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.     

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.     

The theory of polymer adsorption during polymerization reactions

Exact values are obtained for the grand canonical partition function of model chains absorbed at a surface. On a six-choice simple cubic lattice, chains are generated as function of the activity of chain bonds. The molecular weight distribution of adsorbed chains and of solution chains in equilibrium with them is obtained. It is shown that the average molecular weight of adsorbed chains is higher than that of solution chains. The difference is the more pronounced the flatter the chain is when adsorbed at the surface. When the activity of chain bond; increases the surface coverage increases, more segments occur in loops extending into the solution, and the average molecular weight of solution chains approaches that of adsorbed chains. The segment density in loops as function of distance from the surface is shown to be exponential, in agreement with previous results. If polymer adsorption is high and the concentration in solution is small, the number of segments in tails is amall. The dimensions of the adsorbed polymer chain are smaller than those in solution of the same molecular weight. It is suggested that if polymer adsorption during polycondensation reactions with bond interchanges is measured, the bothersome tirne effects, invariably observed, might be eliminated. In this manner a valid comparison between theoretical and experimental results might become feasible.     

Molecular dynamics simulations of polyelectrolyte adsorption

We have performed molecular dynamics simulations of polyelectrolyte adsorption at oppositely charged surfaces from dilute polyelectrolyte solutions. In our simulations, polyelectrolytes were modeled by chains of charged Lennard-Jones particles with explicit counterions. We have studied the effects of the surface charge density, surface charge distribution, solvent quality for the polymer backbone, strength of the short-range interactions between polymers and substrates on the polymer surface coverage, and the thickness of the adsorbed layer. The polymer surface coverage monotonically increases with increasing surface charge density for almost all studied systems except for the system of hydrophilic polyelectrolytes adsorbing at hydrophilic surfaces. In this case the polymer surface coverage saturates at high surface charge densities. This is due to additional monomer-monomer repulsion between adsorbed polymer chains, which becomes important in dense polymeric layers. These interactions also preclude surface overcharging by hydrophilic polyelectrolytes at high surface charge densities. The thickness of the adsorbed layer shows monotonic dependence on the surface charge density for the systems of hydrophobic polyelectrolytes for both hydrophobic and hydrophilic surfaces. Thickness is a decreasing function of the surface charge density in the case of hydrophilic surfaces while it increases with the surface charge density for hydrophobic substrates. Qualitatively different behavior is observed for the thickness of the adsorbed layer of hydrophilic polyelectrolytes at hydrophilic surfaces. In this case, thickness first decreases with increasing surface charge density, then it begins to increase.     

Surface properties of bottle-brush polyelectrolytes on mica: effects of side chain and charge densities

Surface properties of a series of cationic bottle-brush polyelectrolytes with 45-unit-long poly(ethylene oxide) side chains were investigated by phase modulated ellipsometry and surface force measurements. The evaluation of the adsorbed mass of polymer on mica by means of ellipsometry is complex due to the transparency of mica and its birefringence and low dielectric constant. We therefore employed a new method to overcome these difficulties. The charge and the poly(ethylene oxide) side chain density of the bottle-brush polymers were varied from zero charge density and one side chain per segment to one charge per segment and no side chains, thus spanning the realm from a neutral bottle-brush polymer, via a partly charged brush polyelectrolyte, to a linear fully charged polyelectrolyte. The adsorption properties depend crucially on the polymer architecture. A minimum charge density of the polymer is required to facilitate adsorption to the oppositely charged surface. The maximum adsorbed amount and the maximum side chain density at the surface are obtained for the polymer with 50% charged segments and the remaining 50% of the segments carrying poly(ethylene oxide) side chains. It is found that brushlike layers are formed when 25-50% of the segments carry poly(ethylene oxide) side chains. In this paper, we argue that the repulsion between the side chains results in an adsorbed layer that is non-homogeneous on the molecular level. As a result, not all side chains will contribute equally to the steric repulsion but some will be stretched along the surface rather than perpendicular to it. By comparison with linear polyelectrolytes, it will be shown that the presence of the side chains counteracts adsorption. This is due to the entropic penalty of confining the side chains to the surface region.     

Interface effects in nematic liquid-crystalline structures of polymer solutions

A solution of long semirigid linear macromolecules was considered. The liquid-crystalline nematic ordering in the solution was analyzed theoretically using an Onsager-type approach. The orientation entropy was calculated in the frameworks of Lifshits' approach, successfully developed for this system originally by Khokhlov and Semenov. For homogeneous liquid-crystalline phase using the third virial approximation for intersegmental steric interaction the orientation distribution function, the free energy density, the isotropic-nematic coexistence and the spinodal conditions were computed numerically for two types of polymer flexibility mechanism: persistent chains and chains of freely joint segments. For the asymptotically exact second virial approximation the applicability region was analyzed. We considered the general equations, which describe the concentration and orientational segment distribution for a semirigid persistent polymer chain at a surface (or interface) of any shape and orientation. These equations were numerically solved for the case when the nematic director axis was perpendicular to a planar interface boundary between the real coexisting nematic and isotropic phases. The coordinate-dependencies of the polymer concentration and of the order-parameter take the smooth two-steps form in the interface region.     

Structure and molecular motion of poly(ethylene oxide) chains adsorbed on a silica-tethered poly(methyl methacrylate) by the spin label method

The spin-label method was used to study the structure and molecular motion of poly(ethylene oxide) (PEO) chains adsorbed on a silica-tethered poly(methyl methacrylate) (PMMA). Spin-labelled PEO with a narrow molecular weight distribution, having number averaged molecular weight (M _N)=6.0×10³, was adsorbed on the surface of the silica-tethered PMMA with various grafting ratios in carbon tetrachloride solution at 35?°C. ESR spectra were measured at various temperatures after the samples were completely dried. The ESR spectra are composed of two spectra arising from spin-labels attached to “train” and “tail” segments, which are strongly and weakly interacted with the silica surface, respectively. The fractional amount of the “tail” segments increases extremely with the grafting ratio of PMMA. Molecular mobility of the PEO chains estimated from the temperature dependence of the ESR spectra also decreases significantly with the grafting ratio of PMMA. Structure and molecular motion of the PMMA chains tethered on the silica were also studied using the spin-labelled PMMA. Consequently, parts of the PEO segments penetrate into the PMMA chains and is adsorbed on the silica surface (“train” segments), whereas parts of the PMMA segments protrude from the surface. The other PEO segments are entangled with the tethered PMMA chains (“tail” segments).     

Polymer chain of freely‐jointed segments in nematic molecular mean‐field

The distribution function of orientations of a segment, which interacts with the orienting field and is within the chain with given the end‐to‐end distance‐vector, is calculated. The number of segments per chain is finite. The Lagrange method of conditional minimization of the chain free energy (the Helmholtz function) functional is used. Constraints for the segment orientations stem from fixed the chain end‐to‐end distance‐vector. Hence, Lagrange multipliers, energy, free energy, and entropy, for the chain with given the end‐to‐end distance‐vector, are calculated. Then, the distribution function of values of that vector is obtained. Furthermore, an average free energy per chain inside the polymer network with given a topological structure, the system self‐deformation, and modulus of elasticity are calculated and discussed in Gaussian limit, that is, for the number of segments per chain tending to infinity. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 138–144, 2010     

界面层中模型高分子链构象的统计理论

高分子自由链在溶液、熔体或固体中的构象有完整的理论和实验工作 ,已是成熟的知识[1～ 4] .近年来 ,受限大分子构象的探索已成为新的热点[5] .界面是一种最简单的限制 ,将对其附近的高分子产生显著的影响 .除理论家的重视以外 ,界面附近高分子构象问题还与许多应用领域相关 .例如 ,液体高分子的表面张力 ,固体高分子的粘附 ,滑移 ,磨蚀等力学特性 ,甚至它们的光学和电学性质都与界面上的组成和分子构象相关[6] .ｄｅＧｅｎｎｅｓ[7] 最早强调界面附近的分子链构象问题的重要性 .他首先从理论上猜测长链的链端若与界面有强吸引作用 ,则它们…     

Properties of Star-Branched Polymer Chains Near a Surface

We considered two model systems of star-branched polymers near an impenetrable surface. The model chains were constructed on a simple cubic lattice. Each star polymer consisted of f = 3 arms of equal length and the total number of segments was up to 799. The excluded volume effect was included into these models only and therefore the system was studied at good solvent conditions. In the first model system polymer chain was terminally attached with one arm to the surface. The grafted arm could slide along the surface. In the second system the star-branched chain was adsorbed on the surface and the strength of adsorption was were varied. The simulations were performed using the dynamic Monte Carlo method with local changes of chain conformations. The internal and local structures of a polymer layer were determined. The lateral diffusion and internal mobility of star-branched chains were studied as a function of strength of adsorption and the chain length. The lateral diffusion and internal mobility of star-branched chains were studied as a function of strength of adsorption and the chain length. It was shown that the behavior of grafted and weakly adsorbed chains was similar to that of a free three-dimensional polymer, while the strongly adsorbed chains behave as a two-dimensional system.     

Monte Carlo simulation for the adsorption of symmetric triblock copolymers II. Adsorption layer information

By using Monte Carlo simulation, adsorption of both end-adsorbed and middle-adsorbed symmetric triblock copolymers from a non-selective solvent on an impenetrable surface has been studied. Influences of the adsorption energy, the bulk concentration, the chain composition and the chain length on the adsorption behavior including the surface coverage, the adsorption amount and the layer thickness are presented. It is shown that the total surface coverage for both end-adsorbed and middle-adsorbed copolymers increases monotonically as the bulk concentration increases. The higher the adsorption energy and the more the attractive segments, the higher the total surface coverage is exhibited. Surface coverage θ decreases with increasing the length of the non-attractive segments, but the product of θ and the proportion of the non-attractive segments in a triblock copolymer chain is nearly independent of the chain length. The adsorption amount increases almost monotonically with the bulk concentration. The logarithm of the adsorption amount is a linear function of the reciprocal of the reduced temperature. When the adsorption energy is large, the adsorption amount exhibits a maximum as the composition of the attractive segment increases. The adsorption isotherms of copolymers with different length of the non-attractive segments can be mapped onto a single curve under certain energy indicating that copolymers with different chain length have the same adsorption amount. The adsorption layer thickness for the end-adsorbed copolymers decreases as the energy and the number of adsorbing segments increases. The longer non-attractive segments, the larger adsorbed layer thickness is found. The tails mainly governs the adsorption layer thickness.     

Exact density functional theory for ideal polymer fluids with nearest neighbor bonding constraints

We present a new density functional theory of ideal polymer fluids, assuming nearest-neighbor bonding constraints. The free energy functional is expressed in terms of end site densities of chain segments and thus has a simpler mathematical structure than previously used expressions using multipoint distributions. This work is based on a formalism proposed by Tripathi and Chapman [Phys. Rev. Lett. 94, 087801 (2005)]. Those authors obtain an approximate free energy functional for ideal polymers in terms of monomer site densities. Calculations on both repulsive and attractive surfaces show that their theory is reasonably accurate in some cases, but does differ significantly from the exact result for longer polymers with attractive surfaces. We suggest that segment end site densities, rather than monomer site densities, are the preferred choice of "site functions" for expressing the free energy functional of polymer fluids. We illustrate the application of our theory to derive an expression for the free energy of an ideal fluid of infinitely long polymers.     

Conformation of Chains in Cores of Block Copolymer Micelles with Solubilized Homopolymer: a Monte Carlo Study

A system of compatible self‐avoiding polymer chains solubilized in spherical cores of block copolymer micelles was studied by lattice Monte Carlo simulations. The core is modeled as a spherical cavity on a simple cubic lattice, filled in partially by tethered (core‐forming) chains and partially by free (solubilized) chains. Molecular parameters (e.g., the ratio of the contour length of the model chains to the core radius) correspond to those in real micellar systems. The density (the fraction of lattice sites occupied) is 0.6 which corresponds to swollen micellar cores in real micellar systems. Simulations yield a constant segment density profile in the core. Both the tethered and solubilized chains acquire an ellipsoidal shape. The ellipsoids equivalent to both types of chains are more spherical than those in a melt and strongly oriented. The chains in the core show a Gaussian‐like behavior. Minor deviations from Gaussian behavior for tethered chains are due to surface effects.     

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

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

Investigation of the microstructure of micelles formed by hard-sphere chains interacting via size nonadditivity by discontinuous molecular dynamics simulation

Micelle formation by short nonadditive hard surfactant chains was investigated at different size ratios, reduced densities, and nonadditivity parameters using molecular dynamics simulation. It was found that spherical, cylindrical, lamellar, and reverse micelles can form in systems with different head, tail, and solvent characteristics. Hard-core surfactant chains composed of a head segment and three tail segments were simulated in a solvent of hard spheres. The formation of micelles was found to be a strong function of the packing fraction and nonadditivity parameter. Micelles were more stable at higher densities and larger nonadditivity parameters. At lower densities, micelles tended to break into small, dynamic globules.     

Adsorption characteristics of bottle-brush polymers on silica: effect of side chain and charge density

The adsorption behavior of bottle-brush polymers with different charge/PEO ratio on silica was studied using optical reflectometry and QCM-D. The results obtained under different solution conditions clearly demonstrate the existence of two distinct adsorption mechanisms depending on the ratio of charge/PEO. In the case of low-charge density brush polymers (0-10 mol %), the adsorption occurs predominantly through the PEO side chains. However, the presence of a small amount of charge along the backbone (as low as 2 mol %) increases the adsorption significantly above that of the uncharged bottle-brush polymer in pure water. As the charge density of the brush polymers is increased to 25 mol % or larger the adsorption occurs predominantly through electrostatic interactions. The adsorbed layer structure was studied by measuring the layer dissipation using QCM-D. The adsorbed layer formed by the uncharged brush polymer dissipates only a small amount of energy that indicates that the brush lie along the surface, the scenario in which the maximum number of PEO side chains interact with the surface. The adsorbed layers formed by the low-charge density brush polymers (2-10 mol %) in water are more extended, which results in large energy dissipation, whereas those formed by the high-charge density brush polymers (50-100 mol %) have their backbone relatively flat on the surface and the energy dissipation is again low.     

Polymer adsorption and its effect on the stability of hydrophobic colloids

Summary As a first step in a study on the interaction between polymers and hydrophobic colloids we investigated in detail the adsorption of polyvinyl alcohol (PVA) on aqueous silver iodide sols. The adsorption is irreversible. Adsorption isotherms are of the highaffinity type. The amount adsorbed increases with molecular weight and with the fraction of acetate groups in the PVA chain. The effective thickness of the adsorbed layer was determined viscosimetrically and independently checked by an electrophoretic method. Double layer studies enabled the determination of the occupancy of the first layer on the surface by polymer segments. It was found that even at maximal coverage with polymer this layer is still about 30% void. The combination of these data enabled the assessment of the polymer segment distribution. It was found that with not too low coverages the distribution isHoeve- like. The distribution, thus obtained reflects itself in the flocculation of AgI sols by PVA.Presented at the 25th Colloid-Meeting in Munich, October 13–15, 1971     

Elastic behavior of comb-like polymer chains

<正>Three-dimensional Monte Carlo simulations of comb-like polymer chains with various backbone lengths N_b,arm lengths N_a and arm densities m are carried out to study the elastic behavior of comb-like polymer chains.The radius of gyration,the shape factors and bond length in different cases during elastic process are calculated,and it is found that the comb-like polymer molecules with longer backbone or shorter arm are more close to linear chains.But the arm density m affects the chain conformation non-monotonously.Some thermodynamic properties are also studied.Average Helmholtz free energy and elastic force f all increase with elongation ratioλfor all chains.     

Statistics of polymer chains in orienting fields

The theory of a freely jointed polymer chain is modified by introduction of interactions between dipole chain segments and an orienting field. Such a field results either from external forces (e.g. external electric or magnetic fields) or represents interactions between dipole segments of chains (molecular mean-field). The distribution of orientations of chain segments and the free energy of a chain in such orienting fields are calculated and discussed.