At th e in te rfac e b etw een an y two p h a s e s a charge m ay develop. T his charge can arise th ro u g h th e io n isatio n of ionisable groups. For in stan ce , carboxylic acid gro u p s p re se n t on a surface ca n ionize to give rise to a negative charge a t th e interface. If the p h a se s have a different affinity for ions th e n ions can adsorb a t the
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in te rfa c e b etw een th e p h a s e s . A c h a rg e d su rfa c e in a liquid p ro d u ces a diffuse double layer. O ne layer of charge is located a t th e interface an d th e o th e r is d istrib u te d th ro u g h o u t th e liquid p h ase. The driving force for th e p ro d u ctio n of th e double layer is the increase in entropy cau sed by th e dissociation of ions from the surface.
Gouy 3 an d C h ap m an 4 independently developed theories to describe th e electrical double layer in th e 1910's. They a ssu m ed th a t charge a t th e su rface w as uniform (ie sm eared o u t ra th e r th a n p o in t charges) an d th a t th e finite size of ions h a s no effect. The charge density p a t a given potential is given by :
div(gradvF) = -p/e (1)
W h e re £ is th e s ta tic p erm itiv ity of th e m edium . The n u m b e r d ensity an d hen ce th e charge a t a given potential is given by th e B oltzm ann relation. For th e one dim ensional case (i.e. away from a plane or aro u n d a spherical surface) :
ni = ni(°o)exp(-zie'F/kT) (2)
ni is th e is th e local concentration of a n ion of charge z{ and ni(oo) is th e c o n c e n tra tio n of th e ions a t a n infinite d istan ce from th e surface. A com bination of th e above two re su lts (Eqns. 1 and 2) gives th e non-linear Poisson B oltzm ann eq u atio n for a single p la n ar double layer.
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(3)
The so lu tio n to w h ich is
tanh( 'l? r
) = 141111 (“j£r‘)exp('KX) (4)W here k“ 1, th e D ebye length is given by
The su rface charge a, m u st be b a la n ced b y th e charge in the adjacen t solu tion .
W hen tw o sim ilarly charged p la te s app roach e a ch other the overlap o f th e electrica l d ou b le layer p r o d u ce s a rep u lsio n . The co u n te r io n c o n c en tr a tio n at th e m id p la n e (a p la n e eq u id ista n t from b o th su rfaces) in c re a se s ( see fig. 1) . The force b etw een th e su rfa c es ca n th e n b e con sid ered to b e a n o sm o tic p r essu re given
(6)
0 0
by:
d » K
Non interacting surfaces
Interacting surfaces
Figure(l). Two charged surfaces, with surface potential at a large
separation (above) and at a separation d. The potential profile 'F(x) is shown
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The concentration a t th e mid plane nj can be obtained from the B oltzm ann distribution Eqn. 2. an d by su b stitu tio n into Eqn. 7 the p re ssu re becomes:
P = kT Qoo(exp( -ZieT'm/kT) + exp(ZielFm/kT) -2) (8)
= 2kT Cioo (cosh(ZiexFm/kT) - 1) (9)
To solve th is eq u a tio n 'Em m u s t be know n , th is value depends on T'o and so the final solution depends on th e b o u n d ary conditions invoked. C o nstant surface potential an d c o n s ta n t surface charge b o u n d a ry conditions are m o st often used . A sim ple m ass action m odel w hich allows th e su rface charging to change as th e su rfa c e s are b ro u g h t to g e th e r gives p h y sic ally m ore realistic re su lts 5,6
T he th e o re tic a l DLVO c u rv e s w h ic h w ere u s e d for com parison w ith experim ents in P art II were calcu lated u sin g an ex a ct n u m e ric a l so lu tio n to th e n o n -lin e a r P o isso n -B o ltzm an n equation. The pro ced u re selects a p re s s u re a t th e m id-plane and th e n calculates the surface sep aratio n w hich gives th is p re ssu re 7.
At large se p a ra tio n s an d ac ro ss a wide range of p o ten tials th e G ouy-C hapm an th eo ry of th e electrical double layer predicts th e in te ra c tio n s b etw een ch arg ed su rfa c e s extrem ely well. The theory even pred icts th e in teractio n s w here th e a re a p er charge is large by com parison to th e surface sep aratio n , a situ a tio n w here it could be expected th a t th e d isc re te n e ss of su rface charge w~ould influence th e in teractio n . At s h o rt d ista n c e s th e th eo ry becom es com pletely in ad eq u ate in all b u t a few cases. Solvation effects often d o m in a te th e in te ra c tio n in th e s h o r t d ista n c e regim e. O th er
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fa c to rs have in re c e n t tim es b ee n sh o w n to be im p o rta n t in predicting th e interactions. These are:
i) The finite size of the ions.
ii) Image forces and ion-ion correlations. iii) Specific solvent interactions.
More recently other theories an d extensions have b ee n m ade to Poisson-B oltzm ann theory to account for th e se effects. The m ost ac c u ra te m odel of th e double layer in teractio n h a s b een developed by K jellander an d M arcelja. 8 Their m ethod in clu d es th e effects of ion size, im age charge and ion-ion correlations. For th e in teractio n of su rfa c e s in m onovalent electrolytes th e y found th e repulsive p re ssu re to be sm aller th a n the corresponding PB re s u lt except at a sep aratio n of 5Ä w here the p ressu re is larger due to ion size.9 At large s e p a ra tio n s an d dilute electrolyte th e ag ree m en t betw een HNC an d PB theory is very good. At high sa lt co n cen tratio n s (0.5M an d 1M) PB theory fails a t all se p aratio n s an d ion-ion correlation effects need to be invoked to explain th e in teractio n betw een m ica sh e ets in 0 .1 5M CaCl2 and the swelling behaviour of clays 12.
A ttard, M itchell an d N inham 10*11 developed an extension to double-layer theory w hich trea ts van d er W aals and double-layer forces c o n s iste n tly a n d th e effects of ion-ion c o rre la tio n s a n d e le c tro s ta tic im a g e s a re are tr e a te d . T he a s y m p to tic form c a lc u la te d w ith th is ex ten d ed tr e a tm e n t is a n e x p o n e n tia lly decaying in teractio n corresponding to th e PB law for a system w ith a different surface charge.
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