M. D. Whitmore

Self-consistent theory of block copolymer blends : neutral solvent

We present a theoretical study of the lamellar structure of diblock copolymers blended with neutral solvent. The calculations are based on numerical solutions of the equations of the mean field self‐consistent theory of incompressible polymer/solvent blends, from the weak through to the strong segregation limits. For idealized model systems of symmetric copolymers and perfectly nonselective solvent, we examine the variation of the equilibrium lamellar thickness with molecular weight, Flory χ parameter, solvent quality, and copolymer volume fraction, as well as the distribution of solvent and polymer within the subdomains. We also examine a real system which has been extensively studied experimentally, namely PS‐b‐PI with the (nearly) nonselective solvent toluene.

A systematic Monte Carlo study of self-assembling amphiphiles in solution

In this paper, we present a systematic Monte Carlo study of the self-assembly of nonionic, amphiphilic, chainlike molecules in dilute solution. The focus is on the regime in which the molecules form relatively weakly segregated micelles, which are in equilibrium with small submicellar aggregates. We study the size and shape distributions of the aggregates, and the structure of the aggregates’ cores and surfaces. In some cases, spherical micelles, relatively large nonspherical micelles, and submicellar aggregates, all coexist. The size distributions of the spherical micelles are approximately Gaussian, while the nonspherical micelles contribute non-Gaussian tails at relatively large aggregation numbers. The simulation results are interpreted in terms of a simple theory of spherical micelles, and the size distributions are compared with its predictions. For the cases where the agreement is good, we combine the simulations and the theory to calculate the critical micelle concentration as functions of the cha...

Theory of the structure of adsorbed block copolymers: Detailed comparison with experiment

We present a numerical, self‐consistent field study of adsorbed diblock copolymers in thermal solvents, with a detailed and quantitative comparison with recent experiments performed on poly(dimethylsiloxane‐block‐styrene) copolymer spread as a monolayer at the free surface of ethyl benzoate [M. S. Kent, L. T. Lee, B. Farnoux, and F. Rondelez, Macromolecules 25, 6240 (1992); M. S. Kent, L. T. Lee, B. J. Factor, F. Rondelez, and G. S. Smith, J. Chem. Phys. 103, 2320 (1995)]. These neutron reflectivity experiments, for the first time, independently varied both the molecular weight and surface density, and probed the size and nature of the depletion layer at the surface. In the calculations, the polymer and solvent are characterized by realistic values of the pure component densities, the Kuhn length and the Flory interaction parameter. We examine the properties of the dangling block, specifically the depletion layer, the thickness of the brush, the maximum polymer concentration and its location, and the depe...

Numerical self-consistent field study of tethered chains in Θ solvent

We present a numerical, self-consistent field (SCF) study of tethered polymers in Θ solvent, over the range of anchoring densities found in most experiments. We examine the properties of the dangling block, including the thickness of the brush, the density profile, the depletion layer, the maximum polymer concentration and its location, and the dependence of these properties on surface density and molecular weight. In the calculations, the polymer and solvent are characterized by realistic values of the pure component densities, the statistical segment length, and the Flory interaction parameter, and we make quantitative comparison with recent experiments [M. S. Kent, G. S. Smith, J. Majewski, L. T. Lee, and S. Satija, J. Chem. Phys. 108, 5635 (1998)] and with the asymptotic analytic SCF theory. In agreement with these experiments, our results add to the growing body of evidence that many of these systems are not well described by the asymptotic theories.

Monte Carlo and numerical self-consistent field study of end-tethered polymers in good solvent

We present a Monte Carlo (MC) and numerical self-consistent field (NSCF) study of end-tethered polymers in good solvent, for surface coverages found in most experiments. Properties of the system, including the layer thickness, are calculated as functions of the degree of polymerization and surface density of the chains. A lower limit on the range of surface coverage above which the NSCF approach agrees well with the MC simulations is identified. Both approaches indicate that chain stretching begins at very low coverage, but that it remains modest throughout the regime of interest. The layer thickness does not scale linearly with degree of polymerization, and the radius of gyration of a free polymer in solution remains a relevant length scale. The MC and NSCF results are in good agreement with experiment.

Accurate diblock copolymer phase boundaries at strong segregations

We examine the lamellar/cylinder and cylinder/sphere phase boundaries for strongly segregated diblock copolymer melts using self‐consistent‐field theory (SCFT) and the standard Gaussian chain model. Calculations are performed with and without the conventional unit‐cell approximation (UCA). We find that for strongly segregated melts, the UCA simply produces a small constant shift in each of the phase boundaries. Furthermore, the boundaries are found to be linear at strong segregations when plotted versus (χN)−1, which allows for accurate extrapolations to χN=∞. Our calculations using the UCA allow direct comparisons to strong‐segregation theory (SST), which is accepted as the χN=∞ limit of SCFT. A significant discrepancy between the SST and SCFT results indicate otherwise, suggesting that the present formulation of SST is incomplete.

Morphologies of microphase‐separated conformationally asymmetric diblock copolymers

The equilibrium morphological behavior of a series of conformationally asymmetric linear diblock copolymers is characterized via small-angle X-ray scattering (SAXS) and transmission electron microscopy (TEM). The linear diblock molecules of polyisoprene and poly(t-butylmethacrylate) (PtBMA) are prepared anionically over a range of PtBMA volume fractions 0.17 to 0.85. Solution light-scattering experiments are performed on PtBMA homopolymer at theta conditions, and the results were compared with PI data in the literature in order to characterize the degree of conformational asymmetry between the respective blocks. This conformational asymmetry is quantified by an e of 0.75. The experimental results are compared with morphological behavior calculated utilizing self-consistent mean field theory for a diblock system with e = 0.75. At middle to high PtBMA volume fractions, Φ PtBMA > 0.30, the experimental morphological behavior agrees well with the calculated behavior; the microphase boundaries are slightly shifted to higher volume fractions of the PtBMA block due to its larger Kuhn length. At Φ PtBMA < 0.30, however, discrepancies are found in the volume fraction dependence of experimentally determined morphological behavior and that calculated theoretically. Interestingly, extremely well-ordered cylindrical microstructures were observed for PI cylinder domains embedded in PtBMA matrices; these samples were prepared by solvent casting with no treatment, such as shearing, to enhance long-range order. These well-ordered cylinder structures contrast with PtBMA cylinders in a PI matrix on the opposite side of the phase diagram that have very poor long-range order.

Monte Carlo and numerical self-consistent field study of systems with end-grafted and free polymers in good solvent

We present a systematic Monte Carlo and numerical self-consistent field (NSCF) study of thin films consisting of grafted and free polymers in good solvent, for the range of densities found in most experiments. Above the overlap threshold for the grafted polymer, the two approaches agree well. Even at low densities, the agreement is surprisingly good. The NSCF results are also directly compared with experiments. The systematic results are interpreted in the context of the regimes and behavior predicted by scaling and analytic SCF theories. We find that the grafted layer is generally thinner, and the penetration of the free polymer into the grafted layer is generally greater than predicted, and that the overall behavior is not in accord with the earlier theories. We find it useful to introduce and distinguish between two measures of the penetration, and we find that one of them can increase with the concentration of grafted polymer.

Doubly self-consistent field theory of grafted polymers under simple shear in steady state

We present a generalization of the numerical self-consistent mean-field theory of polymers to the case of grafted polymers under simple shear. The general theoretical framework is presented, and then applied to three different chain models: rods, Gaussian chains, and finitely extensible nonlinear elastic (FENE) chains. The approach is self-consistent at two levels. First, for any flow field, the polymer density profile and effective potential are calculated self-consistently in a manner similar to the usual self-consistent field theory of polymers, except that the calculation is inherently two-dimensional even for a laterally homogeneous system. Second, through the use of a modified Brinkman equation, the flow field and the polymer profile are made self-consistent with respect to each other. For all chain models, we find that reasonable levels of shear cause the chains to tilt, but it has very little effect on the overall thickness of the polymer layer, causing a small decrease for rods, and an increase of no more than a few percent for the Gaussian and FENE chains. Using the FENE model, we also probe the individual bond lengths, bond correlations, and bond angles along the chains, the effects of the shear on them, and the solvent and bonded stress profiles. We find that the approximations needed within the theory for the Brinkman equation affect the bonded stress, but none of the other quantities.

Grafted polymers inside cylindrical tubes: Chain stretching vs layer thickness

We present a study of the detailed structure of grafted polymer chains and the layers they form inside cylindrical tubes, using the finitely extensible nonlinear elastic chain model and numerical self-consistent field theory. For very large tube radius, the chain stretching and layer thicknesses are the same as for polymers grafted to a planar surface. For decreasing radius, our calculations indicate that the layer almost always gets thinner, although there can be situations where it is very slightly thicker. However, we find that this thinning is not necessarily due to changes to the polymers: in fact, the root-mean-squared layer thickness would decrease even if the polymers themselves are completely unchanged. Furthermore, we find that the polymer stretching can increase at the same time that the layer thickness decreases. These apparent paradoxes are resolved by analyzing and distinguishing between the volume fraction profiles and monomer number distributions in these systems, including how they change and why. We also find that, in a given system, parts of each polymer move towards the curved surface and parts away from it, and that these differences are key to understanding the behavior.