The stablty of thn (soft) flms Lecture 3 Electrostatc effects November 6, 009 Ian Morrson 009
Revew of Lecture The dsjonng pressure s a jump n pressure at the boundary. It does not vary between the plates. G Π ( h) = A h T P σ σ,,,, N P h P π RR Fs s y Π h dh R R Force between spheres: ( ) ( ) G(h) y ( ) G h Π ( h) = h G 0 = σ σ σ = S 0 G ( ) ( ) = 0 s0 sl lv 3 h Curve Stable Curve Metastable Curve 3 Unstable Ian Morrson 009
Revew of Lecture ( h) Π = G A h σ σ T, P,,, N ( ) = s s μ Ψ σψ σ Π =, r dg S dt dn A d d A dh ( ) = μ σ Ψ σ Ψ Π dg s s S dt Nd A d d A dh =, r s N Π= dμ A μ h h μ,, Ψ, Ψ s a modfed Gbbs equaton. n σ dσ= dμ = Γdμ A N A s Π Γ = A μ h h, dh s the excess adsorpton due to dsjonng pressure. Note that we do not know how much excess s on ether plate! Ian Morrson 009 3
Electrostatc repulson n thn flms Loosely held countercharges form electrc double layers next to each surface. The concentraton of ons s determned by a balance between the attracton to the surface and knetc energy. Electrostatc repulson results from the nteractons of ons around each surface. The overlap of the electrcal double layers s the (electrcal component of ) dsjonng pressure. Ian Morrson 009 4
Contact potentals http://en.wkpeda.org/wk/electrometer In a paper of May, 897 (Royal Insttuton of Great Brtan) Lord Kelvn descrbed a demonstraton experment that nvtes some thoughts, based on hndsght, about the connecton between constant potentals and the voltage of galvanc cells. R.M. Lchtensten n Nneteenth Century Atttudes: Men of Scence, Ross, S. 99. Buld a capactor wth two parallel plates of dssmlar metals, say copper and znc. Ground the znc plate ground. Isolate the copper plate. Connect the copper plate to the hgh termnal of a (hgh mpedance) electrometer. a Ea = Γ ZnCu, When the plates are moved from ( a a ) E( a a ) =Γ contact (at a) to a, the electrometer reads a potental change: Zn, Cu n.b. The contact between plates s broken Ian Morrson 009 5 a > a a
Contact potentals ( a ) E a =Γ Zn, Cu ( a a ) a The experment can be altered slghtly to use Kelvn s null method. The electrc potental at contact s now Ea V =Γ Zn, Cu V V a a ( ) ( ) When the copper plate s rased to a, E a a = ΓZn, Cu V the electrometer wll show the readng a a n.b. The contact between plates s broken Repeated measurements determnes the null Γ potental and hence the contact potental:, a > a Γ = V Zn Cu Ian Morrson 009 6 null
Now a remarkable experment!. Contact potentals 3 The dry metallc contact of s replaced by a wet contact Before the copper plate s rased the electrometer no longer ndcates the metallc zero but a new value, whch h Kelvn called the electrolytc l t zero. Kelvn rased the copper plate, so that the water drop would break, and contnued to rase t. The remarkable effect was that the electrometer started and remaned at ts electrolytc zero! Ths means there are no feld lnes across the gap! And. Γ Zn, Cu = voltage of the cell Cu / water drop / Zn Dry contact: Has electrc feld, but no current, hence no power. Wet contact: No electrc feld but has voltage, hence has current, hence some power. How do you get a lot of power? Ian Morrson 009 7
Lots of power? Make a battery! How? Rubbng or touchng two surfaces, say Cu and Zn, produces some useful power, but not much. Alternatng Cu/Zn plates, wth dry contact. Alternatng Cu/Zn plates, wth wet contacts. 0 V NDG V Equal to just one Cu/Zn set. Ian Morrson 009 Topc 7 Charged nterfaces 8
*R.M. Lchtensten n NneteenthCentury Atttudes: Men of Scence, Ross, S. 99. The Voltac ple Bottom Cu/wet/Zn/dry/Cu/wet/Zn/dry/ Cu/wet/Zn/dry/Cu Top The Voltac ple has n wet contacts and n dry contacts The voltage between the top and the bottom s n tmes the voltage of a sngle wet contact, t V Cu/wet/Zn. There are no voltages across the dry contacts; there are electrc felds, all algned n the same drecton. There are voltages across the wet contacts; there are no electrc felds. Physcs n the dry contacts, chemstry n the wet contacts. Ian Morrson 009 9
Ions near a charged surface Adsorbed ons determne the surface potental, Φ 0. The counterons (valence z ) form a dffuse cloud near the surface. The GuoyChapman model assumes the surface potental s unform and the counterons are pont charges. = n0 exp ze Φ ( x ) kt The concentraton of ons, n, depends on n concentraton of ons far from the surface, n 0, and the local electrc potental, Φ(x): ze ρ Φ The local charge densty, ρ, depends on exp ρ = zen = zen 0 the charges and the concentratons of ons. kt The Posson equaton relates the local charge densty to the Laplacean of the electrc potental. t Φ= Dε ρ 0 Hence the PossonBoltzmann equaton, a dfferental equaton to solve wth boundary condtons for potental as a functon of dstance. Φ= ze Φ zen 0 exp D ε kt Ian Morrson 009 0 0
Ions near a charged surface Φ= ρ D ε 0 Φ=κΦ Φ = ζ exp( κ x) (The surface potental s the zeta potental.) Potental ζ zeta potental Increased onc strength κ = I e cz Dε kt 0 = cz Adsorbed surfactant Dstance layer /κ Ian Morrson 009
σ j = ε0ee j BB j ε0e B δj μ0 μ 0 = Electrostatc component of dsjonng pressure* () r plates dg = SdT VdP N dμ A σ dψ AΠdh j j j () The dsjonng pressure Π ( h ) = ( Eh Eo ) s the excess Maxwell stresses between the feld gradents nsde (E h h) ) and the feld gradents outsde (E 0 ). ε But the gradents are unknown! Internal feld gradents between two flat plates. External walls are assumed to have same potentals as nner. Derjagun, 987, Fg. 6.. () Thermodynamcs gves: (3) To fnd the RHS, use the PB equaton: Π ψ h, ψ σ = h ψ, ψ d ψ ε = = dh (4) A bt of dfferental analyss gves: ( h ) ρ ( ) ρ zen 0 exp Ψ zeψ kt ε dψ Π = Ψ d Ψ π dh 0 8 Ian Morrson 009 For symmetrc surfaces, the second term s zero at mdgap.
Electrostatc component of dsjonng pressure () Frst try: same potental on each plate; bnary electrolytes. ( m ) ( h) ktn φ ( h) Π = cosh Manpulatng the PB equaton produces: Φ 0 κh d Φ = cosh cosh Φ m ( Φ Φ ) m ze where φm = ψm (at the mdplane) kt dφ and = 0 dh h= m These two equatons gve Π(h) parametrcally through φ m. For the approach of surfaces at constant potental, Φ 0 s constant. For the approach of surfaces at constant surface charge densty Φ 0 s: σ s zen = cosh Φ0 cosh Φ κ ( ) m (Wth consderable algebra nvolvng ellptcal ntegrals.) Ian Morrson 009 3
Electrostatc component of dsjonng pressure* (3) Constant surface potental or constant surface charge? If two surfaces approach each other and surface potental reman constant, the charge per unt area must decrease. Ions must ether adsorb or desorb! If two surfaces approach each other and the surface charge reman constant (no on adsorpton or desorpton), the electrc potental must ncrease! Dsjonng pressure as a functon of κh n a symmetrcal electrolyte Ψ Ψ0 lm 4 nkt snh h 0 at constant potental (lower curve) and constant surface charge σ 4nkTπ lm Π = h 0 (upper curve). ( κh) Π = The dfference n behavor s huge! Ian Morrson 009 4 *Followng Derjagun, 987, pp. 8 83.
Complex ons phase dagram Stablty domans as a functon of Al(NO 3 ) 3 or AlCl 3 concentraton and ph for styrenebutadene rubber (SBR). Shaded area desgnates the coagulaton regon; below the c.c.c. lne the sols reman stable, above the c.s.c. lne the sols are restablzed due to charge reversal. (....) The formaton of alumnum hydroxde precptate p n the absence of sol partcles. Matjevc, JCIS, 43, 7, 973. Ian Morrson 009 5