Th Random Phas Appoxmaton: Elctolyts, Polym Solutons and Polylctolyts I. Why chagd systms a so mpotant: thy a wat solubl. A. bology B. nvonmntally-fndly polym pocssng II. Elctolyt solutons standad dvaton A. Posson-Boltzmann quaton B. Dby-Hücl quaton. f ngy. colatons III. Polym solutons A. What a ntactons. chan Hamltonan. monom ntactons 3. chmcal potntal µ () B. Patton functon and f ngy C. Colatons I. Fld-thotcal dvaton of Dby-Hücl thoy. Fld-thotcal dvaton of th Posson-Boltzmann quaton I. Byond Dby-Hücl thoy B. Oscllatoy chag colatons C. Ion assocaton
II. Elctolyt solutons P-B qn A. Posson-Boltzmann quaton () Lt Φ lctostatc potntal at Φ() 4π ε ρ () Posson wh ρ () z + n + () z n valncy of + chags But n + So n () n+ 0 βz + Φ () n 0 βz Φ # dnsty of - chags ρ() z + n 0 + βz + Φ() z n 0 βz Φ ( ) Φ 4π ε zn + 0 βz + Φ ( ) z n 0 βz Φ( ) { } Boltzmann (man fld appoxmaton) B. Dby-Hücl quaton Lnaz th Posson-Boltzmann quaton (assum lctostatc ngy s wa compad to T) Φ 4π ε z n + + 0 βz + Φ ( ) z n 0 + βz Φ { ( ( ) )} { } Φ 4π ε z + n 0 0 ( + z n ) β z + n 0 + + z 0 ( n )Φ()+... but z n 0 + + z n 0 0 du to lctonutalty, so Φ 4π ε T z + n 0 + + z 0 ( n )Φ Lt ε T B Bjum lngth Thn Φ4π B z + n 0 + + z 0 n Φ κ Wh κ Dby scnng lngth---multvalnt ons scn mo ffctvly.
Thus, w hav Φκ Φ Dby-Hücl quaton. F ngy of Dby-Hücl quaton Consd had sphs of damt a wth chags z ±. sph of chag q 0 whos cnt s at th ogn? What s potntal aound a Φκ Φ Φ0 < a a B.C. s: Φ ( ) 0 Φ() s contnuous at a ε φ s contnuous at a (lctc fld) 3
Soluton: Φ() q ε qκ ε +κa κ( a) qε ε( +κa) Yuawa a > a Exclud th slf-ngy tm q ε. Thn φ ( 0 ) du to all th oth changs s ψ( 0) qκ ε +κa Th lctostatc f ngy satsfs df ψ dq wh ψ potntal actng on chag thmodynamc conjugats. q du to all th oth chags, bcaus ψ and q a F N,,T ψ q () So F ψ λ 0 d( λq ) potntal on on whn all ons hav chags λq. q So ψ () λ λ κλ ε( +κλa) F q κ ε 0 dλ λ +κa λ But: q ε ( ε z + n 0 + + z n 0 ) ε z + n 0 + + z 0 n call that κ 4π ε T z + n 0 + + z 0 n 4
f l q T ε 4π κ F l T κ 3 4π So w hav f l κ 3 wh π f κ a f ( κa) 3 κ 3 a 3 0 λ d λ +κ a λ ln( +κa) κa + κ a Not that f ( κa) as κa 0. So, fo dlut solutons f l κ 3 π Ths s ngatv: attactv ntactons btwn oppost chags pdomnat. Osmotc pssu: βp n 0 + +n 0 κ 3 4π Not that th s no scond val coffcnt du to long-angd ntacton.. Colatons n Dby-Hücl thoy: Poo Pson s vson. Loo at chag dstbuton aound a cntal on q 4 π ε ρ() φ() Posson κ φ() So ρ() ε 4π κ φ() ε 4π κ q κ ( a) ε ( +κ a) So th chag q s suoundd by a scnng cloud of oppost sgn that dcays xponntally and monotoncally wth scnng lngth. 5
III. Polym Solutons th andom phas appoxmaton Suppos w hav N chans, ach M monoms long, n a box of volum. Dfn th numb dnsty of monoms: N M () ds δ ( ) n A. What a th ntactons? 0. Th a ntactons that hold th chan togth. s βh 0 + 3 b 0 M ds s Ths s a contnuous vson of βh 0 + 3 b M s s s So, ths cosponds to a bad-spng modl of a chan: Ths has an avag bond lngth b.. In addton, w can nclud a shot-angd ntacton btwn monoms: β H + v d ρ() fo good solvnt v > 0 3. Fnally, allow fo a chmcal potntal µ ( ) (n unts of T). Ths wll allow us to xamn th avag dnsty and dnsty colatons. βh d µ ()ρ (). B. Th patton functon s Z N! D s β H 0 () { s} βh βh 6
ths s th had on bcaus t s a two-body ntacton tm. Hubbad-Statanovch tansfomaton (complt th squa) Not that v ( () d ρ ) v [ () vρ() ] d J - v d ( J() ) + d J ()ρ() So o DJ v ( d ρ() ) v d J( ) vρ () [ ] DJ v d ( J() ) d J()ρ () v d ρ( ) DJ d DJ J()ρ ( ) v v d J ( ( ) d J So, th patton functon s Z N! DJ v d ( J () ) D s ρ H 0 () d [ µ ( )+ J( )]ρ () wh DJ v d J( ) Th advantag of th Hubbad-Statanovch tansfomaton s that w hav placd a two-body ntacton wth a on-body ntacton wth a chmcal potntal J( ). Ths ducs th poblm to on of non-ntactng chans n a spatally vayng chmcal potntal. Not that ρ () ρ () Wh ρ () ds So, w can wt Z: Z N! δ ( s ) ( [ ]) N DJ Z µ + J v d ( J ( ) wh Z [ µ+j] s th patton functon of a sngl chan n chmcal potntal µ ()+J ( ). Lt µ µ+ J. Thn w hav 7
Z [] µ D s βh 0 d µ ( )ρ () *Dgsson on Fou tansfomatons: I wll us th convnton f d f () f () δ ( ) δ, f d. ( ) + ( ) So d µ ( )ρ( ) µ ρ. f Not: w a fxng ρ so 0 µ 0 0. Also, ρ 0 fo 0. Now xpand th xponntal n Z. µ ρ + µ ρ + µ µ ρ ρ +... Z [] µ Z 0 + µ µ ρ ρ +... Dfn Mg ρ ρ. Ths s th fom facto of th chan. R-xponntat: Z µ [] Z 0 M g µ µ +... Substtut bac nto Z: lt c NM b th monom dnsty. Thn th patton functon fo th many-chan systm s Z Z 0 ( ) N N! DJ +... cg µ + J v J Z 8
Z Z 0 ( ) N N! cg µ µ DJ cg µ J cg + v J J Solv by saddlpont appoxmaton: agumnt of xponntal has zo dvatv whn cg µ cg + v J J cg µ cg + v Evaluat agumnt at saddlpont: c g cg + v µ µ + c g µ µ cg + v c g cg + v µ µ Z Z 0 ( ) N N! Th f ngy s cg cg cg + v 0 N µ µ Z N! cg +vcg µ µ. βw ln Z stuff cg +vcg µ µ Th colatons functons satsfy ρ ρ β W cg. µ µ + vcg So w hav ρ ρ v+ cg Ths s th RPA sult fo th colaton functon. Rcall that Mg ρ ρ fo a sngl chan, wh g s th fom facto. g Mf( Mb 6) Mf( Rg ) wh f() x x ( x + x) s th Dby functon (s Do & Edwads, pp. -3). 9
f() x 3 x x x 0 x M g ~ Rg Incasng v just flattns th cuv out. RPA s a good way to dv man-fld f ngs and colaton functons. 0
I. Fld thotcal dvaton of Dby-Hücl thoy. Consd N + ons of valncy Z + N + Z + N Z N - ons of valncy by lctonutalty Z Lt n ± 0 N ± avag dnsty of ach spcs n ± ρ() z + n + and βh l B N ± () δ ( ) local dnsty of ach spcs () z n () ()ρ( ) d d ρ wh B εt Rwt ths n matx fom: s th Bjum lngth. lt n () () () n + n ( ) B z + z + z z + z z Thn ()ρ() ρ n () ( ) n () βh l d d n() ( ) n In Fou spac: βh l n n wh 4 π z B + z + z z + z z
In addton, w can ntoduc µ () µ + ( ) and βh µ () colaton functons. d µ ( ) n( ) n od to calculat Th patton functon s thn gvn by Z N +! N! D + D βh βh Cay out th Hubbad-Statanovch tansfomaton: n v n DJ + DJ + DJ DJ n J J J J J wh J () J + () J () Not that J ± () s l ± βφ() As bfo, potntal actng at du to oth chags Z N +! N! DJ ± J J D ± ( µ +J) n ( [ ]) N± Z µ +J ± wh Z [ µ+j ± ] D ± µ ± n ± fo a +/- on. Agan, dfn µ µ + J and xamn th sngl patcl patton functon. Z [ µ ± ] Z ± µ ± g ± µ ± ( )+ H O T wh g ± s th fom facto of th +/- on. ( Z [ µ ± ]) N ± tuncat n Dby-Hücl thoy. Ths s quvalnt to assumng lctostatc ngy s wa compad to T. N Z ± ± N ± µ ± g ± µ ±
( [ µ + ]) N + So, Z ( Z [ µ ]) N Z + N + Z N µ G µ wh G n 0 + g + 0. 0 n 0 g Substtut bac nto Z: Z Z N + + N +! Z N N! DJ ± ( µ + J) G ( µ + J) DJ ± J J J J Cay out th Gaussan ntgals (s Do and Edwads appndx.i) Dnomnato: Numato: J J DJ ± π DJ ± µ G J f J ( G + ) J DJ ± G µ π f G + µ G G + So Z Z N + + N +! N Z N! µ G µ G + wh G G ( ( G + ) G) G ( + G) o quvalntly, N + N Z Z Z + N +!N! v µ G µ + G / Fo pont patcls, Z, g () ± ±. So Z N + N N +! N! Π + G µ G µ To gt th f ngy, w can st µ 0. 3
βf β F ln Z n 0 + ln n 0 + +n 0 ln n 0 + Now ta th thmodynamc lmt: β f n + 0 ln n + 0 + n 0 ln n 0 + d ln + G ( π) 3 ln + G + G dt + 4π B n + 0 z + n + 0 z + z 4π B n 0 z + z 4π B + 4π B n 0 z + 4π B n 0 + z + +n 0 z + κ. So fo pont patcls, β f n 0 + ln n 0 + + n 0 ln n 0 + d 3 ln + κ ( π) 3 lt x κ subtact slf-ngy κ β f n 0 + ln n 0 + + n 0 ln n 0 + κ 3 d 3 x ln + ( π) 3 x x So th fnal answ s: βf n + 0 lnn + 0 +n 0 lnn 0 κ3 π 6π Sam answ as bfo fom standad appoach! Loo at colatons: n + ()n () δβf δµ + δµ µ 0 4
Th f ngy contbuton fom µ 0 tms s v wh G G ( + G). µ G µ Th advantag of ths appoach s that w can now wt down th Dby-Hücl f ngy fo patcls of abtay stuctu. Rcall β f n + 0 ln n + 0 + n 0 ln n 0 + wh + G dt () + G + κ + 4π B d 3 ln( + G) ( π) 3 n 0 + z + g + () n 0 4π z + z B g () () + 4π B n 0 z g () n + 0 z + z 4π B g + wh κ () 4π B n 0 + z + g + +n 0 z g Fo xampl, suppos th ons a sphs of damt a. Thn κ () κ g wh sna acosa g 3 ( a) 3 s th fom facto of a sph. So a mo gnal xpsson fo th f ngy s β f n 0 + ln n 0 + + n 0 ln n 0 + d 3 ln + κ ( π) 3 κ Fo th gnal cas, th FT of th chag colaton functon s ρ ρ +κ [ 0 g + ()+n 0 g () ] () n + 5
. Fld-thotcal dvaton of Posson-Boltzmann thoy Suppos th a fxd chags wth som dstbuton σ( )< 0. Assum that th N countons a pont chags of valncy z. Thn th patton functon s Z N N! wh D βh βh B z d d n() n ()+z B d d n() σ () + B d d σ() σ () dn()µ B d d [ zn ()+σ ]( )[ zn ( )+σ ( )] n ()µ()d () lt n () zn()+σ. thn βh B d d n () ( ) n ( ) dn ()µ Intoduc th Hubbad-Statanovch tansfomaton:. B n n n J DJ DJ B B J J J J So Z N! D N! N! DJ n J nµ DJ B DJ B J J J J B D ( zn+σ ) σ J wh µ () µ ()+zj (). J J J +nµ D n µ 6
Not that ( ) 4π δ Is th nvs opato fo th Coulomb ntacton. Why? By dfnton, th nvs opato satsfs d ( )( ) δ Fom Posson s quaton ε () 4π ε δ ( ) 4π ( ) δ So d ( )( ) 4π So ( ) 4π δ Now loo at Z [] µ d d dδ d µ ( ) so w hav n µ ( )d µ ( ) Z N N! DJ B J J σ ( ) J()d [ d µ ()+ zj( ) ] N Now loo at th gand canoncal patton functon: [] Ξ λ N Z N µ N J J DJ B σj N! λn Z µ +zj N ( [ ]) N 7
Ξ J J + σj DJ B Loo at λ µ + zj d J J d d J() δ ( )J 4π + 4π d d J () J ( J ()) 4π So th gand canoncal patton functon s Ξ DJ 4π B ( J) + σ J + λ µ +zj Fom th patton functon, w can fnd th avag local dnsty: n () δln Ξ δµ λ zj() () µ 0 Now cay out th saddl pont appoxmaton fo th ntgal ov J. (st µ 0) lt F 8π B Thn w hav Ξ DJ Fd ( J) σ ()J () λ z J( ) Th agumnt of th xponntal s xtmzd whn δf δj 0 F J F J 0 σ () zλ +z J() 4π B J. J 4π B [ σ () zλ z J( ) ] 4π B σ( ) z n() [ ] 8
But call that J() s th ffctv lctostatc potntal du to all of th oth chags. J+β Φ Φ 4π ε [ σ ()+ zn() ] 4 π ε [ σ()+z n ()] Ths s th Posson-Boltzmann quaton. 9