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We analyze the segregation of strongly charged chains of N monomers of size a in the presence of multivalent salts with valence z as a function of the concentration of monomers φ in the solutions. The multivalent ions of opposite charge condense along the monomers and induce monomer attractions that lead to the formation of dense finite size aggregates (micelles) of chains and multivalent ions when their valence z > 1. We compute the number density of chains in aggregates with p = 1, 2, 3, 4, ... chains by equating the chemical potential of all the p-aggregated chains and of the free and condensed ions in the aggregates. At low concentration of multivalent salts m, we observe monomolecular (p = 1) precipitated chains when φ < φ*, where φ* ≅ mz. When φ increases above φ*, the chains redissolve in the solution and adopt stretched conformations. As m increases above a critical value m**, the number density of aggregates with p > 1 chains increases such that there are more aggregates with p* > 1 chains when φ** < φ < φ*. If we only include the surface free energy in the analysis, we find p* = ∞. However, if we include the chain entropy of confining p chains in a region R(p) = (pN)^1/3a in the free energy, p* can be finite and greater than or equal to one. This situation arises when the chains are constrained to stretched conformations, such as in metastable toroidal and in spherical coil aggregates observed in long double-stranded DNA in multivalent ions.

The effect of the thermodynamic selectivity of a solvent on the self-assembly of a styrene−isoprene (SI) diblock copolymer with block molecular weights of 1.1 × 10⁴ and 2.1 × 10⁴ g/mol is investigated. We explore the phase behavior from the melt state to dilute solution in solvents that are of varying selectivities for the two blocks. Bis(2-ethylhexyl) phthalate (DOP) is a neutral good solvent for SI. Di-n-butyl phthalate (DBP) and diethyl phthalate (DEP) are good solvents for PS but marginal and poor, respectively, for PI. Tetradecane (C14) is utilized as a complementary solvent that is good for PI but poor for PS. Small-angle X-ray scattering (SAXS) and static birefringence are used to locate and identify order−order (OOT) and order−disorder transitions (ODT). Dynamic and static light scattering were used to characterize the dilute solution micellar behavior. The neat polymer forms the gyroid (G1) morphology at low temperature, an OOT to hexagonal-packed cylinders (C1) occurs at 185 °C, and the ODT is located at 238 °C. Dilution with DOP decreases the OOT temperature in agreement with the dilution approximation, i.e., χOOT ~ φ^-1, but the ODT follows a stronger dependence of χODT ~ φ^-1.4. The slightly selective solvent DBP stabilizes the ordered state relative to DOP. Rich lyotropic and thermotropic behavior is observed among regions of lamellae (L), inverted gyroid (G2), inverted hexagonal-packed cylinders (C2), and inverted body-centered-cubic spheres (S2^bcc). Solutions in DEP display similar morphological behavior, along with significantly increased ODT temperatures. Because of the asymmetric block copolymer composition, the phase behavior in the isoprene-selective solvent C14 is markedly different, as only G1, C1, and S1^bcc phases are observed. The overall sequence of phases with dilution and/or heating is rationalized on the basis of diagonal trajectories across the phase map (temperature vs composition) for undiluted block copolymers:  addition of a neutral solvent corresponds to increasing the temperature and thus a “vertical” trajectory, whereas the partitioning between microdomains of a selective solvent amounts to a “horizontal” trajectory to a renormalized block volume composition. However, a variety of novel features are observed in DEP:  the formation of face-centered-cubic packed micelles, a reentrant thermotropic ODT, and a large window of L + C2 coexistence. The dependence of the principal length scale, d*, on φ, T, and structure is also considered. The strongly temperature-dependent selectivity produces a crossover in the scaling of d* vs φ for the lamellar phase:  the addition of a selective solvent increases d*, but as the solvent becomes neutral, d* decreases. This phenomenon is captured by self-consistent mean-field calculations.

The concentration correlation length, ξ, and the entanglement spacing, d, have been determined in solutions of hydrogenated polybutadiene in n-alkane solvents at 140 °C by small-angle neutron scattering and rheology, respectively. The former varies as ξ ~ φ^-0.76 over the range 0.04 < φ < 0.7. This result is in agreement with scaling predictions for semidilute solutions in a good solvent, but interestingly this dependence extends to concentrations well beyond the regime where the scaling law should be valid. The entanglement spacing was obtained via measurements of the dynamic loss modulus, G‘ ‘, as a function of frequency, ω, from φ = 0.2 to φ = 1. The maximum value of G‘ ‘ in the terminal dispersion regime varies as φ^1.9, implying that d ~ φ^-0.45. The shape of G‘ ‘(ω) was independent of φ and M, and d was independent of M, as expected for well-entangled solutions. These results indicate that the entanglement spacing and the correlation length need not have the same concentration dependence in concentrated solutions in a good solvent. The ratio ξ/d, termed the constraint porosity parameter in the polymer mode-coupling theory, varies from 0.09 in the melt to 0.15 when φ = 0.2, which may have important implications for the concentration dependence of the molecular weight exponent for chain diffusion.

We present a theoretical study of the influence of solvent on ordered block copolymer solutions. The phase behavior is examined as a function of solvent selectivity, temperature, copolymer concentration, composition, and molecular weight. Phase maps are constructed using self-consistent mean-field (SCMF) theory, via the relative stability of the “classical” phases, lamellae (L), hexagonally packed cylinders (C), and a body-centered cubic array of spheres (S). Solvent selectivity and polymer concentration strongly influence phase transitions in copolymer solutions. When a neutral good solvent is added to a symmetric block copolymer, a direct (lyotropic) transition from L to disordered (D) is expected, analogous to the (thermotropic) L → D transition in melts. Indeed for neutral good solvents the dilution approximation is followed:  the phase map is equivalent to that in the melt, once the interaction parameter is multiplied by the copolymer volume fraction. In contrast, for a symmetric block copolymer in the presence of a slightly selective solvent, the progression L → C → S → micelles → D is expected, although the micellar phase is not treated here. For asymmetric copolymers more elaborate sequences are anticipated, such as the progression CB → L → CA → SA → micelles → D. The stability limit of a homogeneous block copolymer solution is also examined via the random phase approximation (RPA) method. The effect of polymer concentration on the spinodal instability falls into two regimes. When the solvent is not very selective, the stable microphase separation region is reduced as polymer concentration decreases, whereas for very selective solvents, whereas for very selective solvents decreasing polymer concentration broadens the region of stable ordered microstructures.

The distribution of solvent in block copolymer solutions near their order−disorder transitions is examined experimentally, by small-angle neutron scattering (SANS), and theoretically, by the self-consistent mean-field (SCMF) approach. Three lamellar-forming poly(styrene-b-isoprene) diblocks were employed, in toluene, a neutral good solvent, and in cyclohexane, a selective solvent. For a given copolymer concentration, two solutions were prepared, one in protonated and one in perdeuterated solvent, and the scattering profiles compared. For a neutral solvent, one expects a small partitioning of the solvent to the interface between microdomains, to screen unfavorable styrene−isoprene contacts. Such partitioning should be manifest as a difference between the h- and d-solvents in the intensities of the second (and higher order, even) harmonic peaks (i.e., I(2q*), where the lamellar spacing, L, is 2π/q*). This difference is observed experimentally, and is in quantitative agreement with SCMF predictions using literature values for the three interaction parameters. Interestingly, the predicted relative scattering intensity between solutions in h- and d-toluene varies by several orders of magnitude over a small range in copolymer composition or over a small range of relative interaction parameters of the solvent for the two blocks, suggesting that SANS could be used in this way as a very sensitive measure of copolymer composition and/or solvent selectivity. For a selective solvent, one expects partitioning of the solvent between microdomains and a concomitant change in the intensities of the primary (and higher order, odd harmonic) peaks between h- and d-solvents. This effect is seen clearly in cyclohexane, with the partitioning of the solvent into the isoprene domains increasing with decreasing temperature. However, the SCMF calculations can only match the behavior in both solvents by employing polyisoprene−cyclohexane and polystyrene−cyclohexane interaction parameters significantly different from those reported in the literature.