Molecular dynamics simulations of end‐grafted polymers with charged side chains

We report molecular dynamics simulations on bottle‐brush polyelectrolytes end‐grafted to a planar surface. For each bottle‐brush polyelectrolyte, flexible charged side chains are anchored to one neutral main chain. The effects of the counterion valence and the grafting density on the density profiles and the structural characteristics of the brush were studied in this work. It is found that the electrostatic repulsion between charged monomers in the side chains leads an extended conformation of the brush in a solution containing monovalent counterions, while strong electrostatic binding of multivalent counterions to the side chains has a significant contribution to the collapse of the brush. For the trivalent case, the distribution of end monomers in the main chains becomes broader upon decreasing the grafting density, as compared with the monovalent case. However, the position of the distribution for the monovalent case is relatively insensitive to the change of the grafting density. Additionally, with increased counterion valence, enhanced electrostatic correlation between counterions and charged side chains also weakens the diffusive ability of counterions. © 2011 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011     

Monte Carlo simulation and molecular theory of tethered polyelectrolytes

We investigate the structure of end-tethered polyelectrolytes using Monte Carlo simulations and molecular theory. In the Monte Carlo calculations we explicitly take into account counterions and polymer configurations and calculate electrostatic interaction using Ewald summation. Rosenbluth biasing, distance biasing, and the use of a lattice are all used to speed up Monte Carlo calculation, enabling the efficient simulation of the polyelectrolyte layer. The molecular theory explicitly incorporates the chain conformations and the possibility of counterion condensation. Using both Monte Carlo simulation and theory, we examine the effect of grafting density, surface charge density, charge strength, and polymer chain length on the distribution of the polyelectrolyte monomers and counterions. For all grafting densities examined, a sharp decrease in brush height is observed in the strongly charged regime using both Monte Carlo simulation and theory. The decrease in layer thickness is due to counterion condensation within the layer. The height of the polymer layer increases slightly upon charging the grafting surface. The molecular theory describes the structure of the polyelectrolyte layer well in all the different regimes that we have studied.     

Molecular dynamics study of charged dendrimers in salt-free solution: effect of counterions

Polyamidoamine dendrimers, being protonated under physiological conditions, represent a promising class of nonviral, nanosized vectors for drug and gene delivery. We performed extensive molecular dynamics simulations of a generic model dendrimer in a salt-free solution with dendrimer's terminal beads positively charged. Solvent molecules as well as counterions were explicitly included as interacting beads. We find that the size of the charged dendrimer depends nonmonotonically on the strength of electrostatic interactions demonstrating a maximum when the Bjerrum length equals the diameter of a bead. Many other structural and dynamic characteristics of charged dendrimers are also found to follow this pattern. We address such a behavior to the interplay between repulsive interactions of the charged terminal beads and their attractive interactions with oppositely charged counterions. The former favors swelling at small Bjerrum lengths and the latter promotes counterion condensation. Thus, counterions can have a dramatic effect on the structure and dynamics of charged dendrimers and, under certain conditions, cannot be treated implicitly.     

Molecular dynamics simulations of polyelectrolyte brushes: from single chains to bundles of chains

Using molecular dynamics simulations in combination with scaling analysis, we have studied the effects of the solvent quality and the strength of the electrostatic interactions on the conformations of spherical polyelectrolyte brushes in salt-free solutions. The spherical polyelectrolyte brush could be in one of four conformations: (1) a star-like conformation, (2) a "star of bundles" conformation in which the polyelectrolyte chains self-assemble into pinned cylindrical micelles, (3) a micelle-like conformation with a dense core and charged corona, or (4) a conformation in which there is a thin polymeric layer uniformly covering the particle surface. These different brush conformations appear as a result of the fine interplay between electrostatic and monomer-monomer interactions. The brush thickness depends nonmonotonically on the value of the Bjerrum length. This dependence of the brush thickness is due to counterion condensation inside the brush volume. We have also established that bundle formation in poor solvent conditions for the polymer backbone can also occur in a planar polyelectrolyte brush. In this case, the grafted polyelectrolyte chains form hemispherical aggregates at low polymer grafting densities, cylindrical aggregates at an intermediate range of the grafting densities, and vertically oriented ribbon-like aggregates at high grafting densities.     

Simulation Study of the Conformational Properties of Diblock Polyelectrolytes in Salt Solutions

Coarse-grained molecular dynamics simulations are performed to understand the behavior of diblock polyelectrolytes in solutions of divalent salt by studying the conformations of chains over a wide range of salt concentrations. The polymer molecules are modeled as bead spring chains with different charged fractions and the counterions and salt ions are incorporated explicitly. Upon addition of a divalent salt, the salt cations replace the monovalent counterions, and the condensation of divalent salt cations onto the polyelectrolyte increases, and the chains favor to collapse. The condensation of ions changes with the salt concentration and depends on the charged fraction. Also, the degree of collapse at a given salt concentration changes with the increasing valency of the counterion due to the bridging effect. As a quantitative measure of the distribution of counterions around the polyelectrolyte chain, we study the radial distribution function between monomers on different polyelectrolytes and the counterions inside the counterion worm surrounding a polymer chain at different concentrations of the divalent salt. Our simulation results show a strong dependence of salt concentration on the conformational properties of diblock copolymers and indicate that it can tune the self-assembly behaviors of such charged polyelectrolyte block copolymers.     

Ionic effects in collapse of polyelectrolyte brushes

We investigated the effect of counterion valence on the structure and swelling behavior of polyelectrolyte brushes using a nonlocal density functional theory that accounts for the excluded-volume effects of all ionic species and intrachain and electrostatic correlations. It was shown that charge correlation in the presence of multivalent counterions results in collapse of a polyelectrolyte brush at an intermediate polyion grafting density. At high grafting density, the brush reswells in a way similar to that in a monovalent ionic solution. In the presence of multivalent counterions, the nonmonotonic swelling of a polyelectrolyte brush in response to the increase of the grafting density can be attributed to a competition of the counterion-mediated electrostatic attraction between polyions with the excluded-volume effect of all ionic species. While a polyelectrolyte brush exhibits an "osmotic brush" regime at low salt concentration and a "salted brush" regime at high salt concentration regardless of the counterion valence, we found a smoother transition as the valence of the counterions increases. As observed in recent experiments, a quasi-power-law dependence of the brush thickness on the concentration ratio can be identified when the monovalent counterions are replaced with trivalent counterions at a fixed ionic strength.     

Molecular dynamics and neutron scattering study of the dependence of polyelectrolyte dendrimer conformation on counterion behavior

Atomistic molecular dynamics (MD) simulations and contrast variation small angle neutron scattering (SANS) have been combined to investigate the Generation-5 polyelectrolyte polyamidoamine starburst dendrimer. This work reveals the dendrimer conformational dependence on counterion association at different levels of molecular charge. The accuracy of the simulations is verified through satisfactory comparison between modeled results, such as excess intra-dendrimer scattering length density distribution and hydration level, and their experimental counterparts. While the counterion distributions are not directly measureable with SANS, the spatial distribution of the counterions and their dendrimer association are extracted from the validated MD equilibrium trajectories. It is found that the conformation of the charged dendrimer is strongly dependent on the counterion association. Sensitivity of the distribution of counterions around charged amines to the counterion valency is qualitatively explained by adopting Langmuir adsorption theory. Moreover, via extending the concept of electrical double layer for compact charged colloids, we define an effective radius of a charged dendrimer including the spatial distribution of counterions in its vicinity. Within the same framework, the correlation between the strength of intra-dendrimer electrostatic repulsion and the counterion valency and dynamics is also addressed.     

Langevin dynamics of semiflexible polyelectrolytes: rod-toroid-globule-coil structures and counterion distribution

We have investigated the nature of counterion condensation on uniformly charged semiflexible polyelectrolyte chains and the concomitant configurations by monitoring the role of chain stiffness, chain length, counterion valency, and the strength of electrostatic interaction. The counterion condensation is seen to follow the adsorption process and the effective polymer charge increases with chain stiffness. Size and shape, as calculated through the radius of gyration, effective persistence length, and hydrodynamic radius, are studied. Stable coil-like, globular, folded-chain, toroidal, and rodlike configurations are possible at suitable combinations of values of chain stiffness, chain length, electrostatic interaction strength, and the valency of counterion. For high strengths of electrostatic interactions, sufficiently stiff polyelectrolytes form toroids in the presence of multivalent counterions, whereas flexible polyelectrolytes form disordered globules. The kinetic features of the nucleation and growth of toroids are monitored. Several metastable structures are found to frustrate the formation of toroids. The generic pathway involves the nucleation of one primary loop somewhere along the chain contour, followed by a growth process where the rest of the chain is folded continuously on top of the primary loop. The dependence of the average radii of toroids on the chain length is found to be roughly linear, in disagreement with existing scaling arguments.     

Effects of counterion size on the attraction between similarly charged surfaces

Interaction between similarly charged surfaces can be attractive at high electrostatic coupling constants Ξ = l(B)Z(2)/μ(GC), where l(B) is the Bjerrum length, μ(GC) the Gouy-Chapman length, and Z the valency of counterions. While this effect has been studied previously in detail, as a function of surface charge density and valency of the pointlike counterions, much less is known about the effect of counterion size. We apply the Wang-Landau sampling Monte Carlo (MC) simulation method to compute the free energy F as a function of the scaled distance between the plates D?=D/μ(GC) for a range of Ξ and scaled counterion radii R?=R/μ(GC). We find that for large Ξ and small ion radius, there is a global equilibrium distance D?=D?(eq)=2(1+R?), correctly giving the expected value at the point counterion limit. With increasing R? the global minimum in F(D?) changes to a metastable state and finally this minimum vanishes when R? reaches a critical value, which depends on Ξ. We present a state diagram indicating approximate boundaries between these three regimes. The Wang-Landau MC method, as it is applied here, offers a possibility to study a wide spectrum of extended problems, which cannot be treated by the use of contact value theorem.     

Theory of competitive counterion adsorption on flexible polyelectrolytes: divalent salts

The counterion distribution around an isolated flexible polyelectrolyte in the presence of a divalent salt is evaluated using the adsorption model [M. Muthukumar, J. Chem. Phys. 120, 9343 (2004)] that considers the Bjerrum length, salt concentration, and local dielectric heterogeneity as physical variables in the system. Self-consistent calculations of effective charge and size of the polymer show that divalent counterions replace condensed monovalent counterions in competitive adsorption. The theory further predicts that at modest physical conditions for a flexible polyelectrolytes such as sodium polystyrene sulfonate in aqueous solutions polymer charge is compensated and reversed with increasing divalent salt. Consequently, the polyelectrolyte shrinks and reswells. Lower temperatures and higher degrees of dielectric heterogeneity between chain backbone and solvent enhance condensation of all species of ions. Complete diagrams of states for the effective charge calculated as functions of the Coulomb strength and salt concentration suggest that (a) overcharging requires a minimum Coulomb strength and (b) progressively higher presence of salt recharges the polymer due to either electrostatic screening (for low Coulomb strengths) or coion condensation (for high Coulomb strengths). Consideration of ion-bridging by divalent counterions leads to a first-order collapse of polyelectrolytes in modest presence of divalent salts and at higher Coulomb strengths. The authors' theoretical predictions are in agreement with the generic results from experiments and simulations.     

Molecular dynamics simulations of grafted layers of bottle-brush polyelectrolytes

Using molecular dynamics simulations, we study the effect of the brush grafting density and degree of polymerization of the side chains on conformations of brush layers made of charged bottle-brush macromolecules. The thickness of the brush layer first decreases with increasing brush grafting density; then, it saturates and remains constant in the wide interval of the brush grafting densities. The brush layers consisting of the bottle-brush macromolecules with longer side chains have a larger layer thickness. The elongation of the side chains of the bottle-brush macromolecules decreases with increasing brush grafting density. This contraction of the side chains is due to counterion condensation inside the volume occupied by bottle-brushes. Our simulations showed that counterion condensation is a multiscale process reflecting different symmetries of the bottle-brush layer.     

Generalizations for the potential of mean force between two isolated colloidal particles from Monte Carlo simulations

A substantial amount of experimental and numerical evidence has shown that the Derjaguin-Landau-Verwey-Overbeek theory is not suitable for describing those colloidal solutions that contain multivalent counterions. Toward improved understanding of such solutions, the authors report Monte Carlo calculations wherein, following Rouzina and Bloomfield, they postulate that, in the absence of van der Waals forces, the overall force between two isolated charged colloidal particles in electrolyte solutions is determined by a dimensionless parameter Gamma=z(2)l(B)/a, which measures the electrostatic repulsion between counterions adsorbed on the macroion surface, where z = counterion valence, l(B)=Bjerrum length, and a = average separation between counterions on the macroion surface calculated as if the macroion were fully neutralized. The authors find, first, that the maximum repulsion between like-charged macroions occurs at Gamma approximately 0.5 and, second, that onset of attraction occurs at Gamma approximately 1.8, essentially independent of the valence and concentration of the surrounding electrolyte. These observations might provide new understanding of interactions between electrostatic double layers and perhaps offer explanations for some electrostatic phenomena related to interactions between DNA molecules or proteins.     

Electrostatically-driven Ordering in Model Dendrimer Polyelectrolytes: Effects of Concentration

In this study we extend our previous work in the self-organization of dendrimer polyelectrolytes (Macromolecules, 2008 , 41, 225) by examining the effects of dendrimer concentration and/or total volume fraction in the ordering process and the resulting structure, in the arrangement of counterions and dendrimer beads and in the diffusive motion of dendrimers at different strengths of Coulombic interactions. It is found that as long as the total volume fraction remains low (i.e. no jamming phenomena intervene) the symmetry of the resulted cubic phases is unaltered. At a higher volume fraction and at the strong electrostatic regime a kinetic arrest of the dendrimer molecules much in analogy to a colloidal glass-like transition is observed, inhibiting thus the ordering process. Changes in the strength of electrostatic interactions and dendrimer concentration induces a systematic variation of the counterion - counterion and the counterion - charged-dendrimer-bead spatial arrangement. These findings are in qualitative agreement with previous studies in systems with very different structural details of the considered solutes, indicating a more general behaviour in charged macroion/counterion solutions.     

Entropy and enthalpy of polyelectrolyte complexation: Langevin dynamics simulations

We report a systematic study by Langevin dynamics simulation on the energetics of complexation between two oppositely charged polyelectrolytes of same charge density in dilute solutions of a good solvent with counterions and salt ions explicitly included. The enthalpy of polyelectrolyte complexation is quantified by comparisons of the Coulomb energy before and after complexation. The entropy of polyelectrolyte complexation is determined directly from simulations and compared with that from a mean-field lattice model explicitly accounting for counterion adsorption. At weak Coulomb interaction strengths, e.g., in solvents of high dielectric constant or with weakly charged polyelectrolytes, complexation is driven by a negative enthalpy due to electrostatic attraction between two oppositely charged chains, with counterion release entropy playing only a subsidiary role. In the strong interaction regime, complexation is driven by a large counterion release entropy and opposed by a positive enthalpy change. The addition of salt reduces the enthalpy of polyelectrolyte complexation by screening electrostatic interaction at all Coulomb interaction strengths. The counterion release entropy also decreases in the presence of salt, but the reduction only becomes significant at higher Coulomb interaction strengths. More significantly, in the range of Coulomb interaction strengths appropriate for highly charged polymers in aqueous solutions, complexation enthalpy depends weakly on salt concentration and counterion release entropy exhibits a large variation as a function of salt concentration. Our study quantitatively establishes that polyelectrolyte complexation in highly charged Coulomb systems is of entropic origin.     

Overcharging of nanoparticles in electrolyte solutions

Monte Carlo simulations are performed to investigate the effects of salt concentration, valence and size of small ions, surface charge density, and Bjerrum length on the overcharging of isolated spherical nanoparticles within the framework of a primitive model. It is found that charge inversion is most probable in solutions containing multivalent counterions at high salt concentrations. The maximum strength of overcharging occurs near the nanoparticle surface where counterions and coions have identical local concentrations. The simulation results also suggest that both counterion size and electrostatic correlations play major roles for the occurrence of overcharging.     

Conformations and solution structure of polyelectrolytes in poor solvent

Using extensive Molecular Dynamics (MD) simulations we study the behavior of polyelectrolytes in poor solvents, where we take explicitely care of the counterions. The resulting pearl-necklace structures are subject to strong conformational fluctuations, only leading to small signatures in the form factor, which is a severe obstacle for experimental observations. In addition we study how the necklace collapses as a function of Bjerrum length. At last we demonstrate that the position of the first peak in the inter-chain structure factor varies with the monomer density close to for all densities. This is in strong contrast to polyelectrolyte solutions in good solvent.     

New experiments for the quantification of counterion condensation

The condensation of counterions is an important aspect of charged macromolecules. Therefore an experimental characterization of the condensation of counterions is desirable. In this contribution two experimental techniques for the characterization of counterion condensation are introduced and compared: Anomalous Small Angle X-Ray Scattering (ASAXS) is able to probe the spatial distribution of counterions and electrophoresis nuclear magnetic resonance (NMR) measures counterion condensation via the effective charge obtained from the dynamic behaviour of molecules and complexes in an electric field.     

Electrophoretic evidence for a new type of counterion condensation

A systematic capillary electrophoresis study uncovered how polyelectrolyte effective charge density varies with backbone charge spacing and solvent dielectric constant. The study primarily focused on aliphatic ionenes, a special class of polyelectrolytes, which possess regularly spaced quaternary ammonium groups in the backbone. Complete ionization of functional units and good solvency in water or mixtures of water with lower dielectric constant solvents (methanol, acetonitrile) enabled continuous measurements of ionene effective charge density through the onset of counterion condensation. Ionenes with both uniform and alternating charge spacing were examined. As expected, effective charge density rose linearly with fixed charge density to a critical value, above which effective charge density remained constant. Deviating from expectation, the onset of condensation did not occur at a critical fixed charge density. Instead, condensation initiated at a constant critical Bjerrum length. The same onset condition was found for quaternized poly(vinyl pyridine)s. These experimental results suggest a new form of condensation, one driven by ion-pairing of polyelectrolyte with counterions. In support of this hypothesis, the onset of condensation appeared to correlate with counterion size. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3616–3627, 2004     

Conformational Behavior of Bottle‐Brush Polyelectrolytes with Charged and Neutral Side Chains

The conformational transition of a single bottle‐brush polyelectrolyte with charged and neutral side chains is studied through MD simulations. Counterions are included explicitly and no additional salt is added. The structure of the polyelectrolyte and the counterion condensation are found to depend greatly on the Bjerrum length. As the Bjerrum length increases, the neutral side chains in a poor solvent can condense into clusters with variable size. Moreover, the polyelectrolyte forms globular structures at large or very small Bjerrum lengths. This transition is quite different from that in the case of a good solvent, in which there are not observable clusters and a globular structure is only formed at large Bjerrum lengths.

     

Lattice Monte Carlo simulations of three-dimensional charged polymer chains

The configurational properties of strongly charged polyelectrolytes accompanied by neutralizing counterions in dilute solutions are simulated using the cooperative motion algorithm on the face-centered-cubic lattice. The full Coulomb potential and the excluded volume condition between different ions/beads are taken into account and the reduced temperature T* is considered the main, variable parameter. The calculations that have been carried out for solutions of both single and several chains indicate a few regions of their behavior: (1) for T*--> infinity, it corresponds to that of neutral, self-avoiding polymers under good solvent conditions; (2) for T* approximately 1, due to the electrostatic interactions being effectively stronger, the chains are more outstretched compared to their size at other temperatures; (3) for T* well below one, the counterion condensation becomes more and more dominant, which gradually leads to strongly collapsed chains; and (4) at the lowest temperatures the chains and counterions assume low-energy configurations in the form of neutral, compact aggregates.