Publications

Langevin field-theoretic simulations (L-FTSs) have recently evaluated complete phase diagrams for diblock copolymer melts [Matsen et al., Phys. Rev. Lett. 2023, 130, 248101], but they involve a partial saddle-point approximation (PSPA). Although previous complex-Langevin field-theoretic simulations (CL-FTSs) calculated complete phase diagrams without the PSPA [Delaney and Fredrickson, J. Phys. Chem. B 2016, 120, 7615], the diagrams disagree with experiments. We find evidence attributing this to nonuniversal behavior resulting from long-range interactions used to remove an ultraviolet divergence. Therefore, we perform CL-FTSs on symmetric diblocks with contact interactions and use the renormalization procedure employed in the L-FTSs. This successfully removes the divergence and restores the universality, but, without the long-range interactions, the fields have a tendency to form intense delta-like “hot spots”, causing the CL-FTSs to fail. Nevertheless, we find good agreement between the CL-FTSs and L-FTSs over the parameter space where the CL-FTSs are sufficiently well behaved. This implies that the renormalization compensates for the inaccuracy of the PSPA, thereby imparting L-FTSs with the ability to provide accurate universal predictions for block copolymer melts of high molecular weight.

Self-consistent field theory has demonstrated that the homologous series of \((AB)_{M}\) starblock copolymers are promising architectures for the complex network Fddd phase. Nevertheless, it remains to be seen if the level of segregation will be sufficient to survive the fluctuations inevitably present in experiments. Here, we study the effect of fluctuations using field-theoretic simulations, which are uniquely capable of evaluating order-order phase transitions. This facilitates the calculation of complete fluctuation-corrected diagrams for the diblock \((M=1)\), symmetric triblock \((M=2)\), and nine-arm starblock \((M=9)\) architectures. Although fluctuations disorder the Fddd phase at weak segregations, they also stabilize the Fddd phase with respect to its ordered neighbors, which extends the Fddd region to higher segregation. Our results provide strong evidence that Fddd will remain stable in experiments on the side of the phase diagram where the outer \(A\) blocks of the star form the network domain. However, it is doubtful that Fddd will survive fluctuations on the other side where they form the matrix domain.

New developments in field-theoretic simulations (FTSs) are used to evaluate fluctuation corrections to the self-consistent field theory of diblock copolymer melts. Conventional simulations have been limited to the order-disorder transition (ODT), whereas FTSs allow us to evaluate complete phase diagrams for a series of invariant polymerization indices. The fluctuations stabilize the disordered phase, which shifts the ODT to higher segregation. Furthermore, they stabilize the network phases at the expense of the lamellar phase, which accounts for the presence of the Fddd phase in experiments. We hypothesize that this is due to an undulation entropy that favors curved interfaces.

This study examines the ultraviolet (UV) divergence in field-theoretic simulations (FTSs) of block copolymer melts, which causes an unphysical dependence on the grid resolution, \(\Delta\), used to represent the fields. Our FTSs use the discrete Gaussian-chain model and a partial saddle-point approximation to enforce incompressibility. Previous work has demonstrated that the UV divergence can be accounted for by defining an effective interaction parameter, \( \chi = z_{\infty}\chi_{b} + c_{2}\chi_{b}^{2} + c_{3}\chi_{b}^{3} + \ldots \), in terms of the bare interaction parameter, \(\chi_{b}\), used in the FTSs, where the coefficients of the expansion are determined by a Morse calibration. However, the need to use different grid resolutions for different ordered phases generally restricts the calibration to the linear approximation, \(\chi \approx z_{\infty}\chi_{b}\), and prevents the calculation of order-order transitions. Here, we resolve these two issues by showing how the nonlinear calibration can be translated between different grids and how the UV divergence can be removed from free energy calculations. By doing so, we confirm previous observations from particle-based simulations. In particular, we show that the free energy closely matches self-consistent field theory (SCFT) predictions, even in the region where fluctuations disorder the periodic morphologies, and similarly, the periods of the ordered phases match SCFT predictions, provided the SCFT is evaluated with the nonlinear \(\chi\).

Well-tempered metadynamics (WTMD) is applied to field-theoretic simulations (FTS) to locate the order-disorder transition (ODT) in incompressible melts of diblock copolymer with an invariant polymerization index of \(\bar{N} = 10^{4}\). The polymers are modeled as discrete Gaussian chains with \(N = 90\) monomers, and the incompressibility is treated by a partial saddle-point approximation. Our implementation of WTMD proves effective at locating the ODT of the lamellar and cylindrical regions, but it has difficulty with that of the spherical and gyroid regions. In the latter two cases, our choice of order parameter cannot sufficiently distinguish the ordered and disordered states because of the similarity in microstructures. The gyroid phase has the added complication that it competes with a number of other morphologies, and thus, it might be beneficial to extend the WTMD to multiple order parameters. Nevertheless, when the method works, the ODT can be located with impressive accuracy (e.g., \(\Delta \chi N \sim 0.01\)).

Field-theoretic simulations (FTS) provide an efficient technique for investigating fluctuation effects in block copolymer melts with numerous advantages over traditional particle-based simulations. For systems involving two components (i.e., \(A\) and \(B\)), the field-based Hamiltonian, \(H_{f}[W_{-}, W_{+}]\), depends on a composition field, \(W_{-}(r)\), that controls the segregation of the unlike components and a pressure field, \(W_{+}(r)\), that enforces incompressibility. This review introduces researchers to a promising variant of FTS, in which \(W_{-}(r)\) fluctuates while \(W_{+}(r)\) tracks its mean-field value. The method is described in detail for melts of \(AB\) diblock copolymer, covering its theoretical foundation through to its numerical implementation. We then illustrate its application for neat \(AB\) diblock copolymer melts, as well as ternary blends of \(AB\) diblock copolymer with its \(A-\) and \(B-\)type parent homopolymers. The review concludes by discussing the future outlook. To help researchers adopt the method, open-source code is provided that can be run on either central processing units (CPUs) or graphics processing units (GPUs).

The order-disorder transition (ODT) of diblock copolymer melts is evaluated for an invariant polymerization index of \(\bar{N}=10^{4}\), using field-theoretic simulations (FTS) supplemented by a partial saddle-point approximation for incompressibility. For computational efficiency, the FTS are performed using the discrete Gaussian-chain model, and results are then mapped onto the continuous model using a linear approximation for the Flory-Huggins \(\chi\) parameter. Particular attention is paid to the complex phase window. Results are found to be consistent with the well-established understanding that the gyroid phase extends down to the ODT. Furthermore, our simulations are the first to predict that the Fddd phase survives fluctuation effects, consistent with experiments.

This paper improves upon a standard method of determining the Flory-Huggins \(\chi\) parameter, whereby experimental order-disorder transitions (ODTs) of symmetric diblock polymer melts are fit to the mean-field prediction, \((\chi N)_{\textrm{ODT}} = 10.495\). The improvement is achieved by switching to an accurate prediction of \((\chi N)_{\textrm{ODT}}\) from Glaser et al. (Phys. Rev. Lett. 2014, 113, 068302), supplemented with corrections for the small degrees of polydispersity and compositional asymmetry that inevitably exist in real diblock polymers. The first correction is evaluated by simulating polydisperse diblocks over a wide range of invariant polymerization indices, and the second correction is extracted from analogous simulations for compositionally asymmetric diblocks by Ghasimakbari and Morse (Macromolecules 2020, 53, 7399). The resulting calibration method is then demonstrated on 19 different chemical pairs, using previously published experimental data. It provides a considerable increase in accuracy, but yet is nearly as simple to apply as the original version.

Field-theoretic simulations (FTS) provide fluctuation corrections to self-consistent field theory (SCFT) by simulating its field-theoretic Hamiltonian rather than applying the saddle-point approximation. Although FTS work well for ultrahigh molecular weights, they have struggled with experimentally relevant values. Here, we consider FTS for two-component (i.e., AB-type) melts, where the composition field fluctuates but the saddle-point approximation is still applied to the pressure field that enforces incompressibility. This results in real-valued fields, thereby allowing for conventional simulation methods. We discover that Langevin simulations are 1-2 orders of magnitude faster than previous Monte Carlo simulations, which permits us to accurately calculate the order-disorder transition of symmetric diblock copolymer melts at realistic molecular weights. This remarkable speedup will, likewise, facilitate FTS for more complicated block copolymer systems, which might otherwise be unfeasible with traditional particle-based simulations.

The Morse calibration is applied to a lattice model designed for efficient simulations of two-component polymer melts of high molecular weight. The model allows multiple occupancy per site, which results in high invariant polymerization indices, and interactions are limited to monomers within the same site, which enhances the computational speed. The calibration maps the interaction parameter of the lattice model, \(\alpha\), onto the Flory-Huggins \(\chi\) parameter of the standard Gaussian-chain model, by matching the disordered-state structure function, \(S(k)\), of symmetric diblock copolymers to renormalized one-loop predictions. The quantitative accuracy of the calibration is tested by comparing the order-disorder transition of symmetric diblock copolymer melts to the universal prediction obtained from previous simulations. The model is then used to confirm the universality of fluctuation corrections to the critical point of symmetric binary homopolymer blends.

Field-theoretic simulations (FTS) offer a versatile method of dealing with complicated block copolymer systems, but unfortunately they struggle to cope with the level of fluctuations typical of experiments. Although the main obstacle, an ultraviolet divergence, can be removed by renormalizing the Flory-Huggins \(\chi\) parameter, this only works for unrealistically large invariant polymerization indexes, \(\bar{N}\). Here, we circumvent the problem by applying the Morse calibration, where a nonlinear relationship between the bare \(\chi_{b}\) used in FTS and the effective \(\chi\) corresponding to the standard Gaussian-chain model is obtained by matching the disordered-state structure function, \(S(k)\), of symmetric diblock copolymers to renormalized one-loop predictions. This calibration brings the order-disorder transition obtained from FTS into agreement with the universal results of particle-based simulations for values of \(\bar{N}\) characteristic of the experiment. In the limit of weak interactions, the calibration reduces to a linear approximation, \(\chi \approx z_{\infty}\chi_{b}\), consistent with the previous renormalization of \(\chi\) for large \(\bar{N}\).

This study addresses entropic segregation effects at the surfaces of monodisperse and bidisperse melts. For the monodisperse melts, we focus on the segregation of chain ends to the surface, and for the bidisperse melts, we examine the segregation of short polymers to the surface. Universal shapes have been predicted for their concentration profiles, but the derivations rely on the mean-field approximation, which only treats the excluded-volume interactions in an approximate manner. To test whether or not the predictions hold up when the polymers are rigorously prevented from overlapping, we compare mean-field calculations with Monte Carlo simulations performed on the exact same model. Apart from a significant increase in the statistical segment length, the rigorous enforcement of excluded-volume interactions has a relatively small effect on the mean-field predictions. In particular, the universal profiles predicted by mean-field theory are found to be accurate.

Monte Carlo simulations are performed on structurally symmetric binary homopolymer blends over a wide range of invariant polymerization indices, \(\bar{N}\). A finite-size scaling analysis reveals that certain critical exponents deviate from the expected 3D-Ising values as \(\bar{N}\) increases. However, the deviations are consistent with previous simulations and can be attributed to the fact that the system crosses over to mean-field behavior when the molecules become too large relative to the size of the simulation box. Nevertheless, the finite-size scaling techniques provide precise predictions for the position of the critical transition. Using a previous calibration of the Flory-Huggins interaction parameter, \(\chi\), we confirm that the critical point scales as \((\chi N)_{c} = 2 + c \bar{N}^{-1/2}\) for large \(\bar{N}\), and more importantly we are able to extract a reliable estimate, \(c \approx 1.5\), for the universal constant.

The equivalent behavior among analogous block copolymer systems involving chemically distinct molecules or mathematically different models has long hinted at an underlying universality, but only recently has it been rigorously demonstrated by matching results from different simulations. The profound implication of universality is that simple coarse-grained models can be calibrated so as to provide quantitatively accurate predictions to experiment. Here, we provide the first compelling demonstration of this by simulating a polyisoprene-polylactide diblock copolymer melt using a previously calibrated lattice model. The simulation successfully predicts the peak in the disordered-state structure function, the position of the order-disorder transition, and the latent heat of the transition in excellent quantitative agreement with experiment. This could mark a new era of precision in the field of block copolymer research.

Simulations of five different coarse-grained models of symmetric diblock copolymers are compared to demonstrate a universal (i.e., model-independent) dependence of the free energy and order-disorder transition (ODT) on the invariant degree of polymerization \(\bar{N}\). The actual values of \(\chi N\) at the ODT approach predictions of the Fredrickson-Helfand (FH) theory for \(\bar{N} \gtrsim 10^{4}\) but significantly exceed FH predictions at lower values characteristic of most experiments. The FH theory fails for modest \(\bar{N}\) because the competing phases become strongly segregated near the ODT, violating an underlying assumption of weak segregation.

The phase diagram for an AB diblock copolymer melt with polydisperse A blocks and monodisperse B blocks is evaluated using lattice-based Monte Carlo simulations. Experiments on this system have shown that the A-block polydispersity shifts the order-order transitions (OOTs) toward higher A-monomer content, while the order-disorder transition (ODT) moves toward higher temperatures when the A blocks form the minority domains and lower temperatures when the A blocks form the matrix. Although self-consistent field theory (SCFT) correctly accounts for the change in the OOTs, it incorrectly predicts the ODT to shift toward higher temperatures at all diblock copolymer compositions. In contrast, our simulations predict the correct shifts for both the OOTs and the ODT. This implies that polydispersity amplifies the fluctuation-induced correction to the mean-field ODT, which we attribute to a reduction in packing frustration. Consistent with this explanation, polydispersity is found to enhance the stability of the perforated-lamellar phase.

The phase diagram for diblock copolymer melts is evaluated from lattice-based Monte Carlo simulations using parallel tempering, improving upon earlier simulations that used sequential temperature scans. This new approach locates the order-disorder transition (ODT) far more accurately by the occurrence of a sharp spike in the heat capacity. The present study also performs a more thorough investigation of finite-size effects, which reveals that the gyroid (G) morphology spontaneously forms in place of the perforated-lamellar (PL) phase identified in the earlier study. Nevertheless, there still remains a small region where the PL phase appears to be stable. Interestingly, the lamellar (L) phase next to this region exhibits a small population of transient perforations, which may explain previous scattering experiments suggesting a modulated-lamellar (ML) phase.

The effect of polydispersity on an AB diblock copolymer melt is investigated using lattice-based Monte Carlo simulations. We consider melts of symmetric composition, where the B blocks are monodisperse and the A blocks are polydisperse with a Schultz-Zimm distribution. In agreement with experiment and self-consistent field theory (SCFT), we find that polydispersity causes a significant increase in domain size. It also induces a transition from flat to curved interfaces, with the polydisperse blocks residing on the inside of the interfacial curvature. Most importantly, the simulations show a relatively small shift in the order-disorder transition (ODT) in agreement with experiment, whereas SCFT incorrectly predicts a sizable shift towards higher temperatures.