Polyelectrolyte multilayer formation: electrostatics and short-range interactions

0.28 MB·English

by  Adi Shafir

EPJ manuscript No.
(will be inserted by the editor)
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Polyelectrolyte multilayer formation: electrostatics and
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short-range interactions
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n
a Adi Shafira and David Andelmanb
J

3 School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University,Ramat Aviv,
Tel Aviv 69978, Israel

]
t
f 04 December2005
o
s
. Abstract. Weinvestigatethephenomenonofmultilayerformationvialayer-by-layerdepositionofalternat-
t
a ingchargedpolyelectrolytes.Usingmean-fieldtheory,wefindthatastrongshort-rangeattractionbetween
m
thetwotypesofpolymerchainsisessentialfortheformationofmultilayers.Forstrongenoughshort-range
- attraction, the adsorbed amount per layer increases (after an initial decrease), and finally it stabilizes in
d the form of a polyelectrolyte multilayer that can be repeated hundreds of times. For weak short-range
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attractionbetweenanytwoadjacentlayers,theadsorbedamount(peraddedlayer)decaysasthedistance
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from the surface increases, until the chains stop adsorbing altogether. The dependence of the threshold
c
value of the short-range attraction as function of the polymer charge fraction and salt concentration is
[
calculated.

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v PACS. 82.35.Gh Polymers on surfaces; adhesion – 82.35.Rs Polyelectrolytes – 61.41.+e Polymers, elas-
7 tomers, and plastics
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9 1 Introduction
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0 The study of polyelectrolyte (PE) chains interacting with A A A B AB
/ chargedsurfaceshasgeneratedagreatdealofattentionin
t
a recent years. This interest arises, in part, because of the
m numerous biological and industrial applications. The ad- A wash B wash
d- sfaocrepstihoanvaenbdedenepelxettieonnsiovfelpyolsyteuldeicetdroulysitnegs oannaclhyatricgaeld[s1u,2r-, (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
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3,4,5,6] and numerical[7,8,9,10,11] solutions of the non-
o Fig. 1. A schematic illustration of the layer-by-layer PE de-
linear mean-field equations, scaling considerations [7,8,
c position process. Thecharged surface isfirst dippedintoaso-
: 9,10,11,12,13,14,15,16,17], multi-Stern layers of discrete lutionofoppositelychargedPEchains(marked‘A’).Afterthe
v latticemodels[18,19,20,21]andcomputersimulations[22,
i chainsadsorbontothechargedsurface,thesurfaceisremoved
X 23,24,25]. anddippedintoaclearwatersolution(marked‘wash’),remov-
In recent years, formation of polyelectrolyte multilay-
r inganyextranon-adsorbedchains.Thesurfaceisthendipped
a ers has been investigatedexperimentally [26,27,28,29,30, into another solution of PEs (marked ‘B’), carrying a charge
31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47].Theseopposite to the previously adsorbed PE chains, and washed
multilayersarecomposedofalternatingpositivelyandneg- again(‘wash’).Theprocessofdippingandwashing:‘A’,wash,
atively charged PEs and are constructed via a layer-by- ‘B’, wash, ‘A’, wash,... can then be repeated for several hun-
layeradsorptionofpolyelectrolytechains,shownschemat- dred layers.
ically in Fig. 1. The first stage is to dip a charged sur-
face into a solution of oppositely charged polyelectrolytes
and salt. After the polyelectrolyte chains adsorb on the
cesscanberepeatedforseveralhundredtimesandresults
charged surface, the surface is taken out of the solution
in a PE multilayer build-up [32]. More recent studies [30,
and washed in a clear water solution. The washed sur-
31]haveshownthatthesemultilayershaveinterestingand
face and adsorbed layer are then placed in a solution of
potentiallyusefulapplicationsbothforplanarandspheri-
another polyelectrolyte, of an opposite charge to the first
calgeometries,leadingthewaytocreationofmultilayered
PE chain (see Fig. 1), and then washed again. This pro-
and hollow spherical capsules.
a e-mail: [email protected] Theoretical models for multilayer formation have also
b e-mail: [email protected] been considered in recent years [6,11,48,49,50,51,52]. In
2 AdiShafir, David Andelman:Polyelectrolyte multilayer formation: electrostatics and short-range interactions
previousstudies[6,48],itwassuggestedthattheinversion
of the surfacechargeby polyelectrolyte adsorptionoccurs a2d2φ
=vφ3+fζφ+ω2φ5 (2)
under the following conditions: (i) sufficiently large salt 6 dx2
concentration;(ii)thetasolventconditions;and,(iii)weak
φ2isthelocalmonomerconcentration,xthedistancefrom
short-range(SR)interactionsbetweenthesurfaceandthe
the charged surface, ζ = eψ/k T the renormalized (di-
PE chains. The charge inversion, together with complex- B
mensionless) electrostatic potential, a the monomer size,
ation of polyanions and polycations have been used to
andf thechargefractionofthePEmonomers.TheDebye-
model the multilayer formation [48].
In a previous publication [11], we reached different Hu¨ckellengthκ−1 ≡(8πlBcsalt)−1/2isthescreeninglength
for electrostatic interactions in presence of salt ions, and
conclusionsbysolvingnumericallytherelevantmean-field
l ≡ e2/εk T is the Bjerrum length, which is approxi-
equations for the PE adsorption. We found that surface B B
chargeinversionwillnotoccurunderabroadrangeofsys- mately 7˚A for water with ε = 80 and at room tempera-
ture.
tem parameters. More specifically, it was shown that full
chargeinversionoccursonlyforstrongenoughSRsurface- Equation(1)isthePoisson-Boltzmannequation,where
PE interactions. Related results have been reported in a thesaltionsobeytheBoltzmanndistributionandtogether
separate study [51], where the full charge inversion was with the monomer charges they act as charge sources for
foundtooccuronlyforstronglyhydrophobicPE-backbone. the electrostatic potential. Equation (2) is the Edwards
Namely,PEchainsinpoor solventconditions.Thestrong equation for the monomer order parameter φ, where the
hydrophobicity in the bulk creates an effective attraction chains are subject to an external potential composed of
to the surface, replacing the bare surface-PE short-range an electrostatic potential and an excluded volume inter-
attraction. actionbetweenthemonomers.NotethatinEq.(2)wein-
cluded both the second virial term modeled by v and the
In the present work, we study the formation of alter-
third virial one modeled by ω2. In most previous studies
nating charged PE multilayers as a function of polyelec-
of PE adsorption the third virial term has been omitted,
trolytecharge,addedsaltandSRinteractionbetweenthe
because of the dominance of the electrostatic interactions
PE chains. We find that strong SR (non-electrostatic in
and the second virial term. In the case of multilayer for-
origin)interactionsarenecessaryfortheformationofsuch
mation,however,thethirdvirialtermbecomessignificant,
multilayers, and that the adsorbed charge of the alter-
as is explained in the next section. Finally, we note that
nating layers is not necessarily equal, or even close to,
Eqs.(1)and(2)arewrittenforvanishinglysmallbulkcon-
the initialsurfacecharge.Ouranalysisshowsthatthe ad-
centration of monomers, and under the assumption that
sorbedamount(per addedlayer)in the initially adsorbed
the ground state dominance approximationholds.
layers always decreases. If the SR interactions between
the PE chains are too small to attract another PE layer, In order to model the build-up of multilayers,we note
the multilayerformationwillstopaftera smallnumberof that the experimental multilayer build-up is done via a
layers. However, if the SR attraction between the alter- layer-by-layeradsorptionofcationicandanionicPEchains,
nating PE layers is significant, the adsorbed amount (per asisillustratedinFig.1andexplainedintheintroduction.
added layer) starts to increase back, and then saturates During the adsorption of each layer, the chains from pre-
and forms a stable multilayer stack. We also show how viously adsorbed layers are believed not to dissolve back
the multilayer formation depends on the solution salinity into the solution. Note that the layer-by-layerbuild-up is
and PE charge fraction. not a thermodynamically equilibrium process. Any mod-
eling of this phenomenon should take into account these
In the next section, Sec. 2, we present the mean field
specific stages.
equations for multilayer formation and the numerical
Inourmodel,electrostaticandshort-range(SR)inter-
methodusedtosolvethem.Thenumericalresultsformul-
actionswiththePEchainsofthepreviouslayersaretaken
tilayer formation follow in Sec. 3, and their discussion in
intoaccount.TheSRinteractionsbetweentheanionicand
Sec.4.Weendwithconclusionsandsuggestionsforfuture
cationic PEs may have several origins. The repulsive SR
researchin Sec. 5.
interactions include excluded volume interactions, while
an attractive SR interaction (beside the electrostatic at-
traction) arises from polyelectrolyte complexation. We do
2 The Mean Field Equations
not offer in the present work a detailed explanation for
the complexation origin. We rather assume its existence,
Consideranaqueoussolutionincontactwithabulkreser- which yields an effective SR attractive interaction, and
voir of salt ions, and a dilute bulk concentration of long investigate under which conditions it will lead to multi-
polyelectrolyte chains. The solution is in contact with an layer formation. We assume that the electrostatic attrac-
infiniteandplanarsurface.Thesurfaceisoppositelycharged tion and ion pairing between the chains in the adsorbed
and attracts the PE chains. The mean-field equations for multilayerandthe adsorbingPE chains allowthe adsorb-
this system were formulated in Refs. [7,8,9,10], and are ing chains to penetrate the multilayer. This penetration
repeated here. slows down the PE chains dissolution back into the solu-
tion, and creates an effective SR attraction. This interac-
d2ζ tion has a non-equilibrium origin, but for dense layers, it
dx2 =κ2sinhζ−4πlBfφ2 (1) shouldlastlong enoughto allowfor multilayerformation.
AdiShafir, David Andelman:Polyelectrolyte multilayer formation: electrostatics and short-range interactions 3
The electrostatic interaction between the PE chains between the surface and the PE chains, and d > 0 corre-
is taken into account by adding the PE chain charges in sponds to an attractive surface.
the Poisson-Boltzmann treatment, Eq. (1). The SR in-
teractionsbetweenthecationicandanionicPEchainsare For all subsequent layers, i ≥ 2, we expect the PE
takenintoaccountverysimplybyaddingadifferentinter- chains to partially penetrate into the previous layers be-
action parameter χ 6= v in Eq. (4). Hereafter, we assume causeofthecomplexation.Inordertoavoidthepossibility
thatthechargedfractionf ofmonomersforthenegatively of fully interpenetrated layers, we introduce a hard wall
∗
and positively charged PE chains is the same. As the PE for each layer at an arbitrary location x . Otherwise, in
i
adsorptionis done layerby layer,the equations governing our case the cationic and anionic polyelectrolytes would
the adsorption of the ith layer are: form a neutral complex, which will not be able to form
a stable PE multilayer. Because it is known from experi-
ddx2ζ2 =κ2sinhζ−4πlBf(ziSs+zi+1So) (3) mdoenntosttfhualltytmheixawdistohrbtihnegpPreEvicohuasinlasyeorfsawneyisnpterocidfiuccelatyheer
concept of the hard wall. Its justification would require
a2d2φi =vS φ +fz ζφ −χS φ +ω2S2φ (4) further studies.For theith depositedlayer,there is a (ar-
6 dx2 s i i i o i s i tificial) hard wall x∗ inside the previous layers so that no
i
∗
where ζ(x) is the electrostatic potential, and φ2 and fz monomersfromthe ithlayercanreachthe regionx<xi.
i i Inorder to simplify notation,the layerindex i is dropped
denote the monomer concentrationand monomer valency
∗
from the hard wall notation, x .
of the ith layer, respectively. The two above equations
are solved iteratively for the ith layer concentration φ ,
i The adsorptionofeverylayerbringsaboutareversing
whileassumingthatmonomerconcentrationsfromallpre-
ofthe overallchargeof the surface-PE-smallioncomplex.
viously adsorbed layers, i −1,i−2,...,1 are fixed and
WhentheadsorbedamountofionsandPEchainsexactly
knownfrompreviousiterations.TheseSRinteractionsare
balances the total charge of the surface and previous ad-
containedinthetwosumsappearingintherighthandside
sorbed layers, the electrostatic field perpendicular to the
of Eqs. (3) and (4), S and S . The sum
s o surface is exactly zero (Gauss law). The hard wall of the
∗
adsorbing PE layer x is taken somewhat arbitrarily as
Ss ≡ X φ2j(x) (5) the point where the electric field is zero. As an example,
j=i,i−2... the location of x∗ is depicted in Fig. 2.
isthemonomerconcentrationatthepointx,summedover
∗
Our choice of x is motivated by our understanding
all similarly charged layers: j = i,i−2,..., which repel
thecomplexationprocedure.Thedrivingforcesforthead-
the monomers of the ith layer. Similarly, the other sum:
sorbingPEsto penetrate the preceding layerarethe elec-
So ≡ X φ2j(x) (6) tarnodsttahteicioanttpraacirtiinognbbeettwweeeenncohpaprogseidtemlyocnhoamrgeerds.PTEhicshealeincs-
j=i−1,i−3...
trostaticattractionisdrivenbyanattractiveelectricfield
is summed over all oppositely charged layers having an for x > x∗. For x < x∗, the electrostatic field repels the
attractive SR interaction with the newly adsorbing PEs. adsorbingPE,andno significantcomplexationinthat re-
It should be noted that S and S are both functions of gion is expected. Since we assume that the SR attraction
s o
the layernumber i,but the subscriptiis omitted forsim- is a result of non-equilibrium complexation between elec-
plicity.Inthe following,weconsideronlymonovalentPEs trostaticallyattractedPEchains,wedo nottakex∗ tobe
and set the odd layers as negatively charged, z2i+1 =−1, dependentonχ.Itisimportanttonotethatsuchcomplex-
whereas the even ones as positively charged, z2i = 1. We ation cannot occur between similarly charged PE chains,
alsonotethatforhighmonomerconcentrationstheterms because the repulsive interaction between the two chains
S0 and Ss are large, and thus higher orders of both the as well as the excluded volume repulsion would drive the
excluded volume and the attractive interactions may be chains to separate rather than inter-penetrate.
necessary. However, in this simple model we restrict our-
selves to the third order only. The numerical procedure used to solve Eqs. (1) and
The solutionofthe pairof2ndorderdifferentialequa- (2), as applied to a single adsorbing PE layer,is basedon
tions, Eqs. (3) and (4), requires four boundary condi- the relaxationmethod[54],andwaspresentedindetailin
tions. Two of them are for the bulk where we choose a previous publication [10]. Here, we use the same proce-
φ (x → ∞) = 0, corresponding to a negligible amount of dure for the layer-by-layer build-up. After obtaining the
i
PE in the bulk solution (dilute solution), and zero value solutionforthefirstPElayerconcentration,φ2(x),andits
1
fortheelectrostaticpotential,ζ(x→∞)=0.Atthesolid resultingpotential,ζ(x),thelayermonomerconcentration
surface,x=0,weusetheelectrostaticboundarycondition profileφ2(x)isfrozenandaddedasachargedensitysource
1
dζ/dx| =−4πl σ, where σ is the surface charge den- to the right-hand-side of Eqs. (3,4). These equations are
x=0 B ∗
sity. For the first PE layer that adsorbs directly onto the now solved for the second layer using the hard wall x
solid surface (i = 1), we impose the Cahn-de Gennes at- and the two virial coefficients, v and χ. The procedure is
tractiveboundarycondition[6,53] dln(φ)/dx| =−d−1, then repeated iteratively for all following layers in order
x=0
where d is a characteristic length for the SR interactions to obtain a multilayer stack.
4 AdiShafir, David Andelman:Polyelectrolyte multilayer formation: electrostatics and short-range interactions
x* / a 0.4
0.03
0.02
0.3
0.01
3
a
0.2
2
ζ 0 φ
−0.01
0.1
−0.02
0
0 200 400 600 800
−0.03
0 10 20 30 40 50 60
x / a
x / a
Fig.3.Theformationofamultilayerstackisshownforhighχ
Fig. 2. The electrostatic potential ζ (in dimensionless units) value, χ=0.55a3, and the choice of parameters: csalt =1.0M,
ofthefirstfouralternatingPElayersispresentedasafunction a = 10˚A, σ = 2·10−3˚A−2, d = 50˚A. For clarity purpose,
ofthedistancefromthesurfacex.Ineachextremumpoint,the we show only three groups of layers: layers 1-16 (left), 29-33
electrostatic field dζ/dx changes sign, allowing the multilayer (middle) and 46-50 (right). For both polymers v = 0.05a3,
to attract an oppositely charged polyelectrolyte. The location ω2 = 0.5a6, f = 0.5. The aqueous solution has ε = 80 and
of the farthest peak (marked by a dashed line) is taken as T = 300K. The polycation profiles are marked by a solid
x∗5 = x∗ — the hard wall condition for the interpenetration line, while those of the polyanions by a dashed line. Three
of thenext(fifth)layer.This electrostatic potential was taken regions can be seen in the graph. Near the surface the mul-
from the numerical profiles such as in Fig. 3. All parameter tilayer concentration decays rapidly for layers 1-5 (proximity
values are specified in Fig. 3. We note that the layers shown region),andthenincreasesrapidlyinlayers6-10(intermediate
herearetheinitiallyadsorbedlayers,andhencethestrengthof region).Thethirdregioniswherethemultilayerconcentration
the potential oscillations is strongly affected by the existence stabilizes. This stabilization occurs at a higher value than in
ofacharged wall. Theamplitudeof theelectrostatic potential the initially adsorbed layers. The layers in the distal region
oscillationsinthemoredistalregionincreasestowardsthefinal are highly interpenetrating, so that any layer interacts with
layers, due to the increase in the adsorbed layer amount and aboutfiveotherlayersduringitsadsorptionprocess.Notethat
charge. thelowestmonomervolumefraction intheproximityregionis
0.025, whichismuchlowerthaninthedistalregion.However,
thislayerisstronglycomplexatedwiththepreviouslayersand
3 Results
shouldstillbedenseenoughtosurvivethewashingprocedure.
Our calculations show a strong dependence of the mul-
tilayer formation on the value of the SR attraction coef-
ficient χ in the case of a weakly good solvent, modeled ure, the multilayer can be divided into three spatial re-
via the 2nd virial coefficient v = 0.05a3 where a is the gions. In the proximity region, containing the first few
monomersize.Forlowamountsofaddedsaltcsalt =0.1M, layers, the adsorbed amount decreases substantially. In
the formation of multilayers requires very large χ/a3 ∼ 3 the intermediate region (layers 6-10), the monomer con-
values, while for higher amounts of salt csalt = 1M the centrationincreases rapidly to muchhigher values.In the
required χ values drop to more realistic values of χ/a3 ∼ distalregion,(undersomeconditionsdiscussedbelow)the
0.4 − 1. For all salt concentrations, low χ values cause adsorbedamount stabilizes, and the multilayer formation
the adsorbed amount of monomers in each layer to decay continues.The adsorbedlayersareshownto be verywide
strongly with the layer number, so that very few layers (of the order of tens of nanometers) and highly concen-
are formed. For high χ values, the amount of adsorbed trated. The interpenetration between the layers looks to
monomers decreases for the initial layers and then in- bequite significant.Thelocationx∗,wherethe nextlayer
creasesback.The adsorbedamountin eachlayeris found begins to adsorb, is shared by monomers from all four
to reach a stable value because of the third virial term. previous layers. This strong interpenetration is the driv-
The threshold χ value is shown below to depend on the ing force of the multilayer formation, since it allows for a
amountofsaltinthe solutionaswellastheinitialsurface strong interlayer SR attraction. Without it no significant
charge and the monomer charged fraction. overcharging is achieved. The overall charge of each ad-
The numerical solution of Eqs. (3) and (4) yields the sorbedlayerismuchhigherthantheinitialsurfacecharge,
formationofmultilayers,aspresentedinFig.3,under the showing that there is no exact charge reversal in PE ad-
proper choice of parameters.As can be seen from the fig- sorption.
AdiShafir, David Andelman:Polyelectrolyte multilayer formation: electrostatics and short-range interactions 5
800
0.04
600
3
a a
2φ / 400
0.02 D
200
0
0 20 40 60 80 100 0
0 10 20 30 40 50
x / a
layer number
Fig. 4. Themultilayerprofile with lower χ value,χ=0.39a3,
and all other parameters as in Fig. 3. For this low χ, the Fig.5. ThetotalwidthDoftheadsorbedmultilayerinFig.3
decrease in the non-electrostatic interaction between the PE isplottedasafunctionofthelayernumber.The(incremental)
chains causes the adsorbed layers to decay rapidly after four layerwidthisextractedfromthepeakpositioninthemonomer
layers, and nostable multilayerstack is formed. concentration. The total width D is seen to increase mildly
for the initially adsorbed layers (proximity and intermediate
regions), and then increase almost linearly, corresponding to
The multilayer formation is characteristic of high χ thestable multilayer formation, with a constant thickness per
values. In the opposite limit of low χ values, no stable each adsorbed layer.
multilayer stack is formed because the complexation be-
tween the layers is not strong enough. This case is shown
multilayers are indeed stable and reaches very high layer
inFig.4,whichisobtainedforsimilarparametersasFig.3
numbers.
except for a lowerSR interactioncoefficient χ. The figure
In Fig. 6 we present the threshold strength of χ that
shows that the adsorbed amount in each subsequential
is needed to form multilayers as a function of the added
layer decays rapidly, until an additional layer cannot be
salt amount. As can be seen from this graph, an increase
adsorbed, and the formation of a stable multilayer stack
in the amount of added saltcauses the necessaryχ to de-
is not possible.
creasestronglyforlowsaltconcentrations.Forhighersalt
Within our model the formation of a stable multilay-
concentrationstheχvalueisalmostconstant.Thedepen-
eredstackrequires a thirdorder virialterm. Only when a
strongenoughthird-virialcoefficient,oforderω2 ∼0.5a6, denceofχ oncsalt,χ∼cαsalt fitsroughlyapowerlawwith
α≃ −0.8, as can be seen in the inset of Fig. 6. However,
isaddedtotheSRinteractionterm,themultilayerconcen-
this empirical scaling is valid only for small range of salt
tration stabilizes at high, but still physical, values. When
concentrations.
the third virial coefficient is too low, the adsorbed layer
Thedependenceofthethresholdχvalueonthemono-
concentration does not saturate, and rather reaches un-
merchargedfractionf ispresentedinFig.7.Theχthresh-
realistic high values. Our calculations also show that an
old increases with the increase of f. Here, too, the nu-
increase in the second virial coefficient is not enough to
merical results do not imply any simple scaling relation
stabilize the multilayers. It just drives up the threshold
between f and χ (see inset of Fig. 7).
valueofχ.Thespatialregionwherethe adsorbedamount
stabilizes is the multilayer distal region, and it can be
continued for as many as 80 layers (in our calculations)
without any noticeable decay in the adsorbed amount in 4 Discussion
each layer.
We end this section by showing three further results. The spatial behavior in Figs. 3 and 4 can be explained
In Fig. 5 we show the overall thickness of the adsorbed by the following argument. In the proximity region, close
layers from Fig. 3 as a function of layer number. In the to the surface, the adsorbed layers have high monomer
mean-field model, the thickness of the adsorbed layer is concentrationandsmallwidth.Thissmallwidthdoesnot
taken as the position of the last monomer concentration allow for significant complexation between the adjacent
peak,whichisalowerestimateforthelayerwidth.Thead- layers.Therefore,it causes the SRattractionbetweenthe
sorbedlayerwidthincreasesweaklyforthefirstfewlayers already adsorbed layers and the adsorbing PE chains of
(proximity and intermediate regions), and then increases thecurrentlayertobelow.This,inturn,causesadecrease
almost linearly with the layer number (distal region) for intheconcentrationofadsorbingmonomers,accompanied
the entire 50 layers. The linear increase shows that the by an increase in the layer width.
6 AdiShafir, David Andelman:Polyelectrolyte multilayer formation: electrostatics and short-range interactions
3 The behavior of farther layers depends crucially on
the strength of the short-range attraction, χ. For low χ
values, the monomer concentration in the farther layers
100 continues to decay, and the conditions are insufficient for
multilayer formation. This situation is shown in Fig. 4.
2 In the opposite case of large enough χ values (depicted
in Fig. 3), the increase of the layer width causes the ad-
3
a 10−1 100 sorbingpolymerstointeractwithmorethanoneadsorbed

/
layer.The complexationbetweenadjacentlayersbecomes
χ
stronger, allowing the monomer concentration in the ad-
1 sorbing layer to increase beyond that of previous layers
(see intermediate region of growth, layers 6-10 in Fig. 3).
When the monomer concentration in the adsorbing layer
becomes high enough (layers 10 and above in Fig. 3), the
SRattractionbetweenthe differentPEchainsisbalanced
0 by the third virial term of the excluded volume, and the
0 0.5 1 1.5
adsorbed amount in each additional layer stabilizes. This
c [M] stable multilayeris characteristicofthe distalregion,and
salt persists to dozens and even hundreds of layers without
any noticeable decay. We note that the specific built-up
Fig.6.Thethresholdvalueofχneededtocreatestablemulti-
of the first dozen layers is a direct consequence of our
layersisplottedasafunction ofcsalt.Forlow salt weget high simple model of attractive interactions.A more elaborate
values of χ∼3a3, while for high salt the value of χ has lower
model may be needed for quantitative comparison with
values, around 0.5a3. All other parameters are as in Fig. 3,
experimental findings.
except d = 10˚A. The inset shows the same dependence on a
log-logplot.Forlowcsalt values,theslopeofthelinecanbefit Wenowturntotheχthresholdvalueneededtoobtain
to χ ∼ c−0.8, but for higher salt concentrations the exponent astable multilayerformation.The thresholdcomes about
salt
becomes lower. Sincethesechangesoccur overasingle decade because of the competition between the SR and electro-
in csalt values, there does not appear to be a good scaling law static interactions. The decrease of the χ threshold with
for χ as function of csalt. salt, as seen in Fig. 6, can be explained in the following
way.Added salt screens the electrostatic interactions and
1.2 resultsinanincreaseinthePEadsorbedamountandlayer
width [11]. The thicker layers have a larger contribution
to the attractive SR term in Eq. 4, and lead to a lower χ
threshold.
The dependence of the threshold χ on f (Fig. 7) is
0.8
explainedqualitativelyasfollows.Anincreaseinf causes
3 an increase in the monomer-monomer repulsion between
a
the adsorbing PE chains, and an increase in the electro-
/

χ static attraction of the adsorbing PE chains to the al-
0.4 10−1 100 ready adsorbed oppositely charged layer. The increase in
100 the threshold χ with f shows that the main effect of in-
creasing f is to decrease the adsorption, meaning that
the main driving force of multilayer formation is not the
charge reversal caused by the adsorbing polymer layers,
10−1 butrathertheSRinteraction.Itisimportanttonotethat
0
0 0.2 0.4 0.6 0.8 1 in experiment [40] a threshold f value for the multilayer
f formationwasfound.Abovethisthreshold,themultilayer
concentrationdecreaseswithf,whichisinagreementwith
Fig. 7. Thethreshold valueof χ needed to create stable mul-
ourfindings.Athresholdinthef valueisnotfoundinour
tilayers is plotted as a function of f. For low f values, the
calculations, mainly because our SR interactions are ex-
monomer-monomer repulsion is low, and only a weak short-
ternally imposed and do not depend on the value of f.
rangeattractionbetweenthepolymersisneeded.Forhigherf
However, the threshold can be understood qualitatively
valuesthethresholdvalueofχincreases.Allotherparameters
as the value of f for which the polymer chains begin to
areasinFig.3,exceptd=10˚A.Theinsetshowsthesamede-
interpenetrate, giving rise to the SR attraction between
pendence on a log-log plot. No clear scaling law can be found
them. We believe that further studies are needed to bet-
for the dependenceof thethreshold of χ on f.
ter understand the origin of this threshold value.
Our simple model is subject to several limitations.
First, we use the mean-field theory and the ground-state
dominance approximation, valid for long PE chains. The
AdiShafir, David Andelman:Polyelectrolyte multilayer formation: electrostatics and short-range interactions 7
adsorption of short polyelectrolyte chains requires other discussions. Supportfrom theIsraelScienceFoundation(ISF)
treatmentssuchasmoleculardynamics,MonteCarlosim- under grant no. 160/05 and the US-Israel Binational Founda-
ulations or lattice models. Second, during the adsorption tion(BSF)undergrantno.287/02 isgratefully acknowledged.
of each layer, we assume that all preceding layers are
frozen,meaningthattheydonotdissolvebackintotheso-
lutionorchangetheir spatialconformation.This assump- References
tion stems from the fact that within mean-field theory
there is no way to distinguish between an adsorbed chain 1. F.W. Wiegel, J. PhysA: Math. Gen. 10, 299 (1977).
and a chain that is merely ‘stuck’ at the surface vicin- 2. M. Muthukumar,J. Chem. Phys. 86, 7230 (1987).
ity.Thisdeficiency ofmean-fielddoes notallowusto give 3. X.ChatellierandJ.F.Joanny,J.Phys.II(France)6,1669
anaccuratemodelforthewashingprocedure.Experimen- (1996).
tally,weexpectthepolymerconcentrationtodecrease,es- 4. R. Varoqui, A. Johner, A. Elaissari, J. Chem. Phys. 94,
peciallyduringthewashingstep,aswasmodeledbyother 6873 (1991).
techniques [52]. However,due to long relaxationtimes we 5. R. Varoqui,J. Phys.II (France) 3, 1097 (1993).
do not think that the washing will affect drastically the 6. J.F. Joanny, Eur.Phys. J. B. 9, 117 (1999).
structureofthealreadyadsorbedmultilayerstack.Finally, 7. I.Borukhov,D.Andelman,H.Orland,Macromolecules31,
we use a very simple model for the SR attraction, and do 1665 (1998); Europhys.Lett. 32, 499 (1995).
not offer any explanation for its dependence on the PE 8. I.Borukhov,D.Andelman,H.Orland,Eur. Phys.J. B 5,
parameters.Despite theselimitations,webelievethatour 869 (1998).
model gives an insight for the multilayer formation prob- 9. I. Borukhov, D. Andelman, H. Orland, J. Phys. Chem. B
103, 5042 (1999).
lem.
10. A. Shafir, D. Andelman, R.R. Netz, J. Chem. Phys. 119,
Quite recently, we have become aware of a alterna-
2355 (2003).
tive mechanism for multilayer formation suggested by Q.
11. A.Shafir,D.Andelman,Phys.Rev.E.70,061804 (2004).
Wang[55].Thisinvolvesmultilayerformationforpoorsol-
12. M. Manghi and M. Aubouy,cond-mat/0202045 preprint.
ventcondition(balancedbyelectrostatics).Themultilayer
13. O.V. Borisov, E.B. Zhulina, T.M. Birshtein, J. Phys. II
formationcanbeachievedevenifthepolyanionandpoly-
(France) 4, 913 (1994).
cation chains repel each other at short distances, because
14. R.R. Netz and J.F. Joanny, Macromolecules 32, 9013
thepoorsolventconditioninducesstablemultilayerbuilt-
(1999).
up.ThesolventconditioninWang’smodelplaysasimilar
15. A.V. Dobrynin, A. Deshkovski, M. Rubinstein, Macro-
roleastheattractiveinteractionbetweenthecationicand molecules 34, 3421 (2001).
anionic chains in our model. 16. R.R. Netz and D. Andelman, in Encyclopedia of Electro-
chemistry, Eds. M. Urbakh and E. Giladi, Vol. I, (Wiley-
VCH, Weinheim,2002).
5 Conclusions 17. R.R.Netz and D. Andelman,Phys. Rep. 380, 1(2003).
18. H.A. van der Schee and J. Lyklema, J. Phys. Chem. 88,
6661 (1984).
We present a model aiming to explain PE multilayer for-
19. O.A.Evers,G.J. Fleer, J.M.H.M. Scheutjens,J.Lyklema,
mation for marginally good solvents. The model is based
J. Coll. Interface Sci. 111, 446 (1986).
on strong enough short-range interactions between the
20. H.G.M. van de Steeg, M.A. Cohen Stuart, A. de Keizer,
polyanionandpolycation.Thisstrongshort-rangeinterac-
B.H. Bijsterbosch, Langmuir 8, 2538 (1992).
tionisshowntobeindispensableforourmodelingofsuch
21. M.R. B¨ohmer, O.A. Evers, J.M.H.M. Scheutjens, Macro-
stablemultilayers.Weshowthatthemultilayersformeas-
molecules 23, 2288 (1990).
ilyinhighionicstrengthconditions,andthattheirforma-
22. V.Yamakov,A.Milchev,O.Borisov,B.Du¨nweg,J.Phys:
tion does not rely exclusively on the electrostatic attrac- Condens. Matter 11, 9907 (1999).
tion to the previously adsorbed layers. We also calculate 23. R. Messina, E. Gonz´alez-Tovar, M. Lozada-Cassou, C.
what is the threshold strength of the short-range interac- Holm Europhys.Lett. 60, 383 (2002).
tionsneededfortheformationofmultilayersasafunction 24. M. Ellis, C.Y. Kong, M. Muthukumar, J. Chem. Phys.
of the salinity as well as monomer charge fraction. 112, 8723 (2000).
In our model the multilayers are quite thick and in- 25. J. McNamara, C.Y. Kong, M. Muthukumar, J. Chem.
terpenetrating, while in the experiments the layers are Phys. 117, 5354 (2002).
thinner. However,we believe that this simple model gives 26. G. Decher, J.B. Schlenoff, Multilayer Thin Films (Wiley-
good insight on the problem of multilayer formation, and VCH, Weinheim 2002).
can serve as a starting point for more refined models. A 27. Yu. Lvov, H. Haas, G. Decher, H. M¨owald, A. Kalachev,
possibleextensionwillbe to usea morespecific modelfor J. Phys.Chem 97, 12835 (1993).
theshort-rangeinteractionsbetweenthePEchains,which 28. Yu.Lvov,G. Decher,H.Haas, H. M¨ohwald, A.Kalachev,
Physica B 198, 89 (1994).
may give a better explanation to the experimentally ob-
29. B.D. Jung, J.D. Hong, A. Voigt, S. Leporatti, L. D¨ahne,
served multilayer formation.
E. Donath, H. M¨ohwald, Coll. Surf. A: Phys. and Eng.
198-200, 483 (2002).
The authors would like to thank Alexander Grosberg, Jacob 30. I.L.Radtchenko,G.B.Sukhorukov,H.M¨owald,Coll.Surf.
Klein, Henri Orland and Qiang (David) Wang for benefiting A: Phys.and Eng. 202, 127 (2002).
8 AdiShafir, David Andelman:Polyelectrolyte multilayer formation: electrostatics and short-range interactions
31. R. Heuberger, G. Sukhorukov, J. V¨or¨os, M. Textor, H.
M¨owald, Adv.Funct.Mater. 15, 357 (2005).
32. G. Decher,Science 277, 1232 (1997).
33. E. Huebsch, V. Ball, B. Senger, G. Decher, J.-C. Voegel,
P.Schaaf, Langmuir 20, 1980 (2004).
34. J. Schmitt, T. Gru¨nwald, G. Decher, P. S. Pershan, K.
Kjaer, M. L¨osche, Macromolecules 26, 7058 (1993).
35. M. L¨osche, J. Schmitt, G. Decher, W. G. Bouwman, K.
Kjaer, Macromolecules 31, 8893 (1998).
36. P.Scheutz,F.Caruso,Adv.Funct.Mater.13,929(2003).
37. P. Schuetz, F. Caruso, Coll. Surf. A: Phys. and Eng 207,
33 (2002).
38. E. Poptoshev, B. Schoeler, F. Caruso, Langmuir 20, 829
(2004).
39. U. Voigt, V. Khrenov, K. Tauer, M. Halm, W. Jaeger, R.
von Klitzing, J. Phys.: Condens. Matter 15, S213 (2003).
40. U. Voigt, W. Jaeger, G. Findenegg, R. von Klitzing, J.
Phys.Chem B 107, 5273 (2003).
41. R.Steitz,V.Leiner,R.Siebrecht,R.v.Kliting,Coll.Surf.
A.:Phys. and Eng. 163, 63 (2000).
42. R. v. Klitzing, J. E. Wong, W. Jaeger, R. Steitz, Curr.
Opin.Coll. Int.Sci. 9, 158 (2004).
43. Z. Adamczyk, M. Zembala, P. Warszynski, B. Jachimska,
Langmuir 20, 10517 (2004).
44. E. Blomberg, E. Poptoshev, P.M. Claesson, F. Caruso,
Langmuir 20, 5432 (2004).
45. J. Cho, J.F. Quinn, F. Caruso, J. Am. Chem. Soc. 126,
2270 (2004).
46. G. Greene, R. Tannenbaum, App. Surf. Sci. 233, 336
(2004).
47. G. Greene, G. Yau, R. Tannenbaum, Langmuir 20, 2739
(2004).
48. M. Castelnovo, J.F. Joanny,Langmuir 16, 7524 (2000).
49. S.Y.Park,M.F. Rubner,A.M.Mayes,Langmuir18,9600
(2002)
50. R.Messina, Macromolecules 37, 621 (2004).
51. Q.Wang, Macromolecules 38, 8911 (2005).
52. P.A. Patel, J. Jeon, P.T. Marher, A.V. Dobrynin, Lang-
muir 21, 6113 (2005).
53. P.G. deGennes, Macromolecules 14, 1637 (1981).
54. W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetter-
ling, Numerical Recipes in C: The Art of Scientific Com-
puting, (Cambridge University, Cambridge 1992), Chap.
17, p. 762.
55. Q.Wang, cond-mat 0507118 (submitted) .