Verification of topological relationship in 2-D grain growth process by simulation

Journal of University of Science and Technology BejingMaterialsVolume 11, Number 2, April 2004, Page 138Verification of topological relationship in 2-Dgrain growth process by simulationChao Wang, Guoquan Liu, Ya Sun, and Xiangge QinMaterials Science und Engineering School, Universily of Science and Technology Beijing. Bijing 100083. China(Received 2003-04-04)Abstract: Behaviors of the quasi-steady state grain size distribution and the corresponding topological relationship were investigaledusing the Pots Monte Carlo method to simulate the normal grain growth process. The observed quasi-steady state grain size dis-tribution can be well fit by the Weibull function rather than the Hillert distribution. It is also found that the grain size and averagenumber of grain sides are not lincarly rclated. .: reason that the quasi-stcady stale grain size distibution deviates from the Hillertdistribution may contribute to the nonlinearity of the relation of the average number of grain sides with the grain size. The resuls al-so exhibit the reasonability of the relationship deduced by Mullins between the grain size distribution and the average number ofgrain sides.Key words: grain growth; grain size distibution: topology: Monte Carlo simulation: Hillert distribution[This work was suppored by the National Natural Science Foundation of China (No.501710081]1 Introductionlert, Lognormal and Gamma functions [8]. Its specificform is:Grain growth has been defined as the uniform in-crease in grain size that occurs upon annealing a poly-f(r)= A rh-expl-(5)则](2)crystalline aggregate after primary recrstallization iscomplete [I]. It is characterized by the continuous in-where a= I/I7(1+1B), and β a shape parameter.crease in mean grain size. More detailed characteris-Mullins [9] has clarified the reason of the failure oftics of the microstructure, such as the distribution ofgrains in size, orientation, and shape are also impor-the Hillert distribution to agree with simulations or(ant [2-4]. It has been recognized that all these aspectsexperiments. He concluded that in the quasi-steadywould interact and affect the grain growth process.state, the Hillert distribution always follows if the aV-And in DeHof's view, grain growth even can not oC-erage number of grain sides <<(x)> is a linear functionof the normalized grain size x; even a slight nonline-cur without the topological process [5. 6].arity of could cause the drastic deviations fromAmong all the theories on grain growth, the Hillertthe Hillert distribution. He also gave a relation (there-theory [7] has been the most widely accepted or cited.after called Mulins Equation) between the GSD P(x)It predicts that the grain size distribution (GSD) underand <<(x)>: .the quasi-steady state is unique, not related to its ini-tial state. And the GSD is in the form of the so called2Hillert distribution:PP(a)dx=6βx[-6](3)21f(r)= (20)2 (z-rexp(- 2-7)(1)where x。is the cutoff of P(x), β can be derived fromthe parabolic law of the normal grain growth:where r is the relative grain size or normalized grain中国煤化工(4)size and e is the constant 2.7382.... In fact, the Hil-lert distribution has seldom been observed in experi-MHCNMHGsizeattimet,ments and computational simulations. Recently thethe Initial mean grain size, K lne product of the grainWeibull function has been shown in many cases to de-boundary energy and the mobility, and β a constant forscribe the quasi steady state GSD better than the Hil-the quasi -steady state to hold.Corresponding author: Guoquan Liu. E mal: g- 1iu@ustb.cdu.cn.C. Wang et al, Verification of topologcal relationship in 2-D grain growth process by simulation139Actually, Mullins' conclusions depend on the as-a Laguerre tessellation [10] was performed by thesumption of an analytical relation of . But thegrain nucleation and growth followed by varying thenonlinear form of <(r)> (here r is in the place of x asgrowth rate of each grain according to is own size.the normalized grain size) has not been observed. InFigure 1 shows the GSDs of two different initial mi-the present work, Potts Monte Carlo simulations werecrostructures so generated. They can be well fit by the .performed to investigate the form of <(r)> startingWeibull functions with β=l .92 and 2.90, respectively.from the Weibull GSD.The fit goodness is verified by the Chi-squares and thedetermination coefficient better than 0.003 and 0.98,2 Simulation procedureTo generate grain structures with the Weibull GSD、respectively.| (a)Initial structure WI1.2 [b)●Initalstructure WIIWebull: B:-1.92Webull: β " 2.90y 0.80.80.4|%0.00 1.2.01.020-9* 3.0Relative grain size rRelative grain sizc rFigure 1 GSDs for two initial grain structures: (a) structure WI; (b) structure WII.Figure 2 plots the corresponding curves of <(r)>Starting from these initial grain structures, theof the two initial structures. The nonlinearity of process of grain growth was evolved up to 6000can be seen though <8(r)> deviates the linearityMonte Carlo steps (MCS) by the Potts Monte Carloslightly. This may be resulted from the intrinsic cor-Method, details of which can be seen elsewhere [11].relation of the neighbor grains.8t (田), Initial structure WI4[6)●Initial structure WII●Linear ftting line... Linear ftting line4-10-6t.02.0.03.0Rclative grain size rFigure 2 Average number of grain sides s. the relative grain size r for the two initial grain structures: (a) structureWI; (b) structure WII.Figure 3 shows the GSDs of the two evolved mi-by equation (2), the average number of grain sides iscrostructures during grain growth. It shows that thedescribed as:GSDs at time t=3000, 4000, 5000, and 6000 MCS can2Bbe fit by the Weibull function. It should be noticed that<(r)>= 6βr2- 6β B之+6(5)the two final GSDs could be viewed as the sameObviously, this is the specific form of Mullinsthough their corresponding initial ones are quite dif-Equation for grain structures with the Weibul GSD.ferent (figure 1). This suggests that the quasi-steadystate bas reached. For comparison, the Hillert dis中国煤化Idifferent evolved mi-tribution was also shown in figure 3. Very clearly, it iscrost:YHe 4. The cur-not suitable to describe the GSDs obtained in the pre-vesCNM H Gn (5) with the Chi-sent work.square value better than 0.06, and their form keepsLet P(x) in equation (3) be replaced by f(r) definednearly invariant as the simulation time beyond 4000140J. Univ. Sci. Technol Bejing, VoLII, No.2, Apr 2004MCS. This indicates that equation (5) is valid for grainsteady state GSD supports Mullins' view: the slightstructures with he Weibull GSD. The above-deviation from linearity of ~r relation can leadmentioned connection between the nonlinear <5(r)>~rto non-Hillert distributions in grain size distribution.relation and the Weibull rather than the Hillert quasi-1.2(间)0 3000 MCS(6)。3000 MCS4000 MCSE， 5000 MCS▼.5000 MCSWwebull:B-2.43(O0DM_-2.400.8-0.8Webul:Hillert distribution'Hillert distribution.40.4-0.0.01.02.3.01..0Relative grain size rFigure 3 Time evolution of the GSDs in grain growth process, starting from structure WI (a) and WII (b).(a)()888d 。8t6言6， 4000 MCS。4000 MCS4+ g66A 5000 MCS1 5000 MCS- Eq.(5): β3-0.142卜-Eq.(5):0.00.51.5 2.0 2.51.5 2.0 2. 5Figure 4 Evolution of <(r)> vs. r for the simulated grain structure, starting from structure WI (a) and wI (b).4 Conclusionsgrowth [J, Phrys. D, 66(1993), p.50.[4] V.E. Fradkov and D. Udler, Two dimensional grain growth:The Potts Monte Carlo method was performed totopological aspects [J], Adv. Phys, 43 (1994), No.6, p.739.simulate the grain growth starting from two structures5] R.T. DeHoff, Metric and topological contributions to thewith different Weiull grain size distributions. Therate of change of boundary length in two-dimensionalresults show that Mullins Equation for grain structuresgrain growth [], Acta Mater, 46(1998). No.14, p.5175.with the Weibull grain size distribution is valid. That[6] G.Q. Liu, H.B. Yu, X.Y. Song, and X.G. Qin, A newmodel of three-dimensional grain growth: theory andis, the corresponding grain size distribution is deter-computer simulation of topology-dependency of individu-mined by the relation between the average number ofal grain growth rate [J}, Mater. Des., 22(2001), p.33.the grain sides and grain size. It is the nonlinearity in7] M. Hillert, On the theory of normal and abnormal grainthe above-mentioned grain topology-size relation thatgrowth [J]. Acta Metall, 13(1965), No.3, p.227.causes substantial deviations from the Hillert grain[8] W. Fayad, C.V. Thompson, and HJ. Frost, Steady statesize distribution.grain-size distributions resulting from grain growth in twodimensions [J], Scripta Mater, 40(1999), No.10, p.1199.References[9] W.W. Mullins, Grain growh of uniform boundaries with[1] H.V. Akinson, Theories of normal grain growth in purescaling J, Acta Mater. 46( 1998), No.17, p.6219.single phase system [], Acta Metall, 36(1998), No.3,[10] H. Telley, T.M. Liebling, A. Mocellin, The Laguerrep.469. .model of grain growth in two dimensions [J], Philos. Mag.[2] J.A. Glazier, Grain growth in three dimensions depends onB, 73( 1996), No.3, p.395.grain topology 0, Phys. Rev. Lett, 70(1993). No.14.中国煤化工aria, Simulation of curva-p.2170.Fig a modified Monte CarloYHC N M H Grans, 2641995, No.1.3] V.E. Fradkov, ME. Glicksman, M. Palmer, J. Nordbergand K. Rajan, Topological rearrangements during 2D grainp.167.