Diblock Copolymer Melts Research Articles

Monte Carlo simulation of diblock copolymer microphases by means of a fast off-lattice model

We present a mesoscopic off-lattice model for the simulation of diblock copolymer melts by Monte Carlo techniques. A single copolymer molecule is modeled as a discrete Edwards chain consisting of two blocks with vertices of type A and B, respectively. The volume interaction is formulated in terms of coarse-grained densities of the A and B vertices. The model allows the study of equilibrium conformational properties of diblock copolymer microphases for sufficiently large system sizes.

Gravity effects on lamellar pattern formation in confined systems

The microphase separation of a diblock copolymer melt in the presence of a gravitational field is simulated by means of a cell dynamical system model. It is found that the presence of hard walls normal to the gravitational field is important to the formation of well ordered lamellae. Furthermore, it is found that the formation of lamellae parallel to the hard boundaries and normal to the field correspond to the stable configuration and that the field increases the interface width in the bulk, but the lamella in contact with the substrate is strongly segregated.

Block copolymer films between neutral walls: A Monte Carlo study

We study thin, symmetric AB-copolymer films between neutral walls using the bond fluctuation model. Near the surface, the monomer distribution with respect to the chains’ center of mass is nonisotropic: The chains flatten on the surface. The resulting A- and B-lamellae have interfaces practically perpendicular to the walls, as also observed in recent experiments. Our simulations indicate the onset of long-range surface ordering before the bulk orders. The strongest effects take place between the spinodal and the order–disorder transition point of the diblock copolymer melt.

Non‐equilibrium phase behavior of diblock copolymer melts and binary blends in the intermediate segregation regime

Non‐equilibrium phase behavior of diblock copolymer melts and binary blends in the intermediate segregation regime

Non-equilibrium phase behavior of diblock copolymer melts and binary blends in the intermediate segregation regime

The phase behavior of intermediately segregated (χN = 45) poly(ethylene)-poly(ethylethylene) (PE–PEE) diblock copolymers and PE–PEE binary blends are characterized using transmission electron microscopy and small-angle X-ray scattering. Surprisingly, the preparation-dependent, nonequilibrium phase behavior can be overwhelming even at this degree of segregation. A pure diblock with a poly(ethylene) volume fraction of fPE = 0.46 exhibited coexisting lamellae and perforated layers when prepared using a precipitation technique, but contained only the lamellar morphology when solvent cast. This preparation dependence was more dramatic in binary diblock copolymer blends with average compositions of 〈fPE〉 = 0.44, 0.46, and 0.48. Precipitated blends exhibited a microphase separated structure that was disordered and bicontinuous; however, solvent cast samples exhibited either a cylindrical, coexisting cylindrical and lamellar, or lamellar morphology. This nonequilibrium behavior is attributed to the high degree of segregation and the proximity to the cylinder/lamellae phase boundary. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 2229–2238, 1999

Single chain conformations in the system of symmetric and asymmetric diblock copolymers

A theory which describes a local structure and global properties of a diblock copolymer melt has been developed in the framework of the one-loop self-consistent approximation. We have derived expressions for the sizes of a single diblock macromolecule and its parts. Two different behaviors of single macromolecule conformations in the disordered melt have been obtained depending on the asymmetry of chains and morphologies occurring in ordered states after the order-disorder transition (ODT). In the nearly symmetric melt, 0.35 < f ≤ 0.5 (f is a composition), the blocks of both types shrink a little initially as the temperature decreases and then, at some temperature, they begin to swell. In strongly asymmetric melts, f < 0.35, the block of a macromolecule which consists of the monomers of minority type shrinks monotonically, while the other block monotonically swells. We have found nearly Gaussian behavior of the individual blocks and stretching near the chemical bond joining the blocks. Near the ODT the chains are stretched with a magnitude which is of the order of a few percent of their Gaussian sizes. We have calculated the peak position in the scattering curve as a function of the Flory-Huggins interaction parameter, composition and degree of polymerization. Less then 5% change in the size of copolymer molecules lead to a 25% shift of the scattering peak in comparison to the Gaussian limit. We have found a good quantitative agreement of our theoretical results with the experimental neutron scattering data.

Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term.

We are primarily concerned with the variational problem with long-range interaction. This functional represents the Gibbs free energy of the microphase separation of diblock copolymer melts. The critical points of this variational problem can be regarded as the thermodynamic equilibrium state of the phase separation phenomenon. Experimentally it is well-known in the diblock copolymer problem that the final equilibrium state prefers periodic structures such as lamellar, column, spherical, double-diamond geometries and so on. We are interested in the characterization of the periodic structure of the global minimizer of the functional (corresponding to the strong segregation limit). In this paper we completely determine the principal part of the asymptotic expansion of the period with respect to epsilon (interfacial thickness), namely, we estimate the higher order error term of the period with respect to epsilon in a mathematically rigorous way in one space dimension. Moreover, we decide clearly the dependency of the constant of proportion upon the ratio of the length of two homopolymers and upon the quench depth. In the last section, we study the time evolution of the system. We first study the linear stability of spatially homogeneous steady state and derive the most unstable wavelength, if it is unstable. This is related to spinodal decomposition. Then, we numerically investigate the time evolution equation (the gradient flow of the free energy), and see that the free energy has many local minimizers and the system have some kind of sensitivity about initial data. (c) 1999 American Institute of Physics.

Neutron spin echo investigation of the concentration fluctuation dynamics in melts of diblock copolymers

Diblock copolymers in the melt exhibit order–disorder phase transitions (ODT), which are accompanied by strong concentration fluctuations. These transitions are generally described in terms of the random phase approximation (RPA) of Leibler and Fredrickson, which is able to explain small angle scattering results in the neighborhood of the ODT, in particular around the correlation peak at q*. The RPA theory has been extended to include dynamical phenomena, predicting the short time relaxation of the dynamic structure factor in polymeric multicomponent systems. We report small angle neutron scattering and neutron spin echo experiments on polyethylene–block-polyethylethylene (PE-PEE) and poly(ethylene-propylene)–block-polyethylethylene (PEP-PEE) copolymers with molecular weights of 16.500 and 68.000 g/mol, which explore the structure and dynamics of these block copolymers. Studying melts with different hydrogen/deuterium labeling it was possible to observe experimentally the different relaxation modes of such systems separately. In particular the collective relaxation behavior as well as the single chain motion were accessed. The experimental results were quantitatively compared with the RPA predictions, which were based solely on the dynamical properties of the corresponding homopolymers and the static structure factors. The collective dynamics exhibits an unanticipated fast relaxation mode. This mode is most visible at low wave numbers (q⩾q*) but extends to length scales considerably shorter than the radius of gyration. Furthermore, the dynamical RPA yields expressions for the mobilities of chain segments in the block copolymer melt. These combination rules are at variance with the experimental findings for the single chain dynamics, while they hold for the collective response.

On the role of hydrodynamic interactions in block copolymer microphase separation

A melt of linear diblock copolymers (AnBm) can form a diverse range of microphase separated structures. The detailed morphology of the microstructure depends on the length of the polymer blocks An and Bm and their mutual solubility. In this paper, the role of hydrodynamic forces in microphase formation is studied. The microphase separation of block copolymer melts is simulated using two continuum methods: dissipative particle dynamics (DPD) and Brownian dynamics (BD). Although both methods produce the correct equilibrium distribution of polymer chains, the BD simulation does not include hydrodynamic interactions, whereas the DPD method correctly simulates the (compressible) Navier Stokes behavior of the melt. To quantify the mesophase structure, we introduce a new order parameter that goes beyond the usual local segregation parameter and is sensitive to the morphology of the system. In the DPD simulation, a melt of asymmetric block copolymers rapidly evolves towards the hexagonal structure that is predicted by mean-field theory, and that is observed in experiments. In contrast, the BD simulation remains in a metastable state consisting of interconnected tubes, and fails to reach equilibrium on a reasonable time scale. This demonstrates that the hydrodynamic forces play a critical part in the kinetics of microphase separation into the hexagonal phase. For symmetric block copolymers, hydrodynamics appears not to be crucial for the evolution. Consequently, the lamellar phase forms an order of magnitude faster than the hexagonal phase does, and thus it would be reasonable to infer a higher viscosity for the hexagonal phase than for the lamellar phase. The simulations suggest that the underlying cause of this difference is that the hexagonal phase forms via a metastable gyroid-like structure, and therefore forms via a nucleation-and-growth mechanism, whereas the lamellar phase is formed via spinodal decomposition.

Diffusion in Mixtures of Asymmetric Diblock Copolymers with Homopolymers

Tracer diffusion of asymmetric diblock copolymer in mixtures with homopolymers has been investigated as a function of the volume fraction of diblock copolymer and the molecular weight of matrix homopolymer. The depth profile of polydeuteriostyrene-b-2-vinylpyridine (dPS−PVP) diffusing into the mixture of polystyrene-b-2-vinylpyridine (PS−PVP) with polystyrene was measured by forward recoil spectrometry (FRES), and the tracer diffusion coefficient was determined. Secondary ion mass spectrometry (SIMS) was used for depth profiling when better depth resolution (∼15 nm) than that of FRES (∼80 nm) was necessary. The ordered structure of the mixtures was measured by small-angle X-ray scattering (SAXS). In concentrated mixtures of PS−PVP diblock copolymer with polystyrene which exhibit an ordered spherical domain structure, the diffusion coefficients are independent of the molecular weight of polystyrene and are also very close to the diffusion coefficient in a neat diblock copolymer melt. On the other hand, in dilute mixtures exhibiting a disordered arrangement of spherical copolymer micelles in a polystyrene matrix, the diffusion coefficient is strongly dependent on the molecular weight (P) of the matrix polystyrene. In highly dilute systems, the diffusion coefficient, D, scales as D ∼ P-3, which would be expected for Stokes−Einstein diffusion of spherical micelles in a homopolymer whose viscosity scales approximately as P3. The crossover from the “activated hopping” diffusion of single diblock copolymers (the diffusion mechanism in the neat diblock copolymer melt) to the Stokes−Einstein diffusion of whole spherical micelles occurs at the volume fraction of the transition between the ordered spherical domain structure and the disordered spherical domain structure.

Thin Films of Perpendicularly Oriented Diblock Copolymers: Lamellar Distortions Driven by Surface−Diblock Interfacial Tensions

We consider the problem of a diblock copolymer melt thin film on a homogeneous flat surface or confined between two flat plates. When there is significant mismatch between the film thickness and an integer multiple of the lamellar spacing, perpendicular orientation of the lamellae may be observed. However, because of differences in the interfacial tensions between the surface and the two blocks, the lamellae may be distorted, from a straight interface to a curved interface. We examine theoretically this distortion and show that its amplitude may only be a few percent of the lamellar spacing.

Statics and Dynamics of Symmetric Diblock Copolymers: A Molecular Dynamics Study

Extensive molecular dynamics simulations are carried out to study static and dynamic properties of symmetric diblock copolymer melts, both in the disordered and in the lamellar phases. The lamellar phase is constructed using the natural lamellar spacing, determined from constant pressure simulations. The non-Gaussian character of the chains in the disordered phase is demonstrated and quantified. In the lamellar phase, the density profile of the separate blocks, as well as the interface thickness are determined as a function of chain length N and AB interaction parameter e, and compared with experiments and other theoretical results. Single chain and single block form factors indicate that the chains in the lamellar phase are distorted into stick-like objects. Our results in the disordered phase show a stronger dependence of the diffusion constant on the chain length than observed in previous simulations. Diffusion within the lamellar plane for systems with chains of length N ≤ 100 monomers is found to be ...

Diblock Copolymer Thin Film Melts on Striped, Heterogeneous Surfaces: Parallel, Perpendicular and Mixed Lamellar Morphologies

We consider the problem of a diblock copolymer melt thin film on a striped surface. Because of the different surface energies, the different stripes have different preferences for each of the blocks, while the upper free surface may also prefer one of the blocks. As a result of a combination of factors such as the surface tensions and chain stretching, a number of morphologies may appear. We allow for perpendicular, parallel, and mixed type lamellae and study the resulting phase diagrams. These diagrams should provide a crude guide for experimental studies on such systems. Our main conclusions are 2-fold. First the phase diagram is intricate and complicated. Second, to produce a single crystal oriented along the stripes, it is best to use thin films and a diblock lamellar size equal to or larger than the stripe width.

Viscoelastic behavior of cubic phases in block copolymer melts

The viscoelastic behavior of block copolymer melts that exhibit a cubic phase has been examined by oscillatory shear experiments. A low frequency plateau in the measured storage modulus that is absent in the disordered phase is found for both gyroid and body-centered cubic sphere phases of diblock and triblock copolymer melts. The magnitude of the apparent plateau modulus is found to be insensitive to strain amplitude for strains of 0.05%–5%, and so is believed to be characteristic of the true linear response. The value of the apparent terminal relaxation frequency, beneath which G″(ω)>G′(ω) is, however, sensitive to strain amplitude in the same range of strains and decreases steadily with decreasing strain, indicating that the terminal regime is highly sensitive to nonlinear effects. The presence of one or more entangled blocks is found to decrease the terminal frequency, and thus extend the range of linear behavior. Experimental results for the dependence of the plateau modulus and the unit cell size upon molecular volume yield effective power law exponents that are close, but not identical, to those predicted by self-consistent field theory.The viscoelastic behavior of block copolymer melts that exhibit a cubic phase has been examined by oscillatory shear experiments. A low frequency plateau in the measured storage modulus that is absent in the disordered phase is found for both gyroid and body-centered cubic sphere phases of diblock and triblock copolymer melts. The magnitude of the apparent plateau modulus is found to be insensitive to strain amplitude for strains of 0.05%–5%, and so is believed to be characteristic of the true linear response. The value of the apparent terminal relaxation frequency, beneath which G″(ω)>G′(ω) is, however, sensitive to strain amplitude in the same range of strains and decreases steadily with decreasing strain, indicating that the terminal regime is highly sensitive to nonlinear effects. The presence of one or more entangled blocks is found to decrease the terminal frequency, and thus extend the range of linear behavior. Experimental results for the dependence of the plateau modulus and the unit cell size up...

Simultaneous Rheology and Small-Angle Scattering Experiments on Block Copolymer Gels and Melts in Cubic Phases

A new instrument for simultaneous small-angle X-ray scattering and rheology experiments on soft solids is described. This device is based on a commercial rheometer with a shear sandwich geometry in which the sample is subjected to a planar oscillatory deformation. This instrument has been used for time-resolved small-angle X-ray scattering/rheology experiments at the Synchrotron Radiation Source, Daresbury Laboratory, England. The focus has been in particular on the effect of large-amplitude shearing on the orientation of cubic phases in gels of block copolymers formed in concentrated solutions, and on the bicontinuous cubic phase of a block copolymer melt. Representative results are presented for face-centred cubic (f.c.c.) and body-centred cubic (b.c.c.) phases in gels of poly(oxyethylene)–poly(oxybutylene) diblock copolymers, and for the bicontinuous cubic `gyroid' structure in a poly-(ethylene-alt-propylene)–poly(dimethylsiloxane) di-block copolymer melt. The orientations of the micellar b.c.c. phases in the gels and the gyroid structure (belonging to the b.c.c. space group Ia\bar 3d) following large-amplitude shearing are shown to be the same,i.e.directionally oriented crystals are produced in both cases, in which (111) directions are oriented along the shear direction.

Stability study of icosahedral phases in diblock copolymer melt

The stability of three possible icosahedral phases (called here vertex, edge, and face models, respectively) in a diblock copolymer melt is examined, with higher harmonics included in the density-wave mean field theory developed by Leibler. Two cases are examined: (1) diblock copolymer melt with constant coupling vertices, and (2) diblock copolymer melt with general coupling vertices in which, unlike all the previous studies on the effect of higher order harmonics in diblock copolymer melt, the calculation is performed exactly. We find that icosahedral phases are always metastable and the mean-field phase diagram of the diblock copolymer is unchanged.

Self-consistent field theory of diblock copolymer melts at patterned surfaces

Self-consistent field theory is used to investigate the density profile of diblock copolymer melts in a thin film confined between two surfaces, one of which carries a chemically active pattern. The temperature is slowly lowered through the critical point in order to obtain well organized structures. For strong surface interactions, the lamellae for symmetric diblock copolymers comply with the surface pattern. Their orientation depends on the ratio of natural bulk period L0 to surface period L. For L0⩽L, the lamellae tilt at an angle θ=arcsin L0/L with respect to the surface. For L0⩾L, the diblocks are distorted close to the surface and the necessary relaxation off the surface induces parallel oriented lamellae with respect to the surface for films thicker than 2 lamellar periods. Very thin films still support the perpendicular orientation.

Lamellae stability in confined systems with gravity

The microphase separation of a diblock copolymer melt confined by hard walls and in the presence of a gravitational field is simulated by means of a cell dynamical system model. It is found that the presence of hard walls normal to the gravitational field are key ingredients to the formation of well ordered lamellae in BCP melts. To this effect the currents in the directions normal and parallel to the field are calculated along the interface of a lamellar domain, showing that the formation of lamellae parallel to the hard boundaries and normal to the field correspond to the stable configuration. Also, it is found thet the field increases the interface width.

Identification of the Molecular Parameters that Govern Ordering Kinetics in a Block Copolymer Melt

The objective of this paper is to identify the molecular parameters that govern ordering kinetics in a diblock copolymer melt. Experimental data on the ordering kinetics were obtained by time-resolved depolarized light scattering after the sample was quenched from the disordered to the ordered state. The ordered state consisted of cylinders arranged on a hexagonal lattice. During the early stages of the disorder-to-order transition, a dilute suspension of ordered grains grow by invading the surrounding, disordered phase. A time-dependent Ginzburg−Landau model was used to relate the growth rate of these grains to molecular parameters. The growth rate is predicted to be proportional to the quench depth and chain radius of gyration and inversely proportional to the molecular relaxation time. Independent estimates of these parameters were obtained from small angle neutron scattering and rheological measurements. The experimentally determined grain growth rates are in good agreement with theoretical predictions, with no adjustable parameters.

Mixed Parallel−Perpendicular Morphologies in Diblock Copolymer Systems Correlated to the Linear Viscoelastic Properties of the Parallel and Perpendicular Morphologies

Large-amplitude oscillatory shear (LAOS) has been applied to a lamellar, microphase-separated, low molecular weight poly(styrene-b-isoprene) diblock copolymer melt to induce macroscopically aligned morphologies. Small-angle X-ray scattering and dynamic mechanical testing were performed to characterize the states of orientation. The perpendicular orientation was induced by applying LAOS at a fixed frequency (1 rad/s) and strain amplitude (100%) at high temperatures. LAOS applied at the same frequency and strain amplitude but at lower temperatures induced the parallel orientation. Interestingly, intermediate temperatures produced a mixed morphology containing both parallel and perpendicular orientations. The linear viscoelastic responses of the parallel and perpendicular morphologies were related to the LAOS conditions necessary to produce the perpendicular, the parallel, and the mixed parallel−perpendicular morphologies. Specifically, two frequencies were defined using the linear viscoelastic responses: ωPd-Pl‘ where G‘(parallel) = G‘(perpendicular) and ωPd-Pl‘‘ where G‘‘(parallel) = G‘‘(perpendicular). The uniaxial morphology induced by LAOS was found to correspond to that morphology which subsequently produced the lowest linear viscoelastic moduli, both elastic and storage. Analysis also showed that the viscoelastic response of the mixed parallel−perpendicular morphology could be approximated by a linear combination of the responses for samples wholly in the parallel and perpendicular orientations. Similar alignment and rheological results were obtained for a blend containing the diblock copolymer and 10 wt % homopolystyrene.