Formation and structure of the microemulsion phase in two-dimensional ternary AB+A+B polymeric emulsions

Bicontinuous microemulsion in binary blends of complementary diblock copolymers

The phase behavior of binary blends of AB diblock copolymers of compositions f and 1 - f is examined using field-theoretic simulations. Highly asymmetric compositions (i.e., f ≈ 0) behave like homopolymer blends macrophase separating into coexisting A- and B-rich phases as the segregation is increased, whereas more symmetric diblocks (i.e., f ≈ 0.5) microphase separate into an ordered lamellar phase. In self-consistent field theory, these behaviors are separated by a Lifshitz critical point at f = 0.2113. However, its lower critical dimension is believed to be four, which implies that the Lifshitz point should be destroyed by fluctuations. Consistent with this, it is found to transform into a tricritical point. Furthermore, the highly swollen lamellar phase near the mean-field Lifshitz point disorders into a bicontinuous microemulsion (BμE), consisting of large interpenetrating A- and B-rich microdomains. BμE has been previously reported in ternary blends of AB diblock copolymer with its parent A- and B-type homopolymers, but in that system the homopolymers have a tendency to macrophase separate. Our alternative system for creating BμE is free of this macrophase separation.

Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation

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, Hf[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).

Instability of the Microemulsion Channel in Block Copolymer-Homopolymer Blends

Field theoretic simulations are used to predict the equilibrium phase diagram of symmetric blends of AB diblock copolymer with A- and B-type homopolymers. Experiments generally observe a channel of bicontinuous microemulsion (BμE) separating the ordered lamellar (LAM) phase from coexisting homopolymer-rich (A+B) phases. However, our simulations find that the channel is unstable with respect to macrophase separation, in particular, A+B+BμE coexistence at high T and A+B+LAM coexistence at low T. The preference for three-phase coexistence is attributed to a weak attractive interaction between diblock monolayers.

Field theoretic approach for block polymer melts: SCFT and FTS

This perspective addresses the development of polymer field theory for predicting the equilibrium phase behavior of block polymer melts. The approach is tailored to the high-molecular-weight limit, where universality reduces all systems to the standard Gaussian chain model, an incompressible melt of elastic threads interacting by contact forces. Using mathematical identities, this particle-based version of the model is converted to an equivalent field-based version that depends on fields rather than particle coordinates. The statistical mechanics of the field-based model is typically solved using the saddle-point approximation of self-consistent field theory (SCFT), which equates to mean field theory, but it can also be evaluated using field theoretic simulations (FTS). While SCFT has matured into one of the most successful theories in soft condensed matter, FTS are still in its infancy. The two main obstacles of FTS are the high computational cost and the occurrence of an ultraviolet divergence, but fortunately there has been recent groundbreaking progress on both fronts. As such, FTS are now well poised to become the method of choice for predicting fluctuation corrections to mean field theory.

Calibration of the Flory-Huggins interaction parameter in field-theoretic simulations

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 χ parameter, this only works for unrealistically large invariant polymerization indexes, N¯. Here, we circumvent the problem by applying the Morse calibration, where a nonlinear relationship between the bare χ_b used in FTS and the effective χ 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 N¯ characteristic of the experiment. In the limit of weak interactions, the calibration reduces to a linear approximation, χ ≈ z_∞χ_b, consistent with the previous renormalization of χ for large N¯.

The A-B diblock copolymer as a nonordering external field in a three-component A/B/A-B polymer blend

Thermal copolymer fluctuations were explored in a three-component blend consisting of a critical (A/B) homopolymer blend and a symmetric A-B diblock copolymer using the technique of neutron small angle scattering. The copolymer has the function of an external nonordering field and thereby determines phase behavior as well as the regimes of 3d-Ising, isotropic Lifshitz, and Brasovskiî critical universality. It was found that the random phase approximation (RPA) does not correctly describe the copolymer structure function because of strong thermal fluctuations. On the other hand a weak coupling of copolymer and homopolymer was confirmed, consistent with predictions from RPA. Self-assembly of the copolymers was observed prior to the ordering of the "total" blend, e.g. inclusive of the homopolymers, into bicontinuous and lamellar ordered phases.

Microphase morphology in two-dimensional fluids under lateral confinement

We study the effects of confinement between two parallel walls on a two dimensional fluid with competing interactions which lead to the formation of particle microdomains at the thermodynamic equilibrium (microphases or microseparation). The possibility to induce structural changes of the morphology of the microdomains is explored, under different confinement conditions and temperatures. In the presence of neutral walls, a switch from stripes of particles to circular clusters (droplets) occurs as the temperature decreases, which does not happen in bulk. While the passage from droplets to stripes, as the density increases, is a well-known phenomenon, the change of the stripes into droplets as an effect of temperature is rather unexpected. Depending on the wall separation and on the wall-fluid interaction parameters, the stripes can switch from parallel to perpendicular to the walls and also a mixed morphology can be stable.

Phase behavior of symmetric ternary block copolymer-homopolymer blends in thin films and on chemically patterned surfaces

The phase diagram of symmetric ternary blends of diblock copolymers and homopolymers in thin films was determined as a function of increasing volume fraction of homopolymer (phi(H)) and was similar to that for these materials in the bulk. Blends with compositions in the lamellar region of the diagram (phi(H)< or =0.4) could be directed to assemble into ordered lamellar arrays on chemically striped surfaces if the characteristic blend dimension (L(B)) and the period of the stripes (L(S)) were commensurate such that L(S)=L(B)+/-0.10L(B). Blends with compositions in the microemulsion region of the diagram (phi(H) approximately 0.6) assembled into defect-free lamellar phases on patterned surfaces with L(S)> or =L(B), but formed coexisting lamellar (with period L(S)) and homopolymer-rich phases when L(S)<L(B).

Influence of Conformational Asymmetry on the Phase Behavior of Ternary Homopolymer/Block Copolymer Blends around the Bicontinuous Microemulsion Channel

We have developed a new ternary polymeric system, poly(ethylene-alt-propylene) (PEP)/poly(butylene oxide) (PBO)/PEP-PBO, to study the complex phase behavior near the bicontinuous microemulsion phase channel. The molecular weights of the PEP and PBO homopolymers are 2600 and 3050 g/mol, respectively, and the copolymer is 23.4 kg/mol with volume fraction composition fPBO=0.49. A combination of small-angle neutron scattering, small-angle X-ray scattering, rheology, optical microscopy, and visual oil bath measurements was employed to map out the phase diagrams at five fixed homopolymer PBO/PEP ratios, ranging from 40/60 to 60/40 by volume, with copolymer concentrations ranging from 0 to 100%. It was found that the bicontinuous microemulsion channel is consistently cut off at low temperature by a hexagonal phase. We attribute this phenomenon to the effect of the conformational asymmetry between the PEP and PBO species, whereby the more flexible PBO component induces a spontaneous curvature toward the PBO domains. These findings complement previous descriptions of the isopleth phase diagrams for A/B/A-B systems and identify a new design variable for preparing bicontinuous phases.

Composition fluctuations in a homopolymer-diblock copolymer mixture covering the three-dimensional Ising, isotropic Lifshitz, and Brasovskiĭ classes of critical universality

The phase behavior of a three-component polymer blend consisting of a critical mixture of polybutadiene and polystyrene (PB/PS) with varying amount of a symmetric PB-PS diblock copolymer was explored with small-angle neutron scattering. Our focus were thermal composition fluctuations which we discuss in terms of mean field, three-dimensional Ising, isotropic Lifshitz, and Brasovskiĭ classes of critical universality. Particular attention is spent to the observation of a narrow reentrant two-phase regime and double critical point in the Lifshitz critical regime as well as the Lifshitz line. Critical exponents of the isotropic Lifshitz case are proposed in spite of the demonstrated nonexistence of the isotropic Lifshitz critical point. The Ginzburg number (Gi) and Flory-Huggins parameter were determined over the whole diblock concentration range; Gi changes by three orders of magnitude, two orders of magnitude of that change over a 0.03 diblock concentration interval within the isotropic Lifshitz regime.