An unconstrained DFT approach to microphase formation and application to binary Gaussian mixtures

Abstract

Phase behavior and structure of star-polymer-colloid mixtures J. Chem. Phys. 116, 9518 (2002) The formation of microphases in systems of particles interacting by repulsive, bounded potentials is studied by means of density-functional theory (DFT) using a simple, mean-field-like form for the free energy which has already been proven accurate for this class of soft interactions. In an effort not to constrain the configurations available to the system, we do not make any assumption on the functional form of the density profile ρ(r), save for its being periodic. We sample ρ(r) at a large number of points in the unit cell and minimize the free energy with respect to both the values assumed by ρ(r) at these points and the lattice vectors which identify the Bravais lattice. After checking the accuracy of the method by applying it to a one-component generalized exponential model (GEM) fluid with pair potential ϵ exp[−(r/R) 4 ], for which extensive DFT and simulation results are already available, we turn to a binary mixture of Gaussian particles which some time ago was shown to support microphase formation [A. J. Archer, C. N. Likos, and R. Evans, J. Phys.: Condens. Matter 16, L297 (2004)], but has not yet been investigated in detail. The phase diagram which we obtain, that supersedes the tentative one proposed by us in a former study [M. Carta, D. Pini, A. Parola, and L. Reatto, J. Phys.: Condens. Matter 24, 284106 (2012)], displays cluster, tubular, and bicontinuous phases similar to those observed in block copolymers or oil/water/surfactant mixtures. Remarkably, bicontinuous phases occupy a rather large portion of the phase diagram. We also find two non-cubic phases, in both of which one species is preferentially located inside the channels left available by the other, forming helices of alternating chirality. The features of cluster formation in this mixture and in GEM potentials are also compared. C 2015 AIP Publishing LLC. [http://dx

Formation of cluster crystals in an ultra-soft potential model on a spherical surface

“…We remark, however, that the obtained distances are larger than the expected d c = 2πR * . We also observe that, as for the bulk systems 5,32 , the number of particles per cluster grows linearly with the mean density ρ, while the distance between neighbors is nearly independent from the density. These are different aspects connected to the resilience of cluster crystals to compression 1 : in the bulk, the cluster crystal lattice constant does not change upon compression.…”

“…In order to solve the DFT equations we employed a version of the minimization algorithm developed in ref. 5 generalized to the spherical system, which allows us not to make a priori assumptions about the functional form of the density profile. We also performed Monte Carlo simulations to test the theory and assess the full effect of correlations in the fluid, and found quantitative agreement between theory and simulations both in the low-density fluid phase, and in the high-density cluster crystal phases.…”

“…To obtain a local solution of these equations we start from a trial density profile and apply the preconditioned conjugate gradients algorithm with adaptive step size devised in reference 5 .…”

Ordered structures formed by ultrasoft, aspherical particles

“…Minimisation of the functional [ρ], which thus has become a function of these parameters, was then obtained by minimising the function with respect to the parameters. In a recent contribution by Pini et al [16] it was shown that present day computational power in combination with highly efficient and reliable numerical optimisation techniques enable to perform the minimisation of with respect to ρ(r) in an unbiased manner. This could be achieved by representing ρ(r) in a finite volume (e.g.…”

“…Minimisation is then performed by varying the values of ρ(r) on the grid points and -if required -by optimising the shape of the lattice cell. In [16], the authors have shown with this approach that even a spherically symmetric two-body interaction may lead to quite complex structures, which would have likely escaped a biased search. The broad applicability of this method was demonstrated in subsequent work by Roth and coworkers [17][18][19] and again by Pini and Parola [20].…”