Effects of interaction range and compressibility on the microphase separation of diblock copolymers: Mean-field analysis

Using the random-phase approximation and self-consistent field calculations, we have investigated the effects of finite interaction range and compressibility on the order-disorder transition (ODT) and the lamellar structure of symmetric diblock copolymers. While the compressibility does not affect the ODT, both the values of chiN and bulk lamellar period at the ODT increase with increasing interaction range. On the other hand, both the free-energy density and bulk period of the lamellae increase with either increasing interaction range or decreasing compressibility. Even with a finite compressibility, the mean-field ODT is still a second-order phase transition. The scaling exponent of bulk lamellar period with chiN, however, decreases with increasing compressibility. Our mean-field analysis provides a well understood reference for the study of fluctuation effects in diblock copolymers with finite interaction range and compressibility.     

Monte-carlo simulations of the order–disorder transition depression in ABA triblock copolymers with a short terminal block

Although most ABA triblock copolymers are molecularly symmetric (i.e., the terminal blocks possess the same mass), molecularly asymmetric A1BA2 triblock copolymers are of greater fundamental interest in that they can be used to explore the transition from diblock to triblock copolymer in systematic fashion. In this study, we use a lattice Monte Carlo method known as the cooperative motion algorithm to simulate molten ABA triblock copolymers possessing a short terminal block to explore the effect of molecular asymmetry on the copolymer order–disorder transition (ODT). Reduced ODT temperatures, discerned by simultaneously analyzing several features of the simulation results, are found to compare favorably with experimental data. Of particular interest here is the initial depression in the ODT temperature for A1BA2 copolymers possessing a relatively short terminal (A2) block. This signature feature is successfully captured by the simulations and is found to be strongly dependent on composition, but weakly dependent on copolymer chain length. © 2012 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2013     

Ordering transition of block copolymer films

It is well-known that a bulk, symmetric, A-b-B diblock copolymer forms a lamellar morphology, with period L, below an order-disorder transition (T(ODT)) temperature, for chiN < 10.5; chi is the Flory-Huggins interaction parameter and N is the degree of polymerization of the copolymer. The ordering temperatures of poly(styrene-b-methyl methacrylate) (PS-b-PMMA) thin film diblock copolymers of thickness h 相似文献   

Double-Gyroid Network Morphology in Tapered Diblock Copolymers

We report the formation of a double-gyroid network morphology in normal-tapered poly(isoprene-b-isoprene/styrene-b-styrene) [P(I-IS-S)] and inverse-tapered poly(isoprene-b- styrene/isoprene-b-styrene) [P(I-SI-S)] diblock copolymers. Our tapered diblock copolymers with overall poly(styrene) volume fractions of 0.65 (normal-tapered) and 0.67 (inverse-tapered), and tapered regions comprising 30 volume percent of the total polymer, were shown to self-assemble into the double-gyroid network morphology through a combination of small angle X-ray scattering (SAXS) and transmission electron microscopy (TEM). The block copolymers were synthesized by anionic polymerization, where the tapered region between the pure poly(isoprene) and poly(styrene) blocks was generated using a semi-batch feed with programmed syringe pumps. The overall composition of these tapered copolymers lies within the expected network-forming region for conventional poly(isoprene-b-styrene) [P(I-S)] diblock copolymers. Dynamic mechanical analysis (DMA) clearly demonstrated that the order-disorder transition temperatures (T(ODT)'s) of the network-forming tapered block copolymers were depressed when compared to the T(ODT) of their non-tapered counterpart, with the P(I-SI-S) showing the greater drop in T(ODT). These results indicate that it is possible to manipulate the copolymer composition profile between blocks in a diblock copolymer, allowing significant control over the T(ODT), while maintaining the ability to form complex network structures.     

Continuous Thermodynamic Integration in Field‐Theoretic Simulations of Structured Polymers

This work explores the use of continuous thermodynamic integration in field‐theoretic simulations of a symmetric diblock copolymer melt. Free energies of the lamellar and disorder phases are evaluated by thermodynamic integration from a reference state (an Einstein crystal, λ = 0) to a diblock copolymer (λ = 1). This is followed by integration over the interaction parameter, χ_b , to locate the order–disorder transition (ODT). Then the equilibrium lamellar spacing and free energy cost of stretching and compressing lamellae are examined. The ODT, lamellar spacing, and compression modulus are consistent with previous calculations, though found faster and more precisely. The above quantities do not depend on simulation box size, suggesting that finite‐size effects are small and simulating two lamellar periods is sufficient to accurately evaluate bulk behavior. Furthermore, the statistical uncertainty in the ODT increases quickly with system size, suggesting that small systems may lead to more precise results.     

Nonconventional morphologies in two-length scale block copolymer systems beyond the weak segregation theory

The order-disorder and order-order transitions (ODT and OOT) in the linear multiblock copolymers with two-length scale architecture A(fmN)(B(N2)A(N2))(n)B((1-f)mN) are studied under intermediate cooling below the ODT critical point where a nonconventional sequence of the OOTs was predicted previously [Smirnova et al., J. Chem. Phys. 124, 054907 (2006)] within the weak segregation theory (WST). To describe the ordered morphologies appearing in block copolymers (BCs) under cooling, we use the pseudospectral version of the self-consistent field theory (SCFT) with some modifications providing a good convergence speed and a high precision of the solution due to using the Ng iterations [J. Chem. Phys. 61, 2680 (1974)] and a reasonable choice of the predefined symmetries of the computation cell as well as initial guess for the iterations. The WST predicted sequence of the phase transitions is found to hold if the tails of the BCs under consideration are symmetric enough (mid R:0.5-fmid R:0.05, a large region of the face-centered cubic phase stability is found (up to our knowledge, first within the SCFT framework) inside of the body-centered cubic phase stability region. Occurrence of the two-dimensional and three-dimensional phases with the micelles formed, unlike the conventional diblock copolymers, by the longer (rather than shorter) tails, and its relationship to the BC architecture is first described in detail. The calculated spectra of the ordered phases show that nonmonotonous temperature dependence of the secondary peak scattering intensities accompanied by their vanishing and reappearance is rather a rule than exception.     

Relationships among coarse-grained field theories of fluctuations in polymer liquids

Two closely related field-theoretic approaches have been used in previous work to construct coarse-grained theories of corrections to the random phase approximation for correlations in block copolymer melts and miscible polymer blends. The "auxiliary field" (AF) approach is based on a rigorous expression for the partition function Z of a coarse-grained model as a functional integral of an auxiliary chemical potential field. The "effective Hamiltonian" (EH) approach is instead based on an expression for Z as a functional integral of an observable order parameter field. The exact effective Hamiltonian H(eff) in the EH approach is defined as the free energy of a system with a constrained order parameter field. In practice, however, H(eff) has often been approximated by a mean-field free energy functional, yielding what we call a mean-field effective Hamiltonian (MFEH) approximation. This approximation was the starting point of both the Fredrickson-Helfand analysis of fluctuation effects in diblock copolymers and earlier work on the Ginzburg criterion in polymer blends. A more rigorous EH approach by Holyst and Vilgis used an auxiliary field representation of the exact H(eff) and allowed for Gaussian fluctuations of this field. All applications of both AF and EH approaches have thus far relied upon some form of Gaussian, or "one-loop" approximation for fluctuations of a chemical potential and/or order parameter field about a mean-field saddle-point. The one-loop EH approximation of Holyst and Vilgis and the one-loop AF theory are equivalent to one another, but not to the one-loop MFEH theory. The one-loop AF and MFEH theories are shown to yield predictions for the inverse structure factor S(-1)(q) that (in the absence of further approximations to either theory) differ by a function that is independent of the Flory-Huggins interaction parameter χ. As a result, these theories yield predictions for the peak scattering intensity that exhibit a similar χ-dependence near a spinodal. The Fredrickson-Helfand theory for the structure factor in disordered diblock copolymer melts is an asymptotic approximation to the MFEH one-loop theory that captures the dominant asymptotic behavior of very long, symmetric copolymers very near the order-disorder transition.     

Dynamic heterogeneity in diblock copolymer systems

Dynamic light scattering from diblock copolymers in melt and solution in a non-selective solvent reveals different mechanisms for relaxing the composition and orientation fluctuations near the order to disorder transition (ODT). For the former, internal relaxation and copolymer chain diffusion are the main relaxation processes whereas the latter relate to collective orientation of the copolymer chains near the ODT and induced form anisotropy of coherently ordered microstructures below ODT.     

Orientation fluctuations in diblock copolymers

Depolarized light scattering and dielectric relaxation spectroscopy reveal pertinent composition fluctuations effects on the orientation dynamics in diblock copolymers near the ordering transition (ODT). The main evidence stems from the broadening of the block relaxation function and collective chain orientation in the disordered state near ODT as well as a slow relaxation process below ODT.     

Morphological behavior of model poly(ethylene‐alt‐propylene)‐b‐polylactide diblock copolymers

A set of well‐defined poly(ethylene‐alt‐propylene)‐b‐polylactide (PEP‐PLA) diblock copolymers containing volume fractions of PLA (f_PLA) ranging between 0.08 and 0.91 were synthesized by a combination of living anionic polymerization, catalytic hydrogenation, and controlled coordination‐insertion ring‐opening polymerization. The morphological behavior of these relatively low‐molecular‐weight PEP‐PLA diblock copolymers was investigated with a combination of rheology, small‐angle X‐ray scattering, and differential scanning calorimetry. The ordered microstructures observed were lamellae (L), hexagonally packed cylinders (C), spheres (S), and gyroid (G), a bicontinous cubic morphology having Ia3 d space group symmetry. The G morphology existed in only a small region between the L‐C morphologies in close proximity to the order–disorder transition (ODT). Transformations from L to G were observed upon heating in several samples. The efficacy of the reverse G to L transition in one sample was cooling rate dependent. The PEP‐PLA Flory–Huggins interaction parameter as a function of temperature χ_PEP‐PLA(T) was estimated from T_ODT's by mean‐field theory and subsequently used in the construction of the experimental PEP‐PLA morphology diagram (χN versus f_PLA). The resultant morphology diagram was symmetric there were the well‐defined L‐C morphology boundaries. The low molecular weight of the materials imparted no significant deviation from previously documented diblock systems. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 2364–2376, 2002     

Concentration fluctuation effects on the phase behavior of compressible diblock copolymers

A Hartree analysis has been performed for compressible diblock copolymers of incompatible pairs to investigate the concentration fluctuation effects on their microphase separation behavior. The free energy in the Hartree analysis is obtained from the self-consistent correction to its mean-field cousin, which was recently formulated for such copolymer systems. The mean-field phase diagram is shown to be significantly affected by the fluctuation effects as the copolymer chain size N is lowered. An effective interaction chi(cRPA), which carries not only the change in contact interactions but also the compressibility difference between block components, plays a key role in understanding of the phase behavior and the pressure responses of various thermodynamic transitions for the copolymers with finite sizes. In particular, a symmetric copolymer at disorder-to-lamella transition is found to satisfy Nchi(cRPA)(q*)=10.495+41.022N(-1/3) when evaluated at a characteristic wave number q* for ordered microphases.     

对称长链和近对称短链两嵌段聚合物混合体系相行为研究

利用自洽平均场理论(SCMFT)系统地研究了对称长链和近对称短链两嵌段聚合物混合体系在纳米尺度下的自组装行为.体系中具有较高聚合度的对称长链熔体处于层状相,聚合度较低的近对称短链熔体处于无序相,而其混合体系却随着两种成分的不同比例呈现出有序-无序相转变、有序-有序相转变及有序-无序两相共存等复杂的相行为,计算结果与近期类似体系的实验有着较好的吻合.同时与两种对称的两嵌段聚合物混合体系的计算结果进行了比较,得出这两种体系的异同之处.     

Static and dynamic scattering from cyclic diblock copolymer chains in solution

This work discusses the static and dynamic scattering properties of (A-B)_r ring diblock copolymer chains in the semi-dilute solutions. The case of a symmetric diblock is considered in details. The results show substantial differences with respect to linear diblock copolymers. This is mainly due to the effect of connectivity of the chain extremities A and B leading to different conformations for both types of architectures. The additional entropic interaction in cyclic chains induces a decrease of the elastic scattering intensity and a shift of the location of its maximum to a higher q value. As for the dynamics, the fact that the two extremities are connected speeds up the frequency of the interdiffusion mode and shifts its minimum to higher q value. A discussion of hydrodynamic interactions is also included.     

Monte Carlo phase diagram for diblock copolymer melts

A partial phase diagram is constructed for diblock copolymer melts using lattice-based Monte Carlo simulations. This is done by locating the order-disorder transition (ODT) with the aid of a recently proposed order parameter and identifying the ordered phase over a wide range of copolymer compositions (0.2相似文献   

Monte Carlo simulation on symmetric ABA/AB copolymer blends in confined thin films

The effects of blend composition on morphology, order-disorder transition (ODT), and chain conformation of symmetric ABA/AB copolymer blends confined between two neutral hard walls have been investigated by lattice Monte Carlo simulation. Only lamellar structure is observed in all the simulation morphologies under thermodynamic equilibrium state, which is supported by theoretical prediction. When the composition of AB diblock copolymer (phi) increases, both lamellar spacing and the corresponding ODT temperature increase, which can be attributed to the variation of conformation distribution of the diblock and the triblock copolymer chains. In addition, both diblock and triblock copolymer, chains with bridge conformation extend dramatically in the direction parallel to the surface when the system is in ordered state. Finally, the copolymer chain conformation depends strongly on both the blend composition and the incompatibility parameter chiN.     

Bicomponent Block Copolymers Derived from One or More Random Copolymers as an Alternative Route to Controllable Phase Behavior

Block copolymers have been extensively studied due to their ability to spontaneously self‐organize into a wide variety of morphologies that are valuable in energy‐, medical‐, and conservation‐related (nano)technologies. While the phase behavior of bicomponent diblock and triblock copolymers is conventionally governed by temperature and individual block masses, it is demonstrated here that their phase behavior can alternatively be controlled through the use of blocks with random monomer sequencing. Block random copolymers (BRCs), i.e., diblock copolymers wherein one or both blocks are a random copolymer comprised of A and B repeat units, have been synthesized, and their phase behavior, expressed in terms of the order–disorder transition (ODT), has been investigated. The results establish that, depending on the block composition contrast and molecular weight, BRCs can microphase‐separate. We also report that large variation in incompatibility can be generated at relatively constant molecular weight and temperature with these new soft materials. This sequence‐controlled synthetic strategy is extended to thermoplastic elastomeric triblock copolymers differing in chemistry and possessing a random‐copolymer midblock.     

Mesoscopic dynamics of inhomogeneous polymers based on variable cell shape dynamic self-consistent field theory

In this paper, we combine variable cell shape method with dynamic self-consistent field theory and extend to study structure and dynamics under shear for triblock copolymer melts. Due to shear, the calculation cell shape is variable and no longer orthogonal. Pseudospectral method is employed to solve the diffusion equation for chain propagator on the nonorthogonal coordinate and the shear periodical condition can be easily designed in terms of the variable cell shape method. By using this strategy, the shear induced morphology evolution is investigated for topologically complex polymeric systems such as linear and star triblock copolymers; the morphology of linear ABC triblock copolymers is more shear sensitive than that of star triblocks. In particular, once the chain propagator is obtained, the microscopic elastic stress and spatial stress distribution can be derived and thus the dynamic mechanical property can be calculated under shear. By imitating the dynamic storage modulus G' corresponding to any given morphology in the oscillatory shear measurements, we explore the relationship between the morphology and the storage modulus G' and extend to study the mechanism of phase separation dynamics as well as order-disorder transition (ODT) for linear and star triblock copolymers. The results show that the chain architecture can be easily distinguished by investigating the ODT, though the systems such as AB symmetric diblock and ABA triblock copolymers by coupling AB precursors almost exhibit similar microstructures. In addition, the storage modulus G' and loss modulus G" can be simultaneously determined in frequency sweeps of oscillatory shear measurements and the dependence of the moduli on phase separated patterns and the chain topology is investigated. The simulation findings are in qualitatively agreement with the experimental results.     

Influence of electric field on the phase transitions of the hexagonal cylinder phase of diblock copolymers.

We have used the cell dynamic simulations (CDS) method to study the evolution of asymmetric and symmetric diblock copolymers under electric fields. For symmetric diblock copolymers, long-range-ordered lamellar phases form readily under electric fields. For asymmetric diblock copolymers, sphere-to-cylinder phase transitions occur rapidly when strong electric fields are applied, but it takes longer for the system to form hexagonal cylinder structures. The results of these simulations suggest that the sphere phase is stable under weak electric fields, but a threshold electric intensity exists for the sphere-to-cylinder phase transition. In addition, we also studied the kinetic pathways of the transition of the lamellar phase to the hexagonal cylinder phase of the asymmetric diblock copolymers under electric fields. Hexagonal cylinder structures form when the lamellar phase is subjected to a sudden temperature jump. The scattering functions suggest that the hexagonal cylinder structures are very regular and possess very few flaws.     

Diffusion,rheology, and flow birefringence in block copolymer liquids near the ordering transition

Measurements of the diffusion and relaxation of block copolymer chains near the order-disorder transition (ODT) are reported. Forced Rayleigh scattering has been used to determine the diffusivities parallel and perpendicular to the lamellar planes, for poly(ethylene-propylene)-poly(ethylethylene) (PEP-PEE) melts. The anisotropy is relatively weak (i.e., less than a factor of 4), but increases steadily as temperature is decreased. Rheology and flow birefringence have been employed to examine the conformational dynamics in block copolymer solutions. For PEP-PEE in squalane, the stress-optic relation is approximately valid, but the stress-optic coefficient increases in the ordered state. The location of the ODT is consistent with the dilution approximation. For polystyrene-polyisoprene (PS-PI) diblock and triblock copolymers, the stress-optic relation fails completely. There is evidence that the fluctuation regime (i.e., in the disordered state but near the ODT) may be considerably broader in block copolymer solutions than in melts. Furthermore, the onset of structure in the solution is accompanied by substantial form birefringence. In general, the optical signals are more complicated than the Theological ones, but also much more sensitive to small changes in temperature or concentration.     

Tracer diffusion in fluctuating block copolymer melts

We present a theoretical investigation of the tracer diffusion of diblock copolymers and homopolymers in a thermally fluctuating block copolymer melt above the order-disorder transition (ODT) temperature. Entanglement effects and differences in monomeric friction coefficients are ignored; hence, the theory should be most applicable to short copolymers with rheologically similar blocks. Overall, we find that the diffusion rates of both tracer block copolymers and homopolymers in a block copolymer melt are suppressed when compared with diffusivities in a strictly homogeneous medium with the same average composition. This mobility suppression is due to thermally excited composition fluctuations in block copolymer melts near the ODT; the latter result in transient potential barriers to diffusion. We explore the dependence of the tracer diffusion coefficient on molecular weights and compositions of both matrix and tracer, as well as temperature. A comparison of our theoretical predictions to recent experiments by T. Lodge and coworkers shows qualitative agreement. © 1996 John Wiley & Sons, Inc.