Chaos, Solitons and Fractals 42 (2009) 820–825

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Chaos, Solitons and Fractals

journal homepage: www.elsevier .com/locate /chaos

Randomly grain growth in metallic materials

A. Ramírez a,b,*, F. Chávez b, L. Demedices a,b, A. Cruz b, M. Macias b

a Instituto Politécnico Nacional, (SEPI-ESIME), Unidad Profesional Ticoman, Av. Ticoman #600, Del. G.A.M., C.P. 07340 Distrito Federal, México, Mexicob Instituto Politécnico Nacional, (SEPI-ESIQIE), Unidad Profesional Zacatenco, Edif. 6 y Edif. Z planta baja C.P.07300, Distrito Federal, México, Mexico

a r t i c l e i n f o a b s t r a c t

Article history:Accepted 12 February 2009

0960-0779/$ - see front matter � 2009 Elsevier Ltddoi:10.1016/j.chaos.2009.02.011

* Corresponding author. Address: Instituto PolitéDistrito Federal, México, Mexico. Tel.: +55 57 29 60

E-mail addresses: adaramil@yahoo.com.mx, adar

Computational modeling of grain structures is a very important topic in materials science.In this work, the development of the computational algorithms for a mathematical modelto predict grain nucleation and grain growth is presented. The model place a number ofnucleated points randomly in a liquid pool according with the solid and liquid fractions(Xsol and Xliq) of metal solute and the local temperature distribution (SSI,J). Then thesepoints grows isotropically until obtain a grain structure with straight interfaces. Differentgrain morphologies such as columnar and equiaxed can be obtained as a function of thetemperature distributions and growth directions.

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1. Introduction

Many techniques are referred frequently to Monte Carlo methods and comprise direct approaches to stochastic eventswhich can be decomposed in isolated processes and statistical approaches to integrate multidimensional definite integralsnumerically. Monte Carlo simulations are referred with larges series of computational experiments using uncorrelated ran-dom numbers in order to represent a complex problem. The problems related to grain structures and properties of mate-rials are complex and request of computational methods to be described. Nowadays computational methods have becomein a powerful tool to simulated problems related with anisotropy in materials science and engineering. Mathematical algo-rithms can be programmed to be included in sophisticated simulators with graphical display options to represent grainmorphologies in materials (metallic or not metallic). Monte Carlo methods are useful routines to establish a relation be-tween a pre-defined or deterministic model and stochastic processes. These are employed in three steps to solve any prob-lem as follow:

(a) During the first step, the physical problem must be translated in analogous and probabilistic or statistical models. Theintegral expressions for the governing of the differential equations to be solved must be formulated including the rulesto program the algorithm.

(b) In the second step, the model is solved using stochastic techniques including arithmetical and logical operations. Theprocess involves the relation of the original information with random routines.

(c) In the third step, the data obtained must be treated and analyzed statically to be validated with the actual system.

Physical properties of polycrystalline materials depend of the microstructure arising from grain nucleation and growth.The prediction of the topological, crystallographic features is necessary to know the micro, meso and macroscopic behaviorof a material. Particular grain shapes and grain size distributions are the result of solidification or re-crystallization

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cnico Nacional, (SEPI-ESIME), Unidad Profesional Ticoman, Av. Ticoman #600, Del. G.A.M., C.P. 0734000x46133.ami00@latinmail.com (A. Ramírez).

A. Ramírez et al. / Chaos, Solitons and Fractals 42 (2009) 820–825 821

processes. Nevertheless there is no a closed analytical treatment to predict these grain sizes and shapes due to the complex-ity of many of the factors involved.

Geometrical models represent continuum approaches and consist basically of three steps: nucleation, crystal growth toimpingement and in some cases subsequent grain coarsening.

In the present work the nuclei are either initially distributed in a site-saturated as a function of the temperature distri-bution probability. This process is called nucleation. Then all the nuclei grow isotropically at a constant growth rate. Thismeans that all the existing nuclei expand equally forming circles or spheres according with a 2D or 3D model, respectively,until they impinge upon each other. A major population of nucleation points will be placed in the regions where the solidfraction is greater than liquid.

In a similar way topological network and vertex models idealize solid materials as homogeneous continua which containinterconnected boundary segments that meet at vertices (boundary junctions). The grain boundaries appear as lines seg-ments in 2D models and as planes in 3D models. The dynamic of the coupled interfaces determines the evolution of the net-work. Nevertheless these models generate grains structures with a polygonal morphology.

Many authors [1–16] have developed mathematical models to simulate grain growth during solidification. The beginners[2] used basic geometrical models because the computational capacities were limited in 60s and 70s. The development ofnew mathematical methods, more efficient algorithms and the increment in data speed management and storage capacityhave become possible to simulate complex problems in material science. The Monte Carlo method [1,3,5,8] and cellularautomaton [6,9,14–16] are two of the most useful methods employed for these purposes. Many authors [1,5–9,11–13] havedeveloped models for solidification process as a function of the fraction of solid for different metals. Other authors[1–5,7,10,15,16] have been working in models to simulate dendritic growth which is the basic structure in primary metallicproducts obtained after foundry. Some of these authors [6,9,14–16] have been working on developing models to modify anoriginal grain structure to obtain a new one after a manufacturing process.

2. Mathematical model

The grain structure is obtained using the algorithm shown in Fig. 1. The model can be easily programmed using a com-puter language. The present model was done using C++. Here nucleation and growth process are treated separately. Themodel works as a function of a probability for the original temperature distribution field. Different grain morphologiesare generated during solidification of a metallic material due to the heat removal intensity. Near the metallic surfaces a greatnumber of nucleated points appear as a result of very high solidification speed. Here the temperature decreases quickly dueto heat is frequently removed by forced convection techniques. Nevertheless inside the metallic piece the solidification

Fig. 1. Algorithm for simulation of grain growth.

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speed is not as high as is in the surface due to the reduction of the heat removal. Inside the metallic pieces a pool of liquidmetal un-solidified remains and the heat is redistributed only by conduction.

The model considers a 2D rectangular sample; where the temperature distribution is established by the user as boundaryconditions. The first step is described in the column of the left in Fig. 1. An initial distribution of the temperatures in the mate-rial is created using a pre-defined routine. A squared mesh is used to discretizate he sample. In this flow chart (nx) and (ny) arethe number of nodes.

Temperature distribution and heat removal are the factors that determine the solidification speed. The fraction of solid ina node is a very important factor to know if a node can be a nucleated point. The fraction of solid is calculated using Eq. (1).This fraction is very important to know if a node of the sample is solidified in a moment during the simulation or if the nodecan be chosen to be solidified previously during a quench. In this concept if the fraction (XsolI,J) is increased the probability tofind a nuclei in a zone will be also increased.

XsolI;J ¼T � TsolI;J

TsolI;J � T liqI;Jð1Þ

A probability of nucleation is assigned to each node according with the previous initial temperature distribution. The prob-abilities (SS) are defined by the user for the four sides (left, right, upper and lower) of the simulated sample. Then particularprobabilities for all the nodes (PSSI,J) are calculated using Eq. (2). Finally a number is randomly generated and compared toinvolve stochastic process during calculation.

PSSI;J ¼IðSSI¼leftþSSI¼rightÞ

nx þ JðSSJ¼upperþSSJ¼lowÞny

� �

2ð2Þ

The next step for simulating is the execution of the nucleation process described in the column of the middle of the Fig. 1.The process is done for each position of the sample discretizated. The probability for defining a nucleated point is assignedaccording with the previous temperature distribution. Then the nucleated points ‘‘N” are counted while the condition(N 6 nn) was allowed. The node is considered as nucleated if the generated random number and the probability assignedto the node are match.

Finally in the right column of the Fig. 1, the growth process is described. Here the nucleated points are identified in themesh and the growth radius ‘‘r” is increased at each iteration to simulate an isotropic growth.

The algorithm assumes the following facts:Nucleation and growth routines are executed independent and separately in order to avoid unnecessary code.All of the nucleated points are placed in the sample simultaneously at the beginning of the simulation.All of the nucleated points growth increasing the radius at the same rate (r++).The nucleated points becomes in grains. Pixels of colors are used to show the process in the screen.The grain growth of a nucleated point is interrupted if another grain is found in the original growth path. Nevertheless the

radius can continuously be increased to promote the growth in other direction.

Fig. 2. Different temperature distribution in the material.

Fig. 3. Grain structure obtained for a temperature distribution of case (a).

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3. Simulations

In the present work are shown three cases shown in Fig. 2a–c with different temperature distributions defined. The firstcase is an absolutely heterogeneous distribution of the temperature distribution in the sample. This kind of distribution isfound in the central regions of metallic pieces where liquid metal remains a longtime in mushy state. The other cases showninvolve different temperature distributions which have a strong influence on solidification speed. The case shown in Fig. 2bhas a low temperature distribution near the left side. This kind of distribution is used to evidence the grain morphologychanges between zones with high, medium and low solidifications speeds. Finally the case shown in Fig. 2c can be foundin the corner of metallic pieces here low temperature distributions are done in the left and upper sides.

In the Fig. 2a the nucleated points are placed randomly due to that solidification speed is heterogeneous in the material.Although in the Fig. 2b the nucleated points are placed near the side in the left of the metallic material due to the temper-ature distribution defined considered in this wall. A greater number of solidified points appear near the left side due to theintensity of heat removal than in the right side. In consequence after the growth process; the grain size will be smaller in theregions with a low temperature distribution. In the Fig. 2c a great number of solidified points appear near the corner left-upper of the material due to the low temperature distribution and high solidification speed in both these sides.

Different nucleated points distribution are obtained different temperature distribution and in consequence different grainsize structures will be obtained as is shown in Figs. 3–5 which show a pair of grain structures simulated for each case. Fig. 3aand b show a heterogeneous distribution of grain size for the entire sample. In the other hand Figs. 4 and 5, show differentgrain sizes according with the distributions shown in the Fig. 2b and c, respectively. Finer small grains are found near thesurfaces where a greater number of nucleated points appeared. Although bigger grains are found in zones where the tem-peratures distribution are high and solidification speeds were slow.

Fig. 5. Grain structure obtained for a temperature distribution of case (c).

Fig. 4. Grain structure obtained for a temperature distribution of case (b).

Fig. 6. Different kind of grain borders formed during grain growth process.

Fig. 7. Grain structure obtained using different relations (rx/ry) and different nucleated points.

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The interfaces obtained are always straight. The borders are the interface between two or more grains as is shown inFig. 6a–c. In Fig. 6a is shown the junction of three different grains which growth was interrupted. In Fig. 6b the final junctionof three grains is shown but here the junction between them was done at different steps of the growth process. Finally inFig. 6c two different grains can coarsen forming a bigger single one before their growth was interrupted by a third.

4. Conclusions and comments

Different grain structures in grain size and morphology can be created using the algorithm developed as a function of aprevious definition of the solidification speeds.

Grain size is a function of the nucleated points in a zone of the material.Grains have clearly defined interfaces and these can be easily identified.Grain growth can be orientated in order to obtain a grain with a columnar morphology changing the relation (rx/ry).

Where (rx) and (ry) are the radius for growth in the directions (x) and (y), respectively. If the relation (rx/ry = 1) the growthis radial (isotropical). Nevertheless if this relation is modified to be equal to a fraction number the growth will be increasedalong the (y) direction and if the relation is an integer number greater than 1 the growth will be increased along the (x)

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direction as is shown in the Fig. 7a and b using the relations (rx/ry = 5) and (rx/ry = 0.5), respectively. Fig. 7c and d illustrate theinfluence of the number of nucleated points and the previous temperature distribution over the columnar grain structuresimulated.

Acknowledgements

The authors thank to the Instituto Politécnico Nacional (IPN-COFAA).

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