Abstract:A method for calculating pair correlation functions in a crystal is developed. The method is based on separating the one-and two-particle correlation functions into the symmetry conserving and the symmetry broken parts. The conserving parts are calculated using the integral equation theory of homogeneous fluids. The symmetry broken part of the direct pair correlation function is calculated from a series written in powers of order parameters and that of the total pair correlation function from the Ornstein-Zern… Show more
“…Finally, to model the hexagonal structures in three dimensions one should use the more complex form of the excessive energy term and also introduce higher modes to the pair-correlation functions of eight-order or 12-order approximations [16]. The general question here: is it enough to have a higher-order expansion pair-correlation function to describe hexagonal symmetries, or it is necessary to introduce three-point correlations even approximated as combinations of pair-correlation functions [47–51]? Figure 1Distributions of n density field (bulk colour gradient) and ψ concentration field (greyscale mesh) obtained using solutions of dynamical equations (2.20) and (2.21).…”
The phase-field crystal (PFC-model) is a powerful tool for modelling of the crystallization in colloidal and metallic systems. In the present work, the modified hyperbolic phase-field crystal model for binary systems is presented. This model takes into account slow and fast dynamics of moving interfaces for both concentration and relative atomic number density (which were taken as order parameters). The model also includes specific mobilities for each dynamical field and correlated noise terms. The dynamics of chemical segregation with origination of mixed pseudo-hexagonal binary phase (the so-called ‘triangle phase’) is used as a benchmark in two spatial dimensions for the developing model. Using the free energy functional and specific lattice vectors for hexagonal crystal, the structure diagram of co-existence of liquid and three-dimensional hexagonal phase for the binary PFC-model was carried out. Parameters of the crystal lattice correspond to the hexagonal boron nitride (BN) crystal, the values of which have been taken from the literature. The paper shows the qualitative agreement between the developed structure diagram of the PFC model and the previously known equilibrium diagram for BN constructed using thermodynamic functions.
This article is part of the theme issue ‘Transport phenomena in complex systems (part 2)’.
“…Finally, to model the hexagonal structures in three dimensions one should use the more complex form of the excessive energy term and also introduce higher modes to the pair-correlation functions of eight-order or 12-order approximations [16]. The general question here: is it enough to have a higher-order expansion pair-correlation function to describe hexagonal symmetries, or it is necessary to introduce three-point correlations even approximated as combinations of pair-correlation functions [47–51]? Figure 1Distributions of n density field (bulk colour gradient) and ψ concentration field (greyscale mesh) obtained using solutions of dynamical equations (2.20) and (2.21).…”
The phase-field crystal (PFC-model) is a powerful tool for modelling of the crystallization in colloidal and metallic systems. In the present work, the modified hyperbolic phase-field crystal model for binary systems is presented. This model takes into account slow and fast dynamics of moving interfaces for both concentration and relative atomic number density (which were taken as order parameters). The model also includes specific mobilities for each dynamical field and correlated noise terms. The dynamics of chemical segregation with origination of mixed pseudo-hexagonal binary phase (the so-called ‘triangle phase’) is used as a benchmark in two spatial dimensions for the developing model. Using the free energy functional and specific lattice vectors for hexagonal crystal, the structure diagram of co-existence of liquid and three-dimensional hexagonal phase for the binary PFC-model was carried out. Parameters of the crystal lattice correspond to the hexagonal boron nitride (BN) crystal, the values of which have been taken from the literature. The paper shows the qualitative agreement between the developed structure diagram of the PFC model and the previously known equilibrium diagram for BN constructed using thermodynamic functions.
This article is part of the theme issue ‘Transport phenomena in complex systems (part 2)’.
“…(2), using the weighted densities n 3 , n 2 , n 1 , and n 0 augmented by suitable variablesn 2 and n 1 that are specific to inhomogeneous systems, Eq. (11). When applied to the bulk fluid, the functional F reduces to Santos' free energy [25].…”
Section: Discussionmentioning
confidence: 99%
“…These numerical difficulties are part of the reason why the crystal phase (being an extreme realization of an inhomogeneous liquid) is largely unexplored via integral equations; see, however, Ref. [11].…”
Based on Santos' general solution for the scaled-particle differential equation [Phys. Rev. E 86, 040102(R) (2012)], we construct a free-energy functional for the hard-sphere system. The functional is obtained by a suitable generalization and extension of the set of scaled-particle variables using the weighted densities from Rosenfeld's fundamental measure theory for the hard-sphere mixture [Phys. Rev. Lett. 63, 980 (1989)]. While our general result applies to the hard-sphere mixture, we specify remaining degrees of freedom by requiring the functional to comply with known properties of the pure hard-sphere system. Both for mixtures and pure systems, the functional can be systematically extended following the lines of our derivation. We test the resulting functionals regarding their behavior upon dimensional reduction of the fluid as well as their ability to accurately describe the hard-sphere crystal and the liquid-solid transition.
“…For simplicity we assume isotropy in all functions, restricting thereby the applicability of the presented methods to systems in which both the particle pair potential u(r) and the particle pair correlation functions g(r), S(q) are isotropic. Yet, this isotropy assumption is merely a technical simplification that could be lifted in future generalizations of the present work by the use of technically more involved anisotropic Ornstein-Zernike formalisms [18][19][20][21][22][23][24] . These more advanced numerical techniques are expected to extend the applicability range of the present method to systems with anisotropic particle pair interactions 18 as well as systems with anisotropic particle correlations resulting from external fields or confinement [18][19][20][21][22][23] and crystalline systems 24 .…”
Section: Methodsmentioning
confidence: 99%
“…Yet, this isotropy assumption is merely a technical simplification that could be lifted in future generalizations of the present work by the use of technically more involved anisotropic Ornstein-Zernike formalisms [18][19][20][21][22][23][24] . These more advanced numerical techniques are expected to extend the applicability range of the present method to systems with anisotropic particle pair interactions 18 as well as systems with anisotropic particle correlations resulting from external fields or confinement [18][19][20][21][22][23] and crystalline systems 24 . Using the O-Z formalism implies that the presented method is strictly applicable only to systems in thermodynamic equilibrium 25 .…”
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