Theory and simulations of charged polymers: From solution ......chain adopts rings on a string conformation of collapsed toroidal globules connected by the stretched strings of monomers

Current Opinion in Colloid & Interface Science 13 (2008) 376–388

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Current Opinion in Colloid & Interface Science

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Theory and simulations of charged polymers: From solution properties topolymeric nanomaterials

Andrey V. DobryninPolymer Program, Institute of Materials Science and Department of Physics, University of Connecticut, Storrs, Connecticut, 06269-3136, USA

E-mail address: [email protected].

1359-0294/$ – see front matter © 2008 Elsevier Ltd. Aldoi:10.1016/j.cocis.2008.03.006

A B S T R A C T
A R T I C L E I N F O
Article history:
Charged polymers are macr Received 18 March 2008Accepted 24 March 2008Available online 8 April 2008
omolecules with ionizable groups. These polymeric systems demonstrate uniqueproperties that are qualitatively different from their neutral counterparts. In this review I survey the recentprogress made in understanding properties of the solutions of charged polymers, swelling of polyelectrolytegels, conformational transformations of charged dendrimers, complexation between charged macromole-cules, adsorption of charged polymers at surfaces and interfaces, and multilayer assembly in ionic systems.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Charged polymers are macromolecules with ionizable groups[1••,2•,3••–6••]. In polar solvents such as water, these groups candissociate, leaving charges onpolymer chains and releasing counterionsinto the solution. If these polymers carry only acidic or basic groups theyare called polyelectrolytes. Examples of polyelectrolytes include poly-styrene sulfonate, polyacrylic and polymethacrylic acids and their salts,DNA and other polyacids and polybases. Polyampholytes are chargedmacromolecules carrying both acidic and basic groups [4••]. Thus, afterionization, there are positively and negatively charged groups on thepolymer chain. Examples of polyampholytes include denatured proteins(for example gelatin), proteins in their native state (such as bovineserum albumin), and synthetic copolymers made of monomers withacidic and basic groups. At high charge asymmetry these polymersdemonstrate polyelectrolyte-like behavior [4••].

Electrostatic interactions between charged macromolecules controlmolecular processes in different areas of natural sciences ranging frommaterials science to biophysics [1••,2•,3••–12••,13,14,15••,16•]. For example,theelectrostatic attractionbetweenoppositelychargedmacromolecules isa foundation of the electrostatic assembly technique that allowsfabrication of multilayer films from synthetic polyelectrolytes, proteins,DNA, nanoparticles, etc. [9••–11••]. Electrostatic attractions betweennegatively charged DNA and net positively charged histones areresponsible for the packaging of DNA into chromosomes [17••,18]. Thecomplexation of DNA with positively charged polyelectrolytes, dendri-mers, colloidal particles and liposomes facilitates the uptake of the DNAthrough the cell membrane and is utilized for gene therapy [19,20].Electrostatic interactions between multivalent ions and DNA molecules,actin filaments, and tobacco mosaic viruses are the driving forces behindtheir assembly into compact bundle structures [16•,21–26]. The electro-

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statically driven complexation between oppositely charged macromole-cules in solutions is utilized for protein separation [7••,27,28•]. In this case,flexible synthetic polyelectrolytes are added to aqueous protein solutions.Polyelectrolytes form complexes with proteins, which then precipitatefrom the solution. Furthermore charged polymers are essential fordevelopment of lithium batteries and proton exchange membranes forfuel cell technology [29,30].

Over the years molecular simulations and theoretical models ofcharged polymeric systems have proven helpful in elucidating factorscontrolling their properties. In this review I will discuss static anddynamic properties of polyelectrolyte and polyampholyte solutions,swelling of polyelectrolyte gels, conformational transformations ofcharged dendrimers, complexation between charged macromole-cules, adsorption of charged polymers at surfaces and interfaces, andmultilayer assembly in polyelectrolyte systems.

2. Polyelectrolyte solutions

2.1. Salt-free polyelectrolyte solutions

2.1.1. θ and good solvent conditions for the polymer backboneIn dilute salt-free solutions, below the chain overlap concentration,

the intrachain electrostatic interactions dominate over the interchainones. The strength of the electrostatic interactions is controlled by thevalue of the Bjerrum length lB=e2/εkBT defined as the length scale atwhich the Coulomb interaction between two elementary charges e in adielectricmediumwith the dielectric constant ε is equal to the thermalenergy kBT. The deformation of the polyelectrolyte chain in θ or goodsolvent conditions for the polymer backbone is obtained by balancingthe intrachain electrostatic interactions and chain's elasticity. Thisleads to elongation and nonuniformdeformation of the polyelectrolytechainwith backbone tension decreasing towards the chain's ends, andfaster than linear increase of the chain size Rewith the chain degree ofpolymerization N, Re∝N[1n N]1/3 [1••]. The nonuniform stretching of

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Fig. 1. Dependence of the osmotic coefficient of flexible polyelectrolytes on polymerconcentrations for fully f=1 charged chains in a θ-solvent. The arrows show the overlapconcentrations at various chain degrees of polymerizations N. Reproduced withpermission from Liao, Q., Dobrynin, A. V., Rubinstein, M. Macromolecules 36, 3399–3410 (2003). Copyright 2003, American Chemical Society.

377A.V. Dobrynin / Current Opinion in Colloid & Interface Science 13 (2008) 376–388

polyelectrolyte chain was directly tested in the molecular dynamicssimulations by Liao et al. [31]. The chain size monotonically decreaseswith increasing polymer concentration in dilute solution regime. Thisdecrease of the chain size is due to reduction of the net polymericcharge by condensed counterions (see discussion below), whichweakens intrachain electrostatic repulsion.

The crossover from dilute to semidilute solution regime occurs atpolymer concentrations at which the distance between chainsbecomes comparable with the chain size. The simulations data forN-dependence of the overlap concentration c⁎ approach the scalinglaw, c⁎∝N/Re3∝1/ (N2ln N) for large N [1••]. It is important to pointout that the dependence of the chain overlap concentration on thechain degree of polymerization is much stronger than one in solutionsof neutral polymers, c⁎∝1/N1/2 [32]. Thus, the crossover from diluteto semidilute polyelectrolyte solution regime occurs at much lowerpolymer concentrations than that in solution of neutral chains.

In the semidilute solution regime, the electrostatic interactions arescreened at the length scales on the order of the solution correlationslength ξ — the average mesh size of the semidilute polyelectrolytesolution [1••]. The electrostatic interactionsbetween chargedmonomerson the length scales smaller than the correlation length result instretching of the chain sectionswithin correlation blobs. At these lengthscales a section of polyelectrolyte chain with gξ monomer is stronglystretched such that ξ∝gξ. The interactions between correlation volumescan be ignored in the zeroth order approximation because the netpolymeric charge within correlation volume is compensated bycounterions. The concentration dependence of the number of mono-mers in a correlation volume ξ3 can be obtained by imposing the close-packing condition for chain sections of size ξ, c≈gξ/ξ3. The correlationlength of semidilute polyelectrolyte solution is estimated as ξ∝c−1/2.This is in agreement with the results of the molecular dynamicssimulations that show the exponent for the concentration dependenceof the correlation length being close to −1/2 [1••,31].

Since at length scales larger than the correlation length ξ otherchains and counterions screen electrostatic interactions, the statisticsof the chain are those of a Gaussian chain with the effectivepersistence length on the order of the correlation length ξ. Thus, achain in the semidilute salt-free polyelectrolyte solution is a randomwalk of correlation blobs with size Re∝N1/2c−1/4. Simulations by Liaoet al. [31] show that the exponent for the concentration dependence ofthe chain size Re is N-dependent. It changes from the value −0.094 forthe short chains with N=25 to the value −0.22 for the chains withN=300. This result for the longest chains is close to the scalingmodel predictions, −0.25.

The osmotic pressure of the salt-free polyelectrolyte solutions iscontrolled by counterions [1••,33•,34–38]. The osmotic coefficientϕ=π/(kBTc) of flexible polyelectrolytes, defined as a ratio of thesolution osmotic pressure π to the ideal osmotic pressure of allcounterions at concentration c, kBTc, exhibits nonmonotonic depen-dence on polymer concentration (see Fig. 1) [33•]. It decreases withpolymer concentration in dilute solutions. The osmotic coefficientincreases with polymer concentration at high concentrations. Theupturn in the osmotic coefficient occurs around the chain's overlapconcentration c⁎. The nonmonotonic behavior of the osmoticcoefficient on polymer concentration in dilute solution regime is inqualitative agreement with the two-zone model. In the framework ofthis model the volume occupied by a chain is separated into twozones: the cylindrical zone, surrounding stretched polyelectrolytechain, and a spherical zone, located outside the cylindrical region.According to this model the osmotic coefficient is a decreasingfunction of polymer concentration in dilute solutions whereas it isincreasing function of polymer concentration in semidilute solutions.The crossover between these two regimes occurs around the overlapconcentration c⁎, at which the spherical zone of the two-zone modeldisappears and the two-zone model reduces to the classicalKatchalsky's cell model (see for review [1••]).

2.1.2. Poor solvent conditions for the polymer backboneIn poor solvents for the polymer backbone there is an effective

attraction between monomers which causes neutral polymer chainwithout charged groups to collapse into dense spherical globules inorder to maximize the number of favorable polymer–polymer contactsandminimize thenumberof unfavorablepolymer–solvent contacts [32].Molecular dynamics simulations of partially charged polyelectrolyteswith explicit counterions in poor solvent conditions were performed bythe Mainz group [39–42] and by Liao et al. [43]. These simulations haveconfirmed that polyelectrolyte chains at low polymer concentrationsform necklaces of beads connected by strings (see Fig. 2). The necklacestructure optimizes electrostatic repulsion between chargedmonomersand short-range monomer–monomer attraction [1••]. The effectivecharge of the chain decreases with increasing polymer concentrationcausing chain size to decrease by decreasing the length of strings and thenumber of beads per chain. At higher polymer concentrations polymerchains interpenetrate leading to a concentrated polyelectrolyte solution.In this range of polymer concentrations the chain size is observed toincrease toward its Gaussian value. The nonmonotonic dependence ofthe chain size onpolymer concentration is in qualitative agreementwiththeoretical predictions [1••]. A comprehensive study of the effect of theshort-range attractive and of the long-range electrostatic interactions onthe necklace formation in dilute polyelectrolyte solutions was carriedout by Limbach and Holm [41] and by Jeon and Dobrynin [44](see Table 1). These studies have shown that polyelectrolyte chainsadopt necklace-like conformation only in the narrow range of theinteraction parameters. At finite polymer concentrations the necklacestability region is strongly influenced by the counterion condensation(see discussion below). A similar trend was observed in Monte Carlosimulations of titration of hydrophobic polyelectrolytes by Ulrich et al.[45]. Depending on the solvent quality for polymer backbone and pH–pK0 value a polyelectrolyte chain could be in five different conforma-tional states: coil, collapsed spherical globule, necklace globule, sausage-like aggregate, and fully stretched chain. The effect of the chargeannealing on the properties of the polyelectrolyte chain in poor solventcondition was also studied in ref [46].

Explicit solvent simulations of short polyelectrolyte chain in poorsolvents showed that the necklace structure is not very stable because ofthe additional entropic solvent effect weakening effective monomer–monomer attraction [47]. The results of simulations with explicit solventare in good quantitative agreement with the predictions of the solvent-

Fig. 2.Necklace conformation of a polyelectrolyte chain in a poor solvent. The negativelycharged monomers of polyelectrolyte chain are colored in blue and neutral monomersare colored in gray. Positively charged counterions are shown in pink. (Forinterpretation of the references to colour in this figure legend, the reader is referredto the web version of this article.)

Table 1Typical conformations of polyelectrolyte chain with degree of polymerization N=304and fraction of charged monomers f=1/3

The negatively charged monomers of polyelectrolyte chain are colored in blue andneutral monomers are colored in gray. Positively charged counterions are shown inpink. Reproduced with permission from Jeon, J., Dobrynin, A. V. Macromolecules 40,

Table 1Typical conformations of polyelectrolyte chain with degree of polymerization N=304and fraction of charged monomers f=1/3

The negatively chargedmonomers of polyelectrolyte chain are colored in blue and neutralmonomers arecolored in gray. Positively charged counterions are shown in pink. Reproduced with permission fromJeon, J., Dobrynin, A. V. Macromolecules 40, 7695–7706 (2007). Copyright 2007, AmericanChemical Society. (For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

378 A.V. Dobrynin / Current Opinion in Colloid & Interface Science 13 (2008) 376–388

accessible surface area (SASA) model for a polyelectrolyte chain in poorsolvent [48].

The phase separation in poor solvent conditions for the polymerbackbone was observed in MD simulations of Chang and Yethiraj [49].They have found that polyelectrolyte solutions phase separate withincreasing polymer concentration. Polyelectrolytes in the dense phaseform spherical, cylindrical, and lamellar structures depending onpolymer concentration.

Note that the chain rigidity plays an important role in determiningchain conformations in a poor solvent. A semiflexible polyelectrolytechain adopts rings on a string conformation of collapsed toroidalglobules connected by the stretched strings of monomers [50–53].This structure optimizes the short-range monomer–monomer attrac-tive interactions, chain's bending energy and long-range electrostaticrepulsion between charged groups.

2.1.3. Dynamics of polyelectrolyte solutionsThe well-known feature of polyelectrolyte solutions is the

concentration dependence of the solution viscosity called Fuoss law(see for review [1••]). The viscosity of polyelectrolyte solutions isproportional to the square root of polymer concentration in the wideconcentration range, while the viscosity of uncharged polymers in agood solvent scales linearly with polymer concentration in a dilutesolution regime or as c1.3 in semidilute solution regime [32].Furthermore, in this concentration range the chain relaxation timedecreases with increasing polymer concentration indicating that thestress relaxation in polyelectrolyte solutions speeds up as theybecome denser. The physical origin of this unique behavior is thecoupling of electrostatic interactions with conformational transforma-tions of charged macromolecules. The unusually wide Fuoss lawregime supports the conjecture that the entanglements (topologicalconstraints created by surrounding chains) [32] in polyelectrolytesolutions begin to restrict chain motion deep in the semidilutesolution regime. The crossover to entangled polyelectrolyte solutionsoccurs 3–4 decades above the overlap concentration c⁎ [1••]. Note thatin solutions of neutral polymers this crossover takes place closer to thechain overlap concentration at approximately 10c⁎.

Molecular dynamics simulations of dilute and semidilute poly-electrolyte solutions without hydrodynamic interactions were per-formed by Liao et al. [54•] to study Rouse dynamics of polyelectrolytes.These simulations of the Rouse dynamics give qualitatively similarresults to the experimentally observed dynamics of polyelectrolytesolutions. It was observed that the chain relaxation time dependsnonmonotonically on polymer concentration (see Fig. 3). In dilutesolutions this relaxation time exhibits very strong dependence on thechain degree of polymerization, τ ~ N3. The chain relaxation timedecreases with increasing polymer concentration of dilute solutions.This decrease in the chain relaxation time is due to chain contractioninduced by counterion condensation. In the semidilute solution

regime the chain relaxation time decreases with polymer concentra-tion as c−1/2. In this concentration range the chain relaxation timefollows the usual Rouse scaling dependence on the chain degree ofpolymerization, τ ~ N2. At high polymer concentrations the chainrelaxation time begins to increase with increasing polymer concen-tration. The crossover polymer concentration to the new scalingregime does not depend on the chain degree of polymerizationindicating that the increase in the chain relaxation time is due to theincrease of the effective monomeric friction coefficient. The analysis ofthe spectrum and of the relaxation times of Rouse modes confirms theexistence of the single correlation length ξ, which describes bothstatic and dynamic properties of semidilute solutions. These simula-tions also show that the unentangled semidilute solution regime isvery wide. The longest chains with N=373 and 247 start to overlap atabout 10−4 σ−3 and don't show any effect of entanglements up to thehighest polymer concentration 0.15 σ−3. Thus, the unentangledsemidilute solution regime spans three decades above the overlapconcentration. This result is in agreement with the prediction of thescaling model of polyelectrolyte solutions [1••].

The combined effect of the hydrodynamic and electrostatic interac-tions on the nonlinear shear viscosity of the dilute salt-free polyelec-trolyte solutions was studied by Stoltz et al. [55•]. Using Browniandynamics simulationsof the coarse-grainedmodel of thepolyelectrolytechains with explicit counterions they have explored relationshipsbetween flow rate, value of the Bjerrum length and polymer concentra-tion. Hydrodynamic interactions between beads were introduced intosimulation scheme in the framework of the Rotne–Prager–Yamakawatensor. It was found that the polyelectrolyte chains show shear thinningbehavior at high flow rates which is independent on the value of theBjerrum length. However, in the case of the low flow rates the solutionviscosity increases monotonically with the value of the Bjerrum length.The origin of such unusual behavior is the electroviscous effectassociated with the deformation of the counterion clouds surroundingpolyelectrolyte chains by the external flow.


2.2. Counterion condensation and chain collapse

The idea of the counterion condensation was first introduced byManning and Oosawa [1••,56]. They have established that a charge of arigid (rod-like) polyion depends on the fine interplay between theelectrostatic attraction of a counterion to the polyelectrolyte backboneand configurational entropy loss due to counterion localization in thevicinity of the polymer chain. This theory showed that the linearcharge density of a rod-like polyion cannot exceed a critical valuewhich depends on the solution dielectric constant, temperature andcounterion valence determining the value of the Bjerrum length lB.The recent studies of the counterion condensation show that thecondensation is the second order phase transition associated with theformation of the counterion condensate in the vicinity of the polyion[57•,58–60].

The situation is even more complicated in the case of flexiblepolyelectrolytes. In this case chain conformations are directly coupledwith the intrachain electrostatic interactions that are controlled by theamount of the condensed counterions [1••,3••,43,61–66]. Thus, inaddition to counterion configurational entropy and electrostaticinteractions the chain's conformational free energy comes into play.These contributing factors have to be optimized simultaneously todetermine fraction of the condensed counterions and the equilibriumchain size. In the case of the good or θ solvents for the polymerbackbone the counterion condensation results in gradual decrease ofthe chain size with the increase of the polymer or salt concentrations[1••]. A qualitatively different picture of counterion condensation isobserved for polyelectrolytes in poor solvent conditions for thepolymer backbone. In a poor solvent a polyelectrolyte chain forms anecklace globule of dense polymeric beads connected by strings ofmonomers. The counterion condensation on the necklace globule canoccur in avalanche-like fashion. By increasing polymer concentrationor decreasing temperature one can induce a spontaneous condensa-tion of counterions inside beads of the necklace globule. This reducesthe bead's charge and results in increase of the bead mass (size),which initiate further increase of the number of condensed counter-ions inside beads starting the avalanche-like counterion condensationprocess (see for review [1••]).

Note that in addition to the reduction of the net polymeric chargeweakening intrachain electrostatic repulsion the condensed counter-

Fig. 3. Concentration dependence of the chain relaxation time for the system of fullycharged chains. N=25 (dark blue black squares); N=40 (gray circles); N=64 (pinktriangles); N=94 (red triangles); N=124 (black rhombs); N=187 (green triangles);N=247 (blue triangles). Reproduced with permission from Liao, Q., Carrillo, J-M.,Dobrynin, A. V., Rubinstein, M. Macromolecules 40, 7671–7679 (2007). Copyright 2007,American Chemical Society. (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)

ions can also induce effective attractive interactions between chargedmonomers [5••,15••,43,61–63]. In the ion binding and counterionadsorption models [62–64] condensed counterions form ionic pairswith oppositely charged ions on the polymer backbone. The formationof the ionic pairs leads to an additional dipole–dipole and charge–dipole attractive interactions [61–63]. These attractive interactionsdecrease the value of the effective second virial coefficient formonomer–monomer interactions shifting the position of the θ-point. In the case of the strongly charged polyelectrolytes the shiftof the θ-temperature could be significant and change the solventquality for the polymer backbone to poor solvent conditions as thenumber of condensed counterions increases. This can result in a chaincollapse and completely alter scenario of the counterion condensation(see discussion above). The analysis of the effect of the counterioncondensation on conformations of a polyelectrolyte chainwas done bySchiessel and Pincus [61], and by Schiessel [62] in the framework ofthe scaling approach, and by Muthukumar [63] in the framework ofthe variational approach. These theories predict nonmonotonicdependence of the chain size on the solution dielectric constant εand solution temperature T (solution Bjerrum length lB ~ 1/(εT)). Thechain size is first increases with increasing the value of the Bjerrumlength then begin to decrease as the Bjerrum length exceeds thecrossover value. This nonmonotonic dependence of the chain size isthe manifestation of the two-fold role of the electrostatic interactions.At low values of the Bjerrum length the intrachain electrostaticrepulsion controls the chain size. These interactions become strongerwith increasing the value of the Bjerrum and force polyelectrolytechain to expand. At large values of the Bjerrum length the condensedcounterions reduce net polymeric charge weakening the intrachainelectrostatic repulsion, which together with the dipole–dipole andcharge–dipole attractive interactions induce chain contraction.

However, computer simulations of polyelectrolyte solutions showthat the condensed counterions are not permanently attached tooppositely charged groups on the polymer backbone as assumed bythe ion binding and ion adsorption models but rather localized nearthe polymer backbone and are free to move inside the chain volume[1••,40,41,43,44,67].

The localization of counterions inside the chain volume can alsolead to effective attractive interactions [5••,15••]. These interactionsare due to heterogeneous distribution of the charge density along thepolymer backbone. In the case of weak electrostatic attraction theorigin of these interactions is similar to the fluctuation-inducedattraction in two-component plasma and is related to the local chargedensity fluctuations [15••]. In the opposite limit of strong electrostaticinteractions the effect is due to correlation-induced attractionbetween the counterions and the oppositely charged polymer back-bone similar to the interactions in strongly correlated Wigner liquids(see for review [5••,15••]) or in ionic crystals such as NaCl. For example,in the case of the ionic crystal the attractive (negative) lattice energy isdue to the spatial distribution (spatial correlations) of cations andanions over the lattice sites, even though the net charge of the crystalis zero. The crystal will remain stable even if it carries a small nonzerocharge because of the large lattice (correlation) energy.

The effect of the fluctuation/correlation-induced attractive inter-actions on the conformations of a polyelectrolyte chain was studiedtheoretically [43,44] and by molecular dynamics simulations[41,43,44,67] (see Table 1). These studies show that the fluctuation/correlation-induced-attractive interactions can cause additional chaincollapse. In particular, in the case of the poor solvent conditions for thepolymer backbone these studies established existence of the twodifferent mechanisms that could lead to formation of the necklaceglobule [41,43,44,67]. For the values of the Bjerrum length lB=1σ and2σ the necklace structure appears as a result of competition betweenshort-range monomer–monomer attractive interactions and electro-static repulsion between uncompensated charges [44,67]. However,for the value of the Bjerrum length lBN3σ the necklace structure is


controlled by counterion condensation and is due to the optimizationof the correlation-induced attraction between chargedmonomers andcondensed counterions, and electrostatic repulsion between uncom-pensated charges on the polymer backbone. Note, that counterioncondensation on the polymer backbone can also lead to chain collapseand necklace formation even in the good or theta solvent conditionsfor the polymer backbone.

The counterion condensation also influences the dynamic proper-ties of polyelectrolytes [68]. In dilute solutions the chain's transla-tional diffusion coefficient monotonically decreases with the value ofthe Bjerrum length. However, the chain's relaxation time showsnonmonotonic dependence on the Bjerrum length. It first increaseswith increasing the value of the Bjerrum length then it begins todecrease. The decrease in the chain's relaxation time is due tocounterion condensation.

2.3. Salt effects

The electrostatic interactions between chargedmonomers in solutionswith finite salt concentrations are screened by salt ions and their strengthdecreases exponentially with the distance between charges [1••,3••,37].However, the charges still interact through the unscreened Coulombpotential at distances much smaller than the Debye screening length rD=(8πlBcs)−1/2, where cs is the salt concentration. Screening of the intrachainelectrostatic interactions increases chain flexibility by reducing itspersistence length and leads to chain contraction. For semiflexible andstrongly charged flexible polyelectrolytes the electrostatic part of thechain persistence length is proportional to the Debye screening length.[69,70] This result was obtained by evaluating the bending anglefluctuations and in the framework of the Gaussian variational principle.A polyelectrolyte chain with linear dependence of the electrostaticpersistence length on the Debye screening length has lower free energythan that of a chain with quadratic dependence of the electrostaticpersistence so-called Odijk–Skolnick–Fixman electrostatic persistencelength (see for overview [1••,71]).

If multivalent ions are added to the solution, the strong electro-static attraction between multivalent ions and charged monomersfavors condensation of the multivalent ions on the polymer backbone.Note that eachmultivalent ionwith valence Z can neutralize Z chargedmonomers, whichmakes its condensation Z-times more efficient thancondensation of monovalent ions. Using lattice Monte Carlo simula-tions Klos and Pakula [72–74] have studied the effect of the saltconcentration on conformations of single polyelectrolyte chain. Theirsimulations have shown that the presence of multivalent salt ionsresults in chain collapse accompanied by reduction in the systemenergy and effective charge of the polyelectrolyte chain. This indicatesthat the chain collapse is driven by multivalent ion condensation onthe polymer backbone. The sharp reduction in the chain size wasobserved even at relatively low salt concentrations.

In the mixtures of monovalent and tetravalent salts the polyelec-trolyte chain size depends nonmonotonically on salt concentration[75,76]. With increasing salt concentration in dilute polyelectrolytesolutions, the chain size first decreases than it increases again. The sizeof the polyelectrolyte chain reaches a minimum at salt concentrationfor which the total charge of the tetravalent counterions almostexactly neutralizes the net polymeric charge. The degree of the chaincollapse and swelling increases with the ion size. This interestingbehavior is due to the exchange of the monovalent ions by tetravalentcounterions that condense on the polymer backbone. At high saltconcentrations, salt ions can form layered structure around apolyelectrolyte and locally overcompensate the effective polymericcharge. This overcompensation of the polyelectrolyte charge leads tothe chain swelling. Similar reentrant conformational transition wasalso observed in the Monte Carlo simulations by Hsiao and Luijten[77]. A semiflexible polyelectrolyte chain could be either in theelongated or in compact toroid conformation when multivalent ions

are added. The transition between different chain conformations is thefirst order transition [5••,15••,78,79].

The presenceof themultivalent ions can lead to reversible gelation ofthe polyelectrolyte chains [80,81]. In this case the multivalent ions playrole of the cross-linking agents bridging polyelectrolyte chains together.Ermoshkin and de la Cruz [80,81] have applied combination of the Floryapproach and a modified random-phase approximation approach todescribe associations inpolyelectrolyte solutions. The gelation thresholdwas obtained as a function of the ion valence, polymer concentrationand strength of the electrostatic interactions. It alsowas shown that thegelation can only occur in semidilute polyelectrolyte solutionswhen theinterchain electrostatic repulsion is sufficiently screened to allow chainaggregation. The recent studies have extended this approach toassociating polyelectrolyte solutions [82,83].

In salt solutions chain size is very sensitive to the externalperturbations. Using a coarse-grained model of the polyelectrolytechain along with the Debye–Hückel approximation accounting forelectrostatic interactions between charged groups, Pamies et al.[84,85] have performed Brownian dynamics simulations with hydro-dynamic interactions to study deformation of polyelectrolyte chains indilute solutions under shear and elongational flow. Monitoring thechain size and intrinsic viscosity they observed abrupt conformationaltransition of polyelectrolyte molecules in shear γ̇c and elongational ε̇cflow and determined the critical values of the shear and elongationalrates necessary for this transition to occur. The critical values ε̇c and γ̇c

decrease with solution ionic strength. This is a result of the screeningof the intrachain electrostatic interactions with increasing saltconcentrationweakening chain tension and thus requiring theweakerexternal force for chain deformation. For high values of the ionicstrengths, the critical value of the elongational rate ε̇c scales withchain degree of polymerization as, ε̇c∝N−3/2, which is close to that forneutral polymers.

In poor solvent conditions for the polymer backbone addition ofsalt to polyelectrolyte solutions promotes phase separation. At highsalt concentrations, the electrostatic interactions between chargedmonomers are exponentially screened leading to the renormalizationof the second virial coefficient between monomers and changing theeffective solvent quality for the polymer chain. For a recent review ofthe results of the phase separation in polyelectrolyte solutions see refs[1••,86,87].

3. Polyampholyte solutions

Properties of polyampholyte chains in solutions depend on theaverage composition — the fractions of positively charged f+ andnegatively charged f− monomers, and on the distribution of chargedmonomers along the polymer backbone [4••]. This distribution ofcharged monomers is fixed during the polymerization reaction and isknown as a quenched charge distribution. A charge-balanced (netneutral or symmetric) polyampholyte with degree of polymerizationN has equal numbers of positively f+N and negatively f−N chargedgroups. The net charge or charge asymmetryΔfN of a polyampholyte isequal to the absolute value of the difference between numbers ofpositive f+N and of negative f−N charges (ΔfN=|f+− f−|N) [4••].

The effect of the charge distribution along the polymer backboneon conformations of single polyampholyte chain was studiedextensively over the years (see for review [4••]). For example, asymmetric polyampholyte chain with equal numbers of positivelyand negatively charged monomers (ΔfN=0) and random chargedistribution along the polymer backbone collapses forming a denseglobule. The collapse of the symmetric polyampholytes is due tocorrelation-induced-attraction between charged monomers. Thecharge-asymmetric polyampholytes with charge asymmetry ΔfNsmaller than (fN)1/2 behaves similar to polyampholytes with zero netcharge. The strongly charged polyampholytes with the chargeasymmetry form a necklace-like globule of the dense almost neutral


beads connected by charged strings. The ensemble-average chainsize of randomly charged polyampholytes is dominated by elongatedchain sections (strings) and increases with the chain degree ofpolymerization as N1/2.

The details of the collapse transitions in the diblock polyampho-lytes were recently studied by Wang and Rubinstein [88]. Usingcombination of the molecular dynamics simulations and scalinganalysis they have found three different conformational regimes incollapsed diblock polyampholytes with increasing the strength of theelectrostatic interactions controlled by the value of the Bjerrum lengthlB. In the first (folding) regime the electrostatic attraction betweenoppositely charged blocks force chain to fold through the overlap ofthe two blocks while each block is slightly stretched by intrablockelectrostatic repulsion. The second (scramble egg) regime is theclassical globule regime where the chain collapse is driven by thefluctuation-induced attraction between charged monomers. Thestructure of the collapsed chain can be represented as densely packingof the charged chain sections (correlation blobs). The third (strongassociation or ion binding regime) starts with direct binding ofoppositely charged monomers formation of the dipoles, followed bythe cascade of multipole (quadrupole, hexapole, etc) formation withincreasing the value of the Bjerrum length.

In polyampholyte solutions electrostatic interactions betweenoppositely charged monomers promote phase separation [89]. Thecondition for phase separation depends on the chain degree ofpolymerization, charge sequence along the polymer backbone,strength of the electrostatic interactions and solution temperature.The critical temperatures of the diblock polyampholyte solutions aremuch higher and increase more quickly with increasing chain degreeof polymerization than the critical temperatures of solution of randompolyampholyte. In dilute phase diblock polyampholytes form largeraggregates than randomly charged chains.

A theoretical model of aggregation in solutions of diblockpolyampholytes was proposed by Castelnovo and Joanny [90]. Theyhave studied the solution phase diagram at high ionic strengths anddetermined the phase boundary and the critical micelle concentrationas a function of polymer concentration and net polymeric charge ΔfN.The size of the micelle and its aggregation number was found to be astrong function of the fraction of charged monomers on the polymerbackbone. Themicellar core structure in thismodel was assumed to besimilar to the structure of the dense aggregate formed by oppositelycharged polyelectrolyte chains so that the net charge of the core wasvery close to zero. The corona of the block polyampholyte micelle wasformed by charge-unbalanced section of blocks with higher netcharge.

In the framework of the scaling approach Shusharina et al. [91]developed a theory of salt-free solutions of diblock polyampholytes. Acharge-asymmetric block polyampholyte (for example, diblock poly-ampholytewith longer negatively charged block) forms a tadpolewithan almost neutral globular head and a negatively charged polyelec-trolyte tail. With increasing polymer concentration these tadpolesaggregate into micelles. There are two different groups of chains inthese aggregates. Chains in one group are completely confined insidethemicellar core, while chains belonging to the other group have theirentire negatively charged block expelled into the corona. This micellarstructure has lower free energy than micellar structure considered byCaltelnovo and Joanny [90] and corresponds to minimum of theaggregate free energy. It is interesting to point out that similardisproportionation of polyampholyte chains has been observed in thelattice mean-field theory studies and Monte Carlo simulations ofpolyampholyte brushes in both planar and spherical geometries[92,93]. The disproportionation of chains in the systems containingoppositely charged polyelectrolytes has been predicted by Zhang andShklovskii [94].

Molecular dynamics simulations of complexation between poly-ampholyte and polyelectrolyte chains in solutions were performed by

Jeon and Dobrynin [95]. They have established that the complexationbetween polyampholyte and polyelectrolyte chains is due to polariza-tion-induced attractive interactions between molecules. The chargesequence along the polyampholyte backbone has a profound effect onthe resulting complex structure. For example, a diblock polyampho-lyte with a fraction of charged monomers f=1 and degree ofpolymerization N=32 could form a two-arm and three-arm star-likecomplex with a fully charged polyelectrolyte chain of the same degreeof polymerization. The complex structure changes with increasing saltconcentration confirming the dominant role of the electrostaticinteractions in chain's association.

The complex structure depends on polymer concentrations. Smallaggregates consisting of two or three polymer chains dominate at lowpolymer concentrations. As polymer concentration increases, micellaraggregates (see Fig. 4 a) coexist together with small aggregates. Themicellar aggregates persist until the system crosses over to thesemidilute solution regime where all chains form interconnectednetwork of micelles (see Fig. 4 b). Qualitatively different aggregatestructures were observed for random polyampholytes. The structureof multichain polyampholyte–polyelectrolyte complexes in this caseresembles that of branched polymers with polyampholytes bindingpolyelectrolyte chains together (see Fig. 4 c).

Complexation between a polyampholyte and a polyelectrolyte chaincan proceed through the release of counterions condensed on thepolyelectrolyte backbone [67]. These counterions are substituted by thepositively charged monomers of polyampholyte chain. The complexstructure shows strong correlation with the charge sequence along thepolymer backbone. In the case of random polyampholyte the complexhas a flower-like structure of the dense core surrounded by loops(see Fig. 5 a). The core of the aggregate containsmore positively chargedpolyampholyte chain sections and whole polyelectrolyte chain. Themore negatively charge section of the polyampholyte form loopssurrounding the core of the aggregate. The diblock polyampholyte hastadpole like structure with the tail of the tadpole containing part of thenegatively charged block (see Fig. 5 b). The remaining part of thenegatively charged block is wrapped around the head of the tadpole. Inboth systems there are charged density oscillations inside the core. Inthe case of random polyampholytes the center of the core has excess ofpositively charged monomers which is surrounded by the layer of thenegatively charged ones. The core of the complex formed by diblockpolyampholyte has four charge alternating layers with excess of thepositively charged monomers in the core center. These complexstructures persisted throughout the entire studied salt concentrationrange and show no qualitative changes with varying the strength of theelectrostatic interactions (the value of the Bjerrum length).

The more detailed discussion of the polyampholyte–polyelectro-lyte complexes can be found in refs [4••,8••,96,97].

4. Charged dendrimers

Dedrimers are regularly branched molecules with tree-likestructure that can be made by the cascade chemical synthesis startingfrom the core and attaching new branching units to the terminalgroups [98••,99••]. The chemical structure of the dendrimers ischaracterized by the generation number (the number of the focal(branching) points going from the core towards the dendrimersurface) and branching functionality of the monomeric units. Forexample, a dendrimer having four focal points when going from thecenter to the periphery is denoted as the 4th generation dendrimerand abbreviated as G4. The core part of the dendrimer is sometimesdenoted generation “zero”, or “G0”. The charged groups canfunctionalize the terminal groups of the last generation or be a partof the monomeric units used in dendrimer synthesis. The uniquestructure of the dendritic macromolecules may be used to carry lowmolecular substances, e.g. drugs, or may be useful to create alteredproperties of chromophores or fluorophores, etc.

Fig. 5. Snapshots of complexes formed by polyelectrolyte with random polyampholyte(a) and diblock polyampholyte (b) in a salt-free solution. The positively chargedmonomers of a polyampholyte chain are shown in red and negatively charged ones areshown in blue. The negatively chargedmonomers of polyelectrolyte chain are colored ingreen and neutral monomers are colored in gray. (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Complexes formed by polyelectrolyte chains with diblock (a, b) and random(c) polyampholytes. Polyelectrolyte chains are shown in green, positively chargedmonomers on the polyampholyte backbone are shown in red and negatively chargedones are shown in blue. (For interpretation of the references to colour in this figurelegend, the reader is referred to the web version of this article.)


The static and dynamic properties of the dilute dendrimersolutions were studied by Lyulin et al. [100,101]. There was no explicitcounterions in these simulations and electrostatic interactionsbetween charged groups were taken into account on the level of thescreened Coulomb potential. The hydrodynamic interactions wereincluded into simulation scheme through the Rotne–Prager–Yama-kawa tensor. According to these simulations, the dendrimer motionmay be divided into three main types: (1) translational andorientational motions of a dendrimer as a whole (global motions);(2) fluctuations of the dendrimer size and shape (elastic motions);(3) local motions with scales corresponding to the length of only a fewmonomer units. The dendrimer diffusion coefficient D scales with theradius of gyration Rg as D ~ Rg

−0.8 for both charged and neutraldendrimers with different number of generations. The results of themolecular dynamics simulations are in reasonably good qualitative

and in some cases quantitative agreement with simple Flory-likemodel of the dendrimer molecule. However, the results of thesesimulations have to be taken with caution since they do not take intoaccount the counterion condensation.

Simulations with the explicit counterions show that counterioncondensation has a tremendous effect on conformations of chargeddendrimers. Similar to the case of the linear chains, counterioncondensation occurs with increasing value of the Bjerrum length andis accompanied by subsequent swelling and collapse of the dendrimermolecules [102,103]. The existence of such nonmonotonic conforma-tional transformations is due to fine interplay between growingrepulsions between charged end-beads and counterion condensationtendency because of attractive interactions between counterions andthe terminal groups. Furthermore, counterion radial distributionfunctions indicated that counterions occupy not only the dendrimerperiphery but also its interior.

Molecular dynamics simulations of dilute salt-free solutions ofweakly charged dendrimers with spacers of different lengths wereperformed by Lin et al. [104]. In dilute solutions the equilibrium

Fig. 6. The equilibrium swelling behavior of salt-free polyelectrolyte gels in the goodsolvent and close to the θ solvent conditions for the polymer backbone. The data pointsfollow the scaling expression R≈bf eff1-v Nm, where the exponent v is equal to 1/2 in θsolvent and 0.6 in good solvent. feff is the effective fraction of the charged groups on thepolymeric strand with Nm monomers. Reproduced with permission from Mann. B. A.,Kremer, K., Holm, C. Macromol. Symp. 237, 90–107 (2006).


conformation of the chargeddendrimers is determined by balancing thedendrimer elastic free energy either with the electrostatic repulsionbetween charged groups or with the configurational free energy of thelocalized within dendrimer counterions. The crossover between tworegimes (so-called electrostatic and osmotic regimes) takes place atcritical generation number which value depends on the fraction of theionized groups. Note that electrostatic and osmotic regimes are alsoobserved in polyelectrolyte brushes [105••]. The concentration depen-dence of the osmotic coefficient of the solution of charged dendrimers issimilar to that reported for polyelectrolyte solutions [1••].

Atomistic MD simulations of the PAMAM dendrimers at several pHconditions, with explicit water molecules were performed by Lee at al.[106] and by Maiti and Goddard [107]. They obtained structuralcharacteristics of charged dendrimers as function of the solution pHand established the role of the water molecules in determining thedendrimer conformations and monomer density distribution. Thesesimulations showed that the dendrimer size decreases with increasingsolution pH.

5. Polyelectrolyte gels

Networks made from polyelectrolyte chains are quite common inboth nature and industry [108–110]. Polyelectrolyte gels are capable ofswelling to much greater extents than their uncharged counterpartsbecause of high osmotic pressure due to dissociated counterions.Because of such unique properties polyelectrolyte gels are used assuperabsorbent materials, as ion-exchange resins, and as the carrier fornovel drug delivery targeting specific organs. Itwas long recognized thatthe swelling of polyelectrolyte gels is determined by a balance betweenthe osmotic pressure of free ions acting to swell the gel and the elasticityof the gel that restrict the swelling [108,109]. The osmotic part has longbeen understood in terms of the Donnan equilibrium.

Unfortunately only until recently the progress in the computa-tional resources have allowed computer simulations of the polyelec-trolyte networks [111,112,113•,114–118]. The polyelectrolyte networksin such simulations usually had an ideal regular structure. Thesimulations were performed at various network volume fractions,chain's degree of polymerizations, crosslink density, values of theBjerrum length and solvent quality for the polymer backbone.

In a series of papers the Linse group has used Monte Carlosimulations to model salt-free regular polyelectrolyte networks withdiamond-like topology and explicit counterions [116–118]. Thesesimulations have shown that the gel volume increases with increasingthe fraction of charged monomer on the polymeric strands forming agel. It also increases with decreasing the cross-linking density andincreasing chain stiffness. This dependence of the gel swelling on thenetwork parameters is in line with the notion that polyelectrolyte gelswelling is controlled by the osmotic pressure of counterions and bythe elasticity of the polymeric strands forming a gel. The increase inthe fraction of the charged monomers on polymer backbones leads toincrease of the osmotic pressure of the counterions which forces gel toincrease its volume. The decrease in the number of crosslinks(increase in the degree of polymerization of the polymeric standsbetween crosslinks) and increase of the chain stiffness (increase of thechain's Kuhn length) both results in decrease of the gel shearmodulus,which also promotes network swelling. The effect of the counterionosmotic pressure and chain's elasticity on the gel swelling has alsobeen studied by the Mainz group in molecular dynamics simulationsof the salt-free polyelectrolyte networks [113•,114,115]. These simula-tions established that only “free” (osmotically active) counterionscontribute to the gel swelling (see Fig. 6). This was verified for thepolyelectrolyte networks in the good and almost theta solventconditions for the polymer backbone. It is important to point outthat the swollen polyelectrolyte gels deform affinely.

Themultivalent (divalent, trivalent) ions have a strong effect on thegel structure [119]. The exchange of the monovalent ions to multi-

valent ones leads to collapse of the polyelectrolyte network, which ismanifested in discontinuous volume change of the gel. The origin ofthis collapse transition is similar to the collapse of a polyelectrolytechain in the presence of the multivalent ions and is due to thecorrelation-induced attractive interactions between multivalent ionsand charged monomers on the polymer strands forming a gel. Notethat similar effect can be achieved in simulations with monovalentions by increasing the value of the Bjerrum length. Experimentally itcorresponds to immersing polyelectrolyte networks in low dielectricsolvents, lB ~ 1/ε. The network collapse was also observed whenmonovalent counterions were substituted by macroions [112].

The simulations of polyelectrolyte networks with explicit counter-ions and explicit solvent particles were reported by Lu and Hentschke[120•]. Using a “two-box-particle” transfer molecular dynamicssimulation method they modeled an extremely highly crosslinkednetwork in equilibrium with a dipolar Stockmayer solvent. In thesesimulations each bead of the polyelectrolyte strand forming a networkwas carrying a partial charge eq and each corresponding neutralizingcounterion had partial charge − eq. The swelling ratio of thepolyelectrolyte network has a maximum as a function of the partialcharge eq. This nonmonotonic dependence of the swelling ratio wasattributed to a competition between electrostatic repulsion and thenetwork conformational entropy. The maximum becomes lesspronounced with increasing the dipole moment of the Stockmayerfluid.

Edgecombe and Linse have studied the role of the chain'spolydispersity and topological network defects on the polyelectrolytegel swelling [111]. They found that polydisperse networks swell lessthan regular gels. In polydisperse gels the short chains are morestretched and the long ones are less stretched as compared topolyelectrolyte networks made of monodisperse chains. Thus, thedeformation of the short polyelectrolyte chains controls swelling ofthe polydisperse gels. The normal stress in uniaxial stretchingsimulations for neutral and polyelectrolyte gels follows roughlyexponential dependence on the deformation ratio λ. The agreementbetween simulation data and theoretical models of uniaxial geldeformationwas better for the non-Gaussian network theory than forthe Gaussian theory. However, both models deviate significantly fromsimulation data in the limit of large network deformations.

In the poor solvent conditions for the polymer backbone polymericstrands connecting the nodes of the polyelectrolyte network can havenecklace-like conformations similar to those observed in polyelec-trolyte solutions. The detailed study of the network structure in thepoor solvent conditions for the polymer backbone as function of the


fraction of charged monomers f and the value of the Bjerrum lengthwas performed by Mann et al. [113•]. They observed the formation ofthe following chain conformations within polyelectrolyte network:collapsed conformations for small charge fractions or very strongelectrostatic interactions (large values of the Bjerrum length);necklace structure for moderate to high charge fractions and notvery strong electrostatic coupling; stretched structures for largefractions of charged monomers f and moderate values of the Bjerrumlengths lB; and finally the “sausage”-like conformation for largeBjerrum lengths. Note that the diagram of states for polyelectrolytenetworks has regimes similar to those observed at the diagram of statefor the single polyelectrolyte chain. However, the boundaries betweendifferent conformational regimes are shifted due to additional effect ofthe chain connectivity into the network on the backboneconformations.

6. Charged polymers at surfaces and interfaces

6.1. Polyelectrolyte adsorption

Adsorption of charged polymers on charged surfaces and interfacesis a classical problem of polymer physics and has been under extensivetheoretical and experimental studies for the last four decades (see forhistorical overview [1••]). Here I present only recent developments inthis area of polymer science.

An exact analytical solution for adsorption transition of a flexiblepolyelectrolyte chain at oppositely charged spherical particle of theradius R from salt solution was obtained by Winkler and Cherstvy[121•,122]. To obtain this solution they have substituted the Debye–Huckel potential for the charged spherical particle by the Hulthenpotential. It turns out that the difference between two potentials issmall for the wide interval of salt concentrations and particle sizes.The exact analytical solution of the differential equation for theprobability density of a flexible polymer chain in the Hulthen potentialis well known and is given by the Gauss hypergeometric function. Thissolution exactly reproduces results for polyelectrolyte adsorption ontoplanar surface in the limit of the vanishing particle curvature (R≫rD).In this limit the critical charge density σc, determining the adsorptionthreshold, shows strong dependence on the Debye screening lengthσc ~ 1/rD3. In the opposite limit of the small particles (R≪rD) the criticalparticle surface charge density σc is inversely proportional to thesolution Debye screening length, σc ~ 1/rD. These predictions are inagreement with the Monte Carlo simulation results by Chodanowskiand Stoll [123]. Winkler and Cherstvy have extended their approach tostudy the adsorption of polyelectrolytes at oppositely chargedcylinders [124].

Molecular dynamics simulations of polyelectrolyte adsorption atoppositely charged surfaces from dilute polyelectrolyte solutions wereperformed by Carrillo and Dobrynin [125•]. These simulations havestudied the effects of the surface charge density, surface chargedistribution, solvent quality for the polymer backbone, strength ofthe short-range interactions between polymers and substrates on thepolymer surface coverage and the thickness of the adsorbed layer.The solvent quality for the polymer backbone and polymer affinityto the surfacemanifests itself in qualitatively different dependences ofthe thickness of the adsorbed layer on the surface charge density.In the case of adsorption of hydrophobic polyelectrolytes at hydro-philic surfaces, the chain thickness decreases with increasing surfacecharge density. This decrease in the chain thickness is a result of theoptimization of the electrostatic attraction between ionized groups onthe polymer backbone to the oppositely charged substrate and chainsurface energy. Furthermore, these polyelectrolytes form multichainaggregates with increasing surface charge density.

Hydrophobic polyelectrolytes wet a hydrophobic surface forming amonolayer maximizing the number of favorable polymer surfacecontacts. The thickness of the adsorbed layer stays almost constant at

low surface charge densities and is mainly controlled by the strengthof the short-range polymer surface interactions. At high surfacecharge densities, polymers completely cover the surface and thethickness of the adsorbed layer increases linearly with the surfacecharge density. In this range of the surface charge densities theelectrostatic attraction between polyelectrolyte chains and theoppositely charged substrate start to play a dominant role incontrolling the layer thickness.

Adsorption of hydrophilic polymers at hydrophilic surfaces wasaccompanied by a nonmonotonic dependence of the chain thicknesson the surface charge density. This nonmonotonicity is a result of thetwo different mechanisms responsible for the chain thicknessstabilization. At low surface charge densities, the chain thickness isdetermined by the balance of the energy gain due to electrostaticattraction and confinement entropy loss due to chain localization. Inthis regime the chain thickness decreases with increasing the surfacecharge density. At higher surface charge densities, the thickness of theadsorbed layer is determined by the balance between electrostaticattraction of the charged monomers to the substrate and short-rangemonomer–monomer repulsion resulting in an increase in the layerthickness. The results of these simulations are in qualitative agree-ment with the prediction of the scaling models of polyelectrolyteadsorption (see for review [1••]).

The effect of solvent quality on the behavior of a polyelectrolytechain near a charged surface was also studied by Reddy et al. [126].Using molecular dynamics simulation with explicit solvent they havefound that for a given solvent quality, increasing the surface chargedensity causes the chain to adsorb flat on the charged surface. Theshape of the adsorbed polyelectrolyte chain is a complicated functionof the surface charge density, polymer–solvent, polymer–polymer andpolymer–surface interactions. The surprising result of these simula-tions was that the rotational and translational dynamics of the polyionbecome faster with decreasing the solvent quality. This was related tothe decrease in the chain size and decrease in the solvent density atthe surface as the solvent quality is decreased, both of which tend toincrease lateral (along the surface) chain diffusion coefficient.

The conformational transformations in adsorbed polyelectrolytechain in a poor solvent for the polymer backbone were studied byPattanayek1 and Pereira [127]. It was shown by the free energyminimization and by the SCF calculations that an adsorbed polyelec-trolyte chain in the poor solvent conditions for the polymer backbonecan have an elliptical shape or a pearl-necklace shape with deformedelliptical beads depending on the surface charge density and solventquality for the polymer backbone. Adsorption of necklaces was alsostudied in refs [128,129].

Binary mixtures of oppositely charged molecules adsorbed onto asurface (confined to a plane) may exhibit microphase separation [130–132,133•]. The origin of the microphase separation is optimization ofthe short-range interactions, which favor macroscopic segregation,and the long-range electrostatic interactions, favoring componentintermixing. The symmetry of the appearing domain structure(formation of the lamellar or hexagonal patterns) depends on thecharge stoichiometry of the mixture. The typical size of the domainsscales with the parameters of the system as L ∝ (εint/lB f3/2)1/2 whereεint is the cohesive energy and f is the average charge density.

In many experimental situations, such as adsorption of polyelec-trolyte chains from water onto clay, polymer latex particles or at thewater/air interface, the dielectric constant of the solvent ε is largerthan that of the surface εs The presence of the charge in the mediumwith dielectric constant ε near the surface with the dielectric constantεs causes polarization of both media. The result is the appearance ofthe image charge at the symmetric positions with respect to thedielectric boundary.

Monte Carlo simulations of the effect of the dielectric boundary onthe adsorption of strongly charged polyelectrolytes at oppositelycharged planar surface were performed by Messina [134•,135]. These

Fig. 7. Dependence of the polymer surface coverage on the number of deposition stepsfor multiplayer formation from dilute polyelectrolyte solutions. Reproduced withpermission from Patel, P., Jeon, J., Mather, P. T., Dobrynin, A. V. Langmuir 21, 6113–6122(2005). Copyright 2005, American Chemical Society. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version ofthis article.)


simulations have shown that image forces appearing due to thedielectric discontinuity at the adsorbing substrate lead to the decreasein polymer surface coveragewhich precludes the surface overchargingby adsorbed polyelectrolytes. Cheng and Lai [136] studied a singlechain adsorption at the charged substrate with high dielectricconstant. In the framework of the ground state dominance approx-imation they found that adsorption at low ionic strengths is the firstorder transition with the monomer density at the surface scaling

Fig. 8. Evolutionof themultilayerassemblyat charged surface. Snapshots are takenaftercompletshown in green and neutral particles are colored in black. The molecules deposited during difforange (5). Reproduced with permission from Jeon, J., Panchagnula, V., Pan, J., Dobrynin, Ainterpretation of the references to colour in this figure legend, the reader is referred to the we

linearly with the surface charge density. A detailed analysis of theboundary conditions for the polyelectrolyte adsorption at dielectricinterface was done by Cheng [137].

6.2. Layer-by-Layer assembly

Layer-by-layer deposition of charged molecules provides a power-ful tool for fabrication of multicomponent coatings with uniquefunctional properties (see for review [9••–11••]). This technique isbased on the electrostatic attraction between oppositely chargedmolecules, which results in stable multilayer structures. The key to asuccessful deposition of multilayer assemblies in a layer-by-layerfashion is the inversion and subsequent reconstruction of the surfaceproperties. For example, this can be achieved by immersing thesubstrate into a dilute aqueous solution of anionic (or cationic)polyelectrolytes for a period of time required for the adsorption oflayer of given thickness after which the substrate is rinsed. The rinsingstep is necessary to remove the polymers that are not tightly adsorbedto the substrate. During the next step a substrate covered withadsorbed polyelectrolytes is exposed to a dilute solution of cationic(or anionic) macromolecules, followed by a rinsing step to obtain anirreversibly adsorbed layer. Further film growth is achieved bydeposition of polyanions and polycations from their aqueoussolutions. After several dipping cycles the experiments show a linearincrease of multilayer thickness, indicating that the system hasreached a steady state regime.

Unlike the extensive experimental studies of layer-by-layer deposi-tion of charged molecules [9••–11••] the theoretical description of themultilayer formation remains limited. The numerical solutions of theself-consistentfield equationsdescribingmultilayer assemblyhavebeenrecently presented byWang [138,139] andby Shafir andAndelman [140]These calculations have shown that the sufficiently strong short-rangeattraction between oppositely charged polymers is essential for the

ionof thedeposition steps 1 through5.Thepositively chargedparticles on the substrate areerent deposition steps are colored as follows: blue (1), red (2), cyan (3), magenta (4), and. V. Langmuir 22, 4629–4637 (2006). Copyright 2006, American Chemical Society. (Forb version of this article.)

• Of special interest.•• Of outstanding interest.


formation of multilayers. The limitation of the used approach is thatafter completion of each deposition step the polymer density distribu-tion of polyelectrolytes adsorbed during this deposition step was fixedfor the remaining duration of the film growth process. This restrictionprecluded redistribution of the polymer density within the film duringthe deposition of the next polymeric layers. This assumption contradictsto the results of the molecular dynamics simulations (see discussionbelow) and may be the main reason for the periodic film structureobtained in these SCF calculations.

The formation of ionic pairs between polyelectrolyte chainsforming multilayers was recently taken into account by Park et al.[141] and by Lefaux et al. [142]. These models show promising resultsby predicting the correct salt concentration dependence of multilayergrowth by sequential adsorption and by spin coating methods.However, these models neglect strong intermixing between layers.

Monte Carlo simulations of multilayered film assembly frommixtures of oppositely charged polyelectrolytes near charged sphe-rical particles, charged cylinders and uniformly charged surface wereperformed by Messina et al. [143–146] These papers tested thehypothesis that multilayering is an equilibrium process which occursnot only when one proceeds in a step-wise fashion, as done inexperiments [9••–11••], but also when oppositely charged polyelec-trolytes are added together and the resulting solution is exposed to acharged substrate. It was shown that the additional short-rangeattractive interactions between polyelectrolytes and the surface arenecessary to successfully initiate chain adsorption. These simulationsdo not represent the experimental situation in which polyelectrolytesare deposited from a solution in a sequential manner. Thus, it wasimpossible to test the linear increase of the layer thickness and masswith the number of deposition steps as seen in the experiments.This shortcoming was addressed in molecular dynamics simula-tions by the author's research group [147–151] and by Abu-Sharkh[152,153] that studied the sequential adsorption of oppositely chargedpolyelectrolytes.

Molecular dynamics simulations of the sequential adsorption ofoppositely charged polyelectrolytes onto a charged spherical particlewere performed by Panchagnula et al. [147,148] These simulationsconfirmed that the layer build-up proceeds through surface over-charging during each deposition step and with the system reaching asteady state regime after a fewdeposition stepswith nonlinear growthof polymermass in the aggregate. However, despite the steady growth,the spherical symmetry of such amacroion precluded formation of thewell-developed multilayered structures and instead showed non-symmetric oscillation of the local polymer composition — the densitydifference between positively and negatively charged chains withinpolymeric film.

Multilayer formation on the planar surfaces was studied inmolecular dynamic simulations of Patel et al. [149,150]. Thesesimulations have shown that the film build-up follows a linear growthpattern with both the thickness of the adsorbed layer and polymersurface coverage increasing linearly with the number of depositionsteps (see Fig. 7). This steady state linear growth regime is generallyobserved in experiments after deposition of the first few layers. Forpartially charged chains with f=1/2 and 1/3 (here, f is the fraction ofcharged beads in a bead-spring chain), the growth rate of the polymersurface coverage is higher than for the case of fully charged chains. Thisis in agreement with experimental observations of the thicker layersfor partially charged polyelectrolytes compared to very thin layersobtained for the fully charged polymers. While simulations showedstrong intermixing between polyelectrolyte chains adsorbed duringdifferent deposition steps, almost perfect periodic oscillations in localpolymer compositionwere observed, indicating polymer stratification.

Results of the molecular dynamics simulations of layer-by-layerassembly of polyelectrolytes and nanoparticles from dilute solutionshave been reported by Jeon et al. [151]. They have found that formultilayer films consisting of nanoparticles, there is better stratifica-

tion of the layers with almost constant thickness of the layer com-posed of nanoparticles (see Fig. 8). For all studied systems, the processof multilayer formation occurs over several deposition steps: usuallyfour deposition steps are required to complete formation of the twolayers. The film thickness and surface coverage increases almostlinearly with the number of deposition steps, indicating steady statefilm growth. The multilayered films formed by nanoparticles featurehigher roughness than films consisting of flexible polymers.

7. Conclusions and future outlook

In this paper I have presented an overview of the molecularsimulations and theoretical models of polyelectrolyte solutions andgels, polyampholyte solutions, charged dendrimers, and chargedpolymers at charged surfaces. While there is a significant progressin our understanding of these polymeric systems a lot of interestingwork remains to be done as our attention turns to the effect of thesolvent on the properties of ionic systems. Explicit solvent simulationsshould allow us to resolve a problem of coupling between electrostaticand hydrodynamic interactions in polyelectrolyte systems. Thesolution of this problem is extremely important for understandingthe dynamics of polyelectrolyte solutions, dynamics of swelling of thepolyelectrolyte gels, the deformation of polyelectrolyte brushes anddendrimers in external flows, and the dynamics of charged polymersat adsorbing surfaces and within multilayered thin films. The explicitsolvent simulations should also be able to highlight the effect of thelocal dielectric constant fluctuations on the counterion condensation,self-assembly and associations in the polyelectrolyte systems, adsorp-tion of charged polymers at water–air interface, and phase separationin poor solvent conditions for the polymer backbone. All these topicshave a broad spectrum of interesting physical phenomena andtremendous practical applications.

I hope that topics discussed in this review will help readers tograsp the modern developments in the area of ion containingpolymers and will be useful in future studies of more complicatedionic systems.

Acknowledgments

The author is grateful for support of the research to the Donors ofthe American Chemical Society the Petroleum Research Fund underthe Grant PRF#44861-AC7.

References and recommended reading•,••

[1]••

Dobrynin AV, Rubinstein M. Theory of polyelectrolytes in solutions and atsurfaces. Prog Polym Sci 2005;30:1049–118. Comprehensive overview of thetheoretical models and computer simulations of polyelectrolyte solutions andadsorption of charged polymers.

[2]•

Khokhlov AR, Khalatur PG. Solution properties of charged hydrophobic/hydrophilic copolymers. Curr Opin Colloid Interface Sci 2005;10:22–9. Overviewof properties of polyelectrolyte solutions.

[3]••

Holm C, Joanny JF, Kremer K, Netz RR, Reineker P, Seidel C, et al. Polyelectrolytetheory. Adv Polym Sci 2004;166:67–111. Critical overview of the theoreticalmodels of the polyelectrolyte solutions and detailed comparison with computersimulations.

[4]••

Dobrynin AV, Colby RH, RubinsteinM. Polyampholytes. J Polym Sci, B, Polym Phys2004;42:3513–38. Overview of the experimental, theoretical and computersimulation studies of the polyampholyte systems.

[5]••

Grosberg AY, Nguyen TT, Shklovskii BI. Colloquium: the physics of chargeinversion in chemical and biological systems. Rev Mod Phys 2002;74:329–45.Excellent introduction to problem of overcharging.

[6]••

Morawetz H. Revisiting some phenomena in polyelectrolyte solutions. J PolymSci, B, Polym Phys 2002;40:1080–6. Nice overview of the history of polyelec-trolyte solutions.

[7]••

Cooper CL, Dubin PL, Kayitmazer AB, Turksen S. Polyelectrolyte–proteincomplexes. Curr Opin Colloid Interface Sci 2005;10:52–78. Comprehensiveoverview of the experimental studies of polyelectrolyte–protein complexes.


[8]••

de Vries R, Stuart MC. Theory and simulations of macroion complexation. CurrOpin Colloid Interface Sci 2006;11:295–301. Excellent discussion of the currentstate of the art in macroion complexation.

[9]••

Decher G, Schlenoff JB, editors. Multilayer thin films. Wiley-VCH; 2003..Outstanding collection of papers on multilayer assembly.

[10]••

Sukhishvili SA. Responsive polymer films and capsules via layer-by-layerassembly. Curr Opin Colloid Interface Sci 2005;10:37–44. Excellent discussionof the layer-by-layer assembly.

[11]••

Sukhorukov G, Fery A, Mohwald H. Intelligentmicro- and nanocapsules. Prog PolymSci 2005;30:885–97. Comprehensive overview of the layer-by-layer assembly.

[12]••

Boroudjerdi H, Kim YW, Naji A, Netz RR, Schlagberger X, Serr A. Statics anddynamics of strongly charged soft matter. Phys Rep-Rev Sect Phys Lett2005;416:129–99. Nice introduction into electrostatic effects in biological andpolymeric systems.

[13] Belyi VA, Muthukumar M. Electrostatic origin of the genome packing in viruses.Proc Natl Acad Sci U S A 2006;103:17174–8.

[14] Koltover I. Biomolecular self-assembly — stacks of viruses. Nat Mater 2004;3:584–6.

[15]••

Levin Y. Electrostatic correlations: from plasma to biology. Rep Prog Phys2002;65:1577–632. Excellent discussion of problem of overcharging in ionic systems.

[16]•

Wong GCL. Electrostatics of rigid polyelectrolytes. Curr Opin Colloid Interface Sci2006;11:310–5. Discussion of the solution properties of rigid polyelectrolytes.

[17]••

Schiessel H. The physics of chromatin. J Phys, Condens Matter 2003;15:R699–774.Excellent introduction to the problem of complexation in biological polyelectrolytes.

[18] Hayes JJ, Hansen JC. Nucleosomes and the chromatin fiber. Curr Opin Genet Dev2001;11:124–9.

[19] Kabanov AV. Polymer genomics: an insight into pharmacology and toxicology ofnanomedicines. Adv Drug Deliv Rev 2006;58:1597–621.

[20] Kabanov VA. From synthetic polyelectrolytes to polymer-subunit vaccines.Review. Polym Sci, Ser A 2004;46:451–70.

[21] Borukhov L, Bruinsma RF, Gelbart WM, Liu AJ. Structural polymorphism of thecytoskeleton: a model of linker-assisted filament aggregation. Proc Natl Acad SciU S A 2005;102:3673–8.

[22] KwonHJ, KakugoA, Shikinaka K, Osada Y, Gong JP.Morphology of actin assembliesin response to polycation and salts. Biomacromolecules 2005;6:3005–9.

[23] Tang JX, Kas JA, Shah JV, Janmey PA. Counterion-induced actin ring formation. EurBiophys J Biophys Lett 2001;30:477–84.

[24] Zribi OV, Kyung H, Golestanian R, Liverpool TB, Wong GCL. Salt-inducedcondensation in actin-DNA mixtures. Europhys Lett 2005;70:541–7.

[25] ZribiOV,KyungH,GolestanianR, Liverpool TB,WongGCL. CondensationofDNA-actinpolyelectrolyte mixtures driven by ions of different valences. Phys Rev E 2006;73.

[26] Limbach HJ, Sayar M, Holm C. Polyelectrolyte bundles. J Phys, Condens matter2004;16:S2135–44.

[27] Thunemann AF, Muller M, Dautzenberg H, Joanny JF, Lowne H. Polyelectrolytecomplexes. Adv Polym Sci 2004;166:113–71.

[28]•

Ulrich S, SeijoM, Stoll S. Themany facets of polyelectrolytes and oppositely chargedmacroions complex formation. Curr Opin Colloid Interface Sci 2006;11:268–72.Overview of the recent results on polyelectrolyte complexation.

[29] Lutkenhaus JL, Hammond PT. Electrochemically enabled polyelectrolyte multi-layer devices: from fuel cells to sensors. Soft Matt 2007;3:804–16.

[30] Korin E, Siton O, Bettelheim A. Fuel cells and ionically conductive membranes: anoverview. Rev Chem Eng 2007;23:35–63.

[31] Liao Q, Dobrynin AV, Rubinstein M. Molecular dynamics simulations ofpolyelectrolyte solutions: nonuniform stretching of chains and scaling behavior.Macromolecules 2003;36:3386–98.

[32] RubinsteinM, Colby RH. Polymer Physics. New York: Oxford University Press; 2003.[33]•

Liao Q, Dobrynin AV, Rubinstein M. Molecular dynamics simulations ofpolyelectrolyte solutions: osmotic coefficient and counterion condensation.Macromolecules 2003;36:3399–410. Detailed molecular dynamics and theore-tical study of the effect of counterions on osmotic pressure of polyelectrolytesolutions.

[34] Antypov D, Holm C. Osmotic coefficient calculations for dilute solutions of shortstiff-chain polyelectrolytes. Macromolecules 2007;40:731–8.

[35] Chang R, Yethiraj A. Osmotic pressure of salt-free polyelectrolyte solutions: aMonte Carlo simulation study. Macromolecules 2005;38:607–16.

[36] Donley JP, Heine DR. Structure of strongly charged polyelectrolyte solutions.Macromolecules 2006;39:8467–72.

[37] Holm C, Rehahn M, Oppermann W, Ballauff M. Stiff-chain polyelectrolytes. AdvPolym Sci 2004;166:1–27.

[38] Limbach HJ, Holm C, Kremer K. Computer simulations of the “hairy rod” model.Macromol Chem Phys 2005;206:77–82.

[39] Limbach HJ, Holm C, Kremer K. Structure of polyelectrolytes in poor solvent.Europhys Lett 2002;60:566–72.

[40] Limbach HJ, Holm C. Conformational properties of poor solvent polyelectrolytes.Comput Phys Commun 2002;147:321–4.

[41] Limbach HJ, Holm C. Single-chain properties of polyelectrolytes in poor solvent.J Phys Chem B 2003;107:8041–55.

[42] Limbach HJ, Holm C, Kremer K. Conformations and solution structure ofpolyelectrolytes in poor solvent. Macromol Symp 2004;211:43–53.

[43] Liao Q, Dobrynin AV, Rubinstein M. Counterion-correlation-induced attractionand necklace formation in polyelectrolyte solutions: theory and simulations.Macromolecules 2006;39:1920–38.

[44] Jeon J, Dobrynin AV. Necklace globule and counterion condensation. Macro-molecules 2007;40:7695–706.

[45] Ulrich S, Laguecir A, Stoll S. Titration of hydrophobic polyelectrolytes usingMonteCarlo simulations. J Chem Phys 2005;122:094911.

[46] Uyaver S, Seidel C. Pearl-necklace structures in annealed polyelectrolytes. J PhysChem B 2004;108:18804–14.

[47] Chang RW, Yethiraj A. Dilute solutions of strongly charged flexible polyelec-trolytes in poor solvents: molecular dynamics simulations with explicit solvent.Macromolecules 2006;39:821–8.

[48] Reddy G, Yethiraj A. Implicit and explicit solvent models for the simulation ofdilute polymer solutions. Macromolecules 2006;39:8536–42.

[49] Chang RW, Yethiraj A. Strongly charged flexible polyelectrolytes in poor solvents:molecular dynamics simulationswith explicit solvent. J ChemPhys2003;118:6634–47.

[50] Iwaki T. Association–dissociation equilibrium of loop structures in single-chainfolding into a toroidal condensate. J Chem Phys 2006;125:224901.

[51] Iwaki T, Yoshikawa K. Counterion-mediated intra-chain phase segregation on asingle semiflexible polyelectrolyte chain. Europhys Lett 2004;68:113–9.

[52] Sakaue T, Yoshikawa K. On the formation of rings-on-a-string conformations in asingle polyelectrolyte chain: a possible scenario. J Chem Phys 2006;125:074904.

[53] Sakaue T. Emergence of multiple tori structures in a single polyelectrolyte chain.J Chem Phys 2004;120:6299–305.

[54]•

Liao Q, Carrillo JMY, Dobrynin AV, Rubinstein M. Rouse dynamics of polyelelc-trolyte solutions: molecular dynamics study. Macromolecules 2007;40:7671–9.First molecular dynamics study of the Rouse dynamics of polyelectrolytesolutions.

[55]•

Stoltz C, de Pablo JJ, GrahamMD. Simulation of nonlinear shear rheology of dilutesalt-free polyelectrolyte solutions. J Chem Phys 2007;126:124906. First compre-hensive study of the effect of hydrodynamic interactions on rheology of dilutepolyelectrolyte solutions.

[56] Manning GS. The critical onset of counterion condensation: a survey of itsexperimental and theoretical basis. Ber Bunsen-Ges-Phys Chem Chem Phys1996;100:909–22.

[57]•

Deshkovski A, Obukhov S, Rubinstein M. Counterion phase transitions in dilutepolyelectrolyte solutions. Phys Rev Lett 2001;86:2341–4. Exact analyticalsolution for counterion distribution around charged cylinder.

[58] Naji A, Netz RR. Scaling and universality in the counterion–condensationtransition at charged cylinders. Phys Rev E 2006;73:056105.

[59] Naji A, Netz RR. Counterions at charged cylinders: criticality and universalitybeyond mean-field theory. Phys Rev Lett 2005;95:185703.

[60] O'Shaughnessy B, Yang Q. Manning–Oosawa counterion condensation. Phys RevLett 2005;94:048302.

[61] Schiessel H, Pincus P. Counterion-condensation-induced collapse of highlycharged polyelectrolytes. Macromolecules 1998;31:7953–9.

[62] Schiessel H. Counterion condensation on flexible polyelectrolytes: dependenceon ionic strength and chain concentration. Macromolecules 1999;32:5673–80.

[63] Muthukumar M. Theory of counter-ion condensation on flexible polyelectrolytes:adsorption mechanism. J Chem Phys 2004;120:9343–50.

[64] Kramarenko EY, Khokhlov AR, Yoshikawa K. A three-state model for counterionsin a dilute solution of weakly charged polyelectrolytes. Macromol Theory Simul2000;9:249–56.

[65] CastelnovoM, Sens P, Joanny JF. Charge distribution on annealed polyelectrolytes.Eur Phys J, E 2000;1:115–25.

[66] Liu S, Muthukumar M. Langevin dynamics simulation of counterion distribu-tion around isolated flexible polyelectrolyte chains. J Chem Phys 2002;116:9975–9982.

[67] Jeon J, Dobrynin AV. Molecular dynamics simulations of polyelectrolyte–polyam-pholyte complexes. Effect of solvent quality and salt concentration. J Phys Chem B2006;110:24652–65.

[68] Zhou T, Chen SB. Computer simulations of diffusion and dynamics of short-chainpolyelectrolytes. J Chem Phys 2006;124:034904.

[69] Dobrynin AV. Electrostatic persistence length of semiflexible and flexiblepolyelectrolytes. Macromolecules 2005;38:9304–14.

[70] Dobrynin AV. Effect of counterion condensation on rigidity of semiflexiblepolyelectrolytes. Macromolecules 2006;39:9519–27.

[71] Ullner M. Comments on the scaling behavior of flexible polyelectrolytes withinthe Debye–Huckel approximation. J Phys Chem B 2003;107:8097–110.

[72] Klos J, Pakula T. Computer simulations of a polyelectrolyte chainwith amixture ofmultivalent salts. J Phys, Condens Matter 2005;17:5635–45.

[73] Klos J, Pakula T. Monte Carlo simulations of a polyelectrolyte chain with addedsalt: effect of temperature and salt valence. J Chem Phys 2005;123:024903.

[74] Klos J, Pakula T. Lattice Monte Carlo simulations of a charged polymer chain:effect of valence and concentration of the added salt. J Chem Phys 2005;122:134908.

[75] Hsiao PY. Chain morphology, swelling exponent, persistence length, like-chargeattraction, and charge distribution around a chain in polyelectrolyte solutions:effects of salt concentration and ion size studied by molecular dynamicssimulations. Macromolecules 2006;39:7125–37.

[76] Hsiao PY. Linear polyelectrolytes in tetravalent salt solutions. J Chem Phys2006;124:044904.

[77] Hsiao PY, Luijten E. Salt-induced collapse and reexpansion of highly chargedflexible polyelectrolytes. Phys Rev Lett 2006;97:148301.

[78] Miyazawa N, Sakaue T, Yoshikawa K, Zana R. Rings-on-a-string chain structure inDNA. J Chem Phys 2005;122:044902.

[79] Ou ZY, Muthukumar M. Langevin dynamics of semiflexible polyelectrolytes: rod-toroid-globule-coil structures and counterion distribution. J Chem Phys2005;123:154902.

[80] Ermoshkin AV, de la Cruz MO. Polyelectrolytes in the presence of multivalentions: gelation versus segregation. Phys Rev Lett 2003;90:125504.

[81] Ermoshkin AV, de la Cruz MO. Gelation in strongly charged polyelectrolytes.J Polym Sci, B, Polym Phys 2004;42:766–76.


[82] Ermoshkin AV, Kudlay AN, de la Cruz MO. Thermoreversible crosslinking ofpolyelectrolyte chains. J Chem Phys 2004;120:11930–40.

[83] Kudlay A, Ermoshkin AV, de la Cruz MO. Complexation of oppositely chargedpolyelectrolytes: effect of ion pair formation. Macromolecules 2004;37:9231–41.

[84] Pamies R, Cifre JGH, De la Torre JG. Brownian dynamics simulation ofpolyelectrolyte dilute solutions: relaxation time and elongational flow. J PolymSci, B, Polym Phys 2007;45:714–22.

[85] Pamies R, Cifre JGH,De La Torre IG. Browniandynamics simulation of polyelectrolytedilute solutions under shear flow. J Polym Sci, B, Polym Phys 2007;45:1–9.

[86] Muthukumar M. Phase diagram of polyelectrolyte solutions: weak polymereffect. Macromolecules 2002;35:9142–5.

[87] Volk N, Vollmer D, Schmidt M, Oppermann W, Huber K. Conformation and phasediagrams of flexible polyelectrolytes. Adv Polym Sci 2004;166:29–65.

[88] Wang ZW, Rubinstein M. Regimes of conformational transitions of a diblockpolyampholyte. Macromolecules 2006;39:5897–912.

[89] Cheong DW, Panagiotopoulos AZ. Phase behaviour of polyampholyte chains fromgrand canonical Monte Carlo simulations. Mol Phys 2005;103:3031–44.

[90] Castelnovo M, Joanny JF. Phase diagram of diblock polyampholyte solutions.Macromolecules 2002;35:4531–8.

[91] Shusharina NP, Zhulina EB, Dobrynin AV, Rubinstein M. Scaling theory of diblockpolyampholyte solutions. Macromolecules 2005;38:8870–81.

[92] Akinchina A, Shusharina NP, Linse P. Diblock polyampholytes grafted ontospherical particles: Monte Carlo simulation and lattice mean-field theory.Langmuir 2004;20:10351–60.

[93] Akinchina A, Linse P. Diblock polyampholytes grafted onto spherical particles: effectof stiffness, charge density, and grafting density. Langmuir 2007;23:1465–72.

[94] Zhang R, Shklovskii BI. Phase diagram of solution of oppositely chargedpolyelectrolytes. Phys A-Stat Mech Its Applic 2005;352:216–38.

[95] Jeon J, Dobrynin AV. Molecular dynamics simulations of polyampholyte–polyelectrolyte complexes in solutions. Macromolecules 2005;38:5300–12.

[96] da Silva FLB, Lund M, Jonsson B, Akesson T. On the complexation of proteins andpolyelectrolytes. J Phys Chem B 2006;110:4459–64.

[97] Ulrich S, SeijoM, Stoll S. Themany facets of polyelectrolytes and oppositely chargedmacroions complex formation. Curr Opin Colloid Interface Sci 2006;11:268–72.

[98]••

Boas U, Heegaard PMH. Dendrimers in drug research. Chem Soc Rev2004;33:43–63. Excellent introduction to dendrimers.

[99]••

Boas U, Christensen JB, Heegaard PMH. Dendrimers: design, synthesis andchemical properties. J Mater Chem 2006;16:3786–98. Excellent introduction todendrimers.

[100] Lyulin SV, Darinskii AA, Lyulin AV, Michels MAJ. Computer simulation ofthe dynamics of neutral and charged dendrimers. Macromolecules 2004;37:4676–4685.

[101] Lyulin SV, Evers LJ, van der Schoot P, Darinskii AA, Lyulin AV,Michels MAJ. Effect ofsolvent quality and electrostatic interactions on size and structure of dendrimers.Brownian dynamics simulation and mean-field theory. Macromolecules2004;37:3049–63.

[102] Gurtovenko AA, Lyulin SV, Karttunen M, Vattulainen I. Molecular dynamics studyof charged dendrimers in salt-free solution: effect of counterions. J Chem Phys2006;124:094904.

[103] Majtyka M, Klos J. Monte Carlo simulations of a charged dendrimer with explicitcounterions and salt ions. Phys Chem Chem Phys 2007;9:2284–92.

[104] Lin Y, Liao Q, Jin XG. Molecular dynamics simulations of dendritic polyelectrolyteswith flexible spacers in salt free solution. J Phys Chem B 2007;111:5819–28.

[105]••

Ruhe J, Ballauff M, Biesalski M, Dziezok P, Grohn F, Johannsmann D, et al.Polyelectrolyte brushes. Adv Polym Sci 2004;165:79–150. Comprehensive over-view of the theoretical and experimental works on polyelectrolyte brushes.

[106] Lee I, Athey BD, Wetzel AW, Meixner W, Baker JR. Structural molecular dynamicsstudies on polyamidoamine dendrimers for a therapeutic application: effects ofpH and generation. Macromolecules 2002;35:4510–20.

[107] Maiti PK, Goddard WA. Solvent quality changes the structure of G8 PAMAMdendrimer, a disagreement with some experimental interpretations. J PhysChemis B 2006;110:25628–32.

[108] Harland RS, Prud RK, homme, Washington DC, editors. Polyelectrolyte gels:properties, preparation, and applications. American Chemical Society; 1998.

[109] Osada Y, Khokhlov AR, editors. Polymer gels and networks. Marcel Dekker; 2001.[110] Kwon HJ, Osada Y, Gong JP. Polyelectrolyte gels— fundamentals and applications.

Polym J 2006;38:1211–9.[111] Edgecombe S, Linse P. Monte Carlo simulation of polyelectrolyte gels: effects of

polydispersity and topological defects. Macromolecules 2007;40:3868–75.[112] Edgecombe S, Linse P. Monte Carlo simulations of cross-linked polyelectrolyte

gels with oppositely charged macroions. Langmuir 2006;22:3836–43.[113]•

Mann BA, Kremer K, Holm C. The swelling behavior of charged hydrogels.Macromol Symp 2006;237:90–107. Overview of the recent results on computersimulations of polyelectrolyte networks.

[114] Mann BA, Holm C, Kremer K. Swelling of polyelectrolyte networks. J Chem Phys2005;122:154903.

[115] Mann BA, Everaers R, Holm C, Kremer K. Scaling in polyelectrolyte networks.Europhys Lett 2004;67:786–92.

[116] Schneider S, Linse P. Discontinuous volume transitions in cross-linked polyelec-trolyte gels induced by short-range attractions and strong electrostatic coupling.Macromolecules 2004;37:3850–6.

[117] Schneider S, Linse P. Monte Carlo simulation of defect-free cross-linkedpolyelectrolyte gels. J Phys Chemis B 2003;107:8030–40.

[118] Schneider S, Linse P. Swelling of cross-linked polyelectrolyte gels. Eur Phys J, E2002;8:457–60.

[119] Yin DW, Yan QL, de Pablo JJ. Molecular dynamics simulation of discontinuousvolume phase transitions in highly-charged crosslinked polyelectrolyte networkswith explicit counterions in good solvent. J Chem Phys 2005;123:174909.

[120]•

Lu ZY, Hentschke R. Computer simulation study on the swelling of apolyelectrolyte gel by a Stockmayer solvent. Phys Rev E 2003;67:061807. Firstsimulation study of swelling of polyelectrolyte gels with explicit dipolar solvent.

[121]•

Winkler RG, Cherstvy AG. Critical adsorption of polyelectrolytes onto chargedspherical colloids. Phys Rev Lett 2006;96:066103. Exact analytical solution foradsorption of polyelectrolyte chain at charged spherical particle.

[122] Cherstvy AG, Winkler RG. Strong and weak adsorptions of polyelectrolyte chainsonto oppositely charged spheres. J Chem Phys 2006;125:064904.

[123] Stoll S, Chodanowski P. Polyelectrolyte adsorption on an oppositely chargedspherical particle. Chain rigidity effects. Macromolecules 2002;35:9556–62.

[124] Cherstvy AG, Winkler RG. Complexation of semiflexible chains with oppositelycharged cylinder. J Chem Phys 2004;120:9394–400.

[125]•

Carrillo JMY, Dobrynin AV. Molecular dynamics simulations of polyelectrolyteadsorption. Langmuir 2007;23:2472–82. Comprehensive molecular dynamicssimulation study of the effect of short-range and electrostatic interactions onpolyelectrolyte adsorption.

[126] Reddy G, Chang RW, Yethiraj A. Adsorption and dynamics of a singlepolyelectrolyte chain near a planar charged surface: molecular dynamicssimulations with explicit solvent. J Chem Theory Comput 2006;2:630–6.

[127] Pattanayek SK, Pereira GG. Shapes of strongly absorbed polyelectrolytes in poorsolvents. Phys Rev E 2007;75:051802.

[128] Borisov OV, Hakem F, Vilgis TA, Joanny JF, Johner A. Adsorption of hydrophobicpolyelectrolytes onto oppositely charged surfaces. Euro Phys J E 2001;6:37–47.

[129] Dobrynin AV, Rubinstein M. Adsorption of hydrophobic polyelectrolytes atoppositely charged surfaces. Macromolecules 2002;35:2754–68.

[130] Loverde SM, Solis FJ, de la Cruz MO. Charged particles on surfaces: coexistence ofdilute phases and periodic structures at interfaces. Phys Rev Lett 2007;98:237802.

[131] Loverde SM, Velichko YS, de la Cruz MO. Competing interactions in twodimensional Coulomb systems: surface charge heterogeneities in coassembledcationic–anionic incompatible mixtures. J Chem Phys 2006;124:144702.

[132] Velichko YS, de la Cruz MO. Pattern formation on the surface of cationic–anioniccylindrical aggregates. Phys Rev E 2005;72:041920.

[133]•

Solis FJ, Stupp SI, de la Cruz MO. Charge induced pattern formation on surfaces:Segregation in cylindrical micelles of cationic–anionic peptide-amphiphiles.J Chem Phys 2005;122:054905. Theoretical study of microphase separationtransition in adsorbed layers.

[134]•

Messina R. Effect of image forces on polyelectrolyte adsorption at a chargedsurface. Phys Rev E 2004;70:051802. First computer simulation study of the effectof image charges on polyelectrolyte adsorption.

[135] Messina R. Behavior of rodlike polyelectrolytes near an oppositely chargedsurface. J Chem Phys 2006;124:014705.

[136] Cheng CH, Lai PY. Scaling theory of polyelectrolyte adsorption on a repulsivecharged surface. Phys Rev E 2005;72:011808.

[137] Cheng CH. Boundary condition of polyelectrolyte adsorption. Phys Rev E2006;73:012801.

[138] Wang Q. Modeling layer-by-layer assembly of flexible polyelectrolytes (vol 110B,pg 5827, 2006). J Phys Chem B 2006;110:14524.

[139] Wang Q. Modeling layer-by-layer assembly of flexible polyelectrolytes. J PhysChem B 2006;110:5825–8.

[140] Shafir A, Andelman D. Polyelectrolyte multilayer formation: electrostatics andshort-range interactions. Euro Phys J E 2006;19:155–62.

[141] Park SY, Rubner MF, Mayes AM. Free energy model for layer-by-layer processingof polyelectrolyte multilayer films. Langmuir 2002;18:9600–4.

[142] Lefaux CJ, Zimberlin JA, Dobrynin AV, Mather PT. Polyelectrolyte spin assembly:influence of ionic strength on the growth ofmultilayered thin films. J Polym Sci, B,Polym Phys 2004;42:3654–66.

[143] Messina R. Polyelectrolyte multilayering on a charged sphere. Langmuir2003;19:4473–82.

[144] Messina R. Adsorption of oppositely charged polyelectrolytes onto a charged rod.J Chem Phys 2003;119:8133–9.

[145] Messina R, Holm C, Kremer K. Polyelectrolyte adsorption and multilayering oncharged colloidal particles. J Polym Sci, B, Polym Phys 2004;42:3557–70.

[146] Messina R. Polyelectrolyte multilayering on a charged planar surface. Macro-molecules 2004;37:621–9.

[147] Panchagnula V, Jeon J, Dobrynin AV. Molecular dynamics simulations ofelectrostatic layer-by-layer self-assembly. Phys Rev Lett 2004;93:037801.

[148] Panchagnula V, Jeon J, Rusling JF, Dobrynin AV. Molecular dynamics simulationsof polyelectrolyte multilayering on a charged particle. Langmuir 2005;21:1118–25.

[149] Patel PA, Jeon J, Mather PT, Dobrynin AV. Molecular dynamics simulations oflayer-by-layer assembly of polyelectrolytes at charged surfaces: effects of chaindegree of polymerization and fraction of charged monomers. Langmuir2005;21:6113–22.

[150] Patel PA, Jeon J, Mather PT, Dobrynin AV. Molecular dynamics simulations ofmultilayer polyelectrolyte films: effect of electrostatic and short-range interac-tions. Langmuir 2006;22:9994–10002.

[151] Jeon J, Panchagnula V, Pan J, Dobrynin AV. Molecular dynamics simulations ofmutilayer films of polyelectrolytes and nanoparticles. Langmuir 2006;22:4629–37.

[152] Abu-Sharkh B. Stability and structure of polyelectrolyte multilayers depositedfrom salt free solutions. J Chem Phys 2005;123:114907.

[153] Abu-Sharkh B. Structure and mechanism of the deposition of multilayers ofpolyelectrolytes and nanoparticles. Langmuir 2006;22:3028–34.

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Theory and simulations of charged polymers: From solution ......chain adopts rings on a string conformation of collapsed toroidal globules connected by the stretched strings of monomers