Simple model for grafted polymer brushes

The first theories of grafted polymer brushes assumed a step profile for the monomer density. Later, the real density profile was obtained from Monte Carlo or molecular dynamics simulations and calculated numerically using a self-consistent field theory. The analytical approximations of the solutions of the self-consistent field equations provided a parabolic dependence of the self-consistent field, which in turn led to a parabolic distribution for the monomer density in neutral brushes. As shown by numerical simulations, this model is not accurate for dense polymer brushes, with highly stretched polymers. In addition, the scaling laws obtained from the analytical approximations of the self-consistent field theory are identical to those derived from the earlier step-profile-approximation and predict a vanishing thickness of the brush at low graft densities, and a thickness exceeding the length of the polymer chains at high graft densities. Here a simple model is suggested to calculate the monomer density and the interaction between surfaces with grafted polymer brushes, based on an approximate calculation of the partition function of the polymer chains. The present model can be employed for both good and poor solvents, is compatible with a parabolic-like profile at moderate graft densities, and leads to an almost steplike density for highly stretched brushes. While the thickness of the brush depends strongly on solvent quality, it is a continuous function in the vicinity of the temperature. In good and moderately poor solvents, the interactions between surfaces with grafted polymer brushes are always repulsive, whereas in poor solvents the interactions are repulsive at small separations and become attractive at intermediate separation distances, in agreement with experiment. At large separations, a very weak repulsion is predicted.     

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.     

双峰聚合物分子刷的层化机理

建立了用于模拟双峰聚合物分子刷相结构的自洽场理论. 模拟结果表明, 良溶剂条件能够促使双峰聚合物分子刷裂分为内外两个亚分子层, 其中短链居于内分子层, 而长链伸展到外分子层. 体系溶解性的加强不仅使聚合物的密度分布逐渐趋近强分凝理论的解析结果, 而且加大了分子链的伸展和链段的局部取向程度. 分子链接枝密度的增加能够促使分子刷的层化, 并且在良溶剂区域, 不同接枝密度的分子链密度分布可以回归到同一条主线. 在良溶剂条件下, 长链的聚合度对短链的密度分布影响不大, 但能够导致长链向外分子层扩展.     

Interactions between polymer brush-coated spherical nanoparticles: the good solvent case

The interaction between two spherical polymer brushes is studied by molecular dynamics simulation varying both the radius of the spherical particles and their distance, as well as the grafting density and the chain length of the end-grafted flexible polymer chains. A coarse-grained bead-spring model is used to describe the macromolecules, and purely repulsive monomer-monomer interactions are taken throughout, restricting the study to the good solvent limit. Both the potential of mean force between the particles as a function of their distance is computed, for various choices of the parameters mentioned above, and the structural characteristics are discussed (density profiles, average end-to-end distance of the grafted chains, etc.). When the nanoparticles approach very closely, some chains need to be squeezed out into the tangent plane in between the particles, causing a very steep rise of the repulsive interaction energy between the particles. We consider as a complementary method the density functional theory approach. We find that the quantitative accuracy of the density functional theory is limited to large nanoparticle separation and short chain length. A brief comparison to Flory theory and related work on other models also is presented.     

Scaling and structural properties of a polymer in a simple solvent: A study based on integral equations

We present a statistical mechanical theory for polymer–solvent systems based on integral equations derived from the polymer Kirkwood hierarchy. Integral equations for pair monomer–monomer, monomer–solvent, and solvent–solvent correlation functions yield polymer–solvent distribution, chain conformation in three dimensions, and scaling properties associated with polymer swell and collapse in athermal, good, and poor solvents. Variation of polymer properties with solvent density and solvent quality is evaluated for chains having up to 100 bonds. In good solvents, the scaling exponent v has a constant value of about 0.61 at different solvent densities computed. For the athermal solvent case, the gyration radius and scaling exponent decrease with solvent density. In a poor solvent, the chain size scales as N^v with the value of the exponent being about 0.3, compared with the mean field value of ⅓. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 3025–3033, 1998     

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.     

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.     

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.     

Evaluating the accuracy of a density functional theory of polymer solutions with additive hard sphere diameters

We assess the accuracy of a density functional theory for athermal polymer solutions, consisting of solvent particles with a smaller radius than that of the monomers. The monomer and solvent density profiles in a slit bound by hard, flat, and inert surfaces are compared with those obtained by a Metropolis Monte Carlo simulation. At the relatively high density at which the comparison is performed, there are considerable packing effects at the walls. The density functional theory introduces a simple weight function to describe nonlocal correlations in the fluid. A recent study of surface forces in polymer solutions used a different weighting scheme to that proposed in this article, leading to less accurate results. The implications of the conclusions of that study are discussed.     

Semiflexible polymers grafted to a solid planar substrate: changing the structure from polymer brush to "polymer bristle"

Monte Carlo simulations are presented for a coarse-grained model of polymer brushes with polymers having a varying degree of stiffness. Both linear chains and ring polymers grafted to a flat structureless non-adsorbing substrate surface are considered. Applying good solvent conditions, it is shown that with growing polymer stiffness the brush height increases significantly. The monomer density profiles for the case of ring polymers (chain length N(R) = 64) are very similar to the case of corresponding linear chains (N(L) = 32, grafting density larger by a factor of two) in the case of flexible polymers, while slight differences appear with increasing stiffness. Evidence is obtained that the chain dynamics in brushes is slowed down dramatically with increasing stiffness. Very short stiff rings (N(R) ≤ 16) behave like disks, grafted to the substrate such that the vector, perpendicular to the disk plane, is oriented parallel to the substrate surface. It is suggested that such systems can undergo phase transitions to states with liquid crystalline order.     

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.     

Comparison of random-walk density functional theory to simulation for bead-spring homopolymer melts

Density profiles for a homopolymer melt near a surface are calculated using a random-walk polymeric density functional theory, and compared to results from molecular dynamics simulations. All interactions are of a Lennard-Jones form, for both monomer-monomer interactions and surface-monomer interactions, rather than the hard core interactions which have been most investigated in the literature. For repulsive systems, the theory somewhat overpredicts the density oscillations near a surface. Nevertheless, near quantitative agreement with simulation can be obtained with an empirical scaling of the direct correlation function. Use of the random phase approximation to treat attractive interactions between polymer chains gives reasonable agreement with simulation of dense liquids near neutral and attractive surfaces.     

A simple theory of the interaction between polymer brushes immersed in a supercritical fluid

We use a simple two-order parameter model to describe the interaction between the brushes of polymers terminally attached to flat surfaces immersed in a supercritical solvent. Our approach makes it possible to take into account the high compressibility of the supercritical solvent, which proves to give a significant contribution to the disjoining force acting between polymer brushes. Our theory explains why the interaction between brushes can change from repulsive to attractive with decreasing solvent density. This theoretical finding is verified by making a comparison with recent computer simulations. A reasonably good agreement between the results of the present theory and the simulations is found.     

Universal consequences of the presence of excluded volume interactions in dilute polymer solutions undergoing shear flow

The role of solvent quality in determining the universal material properties of dilute polymer solutions undergoing steady simple shear flow is examined. A bead-spring chain representation of the polymer molecule is used, and the influence of solvent molecules on polymer conformations is modelled by a narrow Gaussian excluded volume potential that acts pairwise between the beads of the chain. Brownian dynamics simulations data, acquired for chains of finite length, and extrapolated to the limit of infinite chain length, are shown to be model independent. This feature of the narrow Gaussian potential, which leads to results identical to a delta-function repulsive potential, enables the prediction of both universal crossover scaling functions and asymptotic behavior in the excluded volume limit. Universal viscometric functions, obtained by this procedure, are found to exhibit increased shear thinning with increasing solvent quality. In the excluded volume limit, they are found to obey power law scaling with the characteristic shear rate beta, in close agreement with previously obtained renormalization group results. The presence of excluded volume interactions is also shown to lead to a weakening of the alignment of the polymer chain with the flow direction.     

Simulation and theory of flexible equilibrium polymers under poor solvent conditions

Grand canonical Monte Carlo simulation and simple statistical thermodynamic theory are used to model the aggregation and phase separation of systems of reversibly polymerizing monomers, capable of forming chains with or without the ability to cyclize into rings, with isotropic square-well attractions between nonbonded pairs of monomers. The general trend observed in simulation of chain-only systems, as predicted in a number of published theoretical works, is that the critical temperature for phase separation increases and the critical monomer density decreases with rising polymer bond strength. Introduction of the equilibrium between chains and rings into the theory lowers the predicted critical temperature and increases the predicted critical density. While the chain-only theories predict a vanishing critical density in the limit of complete polymerization, when ring formation is taken into account the predicted critical density in the same limit approaches the density of the onset of the ring-chain transition. The theoretically predicted effect of cyclization on chemical potential is in good qualitative agreement with a subset of simulation results in which chain-only systems were compared with equilibrium mixtures of rings and chains. The influence of attractions on the aggregation number and radius of gyration of chains and rings observed in simulations is also discussed.     

基于密度泛函理论研究二元排斥Yukawa流体的表面结构性质

基于自由能密度泛函理论(DFT)考察了二元排斥Yukawa (HCRY)流体在不同外场下的密度分布. 基于微扰理论, 体系的Helmholtz自由能泛函采用硬球排斥部分和长程色散部分贡献之和, 其中Kierlik和Rosinberg的加权密度近似(WDA)被用来计算硬球排斥部分, 而色散部分采用平均场理论(MFT)进行描述. 为了验证DFT计算结果的合理性, 研究中采用巨正则Monte Carlo(GCMC)模拟计算了在不同主体相密度、硬核直径和位能参数比的条件下二元HCRY混合流体的密度分布. 结果表明, 该DFT计算结果与GCMC模拟值吻合良好.     

Polymer brushes in solvents of variable quality: molecular dynamics simulations using explicit solvent

The structure and thermodynamic properties of a system of end-grafted flexible polymer chains grafted to a flat substrate and exposed to a solvent of variable quality are studied by molecular dynamics methods. The macromolecules are described by a coarse-grained bead-spring model, and the solvent molecules by pointlike particles, assuming Lennard-Jones-type interactions between pairs of monomers (epsilon(pp)), solvent molecules (epsilon(ss)), and solvent monomer (epsilon(ps)), respectively. Varying the grafting density sigma(g) and some of these energy parameters, we obtain density profiles of solvent particles and monomers, study structural properties of the chain (gyration radius components, bond orientational parameters, etc.), and examine also the profile of the lateral pressure P( parallel)(z), keeping in the simulation the normal pressure P( perpendicular) constant. From these data, the reduction of the surface tension between solvent and wall as a function of the grafting density of the brush has been obtained. Further results include the stretching force on the monomer adjacent to the grafting site and its variation with solvent quality and grafting density, and dynamic characteristics such as mobility profiles and chain relaxation times. Possible phase transitions (vertical phase separation of the solvent versus lateral segregation of the polymers into "clusters," etc.) are discussed, and a comparison to previous work using implicit solvent models is made. The variation of the brush height and the interfacial width of the transition zone between the pure solvent and the brush agrees qualitatively very well with corresponding experiments.     

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.     

Theory of heterogeneities in polymer networks

Following Edwards’ ideas we present main experimental results and the theory of random heterogeneities in neutral and charged networks obtained by instantaneous as well as chemical cross-linking of a melt and semidilute solution of linear chains. We study how random monomer density patterns in such networks change after swelling and stretching. We also describe main features of monomer density correlation functions, which determine the neutron and light scattering on spatial heterogeneities. We show that largescale cross-link density patterns written into network structure in the melt or semidilute state, can be revealed upon swelling by monitoring the monomer density patterns. We demonstrate that while isotropic deformations in good solvent yield magnified images of the original pattern, anisotropic deformations distort the image. We study how the monomer density image changes under different solvent conditions and discuss the difference between deformations of the density images in gels and ordinary solids. Possible tests of our predictions and some potential applications are proposed.     

Effect of solvent quality on the dispersibility of polymer-grafted spherical nanoparticles in polymer solutions

In this work, lattice-based self consistent field theory is used to study the structural properties of individual polymer-grafted spherical nanopartices and particle-particle interactions in polymer melts and solutions under variable solvent conditions. Our study has focused on the depth of the minimum in the potential of mean force between the two brush-coated nanoparticles, if such a minimum occurs, and we have also addressed the corresponding radial density profiles of free and grafted chains around a single nanoparticle, in an attempt to clarify the extent of correlation between the depth of the minimum, W(min), and the parameter δ characterizing the interpenetration between the profiles of free and grafted chains. Although one cannot establish a simple one-to-one correspondence between W(min) and δ, we do find common trends, in particular, if the solvent conditions for free and grafted chains differ: varying the volume fraction of the free chains, δ typically exhibits a broad minimum, corresponding to a region where the magnitude of W(min) exceeds thermal energy k(B)T, leading to particle aggregation.