc correlatons 1.66 (11) Bazant εkbt (z e) c εk B T Ι Note that the excess chemcal potental gven by Debye-Hückel theory s negatve. The electrostatc nteracton between an on and ts oppostely charged screenng cloud s attractve, thereby lowerng the total electrostatc energy of the system. The excess chemcal potental of speces can be wrtten n terms of ts actvty coeffcent. Usng the above expresson for the screenng length, the Debye-Hückel actvty coeffcent for dlute electrolytes can be wrtten as z α Ι ln( γ ) 1 + Ba Ι 5. Ionc Lquds Ionc lquds exst both as molten salt (e.g. NaCl at C) and as room temperature onc lquds (e.g. large organc or fatty ons). RTILs can wthstand up to ±6 V, makng them good canddates for supercapactor desgner solvents. In onc lquds, there s no solvent, only hghly crowded ons, so the Gouy-Chapman-Stern model for dlute electrolytes s not vald. In fact, the Debye-Huckel screenng length s smaller than the sze of a sngle on, and the relevant length scale for electrostatc correlatons and double-layer screenng s the on sze. For hghly concentrated electrolytc solutons, the short-range Coulomb correlatons are very strong, and generally lead to overscreenng, whereby an excess of counter-ons are attracted to a central charge, leadng to an excess co-ons n the next layer. The end result s oscllatons n the charge densty untl electroneutralty s reached.

7 Lecture 8: Electrostatc correlatons 1.66 (11) Bazant Overscreenng Crowdng a V 1 kt e b V 1 kt e 4 3 V 1 C _ C + C + + C _ + _ C, C FIG. TOP: Structure of the onc-lqud double layer (n color) predcted by the modfed Posson equaton (*), n agreement wth molecular dynamcs smulatons. (a) At a moderate voltage, V 1 4 1kBTe (.6 V), the surface charge s overscreened by a monolayer of counterons, whch s corrected by an excess of coons n the second monolayer. (b) At a hgh voltage, V 1 4 1kBTe (.6 V), the crowdng of counterons extends across two monolayers and domnates overscreenng, whch now leads to a coon excess n the thrd monolayer. Because of electrostrcton, the dffuse double layer (colored ons) s more dense than the quasneutral bulk lqud (whte ons). BOTTOM: Ion profles obtaned by solvng (*) at hgh voltage, showng the structures of the top fgure. To descrbe these correlaton phenomena, a 4 th order modfed Posson-Boltzmann equaton has recently been proposed [1]: ε(1 l c ) ψ ρ eq (ψ) (*) where l c s an electrostatc correlaton length. The fourth dervatve term gves rse to the oscllatons. Ths theory s consstent wth experments and smulatons wth ρ eq (ψ) usng a lattce gas to account for the excluded volume and s convenent for mathematcal modelng. x/a Image by MIT OpenCourseWare.