Lattice Monte Carlo simulations of a charged polymer chain: effect of valence and concentration of the added salt

The configurational properties of a single polyelectrolyte chain accompanied by counterions and added salt are simulated using the cooperative motion algorithm on the face-centered cubic lattice. In particular, a greater emphasis is put on the effect of valence z(s) and concentration of the added positive (negative) salt ions n(s) on the polymer behavior. This is achieved by inspecting two families of systems with widely varying numbers n(s) of monovalent (z(s)=1) or multivalent (z(s)=4) salt ions at two fixed reduced temperatures T*=0.5, 1. The calculations indicate that especially at the lower temperature the addition of some amount of multivalent salt has a tremendous impact on chain conformations compared to the situation with monovalent salt. Even for relatively low concentrations of the former, the mean radius of gyration (1/2) and the mean end-to-end distance (1/2) decrease sharply, i.e., the polymer exists in strongly collapsed forms. This reduction of polymer size is also accompanied by a drop in the system inner energy e* and the effective mean charge per monomer q*. The analysis of various pair-correlation functions g(ab)(r) indicates that the latter effect-caused by condensation of ions onto the chain-is dominated by the multivalent ones. Furthermore, it is found that for z(s)=4, the uncondensed salt ions tend to group themselves into small clusters.     

Monte Carlo simulations of a charged dendrimer with explicit counterions and salt ions

Static properties of a dendrimer with generation g = 5 with positively charged terminal groups in an athermal solvent are studied by lattice Monte Carlo simulations using the cooperative motion algorithm as the tossing scheme. The calculations are performed both for a salt-free system with neutralizing counterions and for a small amount of added monovalent and divalent salt. The full Coulomb potential and the excluded volume interactions between ions and beads are taken explicitly into account with the reduced temperature tau, the number of salt cations (anions) n(s), and salt valence z(s) as the simulation parameters. The bahaviour of the systems is analyzed by the mean effective charge per end-bead , Coulomb mean energy , mean-square radius of gyration , pair correlation functions g(alphabeta), and charge density rho(ch). The simulations show that for n(s)> or = 0 and decreasing tau: (a) there is encapsulation in the dendrimer and condensation onto the terminal groups of anions accompanied by a monotonic decrease in and and by subsequent swelling and shrinking of the molecule; (b) encapsulation, condensation and shrinking are the most significant and swelling weaker for |z(s)| = 2; (c) penetration of salt cations into the dendrimer is minor when compared to that of anions; (d) rho(ch) is reduced and becomes negative close to the center of mass of the dendrimer and on its periphery; (e) for the considered n(s) > 0, unlike divalent salt ions the monovalent ones cause slight effects when compared to the salt-free case.     

Polyelectrolyte–particle complex formation. Polyelectrolyte linear charge density and ionic concentration effects. Monte Carlo simulations

The influence of the linear charge density (LCD) of a polyelectrolyte on its adsorption on an oppositely charged colloidal particle is investigated by Monte Carlo simulations. Adsorption characteristics are studied at different linear charge densities and ionic concentrations and for a given polyelectrolyte/particle size ratio so that particle curvature has full effect. The isolated polyelectrolyte goes through a smooth transition from a collapsed structure to an extended rod-like conformation with increasing the linear charge density in the low ionic concentration regime. In the high ionic concentration regime, the polyelectrolyte is less sensitive to the increase in the linear charge density and adopts a coil conformation. We found that complex formation is promoted by decreasing the ionic concentration and increasing the linear charge density and that large changes in the polymer dimensions are observed at the adsorption-desorption limit. By adjusting the linear charge density and ionic strength, we demonstrate that the adsorption-desorption limit corresponds to a sharp transition from non-adsorbed to adsorbed conformations and that the mean adsorption energy per monomer has to be less than -0.4 kT to achieve adsorption. We calculated that the linear charge density at the adsorption-desorption limit is related to the Debye-Hückel length according to LCD^crit ~3². At small values of the linear charge density and low ionic strength (no adsorption is observed at high ionic strength), a large amount of monomers are present in loops and tails. By increasing LCD, the amount of monomers in trains reaches a maximum value and the polyelectrolyte adopt flat conformation at the surface of the particle.     

Chain stiffness, salt valency, and concentration influences on titration curves of polyelectrolytes: Monte Carlo simulations

Monte Carlo simulations have been used to study two different models of a weak linear polyelectrolyte surrounded by explicit counterions and salt particles: (i) a rigid rod and (ii) a flexible chain. We focused on the influence of the pH, chain stiffness, salt concentration, and valency on the polyelectrolyte titration process and conformational properties. It is shown that chain acid-base properties and conformational properties are strongly modified when multivalent salt concentration variation ranges below the charge equivalence. Increasing chain stiffness allows to minimize intramolecular electrostatic monomer interactions hence improving the deprotonation process. The presence of di and trivalent salt cations clearly promotes the chain degree of ionization but has only a limited effect at very low salt concentration ranges. Moreover, folded structures of fully charged chains are only observed when multivalent salt at a concentration equal or above charge equivalence is considered. Long-range electrostatic potential is found to influence the distribution of charges along and around the polyelectrolyte backbones hence resulting in a higher degree of ionization and a lower attraction of counterions and salt particles at the chain extremities.     

Lattice Monte Carlo simulations of three-dimensional charged polymer chains. II. Added salt

The configurational properties of strongly charged polyelectrolytes accompanied by counterions and salt ions in dilute solutions are simulated using the cooperative motion algorithm on the face-centered-cubic lattice. The calculations indicate that both positive and negative ions condense on the chains at sufficiently low temperatures and their amount depends on the concentration of added salt. As the temperature decreases from high to low, the chains undergo conformational changes from neutral-like, self-avoiding polymers by more outstretched forms to compact globules. The observed extension of the chains at intermediate temperatures is also affected by the amount of salt. Furthermore, configurations with the lowest energies recorded at the lowest temperatures are aggregates of single or more entangled chains and ions of both types.     

Surface charge density and electrokinetic potential of highly charged minerals: experiments and Monte Carlo simulations on calcium silicate hydrate

In this paper, we are concerned with the charging and electrokinetic behavior of colloidal particles exhibiting a high surface charge in the alkaline pH range. For such particles, a theoretical approach has been developed in the framework of the primitive model. The charging and electrokinetic behavior of the particles are determined by the use of a Monte Carlo simulation in a grand canonical ensemble and compared with those obtained through the mean field theory. One of the most common colloidal particles has been chosen to test our theoretical approach. That is calcium silicate hydrate (C-S-H) which is the main component of hydrated cement and is known for being responsible for cement cohesion partly due to its unusually high surface charge density. Various experimental techniques have been used to determine its surface charge and electrokinetic potential. The experimental and simulated results are in excellent agreement over a wide range of electrostatic coupling, from a weakly charged surface in contact with a reservoir containing monovalent ions to a highly charged one in contact with a reservoir with divalent ions. The electrophoretic measurements show a charge reversal of the C-S-H particles at high pH and/or high calcium concentration in excellent agreement with simulation predictions. Finally, both simulation and experimental results clearly demonstrate that the mean field theory fails not only quantitatively but also qualitatively to describe a C-S-H dispersion under realistic conditions.     

Monte Carlo simulations of a polyelectrolyte chain with added salt: effect of temperature and salt valence

Using the cooperative motion algorithm, the effect of salt valence z(s) and of the reduced temperature T* on a single polyelectrolyte chain as well as on counterions and salt ions themselves is studied. The calculations show that both parameters strongly influence the polymer, causing it to undergo conformational changes. For a given number of the added salt cations (anions) n(s) and temperature T*, the chain takes more and more compact forms as z(s) increases (z(s) > 0). For fixed z(s), in turn, the polymer size reduces sharply as T* drops down from intermediate to low. For high T* configurational the entropy dominates the chain statistics and the mean-square radius of gyration (s2)1/2(T*,n(s),z(s)) approaches its athermal value. The low-temperature polymer collapse is also accompanied by a drop in the effective mean charge per monomer q*(T*,n(s),z(s)) (condensation of ions onto the chain) and the total inner energy e*(T*,n(s),z(s)). Furthermore, the local structure of the system is analyzed by means of pair-correlation functions g(ab)(r,T*,n(s),z(s)). At lower T* they possess sharp local maxima at small interparticle distances r that disappear as T* grows. The former observation indicates that at lower T* the ions tend to group themselves close to each other. In particular, it is concluded that the condensation is dominated by the multivalent salt ions carrying charges of opposite sign to that of monomers.     

Overcharging, charge inversion, and reentrant condensation: using highly charged polyelectrolytes in tetravalent salt solutions as an example of study

We study salt-induced charge overcompensation and charge inversion of flexible polyelectrolytes via computer simulations and demonstrate the importance of ion excluded volume. Reentrant condensation takes place when the ion size is comparable to monomer size, and happens in a middle region of salt concentration. In a high-salt region, ions can overcharge a chain near its surface and charge distribution around a chain displays an oscillatory behavior. Unambiguous evidence obtained by electrophoresis shows that charge inversion does not necessarily appear with overcharging and occurs when the ion size is not big. These findings suggest a disconnection of resolubilization of polyelectrolyte condensates at high salt concentration with charge inversion.     

Phase separation in mixtures of Yukawa and charged Yukawa particles from Gibbs ensemble Monte Carlo simulations and the mean spherical approximation

The phase equilibrium of mixtures of Yukawa and charged Yukawa particles is studied by means of Gibbs ensemble Monte Carlo (GEMC) simulation method and the mean spherical approximation (MSA). The strength of the Coulomb energy compared to that of the Yukawa attraction is characterized by a coupling constant. For low coupling constants a classical vapor--liquid phase separation appears with a good agreement between GEMC and the MSA. For high coupling constant, a phase separation between a salt poor and a salt rich phase occurs that resembles the phase equilibrium behavior of the solvent primitive model.     

Network formation of catanionic vesicles and oppositely charged polyelectrolytes. Effect of polymer charge density and hydrophobic modification

In nonequimolar solutions of a cationic and an anionic surfactant, vesicles bearing a net charge can be spontaneously formed and apparently exist as thermodynamically stable aggregates. These vesicles can associate strongly with polymers in solution by means of hydrophobic and/or electrostatic interactions. In the current work, we have investigated the rheological and microstructural properties of mixtures of cationic polyelectrolytes and net anionic sodium dodecyl sulfate/didodecyldimethylammonium bromide vesicles. The polyelectrolytes consist of two cationic cellulose derivatives with different charge densities; the lowest charge density polymer contains also hydrophobic grafts, with the number of charges equal to the number of grafts. For both systems, polymer-vesicle association leads to a major increase in viscosity and to gel-like behavior, but the viscosity effects are more pronounced for the less charged, hydrophobically modified polymer. Evaluation of the frequency dependence of the storage and loss moduli for the two systems shows further differences in behavior: while the more long-lived cross-links occur for the more highly charged hydrophilic polymer, the number of cross-links is higher for the hydrophobically modified polymer. Microstructure studies by cryogenic transmission electron microscopy indicate that the two polymers affect the vesicle stability in different ways. With the hydrophobically modified polymer, the aggregates remain largely in the form of globular vesicles and faceted vesicles (polygon-shaped vesicles with largely planar regions). For the hydrophilic polycation, on the other hand, the surfactant aggregate structure is more extensively modified: first, the vesicles change from a globular to a faceted shape; second, there is opening of the bilayers leading to holey vesicles and ultimately to considerable vesicle disruption leading to planar bilayer, disklike aggregates. The faceted shape is tentatively attributed to a crystallization of the surfactant film in the vesicles. It is inferred that a hydrophobically modified polyion with relatively low charge density can better stabilize vesicles due to formation of molecularly mixed aggregates, while a hydrophilic polyion with relatively high charge density associates so strongly to the surfactant films, due to strong electrostatic interactions, that the vesicles are more perturbed and even disrupted.     

Molecular dynamics simulations of layer-by-layer assembly of polyelectrolytes at charged surfaces: effects of chain degree of polymerization and fraction of charged monomers

We performed molecular dynamics simulations of the electrostatic assembly of multilayers of flexible polyelectrolytes at a charged surface. The multilayer build-up was achieved through sequential adsorption of oppositely charged polymers in a layer-by-layer fashion from dilute polyelectrolyte solutions. The steady-state multilayer growth proceeds through a charge reversal of the adsorbed polymeric film which leads to a linear increase in the polymer surface coverage after completion of the first few deposition steps. Moreover, substantial intermixing between chains adsorbed during different deposition steps is observed. This intermixing is consistent with the observed requirement for several deposition steps to transpire for completion of a single layer. However, despite chain intermixing, there are almost perfect periodic oscillations of the density difference between monomers belonging to positively and negatively charged macromolecules in the adsorbed film. Weakly charged chains show higher polymer surface coverage than strongly charged ones.     

Nanoparticle adsorption on a weak polyelectrolyte. Stiffness, pH, charge mobility, and ionic concentration effects investigated by Monte Carlo simulations

Monte Carlo simulations have been used to study two different models for a weak linear polyelectrolyte in the presence of nanoparticles: (i) a rodlike and (ii) a flexible polyelectrolytes. The use of simulated annealing has made it possible to simulate a polyelectrolyte chain in the presence of several nanoparticles by improving conformation sampling and avoiding multiple minima problems when dense conformations are produced. Nanoparticle distributions along the polymer backbone were analyzed versus the ionic concentration, polyelectrolyte stiffness, and nanoparticle surface charge. Titration curves were calculated and the influences of the ionic concentration, solution pH, and number of adsorbed nanoparticles on the acid/base polyelectrolyte properties have been systematically investigated. The subtle balance of attractive and repulsive interactions has been discussed, and some characteristic conformations are presented. The comparison of the two limit models provides a good representation of the stiffness influence on the complex formation. In some conditions, overcharging was obtained and presented with respect to both the polyelectrolyte and nanoparticle as the central element. Finally, the charge mobility influence along the polyelectrolyte backbone was investigated by considering annealed and quenched polyelectrolyte chains.     

Lubrication properties of bottle-brush polyelectrolytes: an AFM study on the effect of side chain and charge density

The effect of side chain to charge ratio on the frictional properties of adsorbed layers formed by bottle-brush polyelectrolytes with poly(ethylene oxide) side chains has been investigated. The brush polyelectrolytes were preadsorbed from 0.1 mM NaNO(3) solutions onto mica and silica surfaces; the interfacial friction was then measured in polyelectrolyte-free solutions via AFM (with the silica surface acting as the colloidal probe). It was concluded that the decisive factor for achieving favorable lubrication properties is the concentration of nonadsorbing poly(ethylene oxide) side chains in the interfacial region. However, contrary to what may be expected, the results showed that an ideal brush layer structure with the adsorbed polymers adopting comb-like conformation is not necessary for achieving a low coefficient of friction in the asymmetric mica-silica system. In fact, the lowest coefficient of friction (<0.01) under applied pressures as high as 30 MPa was observed for a system with a side chain to charge ratio of 9:1, incapable of forming brush-like layers.     

Influence of charged intramolecular hydrogen bonds in weak polyelectrolytes: A Monte Carlo study of flexible and extendible polymeric chains in solution and near charged spheres

For weak polyelectrolytes, the interplay between pH, solvent properties, and polymer structure affects the amount of charges, their distribution, and hence their conformations via Coloumb repulsion. Attractive interactions can also develop between charged and neutral sites counteracting the expected Coulomb‐induced expansion. To gauge how such competition affects polyelectrolyte structure and ionization, the titration of a single polyelectrolyte chain, isolated or close to a charged sphere, mimicked with a novel many‐body potential model is simulated with Monte Carlo. Apart from showing a 10‐fold higher ionization than isolated monomers at low pH, interacting species contracted forming short‐range clusters of charged and neutral ionizable groups. The presence of a charged sphere synergically boosted both effects due to monomer interactions, forcing the chains to condense onto its surface at much lower pH. Structural properties, however, seem to be controlled only by the ionization degree despite the presence of the topological restraint represented by the spherical surface. Using Monte Carlo titration results, the equilibrium ionization of isolated chains is also estimated; the results evidence that even weak interactions can easily lead to a doubling of the total charge. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 650–663     

On the contribution of covalent and ionic species to the chain growth. Monte Carlo simulations and theoretical considerations

Contributions of the covalent and ionic species to the chain building in cationic reversible polymerizations are analyzed by means of Monte Carlo simulations and by solving the formulated set of equations. The validity of the equation for the contribution of covalent propagation, derived for reversible polymerization with simplifying assumptions, is discussed. It is shown that repeating units incorporated into the chain by covalent propagations are distributed uniformly only when direct monomer insertion into covalent bonds is not possible, or when intramolecular interconversion of covalent and ionic species is very fast in comparison with propagation.     

The influence of side chains on the properties of simple model biopolymers: Monte Carlo simulations

A model for the simulation of proteins is introduced which is based on a new set of bond vectors and a new method for modeling the side chains of proteins. The drawbacks of united atoms models are summarized and the motivation for this new model is given. Some preliminary results are shown which shall demonstrate the suitability of the model proposed.     

Surface tensions of linear and branched alkanes from Monte Carlo simulations using the anisotropic united atom model

The anisotropic united atoms (AUA4) model has been used for linear and branched alkanes to predict the surface tension as a function of temperature by Monte Carlo simulations. Simulations are carried out for n-alkanes ( n-C5, n-C6, n-C7, and n-C10) and for two branched C7 isomers (2,3-dimethylpentane and 2,4-dimethylpentane). Different operational expressions of the surface tension using both the thermodynamic and the mechanical definitions have been applied. The simulated surface tensions with the AUA4 model are found to be consistent within both definitions and in good agreement with experiments.     

Aqueous suspensions of charged spherical colloids: dependence of the surface charge on ionic strength, acidity, and colloid concentration

We theoretically investigate the dependence of the surface charge developed on charged spherical colloids upon several environmental parameters: the ionic strength of the monovalent added electrolyte, acidity (stabilized by a pH buffer solution), and colloid concentration. In the framework of the mean-field Poisson-Boltzmann spherical cell model, we include the charged colloid-microion correlations into the buffer equation, and we allow for the specific binding of ions to the ionizable groups on the colloid surface. Theoretical predictions are compared to the results obtained under the planar-symmetry Gouy-Chapman approximation and analyzed for the experimental conditions of an aqueous dispersion of the phospholipid dimyristoyl phosphatidylglycerol (DMPG). Experimental measurements of the partition ratio of an aqueous soluble cationic spin label on buffered dispersions of polyanionic unilamellar vesicles of DMPG in the presence of added monovalent salt are theoretically interpreted in terms of ion partition due to electrostatic interactions. We show that the specific binding of the probe must be admitted to explain the experimental results.     

Role of linear charge density and counterion quality in thermodynamic properties of strong acid type polyelectrolytes: divalent transition metal cations

Thermodynamic properties of aqueous solutions of poly[(vinyl alcohol)-co-(vinyl sulfate)] (PVAS) copolymer polyelectrolytes with divalent transition metal (Co(II), Ni(II), and Cu(II)) counterions have been determined by the gel deswelling method in the concentration range of 0.0005-0.12 mol of counterion/kg of water (0.09-9 w/w% of the polymer). The influence of the chemical nature of the counterion as well as the effect of the composition of the copolymer from small to medium linear charge density have been systematically studied. Solvent activity, reduced osmotic pressure, the Flory-Huggins pair interaction parameter, rational osmotic coefficients, and degrees of dissociation were calculated from the measured data. No difference could have been observed between the three counterions. Reduced osmotic pressure curves are found to be convex from above, as for Na+ counterions studied previously, which is contrary to the usual behavior of neutral polymers. Intercepts are increasing, and the calculated apparent molar masses and degrees of dissociation at infinite dilution are decreasing with increasing linear charge density of the polyelectrolytes. The pair interaction parameters show a considerable negative deviation from linearity, except for the high volume fraction region. From the differences, concentration dependence of degrees of dissociation could have been calculated. The values at infinite dilution are in good agreement with those obtained from the intercepts of the reduced osmotic pressure curves. Degrees of dissociation seem to decrease approximately linearly with increasing concentration and reach zero at finite concentrations. Rational osmotic coefficients have been calculated in three different ways, both regarding and neglecting the change in the degrees of dissociation.     

Monte Carlo simulation of polarizable systems: Early rejection scheme for improving the performance of adiabatic nuclear and electronic sampling Monte Carlo simulations

An early rejection scheme for trial moves in adiabatic nuclear and electronic sampling Monte Carlo simulation (ANES-MC) of polarizable intermolecular potential models is presented. The proposed algorithm is based on Swendsen–Wang filter functions for prediction of success or failure of trial moves in Monte Carlo simulations. The goal was to reduce the amount of calculations involved in ANES-MC electronic moves, by foreseeing the success of an attempt before making those moves. The new method was employed in Gibbs ensemble Monte Carlo (GEMC) simulations of the polarizable simple point charge-fluctuating charge (SPC-FQ) model of water. The overall improvement in GEMC depends on the number of swap attempts (transfer molecules between phases) in one Monte Carlo cycle. The proposed method allows this number to increase, enhancing the chemical potential equalization. For a system with 300 SPC-FQ water molecules, for example, the fractions of early rejected transfers were about 0.9998 and 0.9994 at 373 and 423 K, respectively. This means that the transfer moves consume only a very small part of the overall computing effort, making GEMC almost equivalent to a simulation in the canonical ensemble.