Steric and bridging interactions between two plates induced by grafted polyelectrolytes

If colloidal particles are grafted with a polymer, then the grafted chains can provide steric repulsion between them. If some of the grafted polymer chains are also adsorbed to a second particle, then a bridging force is generated as well. For uncharged plates and polymer, the following contributions to the free energy of the system have been taken into account in the calculation of the interaction force: (i) the Flory-Huggins expression for the mixing free energy of the grafted chains with the liquid; (ii) the entropy loss due to the connectivity of the polymeric segments; (iii) the van der Waals interactions between the segments and the plates; and (iv) the free energy of adsorption of the polymer segments of the grafted chains on the other plate. For charged plates, the electrostatic free energy as well as the free energy of the electrolyte are included in the total free energy of the system. By minimizing the free energy with respect to the segment concentration and, when it is the case, with respect to the electrical potential, equations for the segment number density distribution and for the electrical potential are obtained, on the basis of which the interactions between two plates grafted with polymer chains that can be also adsorbed on the other plate were calculated. The interaction thus obtained includes steric and bridging forces.     

Use of numerical methods in the prediction of polymer adsorption

Average polymer segment densities and thermodynamic properties of polymer adsorbed at liquid–solid interfaces were computed by extension of the polymer adsorption theory of Forsman and Hughes. Expressions were derived for the total free energy of adsorbed polymer chains by using the Flory-Huggins theory to represent free energy of mixing. A square-well potential was used to represent segment–surface interaction, and configurational entropy was calculated from the probability density function for the radius of gyration of random-flight chains. For each specified amount of surface coverage the free energy of the adsorbed polymer was minimized by varying the density of segments normal to the adsorbing surface and using a modified gradient search algorithm on a digital computer. Two different segment densities were considered, and they both gave qualitatively the same results. The two densities were (1) the sum of two Gaussian distributions and (2) a two-step density distribution. Isotherms were then calculated by equating the partial molal free energy of polymer at the surface to that of polymer in bulk solution for each specified amount of surface coverage. The results showed that for the initial region of the isotherms the distribution of polymer segments normal to the surface consisted of a high-density layer adjacent to the surface and a low-density “tail” extending far out into the solution. At higher amounts of adsorbed polymer, i.e., in the general concentration range of the pseudo-plateau, the tail of the polymer density distribution was predicted to thicken, and a single Gaussian distribution best described the segment density. Predicted adsorptions were in good agreement with reported experimental values.     

Use of numerical methods in the prediction of polymer adsorption

Average polymer segment densities and thermodynamic properties of polymer adsorbed at liquid–solid interfaces were computed by extension of the polymer adsorption theory of Forsman and Hughes. Expressions were derived for the total free energy of adsorbed polymer chains by using the Flory-Huggins theory to represent free energy of mixing. A square-well potential was used to represent segment–surface interaction, and configurational entropy was calculated from the probability density function for the radius of gyration of random-flight chains. For each specified amount of surface coverage the free energy of the adsorbed polymer was minimized by varying the density of segments normal to the adsorbing surface and using a modified gradient search algorithm on a digital computer. Two different segment densities were considered, and they both gave qualitatively the same results. The two densities were (1) the sum of two Gaussian distributions and (2) a two-step density distribution. Isotherms were then calculated by equating the partial molal free energy of polymer at the surface to that of polymer in bulk solution for each specified amount of surface coverage. The results showed that for the initial region of the isotherms the distribution of polymer segments normal to the surface consisted of a high-density layer adjacent to the surface and a low-density “tail” extending far out into the solution. At higher amounts of adsorbed polymer, i.e., in the general concentration range of the pseudo-plateau, the tail of the polymer density distribution was predicted to thicken, and a single Gaussian distribution best described the segment density. Predicted adsorptions were in good agreement with reported experimental values.     

The bridging force between two plates by polyelectrolyte chains

The interaction between particles in a colloidal system can be significantly affected by their bridging by polyelectrolyte chains. In this paper, the bridging is investigated by using a self-consistent field approach which takes into account the van der Waals interactions between the segments of the polyelectrolyte molecules and the plates, as well as the electrostatic and volume exclusion interactions. A positive contribution to the force between two plates is generated by the van der Waals interactions between the segments and the plates. This positive (repulsive) contribution plays an important role in the force when the distances between the plates are small. With increasing van der Waals interaction strength between segments and plates, the force between the plates becomes more repulsive at small distances and more attractive at large distances. When the surfaces of the plates have a constant surface electrical potential and a charge sign opposite to that of the polyelectrolyte chains, the force between the two plates becomes less attractive as the bulk polyelectrolyte concentration increases. This behavior is due to a higher bulk counterion concentration dissociated from the polyelectrolyte molecules. At short distances, the force between plates is more repulsive for stiffer chains. A comparison between theoretical and experimental results regarding the contraction of the interlayer separation between the platelets of vermiculite clays against the concentration of poly(vinyl methyl ether) was made.     

Interaction between small colloidal particles and polymer chains in a semidilute solution: Monte Carlo simulation

 Interaction between flexible-chain polymers and small (nanometric) colloidal particles is studied by Monte Carlo simulation using two-dimensional and three-dimensional lattice models. Spatial distribution of colloidal particles and conformational characteristics of chains in a semidilute solution are considered as a function of the segment adsorption energy, ɛ. When adsorption is sufficiently strong, it induces effective attraction of polymer segments, which results in contraction of macromolecular coils. The strongly adsorbing polymer chains affect the equilibrium spatial distribution of the colloidal particles. The average size of colloidal aggregates <m> exhibits a nontrivial behavior: with ɛ increasing, the value of <m> first decreases and then begins to grow. The adsorption polycomplex formed at strong adsorption exhibits a mesoscopic scale of structural heterogeneity. The results of computer simulations are in a good agreement with predictions of the analytic theory [P.G. Khalatur, L.V. Zherenkova and A.R. Khokhlov (1997) J Phys II (France) 7:543] based on the integral RISM equation technique. Received: 4 August 1997 Accepted: 16 April 1998     

SEGMENT DENSITY PROFILES OF COMPACT POLYMER CHAINS CONFINED BETWEEN TWO PARALLEL PLANE WALLS

We use the pruned-enriched Rosenbluth method to investigate systematically the segment density profiles of compact polymer chains confined between two parallel plane walls.The non-adsorption case of adsorption interaction energyε=0 and the weak adsorption case ofε=-1 are considered for the compact polymer chains with different chain lengths N and different separation distances between two walls D.Several special entropy effects on the confined compact polymer chains,such as a damped oscillation in the segment density profile for the large separation distance D,are observed and discussed for different separation distances D in the non-adsorption case.In the weak adsorption case,investigations on the segment density profiles indicate that the competition between the entropy and adsorption effects results in an obvious depletion layer.Moreover,the scaling laws of the damped oscillation period T_d and the depletion layer width L_d are obtained for the confined compact chains.Most of these results are obtained for the first time so far as we know,which are expected to understand the properties of the confined compact polymer chains more completely.     

Simulation of interaction forces between nanoparticles in the presence of Lennard-Jones polymers: freely adsorbing homopolymer modifiers

The force between two nanoscale colloidal particles dispersed in a solution of freely adsorbing Lennard-Jones homopolymer modifiers is calculated using the expanded grand canonical Monte Carlo simulation method. We investigate the effect of polymer chain length (N), nanoparticle diameter (sigma(c)), and colloid-polymer interaction energy (epsilon(cp)) on polymer adsorption (Gamma) and polymer-induced forces (F(P)(r)) between nanoparticles in the full thermodynamic equilibrium condition. There is a strong correlation between polymer adsorption and the polymer-mediated nanoparticle forces. When the polymer adsorption is weak, as in the case of smaller diameters and short polymer chain lengths (sigma(c) = 5, N = 10), the polymers do not have any significant effect on the bare nanoparticle interactions. The adsorbed amount increases with increasing particle diameter, polymer chain length, and colloid-polymer interaction energy. In general, for strong polymer-particle adsorption the polymer-governed force profiles between nanoparticles show short-range repulsion and long-ranged attraction, suggesting that homopolymers would not be ideal for achieving stabilization in nanoparticle dispersions. The attraction is likely due to bridging, as well as polymer segment-segment interactions. The location and magnitude of attractive minimum in the force profile can be controlled by varying N and epsilon(cp). The results show partial agreement and some marked differences with previous theoretical and experimental studies of forces in the limit of flat walls in an adsorbing polymer solution. The difference could be attributed to incorporation of long-ranged colloid-polymer potential in our simulations and the influence of the curvature of the nanoparticles.     

平行板间高分子链构象熵的动力学Monte Carlo模拟

用动力学Monte Carlo方法模拟了受限于两平行板之间的高分子链,并用扫描法计算了链的构象熵S,研究了构象熵相对于自由链的减小量(S0-S)与平行板间距D和高分子链长n的关系.结果证实了de Gennes的自由能标度关系,并给出了标度关系适用的范围.当D非常小时,高分子链受到强烈限制,S0-S与n成正比,表明单链节受到平行板的平均排斥作用力与链长无关.随着D增大,平行板对构象熵的影响越来越弱,单链节受到平行板的平均排斥作用力随链长的增长而增大.当D比较大时,平行板对构象熵的影响近似可以忽略,高分子链构象熵与自由空间中的结果一致.     

Interaction between small colloidal particles and molecular chains with selectively adsorbing groups: computer simulation study

Using the Monte Carlo simulation technique and the method of simulated annealing, we study interactions between small (nanometric) particles and flexible‐chain polymers with sticker groups which selectively adsorb on the particles and also can strongly attract each other. For the chains with two end stickers (telechelic polymers), we find that the colloidal particles adsorbing on the polymers play the role of junction points (locks) which bind together the ends of different chains. This direct or indirect binding leads to the formation of a web‐like structure throughout the sample: colloidal particles and chain stickers group into mixed clusters – “drops of a fog” – which are wrapped by polymer chains and connected by bridges. Analyzing static structure factors, we show that the selectively adsorbing telechelic polymers can affect the equilibrium spatially homogeneous distribution of colloidal particles that results in the appearance of a quasiregular structure on the intermediate scale related to the average intercluster distance. At sufficiently strong attraction between particles and chain end‐groups, most of the telechelic chains (>90%) adopt either a loop‐like or a stretched bridge‐like conformation, the most typical morphology of the system being a combination of these two structural elements. In the mixed clusters, the colloidal particles and the chain ends pack locally on a binary grid corresponding to a local crystal‐like arrangement. For the chains without attracting end‐groups, we observe the formation of elongated, rugby‐ball‐like clusters having alternate layers of particles and adsorbing chain groups.     

Particle-induced phase separation in mixed polymer solutions

The influence of added colloidal particles on the phase separation of mixed aqueous polymer solutions is investigated. Two types of particles (polystyrene latex or silica) and different combinations of segregating polymers (dextran of varying molar mass combined with poly(ethylene oxide) (PEO) of varying molar mass, or Ucon, a copolymer of ethylene oxide and propylene oxide) were used. All systems displayed particle-induced instability effects, but the extent of the effect varied strongly between the various combinations and with the amount of added salt. Very large instability effects were seen in certain mixtures. Two mechanisms, both relying on the adsorption of at least one of the polymers to the particle surface, seem to operate. Close to the cloud-point curve of the particle-free polymer1/polymer2/water mixture, adsorption of PEO or Ucon to the particles gives rise to a capillary-induced phase separation. Close to the dextran/water axis of the phase diagram, the adsorbing polymer gives rise to a surface modification of the particles, which then interacts repulsively with the surrounding dextran solution.     

Polymer modified colloidal dispersions

The effect of added polyvinyl alcohol (PVA) and polyvinyl pyrrolidone (PVP) on a dispersion of polystyrene particles stabilised with grafted methoxy PEG chains is discussed. PVA adsorbed more strongly on the particles compared to PVP. Polymer addition led to stable mixtures in the case of PVA compared to depletion phase separation observed in the case of PVP. Rheological measurements showed thickening of the dispersion and absence of any structure in the case of PVA, in contrast to structure formation in the case of PVP due to depletion phase separation. A weak gel state was reached for ca. 7% w/w PVP. The observed behaviour is in accord with the relative propensity of PVA and PVP to interact with the particle surface, the grafted chains and their solubility in water. The solvency of the free polymer chains dominated the overall behaviour while the contribution from the incompatibility between free and grafted chains was conterbalanced by differences in the free polymer adsorption on the particles.     

The study of orientational relaxation of rodlike molecules in the presence of an adsorbing surface by the molecular dynamics method

Coefficients of translational and rotational self-diffusion of rigid-chain rodlike molecules formed from four spherical particles are determined by the molecular dynamics method. Simulations are performed for a three-dimensional canonical ensemble of 4096 Lennard-Jones particles within the range of chain concentration in its monomer varying from 2 to 100 mol % with allowance for the adsorption of chains on two parallel walls confining the system. Changes in the concentration profiles of chains and solvent particles over the normal to walls during variations in adsorption energy are considered. It is shown that the dependences of translational and rotational self-diffusion coefficients on the concentration and adsorption energy govern the changes in the characteristic times of the orientation-disorientation processes of molecules. All specific features of the establishment of orientational order and its relaxation are determined mainly by the degree of coverage of the adsorption monolayer. The contributions of the second and third monolayers to the weighted-mean mobility of chains begin to be pronounced with an increase in concentration. The exchange of chains between the adsorption monolayer and bulk solution is suppressed with an increase in the adsorption energy, and the monolayer is transformed into a set of two-dimensional “crystallites.” These crystallites form a typical domain structure on the adsorbing surface. The orientation and, hence, the ordering of domains by the external field occur a little more slowly than the orientation of molecules in solution. The disorientation requiring asynchronous rotations of chains is impeded, thus resulting in noticeable retardation of this process relative to the orientation, and upon achievement of a certain value of adsorption energy, the orientation of chains induced in the first adsorption monolayer becomes stable.     

Incorporating water-release and lateral protein interactions in modeling equilibrium adsorption for ion-exchange chromatography

The equilibrium adsorption of two albumin proteins on a commercial ion exchanger has been studied using a colloidal model. The model accounts for electrostatic and van der Waals forces between proteins and the ion exchanger surface, the energy of interaction between adsorbed proteins, and the contribution of entropy from water-release accompanying protein adsorption. Protein-surface interactions were calculated using methods previously reported in the literature. Lateral interactions between adsorbed proteins were experimentally measured with microcalorimetry. Water-release was estimated by applying the preferential interaction approach to chromatographic retention data. The adsorption of ovalbumin and bovine serum albumin on an anion exchanger at solution pH>pI of protein was measured. The experimental isotherms have been modeled from the linear region to saturation, and the influence of three modulating alkali chlorides on capacity has been evaluated. The heat of adsorption is endothermic for all cases studied, despite the fact that the net charge on the protein is opposite that of the adsorbing surface. Strong repulsive forces between adsorbed proteins underlie the endothermic heat of adsorption, and these forces intensify with protein loading. It was found that the driving force for adsorption is the entropy increase due to the release of water from the protein and adsorbent surfaces. It is shown that the colloidal model predicts protein adsorption capacity in both the linear and non-linear isotherm regions, and can account for the effects of modulating salt.     

Free energy of adsorption of water and metal ions on the [1014] calcite surface

We calculated the free energy profiles of water and three metal ions (magnesium, calcium, and strontium) adsorbing on the [1014] calcite surface in aqueous solution. The approach uses molecular dynamics with parametrized equations to describe the interatomic forces. The potential model is able to reproduce the interactions between water and the metal ions regardless of whether they are at the mineral surface or in bulk water. The simulations predict that the free energy of adsorption of water is relatively small compared to the enthalpy of adsorption calculated in previous papers. This suggests a large change in entropy associated with the water adsorption on the surface. We also demonstrate that the free energy profile of a metal ion adsorbing on the surface correlates with the solvent density and that the rate of formation of an innersphere complex depends on overcoming a large free energy barrier, which is mainly electrostatic in nature. Furthermore, comparison among the rates of desorption of magnesium, calcium, and strontium from the calcite surface suggests that magnesium has a much lower rate of desorption due to its strong interactions with both water and the surface.     

Monte Carlo simulation of structure and nanoscale interactions in polymer nanocomposites

Off-lattice Monte Carlo simulations in the canonical ensemble are used to study polymer-particle interactions in nanocomposite materials. Specifically, nanoscale interactions between long polymer chains (N=550) and strongly adsorbing colloidal particles of comparable size to the polymer coils are quantified and their influence on nanocomposite structure and dynamics investigated. In this work, polymer-particle interactions are computed from the integrated force-distance curve on a pair of particles approaching each other in an isotropic polymer medium. Two distinct contributions to the polymer-particle interaction potential are identified: a damped oscillatory component that is due to chain density fluctuations and a steric repulsive component that arises from polymer confinement between the surfaces of approaching particles. Significantly, in systems where particles are in a dense polymer melt, the latter effect is found to be much stronger than the attractive polymer bridging effect. The polymer-particle interaction potential and the van der Waals potential between particles determine the equilibrium particle structure. Under thermodynamic equilibrium, particle aggregation is observed and there exists a fully developed polymer-particle network at a particle volume fraction of 11.3%. Near-surface polymer chain configurations deduced from our simulations are in good agreement with results from previous simulation studies.     

Computer Simulation of Bridging Flocculation Processes: The Role of Colloid to Polymer Concentration Ratio on Aggregation Kinetics

The flocculation of colloidal particles by adsorbing polymers is one of the central issues of colloid science and a very important topic in many industrial, biological, and environmental processes. We report a computer simulation study of a 2- and 3-dimensional model for bridging flocculation betweenlarge linear polymer chains and comparatively small colloidal particles,where the structure and growth kinetics of cluster formation are investigated. This model was developed within the framework of the cluster–cluster aggregation model using mass and fractal dimension dependent diffusion constants, where bridging flocculation is seen as a case of heterocoagulation in which, in addition, macromolecule configurations and lengths play an important role. The simulation of aggregate structure and formation kinetics obtained at different (i) relative particle concentrations, (ii) polymer chain conformations, and (iii) sticking probabilities are described from a qualitatively and quantitative point of view. The results suggest that the formation of large aggregates is a slow process, controlled by the reactivity of the clusters, even when the reaction between microcolloids and macrochains is very fast. Aggregation kinetics are strongly dependent on the particle/chain concentration ratio and on the configurational properties of the chains. It is shown that the scaling laws which are valid for homocoagulation processes are also applicable to the kinetics of bridging flocculation. The corresponding scaling exponents have been calculated.     

Interactions between colloidal particles induced by polymer brushes grafted onto the substrate

We investigate the interaction energy between two colloidal particles on or immersed in nonadsorbing polymer brushes grafted onto the substrate as a function of the separation of the particles by the use of a self-consistent-field theory calculation. Depending on the colloidal size and the penetration depth, we demonstrate the existence of a repulsive energy barrier of several kBT, which can be interpreted by separating the interaction energy into three parts: colloid-polymer interfacial energy, entropic contribution due to "depletion zone" overlap of colloidal particles, and entropic elastic energy of grafted chains by the compression of particles. The existence of a repulsive barrier which is of entirely entropic origin can lead to kinetic stabilization of the mixture rather than depletion flocculation or phase separation. Therefore, the present result may suggest an approach for controlling the self-assembling behavior of colloids for the formation of target structures, by tuning the colloidal interaction on the grafting substrate under appropriate selection of colloidal size, effective gravity (influencing the penetration depth), and brush coverage density.     

Adsorption of interacting oligomers and polymers at an interface

We discuss in a qualitative way the physical background of a recently developed polymer adsorption theory, in which all the possible chain conformations for interacting chain molecules near an adsorbing interface are taken into account. Any conformation is described as a step-weighted random walk in a lattice. Each step is weighted according to a segmental weighting factor that contains the adsorption energy (for segments in contact with the surface), the entropy of mixing, and the attraction or repulsion between segments and solvent molecules. A suitable computing method is used to calculate the contribution of all chain conformations to the concentration profile, to the adsorbed amount, to the fraction of trains, loops and tails, to the layer thickness, etc. The theory is valid for any chain length and any concentration in the solution.Results for various chain lengths are given. Oligomers have a low affinity for the surface, whereas polymer adsorption isotherms are of the well known high affinity type. Three concentration regimes can be distinguished. In (extremely) dilute solutions the molecules on the surface adsorb as isolated chains (the Henry region).     

Influence of added particles on the phase behavior of polymer solutions. Analysis by mean-field lattice theory

The influence of added colloidal particles on the phase stability of polymer solutions is investigated theoretically. The polymer has an affinity to the particle surface. A mean-field lattice theory based on the Flory-Huggins theory is used to calculate the phase behavior in solutions containing a single polymer component and particles. The particles are described in two different ways. The first approach considers the surface free energy associated with added solid particles and the mixing entropy of the particles. In the second approach, the particles are simply modeled as large polymers. Both ways of describing the added particles show that the added particles decrease the stability of the polymer solution when the polymer-particle attraction is strong. A higher particle concentration enhances the effect. Experiments where polystyrene latex particles are added at different concentrations to aqueous dispersions of ethyl(hydroxyethyl)cellulose (EHEC) support the theoretical findings.     

Internal and free energy in a pair of like-charged colloids: Monte Carlo simulations

The effective interaction between two colloidal particles in a bath of monovalent co- and counterions is studied by means of lattice Monte Carlo simulations with the primitive model. The internal electrostatic energy as a function of the colloid distance is studied fixing the position of the colloids. The free energy of the whole system is obtained introducing a bias parabolic potential, that allows us to sample efficiently small separations between the colloidal particles. For small charges, both the internal and free energy increase when the colloids approach each other, resulting in an effective repulsion driven by the electrostatic repulsion. When the colloidal charge is large enough, on the other hand, the colloid-ion coupling is strong enough to form double layers. The internal energy in this case decreases upon approaching the colloids because more ions enter the double layer. This attractive contribution to the interaction between the colloids is stronger for larger charges and larger ionic concentrations. However, the total free energy increases due to the loss of ionic entropy, and resulting finally in a repulsive interaction potential driven by the entropic contributions. The loss of ionic entropy can be almost quantitatively reproduced with the ideal contribution, the same level of approximation as the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. The overall behavior is captured by the DLVO theory qualitatively, and a comparison is made with the functional form predicted by the theory, showing moderate agreement.