Interactions between polymer brushes in solvents of variable quality: a density functional theory study

We present a density functional theory study of interactions between sterically stabilized colloidal particles in solvents of variable quality. Both flat and spherical polymer brushes are considered, as well as both monatomic and polymeric solvents. It is shown that the interaction between sterically stabilized particles can be tuned from repulsive to attractive by varying the solvent quality, the relative length of free and grafted chains, and by employing a mixed brush consisting of both well and poorly solvated chains.     

Steric stabilization of spherical colloidal particles: Implicit and explicit solvent

We present the results of Monte Carlo simulations and density functional theory treatment of interactions between spherical colloidal brushes both in implicit (good) solvent and in an explicit polymeric solution. Overall, theory is seen to be in good agreement with simulations. We find that interactions between hard-sphere particles grafted with hard-sphere chains are always repulsive in implicit solvent. The range and steepness of the repulsive interaction is sensitive to the grafting density and the length of the grafted chains. When the brushes are immersed in an explicit solvent of hard-sphere chains, a weak mid-range attraction arises, provided the length of the free chains exceeds that of the grafted chains.     

Spherical polymer brushes under good solvent conditions: molecular dynamics results compared to density functional theory

A coarse grained model for flexible polymers end-grafted to repulsive spherical nanoparticles is studied for various chain lengths and grafting densities under good solvent conditions by molecular dynamics methods and density functional theory. With increasing chain length, the monomer density profile exhibits a crossover to the star polymer limit. The distribution of polymer ends and the linear dimensions of individual polymer chains are obtained, while the inhomogeneous stretching of the chains is characterized by the local persistence lengths. The results on the structure factor of both single chain and full spherical brush as well as the range of applicability of the different theoretical tools are presented. Finally, a brief discussion of the experiment is given.     

Monomer density profiles for polymer chains in confined geometries: massive field theory approach

Taking into account the well known correspondence between the field theoretical ?(4) O(n)-vector model in the limit n → 0 and the behavior of long flexible polymer chains in a good solvent, the universal density-force relation is analyzed and the corresponding universal amplitude ratio B(real) is obtained using the massive field theory approach in fixed space dimensions d < 4. The monomer density profiles of ideal chains and real polymer chains with excluded volume interaction in a good solvent between two parallel repulsive walls, one repulsive and one inert wall, are obtained in the framework of the massive field theory approach up to one-loop order. Besides, the monomer density profiles for the dilute polymer solution confined in semi-infinite space containing mesoscopic spherical particle of big radius are calculated. The obtained results are in qualitative agreement with previous theoretical investigations and with the results of Monte Carlo simulations.     

Interactions between colloidal particles in amphiphilic mixtures: a density functional theory study

We present a density functional theory study of interactions between spherical colloidal particles in amphiphile solutions. Theory is found to be in good agreement with previously published molecular dynamics simulations. It is used to analyze the effect of the amphiphile solution bulk density, the chain length, and the solvent mole fraction on the potential of mean force between the particles. The general features of the potential of mean force are rationalized in terms of formation of layers and bilayers of amphiphilic molecules in the intercolloidal gap. Theory yields the same general trends as observed in simulations and in experiments. In particular, the computed mean force changes its character from repulsive to attractive and back to repulsive as the solvent mole fraction is gradually increased.     

Density functional theory of homopolymer mixtures confined in a slit

A density functional theory (DFT) is developed for polymer mixtures with shorted-ranged attractive interparticle interactions confined in a slit. Different weighting functions are used separately for the repulsive part and the attractive part of the excess free energy functional by applying the weighted density approximation. The predicted results by DFT are in good agreement with the corresponding simulation data indicating the reliability of the theory. Furthermore, the center-of-mass profiles and the end-to-end distance distributions are obtained by the single chain simulation; the predictions also agree well with simulation data. The results reveal that both the attraction of the slit wall and the temperature has stronger effect on longer chains than on shorter ones because the intrasegment correlation of chains increases with increasing chain length.     

Collapse transitions of a supersized neutral chain due to irreversibly adsorbed small colloidal particles

 We performed Monte Carlo simulations to study the destabilization processes of large neutral and flexible polymer chains due to irreversibly adsorbed colloidal particles attached to the chains like beads on a necklace. The particles are modeled as charged spherical units which interact with each other via repulsive electrostatic and attractive van der Waals (vdW) potentials. The usual Monte Carlo search procedure is extended and carefully checked to completely sample the chain conformational space and achieve dense conformations in the limit of both strong attractive and repulsive interaction potentials. Configurational properties, such as the radius of gyration, the end-to-end length, and the Kuhn length, are calculated as a function of the intensity of the vdW interactions and ionic strength values. It is observed that chains exhibit a new range of possible conformations compared to the classical random walk and self avoiding walk chains or polyelectrolytes. In the limit of low salt concentration, by gradually increasing vdW interactions, chains undergo a cascade of transitions from extended structures to dumbbells, from dumbbells to pearl necklaces, and from pearl necklaces to collapsed coils. Because of strong competition between the vdW and electrostatic forces, the distance along the chain between the interacting particles, and the sampling limitations, these transitions are found to sample metastable domains and to depend on the initial conformations. To gain insight into the spatial organization of the collapsed conformations, the pair correlation functions of both monomers and particles are calculated. It is shown that collapsed conformations which are the result of strong particle–particle interactions exhibit two distinct parts: a hard core mainly composed of particles and a surrounding polymeric shell composed of loops and tails. Possible effects of such a collapsed transition on the kinetics of flocculation of a mixture containing large flexible chains and small adsorbing colloidal particles are discussed. Received: 26 July 1999 Accepted in revised form: 9 November 1999     

Solvent response of mixed polymer brushes

We have performed classical density functional theory calculations to study the behavior of mixed polymer brushes tethered to a planar surface. We assume no lateral segregation of the polymer at the grafting density studied and consider an implicit solvent. For a binary mixture of short and long athermal polymer chains, the short chain is compressed while the long chain is stretched compared with corresponding pure polymer chains at the same grafting density, which is consistent with simulation. This results from configurational entropy effects. Furthermore, we add a mean-field interaction for each polymer brush to simulate their different response towards a solvent. The long chain is forced to dislike the solvent more than the short chain. Through the interplay between the solvent effects and configurational entropy effects, a switch of the polymer brush surface (or outer) layer is found with increasing chain length of the long chain. The transition chain length (long chain) increases with increasing the solvent selectivity, and decreases with increasing the grafting density of the long chain. These results can provide guidance for the design of smart materials based on mixed polymer brushes.     

Structure of short polymers at interfaces: a combined simulation and theoretical study

The structure of polymers confined between surfaces is studied using computer simulation and a density functional approach. The simple model system considers the polymer molecule as a pearl necklace of freely jointed hard spheres, having attractions among the beads, confined between attractive surfaces. This approach uses the universality of the free-energy functional to obtain the self-consistent field required in the single chain simulation. The second-order direct correlation function for the uniform bulk fluid required as input has been calculated from the reference interaction site model integral equation theory using mean spherical approximation. The theoretical results are shown to compare well with the Monte Carlo simulation results for varying densities, chain lengths, and with different attractive interaction parameters. The simulation results on the conformational properties give important indications regarding the behavior of chains as they approach the surfaces.     

Hybrid density functional theory for homopolymer mixtures confined in a selective nanoslit

By integrating polymer density function theory (DFT) and single-chain molecular simulation, a hybrid DFT is developed for homopolymer mixtures confined in a selective nanoslit. Two weighting functions are adopted separately in the polymer DFT for repulsive and attractive contributions to the excess free energy functional. The theoretical results agree well with simulation data for the density profiles, configurations (tail, loop and train), adsorption amounts, layer thicknesses, and partition coefficients. The polymer-slit interaction is found to have a large effect on the density profiles and partition coefficients but is found to have a small effect on the average sizes and percentages of the configurations. Nearly half of the polymer segments form tails, and the other half form trains. In addition, bridges are observed to form for sufficiently long polymer chains. As the length difference between two polymers increases, the effect of chain connectivity becomes increasingly important.     

Interacting nanoparticles with functional surface groups

Functionalized nanoparticles with ionizable groups have generated a large variety of structures with important potential applications in technology. The nature of their interactions is crucial to determining their solubility and to exploring assemblies with diverse symmetries. Here, we use a molecular theory to describe the interactions between two nanoparticles coated with short polymer chains that contain ionizable (functional) end‐groups immersed in aqueous salt solution. It is shown here that the fraction of ionized functional groups in the system depends on factors such as the ionic strength and pH of solution, grafting density of polymer chains, the chain length, as well as the separation distance between the nanoparticles. The interactions between two neighboring nanoparticles influence the charge regulation of the end‐groups, which consequently induces an asymmetric distribution of these charged end‐groups on the nanoparticles, and thus confers a preferred directionality in nanoparticle–nanoparticle interactions. We show that the charge regulating system is less repulsive than an equivalent system with a fixed charge distribution. This is due to a decrease in the charge density of the weak acid end‐groups, to avoid a local increase in counterion confinement (condensation) in the region between neighboring nanoparticles, when their separation decreases. The anisotropic degree of ionization found in our results can be used to design aggregates of nanoparticles with reduced symmetries. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2012     

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.     

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.     

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.     

Density and chain conformation profiles of square-well chains confined in a slit by density-functional theory

Density and chain conformation profiles of square-well chains between two parallel walls were studied by using density-functional theory. The free energy of square-well chains is separated into two contributions: the hard-sphere repulsion and the attraction. The Heaviside function is used as the weighting function for both of the two parts. The equation of state of Hu et al. is used to calculate the excess free energy of the repulsive part. The equation of state of statistical associating fluid theory for chain molecules with attractive potentials of variable range [A. Gil-Villegas et al. J. Chem. Phys. 106, 4168 (1997)] is used to calculate the excess free energy of the attractive part. Because the wall is inaccessible to a mass center of a longer chain, there exists a sharp fall in the distribution of end-to-end distance near the wall as the chain length increases. When the average density of the system is not too low, the prediction of this work is in good agreement with computer simulation results for the density profiles and the chain conformation over a wide range of chain length, temperature, and attraction strength of the walls. However, when the average density and the temperature are very low, the prediction deviates to a certain degree from the computer simulation results for molecules with long chain length. A more accurate functional approximation is needed.     

Forces between nanorods with end-adsorbed chains in a homopolymer melt

Adsorbed or grafted polymers are often used to provide steric stabilization of colloidal particles. When the particle size approaches the nanoscale, the curvature of the particles becomes relevant. To investigate this effect for the case of cylindrical symmetry, I use a classical fluids density functional theory applied to a coarse-grained model to study the polymer-mediated interactions between two nanorods. The rods are coated with end-adsorbing chains and immersed in a polymer melt of chemically identical, nonadsorbing chains. The force between the nanorods is found to be nonmonotonic, with an attractive well when the two brushes come into contact with each other, followed by a steep repulsion at shorter distances. The attraction is due to the entropic phenomenon of autophobic dewetting, in which there is a surface tension between the brush and the matrix chains. These results are similar to previous results for planar and spherical polymer brushes in melts of the same polymer. The depth of the attractive well increases with matrix chain molecular weight and with the surface coverage. The attraction is very weak when the matrix chain molecular weight is similar to or smaller than the brush molecular weight, but for longer matrix chains the magnitude of the attraction can become large enough to cause aggregation of the nanorods.     

Polymer-particle mixtures: depletion and packing effects

The structure of polymers in the vicinity of spherical colloids is investigated by Monte Carlo simulations and integral equation theory. Polymers are represented by a simple bead-spring model; only repulsive Lennard-Jones interactions are taken into account. Using advanced trial moves that alter chain connectivity, depletion and packing effects are analyzed as a function of chain length and density, both at the bond and the chain level. Chain ends segregate to the colloidal surface and polymer bonds orient parallel to it. In the dilute regime, the polymer chain length governs the range of depletion and has a negligible influence on monomer packing in dense polymer melts. Polymers adopt an ellipsoidal shape, with the larger axis parallel to the surface of the particle, as they approach larger colloids. The dimensions are perturbed within the range of the depletion layer.     

Phase transition of short linear molecules adsorbed on solid surfaces from a density functional approach

A microscopic density functional theory is used to investigate the adsorption of short chains on strongly attractive solid surfaces. We analyze the structure of the adsorbed fluid and investigate how the layering transitions change with the change of the chain length and with relative strength of the fluid-solid interaction. The critical temperature of the first layering transition, rescaled by the bulk critical temperature, increases slightly with an increase of the chain length. We have found that for longer chains the layering transitions within consecutive layers are shifted toward very low temperatures and that their sequence is finally replaced by a single transition.     

An enhanced entangled polymer model for dissipative particle dynamics

We develop an alternative polymer model to capture entanglements within the dissipative particle dynamics (DPD) framework by using simplified bond-bond repulsive interactions to prevent bond crossings. We show that structural and thermodynamic properties can be improved by applying a segmental repulsive potential (SRP) that is a function of the distance between the midpoints of the segments, rather than the minimum distance between segments. The alternative approach, termed the modified segmental repulsive potential (mSRP), is shown to produce chain structures and thermodynamic properties that are similar to the softly repulsive, flexible chains of standard DPD. Parameters for the mSRP are determined from topological, structural, and thermodynamic considerations. The effectiveness of the mSRP in capturing entanglements is demonstrated by calculating the diffusion and mechanical properties of an entangled polymer melt.     

Effect of attractions on the structure of polymer solutions confined between surfaces: a density functional approach

A density functional theory is presented to study the effect of attractions on the structure of polymer solutions confined between surfaces. The polymer molecules have been modeled as a pearl necklace of freely jointed hard spheres and the solvent as hard spheres, both having Yukawa-type attractions and the mixture being confined between attractive Yukawa-type surfaces. The present theory treats the ideal gas free energy functional exactly and uses weighted density approximation for the hard chain and hard sphere contributions to the excess free energy functional. The attractive interactions are calculated using the direct correlation function obtained from the polymer reference interaction site model theory along with the mean spherical approximation closure. The theoretical predictions on the density profiles of the polymer and the solvent molecules are found to agree quite well with the Monte Carlo simulation results for varying densities, chain lengths, wall separations, and different sets of interaction potentials.