Self-consistent Field Lattice Model Research Articles

Self-Consistent Field Lattice Model for Polymer Networks.

A lattice model based on polymer self-consistent field theory is developed to predict the equilibrium statistics of arbitrary polymer networks. For a given network topology, our approach uses moment propagators on a lattice to self-consistently construct the ensemble of polymer conformations and cross-link spatial probability distributions. Remarkably, the calculation can be performed “in the dark”, without any prior knowledge on preferred chain conformations or cross-link positions. Numerical results from the model for a test network exhibit close agreement with molecular dynamics simulations, including when the network is strongly sheared. Our model captures nonaffine deformation, mean-field monomer interactions, cross-link fluctuations, and finite extensibility of chains, yielding predictions that differ markedly from classical rubber elasticity theory for polymer networks. By examining polymer networks with different degrees of interconnectivity, we gain insight into cross-link entropy, an important quantity in the macroscopic behavior of gels and self-healing materials as they are deformed.

Self-consistent field theoretic simulations of amphiphilic triblock copolymer solutions: Polymer concentration and chain length effects

Using the self-consistent field lattice model, polymer concentration $\bar{\phi}_{{P}}$ and chain length $N$ (keeping the length ratio of hydrophobic to hydrophilic blocks constant) the effects on temperature-dependent behavior of micelles are studied, in amphiphilic symmetric ABA triblock copolymer solutions. When chain length is increased, at fixed $\bar{\phi}_{{P}}$, micelles occur at higher temperature. The variations of average volume fraction of stickers $\bar{\phi}_\textrm{co}^\textrm{s}$ and the lattice site numbers $N_\textrm{co}^\textrm{ls}$ at the micellar cores with temperature are dependent on $N$ and $\bar{\phi}_{{P}}$, which demonstrates that the aggregation of micelles depends on $N$ and $\bar{\phi}_{{P}}$. Moreover, when $\bar{\phi}_{{P}}$ is increased, firstly a peak appears on the curve of specific heat $C_V$ for unimer-micelle transition, and then in addition a primary peak, the secondary peak, which results from the remicellization, is observed on the curve of $C_V$. For a long chain, in intermediate and high concentration regimes, the shape of specific heat peak markedly changes, and the peak tends to be a more broad peak. Finally, the aggregation behavior of micelles is explained by the aggregation way of amphiphilic triblock copolymer. The obtained results are helpful in understanding the micellar aggregation process.

Effect of polymer concentration and length of hydrophobic end block on the unimer-micelle transition broadness in amphiphilic ABA symmetric triblock copolymer solutions

The effects of the length of each hydrophobic end block N_{st} and polymer concentration \bar{\phi}_{P} on the transition broadness in amphiphilic ABA symmetric triblock copolymer solutions are studied using the self-consistent field lattice model. When the system is cooled, micelles are observed, i.e.,the homogenous solution (unimer)-micelle transition occurs. When N_{st} is increased, at fixed \bar{\phi}_{P}, micelles occur at higher temperature, and the temperature-dependent range of micellar aggregation and half-width of specific heat peak for unimer-micelle transition increase monotonously. Compared with associative polymers, it is found that the magnitude of the transition broadness is determined by the ratio of hydrophobic to hydrophilic blocks, instead of chain length. When \bar{\phi}_{P} is decreased, given a large N_{st}, the temperature-dependent range of micellar aggregation and half-width of specific heat peak initially decease, and then remain nearly constant. It is shown that the transition broadness is concerned with the changes of the relative magnitudes of the eductions of nonstickers and solvents from micellar cores.

Effect of distribution of stickers along backbone on temperature-dependent structural properties in associative polymer solutions

Effect of distribution of stickers along the backbone on structural properties in associating polymer solutions is studied using self-consistent field lattice model. Only two inhomogeneous morphologies, i.e., microfluctuation homogenous (MFH) and micelle morphologies, are observed. If the system is cooled, the solvent content within the aggregates decreases. When the spacing of stickers along the backbone is increased the temperature-dependent range of aggregation in MFH morphology and half-width of specific heat peak for homogenous solutions-MFH transition increase, and the symmetry of the peak decreases. However, with increasing spacing of stickers, the above three corresponding quantities related to micelles behave differently. It is demonstrated that the broad nature of the observed transitions can be ascribed to the structural changes which accompany the replacement of solvents in aggregates by polymer, which is consistent with the experimental conclusion. It is found that different effect of spacing of stickers on the two transitions can be interpreted in terms of intrachain and interchain associations.

Effects of polymer concentration and chain length on aggregation in physically associating polymer solutions

The effects of polymer concentration and chain length on aggregation in associative polymer solutions, are studied using self-consistent field lattice model. Only two inhomogenous morphologies, i.e. microfluctuation homogenous (MFH) and micelle morphologies, are observed in the systems with different chain lengths. The temperatures at which the above two inhomogenous morphologies first appear, which are denoted by T_MFH and T_m, respectively, are dependent on polymer concentration and chain length. The variation of the logarithm of critical MFH concentration with the logarithm of chain length fulfils a linear-fitting relationship with a slope equaling -1. Furthermore, the variation of the average volume fraction of stickers at the micellar core (AVFSM) with polymer concentration and chain length is focused in the system at T_m. It is founded by calculations that the above behavior of AVFSM, is explained in terms of intrachain and interchain associations.

Self-consistent field lattice model study on the phase behavior of physically associating polymer solutions

The phase behavior of physically associating polymer solutions, where the polymer chain contains a small fraction of "stickers" regularly placed along the backbone, is studied using self-consistent field lattice model. Two inhomogenous morphologies are observed. One is a microfluctuation homogenous (MFH) morphology, where the mean-field values of the local average concentrations of polymers phi(P)(r) and stickers phi(st)(r) slightly fluctuate around their respective bulk average values phi(P) and phi(st) and regularly from site to site. The other is a randomly close-packed micelle (RCPM) morphology. The structure of the micelle in RCPM morphology is similar to that of the "flower micelle" in the telechelic associative polymer system, where stickers are located in the core of the micelle and nonsticky groups in the corona. When phi(P) approximately or > 0.08, if homogenous associating polymer solutions are cooled, MFH morphology appears, and the system entirely changes from homogenous solutions (HS) to MFH morphology; If the solutions are cooled further, RCPM morphology appears. When phi(P) < 0.08, however, RCPM morphology appears immediately. If phi(P) < 0.53, a macroscopic phase separation, where the polymer rich phase is RCPM morphology, occurs. If phi(P) approximately or > 0.53, only RCPM morphology is found in the system. A peak appears in the temperature-dependent specific-heat curve C(V)(chi) at each transition point. For the HS-MFH transition, C(V)(chi) has an abrupt increase and a slow decrease, whereas for the MFH-RCPM transition, both the increase and the decrease in C(V)(chi) are slow. Furthermore, the system with only MFH morphology may be trapped in one of the two energy basins in a experimental time scale. However, the appearance of RCPM morphology means that the system is trapped in one of a series of "deeper" energy basins, and it is very difficult to jump off this deep basin into the one of MFH morphology or one of the other RCPM morphologies through thermal fluctuations.

Entanglement Network of the Polypropylene/Polyamide Interface. 3. Deformation to Fracture

Recently (Macromolecules 2000, 33, 1385, 1396), we have proposed a novel algorithm for generating entanglement network specimens of interfacial polymeric systems. The specimens are created by sampling the configurational distribution functions derived from a self-consistent mean field lattice model. Although overstretched strands can be relaxed by a Monte Carlo (MC) method, the specimens generated are not in detailed mechanical equilibrium. In this paper, we develop a method for relaxing the network with respect to its density distribution, and thereby imposing the condition of mechanical equilibrium, without changing the network topology. Our method rests on minimizing a free energy function characterizing the network, with respect to the coordinates of all entanglement points and chain ends. Contributions to the free energy include (a) the elastic energy due to stretching of the chain strands and (b) the free energy due to the repulsive and attractive (cohesive) interactions between segments. To calculate the interaction energy, a simple cubic grid is superimposed on the network. The repulsive interactions are calculated within each grid cell. The attractive interactions are calculated from contributions between cells and within each cell. We first apply the free energy minimization procedure to a simple system (bulk polypropylene) and obtain results which agree satisfactorily with experimental data and serve as a basis for parametrizing the interaction model. The free energy minimization is then applied to mechanically relax computer specimens of a polypropylene/polyamide6 (PP/PA6) interfacial system, strengthened by PP chains grafted onto the PA6 surface. The fully relaxed networks serve as a starting point for the mesoscopic simulation of fracture phenomena, caused by the application of tensile stress perpendicular to the interface. The network is deformed at a constant strain rate and the network topology evolves according to elementary mechanical processes of chain scission, chain slippage, disentanglement, and reentanglement. Chain slippage across an entanglement point occurs according to a Zhurkov activated rate equation with parameters derived from viscosity data. Each cycle of the kinetic MC algorithm used to track the deformation process consists of the imposition of a small incremental strain on the network, relaxation to mechanical equilibrium, introduction of the micromechanical processes mentioned above, and again relaxation to mechanical equilibrium. The MC cycles are repeated until fracture occurs. Results of the fracture experiments confirm the assertion, previously founded on structural grounds, that optimal adhesion in the PP/PA6 system examined is achieved for a relatively low surface grafting density of 0.1 chains/nm2.

Nanoscale clustering of RGD peptides at surfaces using Comb polymers. 1. Synthesis and characterization of Comb thin films.

Theoretical and experimental studies were conducted to elucidate the structure and properties of amphiphilic comb polymer thin films presenting nanoscale clusters of Arg-Gly-Asp (RGD) peptides for control of cell adhesion on biomaterials. Combs comprised of a poly(methyl methacrylate) backbone and short poly(ethylene oxide) side chains were synthesized, and peptides were tethered to the side chain ends to create nanoscale peptide clusters. In thin films, comb polymers containing >or = 30 wt % six to nine unit PEO side chains completely resisted adhesion of a model fibroblast cell line in the presence of 7.5% serum over 24 h. These same polymers modified with RGD peptides elicited tunable cell adhesion when mixed with unmodified combs in varying proportion. A self-consistent field lattice model of the interface between comb polymer films and water predicts an organization of the top molecular layer of comb polymer with the backbone oriented parallel to the interface in quasi-two-dimensional confinement and hydrophilic side chains extended in a brushlike layer into solution. This picture of a quasi-2D configuration is consistent with the observed surface properties of comb films in water as well as measurements of the RGD cluster density on mixed comb surfaces using fluorescent nanosphere labeling of ligand clusters.

Elastic Modulus at High Frequency of Polymerically Stabilized Suspensions

The elastic moduli of polymerically stabilized suspensions consisting of colloidal silica particles coated with endgrafted PDMS (Mn = 80 000) in heptane, were measured as a function of concentration. And the elastic modulus at high frequency was quantitatively described by model calculations with microscopic parameters. In the modeling of we essentially followed the method of Elliott and Russel [Elliott, S. L.; Russel, W. B. J. Rheology 1998, 42, 361]. To suit our case of strongly curved polymer layers, we made adaptations in the description of both the pair potential and hydrodynamic interactions. Anticipating that the predicted is sensitive to the modeling of the polymer brush, we explored two different models to predict the pair potential and the polymer layer thickness: a numerical self-consistent field lattice model [Wijmans, C. M.; Zhulina, E. B. Macromolecules 1993, 26, 7214] and an analytical method based on the approach of Li and Witten [Li, H.; Witten, T. A. Macromolecules 1994, 27, 449]. The polymer models were used separately in the model calculations for with the magnitude of their parameters obtained from (elaborate) characterizations of the particles. Both models result in a quantitative description of the experimental and thickness of the polymer layer.

Adhesion at Polymer-Solid Interfaces: Influence of Sticker Groups on Structure, Chain Connectivity and Strength

The influence of sticker groups (COOH) on the adhesion energy (G 1C) of a model carboxylated polybutadiene-amine terminated silane modified aluminum (cPBD-AIS) interface was examined using a T-peel test. An optimal COOH concentration of around 0.5 mol% was observed. At this concentration the interface possessed a maximum peel energy of 600 J/m2, which is approximately equal to the cohesive peel energy of PBD. The adhesion strength of the interface increased as the interaction parameter (°CHP-S) between COOH and the solid substrate increased. Additionally, G IC was strongly dependent on the annealing time, resulting from slow surface reorganization processes. The interface structure was explored using the self-consistent field lattice model (SCFLM) developed by Theodorou. An interphase region adjacent to the solid substrate was found as a consequence of a gradient in chain connectivity and a strong sticker group segregation near the solid surface. The modeling study provided information about how the interphase structure changed with φ and °CHP-S, explaining the experimental results well.

Polyelectrolyte-Induced Micellization of Charged Surfactants. Calculations Based on a Self-Consistent Field Lattice Model

The complexation of charged surfactant molecules and oppositely charged polyelectrolytes was studied by the use of a mean-field lattice theory for flexible polyelectrolytes and surfactants in solution. The effect on the critical aggregation concentration, cac, and the surfactant aggregation number, Nagg, of (i) salt concentration, (ii) polyelectrolyte concentration, (iii) linear charge density of the polyelectrolyte, and (iv) hydrophobicity of the polyelectrolyte was investigated. The changes in the cac and Nagg upon changes of polyelectrolyte properties were found to qualitatively agree with experimental results, but the predicted cac was slightly decreased at salt addition instead of being increased as experimentally observed. The results showed that the solvency of the polymer backbone affected the structure of the aggregate. A polyelectrolyte with a hydrophilic backbone was accumulated in the headgroup region and slightly outside it but did not otherwise penetrate the micelle. For the case with a hydr...

Effect of Copolymer Sequence Distribution upon Surface Structure and Interfacial Tension As Predicted by a Self-Consistent Field Lattice Model

Effect of Copolymer Sequence Distribution upon Surface Structure and Interfacial Tension As Predicted by a Self-Consistent Field Lattice Model

Adsorption of ionic surfactants on constant charge surfaces. Analysis based on a self-consistent field lattice model

Adsorption of ionic surfactants on constant charge surfaces. Analysis based on a self-consistent field lattice model

Micellization of ionic surfactants: calculations based on a self-consistent field lattice model

The equilibrium structure of micelles formed in aqueous solutions of ionic surfactants is studied by using the self-consistent field lattice approach of Scheutjens, Fleer, and Leermarkers. Equilibrium volume fractions profiles of micelles are obtained, and it appears that the surfactant head groups have a rather wide distribution. The boundary between the hydrocarbon core of the micelle and the solution is rather sharp and does not vary with salt concentration. An electrostatic potential develops, which causes accumulation of indifferent counterions in between the head groups and, consequently, leads to partial neutralization of the charges. The remaining net charge in the head group region is independent of the salt concentration. The calculated effects of chain length, salt concentration, and branching on the cmc and the aggregation numbers show a good agreement with experimental data. By use of lattices of different geometry, transitions from small spherical micelles to large rodlike aggregates can be predicted.

Characteristics of macromolecular monolayers grafted onto an impermeable boundary calculated by the self-consistent field method

The mean thickness and its fluctuation have been calculated by means of the self-consistent field lattice model for monolayers formed by flexible polymer chains grafted onto a planar insert surface at different graft densities σ −1. At σ⪆4 the mean layer height z̄ obeys the relationship z ̄ ≌0·52 Nσ −0·38 where N is the number of repeat units in the chain), in close agreement with the result of the scaling theory. Deviations from this formula appear in densely grafted systems, i.e., when the distance between the neighbouring chains drops below the length covered by two or three monomer units. Tke fluctuations in monolayer height are described by ( z ̄ 2- z ̄ 2) 1 2 ≌0·18 Nσ −0·325 a nd substantially exceed the fluctuations in size of an isolated macromolecule.