Current Opinion in Colloid & Interface Science

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1 Current Opinion in Colloid & Interface Science 13 (2008) Contents lists available at ScienceDirect Current Opinion in Colloid & Interface Science journal homepage: Theory and simulations of charged polymers: From solution properties to polymeric nanomaterials Andrey V. Dobrynin Polymer Program, Institute of Materials Science and Department of Physics, University of Connecticut, Storrs, Connecticut, , USA ARTICLE INFO ABSTRACT Article history: Received 18 March 2008 Accepted 24 March 2008 Available online 8 April 2008 Charged polymers are macromolecules with ionizable groups. These polymeric systems demonstrate unique properties that are qualitatively different from their neutral counterparts. In this review I survey the recent progress made in understanding properties of the solutions of charged polymers, swelling of polyelectrolyte gels, conformational transformations of charged dendrimers, complexation between charged macromolecules, adsorption of charged polymers at surfaces and interfaces, and multilayer assembly in ionic systems Elsevier Ltd. All rights reserved. 1. Introduction address: avd@ims.uconn.edu. Charged polymers are macromolecules with ionizable groups [1,2,3 6]. In polar solvents such as water, these groups can dissociate, leaving charges on polymer chains and releasing counterions into the solution. If these polymers carry only acidic or basic groups they are called polyelectrolytes. Examples of polyelectrolytes include polystyrene sulfonate, polyacrylic and polymethacrylic acids and their salts, DNA and other polyacids and polybases. Polyampholytes are charged macromolecules carrying both acidic and basic groups [4]. Thus, after ionization, there are positively and negatively charged groups on the polymer chain. Examples of polyampholytes include denatured proteins (for example gelatin), proteins in their native state (such as bovine serum albumin), and synthetic copolymers made of monomers with acidic and basic groups. At high charge asymmetry these polymers demonstrate polyelectrolyte-like behavior [4 ]. Electrostatic interactions between charged macromolecules control molecular processes in different areas of natural sciences ranging from materials science to biophysics [1,2,3 12,13,14,15,16].Forexample, the electrostatic attraction between oppositely charged macromolecules is a foundation of the electrostatic assembly technique that allows fabrication of multilayer films from synthetic polyelectrolytes, proteins, DNA, nanoparticles, etc. [9 11]. Electrostatic attractions between negatively charged DNA and net positively charged histones are responsible for the packaging of DNA into chromosomes [17,18].The complexation of DNA with positively charged polyelectrolytes, dendrimers, colloidal particles and liposomes facilitates the uptake of the DNA through 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 behind their assembly into compact bundle structures [16,21 26]. The electrostatically driven complexation between oppositely charged macromolecules 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 precipitate from the solution. Furthermore charged polymers are essential for development of lithium batteries and proton exchange membranes for fuel cell technology [29,30]. Over the years molecular simulations and theoretical models of charged polymeric systems have proven helpful in elucidating factors controlling their properties. In this review I will discuss static and dynamic properties of polyelectrolyte and polyampholyte solutions, swelling of polyelectrolyte gels, conformational transformations of charged dendrimers, complexation between charged macromolecules, adsorption of charged polymers at surfaces and interfaces, and multilayer assembly in polyelectrolyte systems. 2. Polyelectrolyte solutions 2.1. Salt-free polyelectrolyte solutions θ and good solvent conditions for the polymer backbone In dilute salt-free solutions, below the chain overlap concentration, the intrachain electrostatic interactions dominate over the interchain ones. The strength of the electrostatic interactions is controlled by the value of the Bjerrum length l B =e 2 /εk B T defined as the length scale at which the Coulomb interaction between two elementary charges e in a dielectric medium with the dielectric constant ε is equal to the thermal energy k B T. The deformation of the polyelectrolyte chain in θ or good solvent conditions for the polymer backbone is obtained by balancing the intrachain electrostatic interactions and chain's elasticity. This leads to elongation and nonuniform deformation of the polyelectrolyte chain with backbone tension decreasing towards the chain's ends, and faster than linear increase of the chain size R e with the chain degree of polymerization N, R e N[1n N] 1/3 [1]. The nonuniform stretching of /$ see front matter 2008 Elsevier Ltd. All rights reserved. doi: /j.cocis

2 A.V. Dobrynin / Current Opinion in Colloid & Interface Science 13 (2008) polyelectrolyte chain was directly tested in the molecular dynamics simulations by Liao et al. [31]. The chain size monotonically decreases with increasing polymer concentration in dilute solution regime. This decrease of the chain size is due to reduction of the net polymeric charge by condensed counterions (see discussion below), which weakens intrachain electrostatic repulsion. The crossover from dilute to semidilute solution regime occurs at polymer concentrations at which the distance between chains becomes comparable with the chain size. The simulations data for N-dependence of the overlap concentration c approach the scaling law, c N/R e 3 1/ (N 2 ln N) for large N [1]. It is important to point out that the dependence of the chain overlap concentration on the chain degree of polymerization is much stronger than one in solutions of neutral polymers, c 1/N 1/2 [32]. Thus, the crossover from dilute to semidilute polyelectrolyte solution regime occurs at much lower polymer concentrations than that in solution of neutral chains. In the semidilute solution regime, the electrostatic interactions are screened at the length scales on the order of the solution correlations length ξ the average mesh size of the semidilute polyelectrolyte solution [1]. The electrostatic interactions between charged monomers on the length scales smaller than the correlation length result in stretching of the chain sections within correlation blobs. At these length scales a section of polyelectrolyte chain with g ξ monomer is strongly stretched such that ξ g ξ. The interactions between correlation volumes can be ignored in the zeroth order approximation because the net polymeric charge within correlation volume is compensated by counterions. The concentration dependence of the number of monomers in a correlation volume ξ 3 can be obtained by imposing the closepacking condition for chain sections of size ξ, c g ξ /ξ 3. The correlation length of semidilute polyelectrolyte solution is estimated as ξ c 1/2. This is in agreement with the results of the molecular dynamics simulations that show the exponent for the concentration dependence of the correlation length being close to 1/2 [1,31]. Since at length scales larger than the correlation length ξ other chains and counterions screen electrostatic interactions, the statistics of the chain are those of a Gaussian chain with the effective persistence length on the order of the correlation length ξ. Thus, a chain in the semidilute salt-free polyelectrolyte solution is a random walk of correlation blobs with size R e N 1/2 c 1/4. Simulations by Liao et al. [31] show that the exponent for the concentration dependence of the chain size R e is N-dependent. It changes from the value for the short chains with N=25 to the value 0.22 for the chains with N=300. This result for the longest chains is close to the scaling model predictions, The osmotic pressure of the salt-free polyelectrolyte solutions is controlled by counterions [1,33,34 38]. The osmotic coefficient ϕ=π/(k B Tc) of flexible polyelectrolytes, defined as a ratio of the solution osmotic pressure π to the ideal osmotic pressure of all counterions at concentration c, k B Tc, exhibits nonmonotonic dependence on polymer concentration (see Fig. 1) [33]. It decreases with polymer concentration in dilute solutions. The osmotic coefficient increases with polymer concentration at high concentrations. The upturn in the osmotic coefficient occurs around the chain's overlap concentration c. The nonmonotonic behavior of the osmotic coefficient on polymer concentration in dilute solution regime is in qualitative agreement with the two-zone model. In the framework of this model the volume occupied by a chain is separated into two zones: the cylindrical zone, surrounding stretched polyelectrolyte chain, and a spherical zone, located outside the cylindrical region. According to this model the osmotic coefficient is a decreasing function of polymer concentration in dilute solutions whereas it is increasing function of polymer concentration in semidilute solutions. The crossover between these two regimes occurs around the overlap concentration c, at which the spherical zone of the two-zone model disappears and the two-zone model reduces to the classical Katchalsky's cell model (see for review [1]) Poor solvent conditions for the polymer backbone In poor solvents for the polymer backbone there is an effective attraction between monomers which causes neutral polymer chain without charged groups to collapse into dense spherical globules in order to maximize the number of favorable polymer polymer contacts and minimize the number of unfavorable polymer solvent contacts [32]. Molecular dynamics simulations of partially charged polyelectrolytes with explicit counterions in poor solvent conditions were performed by the Mainz group [39 42] and by Liao et al. [43]. These simulations have confirmed that polyelectrolyte chains at low polymer concentrations form necklaces of beads connected by strings (see Fig. 2). The necklace structure optimizes electrostatic repulsion between charged monomers and short-range monomer monomer attraction [1]. The effective charge of the chain decreases with increasing polymer concentration causing chain size to decrease by decreasing the length of strings and the number of beads per chain. At higher polymer concentrations polymer chains interpenetrate leading to a concentrated polyelectrolyte solution. In this range of polymer concentrations the chain size is observed to increase toward its Gaussian value. The nonmonotonic dependence of the chain size on polymer concentration is in qualitative agreement with theoretical predictions [1]. A comprehensive study of the effect of the short-range attractive and of the long-range electrostatic interactions on the necklace formation in dilute polyelectrolyte solutions was carried out by Limbach and Holm [41] and by Jeon and Dobrynin [44] (see Table 1). These studies have shown that polyelectrolyte chains adopt necklace-like conformation only in the narrow range of the interaction parameters. At finite polymer concentrations the necklace stability region is strongly influenced by the counterion condensation (see discussion below). A similar trend was observed in Monte Carlo simulations of titration of hydrophobic polyelectrolytes by Ulrich et al. [45]. Depending on the solvent quality for polymer backbone and ph pk 0 value a polyelectrolyte chain could be in five different conformational states: coil, collapsed spherical globule, necklace globule, sausagelike aggregate, and fully stretched chain. The effect of the charge annealing on the properties of the polyelectrolyte chain in poor solvent condition was also studied in ref [46]. Explicit solvent simulations of short polyelectrolyte chain in poor solvents showed that the necklace structure is not very stable because of the additional entropic solvent effect weakening effective monomer monomer attraction [47]. The results of simulations with explicit solvent are in good quantitative agreement with the predictions of the solvent- Fig. 1. Dependence of the osmotic coefficient of flexible polyelectrolytes on polymer concentrations for fully f =1 charged chains in a θ-solvent. The arrows show the overlap concentrations at various chain degrees of polymerizations N. Reproduced with permission from Liao, Q., Dobrynin, A. V., Rubinstein, M. Macromolecules 36, (2003). Copyright 2003, American Chemical Society.

3 378 A.V. Dobrynin / Current Opinion in Colloid & Interface Science 13 (2008) Fig. 2. Necklace conformation of a polyelectrolyte chain in a poor solvent. The negatively charged monomers of polyelectrolyte chain are colored in blue and neutral monomers are colored in gray. Positively charged counterions are shown in pink. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) accessible surface area (SASA) model for a polyelectrolyte chain in poor solvent [48]. The phase separation in poor solvent conditions for the polymer backbone was observed in MD simulations of Chang and Yethiraj [49]. They have found that polyelectrolyte solutions phase separate with increasing polymer concentration. Polyelectrolytes in the dense phase form spherical, cylindrical, and lamellar structures depending on polymer concentration. Note that the chain rigidity plays an important role in determining chain conformations in a poor solvent. A semiflexible polyelectrolyte chain adopts rings on a string conformation of collapsed toroidal globules connected by the stretched strings of monomers [50 53]. This structure optimizes the short-range monomer monomer attractive interactions, chain's bending energy and long-range electrostatic repulsion between charged groups Dynamics of polyelectrolyte solutions The 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 is proportional to the square root of polymer concentration in the wide concentration range, while the viscosity of uncharged polymers in a good solvent scales linearly with polymer concentration in a dilute solution regime or as c 1.3 in semidilute solution regime [32]. Furthermore, in this concentration range the chain relaxation time decreases with increasing polymer concentration indicating that the stress relaxation in polyelectrolyte solutions speeds up as they become denser. The physical origin of this unique behavior is the coupling of electrostatic interactions with conformational transformations of charged macromolecules. The unusually wide Fuoss law regime supports the conjecture that the entanglements (topological constraints created by surrounding chains) [32] in polyelectrolyte solutions begin to restrict chain motion deep in the semidilute solution regime. The crossover to entangled polyelectrolyte solutions occurs 3 4 decades above the overlap concentration c [1]. Note that in solutions of neutral polymers this crossover takes place closer to the chain overlap concentration at approximately 10c. Molecular dynamics simulations of dilute and semidilute polyelectrolyte solutions without hydrodynamic interactions were performed by Liao et al. [54] to study Rouse dynamics of polyelectrolytes. These simulations of the Rouse dynamics give qualitatively similar results to the experimentally observed dynamics of polyelectrolyte solutions. It was observed that the chain relaxation time depends nonmonotonically on polymer concentration (see Fig. 3). In dilute solutions this relaxation time exhibits very strong dependence on the chain degree of polymerization, τ ~ N 3. The chain relaxation time decreases with increasing polymer concentration of dilute solutions. This decrease in the chain relaxation time is due to chain contraction induced by counterion condensation. In the semidilute solution regime the chain relaxation time decreases with polymer concentration as c 1/2. In this concentration range the chain relaxation time follows the usual Rouse scaling dependence on the chain degree of polymerization, τ ~ N 2. At high polymer concentrations the chain relaxation time begins to increase with increasing polymer concentration. The crossover polymer concentration to the new scaling regime does not depend on the chain degree of polymerization indicating that the increase in the chain relaxation time is due to the increase of the effective monomeric friction coefficient. The analysis of the spectrum and of the relaxation times of Rouse modes confirms the existence of the single correlation length ξ, which describes both static and dynamic properties of semidilute solutions. These simulations also show that the unentangled semidilute solution regime is very wide. The longest chains with N=373 and 247 start to overlap at about 10 4 σ 3 and don't show any effect of entanglements up to the highest polymer concentration 0.15 σ 3. Thus, the unentangled semidilute solution regime spans three decades above the overlap concentration. This result is in agreement with the prediction of the scaling model of polyelectrolyte solutions [1]. The combined effect of the hydrodynamic and electrostatic interactions on the nonlinear shear viscosity of the dilute salt-free polyelectrolyte solutions was studied by Stoltz et al. [55]. Using Brownian dynamics simulations of the coarse-grained model of the polyelectrolyte chains with explicit counterions they have explored relationships between flow rate, value of the Bjerrum length and polymer concentration. Hydrodynamic interactions between beads were introduced into simulation scheme in the framework of the Rotne Prager Yamakawa tensor. It was found that the polyelectrolyte chains show shear thinning behavior at high flow rates which is independent on the value of the Bjerrum length. However, in the case of the low flow rates the solution viscosity increases monotonically with the value of the Bjerrum length. The origin of such unusual behavior is the electroviscous effect associated with the deformation of the counterion clouds surrounding polyelectrolyte chains by the external flow. Table 1 Table 1 Typical conformations of polyelectrolyte chain with degree of polymerization N=304 Typical conformations of polyelectrolyte chain with degree of polymerization N=304 and fraction of charged monomers f =1/3 and fraction of charged monomers f =1/3 The negativelycharged monomersof polyelectrolytechainare coloredin blue and neutralmonomers are The negatively charged monomers of polyelectrolyte chain are colored in blue and colored in gray. Positively charged counterions are shown in pink. Reproduced with permission from neutral monomers are colored in gray. Positively charged counterions are shown in Jeon, J., Dobrynin, A. V. Macromolecules 40, (2007). Copyright 2007, American pink. Reproduced with permission from Jeon, J., Dobrynin, A. V. Macromolecules 40, Chemical Society. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

4 A.V. Dobrynin / Current Opinion in Colloid & Interface Science 13 (2008) Counterion condensation and chain collapse Fig. 3. Concentration dependence of the chain relaxation time for the system of fully charged chains. N=25 (dark blue black squares); N=40 (gray circles); N=64 (pink triangles); 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, (2007). Copyright 2007, American Chemical Society. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) The idea of the counterion condensation was first introduced by Manning and Oosawa [1,56]. They have established that a charge of a rigid (rod-like) polyion depends on the fine interplay between the electrostatic attraction of a counterion to the polyelectrolyte backbone and configurational entropy loss due to counterion localization in the vicinity of the polymer chain. This theory showed that the linear charge density of a rod-like polyion cannot exceed a critical value which depends on the solution dielectric constant, temperature and counterion valence determining the value of the Bjerrum length l B. The recent studies of the counterion condensation show that the condensation is the second order phase transition associated with the formation of the counterion condensate in the vicinity of the polyion [57,58 60]. The situation is even more complicated in the case of flexible polyelectrolytes. In this case chain conformations are directly coupled with the intrachain electrostatic interactions that are controlled by the amount of the condensed counterions [1,3,43,61 66]. Thus, in addition to counterion configurational entropy and electrostatic interactions the chain's conformational free energy comes into play. These contributing factors have to be optimized simultaneously to determine fraction of the condensed counterions and the equilibrium chain size. In the case of the good or θ solvents for the polymer backbone the counterion condensation results in gradual decrease of the chain size with the increase of the polymer or salt concentrations [1]. A qualitatively different picture of counterion condensation is observed for polyelectrolytes in poor solvent conditions for the polymer backbone. In a poor solvent a polyelectrolyte chain forms a necklace globule of dense polymeric beads connected by strings of monomers. The counterion condensation on the necklace globule can occur in avalanche-like fashion. By increasing polymer concentration or decreasing temperature one can induce a spontaneous condensation of counterions inside beads of the necklace globule. This reduces the bead's charge and results in increase of the bead mass (size), which initiate further increase of the number of condensed counterions inside beads starting the avalanche-like counterion condensation process (see for review [1]). Note that in addition to the reduction of the net polymeric charge weakening intrachain electrostatic repulsion the condensed counterions can also induce effective attractive interactions between charged monomers [5,15,43,61 63]. In the ion binding and counterion adsorption models [62 64] condensed counterions form ionic pairs with oppositely charged ions on the polymer backbone. The formation of the ionic pairs leads to an additional dipole dipole and charge dipole attractive interactions [61 63]. These attractive interactions decrease the value of the effective second virial coefficient for monomer monomer interactions shifting the position of the θ- point. In the case of the strongly charged polyelectrolytes the shift of the θ-temperature could be significant and change the solvent quality for the polymer backbone to poor solvent conditions as the number of condensed counterions increases. This can result in a chain collapse and completely alter scenario of the counterion condensation (see discussion above). The analysis of the effect of the counterion condensation on conformations of a polyelectrolyte chain was done by Schiessel and Pincus [61], and by Schiessel [62] in the framework of the scaling approach, and by Muthukumar [63] in the framework of the variational approach. These theories predict nonmonotonic dependence of the chain size on the solution dielectric constant ε and solution temperature T (solution Bjerrum length l B ~ 1/(εT)). The chain size is first increases with increasing the value of the Bjerrum length then begin to decrease as the Bjerrum length exceeds the crossover value. This nonmonotonic dependence of the chain size is the manifestation of the two-fold role of the electrostatic interactions. At low values of the Bjerrum length the intrachain electrostatic repulsion controls the chain size. These interactions become stronger with increasing the value of the Bjerrum and force polyelectrolyte chain to expand. At large values of the Bjerrum length the condensed counterions reduce net polymeric charge weakening the intrachain electrostatic repulsion, which together with the dipole dipole and charge dipole attractive interactions induce chain contraction. However, computer simulations of polyelectrolyte solutions show that the condensed counterions are not permanently attached to oppositely charged groups on the polymer backbone as assumed by the ion binding and ion adsorption models but rather localized near the 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 also lead to effective attractive interactions [5,15]. These interactions are due to heterogeneous distribution of the charge density along the polymer backbone. In the case of weak electrostatic attraction the origin of these interactions is similar to the fluctuation-induced attraction in two-component plasma and is related to the local charge density fluctuations [15]. In the opposite limit of strong electrostatic interactions the effect is due to correlation-induced attraction between the counterions and the oppositely charged polymer backbone 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 is due to the spatial distribution (spatial correlations) of cations and anions over the lattice sites, even though the net charge of the crystal is zero. The crystal will remain stable even if it carries a small nonzero charge because of the large lattice (correlation) energy. The effect of the fluctuation/correlation-induced attractive interactions on the conformations of a polyelectrolyte chain was studied theoretically [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 chain collapse. In particular, in the case of the poor solvent conditions for the polymer backbone these studies established existence of the two different mechanisms that could lead to formation of the necklace globule [41,43,44,67]. For the values of the Bjerrum length l B =1σ and 2σ the necklace structure appears as a result of competition between short-range monomer monomer attractive interactions and electrostatic repulsion between uncompensated charges [44,67]. However, for the value of the Bjerrum length l B N3σ the necklace structure is

5 380 A.V. Dobrynin / Current Opinion in Colloid & Interface Science 13 (2008) controlled by counterion condensation and is due to the optimization of the correlation-induced attraction between charged monomers and condensed counterions, and electrostatic repulsion between uncompensated charges on the polymer backbone. Note, that counterion condensation on the polymer backbone can also lead to chain collapse and necklace formation even in the good or theta solvent conditions for the polymer backbone. The counterion condensation also influences the dynamic properties of polyelectrolytes [68]. In dilute solutions the chain's translational diffusion coefficient monotonically decreases with the value of the Bjerrum length. However, the chain's relaxation time shows nonmonotonic dependence on the Bjerrum length. It first increases with increasing the value of the Bjerrum length then it begins to decrease. The decrease in the chain's relaxation time is due to counterion condensation Salt effects The electrostatic interactions between charged monomers in solutions with finite salt concentrations are screened by salt ions and their strength decreases exponentially with the distance between charges [1,3,37]. However, the charges still interact through the unscreened Coulomb potential at distances much smaller than the Debye screening length r D = (8πl B c s ) 1/2,wherec s is the salt concentration. Screening of the intrachain electrostatic interactions increases chain flexibility by reducing its persistence length and leads to chain contraction. For semiflexible and strongly charged flexible polyelectrolytes the electrostatic part of the chain persistence length is proportional to the Debye screening length. [69,70] This result was obtained by evaluating the bending angle fluctuations and in the framework of the Gaussian variational principle. A polyelectrolyte chain with linear dependence of the electrostatic persistence length on the Debye screening length has lower free energy than that of a chain with quadratic dependence of the electrostatic persistence so-called Odijk Skolnick Fixman electrostatic persistence length (see for overview [1,71]). If multivalent ions are added to the solution, the strong electrostatic attraction between multivalent ions and charged monomers favors condensation of the multivalent ions on the polymer backbone. Note that each multivalent ion with valence Z can neutralize Z charged monomers, which makes its condensation Z-times more efficient than condensation of monovalent ions. Using lattice Monte Carlo simulations Klos and Pakula [72 74] have studied the effect of the salt concentration on conformations of single polyelectrolyte chain. Their simulations have shown that the presence of multivalent salt ions results in chain collapse accompanied by reduction in the system energy and effective charge of the polyelectrolyte chain. This indicates that the chain collapse is driven by multivalent ion condensation on the polymer backbone. The sharp reduction in the chain size was observed even at relatively low salt concentrations. In the mixtures of monovalent and tetravalent salts the polyelectrolyte chain size depends nonmonotonically on salt concentration [75,76]. With increasing salt concentration in dilute polyelectrolyte solutions, the chain size first decreases than it increases again. The size of the polyelectrolyte chain reaches a minimum at salt concentration for which the total charge of the tetravalent counterions almost exactly neutralizes the net polymeric charge. The degree of the chain collapse and swelling increases with the ion size. This interesting behavior is due to the exchange of the monovalent ions by tetravalent counterions that condense on the polymer backbone. At high salt concentrations, salt ions can form layered structure around a polyelectrolyte and locally overcompensate the effective polymeric charge. This overcompensation of the polyelectrolyte charge leads to the chain swelling. Similar reentrant conformational transition was also observed in the Monte Carlo simulations by Hsiao and Luijten [77]. A semiflexible polyelectrolyte chain could be either in the elongated or in compact toroid conformation when multivalent ions are added. The transition between different chain conformations is the first order transition [5,15,78,79]. The presence of the multivalent ions can lead to reversible gelation of the polyelectrolyte chains [80,81]. In this case the multivalent ions play role of the cross-linking agents bridging polyelectrolyte chains together. Ermoshkin and de la Cruz [80,81] have applied combination of the Flory approach and a modified random-phase approximation approach to describe associations in polyelectrolyte solutions. The gelation threshold was obtained as a function of the ion valence, polymer concentration and strength of the electrostatic interactions. It also was shown that the gelation can only occur in semidilute polyelectrolyte solutions when the interchain electrostatic repulsion is sufficiently screened to allow chain aggregation. The recent studies have extended this approach to associating polyelectrolyte solutions [82,83]. In salt solutions chain size is very sensitive to the external perturbations. Using a coarse-grained model of the polyelectrolyte chain along with the Debye Hückel approximation accounting for electrostatic interactions between charged groups, Pamies et al. [84,85] have performed Brownian dynamics simulations with hydrodynamic interactions to study deformation of polyelectrolyte chains in dilute solutions under shear and elongational flow. Monitoring the chain size and intrinsic viscosity they observed abrupt conformational transition of polyelectrolyte molecules in shear γ c and elongational ε c flow and determined the critical values of the shear and elongational rates necessary for this transition to occur. The critical values ε c and γ c decrease with solution ionic strength. This is a result of the screening of the intrachain electrostatic interactions with increasing salt concentration weakening chain tension and thus requiring the weaker external force for chain deformation. For high values of the ionic strengths, the critical value of the elongational rate ε c scales with chain degree of polymerization as, ε c N 3/2, which is close to that for neutral polymers. In poor solvent conditions for the polymer backbone addition of salt to polyelectrolyte solutions promotes phase separation. At high salt concentrations, the electrostatic interactions between charged monomers are exponentially screened leading to the renormalization of the second virial coefficient between monomers and changing the effective solvent quality for the polymer chain. For a recent review of the results of the phase separation in polyelectrolyte solutions see refs [1,86,87]. 3. Polyampholyte solutions Properties of polyampholyte chains in solutions depend on the average composition the fractions of positively charged f + and negatively charged f monomers, and on the distribution of charged monomers along the polymer backbone [4 ]. This distribution of charged monomers is fixed during the polymerization reaction and is known as a quenched charge distribution. A charge-balanced (net neutral or symmetric) polyampholyte with degree of polymerization N has equal numbers of positively f + N and negatively f N charged groups. The net charge or charge asymmetry ΔfN of a polyampholyte is equal to the absolute value of the difference between numbers of positive f + N and of negative f N charges (ΔfN= f + f N) [4]. The effect of the charge distribution along the polymer backbone on conformations of single polyampholyte chain was studied extensively over the years (see for review [4]). For example, a symmetric polyampholyte chain with equal numbers of positively and negatively charged monomers (ΔfN=0) and random charge distribution along the polymer backbone collapses forming a dense globule. The collapse of the symmetric polyampholytes is due to correlation-induced-attraction between charged monomers. The charge-asymmetric polyampholytes with charge asymmetry ΔfN smaller than (fn) 1/2 behaves similar to polyampholytes with zero net charge. The strongly charged polyampholytes with the charge asymmetry form a necklace-like globule of the dense almost neutral

6 A.V. Dobrynin / Current Opinion in Colloid & Interface Science 13 (2008) beads connected by charged strings. The ensemble-average chain size of randomly charged polyampholytes is dominated by elongated chain sections (strings) and increases with the chain degree of polymerization as N 1/2. The details of the collapse transitions in the diblock polyampholytes were recently studied by Wang and Rubinstein [88]. Using combination of the molecular dynamics simulations and scaling analysis they have found three different conformational regimes in collapsed diblock polyampholytes with increasing the strength of the electrostatic interactions controlled by the value of the Bjerrum length l B. In the first (folding) regime the electrostatic attraction between oppositely charged blocks force chain to fold through the overlap of the two blocks while each block is slightly stretched by intrablock electrostatic repulsion. The second (scramble egg) regime is the classical globule regime where the chain collapse is driven by the fluctuation-induced attraction between charged monomers. The structure of the collapsed chain can be represented as densely packing of the charged chain sections (correlation blobs). The third (strong association or ion binding regime) starts with direct binding of oppositely charged monomers formation of the dipoles, followed by the cascade of multipole (quadrupole, hexapole, etc) formation with increasing the value of the Bjerrum length. In polyampholyte solutions electrostatic interactions between oppositely charged monomers promote phase separation [89]. The condition for phase separation depends on the chain degree of polymerization, charge sequence along the polymer backbone, strength of the electrostatic interactions and solution temperature. The critical temperatures of the diblock polyampholyte solutions are much higher and increase more quickly with increasing chain degree of polymerization than the critical temperatures of solution of random polyampholyte. In dilute phase diblock polyampholytes form larger aggregates than randomly charged chains. A theoretical model of aggregation in solutions of diblock polyampholytes was proposed by Castelnovo and Joanny [90]. They have studied the solution phase diagram at high ionic strengths and determined the phase boundary and the critical micelle concentration as a function of polymer concentration and net polymeric charge ΔfN. The size of the micelle and its aggregation number was found to be a strong function of the fraction of charged monomers on the polymer backbone. The micellar core structure in this model was assumed to be similar to the structure of the dense aggregate formed by oppositely charged polyelectrolyte chains so that the net charge of the core was very close to zero. The corona of the block polyampholyte micelle was formed by charge-unbalanced section of blocks with higher net charge. In the framework of the scaling approach Shusharina et al. [91] developed a theory of salt-free solutions of diblock polyampholytes. A charge-asymmetric block polyampholyte (for example, diblock polyampholyte with longer negatively charged block) forms a tadpole with an almost neutral globular head and a negatively charged polyelectrolyte tail. With increasing polymer concentration these tadpoles aggregate into micelles. There are two different groups of chains in these aggregates. Chains in one group are completely confined inside the micellar core, while chains belonging to the other group have their entire negatively charged block expelled into the corona. This micellar structure has lower free energy than micellar structure considered by Caltelnovo and Joanny [90] and corresponds to minimum of the aggregate free energy. It is interesting to point out that similar disproportionation of polyampholyte chains has been observed in the lattice mean-field theory studies and Monte Carlo simulations of polyampholyte brushes in both planar and spherical geometries [92,93]. The disproportionation of chains in the systems containing oppositely charged polyelectrolytes has been predicted by Zhang and Shklovskii [94]. Molecular dynamics simulations of complexation between polyampholyte and polyelectrolyte chains in solutions were performed by Jeon and Dobrynin [95]. They have established that the complexation between polyampholyte and polyelectrolyte chains is due to polarization-induced attractive interactions between molecules. The charge sequence along the polyampholyte backbone has a profound effect on the resulting complex structure. For example, a diblock polyampholyte with a fraction of charged monomers f = 1 and degree of polymerization N=32 could form a two-arm and three-arm star-like complex with a fully charged polyelectrolyte chain of the same degree of polymerization. The complex structure changes with increasing salt concentration confirming the dominant role of the electrostatic interactions in chain's association. The complex structure depends on polymer concentrations. Small aggregates consisting of two or three polymer chains dominate at low polymer concentrations. As polymer concentration increases, micellar aggregates (see Fig. 4 a) coexist together with small aggregates. The micellar aggregates persist until the system crosses over to the semidilute solution regime where all chains form interconnected network of micelles (see Fig. 4 b). Qualitatively different aggregate structures were observed for random polyampholytes. The structure of multichain polyampholyte polyelectrolyte complexes in this case resembles that of branched polymers with polyampholytes binding polyelectrolyte chains together (see Fig. 4 c). Complexation between a polyampholyte and a polyelectrolyte chain can proceed through the release of counterions condensed on the polyelectrolyte backbone [67]. These counterions are substituted by the positively charged monomers of polyampholyte chain. The complex structure shows strong correlation with the charge sequence along the polymer backbone. In the case of random polyampholyte the complex has a flower-like structure of the dense core surrounded by loops (see Fig. 5 a). The core of the aggregate contains more positively charged polyampholyte chain sections and whole polyelectrolyte chain. The more negatively charge section of the polyampholyte form loops surrounding the core of the aggregate. The diblock polyampholyte has tadpole like structure with the tail of the tadpole containing part of the negatively charged block (see Fig. 5 b). The remaining part of the negatively charged block is wrapped around the head of the tadpole. In both systems there are charged density oscillations inside the core. In the case of random polyampholytes the center of the core has excess of positively charged monomers which is surrounded by the layer of the negatively charged ones. The core of the complex formed by diblock polyampholyte has four charge alternating layers with excess of the positively charged monomers in the core center. These complex structures persisted throughout the entire studied salt concentration range and show no qualitative changes with varying the strength of the electrostatic interactions (the value of the Bjerrum length). The more detailed discussion of the polyampholyte polyelectrolyte complexes can be found in refs [4,8,96,97]. 4. Charged dendrimers Dedrimers are regularly branched molecules with tree-like structure that can be made by the cascade chemical synthesis starting from the core and attaching new branching units to the terminal groups [98,99]. The chemical structure of the dendrimers is characterized by the generation number (the number of the focal (branching) points going from the core towards the dendrimer surface) and branching functionality of the monomeric units. For example, a dendrimer having four focal points when going from the center to the periphery is denoted as the 4th generation dendrimer and abbreviated as G4. The core part of the dendrimer is sometimes denoted generation zero, or G0. The charged groups can functionalize the terminal groups of the last generation or be a part of the monomeric units used in dendrimer synthesis. The unique structure of the dendritic macromolecules may be used to carry low molecular substances, e.g. drugs, or may be useful to create altered properties of chromophores or fluorophores, etc.

7 382 A.V. Dobrynin / Current Opinion in Colloid & Interface Science 13 (2008) and in some cases quantitative agreement with simple Flory-like model of the dendrimer molecule. However, the results of these simulations have to be taken with caution since they do not take into account the counterion condensation. Simulations with the explicit counterions show that counterion condensation has a tremendous effect on conformations of charged dendrimers. Similar to the case of the linear chains, counterion condensation occurs with increasing value of the Bjerrum length and is accompanied by subsequent swelling and collapse of the dendrimer molecules [102,103]. The existence of such nonmonotonic conformational transformations is due to fine interplay between growing repulsions between charged end-beads and counterion condensation tendency because of attractive interactions between counterions and the terminal groups. Furthermore, counterion radial distribution functions indicated that counterions occupy not only the dendrimer periphery but also its interior. Molecular dynamics simulations of dilute salt-free solutions of weakly charged dendrimers with spacers of different lengths were performed by Lin et al. [104]. In dilute solutions the equilibrium Fig. 4. Complexes formed by polyelectrolyte chains with diblock (a, b) and random (c) polyampholytes. Polyelectrolyte chains are shown in green, positively charged monomers on the polyampholyte backbone are shown in red and negatively charged ones are shown in blue. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) The static and dynamic properties of the dilute dendrimer solutions were studied by Lyulin et al. [100,101]. There was no explicit counterions in these simulations and electrostatic interactions between charged groups were taken into account on the level of the screened Coulomb potential. The hydrodynamic interactions were included into simulation scheme through the Rotne Prager Yamakawa tensor. According to these simulations, the dendrimer motion may be divided into three main types: (1) translational and orientational 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 few monomer units. The dendrimer diffusion coefficient D scales with the radius of gyration R g as D ~ R 0.8 g for both charged and neutral dendrimers with different number of generations. The results of the molecular dynamics simulations are in reasonably good qualitative Fig. 5. Snapshots of complexes formed by polyelectrolyte with random polyampholyte (a) and diblock polyampholyte (b) in a salt-free solution. The positively charged monomers of a polyampholyte chain are shown in red and negatively charged ones are shown in blue. The negatively charged monomers of polyelectrolyte chain are colored in green and neutral monomers are colored in gray. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

8 A.V. Dobrynin / Current Opinion in Colloid & Interface Science 13 (2008) conformation of the charged dendrimers is determined by balancing the dendrimer elastic free energy either with the electrostatic repulsion between charged groups or with the configurational free energy of the localized within dendrimer counterions. The crossover between two regimes (so-called electrostatic and osmotic regimes) takes place at critical generation number which value depends on the fraction of the ionized groups. Note that electrostatic and osmotic regimes are also observed in polyelectrolyte brushes [105 ]. The concentration dependence of the osmotic coefficient of the solution of charged dendrimers is similar to that reported for polyelectrolyte solutions [1 ]. Atomistic MD simulations of the PAMAM dendrimers at several ph conditions, with explicit water molecules were performed by Lee at al. [106] and by Maiti and Goddard [107]. They obtained structural characteristics of charged dendrimers as function of the solution ph and established the role of the water molecules in determining the dendrimer conformations and monomer density distribution. These simulations showed that the dendrimer size decreases with increasing solution ph. 5. Polyelectrolyte gels Networks made from polyelectrolyte chains are quite common in both nature and industry [ ]. Polyelectrolyte gels are capable of swelling to much greater extents than their uncharged counterparts because of high osmotic pressure due to dissociated counterions. Because of such unique properties polyelectrolyte gels are used as superabsorbent materials, as ion-exchange resins, and as the carrier for novel drug delivery targeting specific organs. It was long recognized that the swelling of polyelectrolyte gels is determined by a balance between the osmotic pressure of free ions acting to swell the gel and the elasticity of the gel that restrict the swelling [108,109].The osmotic part has long been understood in terms of the Donnan equilibrium. Unfortunately only until recently the progress in the computational resources have allowed computer simulations of the polyelectrolyte networks [111,112,113, ]. The polyelectrolyte networks in such simulations usually had an ideal regular structure. The simulations were performed at various network volume fractions, chain's degree of polymerizations, crosslink density, values of the Bjerrum length and solvent quality for the polymer backbone. In a series of papers the Linse group has used Monte Carlo simulations to model salt-free regular polyelectrolyte networks with diamond-like topology and explicit counterions [ ]. These simulations have shown that the gel volume increases with increasing the fraction of charged monomer on the polymeric strands forming a gel. It also increases with decreasing the cross-linking density and increasing chain stiffness. This dependence of the gel swelling on the network parameters is in line with the notion that polyelectrolyte gel swelling is controlled by the osmotic pressure of counterions and by the elasticity of the polymeric strands forming a gel. The increase in the fraction of the charged monomers on polymer backbones leads to increase of the osmotic pressure of the counterions which forces gel to increase its volume. The decrease in the number of crosslinks (increase in the degree of polymerization of the polymeric stands between crosslinks) and increase of the chain stiffness (increase of the chain's Kuhn length) both results in decrease of the gel shear modulus, which also promotes network swelling. The effect of the counterion osmotic pressure and chain's elasticity on the gel swelling has also been studied by the Mainz group in molecular dynamics simulations of the salt-free polyelectrolyte networks [113,114,115]. These simulations established that only free (osmotically active) counterions contribute to the gel swelling (see Fig. 6). This was verified for the polyelectrolyte networks in the good and almost theta solvent conditions for the polymer backbone. It is important to point out that the swollen polyelectrolyte gels deform affinely. The multivalent (divalent, trivalent) ions have a strong effect on the gel structure [119]. The exchange of the monovalent ions to multivalent ones leads to collapse of the polyelectrolyte network, which is manifested in discontinuous volume change of the gel. The origin of this collapse transition is similar to the collapse of a polyelectrolyte chain in the presence of the multivalent ions and is due to the correlation-induced attractive interactions between multivalent ions and charged monomers on the polymer strands forming a gel. Note that similar effect can be achieved in simulations with monovalent ions by increasing the value of the Bjerrum length. Experimentally it corresponds to immersing polyelectrolyte networks in low dielectric solvents, l B ~ 1/ε. The network collapse was also observed when monovalent counterions were substituted by macroions [112]. The simulations of polyelectrolyte networks with explicit counterions and explicit solvent particles were reported by Lu and Hentschke [120 ]. Using a two-box-particle transfer molecular dynamics simulation method they modeled an extremely highly crosslinked network in equilibrium with a dipolar Stockmayer solvent. In these simulations each bead of the polyelectrolyte strand forming a network was carrying a partial charge eq and each corresponding neutralizing counterion had partial charge eq. The swelling ratio of the polyelectrolyte network has a maximum as a function of the partial charge eq. This nonmonotonic dependence of the swelling ratio was attributed to a competition between electrostatic repulsion and the network conformational entropy. The maximum becomes less pronounced with increasing the dipole moment of the Stockmayer fluid. Edgecombe and Linse have studied the role of the chain's polydispersity and topological network defects on the polyelectrolyte gel swelling [111]. They found that polydisperse networks swell less than regular gels. In polydisperse gels the short chains are more stretched and the long ones are less stretched as compared to polyelectrolyte networks made of monodisperse chains. Thus, the deformation of the short polyelectrolyte chains controls swelling of the polydisperse gels. The normal stress in uniaxial stretching simulations for neutral and polyelectrolyte gels follows roughly exponential dependence on the deformation ratio λ. The agreement between simulation data and theoretical models of uniaxial gel deformation was better for the non-gaussian network theory than for the Gaussian theory. However, both models deviate significantly from simulation data in the limit of large network deformations. In the poor solvent conditions for the polymer backbone polymeric strands connecting the nodes of the polyelectrolyte network can have necklace-like conformations similar to those observed in polyelectrolyte solutions. The detailed study of the network structure in the poor solvent conditions for the polymer backbone as function of the Fig. 6. The equilibrium swelling behavior of salt-free polyelectrolyte gels in the good solvent and close to the θ solvent conditions for the polymer backbone. The data points follow the scaling expression R bf 1-v eff N m, where the exponent v is equal to 1/2 in θ solvent and 0.6 in good solvent. f eff is the effective fraction of the charged groups on the polymeric strand with N m monomers. Reproduced with permission from Mann. B. A., Kremer, K., Holm, C. Macromol. Symp. 237, (2006).

9 384 A.V. Dobrynin / Current Opinion in Colloid & Interface Science 13 (2008) fraction of charged monomers f and the value of the Bjerrum length was performed by Mann et al. [113]. They observed the formation of the following chain conformations within polyelectrolyte network: collapsed conformations for small charge fractions or very strong electrostatic interactions (large values of the Bjerrum length); necklace structure for moderate to high charge fractions and not very strong electrostatic coupling; stretched structures for large fractions of charged monomers f and moderate values of the Bjerrum lengths l B ; and finally the sausage -like conformation for large Bjerrum lengths. Note that the diagram of states for polyelectrolyte networks has regimes similar to those observed at the diagram of state for the single polyelectrolyte chain. However, the boundaries between different conformational regimes are shifted due to additional effect of the chain connectivity into the network on the backbone conformations. 6. Charged polymers at surfaces and interfaces 6.1. Polyelectrolyte adsorption Adsorption of charged polymers on charged surfaces and interfaces is a classical problem of polymer physics and has been under extensive theoretical and experimental studies for the last four decades (see for historical overview [1]). Here I present only recent developments in this area of polymer science. An exact analytical solution for adsorption transition of a flexible polyelectrolyte chain at oppositely charged spherical particle of the radius 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 Hulthen potential. It turns out that the difference between two potentials is small for the wide interval of salt concentrations and particle sizes. The exact analytical solution of the differential equation for the probability density of a flexible polymer chain in the Hulthen potential is well known and is given by the Gauss hypergeometric function. This solution exactly reproduces results for polyelectrolyte adsorption onto planar surface in the limit of the vanishing particle curvature (R r D ). In this limit the critical charge density σ c, determining the adsorption threshold, shows strong dependence on the Debye screening length σ c ~1/r D 3. In the opposite limit of the small particles (R r D ) the critical particle surface charge density σ c is inversely proportional to the solution Debye screening length, σ c ~1/r D. These predictions are in agreement with the Monte Carlo simulation results by Chodanowski and Stoll [123]. Winkler and Cherstvy have extended their approach to study the adsorption of polyelectrolytes at oppositely charged cylinders [124]. Molecular dynamics simulations of polyelectrolyte adsorption at oppositely charged surfaces from dilute polyelectrolyte solutions were performed by Carrillo and Dobrynin [125 ]. These simulations have studied the effects of the surface charge density, surface charge distribution, solvent quality for the polymer backbone, strength of the short-range interactions between polymers and substrates on the polymer surface coverage and the thickness of the adsorbed layer. The solvent quality for the polymer backbone and polymer affinity to the surface manifests itself in qualitatively different dependences of the thickness of the adsorbed layer on the surface charge density. In the case of adsorption of hydrophobic polyelectrolytes at hydrophilic surfaces, the chain thickness decreases with increasing surface charge density. This decrease in the chain thickness is a result of the optimization of the electrostatic attraction between ionized groups on the polymer backbone to the oppositely charged substrate and chain surface energy. Furthermore, these polyelectrolytes form multichain aggregates with increasing surface charge density. Hydrophobic polyelectrolytes wet a hydrophobic surface forming a monolayer maximizing the number of favorable polymer surface contacts. The thickness of the adsorbed layer stays almost constant at low surface charge densities and is mainly controlled by the strength of the short-range polymer surface interactions. At high surface charge densities, polymers completely cover the surface and the thickness of the adsorbed layer increases linearly with the surface charge density. In this range of the surface charge densities the electrostatic attraction between polyelectrolyte chains and the oppositely charged substrate start to play a dominant role in controlling the layer thickness. Adsorption of hydrophilic polymers at hydrophilic surfaces was accompanied by a nonmonotonic dependence of the chain thickness on the surface charge density. This nonmonotonicity is a result of the two different mechanisms responsible for the chain thickness stabilization. At low surface charge densities, the chain thickness is determined by the balance of the energy gain due to electrostatic attraction and confinement entropy loss due to chain localization. In this regime the chain thickness decreases with increasing the surface charge density. At higher surface charge densities, the thickness of the adsorbed layer is determined by the balance between electrostatic attraction of the charged monomers to the substrate and short-range monomer monomer repulsion resulting in an increase in the layer thickness. The results of these simulations are in qualitative agreement with the prediction of the scaling models of polyelectrolyte adsorption (see for review [1]). The effect of solvent quality on the behavior of a polyelectrolyte chain near a charged surface was also studied by Reddy et al. [126]. Using molecular dynamics simulation with explicit solvent they have found that for a given solvent quality, increasing the surface charge density causes the chain to adsorb flat on the charged surface. The shape of the adsorbed polyelectrolyte chain is a complicated function of the surface charge density, polymer solvent, polymer polymer and polymer surface interactions. The surprising result of these simulations was that the rotational and translational dynamics of the polyion become faster with decreasing the solvent quality. This was related to the decrease in the chain size and decrease in the solvent density at the surface as the solvent quality is decreased, both of which tend to increase lateral (along the surface) chain diffusion coefficient. The conformational transformations in adsorbed polyelectrolyte chain in a poor solvent for the polymer backbone were studied by Pattanayek1 and Pereira [127]. It was shown by the free energy minimization and by the SCF calculations that an adsorbed polyelectrolyte chain in the poor solvent conditions for the polymer backbone can have an elliptical shape or a pearl-necklace shape with deformed elliptical beads depending on the surface charge density and solvent quality for the polymer backbone. Adsorption of necklaces was also studied in refs [128,129]. Binary mixtures of oppositely charged molecules adsorbed onto a surface (confined to a plane) may exhibit microphase separation [ ,133]. The origin of the microphase separation is optimization of the short-range interactions, which favor macroscopic segregation, and the long-range electrostatic interactions, favoring component intermixing. The symmetry of the appearing domain structure (formation of the lamellar or hexagonal patterns) depends on the charge stoichiometry of the mixture. The typical size of the domains scales with the parameters of the system as L (ε int /l B f 3/2 ) 1/2 where ε int is the cohesive energy and f is the average charge density. In many experimental situations, such as adsorption of polyelectrolyte chains from water onto clay, polymer latex particles or at the water/air interface, the dielectric constant of the solvent ε is larger than that of the surface ε s The presence of the charge in the medium with dielectric constant ε near the surface with the dielectric constant ε s causes polarization of both media. The result is the appearance of the image charge at the symmetric positions with respect to the dielectric boundary. Monte Carlo simulations of the effect of the dielectric boundary on the adsorption of strongly charged polyelectrolytes at oppositely charged planar surface were performed by Messina [134,135]. These

10 A.V. Dobrynin / Current Opinion in Colloid & Interface Science 13 (2008) linearly with the surface charge density. A detailed analysis of the boundary conditions for the polyelectrolyte adsorption at dielectric interface was done by Cheng [137] Layer-by-Layer assembly Fig. 7. Dependence of the polymer surface coverage on the number of deposition steps for multiplayer formation from dilute polyelectrolyte solutions. Reproduced with permission from Patel, P., Jeon, J., Mather, P. T., Dobrynin, A. V. Langmuir 21, (2005). Copyright 2005, American Chemical Society. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) simulations have shown that image forces appearing due to the dielectric discontinuity at the adsorbing substrate lead to the decrease in polymer surface coverage which precludes the surface overcharging by adsorbed polyelectrolytes. Cheng and Lai [136] studied a single chain adsorption at the charged substrate with high dielectric constant. In the framework of the ground state dominance approximation they found that adsorption at low ionic strengths is the first order transition with the monomer density at the surface scaling Layer-by-layer deposition of charged molecules provides a powerful tool for fabrication of multicomponent coatings with unique functional properties (see for review [9 11]). This technique is based on the electrostatic attraction between oppositely charged molecules, which results in stable multilayer structures. The key to a successful deposition of multilayer assemblies in a layer-by-layer fashion is the inversion and subsequent reconstruction of the surface properties. For example, this can be achieved by immersing the substrate into a dilute aqueous solution of anionic (or cationic) polyelectrolytes for a period of time required for the adsorption of layer of given thickness after which the substrate is rinsed. The rinsing step is necessary to remove the polymers that are not tightly adsorbed to the substrate. During the next step a substrate covered with adsorbed polyelectrolytes is exposed to a dilute solution of cationic (or anionic) macromolecules, followed by a rinsing step to obtain an irreversibly adsorbed layer. Further film growth is achieved by deposition of polyanions and polycations from their aqueous solutions. After several dipping cycles the experiments show a linear increase of multilayer thickness, indicating that the system has reached a steady state regime. Unlike the extensive experimental studies of layer-by-layer deposition of charged molecules [9 11] the theoretical description of the multilayer formation remains limited. The numerical solutions of the self-consistent field equations describing multilayer assembly have been recently presented by Wang [138,139] and by Shafir and Andelman [140] These calculations have shown that the sufficiently strong short-range attraction between oppositely charged polymers is essential for the Fig. 8. Evolutionofthe multilayerassemblyatchargedsurface. Snapshots are takenaftercompletion ofthe deposition steps 1 through 5. The positivelychargedparticles onthe substrateare shown in green and neutral particles are colored in black. The molecules deposited during different deposition steps are colored as follows: blue (1), red (2), cyan (3), magenta (4), and orange (5). Reproduced with permission from Jeon, J., Panchagnula, V., Pan, J., Dobrynin, A. V. Langmuir 22, (2006). Copyright 2006, American Chemical Society. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)