We summarize our knowledge of the phase behavior of polymer solutions and blends using a unified approach. We begin with a derivation of the Flory–Huggins expression for the Gibbs free energy of mixing two chemically dissimilar polymers. The Gibbs free energy of mixing of polymer solutions is obtained as a special case. These expressions are used to interpret observed phase behavior of polymer solutions and blends. Temperature- and pressure-dependent phase diagrams are used to determine the Flory–Huggins interaction parameter, χ. We also discuss an alternative approach for measuring χ due to de Gennes, who showed that neutron scattering from concentration fluctuations in one-phase systems was a sensitive function of χ. In most cases, the agreement between experimental data and the standard Flory–Huggins–de Gennes approach is qualitative. We conclude by summarizing advanced theories that have been proposed to address the limitations of the standard approach. In spite of considerable effort, there is no consensus on the reasons for departure between the standard theories and experiments.

The underpinnings of microphase separation in symmetric poly(styrenesulfonate-block-methylbutylene) (PSS−PMB) copolymer melts were examined by Monte Carlo lattice simulations. The main challenge is understanding the effect of ion pairs in the PSS block on thermodynamics. We assume that experimentally determined Flory−Huggins interaction parameters are adequate for describing intermonomer interactions. Our model does not account for either electrostatic or dipolar interactions. This enables comparisons between simulated and experimentally observed microphases reported by Park and Balsara [Macromolecules 2008, 41, 3678] without resorting to any adjustable parameters. The PSS block in both experiments and theory is partially sulfonated. We quantified the effect of sequence distribution on phase behavior by using alternating and blocky PSS chains in the simulations. Depending on temperature and sequence distribution, simulations show perforated lamellae, gyroid, and hexagonally packed cylinders in addition to the lamellar phase found in simple symmetric block copolymers that do not contain ions. This is driven by extremely repulsive interactions between styrenesulfonate monomers and the uncharged species in the melts. The symmetry of the microphases and the locations of the order−disorder and order−order phase transitions are in qualitative agreement with experimental observations.

We present both lattice and off‐lattice Monte Carlo simulations for multiblock copolymer chains of two lengths, N = 64 and N = 128, with microarchitectures (8–8)4 and (16–16)4, respectively. The simulations demonstrate that a variety of lattice and off‐lattice Monte Carlo methods gives the same protein‐like behavior, showing that the multiblock chains undergo a two‐step transition, first from a swollen state to a secondary “pearl‐necklace” state, and then to a tertiary superglobular state as the solvent quality decreases, that is upon cooling. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

With a lattice Monte Carlo method, we investigate 16-16 symmetric diblock in selective solvent, A-b-B/A, at 10 volume fractions from 1.0 to 0.1, and for each volume fraction, we perform simulations at up to 54 temperatures, using simulation boxes of different sizes. We report temperature dependencies for a number of quantities such as energy, specific heat, and mean-squared end-to-end distances and construct a phase diagram using the thermodynamic and structural quantities as well as snapshots of the selected configurations. The simulated phase diagram is compared with the experimental data of Lodge and co-workers for nearly symmetric poly(styreneb-isoprene) mixed with dimethyl phthalate.

We present the results of Monte Carlo lattice simulations of a model symmetric diblock copolymer wherein a fraction of segments of one block, p, corresponds to ionic species, and the other block does not contain ions. We use experimentally determined Flory−Huggins interaction parameters, χ, to quantify the interactions between ionic and nonionic monomers. Analysis of the experimental data indicate that χ between poly(styrenesulfonate) and polystyrene is about 5, a value that is orders of magnitude larger than that obtained in mixtures of nonionic polymers. Our model predicts that clustering of ionic monomers in the disordered state results in stabilization of the disordered phase and the product p 2 χN is well above the mean-field value of 10.5 at the order−disorder transition (N is the total number of monomers per chain). Network morphologies and hexagonally packed cylinders are observed in the ordered state at large p values while more traditional lamellar phases are found at small values of p.

We performed Monte Carlo simulations using a minimal lattice model with short-range interactions modeled using Flory−Huggins interactions parameters, χ, to investigate morphology of ioncontaining A−B diblock copolymers. A fraction of the segments in the A block, p, were ionic (labeled S) while the B block segments were nonionic (p was held fixed at 0.588). The dielectric constants of the polymers is assumed to be low, and thus charge dissociation effects are negligible. The magnitude of the χ between ion and nonionic species, determined in previous experiments on poly(styrenesulfonate)-bpoly(methyl butylene), PSS−PMB, is an order of magnitude larger than that between the nonionic segments. Simulations indicate that complex morphologies such as gyroid and perforated lamellae are obtained in symmetric block copolymers wherein the volume fraction of the B block, ϕ B , is about 0.5, while simple unperforated lamellae are obtained in asymmetric block copolymers wherein ϕ B is about 0.25. This result is very different from the well-established phase behavior of nonionic block copolymers but consistent with experimental results of Wang et al. [Macromolecules 2010, 43, 5306]. We also make a number of additional predictions, still awaiting an experimental verification, such as the emergence of the hexagonal phase in the weak segregation limit, and a remarkable insensitivity of the product p 2 χN (N is the total number of segments in a copolymer chain) at the order−disorder transition to ϕ B .

Systems kept out of equilibrium in stationary states by an external source of energy store an energy ∆U = U − U 0 . U 0 is the internal energy at equilibrium state, obtained after the shutdown of energy input.We determine ∆U for two model systems: ideal gas and Lennard-Jones fluid. ∆U depends not only on the total energy flux, J U , but also on the mode of energy transfer into the system. We use three different modes of energy transfer where: the energy flux per unit volume is (i) constant; (ii) proportional to the local temperature (iii) proportional to the local density. We show that ∆U/J U = τ is minimized in the stationary states formed in these systems, irrespective of the mode of energy transfer. τ is the characteristic time scale of energy outflow from the system immediately after the shutdown of energy flux. We prove that τ is minimized in stable states of the Rayleigh-Benard cell.Systems out of equilibrium are notoriously difficult to describe in a single coherent methodology based on variational principles. Principles such as Prigogine minimum entropy production [1], Attard second entropy variation [2] or, Ziegler maximum entropy production [3] etc. suggested over the last 100 years, have not reached the same status as the maximum entropy principle known from equilibrium thermodynamics [4][5][6]. A new paradigm, such as the driven lattice gas system, is believed to become an "Ising model" for non-equilibrium statistical physics [7][8][9][10][11][12]. Steady State Thermodynamics (SST) is yet another description framework for non-equilibrium stationary states, which is still being developed [13][14][15][16]. Here we present a different approach to stationary states, based on two quantities: the energy stored in non-equilibriums states, ∆U, and the total energy flux, J U in these states.The second law of thermodynamics states that the entropy of a system has its maximum value at the equilibrium state. Entropy, S , is a function of state, thus for an isolated system of N molecules of total internal energy U enclosed in a volume, V , the entropy has a fixed value S = S (U, V, N).

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