Symmetric Diblock Copolymer Melts Research Articles

Photomodulated phase-separation kinetics in block copolymer melts: a DPD simulation study

ABSTRACT We study phase-separation kinetics of symmetric diblock copolymer (BCP) melt in 3 d subjected to sequential light on/off cycles. The incompatible blocks of each BCP chain are connected by a photosensitive bond, which undergoes breakage during photo-illumination (on-cycle) and recombination during the off-cycle. We set a shorter fixed time limit for each on-cycles and focus on analysing how the much longer duration of the light off-cycles affects the segregation kinetics. We explore two scenarios for each off-cycle time variation: one with equal and another with increasing durations. For both cases, we determine the rate constants for bond formation and degradation reactions and assess their influence on the dynamical scaling functions and the growth scale of evolving morphologies. The average domain size typically exhibits diffusive growth ( R ( t ) ∼ t 1 / 3 ) for a couple of cycles in the beginning. However, the domain growth nearly freezes during the rest of the cycles, resulting in bicontinuous and nearly frozen isotropic morphologies.

Topological Effects Near Order–Disorder Transitions in Symmetric Diblock Copolymer Melts

Topological Effects Near Order–Disorder Transitions in Symmetric Diblock Copolymer Melts

Simulation of diblock copolymer surfactants. III. Equilibrium interfacial adsorption.

Monte Carlo simulations are used to study adsorption of highly asymmetric diblock copolymers to a polymer-polymer interface, and the results compared to self-consistent field theory (SCFT) predictions. The simulation model used here is a bead-spring model that has been used previously to study equilibrium and kinetic properties of spherical micelles [J. A. Mysona et al., Phys. Rev. E 100, 012602 (2019)2470-004510.1103/PhysRevE.100.012602; Phys. Rev. E 100, 012603 (2019)10.1103/PhysRevE.100.012603; Phys. Rev. Lett. 123, 038003 (2019)10.1103/PhysRevLett.123.038003]. Interfacial copolymer concentration Γ and interfacial tension γ are measured as functions of bulk copolymer concentration at concentrations up to the critical micelle concentration over a range of values of the Flory-Huggins χ parameter. The dependence of interfacial pressure Π = γ_{0}-γ on Γ (where γ_{0} is the interfacial tension in the absence of copolymer) is found to be almost independent of χ and to be accurately predicted by SCFT. The bare interfacial tension γ_{0} and total interfacial tension γ(Γ) can also be accurately predicted by SCFT using an estimate of χ obtained from independent analysis of properties of symmetric diblock copolymer melts. SCFT predictions obtained with this estimate of χ do not, however, adequately describe the thermodynamics of the coexisting bulk copolymer solution, as a result of contraction of the strongly interacting core block of dissolved copolymers. Accurate predictions of the relationship between bulk and interfacial properties can thus only be obtained for this system by combining SCFT predictions of the interfacial equation of state with a fit to the measured equation of state for the bulk solution.

Structure-based coarse graining of diblock copolymer melts using PRISM theory

We applied our recently proposed systematic and simulation-free strategy for the structure-based coarse graining of multicomponent polymeric systems (Wang, Q., Polymer2017,117, 315) to diblock copolymer melts, where we use the well-developed polymer reference interaction site model (PRISM) theory, instead of the commonly used many-chain molecular simulations, to obtain the structural and thermodynamic properties of both the original and coarse-grained (CG) systems, and to quantitatively examine how the effective non-bonded pair potentials between CG segments and the thermodynamic properties of CG systems vary with the coarse-graining level. We proved that our strategy does not change the spinodal curve, regardless of the original model system, closures, and coarse-graining levels for the two blocks. Using a simple original model system for symmetric diblock copolymer melts, we coarse-grained each block as N/2 segments and examined CG systems with N ranging from 2 to 100. We found that coarse graining (i.e., decreasing N) increases both the peak value and peak location of the partial structure factor characterizing the composition fluctuations in the CG system. Contrary to the common practice in the literature, CG potentials obtained from short-chain systems cannot be directly used for long-chain systems; this is in fact the transferability problem in coarse graining (Louis, A. A., J. Phys.: Condens. Matter2002,14, 9187). We also found that the structure-based coarse graining cannot give thermodynamic consistency at any coarse-graining level, consistent with our previous work on homopolymer melts (Yang, D. L.; Wang, Q., J. Chem. Phys.2015,142, 054905; Wang, Q.; Yang, D. L., Polymer2017,111, 103) and binary polymer blends (Wang, Q., Polymer2017,117, 315).

Network Model of the Disordered Phase in Symmetric Diblock Copolymer Melts.

We present a model for the order-disorder transition of symmetric A-B diblock copolymer melts in which the disordered phase is treated as a bicontinuous network, and in which self-consistent field predictions of properties of an analogous ordered network are used to estimate some properties. Such a model is shown to accurately predict the latent heat of this transition. The dependence of the location of the transition upon the invariant degree of polymerization N[over ¯] is shown to be consistent with a simple hypothesis that the disordered bicontinuous structure is stabilized relative to an analogous ordered network by a nearly constant entropy per network junction.

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, χb, 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.

Ordering Lamellar-Forming Copolymer Thin Films in 3D Bicontinuous Morphologies via Lamellar Patterned Substrate

The formation of ordered morphologies in thin films of symmetric diblock copolymer melts is considered theoretically. Somewhat surprisingly, under proper boundary conditions the presence of a lamellar chemical pattern on the substrate, being sufficiently pronounced and with the right period, is found to induce the formation of diamond-like morphologies. The phase diagram of the most stable phases on the plane (the substrate period Lx–the film width H) is built within the self-consistent field theory numerical procedure. We also discuss the behavior of the order parameter Fourier spectrum at the transitions between the various morphologies.

Shear-Induced Ordering in Thin Films of Diblock Copolymer Melts.

By means of molecular dynamics simulations we investigate the response of thin, symmetric diblock copolymer melts under shear in the limit of strong segregation with nonselective substrates, where vertically oriented lamellae form. Under small shear perpendicular to the lamellar orientation, we observe an inclination of the lamellar layers. At a critical shear rate, the lamellar layers become distorted and, for very large shear, recombine with a new orientation along the direction of shear. Our simulations are accompanied by a novel, easily understandable theoretical approach to predict the critical shear rate, at which the interfaces become distorted and shear-induced reorientation sets in. This allows one to calculate quantities such as the inclination angle or the pair interaction energy as a function of applied shear rate. Our results are relevant for many technical applications, where defect-free, long-range ordered structures are needed.

Test of a scaling hypothesis for the structure factor of disordered diblock copolymer melts

Coarse-grained theories of dense polymer liquids such as block copolymer melts predict a universal dependence of equilibrium properties on a few dimensionless parameters. For symmetric diblock copolymer melts, such theories predict a universal dependence on only χN and , where χ is an effective interaction parameter, N is the degree of polymerization, and is a measure of overlap. We test whether simulation results for the structure factor S(q) obtained from several different simulation models are consistent with this two-parameter scaling hypothesis. We compare results from three models: (1) a lattice Monte Carlo model, the bond-fluctuation model, (2) a bead–spring model with harsh repulsive interactions, similar to that of Kremer and Grest, and (3) a bead–spring model with very soft repulsion between beads, and strongly overlapping beads. We compare results from pairs of simulations of different models that have been designed to have matched values of , over a range of values of χN and N, and devise methods to test the scaling hypothesis without relying on any prediction for how the phenomenological interaction parameter χ depends on more microscopic parameters. The results strongly support the scaling hypothesis, even for rather short chains, confirming that it is indeed possible to give an accurate universal description of simulation models that differ in many details.

Localization of spherical nanoparticles within lamellar AB diblock copolymer melts through self-consistent field theory

Here we report the results of a three dimensional hybrid self-consistent field theoretic (HSCFT) model describing the equilibrium particle distribution of spherical nanoparticles within symmetric AB diblock copolymer melts. Holding the polymer composition and morphology fixed, we consider a comprehensive parameter space comprised of the Flory interaction parameter describing interactions between B segments and the particle surface compared to the segment–segment interaction parameter ( χ B P / χ A B ) , the particle volume fraction ( ɸ P ), and the ratio of the particle diameter to block copolymer domain spacing ( d P / d A B ) . Analysis of the free energy over this parameter space yields phase diagrams showing the conditions under which particles segregate to the intermaterial diving surface (IMDS) or the center of the domain. Interestingly, w e predict a particle concentration-dependent “reentrant” phase transition in which particles move from the domain interior, to the IMDS, and back as ɸ P increases. These results are interpreted as a subtle consequence of the competition between enthalpic polymer–particle interactions and the chain packing frustration imposed by the particulate inclusion. These results are consistent with recent experiments on block copolymer nanocomposites.

Dynamical self-arrest in symmetric and asymmetric diblock copolymer melts using a replica approach within a local theory

We investigate dynamical self-arrest in a diblock copolymer melt using a replica approach within a self-consistent local method based on dynamical mean-field theory (DMFT). The local replica approach effectively predicts (chiN)_{A} for dynamical self-arrest in a block copolymer melt for symmetric and asymmetric cases. We discuss the competition of the cubic and quartic interactions in the Landau free energy for a block copolymer melt in stabilizing a glassy state depending on the chain length. Our local replica theory provides a universal value for the dynamical self-arrest in block copolymer melts with (chiN)_{A} approximately 10.5+64N;{-3/10} for the symmetric case.

Computing free energies of interfaces in self-assembling systems

Using molecular simulation of a coarse-grained model for a symmetric diblock copolymer melt, we calculate the free energy of interfaces between lamellar morphologies with different orientations. Two examples are considered: (a) a T-junction between two lamellar structures with perpendicular orientation and (b) a surface reconstruction that arises when lamella-forming diblock copolymers assemble on a stripe-patterned surface, where the pattern period is significantly larger than the lamellar spacing in the bulk. The computational scheme relies on reversibly relating the defect structure to a reference state by an external ordering field. Free energy differences are accurately obtained by using a combination of expanded-ensemble and replica-exchange Monte Carlo techniques.

Calculating the free energy of self-assembled structures by thermodynamic integration

We discuss a method for calculating free energy differences between disordered and ordered phases of self-assembling systems utilizing computer simulations. Applying an external, ordering field, we impose a predefined structure onto the fluid in the disordered phase. The structure in the presence of the external, ordering field closely mimics the structure of the ordered phase (in the absence of an ordering field). Self-consistent field theory or density functional theory provides an accurate estimate for choosing the strength of the ordering field. Subsequently, we gradually switch off the external, ordering field and, in turn, increase the control parameter that drives the self-assembly. The free energy difference along this reversible path connecting the disordered and the ordered state is obtained via thermodynamic integration or expanded ensemble simulation techniques. Utilizing Single-Chain-in-Mean-Field simulations of a symmetric diblock copolymer melt we illustrate the method and calculate the free energy difference between the disordered phase and the lamellar structure at an intermediate incompatibility chiN=20. Evidence for the first-order character of the order-disorder transition at fixed volume is presented. The transition is located at chi(ODT)N=13.65+/-0.10 for an invariant degree of polymerization of N=14 884. The magnitude of the shift of the transition from the mean field prediction qualitatively agrees with other simulations.

The nature of phase transitions of symmetric diblock copolymer melts under confinement

The effects of confinement, both the structure frustration and the surface field, on phase transitions of symmetric diblock copolymer melts are investigated within several theoretical methods on the mean-field level. Confinements are applied by restricting polymer chains in the finite spaces of slabs. The surface can be neutral or preferential depending on the strength of the surface field. Within the one-dimensional self-consistent mean-field theory, for the neutral surface case, an oscillative behavior is observed for the size dependence of the order–disorder transition (ODT) point ( χN) t due to the structure frustration. The spinodal ( χN) s for this corresponding confined system is also calculated using the Gaussian fluctuation theory and the Landau–Brazovskii theory, and ( χN) s coincides exactly with ( χN) t. On the other hand, the surface effect plays the role to decrease ( χN) t due to the surface-induced spatial oscillation for the preferential surface case. In all confined systems considered, the ODT for symmetric diblock copolymer melts is a continuous second-order phase transition in the present mean-field calculation.

Diblock Copolymer Thin Films: A Field-Theoretic Simulation Study

Using field-theoretic simulations, we study a symmetric diblock copolymer melt confined between two parallel neutral walls separated by a distance L. We consider two scenarios: a mean-field regime and a fluctuating case corresponding to a polymerization index N ∼ 10 5 . For both cases, we analyze the behavior of this system in the disordered and the ordered phases as a function of the film thickness L. In particular, we compute the structure factor for each case, and compare it to the predicted structure factor for a bulk system using the random phase approximation. In the disordered phase, we find qualitative agreement with the bulk system, except that the amplitudes are dependent on the modes perpendicular to the slit. The ordered structure in all cases was found to be a lamellar phase oriented perpendicular to the walls. Also, we studied the location of the order-disorder transition (ODT) in both cases. In the mean-field regime, it was found that the transition is insensitive to the width of the cavity, and occurs, as expected, at ΧN* = 10.5. For the fluctuating case, we find a shift in the ODT that becomes essentially constant for slit widths larger than the unperturbed radius of gyration of the copolymers. For thicknesses below this value, we observe a confinement induced melting of the film, for which a theoretical explanation is proposed.

Fluctuation Effects in the Theory of Microphase Separation of Diblock Copolymers in the Presence of an Electric Field

We generalize the Fredrickson−Helfand theory of the microphase separation in symmetric diblock copolymer melts by taking into account the influence of a time-independent homogeneous electric field on the composition fluctuations within the self-consistent Hartree approximation. We predict that electric fields suppress composition fluctuations, and consequently weaken the first-order transition. In the presence of an electric field the critical temperature of the order−disorder transition is shifted toward its mean-field value. The collective structure factor in the disordered phase becomes anisotropic in the presence of the electric field. Fluctuational modulations of the order parameter along the field direction are strongest suppressed. The latter is in accordance with the parallel orientation of the lamellae in the ordered state.

Electric field versus surface alignment in confined films of a diblock copolymer melt

The dynamics of alignment of microstructure in confined films of diblock copolymer melts in the presence of an external electric field was studied numerically. We consider in detail a symmetric diblock copolymer melt, exhibiting a lamellar morphology. The method used is a dynamic mean-field density functional method, derived from the generalized time-dependent Ginzburg-Landau theory. The time evolution of concentration variables and therefore the alignment kinetics of the morphologies are described by a set of stochastic equations of a diffusion form with Gaussian noise. We investigated the effect of an electric field on block copolymers under the assumption that the long-range dipolar interaction induced by the fluctuations of composition pattern is a dominant mechanism of electric-field-induced domain alignment. The interactions with bounding electrode surfaces were taken into account as short-range interactions resulting in an additional term in the free energy of the sample. This term contributes only in the vicinity of the surfaces. The surfaces and the electric field compete with each other and align the microstructure in perpendicular directions. Depending on the ratio between electric field and interfacial interactions, parallel or perpendicular lamellar orientations were observed. The time scale of the electric-field-induced alignment is much larger than the time scale of the surface-induced alignment and microphase separation.

Correlated disorder in random block copolymers

We study the effect of a random Flory-Huggins parameter in a symmetric diblock copolymer melt which is expected to occur in a copolymer where one block is near its structural glass transition. In the clean limit the microphase segregation between the two blocks causes a weak, fluctuation induced first order transition to a lamellar state. Using a renormalization group approach combined with the replica trick to treat the quenched disorder, we show that beyond a critical disorder strength, which depends on the length of the polymer chain, the character of the transition is changed. The system becomes dominated by strong randomness and a glassy rather than an ordered lamellar state occurs. A renormalization of the effective disorder distribution leads to nonlocal disorder correlations that reflect strong compositional fluctuation on the scale of the radius of gyration of the polymer chains. The reason for this behavior is shown to be the chain length dependent role of critical fluctuations, which are less important for shorter chains and become increasingly more relevant as the polymer length increases and the clean first order transition becomes weaker.

Effect of particle size and shape on the order–disorder phase transition in diblock copolymers

The effect of spherical and sphero-cylindrical fillers on the order–disorder transition (ODT) of a symmetric diblock copolymer melt is investigated. Self-consistent equations describing the copolymer density distribution in the presence of fillers are derived. Using these equations, we calculate the excess free energy due to the presence of the particles in the diblocks. The critical value of the segregation factor χN is recalculated with the effect of the fillers taken into account. We find that a relatively small volume fraction of fillers can cause a significant suppression of the ODT temperature. It is shown that smaller particles cause a greater suppression of the ODT temperature provided a constant particle volume fraction is maintained. The effect of the particle shape on the ODT is investigated. The ODT temperature shift is calculated for the sphero-cylindrical particles as a function of their aspect ratio at a given particle volume. It is found that sphero-cylinders with the smaller aspect ratio produce the bigger effect on the ODT. A scaling analysis of the presented results and a comparison with the experimental work are given.

Fluctuation effects in block copolymer melts

Fluctuation effects in symmetric diblock copolymer melts are quantitatively examined with the first direct comparison between Monte Carlo simulations and mean-field theory where both are performed on the same identical model. The simulations provide the most conclusive evidence to date that fluctuations transform the continuous mean-field order–disorder transition (ODT) into a discontinuous transition. Furthermore, the fluctuations shift the ODT toward high segregation, χN, by a factor of ∼2.7, which is much greater than that suggested by the Fredrickson–Helfand fluctuation theory. For the range of small molecules examined in our study, this shift is nearly independent of molecular weight. In the disordered phase, we find the usual large deviations from mean-field theory, but we find no indication whatsoever that they set in abruptly at some well-defined temperature as previously claimed.