{"title":"Stereological description of grain boundary driving and resisting pressures -Solution to the Smith/Zener model","authors":"Burton R. Patterson, Yixiong Liu","doi":"10.1016/j.actamat.2025.121360","DOIUrl":null,"url":null,"abstract":"This paper presents a model for predicting second phase pinned grain size using a ratio of simple stereological terms- the length of grain boundary-second phase triple-line, <ce:italic>L<ce:inf loc=\"post\">V</ce:inf></ce:italic>, per grain boundary area, <mml:math altimg=\"si20.svg\"><mml:mrow><mml:mspace width=\"0.33em\"></mml:mspace><mml:msubsup><mml:mi>S</mml:mi><mml:mi>V</mml:mi><mml:mi>g</mml:mi></mml:msubsup></mml:mrow></mml:math>. This pinning potential <ce:italic>Z = L<ce:inf loc=\"post\">V</ce:inf> /</ce:italic><mml:math altimg=\"si20.svg\"><mml:mrow><mml:mspace width=\"0.33em\"></mml:mspace><mml:msubsup><mml:mi>S</mml:mi><mml:mi>V</mml:mi><mml:mi>g</mml:mi></mml:msubsup><mml:mspace width=\"0.33em\"></mml:mspace></mml:mrow></mml:math>= <ce:italic>P<ce:inf loc=\"post\">A</ce:inf> /</ce:italic><mml:math altimg=\"si21.svg\"><mml:mrow><mml:mspace width=\"0.33em\"></mml:mspace><mml:msubsup><mml:mi>P</mml:mi><mml:mi>L</mml:mi><mml:mi>g</mml:mi></mml:msubsup></mml:mrow></mml:math> is straightforward to measure from counting triple points, <ce:italic>P<ce:inf loc=\"post\">A,</ce:inf></ce:italic> and test line intercepts <mml:math altimg=\"si21.svg\"><mml:mrow><mml:mspace width=\"0.33em\"></mml:mspace><mml:msubsup><mml:mi>P</mml:mi><mml:mi>L</mml:mi><mml:mi>g</mml:mi></mml:msubsup></mml:mrow></mml:math> on the section plane. The measure of triple line per boundary area is the quantity Smith/Zener sought to quantify indirectly in their <ce:italic>f/r</ce:italic> model, these direct measurements not being available at the time. The above expression for <ce:italic>Z</ce:italic> encompasses the generally overlooked degree of non-random contact between the grain boundary and second phase, <ce:italic>R</ce:italic>. The final predictive model includes the stereologically measured boundary curvature<mml:math altimg=\"si22.svg\"><mml:mrow><mml:mo>,</mml:mo><mml:mspace width=\"0.33em\"></mml:mspace><mml:mover accent=\"true\"><mml:mi>H</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:mrow></mml:math>, that drives grain growth, which Smith/Zener approximated as inverse grain size. These terms, which are easier to measure experimentally than <ce:italic>f</ce:italic> and <ce:italic>r</ce:italic>, were found to exert far greater effect on pinned grain size than the specific formulation of <ce:italic>Z</ce:italic> that has been the focus of many prior models. Testing was performed on sintered alumina with three levels of MgO doping causing widely different levels of grain boundary curvature and contact. Using this model the predicted grain sizes for this large variation in pinning conditions condensed to a single trend within experimental measurement error of the actual pinned grain size.","PeriodicalId":238,"journal":{"name":"Acta Materialia","volume":"659 1","pages":""},"PeriodicalIF":8.3000,"publicationDate":"2025-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Materialia","FirstCategoryId":"88","ListUrlMain":"https://doi.org/10.1016/j.actamat.2025.121360","RegionNum":1,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATERIALS SCIENCE, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
This paper presents a model for predicting second phase pinned grain size using a ratio of simple stereological terms- the length of grain boundary-second phase triple-line, LV, per grain boundary area, SVg. This pinning potential Z = LV /SVg= PA /PLg is straightforward to measure from counting triple points, PA, and test line intercepts PLg on the section plane. The measure of triple line per boundary area is the quantity Smith/Zener sought to quantify indirectly in their f/r model, these direct measurements not being available at the time. The above expression for Z encompasses the generally overlooked degree of non-random contact between the grain boundary and second phase, R. The final predictive model includes the stereologically measured boundary curvature,H¯, that drives grain growth, which Smith/Zener approximated as inverse grain size. These terms, which are easier to measure experimentally than f and r, were found to exert far greater effect on pinned grain size than the specific formulation of Z that has been the focus of many prior models. Testing was performed on sintered alumina with three levels of MgO doping causing widely different levels of grain boundary curvature and contact. Using this model the predicted grain sizes for this large variation in pinning conditions condensed to a single trend within experimental measurement error of the actual pinned grain size.
期刊介绍:
Acta Materialia serves as a platform for publishing full-length, original papers and commissioned overviews that contribute to a profound understanding of the correlation between the processing, structure, and properties of inorganic materials. The journal seeks papers with high impact potential or those that significantly propel the field forward. The scope includes the atomic and molecular arrangements, chemical and electronic structures, and microstructure of materials, focusing on their mechanical or functional behavior across all length scales, including nanostructures.