scholarly journals The 2-D Cluster Variation Method: Topography Illustrations and Their Enthalpy Parameter Correlations

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 319
Author(s):  
Alianna J. Maren
Keyword(s):  
Free Energy ◽  
Nearest Neighbor ◽  
Activation Enthalpy ◽  
Cluster Variation Method ◽  
Variation Method ◽  
Energy Minimum ◽  
Dimensional Parameter ◽  
Free Energy Minimum ◽  
Interaction Enthalpy ◽  
Cluster Variation

One of the biggest challenges in characterizing 2-D image topographies is finding a low-dimensional parameter set that can succinctly describe, not so much image patterns themselves, but the nature of these patterns. The 2-D cluster variation method (CVM), introduced by Kikuchi in 1951, can characterize very local image pattern distributions using configuration variables, identifying nearest-neighbor, next-nearest-neighbor, and triplet configurations. Using the 2-D CVM, we can characterize 2-D topographies using just two parameters; the activation enthalpy (ε0) and the interaction enthalpy (ε1). Two different initial topographies (“scale-free-like” and “extreme rich club-like”) were each computationally brought to a CVM free energy minimum, for the case where the activation enthalpy was zero and different values were used for the interaction enthalpy. The results are: (1) the computational configuration variable results differ significantly from the analytically-predicted values well before ε1 approaches the known divergence as ε1→0.881, (2) the range of potentially useful parameter values, favoring clustering of like-with-like units, is limited to the region where ε0<3 and ε1<0.25, and (3) the topographies in the systems that are brought to a free energy minimum show interesting visual features, such as extended “spider legs” connecting previously unconnected “islands,” and as well as evolution of “peninsulas” in what were previously solid masses.

scholarly journals The 2-D Cluster Variation Method: Initial Findings and Topography Illustrations

2021 ◽  
Author(s):  
Alianna J. Maren
Keyword(s):  
Free Energy ◽  
Energy Minimization ◽  
Nearest Neighbor ◽  
Temporal Correlation ◽  
Cluster Variation Method ◽  
Variation Method ◽  
Dimensional Parameter ◽  
Free Energy Minimization ◽  
Cluster Variation ◽  
Two Parameters

One of the biggest challenges in characterizing 2-D image topographies is finding a low-dimensional parameter set that can succinctly describe, not so much image patterns themselves, but the nature of these patterns. The 2-D Cluster Variation Method (CVM), introduced by Kikuchi in 1951, can characterize very local image pattern distributions using configuration variables, identifying nearest-neighbor, next-nearest-neighbor, and triplet configurations. Using the 2-D CVM, we can characterize 2-D topographies using just two parameters; the activation enthalpy and the interaction enthalpy. Initial investigations with two different representative topographies (``scale-free-like'' and ``rich club-like'') produce interesting results when brought to a CVM free energy minimum. Additional phase space investigations, where one of these two parameters has been set to zero, identify useful parameter ranges. Careful comparison of the analytically-predicted configuration variables versus those obtained when performing computational free energy minimization on a 2-D grid show that the computational results differ significantly from the analytic solution. The 2-D CVM can potentially function as a secondary free energy minimization within the hidden layer of a neural network, providing a basis for extending node activations over time and allowing temporal correlation of patterns.

Phase Field Calculations with CVM Free Energy within Square Approximation

2007 ◽  
Vol 561-565 ◽  
pp. 1935-1940
Author(s):  
Tetsuo Mohri ◽  
Nao Fujihashi ◽  
Ying Chen
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Stoichiometric Composition ◽  
Sharp Decrease ◽  
Cluster Variation Method ◽  
Field Method ◽  
Phase Field Method ◽  
Square Lattice ◽  
Variation Method ◽  
Cluster Variation

Phase Field Method is combined with the Cluster Variation Method within the square approximation, and the multiscale ordering behavior from atomistic to microstructural evolution process of ordered domains in the two dimensional square lattice is investigated. The transition temperature is determined at 1:1 stoichiometric composition and it is confirmed that the transition is of the second order. The growth process of the ordered domains is visualized and it is revealed that the sharp decrease of the free energy takes place during the process.

Thermodynamics of mixing and ordering in the diopside–jadeite system: I. A CVM model

2002 ◽  
Vol 66 (4) ◽  
pp. 513-536 ◽  
Author(s):  
V. L. Vinograd
Keyword(s):  
Free Energy ◽  
Cluster Variation Method ◽  
Solution Calorimetry ◽  
Range Order ◽  
Variation Method ◽  
Energy Of Mixing ◽  
Equilibrium Data ◽  
Thermodynamics Of Mixing ◽  
Cluster Variation ◽  
Free Energy Of Mixing

AbstractSolution calorimetry and phase equilibrium data for the diopside–jadeite system are assessed using a combination of the cluster variation method (CVM) and Redlich-Kister (RK) polynomial expansion. The CVM is used to model part of the free energy of mixing which depends on short-range order (SRO) and long-range order (LRO) effects. The SRO/LRO independent part of the free energy is modelled using an RK polynomial. The parameters of the RK and CVM models are obtained through the fit to the experimental data. The best-fit parameters are used to calculate activity-composition relations and a temperature-composition phase diagram of the diopside–jadeite system.

Phase Transformations in Hexagonal-Close-Packed Alloys: Analysis with the Cluster Variation Method

1992 ◽  
Vol 291 ◽  
Author(s):  
Ryan Mccormack ◽  
Mark Asta ◽  
Gerbrand Ceder ◽  
Didier de Fontaine
Keyword(s):  
Nearest Neighbor ◽  
Binary Alloys ◽  
Cluster Variation Method ◽  
Variation Method ◽  
Temperature Phase ◽  
Hexagonal Close Packed ◽  
Cluster Configuration ◽  
Cluster Variation ◽  
Polyhedron Method ◽  
Pair Interactions

ABSTRACTWe present a study of the hexagonal close-packed (hcp) Ising model for binary alloys within the cluster variation approximation. Groundstates of order stabilized by nearest-neighbor (NN) pair, triplet, and tetrahedron interactions were determined using the cluster configuration polyhedron method; no previous hcp ground-state study has considered all of these interactions. We predict physically realizable groundstates with stoichiometries A, AB (3 distinct structures), A2B, A3B, and A4B3. The previously unpredicted A4B3 structure is stabilized by multiatom (i.e. triplet, tetrahedron) interactions, while the others are stabilized by the two NN pair interactions. The Cluster Variation Method (CVM) was used to calculate the finite-temperature phase-equilibria for prototypical binary alloys. We present the first ordering phase diagrams computed with the CVM which contain all relevant groundstates for both isotropic and anisotropic NN pair interactions.

Glass Transition within the Cluster Variation Approximation

2007 ◽  
Vol 26-28 ◽  
pp. 723-726
Author(s):  
Tetsuo Mohri
Keyword(s):  
Phase Diagram ◽  
Free Energy ◽  
Glass Transition ◽  
Cluster Variation Method ◽  
Variation Method ◽  
Cluster Variation ◽  
Ordering Transition ◽  
Limiting Case ◽  
The Ideal

The detailed behavior of the free energy of Cluster Variation Method in the vicinity of spinodal ordering transition is examined. The generalized phase diagram proposed in the previous study is modified and spinodal ordering transition is reinterpreted as a limiting case of the ideal glass transition.

Theoretical Study of Glass Transition Based on Cluster Variation Method

2007 ◽  
Vol 539-543 ◽  
pp. 2425-2430 ◽  
Author(s):  
Tetsuo Mohri ◽  
Yoshitaka Kobayashi
Keyword(s):  
Free Energy ◽  
Glass Transition ◽  
Order Parameter ◽  
Cluster Variation Method ◽  
Variation Method ◽  
Bravais Lattice ◽  
Probability Method ◽  
Temperature Dependent Viscosity ◽  
Viscosity Term ◽  
Cluster Variation

Modeling of Glass transition is attempted based on the Cluster Variation Method. Free energy functional of an L10 ordered phase is employed to describe the first order nature of the transition. Free energy contour surface calculated as a function of temperature and an order parameter which simulates an amount of defects provides a generalized stability diagram in which the ideal glass transition temperature is identified as a critical point. Transition kinetics is investigated by Path Probability Method which is the kinetics version of the CVM to time domain. Continuous cooling behavior is calculated by explicitly incorporating the temperature dependent viscosity term based on VFT (Vogel-Fulcher-Tamman) formula. The glass transition is realized as the freezing of the order parameter due to the enhanced viscosity. The extension of the present theoretical scheme to non-Bravais lattice is attempted by Continuous Cluster Variation Method.

LMTO/CVM and LAPW/CVM Calculations of the Nial-Niti Pseudobinary Phase Diagram

1990 ◽  
Vol 213 ◽  
Author(s):  
Benjamin P. Burton ◽  
Jean E. Osburn ◽  
Alain Pasturel
Keyword(s):  
Nearest Neighbor ◽  
Equations Of State ◽  
Cluster Variation Method ◽  
Variation Method ◽  
Two Phase ◽  
Pairwise Interactions ◽  
Phase Fields ◽  
Pseudobinary Phase Diagram ◽  
Total Energies ◽  
Cluster Variation

ABSTRACTLinear Muffin Tin Orbital and Linearly Augmented Plane Wave calculations of equations of state were performed for observed and hypothetical ordered structures in the NiAl-NiTi system. Total energies were parameterized in both the Connolly-Williams and ∈-G approximations, and the resulting parameters were used to calculate theoretical phase diagrams by the cluster variation method. Third nearest neighbor Al-Ti pairwise interactions are predicted to be strongly repulsive, and to be a major cause of observed B2+L21 two phase fields.

Phase Stabilities in Some B2 Type Ordered Ni-Al-X Alloys

1994 ◽  
Vol 364 ◽  
Author(s):  
Tetsumori Shinoda ◽  
Y. Mishima ◽  
T. Suzuki
Keyword(s):  
Free Energy ◽  
Chemical Potential ◽  
Cluster Variation Method ◽  
Phase Region ◽  
Variation Method ◽  
Isothermal Phase Diagram ◽  
B2 Phase ◽  
The Difference ◽  
Cluster Variation ◽  
Ordered State

AbstractThe difference in developing of solubility lobe (s.l.) of B2 phase from the B2 NiAl point on each isothermal phase diagram of Ni-Al-Co, Ni-Al-Fe and Ni-Al-Mn is expected to be explained in terms of the pattern of free energy minimum (f.e.m.) line in the B2 phase region of these phase diagrams. To calculate such the f.e.m. line, a simple way has been proposed based on the tetrahedron approximation of the cluster variation method (CVM) making use of a relationship to correlate the conventional chemical potential of a species with some kinds of the CVM variables. The calculated results show that the f.e.m. line well reflects the solubility lobe developing of the B2 phase for both Ni-Al-Fe and Ni-Al-Co while some disagreement arises with Ni-Al-Mn. The results also show a theoretical possibility of transition from the high ordered state B2 to a low ordered one under a relatively low temperature regime so far as the present free energy model are concerned.

CVM-based first-principles calculations for Fe-based alloys

2011 ◽  
Vol 1296 ◽  
Author(s):  
Tetsuo Mohri
Keyword(s):  
Free Energy ◽  
First Principles ◽  
Cluster Variation Method ◽  
Phase Field Method ◽  
Variation Method ◽  
Probability Method ◽  
First Principles Calculations ◽  
Wide Range ◽  
Cluster Variation ◽  
Alloy Systems

ABSTRACTCluster Variation Method (CVM) has been recognized as one of the most reliable theoretical tools to incorporate wide range of atomic correlations into a free energy formula. By combining CVM with electronic structure total energy calculations, one can perform first-principles calculations of alloy phase equilibria. The author attempted such CVM-based first-principles calculations for various alloy systems including noble metal alloys, transition-noble alloys, III-V semiconductor alloys and Fe-based alloy systems. Furthermore, CVM can be extended to two kinds of kinetics calculations. One is Path Probability Method (PPM) which is the natural extension of the CVM to time domain and is quite powerful to investigate atomistic kinetic phenomena. The other one is Phase Field Method (PFM) with the CVM free energy as a homogeneous free energy density term in the PFM. The author’s group applied the latter procedure to study time evolution process of ordered domains associated with disorder-L10 transition in Fe-Pd and Fe-Pt systems. CVM has, therefore, a potential applicability for the systematic studies covering atomistic to microstructural scales. It has been, however, pointed out that the conventional CVM is not able to include local lattice relaxation effects and that the resulting order-disorder transition temperatures are overestimated. In order to circumvent such inconveniences, Continuous Displacement Cluster Variation Method (CDCVM) has been developed. Since first-principles CDCVM calculations are still beyond the scope at the present stage, preliminary results on the two dimensional square lattice and an fcc lattice with primitive Lennard-Jones type potentials are demonstrated in the last section.

Sign in / Sign up

Export Citation Format

Share Document