Strong Segregation Limit Research Articles

Response and instabilities of the lamellar phase of diblock copolymers under uniaxial stress

We study the quasistatic behavior of the lamellar phase of diblock copolymers under uniaxial compression and tension along the normal direction of the layers, in both the weak segregation limit (WSL) and the strong segregation limit (SSL). In the SSL, we derive a (nonlinear) continuum free energy description of the system in terms of local displacement of the lamellar layers, and use this free energy to study the mechanical behaviors. While compression induces the usual Hookian elastic response (for strains or stresses that are not too large), tension leads to square-lattice wave undulations in the transverse directions when the strain exceeds a critical value. In the WSL close to the order–disorder transition temperature, compression can ‘‘melt’’ the lamellar phase to the isotropic phase; such a melting can take the form of three types of instabilities, a quasithermodynamic instability, a spinodal at controlled strain, and a mechanical instability at controlled stress. It is shown that the third instability always precedes the second one under controlled-stress conditions. For a weakly first-order transition, the quasithermodynamic instability precedes the mechanical instability; but for a (hypothetical) second-order transition, the mechanical instability appears first as the stress is increased. In the case of tension, a transverse square-lattice wave deformation again develops at a critical strain. This deformation can be followed by a subsequent melting of types similar to the compressional case, upon further increase of the stress or strain. In both the SSL and WSL, the modulus undergoes an abrupt decrease when layer undulation develops, to a value 7/15 of that before the onset of undulation. Because the critical strain for the onset of undulation is usually very small, the modulus for tension will appear different from the modulus for compression, the former being 7/15 of the latter. As a result of this decrease in the modulus, melting of the lamellar phase in the WSL will occur at larger strains under tension than under compression.

Lamellar phase of a symmetric triblock copolymer

We study the lamellar phase formed by symmetric ABA triblock copolymers with degree of polymerization N and A monomer fraction f. Employing a mean-field, lattice formalism, we calculate the fraction of copolymers which bridge the B-rich lamellae. This fraction is found to be significant, although it is strictly bounded above by l/~ within the formalism. Once the lamellar phase is strongly segregated, the fraction of bridges decreases slowly with increasing N, increasing Flory parameter x, and decreasing f. In the strong-segregation limit, the lamellar spacing D is found to scale as D - N0xb with a = 0.68 and b = 0.19. The interfacial width 5 does not scale as a power of N but does scale with x as 5 - xc, where c = -0.51. Both D and 5 are nearly independent off. 187

Theory of block copolymer interfaces in the strong segregation limit

An extension of the Helfand theory of equilibrium properties of interfaces between different polymers is presented. The corrections to the interface width due to higher-order terms in gradient expansion of the free energy of the system are shown to be negligible in comparison with «end» corrections due to the connectivity of blocks. It is also shown that the latter corrections that are specific for block copolymers are typically much larger than end corrections for a system of immiscible homopolymers. The effect of fluctuations of concentration profiles is also analyzed. It is shown that the effect results in an appreciable increase of the apparent interface width

Microphase separation of ABC-type triblock copolymers

Microphase separation of ABC-type triblock copolymers in the strong segregation limit was investigated by generalizing our previous theory for diblock copolymers. The free energy functional in terms of local monomer concentrations is derived via a mean field approximation and a local approximation for higher order coupling. In the special case where the monomer number of the A block is the same as that of the C block, we calculate the critical values of the volume fraction for morphological transitions of ordered domains from lamellar to cylinder and from cylinder to sphere. It is emphasized that in the ABC-tpye copolymer, a square lattice is predicted to be more stable than a hexonal one for cylindrical domains

Noncircular Wigner-Seitz cells in strongly segregated block copolymers

A perturbation method is developed that allows for the systematic calculation of the elastic free energy of block copolymer microphases and goes beyond the approximation of a circularly (or spherically) symmetric Wigner-Seitz cell. The method is applicable in the limit of strong segregation between dissimilar blocks and to any the of microphase or copolymer architecture. As a demonstration of its utility, I perform an explicit calculation of the elastic free energy for the hexagonal microphase of diblock copolymers. In the composition window there tits microphase is observed, I show that the circular cell approximation of Semenov underestimates the elastic energy of the corona by no more than 6%

Phase transitions and shapes of two component membranes and vesicles I: strong segregation limit

Phase transitions and shapes of two component membranes and vesicles I: strong segregation limit

Elastic properties of an interface of diblock copolymer

A ternary mixture is studied wich consists of A and B homopolymers of equal polymerization indices and a symmetric AB diblock copolymer of a smaller index. In addition to A- and B-rich phases, there is a disordered phase in wich the A and B homopolymers are solubilized by the copolymer and wich coexists with the other two. The three interfacial tensions are calculated in the weak-segregation limit, as weel as the elastic properties of the A/B interface when there is only a microscopic thickness of copolymer adsorbed there. It is found that the saddle-splay modulus is positive, in contrast to results applicable to the strong segregation limit

Lamellar ordering in symmetric diblock copolymers

The lamellar period L and interphase thickness a of an incompressible melt of symmetric diblock copolymers from the onset of phase segregation (weak segregation limit) to the limit where segregation is almost complete (strong segregation limit) are calculated by numerically solving a mean field lattice model where the lattice spacing is taken small enough to approximate a continuum. The results for L and a agree with previously derived theoretical forms in both limits. Based on the first few terms of a Fourier series expansion of the density profile, it is shown analytically that L= 0.844R(χN) 0.571 at the weak segregation limit, where R is the unperturbed molecular radius of gyration, χ the Flory interaction parameter and N the number of statistical segments per molecule

Lamellar phase in a model for block copolymers.

We report results from a simulated annealing study of a coarse-grained model for block copolymers. For sufficiently slow cooling rate, the system develops a well-defined lamellar structure by avoiding the formation of defects. We show that the strong segregation limit of the model is computationally accessible when the interface thickness is small enough.

Ellipsometric studies on immiscible polymer-polymer interfaces

Ellipsometry was employed to estimate the thickness of the interface λ between two bulk layers of dissimilar polymers. The polymer/polymer combinations investigated were poly(methyl methacrylate) (PMMA)/poly(styrene- co-acrylonitrile) (SAN) (AN content < 10 and > 33 wt%), PMMA/poly(methyl methacrylate- co-styrene) (MS) and polystyrene/MS. They provide a series of immiscible combinations with various values of the Flory interaction parameter χ ranging from 1.72 × 10 −3 to 8.71 × 10 −2. The bilayer specimen prepared by the spin-coating technique was annealed at 130°C (> T g) and the time variation of λ was observed during annealing. The observed λ attained a constant value after annealing for ∼1 h and then remained constant. The constant value was assumed to be the equilibrium thickness. The thickness ranged from ∼5 to 40 nm, depending on the magnitude of χ. The thin interface (< ∼10 nm) was nicely explained by the recent theory by Broseta et al., which deals with the effect of molecular weight on λ within the strong segregation limit.

Thermodynamic considerations of triblock copolymers with a random middle block

A model based on the confined-chain statistics originally developed for diblock copolymers is presented to (i) identify the molecular characteristics responsible for producing a three-microphase A(A/B)B block copolymer and (ii) elucidate the relationships existing between molecular and microstructural characteristics within this regime. Boundary conditions associated with the strong segregation limit were used to determine the range of model applicability in terms of block and molecular composition and molecular weight.

Structure of symmetric polyolefin block copolymer thin films

The microstructure of thin films of nearly symmetric poly(ethylene–propylene)–poly(ethylethylene) (PEP-PEE) diblock copolymers (f=0.55, where f is the volume fraction of PEP) was characterized by neutron reflectometry (NR). A symmetric film structure in which the PEE block segregates preferentially to both interfaces is observed above and below the bulk order–disorder transition (ODT). Measurements at room temperature for several chain lengths, N, provide a real-space picture of the change in interdomain interfacial profiles associated with the crossover between the strong and weak segregation limits. The polymer/air and substrate/polymer interfaces are observed to induce an ordered microstructure even when the center of the film is disordered. The characteristic dimension of this near surface microstructure is larger than the corresponding bulk value for values of χN lying between those of the bulk Gaussian-to-stretched-coil and order–disorder transitions, where χ is the segment–segment interaction parameter. This behavior is attributed to the correlation of large amplitude composition fluctuations in the film with the interfaces. A mean-field behavior prevails for χN&amp;lt;(χN)GST,bulk, where some preferential segregation occurs at the interfaces, but the characteristic dimension once again matches that in the bulk.

Theory of supermolecular structures in polydisperse block copolymers: 4. Cylindrical domains in binary mixtures of diblock copolymers and cylinder-lamellae transition

The theory of well defined superstructures formed in binary mixtures of diblock copolymers with different molecular weights and composition in a strong segregation limit has been developed. Three types of binary mixtures were considered: a mixture of two cylinder-forming block copolymers; a mixture of cylinder- and lamellae-forming block copolymers; and a mixture of two lamellae-forming block copolymers. The formation (conditions and the thermodynamic characteristics) of mixed lamellar and cylindrical superstructures and their thermodynamic stability were investigated. It was shown that in the case of non-isomorphous block copolymers the change in mixture composition leads to a first-order phase transition from one morphology of the superstructure to another, i.e. to real phase segregation (each phase remains microphase segregated). The theory was compared with experimental data. The possibility of the formation of cylindrical domains in the mixture of two lamellae-forming block copolymers was predicted.

Theory of supermolecular structures in polydisperse block copolymers: 5. New double cylindrical structure in binary mixture of cylinder-forming diblock copolymers

Analytical self-consistent field theory of superstructure formation in a binary mixture of cylinder-forming diblock copolymers in the strong segregation limit is developed. It is shown that for the case when both components of the mixture are ‘quasi-isomorphous’, i.e. when the domain-forming blocks of the components are different, several types of supermolecular structure are possible depending on the composition of the mixture. The possible types of structure are: mixed cylindrical structure (C), mixed lamellar structure (L), two cylindrical structures (C + C′) formed by individual components of the mixture, and, finally, a new double cylinder structure (CC′), formed by cylindrical domains of one component coated by cylindrical layers of another component. Stability ranges of these structures are investigated for different compositions of block copolymers and mixture content.

Interfacial studies of incompressible binary blends

Density functional theory is used to investigate two of the possible reasons for the discrepancy between theory and experiment for the interfacial width and tension of phase separated binary blends. The two reasons are the inadequacy of the density functionals used previously and the neglect of the composition dependence of the Flory interaction parameter χ. For the sake of algebraic simplicity, our computations are restricted to symmetric, incompressible binary blends with correlation functions taken in the random phase approximation. We extend the analysis in Paper I [Tang and Freed, J. Chem. Phys. 94, 1572 (1991)] on the validy of density functionals for binary blends to the strong segregation limit with narrow interfaces by studying the conditions under which the Roe free energy functional is appropriate. Our derivation in this limit resums a rigorous Landau expansion to show that the Roe functional is appropriate only for infinite molecular weights. Nevertheless, even in the strong segregation limit this functional becomes incorrect in the wings of the concentration profile where a Landau expansion is valid despite the current unavailability of a local density functional. Interfacial properties in the theoretically simplest weak segregation limit are in accord with those of previous workers, but we are able to suggest an interpolation form for the interfacial width and tension of symmetric blends that crosses over between the weak and strong segregation limits. The composition dependence of the Flory interaction parameter χ poses several problems when experimental χeff are substituted into theories that ignore this composition dependence. We therefore investigate the errors incurred by the use of different experimental measures of χ and by the use of χ measured at different compositions for two simple models in which χ varies linearly or quadratically with blend composition. Situations arise where there exists an experimental χeff that yields the correct interfacial width in the composition independent theory, but other examples produce substantial errors. Dimensionless combinations of the interfacial width and tension are constructed that should be weakly dependent on molecular weight and temperature unless ignored factors, such as blend compressibility, have a significant influence on interfacial properties. The relevance of the composition dependece of χ to interfacial profiles suggests a similar relevance of blend compressibility because compressibility influences χ and its composition dependence.

Stability of a catenoid-lamellar phase for strongly stretched block copolymers

The free energy of the catenoid lamellar structure in the so-called «strong segregation limit» of low temperature and high molecular weight for a diblock copolymer is calculated and compared with the free energies of the lamellar, cylindrical, and spherical phases already calculated

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.

Comment on the free energy functional of block copolymer melts in the strong segregation limit

An elastic theory was previously developed for gently deformed periodic structures. However the elastic constants obtained are not consistent with those of Wang and Safran calculated recently by a different method. The main difference is the block ratio dependence of the equilibrium period D. Our result show a rather strong dependence of the block ratio f while Wang and Safran have derived D independent of f. The origin of the above discrepancy is explored

Elastic property of the lamellar structure in block copolymer melt added selective homopolymer chains

Statistical property of the microphase separated phase in copolymer-homopolymer mixtures is investigated in the strong segregation limit. The elastic constant of the lamellar structure influenced by the selective homopolymer chains is calculated for the block ratio f=1/2 of the block copolymer. The concentration profile of the added homopolymer chains is also obtained. The approximation used is the volume fraction expansion of the homopolymer and the weak interaction between the copolymer and the homopolymer is assumed.

Geometrical constraints on microstructural development in block copolymer ultrathin films

The fact that microstructures form in microphase-separated block copolymers, endowing the materials with unique thermo-mechanical properties, is well-established. However, no thermodynamic theories directly address the problem of microstructural development in ultrathin films, which might be useful as adhesives or resists. To predict the microstructural dimensions in such films, a model based on unit-cell geometries and material/volume balances is developed. Predictions of the ratio of the characteristic length of the domain core to domain repeat distance are obtained for poly (styrene-butadiene) diblock (SB) copolymers at 298 K. The finite interphase region is included in the volume balances with the parameterf, the volume fraction of interphase material, obtained from a modified version of the Leary-Henderson-Williams thermodynamic theory and shown here not to be a strong function of composition. An approach, implementingf, for accurately estimating the critical molecular weight of microphase separation in the strong-segregation limit (Mc), as a function of bulk composition for monodisperse SB copolymers at 298 K, is also presented.