Mean-field Free Energy Research Articles

Theoretical analysis of the elastic free energy contributions to polymer brushes in poor solvent: A refined mean-field theory.

We consider polymer brushes in poor solvent that are grafted onto planar substrates and onto the internal and external surfaces of a cylinder using molecular dynamics simulation, self-consistent field (SCF), and mean-field theory. We derive a unified expression for the mean field free energy for the three geometrical classes. While for low grafting densities, the effect of chain elasticity can be neglected in poor solvent conditions, it becomes relevant at higher grafting densities and, in particular, for concave geometries. Based on the analysis of the end monomer distribution, we introduce an analytical term that describes the elasticity as a function of grafting density. The accuracy of the model is validated with molecular dynamics simulations as well as SCF computations and shown to yield precise values for the layer thickness over a wide range of system parameters. We further apply this model to analyze the gating behavior of switchable brushes inside nanochannels.

Emergence of a finite-size-scaling function in the supervised learning of the Ising phase transition

We investigate the connection between the supervised learning of the binary phase classification in the ferromagnetic Ising model and the standard finite-size-scaling theory of the second-order phase transition. Proposing a minimal one-free-parameter neural network model, we analytically formulate the supervised learning problem for the canonical ensemble being used as a training data set. We show that just one free parameter is capable enough to describe the data-driven emergence of the universal finite-size-scaling function in the network output that is observed in a large neural network, theoretically validating its critical point prediction for unseen test data from different underlying lattices yet in the same universality class of the Ising criticality. We also numerically demonstrate the interpretation with the proposed one-parameter model by providing an example of finding a critical point with the learning of the Landau mean-field free energy being applied to the real data set from the uncorrelated random scale-free graph with a large degree exponent.

TAP free energy, spin glasses and variational inference

We consider the Sherrington–Kirkpatrick model of spin glasses with ferromagnetically biased couplings. For a specific choice of the couplings mean, the resulting Gibbs measure is equivalent to the Bayesian posterior for a high-dimensional estimation problem known as “${\mathbb{Z}}_{2}$ synchronization.” Statistical physics suggests to compute the expectation with respect to this Gibbs measure (the posterior mean in the synchronization problem), by minimizing the so-called Thouless–Anderson–Palmer (TAP) free energy, instead of the mean field (MF) free energy. We prove that this identification is correct, provided the ferromagnetic bias is larger than a constant (i.e., the noise level is small enough in synchronization). Namely, we prove that the scaled $\ell _{2}$ distance between any low energy local minimizers of the TAP free energy and the mean of the Gibbs measure vanishes in the large size limit. Our proof technique is based on upper bounding the expected number of critical points of the TAP free energy using the Kac–Rice formula.

On the Singularity of the Liquid-Gas Coexistence Curve Diameter

A simplified Anisimov–Wang variant of the complete scaling approach makes it possible to determine the amplitudes of singularities for the diameter of the phase coexistence curve (CXC) on the basis of the coefficients in the power series expansion of the mean-field free energy in the reduced temperature and pressure near the critical point. This method is applied to obtain the amplitudes for the leading critical singularities of the CXC diameter in the case of a fluid described in the framework of the mesoscopic mean-field model. The results obtained demonstrate that the amplitudes of leading singularities of the CXC diameter are determined by the mesoscopic asymmetry parameters of the heterophase fluid.

D -wave superconductivity and Bogoliubov-Fermi surfaces in Rarita-Schwinger-Weyl semimetals

We uncover the properties of complex tensor (d-wave) superconducting order in three-dimensional Rarita-Schwinger-Weyl semimetals that host pseudospin-3/2 fermions at a fourfold linear band crossing point. Although the general theory of d-wave order was originally developed for materials displaying quadratic band touching, it directly applies to the case of semimetals with linear dispersion, several candidate compounds of which have been discovered experimentally very recently. The spin-3/2 nature of the fermions allows for the formation of spin-2 Cooper pairs which may be described by a complex second-rank tensor order parameter. In the case of linear dispersion, for the chemical potential at the Fermi point and at strong coupling, the energetically preferred superconducting state is the uniaxial nematic state, which preserves time-reversal symmetry and provides a full (anisotropic) gap for quasiparticle excitations. In contrast, at a finite chemical potential, we find that the usual weak-coupling instability is towards the "cyclic state", well known from the studies of multicomponent Bose-Einstein condensates, which breaks time reversal symmetry maximally, has vanishing average value of angular momentum, and features 16 small Bogoliubov--Fermi surfaces. The Rarita-Schwinger-Weyl semimetals provide therefore the first example of weakly coupled, three-dimensional, isotropic d-wave superconductors where the ground state is uniquely determined by the quartic expansion of the mean-field free energy, and not afflicted by the accidental degeneracy first noticed by Mermin over 40 years ago. We discuss the appearance and stability of the Bogoliubov-Fermi surfaces in absence of inversion symmetry in the electronic Hamiltonian, as in the case at hand.

Electron-hole superfluidity controlled by a periodic potential

We propose to control of an electron-hole superfluid in semiconductor coupled quantum wells and double layers of two-dimensional (2D) material by an external periodic field. This can either be created by the gates periodically located and attached to the quantum wells or double layers of 2D material or by the Moir\'e pattern of two twisted layers. The dependence of the electron-hole pairing order parameter on the temperature, the charge carrier density, and the gate parameters is obtained by minimization of the mean-field free energy. The second order phase transition between superfluid and electron-hole plasma, controlled by the external periodic gate field, is analyzed for different parameters.

Electrostatic depletion effects on the stability of colloidal dispersions of sepiolite and natural rubber latex

Biocomposites based on sepiolite (Sep) clays and natural rubber latex (NRL) are novel green synthetic materials with significant mechanical performance obtained by an eco-friendly and sustainable mixing procedure, without the use of surfactants. In this work, experiments and theory are combined to investigate the stability of colloidal dispersions formed by a mixture of both negatively charged Sep fibers and non-adsorbing NRL particles. Experiments were performed by adding Sep fibers to NRL dispersions with different Sep/NRL volume fractions to evaluate the effect of Sep dispersion and NRL loading on the flocculation process. In order to theoretically understand the experimental results on colloidal stability, a density functional approach was applied to calculate the depletion interaction between two Sep fibers induced by the presence of naturally charged NRL, and an effective one-component mean-field free energy was developed to predict the phase behavior of the Sep/NRL mixture. The existence of a depletion attraction, enhanced by the electrostatic repulsion between Sep and NRL, is shown to be strong enough to induce the flocculation of the mixture at determined Sep and NRL volume fractions. The theoretical predicted phase diagram is in excellent qualitative and quantitative agreement with the experimental results, indicating that this electrostatically-enhanced depletion effect plays a key role in the colloidal stability of this system. To the best of our knowledge, this study represents the first attempt to tackle how depletion effects can be exploited to produce and control Sep/NR biocomposites.

Phase transition in time-reversible Navier-Stokes equations.

We present a comprehensive study of the statistical features of a three-dimensional (3D) time-reversible truncated Navier-Stokes (RNS) system, wherein the standard viscosity ν is replaced by a fluctuating thermostat that dynamically compensates for fluctuations in the total energy. We analyze the statistical features of the RNS steady states in terms of a non-negative dimensionless control parameter R_{r}, which quantifies the balance between the fluctuations of kinetic energy at the forcing length scale ℓ_{f} and the total energy E_{0}. For small R_{r}, the RNS equations are found to produce "warm" stationary statistics, e.g., characterized by the partial thermalization of the small scales. For large R_{r}, the stationary solutions have features akin to standard hydrodynamic ones: they have compact energy support in k space and are essentially insensitive to the truncation scale k_{max}. The transition between the two statistical regimes is observed to be smooth but rather sharp. Using insights from a diffusion model of turbulence (Leith model), we argue that the transition is in fact akin to a continuous second-order phase transition, where R_{r} indeed behaves as a thermodynamic control parameter, e.g., a temperature. A relevant order parameter can be suitably defined in terms of a (normalized) enstrophy, while the symmetry-breaking parameter h is identified as (one over) the truncation scale k_{max}. We find that the signatures of the phase transition close to the critical point R_{r}^{★} can essentially be deduced from a heuristic mean-field Landau free energy. This point of view allows us to reinterpret the relevant asymptotics in which the dynamical ensemble equivalence conjectured by Gallavotti [Phys. Lett. A 223, 91 (1996)PYLAAG0375-960110.1016/S0375-9601(96)00729-3] could hold true. We argue that Gallavotti's limit is precisely the joint limit R_{r}→[over >]R_{r}^{★} and h→[over >]0, with the overset symbol ">" indicating that those limits are approached from above. The limit therefore relates to the statistical features at the critical point. In this regime, our numerics indicate that the low-order statistics of the 3D RNS are indeed qualitatively similar to those observed in direct numerical simulations of the standard Navier-Stokes equations with viscosity chosen so as to match the average value of the reversible thermostat. This result suggests that Gallavotti's equivalence conjecture could indeed be of relevance to model 3D turbulent statistics, and provides a clear guideline for further numerical investigations involving higher resolutions.

Numerical algorithms for solving self-consistent field theory reversely for block copolymer systems.

Besides dictating the equilibrium phase diagram, the rugged free-energy landscape of AB block copolymers gives rise to a multitude of non-equilibrium phenomena. Self-consistent field theory (SCFT) can be employed to calculate the mean-field free energy, , of a non-equilibrium unstable state that is characterized by a given spatial density distribution, , in the incompressible system. Such a free-energy functional is the basis of describing the structure formation by dynamic SCFT techniques or the identification of minimum free-energy paths via the string method. The crucial step consists in computing the external potential fields that generate the given density distribution in the corresponding system of non-interacting copolymers, i.e., the potential-to-density relation employed in equilibrium SCFT calculations has to be inverted (reverse SCFT calculation). We describe, generalize, and evaluate the computational efficiency of two different numerical algorithms for this reverse SCFT calculation-the Debye-function algorithm based on the structure factor and the field-theoretic umbrella-potential (FUP) algorithm. In contrast to the Debye-function algorithm, the FUP algorithm only yields the exact mean-field values of the given target densities in the limit of a strong umbrella potential, and we devise a two-step variant of the FUP algorithm that significantly mitigates this issue. For Gaussian copolymers, the Debye-function algorithm is more efficient for highly unstable states that are far away from the equilibrium, whereas the improved FUP algorithm outperforms the Debye-function algorithm closer to metastable states and is easily transferred to more complex molecular architectures.

Tension-Induced Nematic Phase Separation in Bidisperse Homopolymer Melts.

We use an analytical mean-field theory and all-atom molecular dynamics (MD) simulations to predict that external tension, together with the nematic coupling interactions, can drive phase separation of long chains from short ones in bidisperse homopolymer melts. The nematic coupling parameter α for polyethylene (PE) oligomers under applied tension is extracted from the MD simulations and used in the mean-field free energy to predict the phase boundary for bidisperse melts in which the longer chains are stretched by uniaxial tension. The predicted phase diagram is validated by direct MD simulations. We also show that extensional flow, and possibly even shear flow, may lead to nematic phase separation in molten PE oligomers, because the flow can impose a stronger tension on the longer chains than the short ones.

Convincing evidence for the Halperin-Lubensky-Ma effect at the N-SmA transition in alkyloxycyanobiphenyl binary mixtures via a high-resolution birefringence study.

We present new high-resolution experimental data for the temperature behavior of optical birefringence for a series of mixture of the liquid crystals octyloxycyanobiphenyl (8OCB) and nonyloxycyanobiphenyl (9OCB) by using a rotating analyzer technique. The birefringence data have been used to probe the temperature dependence of the nematic order parameter [Formula: see text]. We have then arrived at values for possible entropy discontinuities at the nematic-smectic A transition temperature [Formula: see text] from the detailed inspection of [Formula: see text] data in the immediate vicinity of [Formula: see text]. The 9OCB mole fraction dependence of the obtained reduced entropy discontinuities has been shown to be well fitted with a crossover function which is itself consistent with the mean-field free energy expression with a non-zero cubic term arising from the Halperin-Lubensky-Ma (HLM) coupling. The obtained results are in good accordance with existing results from adiabatic scanning calorimetry (ASC). Our birefringence results and determined entropy discontinuities (consistent with calorimetry results) are in striking contrast with the recent birefringence results of Barman et al. (Phase Transit. 91, 58 (2018) published online 16 Aug. 2017) claiming second-order nematic-to-smectic A transitions for all mixtures. In this paper we present a possible explanation for this discrepancy. We have also extracted the effective critical exponent values [Formula: see text] characterizing the critical fluctuations near the N-SmA transition for all compositions by using the fact that the temperature derivative of the order parameter [Formula: see text] near [Formula: see text] exhibits the same power-law divergence as the specific heat capacity. Measurable latent heat values were extracted from optical birefringence data for mole fractions of 9OCB where the [Formula: see text] values are as low as 0.2, which is substantially lower than the tricritical value [Formula: see text]. This is qualitatively different from what has been observed so far in other liquid-crystal systems. Together with ASC data, these pecuilarities of the 8OCB+9OCB system render further convincing evidence for the presence of the HLM coupling effect at the N-SmA transition phase transition line.

Oseen–Frank-type theories of ordered media as the Γ-limit of a non-local mean-field free energy

In this work we recover the Oseen–Frank theory of nematic liquid crystals as a [Formula: see text]-limit of a particular mean-field free energy as the sample size becomes infinitely large. The Frank constants are not necessarily all equal. Our analysis takes place in a broader framework however, also providing results for more general systems such as biaxial or polar molecules. We start from a mesoscopic model describing a competition between entropy and a non-local pairwise interaction between molecules. We assume the interaction potential is separable so that the energy can be reduced to a model involving a macroscopic order parameter. We assume only integrability and decay properties of the macroscopic interaction, but no regularity assumptions are required. In particular, singular interactions are permitted. The analysis becomes significantly simpler on a translationally invariant domain, so we first consider periodic domains with increasing domain of periodicity. Then we tackle the problem on a Lipschitz domain with non-local boundary conditions by considering an asymptotically equivalent problem on periodic domains. We conclude by applying the results to a case which reduces to the Oseen–Frank model of elasticity, and give expressions for the Frank constants in terms of integrals of the interaction kernel.

Effective-range corrections to the ground-state energy of the weakly-interacting Bose gas in two dimensions

Nonuniversal effects due to leading effective-range corrections are computed for the ground-state energy of the weakly-coupled repulsive Bose gas in two spatial dimensions. Using an effective field theory of contact interactions, these corrections are computed first by considering fluctuations around the mean-field free energy of a system of interacting bosons. This result is then confirmed by an exact calculation in which the energy of a finite number of bosons interacting in a square with period boundary conditions is computed and the thermodynamic limit is explicitly taken.

Analysis of the Mean Field Free Energy Functional of Electrolyte Solution with Nonhomogenous Boundary Conditions and the Generalized PB/PNP Equations with Inhomogeneous Dielectric Permittivity

The energy functional, the governing partial differential equation(s) (PDE), and the boundary conditions need to be consistent with each other in a modeling system. In electrolyte solution study, people usually use a free energy form of an infinite domain system (with vanishing potential boundary condition) and the derived PDE(s) for analysis and computing. However, in many real systems and/or numerical computing, the objective domain is bounded, and people still use the similar energy form, PDE(s), but with different boundary conditions, which may cause inconsistency. In this work, (1) we present a mean field free energy functional for the electrolyte solution within a bounded domain with either physical or numerically required artificial boundary. Apart from the conventional energy components (electrostatic potential energy, ideal gas entropy term, and chemical potential term), new boundary interaction terms are added for both Neumann and Dirichlet boundary conditions. These new terms count for physical...

Stability of Soft Quasicrystals in a Coupled-Mode Swift-Hohenberg Model for Three-Component Systems

AbstractIn this article, we discuss the stability of soft quasicrystalline phases in a coupled-mode Swift-Hohenberg model for three-component systems, where the characteristic length scales are governed by the positive-definite gradient terms. Classic two-mode approximation method and direct numerical minimization are applied to the model. In the latter approach, we apply the projection method to deal with the potentially quasiperiodic ground states. A variable cell method of optimizing the shape and size of higher-dimensional periodic cell is developed to minimize the free energy with respect to the order parameters. Based on the developed numerical methods, we rediscover decagonal and dodecagonal quasicrystalline phases, and find diverse periodic phases and complex modulated phases. Furthermore, phase diagrams are obtained in various phase spaces by comparing the free energies of different candidate structures. It does show not only the important roles of system parameters, but also the effect of optimizing computational domain. In particular, the optimization of computational cell allows us to capture the ground states and phase behavior with higher fidelity. We also make some discussions on our results and show the potential of applying our numerical methods to a larger class of mean-field free energy functionals.

Registered and Antiregistered Phase Separation of Mixed Amphiphilic Bilayers

We derive a mean-field free energy for the phase behavior of coupled bilayer leaflets, which is implicated in cellular processes and important to the design of artificial membranes. Our model accounts for amphiphile-level structural features, particularly hydrophobic mismatch, which promotes antiregistration, in competition with the direct transmidplane coupling usually studied, which promotes registration. We show that the phase diagram of coupled leaflets allows multiple metastable coexistences, and we illustrate the kinetic implications of this with a detailed study of a bilayer of equimolar overall composition. For approximate parameters estimated to apply to phospholipids, equilibrium coexistence is typically registered, but metastable antiregistered phases can be kinetically favored by hydrophobic mismatch. Thus, a bilayer in the spinodal region can require nucleation to equilibrate, in a novel manifestation of Ostwald’s rule of stages. Our results provide a framework for understanding disparate existing observations in the literature, elucidating a subtle competition of couplings and a key role for phase-transition kinetics in bilayer phase behavior.

Theory of Registered and Antiregistered Phase Separation in Mixed Amphiphilic Bilayers

We derive a mean-field free energy for the phase behaviour of coupled bilayer leaflets, which is implicated in cellular processes and important to the design of artificial membranes. Our model accounts for amphiphile-level structural features, particularly hydrophobic mismatch, which promotes antiregistration (AR), in competition with the `direct' trans-midplane coupling usually studied, promoting registration (R). We show that the phase diagram of coupled leaflets allows multiple \textit{metastable} coexistences, then illustrate the kinetic implications with a detailed study of a bilayer of equimolar overall composition. For approximate parameters estimated to apply to phospholipids, equilibrium coexistence is typically registered, but metastable antiregistered phases can be kinetically favoured by hydrophobic mismatch. Thus a bilayer in the spinodal region can require nucleation to equilibrate, in a novel manifestation of Ostwald's `rule of stages'. Our results provide a framework for understanding disparate existing observations, elucidating a subtle competition of couplings, and a key role for phase transition kinetics in bilayer phase behaviour.

Accurate phase diagram of tetravalent DNA nanostars

We evaluate, by means of molecular dynamics simulations employing a realistic DNA coarse-grained model, the phase behaviour and the structural and dynamic properties of tetravalent DNA nanostars, i.e., nanoconstructs completely made of DNA. We find that, as the system is cooled down, tetramers undergo a gas–liquid phase separation in a region of concentrations which, if the difference in salt concentration is taken into account, is comparable with the recently measured experimental phase diagram [S. Biffi, R. Cerbino, F. Bomboi, E. M. Paraboschi, R. Asselta, F. Sciortino, and T. Bellini, Proc. Natl. Acad. Sci. U.S.A. 110, 15633 (2013)]. We also present a mean-field free energy for modelling the phase diagram based on the bonding contribution derived by Wertheim in his studies of associating liquids. Combined with mass-action law expressions appropriate for DNA binding and a numerically evaluated reference free energy, the resulting free energy qualitatively reproduces the numerical data. Finally, we report information on the nanostar structure, e.g., geometry and flexibility of the single tetramer and of the collective behaviour, providing a useful reference for future small angle scattering experiments, for all investigated temperatures and concentrations.

Prediction of A2 to B2 Phase Transition in the High-Entropy Alloy Mo-Nb-Ta-W

In this article, we show that an effective Hamiltonian fit with first-principles calculations predicts that an order/disorder transition occurs in the high-entropy alloy Mo-Nb-Ta-W. Using the Alloy Theoretic Automated Toolkit, we find T = 0 K enthalpies of formation for all binaries containing Mo, Nb, Ta, and W, and in particular, we find the stable structures for binaries at equiatomic concentrations are close in energy to the associated B2 structure, suggesting that at intermediate temperatures, a B2 phase is stabilized in Mo-Nb-Ta-W. Our previously published hybrid Monte Carlo (MC)/molecular dynamics (MD) results for the Mo-Nb-Ta-W system are analyzed to identify certain preferred chemical bonding types. A mean field free energy model incorporating nearest-neighbor bonds is derived, allowing us to predict the mechanism of the order/disorder transition. We find the temperature evolution of the system is driven by strong Mo-Ta bonding. A comparison of the free energy model and our MC/MD results suggests the existence of additional low-temperature phase transitions in the system likely ending with phase segregation into binary phases.

An analytic mean-field model for the magnetic response of a ferrofluid monolayer

We extend the analysis of the mean-field model introduced in a previous paper [Piastra and Virga, Soft Matter, 2012, 8, 10969] for a monolayer of spherical colloidal particles with shifted point dipoles studied in previous work [Kantorovich et al., Soft Matter, 2011, 7, 5217]. This system was predicted to exhibit a spontaneous magnetization in the absence of any magnetic field, provided that the temperature is sufficiently low or the particles' density is sufficiently high. A ferromagnetic behavior is confirmed here by a solution in closed form for the mean-field free energy, reminiscent of (but not identical to) the classical solutions of Langevin and Weiss; the closed-form solution also applies in the presence of an external magnetic field, however oriented relative to the layer's normal. Such an analytic solution has perhaps its most telling application to the case where the external magnetic field is applied normal to the layer, where a mixed ferromagnetic and paramagnetic response is predicted. In that case, if the field's strength is not too high, then for high enough temperatures (or low enough densities) the magnetic response is paramagnetic (and with the magnetization aligned to the field), whereas for low enough temperatures (or high enough densities) an extra magnetization grows in the plane of the layer (the vertical component remaining frozen), a remnant of the ferromagnetic transition in the absence of a field. If instead the field's strength exceeds a critical value, any ferromagnetic behavior is suppressed, and the layer becomes paramagnetic at all temperatures and densities.