General framework for fluctuating dynamic density functional theory

Durán-Olivencia, Miguel A.; Yatsyshin, Peter; Goddard, Benjamin D.; Kalliadasis, Serafim

doi:10.1088/1367-2630/aa9041

Cited by 24 publications

(56 citation statements)

References 71 publications

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“…Text on arrows give brief descriptions of the approximations/manipulations made. Nomenclature:  projection operator, f neq nonequilibrium distribution, γ/m?1 overdamped limit, and  ò á´ñ t d time integration and average over multiple realizations [44]. where Einstein's summation convention was used for simplicity.…”

Section: Discussionmentioning

confidence: 99%

“…Within this framework, commonly known as fluctuating hydrodynamics (FH), the system is described by its timedependent hydrodynamic fields. In the prototypical case of an isothermal system with strong dissipation due to the thermal bath, the only relevant quantity is the density field [44]. In principle, the equations of FH provide us with a full and self-contained description of the system's dynamics, including rare events, such as nucleation.…”

Section: Introductionmentioning

confidence: 99%

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General framework for nonclassical nucleation

et al. 2018

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A great deal of experimental evidence suggests that a wide spectrum of phase transitions occur in a multistage manner via the appearance and subsequent transformation of intermediate metastable states. Such multistage mechanisms cannot be explained within the realm of the classical nucleation framework. Hence, there is a strong need to develop new theoretical tools to explain the occurrence and nature of these ubiquitous intermediate phases. Here we outline a unified and self-consistent theoretical framework to describe both classical and nonclassical nucleation. Our framework provides a detailed explanation of the whole multistage nucleation pathway showing in particular that the pathway involves a single energy barrier and it passes through a dense phase, starting from a lowdensity initial phase, before reaching the final stable state. Moreover, we demonstrate that the kinetics of matter inside subcritical clusters favors the formation of nucleation clusters with an intermediate density, i.e. nucleation precursors. Remarkably, these nucleation precursors are not associated with a local minimum of the thermodynamic potential, as commonly assumed in previous phenomenological approaches. On the contrary, we find that they emerge due to the competition between thermodynamics and kinetics of cluster formation. Thus, the mechanism uncovered for the formation of intermediate phases can be used to explain recently reported experimental findings in crystallization: up to now such phases were assumed a consequence of some complex energy landscape with multiple energy minima. Using fundamental concepts from kinetics and thermodynamics, we provide a satisfactory explanation for the so-called nonclassical nucleation pathways observed in experiments.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

General framework for nonclassical nucleation

et al. 2018

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“…DDFT is based on equilibrium density functional theory (DFT) [25][26][27] and for an equilibrium system, DDFT is equivalent to DFT. DDFT was originally developed as a theory for Brownian particles with over-damped stochastic equations of motion [28][29][30][31], but it has also been extended to describe the dynamics of under-damped systems and atomic or molecular systems where the particle dynamics is governed by Newton's equations of motion [32][33][34][35][36][37]. This body of work shows that when such systems are not too far from equilibrium, then the dynamics predicted by the original DDFT is still often correct in the long-time limit where the particle dynamics is dominated by diffusive processes.…”

Section: Introductionmentioning

confidence: 99%

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

et al. 2019

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Phase field crystal (PFC) theory is extensively used for modelling the phase behaviour, structure, thermodynamics and other related properties of solids. PFC theory can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Here, we carefully identify all of these approximations and explain the consequences of each. One approximation that is made in standard derivations is to neglect a term of form ∇ · [n∇Ln], where n is the scaled density profile and L is a linear operator. We show that this term makes a significant contribution to the stability of the crystal, and that dropping this term from the theory forces another approximation, that of replacing the logarithmic term from the ideal gas contribution to the free energy with its truncated Taylor expansion, to yield a polynomial in n. However, the consequences of doing this are: (i) the presence of an additional spinodal in the phase diagram, so the liquid is predicted first to freeze and then to melt again as the density is increased; and (ii) other periodic structures, such as stripes, are erroneously predicted to be thermodynamic equilibrium structures. In general, L consists of a nonlocal convolution involving the pair direct correlation function. A second approximation sometimes made in deriving PFC theory is to replace L by a gradient expansion involving derivatives. We show that this leads to the possibility of the density going to zero, with its logarithm going to −∞ whilst being balanced by the fourth derivative of the density going to +∞. This subtle singularity leads to solutions failing to exist above a certain value of the average density. We illustrate all of these conclusions with results for a particularly simple model two-dimensional fluid, the generalised exponential model of index 4 (GEM-4), chosen because a DDFT is known to be accurate for this model. The consequences of the subsequent PFC approximations can then be examined. These include the phase diagram being both qualitatively incorrect, in that it has a stripe phase, and quantitatively incorrect (by orders of magnitude) regarding the properties of the crystal phase. Thus, although PFC models are very successful as phenomenological models of crystallisation, we find it impossible to derive the PFC as a theory for the (scaled) density distribution when starting from an accurate DDFT, without introducing spurious artefacts. However, we find that making a simple one-mode approximation for the logarithm of the density distribution log ρ(x) (rather than for ρ(x)), is surprisingly accurate. This approach gives a tantalising hint that accurate PFC-type theories may instead be derived as theories for the field log ρ(x), rather than for the density profile itself.

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“…(8) and 20) and liquid film theory (Grün et al, 2006, Eq. (19)); see (Durán-Olivencia et al, 2017) for connections to dynamic density functional theory.…”

Section: Outline Of the Approachmentioning

confidence: 99%

Harnessing fluctuations to discover dissipative evolution equations

Dirr

Embacher

et al. 2019

Journal of the Mechanics and Physics of Solids

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Continuum modeling of dissipative processes in materials often relies on strong phenomenological assumptions, as their derivation from underlying atomistic/particle models remains a major long-standing challenge. Here we show that the continuum evolution equations of a wide class of dissipative phenomena can be numerically obtained (in a discretized form) from fluctuations via an infinite-dimensional fluctuation-dissipation relation. A salient feature of the method is that these continuum equations can be fully pre-computed, enabling macroscopic simulations of arbitrary admissible initial conditions, without the need of any further microscopic simulations. We test this coarse-graining procedure on a one-dimensional non-linear diffusive process with known analytical solution, and obtain an excellent agreement for the density evolution. This illustrative example serves as a blueprint for a new multiscale paradigm, where full dissipative evolution equations-and not only parameters-can be numerically computed from lower scale data.

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General framework for fluctuating dynamic density functional theory

Cited by 24 publications

References 71 publications

General framework for nonclassical nucleation

General framework for nonclassical nucleation

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

Harnessing fluctuations to discover dissipative evolution equations

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