ALBERT

Notes: A new mean-field theory for dilute polymer solutions is presented. The approach is based on looking at a polymer molecule in the mean-field of the solution. This has the advantage of allowing the conformational behavior of the chain molecules and the thermodynamic properties of the solution to be studied within the same theoretical framework. The phase diagram predicted by the theory has a critical temperature which is lower, and a width of its coexistence curve which is larger, than those obtained from the simpler mean-field (Flory) theory. The critical volume fraction is found to scale with the degree of polymerization n as φc∼n−0.40, in excellent agreement with experimental results. The theory allows one to study the size and shape of the polymer molecules as functions of the thermodynamic state of the solution. It is found that on the concentrated side of the coexistence curve the average shape of the molecules is an elongated ellipsoid, while in the dilute side the molecules are smaller in size and their shape becomes more sphere-like as the concentration is reduced. The theory is extended to study the properties of the interface separating two phases at coexistence. In particular, it is applied to study the variation of the sizes, shapes and orientations of the polymer molecules due to the inhomogeneous density in the interfacial region. It is found that as the interface is approached from the polymer-rich side the molecules tend to slightly decrease their size and their longest axes tend to orient parallel to the interface. At the edge of the polymer-poor side of the interface the molecules are very elongated and their long axes are oriented almost exclusively perpendicular to the interface. It is argued that the orientation profile in the interfacial region is a general property of nonspherical molecules.

Notes: We treat a lattice model of phase separation in polymer solutions in mean-field (Flory) approximation, taking account of anisotropic biases in the chain conformations in the interface, as in an earlier theory of Helfand. Near the critical point of the phase separation those biases lead to a square-gradient contribution to the free-energy density that is of de Gennes' form, and thus to an interfacial tension that varies with polymerization index N and temperature distance below the critical point Tc−T as in the theories of Nose and of Vrij and Roebersen. In the scaling regime N→∞ and Tc−T→0 at fixed x=const N1/2(Tc−T) the surface tension σ is of the form N−1 Σ(x), with Σ(x) a scaling function that we display and that has the asymptotic behavior Σ(x)∼const x3/2 for x→0 and Σ(x)∼const x2 for x→∞. The latter contrasts with the x5/2 found earlier when no account was taken of the chain-conformation biases. These biases are displayed as functions of location in the interface. On the concentrated-phase side of the interface the concentration of horizontal links is greater, and on the dilute-phase side it is less, than in a random chain.

Notes: The conformational and thermodynamic characteristics of molecular organization in mixed amphiphilic aggregates of different compositions and geometries are analyzed theoretically. Our mean-field theory of chain conformational statistics in micelles and bilayer membranes is extended from pure to mixed aggregates, without invoking any additional assumptions or adjustable parameters. We consider specifically binary aggregates comprised of long-chain and short-chain surfactants, packed in spherical micelles, cylindrical rods, and planar bilayers. Numerical results are presented for mixtures of 11- and 5-carbon chain amphiphiles. The probability distribution functions (pdfs) of the (different types of) chains are determined by minimizing the conformational free energy, subject to packing constraints which reflect the segment density distribution within the hydrophobic core. In order to analyze the relative thermodynamic stabilities of mixed aggregates of different compositions (long/short chain ratios) and different geometries, the aggregate's free energy is expressed as a sum of conformational, surface, and mixing contributions. The conformational free energy is determined by the pdfs of the chains and the surface term is modeled in terms of the "opposing forces'' operative at the hydrocarbon–water interface. An interesting coupling between these terms arises from the special geometric (surface/volume) limitations associated with packing short and long chains in a given ratio within a given aggregate. In particular, it is found that the minimal area per surfactant head group in a mixed spherical micelle is significantly lower than that in a pure micelle (similarly, though less drastically so, for cylindrical micelles). The most important qualitative conclusion of our thermodynamic analysis is that the preferred aggregation geometry may vary with composition. For example, we find that under certain conditions (areas per head group, chain lengths) the preferred micellar geometry of pure long or short-chain aggregates is that of a planar bilayer, whereas at intermediate compositions spherical micelles are more stable. Our analysis of chain conformational properties provides quantitativeinformation on the extent of long (or short) chain distortion attendant upon chain mixing. For example, the results for bond order parameter profiles and segment density distributions reveal enhanced stretching of the long chain towards the central regions of the hydrophophic core as the fraction of short chains is increased.

Notes: A recently developed mean field ("single-chain'') theory for amphiphile chain organization and thermodynamics in micellar aggregates is applied to rotational isomeric state, model chains. The theory provides explicit, simple expressions for the probability distribution of chain conformations and related molecular and thermodynamic properties applicable to aggregates of arbitrary geometries. Bond order parameter profiles calculated from the theory for a planar bilayer, assuming a compact hydrophobic core, show very good agreement with experimental data and molecular dynamics simulations. For small spherical micelles comparison between theory and experiment suggest the existence of a somewhat rough (few angstroms wide) hydrocarbon–water interfacial region. Cylindrical aggregates reveal intermediate behavior. The extent of "surface roughness'' is introduced into the theory via a density profile of the hydrophobic core which decreases gradually from the bulk liquid (compact core) density to zero. A series of calculations is presented to analyze the effects of internal chain (gauche/trans) energy and micellar geometry on the conformational and thermodyamic properties of the hydrocarbon chains. It is found that the internal energy plays only a secondary role, compared to the primary role of the packing constraints. (This is qualitatively consistent with our previous findings for approximate, "cubic,'' model chains.) The conformational free energy cost associated with chain packing in aggregates is shown to depend on the micellar geometry (i.e., on the curvature of, and the average area per head group at, the hydrocarbon–water interface) and to be comparable with the surface (head group) contributions treated exclusively in the prevailing theories of surfactant self-assembly. Finally, a "corresponding-states'' behavior is demonstrated for packed chains (in planar bilayers) by referencing all thermodynamic functions and configurational properties to those of the associated "free'' chain.

Notes: The conformational and thermodynamic properties of lattice chain molecules in confined environments have been studied by Monte Carlo simulations. In the case of homopolymers with purely repulsive interactions with the walls the scaling laws proposed by Daoud and deGennes for the free energy of confinement, from three dimensions to one dimension and for two dimensions to one dimension, have been confirmed. The number of self-avoiding walks (SAW) of a chain with n segments for each film thickness d has been found to follow an effective relationship of the form NSAW(d)∝zeffn (d)nγ (d)−1, where zeff (d) is a thickness dependent effective lattice coordination number and γ (d) is the d dependent enhancement exponent. In the case of triblock copolymers in which the segments of the end blocks have attractions with the walls of the confined media it was found that for strong enough attractive interactions there is a minimum in the effective interactions between parallel walls. The distance at which the interaction potential has its minimum corresponds to the bulk end-to-end distance of the middle (nonattractive) block. This turns out to also be the distance at which the chains have the maximum number of bridging configurations between the two surfaces. © 1995 American Institute of Physics.

Notes: The conformational and thermodynamic behavior of chain molecules tethered to a planar surface are studied for a variety of solvent qualities with a recently developed single-chain mean-field theory. The lateral pressure isotherms calculated from the theory for chains of n=50 segments show very good quantitative agreement with the recent molecular dynamics simulations of Grest, without the use of any adjustable parameter, for good and aitch-theta solvents. The behavior of the pressure isotherms is analyzed in terms of a virial expansion and it is shown that the regimes where there is scaling of the pressure with surface coverage σ are very narrow for this chain length. Moreover, comparisons with analytical self-consistent field (SCF) theory show good agreement only when the parabolic density profile is used in the full virial equation. In the bad solvent regime the pressure isotherms for grafted chains show negative values of the pressure and also a negative compressibility for some range of surface coverages for temperatures below the aitch-theta temperature. This indicates the possibility of microphase separation in this regime of temperature and surface coverages.For chains with translational degrees of freedom there is a first-order phase separation at temperatures below the aitch-theta temperature into a very dilute and more concentrated polymer phase. From results of two different chain lengths it seems that there is a universal coexistence phase diagram in the scaling variables n0.5(T/aitch-theta−1) vs nσ. In the bad solvent regime it is found that the chains are highly collapsed for very low surface coverage. As the surface coverage is increased the chains tend to stretch laterally in order to gain as much contact as possible with the other polymer chains. This is in contrast to the good solvent regime where the chains always stretch perpendicular to the surface in order to avoid as much contact as possible with the other polymer chains. In the intermediate surface coverage regime it is found that the density profiles look parabolic-like for good solvents becoming more steplike as the quality of the solvent becomes poorer, in agreement with recent neutron scattering experiments. For very high surface coverages the density profiles in the good solvent regime also look steplike.

Notes: The tricritical point of polymer solutions composed by two polymer homologs of different molecular weight in a solvent is studied using the single-chain mean-field theory. The tricritical point is found for a ratio of molecular weights of the two polymers r=N1/N2, which decreases as a function of the short chain length N2 from 51 to approximately 40 for N2 increasing from 1 to 8. This last value is still much larger than the experimentally measured value for mixtures of polystyrene in cyclohexane, where it was found that r(approximately-equal-to)25. Although a further reduction of r as a function of N2 seems likely, the chain lengths of the corresponding long chains are beyond the reliability limit of the SAW's simulation involved. The addition of a small effective repulsion between the polymer homologs, is found to bring down the predicted value to 25. © 1996 American Institute of Physics.