Electrostatics. . where,.(1.1) Maxwell Eqn. Total Charge. Two point charges r 12 distance apart in space

Transcription

1 Maxwell Eq. E ρ Electrstatics e. where,.(.) first term is the permittivity i vacuum 8.854x0 C /Nm secd term is electrical field stregth, frce/charge, v/m r N/C third term is the charge desity, C/m 3 E 0 Ttal Charge e dv V V Diverget therem. E dv E ds (.3) Q ρ. E dv E ds (.) pherical surface arud a pit charge Q E ds Er ds Er 4 π r.(.4) r s v If Q..(.5) r, E r Q/ ( ο 4 π r ); Ē is the filed stregth f ir, E ir 4π r the pit charge. Tw pit charges r distace apart i space r Q Q F E Q r 3 4 π r (.6) Q Q F 4 π r Take abslute value Culmb s law: F (Q Q ) / (4 π r ο )...(.7) Uits: F E Q N (N/C). C N r N (V/m).C J/m

2 Electrical field stregth is cservative i ature E t dl 0 where, t is the taget t the clse curve as shw abve. Prve the same: y C B E x D Curl f gradiet f a scalar is zer. 0 Thus, Electric Field Ptetial E ad (.8) which satisfies E 0, idetically Usig (.) i (.8) e ρ..(.9) e Zer Charge, ρ 0, 0..(.0)

3 Dielectric Medium : Ncductive f Electr plastics, water, rgaic liquids Diple Plarizati Mmet E (electric field stregth) P χ E (.), where χ is the susceptibility ad χ is a cstat. P χ E ρ p where ρ p is the plarizati charge desity..(.) dd (.) ad (.) Use (.) P e p E ρ ρ.(.3) e P f [( χ ) ] ρ ρ E ρ.(.4) χ, called dielectric cstat Maxwell s Eq. fr dielectric with a free charge desity (ρ f ) f ( ) (.5) E ρ Use (.8) that is, E, i (.5) t get Piss s Eq. f ρ (.6) Laplace Eq. 0, fr charge (.7)

4 Tw pit charges i dielectric medium F Q Q.(.8) 4π r Isptetial fr a pit charge Frm (.) ad csiderig sice with E θ E φ 0 E ir Er iθ Eθ iφ Eφ, dr d r d d r r 0 pply B.C : r, 0 ad eq (.4) shw that the ptetial, Q (.9) 4 π r Remember E field stregth,, it ca be shw that the Q E i (3.0) r 4 π r Charge f spherical cductr f radius a ad surface ptetial, s is give by, (i vacuum) Q 4 π a s.(3.)

5 I dielectric, Q dr d r r 4 π a s..(3.) d 0B.C : r, 0 d r d l: ad dr r pplyig B.C. 0 We already kw that Q E ds ds ds Therefre. Q 4 π d d r r r Q ad substitutig it i the sluti with itg. csts leads t Q 4 π r

6 Origi f Iterfacial Charge:. Iizati f surface grup. Differetial sluti f is frm sparigly sluble crystal 3. Charged crystal surfaces 4. pecific i adsrpti

7 Particle _ luti Electrical Duble Layer rud a Particle 0

8 Mathematical alyses fr duble layer fr a flat surface: Ptetial distributi with diffuse duble layer (Piss Eq.) f ρ Oedimesial case: f ρ x free charge desity f ρ e p ρ ρ i i z i e (3.3) x i i z i e.(3.4) Wheever thermal equilibrium exist, the prbability that a state with a eergy W will be ccupied has shw by Bltzma Prbability fucti, P e W/kT, W Z i e W : wrk required t brig e i f valecy Z i frm ifiity (0) t a lcati x where ptetial is. e : electric charge P cst e z i e k T k is the Bltzma cstat (J/K) (R/ a ), R gas cstat, a vgadr s. P P i e zi e k T..(3.5) Prbability f fidig is at lcati x z i e i i exp k T This is called Bltzma distributi fucti..(3.6)

9 i is the umber f is i the eutral state (bulk) at 0 N d i zi e exp d x i PissBltzma Equati z i e k T (3.7) NaCl: ; z z z Catis: d z e z e sih d x k T B.C. : x 0, (surface ptetial); x, 0 luti: exp Ψ l exp Where, 4 ( κ x) tah Ψ ( κ x) tah Ψ (3.8) z e k T 4 Ψ (3.9) this is kw as dimesiless ptetial κ e k T z duble layer thickess;..(4.0); κ is iverse Debye legth r DebyeHuckel pprximati

10 If z e k T Ψ is less tha z e z e sih k T k T therefre, d d x e z κ k T pplyig B.C.: x 0, (surface ptetial); x, 0 luti: s exp ( κ x ).(4.) Dimesiless frm Ψ Ψs exp ( κ x).(4.) Dimesiless Ptetial (Ψ) DebyeHuckel Exact κ x

11 Nrmalized Duble Layer Ptetial (/ s ) : electrlyte (/ s ) 0.00 M 0.M 0.0 M X (/ s ) : 3:3 : X

12 Debye legth(κ ) decreases with the icrease i electrlyte ccetratis. Values f κ fr electrlyte fr several differet electrlyte ccetrati ad valeces Mlarity Z : Z κ Z : Z κ 0.00 : : 3:3 9.6x x0 9 :, : 3:, :3 :3, 3: 0.0 : : 3:3 0. : : 3:3 3.0x x0 9.5x0 9.0x x x x0 0 :, : 3:, :3 :3, 3: :, : 3:, :3 :3, 3: 5.56x x0 9.49x0 9.76x0 9.4x x x x0 0.49x0 0 Calculate the thickess f the duble layer? (ifrmal ssigmet)

13 Iccetrati Rati Remember: Z Z Z; Frm Bltzma distributi law z i e i i exp k T exp Ψ exp Ψ exp exp exp imilarly, exp (4.3) [ ] [ ( κ x) ] [ ( κ x ] [ exp( κ x ] ) (4.4a) )...(4.4b) / cteris / cis x

14 Diffused Electric Duble Layer arud a phere Csiderig PissBltzma eq Electrlyte sluti (Z:Z) r d r dr d d r e z z e sih k T (4.5) agular variati f subject t x a, (sphere surface ptetial r at shear plae); x, 0 N aalytical sluti exist with b.c. Get umerical sluti! (ssigmet) ssumig thi duble layer (ka >> ) ad r a (y/(ka)) ad we kw dimesiless ptetial, z e Ψ k T d Ψ d y dψ Ψ sih ka y d y k a y 0, Ψ Ψ (sphere surface ptetial r at shear plae); y, Ψ 0 usig z e k T Ψ << (DebyeHuckel pprximati)

15 r Ψ Ψ exp k a a.(4.6) The decay i Ψ is expetial. urface charge desity ad Ptetial Because f electreutrality excess charge i a sluti must balace the charge the particle surface. Fr flat surface: q f d ρ dx dx d x 0 0 d 0 d x 0 d d dx d x d x ( ) Differetiatig exp κ x tah Ψ 4 (3.8) ad Ψ l exp 4 ( κ x) tah Ψ substitutig ptetial gradiet at the surface (abve eq.), q [ 0 k T ] sih Ψ..(4.7) usig defiiti κ q 0 k T κ sih Ψ e z

16 Fr Ψ <<, (islated flat surface) ad usig q Ψ z e k T [ κ ] 0 s (4.8) The abve relatiship relates the surface charge ad the surface ptetial fr a flat surface. Fr the case f sphere, Leb et al. (96) gave apprximate frmula fr the surface charge Q kt 4 q κ sih tah, C / m Ψ Ψ.(4.9) 4π a e z Fr Ψ << κ a k T z e q κ e z κ a k T.(5.0) Fr ka>>, the abve eq. wuld becme that fr the flat plate. This is because the duble layer is s thi that the surface curvature effect is uimprtat. 4 Read: urface Ptetial Mdels Frce betwee tw charged bdies

17 b a X h/ X h/ f Piss Eq. ρ x DebyeHuckel pprximati d e z κ k T d x (frm 4.) Mmetum Equati with electrical frce (with iterial terms as the flw is viscus) µ u f ρ g ρ E bsece f gravity ad fluid flw: f ρ E p d P d d 0 d x dx dx p (5.) There are tw terms:. Osmtic Pressure (because differet i i ccetratis i betwee plates ad the surrudigs). Electrical frces Itegratig,

18 d P d x F d P P d x.(5.) We eed P ad i rder t fid ut frce/area C luti f the eq. (Piss eq) is e z κ d d x k T cshκ x B sihκ x.(5.3) Use B.C. (x h/; a ad b ) t fid ut a b a b ; B κ h κ h csh sih (5.4) Fr special case, a b ad hece B 0. Osmtic pressure fr dilute system is give by va t Hff s law p ( k ) kt Excess is ccetrati Usig (3.6) Bltzma s distributi law fr (z:z)

19 kt ze kt ze k T p exp exp (5.5) Fr Ψ<< T k e z kt p..(5.6) Frm (5.) usig (5.6) x d d kt ze k T F p Makig use f the sluti fr ad the expressi fr ad B, ad a b s, e btais h h T k z e F s p κ κ sih csh (5.7a) Fr a b h h T k z e F b a b a p κ κ sih csh..(5.7b) s κh<<, csh κh (κh) / ad sih κh κh Eq. 5.7 becmes

20 s s p T k z e F κ.(5.8) Nte that frce is idepedet f separati gap ad it ly depeds s. Fr κh>>, csh κh sih κh ½ e κh h s p e T k z e F κ 4 r h s p e F κ κ.(5.9) Usig eq. (5.7b), a Repulsi k T F P b 0.0 κ h ttracti a. a 0.5; b.0 b. a b 0.5

21 Calculate ad B whe the surface charge are held cstat. ( q q ) b / κ sih κ h a ( q q ) a b / κ csh κ h Istead f plates if tw spheres are take ad Derjagui pprximati was applied Cstat Ptetial F π k T z e κ a Ψ exp exp ( κ h) ( κ h) Cstat Charge F where, k T * exp π κ a q z e exp * q q k T κ z e ( κ h) ( κ h) Refer fllwig fr details: McCarthy, L. N. ad Levie,. (969), J. Cllid Iterface ci., 30, Derjagui pprximati. useful apprximati is t replace particles with the curved budaries by a series f flat strips, sice fr each f these, the frce per uit area is kw.

22 H a ds H Iteracti betwee tw spheres: F sp is the ttal frce the sphere whereas F p is frce per uit area the plate. F sp Fp π 0 sds Frm gemetry: H H a a s differetiatig, applyig a >> s i the regi f iterest dh s/a ds. s ds dh ad a s The frce a sphere: a dh Fsp Fp s a s π π H F dh 0 p It is apprximated here that duble layer is thi cmpared t size f the sphere ad F p F/ P. pplyig eq. (5.9) fr flat plate ad κh>>, i the abve expressi κ h F π κ a e sp s This expressi implicitly assumes:. GuyChapma duble layer mdel. DebyeHuckel apprximati 3. κ a >> that is thi duble layer Electrstatic ptetial eergy: h φ ( frce) dh va der Waals frce: Iheretly preset betwee all atms ad mlecules ad bdies (summig up atmatm r mledulemlecule iteractis). This is rigiates due t

23 istataeus diple mmet preset i all atms as electric clud (r charge) is revlvig arud ucleus. ummig up diplediple iteracti ver time betwee atms wuld result i fiite attracti betwee like atms. This may result i repulsi depedig iterveig medium betwee tw bdies r tw dissimilar bdies. Expressi fr va der Waals iteracti eergy r ptetial is give belw. Oe ca get frce by takig derivative with respect t distace betwee tw bdies. W (/ (6 H))*(R R )/(R R ) tw spheres f R ad R radius ad H distace apart. W R / (6 H) sphere ad flat surface W ( L / ( H.5 ))*[(R R )/(R R )] 0.5 tw cyliders f R ad R radius ad legth L parallel t each ther W /( π H ) per uit area tw flat surfaces is the Hamaker cstat (typical value f 0 0 J). Oe fid ut frm the refractive idices ad dielectric cstats f the ivlvig medium, ad absrpti frequecy Lifshitz thery f va der Waals frce.