Continuum theory of polymer crystallization

We present a kinetic model of crystal growth of polymers of finite molecular weight. Experiments help to classify polymer crystallization broadly into two kinetic regimes. One is observed in melts or in high molar mass polymer solutions and is dominated by nucleation control with G approximately exp(1/TDeltaT), where G is the growth rate and DeltaT is the supercooling. The other is observed in low molar mass solutions (as well as for small molecules) and is diffusion controlled with G approximately DeltaT, for small DeltaT. Our model unifies these two regimes in a single formalism. The model accounts for the accumulation of polymer chains near the growth front and invokes an entropic barrier theory to recover both limits of nucleation and diffusion control. The basic theory applies to both melts and solutions, and we numerically calculate the growth details of a single crystal in a dilute solution. The effects of molecular weight and concentration are also determined considering conventional polymer dynamics. Our theory shows that entropic considerations, in addition to the traditional energetic arguments, can capture general trends of a vast range of phenomenology. Unifying ideas on crystallization from small molecules and from flexible polymer chains emerge from our theory.     

Density functional theory for inhomogeneous associating chain fluids

We propose a nonlocal density functional theory for associating chain molecules. The chains are modeled as tangent spheres, which interact via Lennard-Jones (12,6) attractive interactions. A selected segment contains additional, short-ranged, highly directional interaction sites. The theory incorporates an accurate treatment of the chain molecules via the intramolecular potential formalism and should accurately describe systems with strongly varying external fields, e.g., attractive walls. Within our approach we investigate the structure of the liquid-vapor interface and capillary condensation of a simple model of associating chains with only one associating site placed on the first segment. In general, the properties of inhomogeneous associating chains depend on the association energy. Similar to the bulk systems we find the behavior of associating chains of a given length to be in between that for the nonassociating chains of the same length and that for the nonassociating chains twice as large.     

Integral equation theory for two-dimensional polymer melts

The polymer reference interaction site model theory is investigated for two-dimensional polymer melts composed of freely-jointed hard disk chains and tangent-disk rods. Exact results for the intramolecular pair correlation functions are input into the theory, and predictions of the theory for the intermolecular pair correlation functions are tested via comparison with simulation. The theory is not as accurate for this system as it is for three-dimensional polymer melts, and the quantitative predictions are not good except at the highest area fractions. Possible reasons for the deficiency in the theory are discussed.     

Comprehensive theory for star-like polymer micelles; combining classical nucleation and polymer brush theory

A comprehensive theory is proposed that combines classical nucleation and polymer brush theory to describe star-like polymer micelles. With a minimum of adjustable parameters, the model predicts properties such as critical micelle concentrations and micellar size distributions. The validity of the present theory is evidenced in direct comparison to experiments; this revealed that the proportionality constant in the Daoud-Cotton model is of the order of unity and that the star-limit is valid down to relatively short corona chains. Furthermore, we show that the predicted saddle points in the free energy correspond to those solutions that are accessible with self-consistent field methods for self-assembly.     

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.     

Modified interfacial statistical associating fluid theory: a perturbation density functional theory for inhomogeneous complex fluids

A density functional theory based on Wertheim's first order perturbation theory is developed for inhomogeneous complex fluids. The theory is derived along similar lines as interfacial statistical associating fluid theory [S. Tripathi and W. G. Chapman, J. Chem. Phys. 122, 094506 (2005)]. However, the derivation is more general and applies broadly to a range of systems, retaining the simplicity of a segment density based theory. Furthermore, the theory gives the exact density profile for ideal chains in an external field. The general avail of the theory has been demonstrated by applying the theory to lipids near surfaces, lipid bilayers, and copolymer thin films. The theoretical results show excellent agreement with the results from molecular simulations.     

Graph theory of viscoelastic and configurational properties of Gaussian chains

The spring-and-bead model proposed by Rouse and Zimm can theoretically treat the viscoelastic behaviour of polymers. In this paper, we first point out that the Rouse and Zimm matrices in the molecular theory of polymer viscoelasticities are equivalent to the adjacency matrix and admittance (or Kirchhoff) matrix in graph theory, respectively. In order to solve the eigen-value problems of Rouse and Zimm matrices, the matrices are first represented by their corresponding eigen-graphs, which reflect the topological structure of the real chain. Starting from the eigen-graph, instead of tedious mathematics, the eigen-value problems are solved by a series of simple graphic operations, such as cutting-off the bonds, removing the closed pathways, etc. The eigen-polynomial of Rouse and Zimm matrices and the viscoelastic properties of the chain are obtained by using the theorems given in this paper. It is also shown that the eigen-polynomial of the chain can be greatly reduced if the chain graph has elements of symmetry. As the Rouse theory of viscoelasticity is closely related to the conformational statistics of Gaussian chains, it is demonstrated that the graph-theoretic approach developed here can also be applied to solve the configurational properties of Gaussian chains, such as the distribution function of the radius of gyration and its moments, the shape of a Gaussian chain, etc. We have also demonstrated that the graph-theoretic approach developed here is also applicable to copolymeric chains.     

Analytical theory of ideal polydisperse polymers at interfaces

We use a recently developed continuum theory to present an exact treatment of the interfacial properties of ideal polymers displaying Schulz-Flory polydispersity. Our results are remarkably compact and can be derived from the properties of equilibrium, ideal polymers at interfaces. We apply our theory to a number of cases, including, non-adsorbing and adsorbing surfaces, as well as telechelic chains.     

Density-functional theory and Monte Carlo simulation for the surface structure and correlation functions of freely jointed Lennard-Jones polymeric fluids

We present a nonlocal density-functional theory of polymeric fluids consisting of freely jointed Lennard-Jones chains with explicit consideration of the segment size, van der Waals attraction, and structural correlations due to chain connectivity. The excess Helmholtz energy functional is derived from a modified fundamental measure theory for the short-ranged repulsion and the first-order thermodynamic perturbation theory for chain connectivity. The contribution of the long-ranged attraction to the Helmholtz energy functional is taken into account using a quadratic density expansion with the direct correlation function obtained from the first-order mean-spherical approximation. The numerical performance of the density-functional theory is compared well with the simulation results from this work as well as those from the literature for the segment-level density profiles and correlation functions of Lennard-Jones chains in slit pores, near isolated nanoparticles, or in bulk.     

Integral equation theory for athermal solutions of linear polymers

An integral equation model is developed for athermal solutions of flexible linear polymers with particular reference to good solvent conditions. Results from scaling theory are used in formulating form factors for describing the single chain structure, and the impact of solvent quality on the chain fractal dimension is accounted for. Calculations are performed within the stringlike implementation of the polymer reference interaction site model with blobs (as opposed to complete chains) treated as the constituent structural units for semidilute solutions. Results are presented for the second virial coefficient between polymer coils and the osmotic compressibility as functions of the chain length and polymer volume fraction, respectively. Findings from this model agree with results from scaling theory and experimental measurements, as well as with an earlier investigation in which self-avoiding chains were described using Gaussian form factors with a chain length and concentration-dependent effective statistical segment length. The volume fractions at the threshold for connectedness percolation are evaluated within a coarse-grained closure relation for the connectedness Ornstein-Zernike equation. Results from these calculations are consistent with the usual interpretation of the semidilute crossover concentration for model solutions of both ideal and swollen polymer coils.     

Dynamical problems in the theory of polymer solutions

Qualitative discrepancies are found between what is predicted by available theory and what is actually observed, for several concentration regimes of the dynamical properties of polymer solutions. The difficulties are most severe, from the standpoint of experiment or simulation as well as theory, for the entanglement concentration regime. However, the classical problems of chain polymers in dilute solution are not fully understood. For example, the constants of proportionality that relate hydrodynamic radii to the radius of gyration, in the nondraining limit and in theta solvents, may not be universal constants. That is, the proportionality constants may vary with polymer and solvent species. Discrepancies between theory and experiment are discussed for the two different systems, dilute chains and semidilute rods. Speculation is offered on the resolution of these difficulties.     

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.     

Crystallization under stress: A theory of fibrillar crystallization

A theory of crystallization in stretched polymer networks is developed. In it, four principal features are incorporated: (i) crosslinks are displaced by growing crystallites, (ii) network chains are constrained to positions compatible with fixed sample shape and volume, (iii) some network chains remain amorphous, and (iv) the relative direction of a chain through a crystallite may not be the same for all chains. The derived network force exhibits a V or U shape with changing temperature in the crystallization zone that is a close replica of the behavior of gutta percha networks. Postulates of fibrillar-lamellar transitions are not introduced into the calculations.     

Modified interfacial statistical associating fluid theory: application to tethered polymer chains

Modified interfacial statistical associating fluid theory density functional theory is extended to tethered polymer chains in the absence or presence of free polymer chains. The structures of the "dry" and "wet" polymer brushes have been calculated and compared with simulation results available in the literature. The comparisons show that the theory accurately predicts the structure of the tethered polymer brush. The average brush heights calculated from the theory agree with well-established scaling theories for tethered polymers. However, these scaling theories cannot predict the detailed structure, accurately. The effects of the segment-segment interactions of the tethered polymer and the free polymer have been effectively captured by the theory.     

Experimental analysis of the molecular theory of rubber elasticity

Stress-strain and birefringence-strain experiments, on polybutadiene and poly(methyl-3,3,3, trifluoropropyl siloxane) elastomeric networks of various degrees of crosslinking have been carried out in order to evaluate the recent molecular theory of rubber elasticity of Erman and Flory, which is based on the idea of constraints imposed by entangled network chains on crosslink fluctuations. Previously published results on polyoxypylene and poly(dimethylsiloxane) networks have also been analyzed. In general, the theory describes the elastic behavior of all these systems satisfactorily. Stress-strain and birefringence data can be interpreted using the same set of parameters. The most important theoretical parameter κ, which is a measure of the severity of constraints, varies with the elastic modulus as predicted by theory. However, the relationship is not a universal one, as was originally suggested.     

Adsorption of polymer chains onto charged spheres: Experiment and theory

A self-consistent field (SCF) theory for the adsorption of polyelectrolyte chains onto oppositely charged spheres is considered. It is demonstrated that the criterion for critical adsorption shows a different behavior for small and large curvature of the surface. Experiments give indeed evidence for the power-law behavior as theoretically predicted for large curvature.     

Simulation and theory of self-assembly and network formation in reversibly cross-linked equilibrium polymers

A simulation model of hard spheres capable of reversible assembly into chains, which then may reversibly cross-link into networks, has been studied through grand canonical Monte Carlo simulation. Effects of varying intra- and interchain bond strengths, chain flexibilities, and restrictions on cross-linking angle were investigated. Observations including chain-length distributions and phase separation could be captured in most cases using a simple model theory. The coupling of chain growth to cross-linking was shown to be highly sensitive to the treatment of cross-linking by chain ends. In some systems, ladderlike domains of several cross-links joining two chains were common, resulting from cooperativity in the cross-linking. Extended to account for this phenomenon, the model theory predicts that such cooperativity will suppress phase separation in weakly polymerizing chains and at high cross-link concentration. In the present model, cross-linking stabilizes the isotropic phase with respect to the nematic phase, causing a shift in the isotropic-nematic transition to higher monomer concentration than in simple equilibrium polymers.     

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.     

Effect of end-group sticking energy on the properties of polymer brushes: comparing experiment and theory

Using surface force balance measurements we have established that polystyrene chains bearing three zwitterionic groups have a higher end-group sticking energy than equivalent chains bearing a single zwitterionic group. In a good solvent, polystyrene chains end-functionalized with three zwitterionic groups form brushes of a higher surface coverage than those bearing a single zwitterion. The increase in surface coverage is slow compared with the initial formation of the brush. Measurements of the refractive index allow us to directly quantify the variation of surface coverage, permitting comparison with models for the kinetics of brush formation based on scaling theory and an analytical self-consistent field. We find qualitative support for associating the kinetic barrier with the energy required for an incoming chain to stretch as it penetrates the existing brush.     

The molecular structure and orientation of antiferroelectric liquid crystal using the density-functional theory and two-dimensional correlation-polarized infrared spectroscopy

The structure and vibrational frequencies of the chiral antiferroelectric liquid-crystal molecule, 4-(1-methyheptyloxycarbonyl) phenyl-4-(4'-octyloxy) benzoate (MHOCPOOB), have been calculated using the density-functional theory (DFT) with the Becke-3 Lee-Yang-Parr/6-31G(d,p) level. The observed vibrational spectra have been resolved and assigned in detail by comparison to the computed values. The results indicate that the computed and observed spectra are in good agreement with each other. The stable molecular structure obtained with the DFT theory shows that the two hydrocarbon chains are all-trans zigzag conformer and nearly perpendicular to each other. The orientation of the mesogen part and the hydrocarbon chains for MHOCPOOB in the Sm-C*A phase are investigated by employing the polarization-angle-dependent infrared spectra in the electric-field induced and the two-dimensional correlation spectroscopy. After combining the experimental and theoretical results, it can be concluded that the azimuth of the achiral and chiral chains is opposite to each other, the orientation of the achiral chain is almost the same direction as the mesogen core, and the orientation of the chiral chain is nearly perpendicular to the mesogen part. The achiral and chiral CH2 chains are both a probable all-trans zigzag conformer.