Fluid Phase Equilibria 358 (2013) 156–160 Contents lists available at ScienceDirect Fluid Phase Equilibria j our na l ho me pa ge: www.elsevier.com/locate/fluid Monte Carlo binodals for the order–disorder transition in A-B-A copolymer melts S. Wołoszczuk, M. Banaszak ∗ Faculty of Physics, A. Mickiewicz University, ul. Umultowska 85, 61-614 Poznan, Poland a r t i c l e i n f o Article history: Received 2 July 2013 Received in revised form 6 August 2013 Accepted 8 August 2013 Available online 28 August 2013 Keywords: Triblock copolymer melt Binodals Monte Carlo Parallel tempering a b s t r a c t The ABA triblock copolymer melts are simple and illuminating prototypes for soft self-assembling systems because they exhibit a remarkable richness of nanostructures and useful molecular features, such as B-bridges connecting the neighboring A-nanodomains. The transition from disordered phase to ordered phase is of particular interest and therefore we determine binodals for the order–disorder transition (in terms of the thermodynamic incompatibility and the triblock asymmetry), using lattice Monte Carlo method, known as cooperative motion algorithm, and also employing the parallel temper- ing method which is known to be efficient at lower temperatures. The simulated binodals are presented as a 3-dimensional phase diagram and confronted with earlier mean-field (self-consistent field theory) calculations. While the Monte Carlo binodals show qualitatively similar trends to those observed in the mean-field calculations, there is a pronounced numerical difference between them which indicates the significant role of fluctuations in the vicinity of the order–disorder transition. © 2013 Elsevier B.V. All rights reserved. 1. Introduction Triblock copolymer melts can self-assemble into a plethora of nanostructures (nanophases) [1]. This self-assembly is mostly gov- erned by a competition of entropic stretching energy and enthalpic interfacial energy. A linear triblock ABC copolymer chain consists, in general, of three the distinct blocks, A, B, and C, which are con- nected sequentially. Terminal blocks A and C are often built from the same type of segments, resulting in a triblock copolymer ABA which has only 2 types of segments, A and B, as the diblock. In this paper we focus on such ABA triblocks. While AB-diblock copoly- mer melts are known to form only a few stable nanophases (such as layers, L, hexagonally packed cylinders, C, gyroid nanostruc- tures, G, with the Ia 3d symmetry, cubically packed spherical cells S, and the O 70 -phase [2,3]), the triblock melts form tens of differ- ent phases [1]. In case of a diblock AB copolymer melt, the phase behavior is controlled by the chain composition, f (volume fraction of segments of type A), degree of polymerization (total number of segments), N, and the temperature-related  parameter [4,5]. The ordered nanophase can be dissolved into a disordered phase, for example, upon heating. Phase diagrams of such melts exhibiting order–disorder transition (ODT) lines, also referred to as binodals, and order–order transition (OOT) lines are known from experiment [6] and are successfully predicted by mean-field (MF) theories [7,8], ∗ Corresponding author. Tel.: +48 618295065. E-mail address: banaszak4@gmail.com (M. Banaszak). such as self-consistent field theory (SCFT) which is based on the standard Gaussian chain model [9], or theories including fluctua- tions [10,11]. Because, in the MF theories, it is sufficient to know the composition, f, and the product N (known also as thermodynamic incompatibility parameter) in order to determine the nanophase [7,12,13]. Similarly, the MF phase behavior of the ABA triblock melt is governed by N and the triblock composition, which can be parametrized by 2 convenient numbers, ˛ and ˇ, as follows f 1 = 1 4 - ˇ 2 - ˛, (1) f 2 = 1 2 + ˇ, (2) f 3 = 1 4 - ˇ 2 + ˛. (3) where f i (i = 1, 2 or 3) is the volume fraction of the block of type i (labels 1 and 3 correspond to the terminal A-blocks, and label 2 corresponds to the middle B-block); f 1 + f 2 + f 3 = 1. In most previous studies [14–17] a different parametrization was used, as shown below f 1 = f A , (4) f 2 = 1 - f A , (5) f 3 = (1 - )f A . (6) The first parameter, ˛, provides a measure of asymmetry between terminal A-blocks (f 1 and f 3 ); ˛ = 0 corresponds to a sym- metric triblock with equal terminal blocks. The second parameter, 0378-3812/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fluid.2013.08.010