A note on grain topology-size relationship of three-dimensional polycrystalline microstructures

Luan, Junhua; Liu, G. Q.; Wang, Hao Yu; Ullah, Asad

doi:10.1209/0295-5075/99/28001

Cited by 5 publications

(6 citation statements)

References 29 publications

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“…The results revealed that a strong correlation exist between grains and their first nearest neighbors, while there exist very little (negligible) correlation between grains and their kth (k2) nearest neighbors (on average). This confirms that the topological correlations of 3D grains with their kth layers (k2) of nearest-neighbors may have trivial effect on the topologysize relationship, and thus provides support to a new grain topology-size equation [17] that only considered the nonrandom first nearest-neighbor interactions.…”

Section: Discussionsupporting

confidence: 65%

“…(4) takes the form of eq. (1), the proposed grain topology-size equation in [17]; which can serve as a special case of this general grain topology-size equation (eq. (4)).…”

Section: Methods and Parametersmentioning

confidence: 99%

“…Recent studies strongly suggested the importance of considering the effects of first nearest-neighbor grains when studying grain growth [10,16] and also predicting the grain topology-size relationships. More recently, Luan et al [17] introduced a new grain topology-size equation in three dimensions by considering the interactions of first nearest neighbor grains, which states that the topologically averaged relative grain size R f /<R> depends on the difference between the number of faces of a grain (f) and the mean face number of its nearest neighbors (m 1 ), that is…”

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confidence: 99%

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Neighborhood topological effect on grain topology-size relationship in three-dimensional polycrystalline microstructures

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A new grain topology-size relationship in three-dimensional (3D) polycrystalline microstructures has recently been established by considering the effects of non-random first nearest neighbor grains. In this contribution, a generalized form for this relationship is presented by considering the interactions of kth (k=1, 2, 3…) nearest neighbor grains, and large scale Monte Carlo-Potts model simulation is used to investigate the general neighborhood topological effect on the topology-size relationship. The results show that, unlike their first nearest neighbors (k=1), the topological correlations of 3D grains with their kth layers (k2) of nearest-neighbors may have trivial effect on the topology-size relationship. Many macroscopic properties such as mechanical, thermal, magnetic and conductive properties of the material can be directly linked to microstructure; therefore, it is very important to predict the grain microstructure and its evolution [1][2][3][4]. The size and the shape of grains and their spatial correlation are likely to play a significant role in the microstructural evolution. Therefore, the grain topology-size relationship is of fundamental importance for a better understanding of polycrystalline materials. In 1985, by applying the methods of statistical mechanics to the structure of random, space-filling cellular structures, Rivier [5] revealed that the maximum entropy inference under a few constraints yields structural equations of state, relating the topology of cells to their size. These equations of state are namely Perimeter law (for metallurgical grains) and Lewis's law (for ideal soap froths). According to Perimeter law, the average radius of grains is proportional to the number of their edges. While, in three dimensions (3D), a parabolic relationship exists between the grain size and grain topology. DeHoff and Liu [6] have proposed a linear relationship between the number of grain faces and the mean tangent diameter of individual grains in 3D. Thereafter, Abbruzzese and Compopiano [7] and Thorvaldsen [8] proposed two independent forms of quadratic relationships between the number of grain faces and the sphere-equivalent radius of grains in 3D. The DeHoff-Liu's linear model has been verified experimentally by Liu et al. [9] and strongly recommended it when the mean tangent diameter is used for grain size. Meanwhile, Abbruzzese-Compopiano's quadratic model [7] also agrees well with the experimental results of Liu et al. [9] when the sphere equivalent radius is used for grain size instead of mean tangent diameter. The Abbruzzese-Compopiano's [7] and Thorvaldsen's [8] quadratic models are also consistent with the experimental results of -titanium

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Section: Discussionsupporting

confidence: 65%

“…(4) takes the form of eq. (1), the proposed grain topology-size equation in [17]; which can serve as a special case of this general grain topology-size equation (eq. (4)).…”

Section: Methods and Parametersmentioning

confidence: 99%

mentioning

confidence: 99%

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Neighborhood topological effect on grain topology-size relationship in three-dimensional polycrystalline microstructures

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“…We also experimentally analyzed the topological characteristics of 3D grain microstructures generated by Monte Carlo-Potts simulation [12][13][14]. A continuum microstructure was mapped onto a cubic lattice with 300 × 300 × 300 sites and with periodic boundary conditions in three directions (see details of the method in our previous papers [1,2,15,16]). 6093 grains at different simulation time were sampled.…”

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confidence: 99%

Topological correlations of grain faces in polycrystal with experimental verification

Liu

Wang

et al. 2013

EPL

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“…-Grains structures, biological tissues and common soap froths bear a similar nature in that they are space-filling celluar structures [1,2]. Topological properties of such cellular structures are subjects of longstanding interest which despite intensive studies is still not fully understood [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Among several topological relations summarized in the past (including Aboav-Weaire law on nearest-neighbor correlations [3,4], the von Neumann law on the growth rate of individual cells [1,5], the Lewis law on the statistics of cell area [6], and the scaling law on the probability distribution of the cells [7]), the Aboav-Weaire law [3,4] is a fundamentally empirical expression to describe the short-range topological correlations between a center grain and its first nearest neighbors in two dimensions.…”

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confidence: 99%

Generalization of the Aboav-Weaire law

Wang

Liu

2012

EPL

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How cells are arranged in a three-dimensional celluar network is one of the classic problems of physics, biology and materials science. Based on the topological analyses on steady-state grain-growth structure, a generalization of the Aboav-Weaire law to layers beyond the first is suggested that the average number of faces per grain, mj(f), is related to the number of grains, qj(f), in thej-th layer of a center grain with f faces (on average). This result is verified by the data from the curvature flow simulation and assists in better understanding the arrangement of cells in three dimensions.

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A note on grain topology-size relationship of three-dimensional polycrystalline microstructures

Cited by 5 publications

References 29 publications

Neighborhood topological effect on grain topology-size relationship in three-dimensional polycrystalline microstructures

Neighborhood topological effect on grain topology-size relationship in three-dimensional polycrystalline microstructures

Topological correlations of grain faces in polycrystal with experimental verification

Generalization of the Aboav-Weaire law

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