Interaction between like-charged polyelectrolyte-colloid complexes in electrolyte solutions: a Monte Carlo simulation study in the Debye-Hückel approximation
D. Truzzolillo, F. Bordi, F. Sciortino, S. Sennato
We study the effective interaction between differently charged polyelectrolyte-colloid complexes in electrolyte solutions via Monte Carlo simulations. These complexes are formed when short and flexible polyelectrolyte chains adsorb onto oppositely charged colloidal spheres, dispersed in an electrolyte solution. In our simulations the bending energy between adjacent monomers is small compared to the electrostatic energy, and the chains, once adsorbed, do not exchange with the solution, although they rearrange on the particles surface to accomodate further adsorbing chains or due to the electrostatic interaction with neighbor complexes. Rather unexpectedly, when two interacting particles approach each others, the rearrangement of the surface charge distribution invariably produces anti-parallel dipolar doublets, that invert their orientation at the isoelectric point. These findings clearly rule out a contribution of dipole-dipole interactions to the observed attractive interaction between the complexes, pointing out that such suspensions can not be considered dipolar fluids. On varying the ionic strength of the electrolyte, we find that a screening length, short compared with the size of the colloidal particles, is required in order to observe the attraction between like charged complexes due to the non-uniform distribution of the electric charge on their surface ('patch attraction'). On the other hand, by changing the polyelectrolyte/particle charge ratio, the interaction between like-charged polyelectrolyte-decorated (pd) particles, at short separations, evolves from purely repulsive to strongly attractive. Hence, the effective interaction between the complexes is characterized by a potential barrier, whose height depends on the net charge and on the non-uniformity of their surface charge distribution.
We present a numerical study of polyelectrolytes electrophoresing in free solution while squeezed by an axisymmetric confinement force transverse to their net displacement. Hybrid multi-particle collision dynamics and molecular dynamics simulations with mean-field finite Debye layers show that even though the polyelectrolyte chains remain "free-draining", their electrophoretic mobility increases with confinement in nanoconfining potential wells. The primary mechanism leading to the increase in mobility above the free-solution value, despite long-range hydrodynamic screening by counterion layer…
This paper discusses the elastic behavior of a very long crosslinked polyelectrolyte chain (Debye-Hückel chain), which is weakly charged. Therefore the response of the crosslinked chain (network) on an external constant force $f$ acting on the ends of the chain is considered. A selfconsistent variational computation of an effective field theory is employed. It is shown, that the modulus of the polyelectrolyte network has two parts: the first term represents the usual entropy elasticity of connected flexible chains and the second term takes into account the electrostatic interaction of the mono…
We present a field-theoretic renormalization group (RG) analysis of a dilute solution of flexible, screened polyelectrolyte chains (Debye-Hueckel chains) in a polar solvent. We perform a virial expansion to calculate the scaling dependence of the Osmotic pressure on the chain density, chain size and screening length. The formalism is developed for general chain distributions and we give explicit forms for a monodisperse chain size distribution. We utilize the fact that a Debye-Hueckel chain may be mapped to a local field theory with TWO length-scales requiring the calculation of scaling functi…
Brownian dynamics simulations of a coarse-grained bead-spring chain model, with Debye-Hückel electrostatic interactions between the beads, are used to determine the root-mean-square end-to-end vector, the radius of gyration, and various shape functions (defined in terms of eigenvalues of the radius of gyration tensor) of a weakly-charged polyelectrolyte chain in solution, in the limit of low polymer concentration. The long-time diffusivity is calculated from the mean square displacement of the centre of mass of the chain, with hydrodynamic interactions taken into account through the incorporat…
The effective elasticity of highly charged stiff polyelectrolytes is studied in the presence of counterions, with and without added salt. The rigid polymer conformations may become unstable due to an effective attraction induced by counterion density fluctuations. Instabilities at the longest, or intermediate length scales may signal collapse to globule, or necklace states, respectively. In the presence of added-salt, a generalized electrostatic persistence length is obtained, which has a nontrivial dependence on the Debye screening length. It is also found that the onset of conformational ins…
We present a field-theoretic renormalization group (RG) analysis of a single flexible, screened polyelectrolyte chain (a Debye-Hückel chain) in a polar solvent. We point out that the Debye-Hückel chain may be mapped onto a local field theory which has the same fixed point as a generalised $n \to 1$ Potts model. Systematic analysis of the field theory shows that the system is one with two interplaying length-scales requiring the calculation of scaling functions as well as exponents to fully describe its physical behaviour. To illustrate this, we solve the RG equation and explicitly calculate th…
We provide numerical evidence that hydrodynamic interactions are screened for charged polymers in salt solution on time scales below the Zimm time. At very short times, a crossover to hydrodynamic behavior is observed. Our conclusions are drawn from extensive coarse-grained computer simulations of polyelectrolytes in explicit solvent and explicit salt, and discussed in terms of analytical arguments based on the Debye-Hueckel approximation.
In a previous paper [S. Ghosal, Phys. Rev. E 74, 041901 (2006)] a hydrodynamic model for determining the electrophoretic speed of a polyelectrolyte through an axially symmetric slowly varying nanopore was presented in the limit of a vanishingly small Debye length. Here the case of a finite Debye layer thickness is considered while restricting the pore geometry to that of a cylinder of length much larger than the diameter. Further, the possibility of a uniform surface charge on the walls of the nanopore is taken into account. It is thereby shown that the calculated transit times are consistent …
We investigate a system of dense polyelectrolytes in solution. The Langevin dynamics of the system with linearized hydrodynamics is formulated in the functional integral formalism and a transformation made to collective coordinates. Within a dynamical Random Phase Approximation (RPA) integration over the counter- and salt ions produces the Debye-Hückel-like screening of the Coulomb interactions with dependence on the frequency only as part of a more complicated coupling structure. We investigate the dynamics of the structure factor as well as the collective diffusion coefficient and comment up…
The electric double layer (EDL) that forms at the interface between a polyelectrolyte gel and a salt bath is studied using asymptotic and numerical methods. Specifically, matched asymptotic expansions, based on the smallness of the Debye length relative to the typical gel dimensions, are used to construct solutions of the governing equations and derive electroneutral models with consistent jump conditions across the gel-bath interface. A general approach for solving the equations of incompressible nonlinear elasticity in a curved boundary layer is developed and used to resolve the gel mechanic…
We study the adsorption-desorption transition of polyelectrolyte chains onto planar, cylindrical and spherical surfaces with arbitrarily high surface charge densities by massive Monte Carlo computer simulations. We examine in detail how the well known scaling relations for the threshold transition-demarcating the adsorbed and desorbed domains of a polyelectrolyte near weakly charged surfaces-are altered for highly charged interfaces. In virtue of high surface potentials and large surface charge densities, the Debye-Hueckel approximation is often not feasible and the nonlinear Poisson-Boltzmann…
Monte Carlo simulations are used to study the non-uniform equilibrium charge distribution along a single annealed polyelectrolyte chain under theta-solvent conditions and with added salt. Within a range of the order of the Debye length charge accumulates at chain ends while a slight charge depletion appears in the central part of the chain. The simulation results are compared with theoretical predictions recently given by Castelnovo et al. In the parameter range where the theory can be applied we find almost perfect quantitative agreement.
By resorting to the thick-chain model we discuss how the stretching response of a polymer is influenced by the self-avoidance entailed by its finite thickness. The characterization of the force versus extension curve for a thick chain is carried out through extensive stochastic simulations. The computational results are captured by an analytic expression that is used to fit experimental stretching measurements carried out on DNA and single-stranded RNA (poly-U) in various solutions. This strategy allows us to infer the apparent diameter of two biologically-relevant polyelectrolytes, namely DNA…
The build up of polyelectrolyte multilayers (PEMs) was observed by a silicon-on-insulator (SOI) based thin film resistor. Differently charged polyelectrolytes adsorbing to the sensor surface result in defined potential shifts, which decrease with the number of layers deposited. We model the response of the device assuming electrostatic screening of polyelectrolyte charges by mobile ions within the PEMs. The Debye screening length inside the PEMs was found to be increased compared to the value corresponding to the bulk solution. Furthermore the partitioning of mobile ions between the bulk phase…
The effective elasticity of highly charged stiff polyelectrolytes is studied in the presence of counterions, with and without added salt. The rigid polymer conformations may become unstable due to an effective attraction induced by counterion density fluctuations. Instabilities at the longest, or intermediate length scales may signal collapse to globule, or necklace states, respectively. In the presence of added-salt, a generalized electrostatic persistence length is obtained, which has a nontrivial dependence on the Debye screening length.
We investigate theoretically polyelectrolyte bridging interactions on the two-body level. The model system is composed of two macroions with two oppositely charged flexible chains. The electrostatic interactions are treated on the Debye - H\" uckel level. The formal level of the theory is provided by the Feynman-Kleinert variational method generalized to include also self-interactions between polyelectrolyte segments. The variational equations are shown to exhibit two solution branches corresponding to {\sl strong} and {\sl weak} coupling, whereas conformations of the chain can be described as…
We investigate the equilibrium charge distribution along a single annealed polyelectrolyte chain under different conditions. The coupling between the conformation of the chain and the local charge distribution is described for various solvent qualities and salt concentration. In salt free solution, we find a slight charge depletion in the central part of the chain: the charges accumulate at the ends. The effect is less important if salt is added to the solution since the charge inhomogeneity is localized close to the chain ends over a distance of order of the Debye length. In the case of poor …
We present a theory for the equilibrium structure of polyelectrolyte solutions. The main element is a simple, new optimization scheme that allows theories such as the random phase approximation (RPA) to handle the harsh repulsive forces present in such systems. Comparison is made with data from recent neutron scattering experiments of randomly charged, hydrophilic polymers in salt-free, semi-dilute solution at various charge densities. Models with varying degrees of realism are examined. The usual explanation of the invariance observed at high charge density has been counterion condensation. H…
We address the long standing problem of the dependence of the electrostatic persistence length $l_e$ of a flexible polyelectrolyte (PE) on the screening length $r_s$ of the solution within the linear Debye-Huckel theory. The standard Odijk, Skolnick and Fixman (OSF) theory suggests $l_e \propto r_s^2$, while some variational theories and computer simulations suggest $l_e \propto r_s$. In this paper, we use Monte-Carlo simulations to study the conformation of a simple polyelectrolyte. Using four times longer PEs than in previous simulations and refined methods for the treatment of the simulatio…
We study the effect of intersegment interactions on the effective bending and stretching moduli of a semiflexible polymer chain with a finite stretching modulus. For an interaction potential of a screened Debye-H\" uckel type renormalization of the stretching modulus is derived on the same level of approximation as the celebrated Odijk-Skolnick-Fixman result for the bending modulus. The presence of mesoscopic intersegment interaction potentials couples the bending and stretching moduli in a manner different from that predicted by the macroscopic elasticity theory. We advocate a fundamental cha…
A self-consistent mode-coupling theory is presented for the viscosity of solutions of charged rod-like polymers. The static structure factor used in the theory is obtained from polymer integral equation theory; the Debye-Hückel approximation is inadequate even at low concentrations. The theory predicts a non-monotonic dependence of the reduced excess viscosity, $η_R$, on concentration from the behaviour of the static structure factor in polyelectrolyte solutions. The theory predicts that the peak in $η_R$ occurs at concentrations slightly lower than the overlap threshold concentration, $c^\ast…
We theoretically investigate electrostatic properties between two charged surfaces with a grafted polyelectrolyte layer in an aqueous electrolyte solution by using the Poisson-Boltzmann approach accounting for ion partitioning. In order to consider the ion partitioning effect, we focus on changes of electrostatic properties due to the difference in dielectric permittivity of the polyelectrolyte layer and the aqueous electrolyte solution. We find that ion partitioning enhances electrostatic potential in the region between two charged soft surfaces and hence increases the electrostatic interacti…
Macromolecules based on dendritic or hyperbranched polyelectrolytes have been emerging as high potential candidates for biomedical applications. Here we study the charge and solvation structure of dendritic polyglycerol sulphate (dPGS) of generations 0 to 3 in aqueous sodium chloride solution by explicit-solvent molecular dynamics computer simulations. We characterize dPGS by calculating several important properties such as relevant dPGS radii, molecular distributions, the solvent accessible surface area, and the partial molecular volume. In particular, as the dPGS exhibits high charge renorma…
We develop a first-principle equation of state of salt-free polyelectrolyte solution in the limit of infinitely long flexible polymer chains in the framework of a field-theoretical formalism beyond the linear Debye-Hueckel theory and predict a liquid-liquid phase separation induced by a strong correlation attraction. As a reference system we choose a set of two subsystems - charged macromolecules immersed in a structureless oppositely charged background created by counterions (polymer one component plasma) and counterions immersed in oppositely charged background created by polymer chains (har…
We report an investigation of the equilibrium and dynamic properties of polyelectrolyte solutions confined between platinum surfaces with a dynamic Surface Force Apparatus. The polyelectrolyte adsorbs on the surfaces in a dense compact layer bearing a surface charge in good agreement with the theoretical predictions. The flow of the solution on this charged adsorbed layer is probed over four decades of spatial scales and one decade of frequency by dynamic measurements. At distances larger than the hundredth of nanometers, the flow of the viscoelastic solution is well described by a partial sli…
The association between polyelectrolytes (PEs) of the same size but opposite charge is systematically studied in terms of the potential of mean force (PMF) along their center-of-mass reaction coordinate via coarse-grained, implicit-solvent, explicit-salt computer simulations. The focus is set on the onset and the intermediate, transient stages of complexation. At conditions above the counterion-condensation threshold, the PE association process exhibits a distinct sliding-rod-like behavior where the polymer chains approach each other by first stretching out at a critical distance close to thei…
Using self-assembly techniques, x-ray reflectivity measurements, and computer simulations, we study the effective interaction between charged polymer rods and surfaces. Long-time Brownian dynamics simulations are used to measure the effective adhesion force acting on the rods in a model consisting of a planar array of uniformly positively charged, stiff rods and a negatively charged planar substrate in the presence of explicit monovalent counterions and added monovalent salt ions in a continuous, isotropic dielectric medium. This electrostatic model predicts an attractive polymer-surface adhes…
The behavior of polyelectrolytes between charged surfaces immersed in semi-dilute solutions is investigated theoretically. A continuum mean field approach is used for calculating numerically concentration profiles between two electrodes held at a constant potential. A generalized contact theorem relates the inter-surface forces to the concentration profiles. The numerical results show that over-compensation of the surface charges by adsorbing polyelectrolytes can lead to effective attraction between equally charged surfaces. Simple scaling arguments enable us to characterize qualitatively the …
Monte Carlo simulations and variational calculations using a Gaussian ansatz are applied to a model consisting of a flexible linear polyelectrolyte chain as well as to an intrinsically stiff chain with up to 1000 charged monomers. Addition of salt is treated implicitly through a screened Coulomb potential for the electrostatic interactions. For the flexible model the electrostatic persistence length shows roughly three regimes in its dependence on the Debye-Hückel screening length, $κ^{-1}$.As long as the salt content is low and $κ^{-1}$ is longer than the end-to-end distance, the electrostati…
We study the interactions between two negatively charged macroscopic surfaces confining positive counterions. A density-functional approach is introduced which, besides the usual mean-field interactions, takes into account the correlations in the positions of counterions. The excess free energy is derived in the framework of the Debye-H{ü}ckel theory of the one-component plasma, with the homogeneous density replaced by a weighted density. The minimization of the total free energy yields the density profile of the microions. The pressure is calculated and compared with the simulations and the r…
What are the fundamental laws for the adsorption of charged polymers onto oppositely charged surfaces, for convex, planar, and concave geometries? This question is at the heart of surface coating applications, various complex formation phenomena, as well as in the context of cellular and viral biophysics. It has been a long-standing challenge in theoretical polymer physics; for realistic systems the quantitative understanding is however often achievable only by computer simulations. In this study, we present the findings of such extensive Monte-Carlo in silico experiments for polymer-surface a…
We present an extended structural study of globular complexes made by mixing a positively charge protein (lysozyme) and a negatively charged polyelectrolyte (PSS). We study the influence of all the parameters that may act on the structure of the complexes (charge densities and concentration of the species, partial hydrophobicity of the polyion chain, ionic strength). The structures on a 15 scale range lying from 10Å to 1000Å are measured by SANS. Whatever the conditions, the same structure is found, based on the formation of dense globules of ~ 100Å with a neutral core and a volume fraction of…
Ion mixtures require an exclusion core to avoid collapse. The Debye Hueckel theory, where ions are point charges, is accurate only in the limit of infinite dilution. The MSA is the embedding of hard cores into DH, is valid for higher densities. In the MSA the properties of any ionic mixture can be represented by a single screening parameter $Γ$. For equal ionic size restricted model is obtained from the Debye parameter $κ$. This one parameter representation (BIMSA) is valid for complex and associating systems, such as the general n-polyelectrolytes. The BIMSA is the only theory that satisfies …
We revisit the popular notion of effective or renormalized charge, which is a concept of central importance in the field of highly charged colloidal or polyelectrolyte solutions. Working at the level of a linear Debye-Hückel like theory only, we propose a simple, efficient and versatile method to predict the saturated amount of charge renormalization, which is however a non-linear effect arising at strong electrostatic coupling. The results are successfully tested against the numerical solutions of Poisson-Boltzmann theory for polyions of various shapes (planar, cylindrical and spherical), bot…
Random polyampholytes (PAs) contain positively and negatively charged monomers that are distributed randomly along the polymer chain. The interaction between charges is assumed to be given by the Debye-Huckel potential. We show that the size of the PA is determined by an interplay between electrostatic interactions, giving rise to the polyelectrolyte (PE) effect due to net charge per monomer ($σ$), and an effective attractive PA interaction due to charge fluctuations, $δσ$. The interplay between these terms gives rise to non-monotonic dependence of the radius of gyration, $R_g$ on the inverse …
Charged colloidal monolayers at the interface between water and air (or oil) are used in a large number of chemical, physical and biological applications. Although a considerable experimental and theoretical effort has been devoted in the past few decades to investigate such monolayers, some of their fundamental properties are not yet fully understood. In this paper, we model charged colloidal monolayers as a continuum layer of finite thickness, with separate charge distribution on the water and air sides. The electrostatic surface free-energy and surface pressure are calculated via the chargi…
We present a theory of effective electrostatic interactions in polydisperse suspensions of charged macroions, generalizing to mixtures a theory previously developed for monodisperse suspensions. Combining linear response theory with a random phase approximation for microion correlations, we coarse-grain the microion degrees of freedom to derive general expressions for effective macroion-macroion pair potentials and a one-body volume energy. For model mixtures of charged hard-sphere colloids, we give explicit analytical expressions. The resulting effective pair potentials have the same general …
Charged surfaces in contact with liquids containing ions are accompanied in equilibrium by an electric double layer consisting of a layer of electric charge on the surface that is screened by a diffuse ion cloud in the bulk fluid. This screening cloud determines not only the interactions between charged colloidal particles or polyelectrolytes and their self-assembly into ordered structures, but it is also pivotal in understanding energy storage devices, such as electrochemical cells and supercapacitors. However, little is known to what spatial complexity the electric double layers can be desig…
Conformational characteristics of polyzwitterionic molecules in aqueous solutions are investigated using the variational method. Analytical relations are derived for the radius of gyration of a single polyzwitterionic chain as a function of the chain length, electrostatic interaction strength, added salt concentration, dipole moment and degree of ionization of the zwitterionic monomers. In the absence of the small ions (counterions and coions) near the polyzwitterionic chain, attractive dipole-dipole interactions are shown to induce a collapse of the polyzwitterionic chain. However, in the pre…
