Development of a Computer Application to Simulate Porous Structures

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1 Vol. Materials 5, No. Research, 3, 00Vol. 5, No. 3, 75-79, 00. Developent of a Coputer Application to Siulate Porous Structures Developent of a Coputer Application to Siulate Porous Structures S.C. Reis a, V. Vasconcelos* b, M.C. Leite b and W.L. Vasconcelos b a Nuclear Technology Developent Center, CDTN, Nuclear Energy National Coission, CNEN Rua Prof. Mário Werneck, s/n, Belo Horizonte - MG, Brazil b Dept. of Metallurgical and Materials Engineering, UFMG Rua Espírito Santo, 35, andar, Belo Horizonte - MG, Brazil Received: Septeber 7, 001; Revised: July 10, 00 Geoetric odeling is an iportant tool to evaluate structural paraeters as well as to follow the application of stereological relationships. The obtention, visualization and analysis of voluetric iages of the structure of aterials, using coputational geoetric odeling, facilitates the deterination of structural paraeters of difficult experiental access, such as topological and orphological paraeters. In this work, we developed a geoetrical odel ipleented by coputer software that siulates rando pore structures. The nuber of nodes, nuber of branches (connections between nodes) and the nuber of isolated parts, are obtained. Also, the connectivity (C) is obtained fro this application. Using a list of eleents, nodes and branches, generated by the software, in AutoCAD coand line forat, the obtained structure can be viewed and analyzed. Keywords: geoetric odeling, pore structure, connectivity, nanostructures 1. Introduction The deterination of topological paraeters is hard to get in laboratory because it requires the aking of serial sections on the aterial 1,. Considering extreely sall structures, like the ones obtained via sol-gel ethod, the deterination of topological paraeters can be even ore difficult. In this case, it can be necessary to assue soe shapes and coordination nubers for the structural eleents and estiate these paraeters based on easurable paraeters, by using specific stereological relationships 3,4. In order to ake it easier to analyze porous nanostructures and to allow visualization of topological paraeters, we developed a coputer software that ipleented a geoetric odel, capable of siulate tridiensional porous structures.. Developent According to the icrostructural description ethod developed by Vasconcelos 5, we developed a odeling taking into consideration that the structural level of interest is the nanostructure and the structural eleents to be analyzed *e-ail: wlv@deet.ufg.br are the pores, for which the connectivity is deterinated. Modeling aied to siulate porous structures within a reasonable coputational cost, to which the nuber, the diension and the coordination of the structural eleents are deterinated. This was ade possible by the applying of contour conditions, where the nodes of the structure are spherical and the connections between the, the branches, are cylindrical. In this way the diaeter of the nodes, Dn, and the diaeter of the branches, Db, are equal and are calculated according to the following stereological relationship 6,7 : 4 4Vv Dn = Db = [ λ 3 ] = (1) Sv where Vv is the volue fraction of pores, Sv is the surface area per unit volue of pores and λ 3 is the ean lineal intercept. The average length of the branches, Hed, is obtained by the geoetrical relationship Eq., St Hed = () π.4. Vt. Nb

76 Reis et al. Materials Research where St is the total surface area of pores, Vt is the total pore volue, Nb is the nuber of branches and is given by the eq. 3. Nn.

2 76 Reis et al. Materials Research where St is the total surface area of pores, Vt is the total pore volue, Nb is the nuber of branches and is given by the eq. 3. Nn.K1 Nb = (3) Where K1 is obtained for equation (4), 4. Cn Cn3 +. Cn + Cn1 K1 = (4) 100 where Cn4, Cn3, Cn and Cn1 are the percentage of nodes connected to 4, 3, and 1 neighboring nodes (node coordination nuber), respectively. The average length of the branches, Hed, is also obtained by the following epiric relationship: Hed = Di Nn (5) where Di is the length of the edge of the cube (siulated volue). This epiric relationship was obtained by generating rando nodes inside a cube with the edge equal to Di and by calculating the average of the four closest neighborhood nodes distance to each generated node. At last, the nuber of nodes, eq. 6, of the siulated structure, Nn, is estiated by using eqs and 5. St Nn =.0, Di. π Vt K 1 0,6406. The following equations show the relationships between: the specific surface area of the pores, Sg, and the total surface area of the pores, St, (eq. 7); the specific pore volue, Vp, and the total pore volue, Vt, (eq. 8); and the length of the edge of the cube, Di, and the bulk density (eq. 9). (6) 3. Application The software developed aied to siulate structures where the pores are settled in rando positions. The ain structural eleents of this structure, such as nuber of nodes and nuber of branches are known. The connectivity (C) is also obtained. For the ipleentation of the representation and project steps we used the prograing environent Delphi, based on Object Pascal prograing language, which allowed us to develop low coputational cost algoriths 8. On the visualization step we used the AutoCAD software as a graphic coputer and solid odeling environent 9. Data Input The software is fed by the user with data that will deterine the contour conditions used in the siulation. The necessary data, as well as their easure units, are described below. One ust provide as data input: specific pore volue (c 3 /g); specific pore surface area ( /g); density of the aterial (g/c 3 ); ass of the saple (g); distribution of pore coordination nubers. In the distribution of the coordination nubers, the user defines the percentage of connected pores to four, three, two or one neighbors (Cn4, Cn3, Cn and Cn1, respectively). The chosen values of connectivity will be used by the software to try to reach the initial ais, however, these values ay be not reached. Figure 1 presents a structure in which one can visualize nodes connected to none, one, two, three or four neighboring nodes (NC = 0, NC = 1, NC =, NC = 3, NC = 4, respectively). Definition of the siulated volue Taking into account that it is desirable to achieve a low NC=0 NC=1 St Sg = (7) ( ρt ρv ) ( ρ. ρ ) Vt Vp = = (8) t v NC=4 NC= NC=3 Di ρ = 3 (9) v where ρ t is the true density, ρ v is the bulk density and is the ass of the saple. Figure 1. Structure showing nodes with different coordination nubers (NC).

Vol. 5, No. 3, 00 Developent of a Coputer Application to Siulate Porous Structures 77 coputational cost, the user can stablish a liit in the nuber of nodes to be generated by the software.

3 Vol. 5, No. 3, 00 Developent of a Coputer Application to Siulate Porous Structures 77 coputational cost, the user can stablish a liit in the nuber of nodes to be generated by the software. Therefore, we liited the nuber of nodes that can be generated by the progra fro 10 to 100. After defining the length of the edge of the cube, Di, that deliitates the physical volue of the siulated structure, and the other required data input, the software is capable of siulate the porous structure. The software identifies autoatically a valid range for Di, based on the nuber of nodes. Generation of the structure The generation of the structure is detailed in Fig., where the steps of node generation, deterination of the neighborhood and the branches generation are presented. Node generation The software considers that all nodes are spherical and have the sae diaeter. Therefore, they can be represented by the coordinates of a sphere center and its diaeter. The coordinates of the center of the node are generated randoly but the software does not allow that two different nodes intercept each other in the space. By the end of the process, it is obtained a data file containing all the inforation about the generated structure that is useful as a source for the generation of output files that are iported by AutoCAD. Deterination of the neighborhood This step deterinates the eight closest neighbors nodes of each generated node. The neighbors are deterinated consulting the data file and storing the identification of the closest nodes, in increasing order of distance, for each node, deterining its neighborhood region, Fig. 3. Branch generation The step of branch generation originates the branches that will be created when nodes are connected. These branches will be iported later by AutoCAD. The stage of branch generation verifies aong the neighbors (neighborhood region) of a given node those with which it is possible to establish a connection. In order to be considered valid, a given connection ust pass two tests: the shadow test and the intersection test. The shadow test verifies if the branch that is being generated has a coon region with an interediate node, Fig. 4a, while the intersec- Figure. Detailing the stage of generating the structure.

78 Reis et al. Materials Research Figure 5. a) Structure of pores; b) structure of aterial (solid). Figure 3.

Exaples of situations not allowed by the progra: a) intersection between a branch and an interediate node; b) intersection between two branches. Figure 6.

Coparisson between the real data, data fro the proposed structure and, data fro the siulated structure. All the ites were calculated considering a cubic eter.

4 78 Reis et al. Materials Research Figure 5. a) Structure of pores; b) structure of aterial (solid). Figure 3. Neighborhood region defined by the sphere with center in the coordinates of the central node and radius equal to the distance between the central node and the eighth closest node. Figure 4. Exaples of situations not allowed by the progra: a) intersection between a branch and an interediate node; b) intersection between two branches. Figure 6. Siulated porous structures, for the sae aterial, with Di of 100 n, 00 n, 50 n and 300 n. Table 1. Coparisson between the real data, data fro the proposed structure and, data fro the siulated structure. All the ites were calculated considering a cubic eter. Ite Real Structure Proposed Structure Siulated Structure Nodes per unit volue (Nv) E E+ Branches per unit volue (Bv) E+3 1.8E+3 Genus per unit volue (Gv) E+ 6.0E+ Volue fraction of pores (Vv) Surface area per unit volue of pores (Sv) 1.379E E E+8 Average connectivity (Cn) tion test verifies if this branch intercepts another branch, that already exists, deliitated by two others nodes, Fig 4b. Text file generation After the establishent of connection aong the nodes, the software executes the text file generator, which saves to disk the files in TXT forat, storing the coands and the inforation about the nodes and branches to be iported by AutoCAD. These files can be accessed later on, allowing the user to load in the software, data pertinent to a previous siulation and change its execution conditions. To exeplify the application of the ipleented odel, we siulated porous structures of a aterial obtained via sol-gel ethod. The data referring to this aterial is shown in Table 1, in which we also present the data fro the siulated structure.

5 Vol. 5, No. 3, 00 Developent of a Coputer Application to Siulate Porous Structures 79 Figure 5 presents the structure siulated referring to the aterial analyzed, while Fig. 6 presents obtained structures, of the sae aterial, in several different sizes of Di. 4. Conclusions We elaborated a odel using software developent written in Object Pascal prograing language (Delphi ), to siulate porous structures fro which the nuber of nodes, the nuber of separated parts, the genus and the coordination nubers of pores can be estiated. The software generates a list of structural eleents, nodes and branches, which are iported by AutoCAD, in a way that this structure can be visualized and analyzed. We copared the data fro a real aterial, obtained via sol-gel, with the data of a siulated structure, and the results are satisfactory. Acknowledgeents The authors thank the financial support fro CNPq. References 1. Aigeltinger, E.H.; DeHoff, R.T. Quantitative deterination of topological and etric properties during sintering of copper, Metallurgical Transactions A., v. 6A, n. 10, p , Kurzydlowski, K.J.; Ralph, B. The quantitative description of the icrostructure of aterials. Boca Raton: CRC Press, Mann, R.; Allay, A.; Holt, A. Visualized porosietry for pore structure characterization of a nickel aluina reforing catalyst. Transactions of The Institution of Cheical Engineers, v. 73, p , Vasconcelos, W.L.; Hench, L.L. Análise da evolução topológica durante a densificação de gel de sílica. Cerâica, v. 38, n. 54, p. 17-0, Vasconcelos, V. Desenvolviento e aplicação de ua etodologia para a descrição icroestrutural. Tese de Doutorado. Belo Horizonte: DEMM/UFMG, Rhines F.N.; DeHoff, R.T.; Kronsbein, J. A topological study of the sintering process. Gainesville, Florida: U.S. Atoic Energy Coission, Vasconcelos, W.L. Topological Evolution and Properties of Sol-Gel Silica Monoliths. Tese de Doutorado. Gainesville, USA: University of Florida, Côrtes, P.L.; Shiraishi, K. Conhecendo e Trabalhando co o Delphi 4. São Paulo: Editora Érica, Mortenson, M.E. Geoetric odeling. New York, USA: John Wiley & Sons, 1985.

Medical Biophysics 302E/335G/ st1-07 page 1

Medical Biophysics 302E/335G/500 20070109 st1-07 page 1 STEREOLOGICAL METHODS - CONCEPTS Upon copletion of this lesson, the student should be able to: -define the ter stereology -distinguish between quantitative