3D KMC simulation on the precipitation in the annealed ternary alloy system

Transcription

1 3D KM simulatin n the precipitatin in the annealed ternary ally system Xuan Zhang, Mengqi Huang Abstract Kinetic Mnte arl methd is used t study the precipitatin phenmenn in binary and ternary ally system, crrespnding t sme materials we are studying in experiment, e.g. u-w, u-nb, u-nb-w, etc. It is und that in binary system (90 at% A and 10 at% B), huge particles are always bserved. By adding just 1% a third element, the particle size has been well under cntrl, and urther changing the kinetic parameters this element, mainly the pint energy the third element, dierent precipitatin behavirs are bserved. T have better understanding, we use the ive-requency mdel t shw the kinetics. By chsing the value the pint energy the third element t be mst representative the real materials, dierent cmputatinal experiments were perrmed, and their results are in general in gd agreement with experimental results, thus prviding us sme qualitative explanatins the kinetic mechanism. 1. Intrductin Precipitate hardening is widely used t increase the yield strength allys. Hwever, this strengthening mechanism depends n the size the precipitates. The critical radius is typically 5~30 nm [ 1 ]. Thse larger precipitates will bent the dislcatins rather than cutting thrugh them, which decreases the strength the material [2]. nsidered the act that precipitates will grw during heat treatment, nrmal allys will lse their strength at high temperature. Therere, reducing the size the precipitates, especially at high temperature, is desired t imprve the mechanical prperty the allys. u-nb ally (10 at.% Nb) were und t lse the strength because the rmatin large precipitates (40~80 nm) at 600 [3]. In rder t reduce the size the precipitates at high temperature, we added a third element - Tungsten - t the u- Nb binary system t rm a u 88.5 Nb 10 W 1.5 ternary ally. The XRD results in Fig. 1are kind interesting, and it is pssible t get sme imprtant inrmatin abut hw the kinetics is when this material is subjected t therm annealing. Samples were treated in three dierent cnditins: (1) direct annealing at 600 r 2 hurs; (2) irst annealing at 600 r 2hrs, then annealing at 700 r 2 hurs; (3) direct annealing at 700 r 2 hurs. Under cnditin (1), tw peaks besides the u peak are bserved: ne at pure Nb psitin, and the ther is in between the peak psitin Nb and W, which suggests Nb-W ally particles are rmed, and the average particle size and cncentratin can be get rm the halmaximum width and psitin the peak, respectively. Under cnditin (2), tw 1

2 things happened cmpared with cnditin (1), which are, irst, the ally particle peak shit signiicantly t W peak side, and secnd, pure Nb peak grew. Under cnditin (3), nly the ally particle peak is bserved besides the u peak. Since the tw acts that (1) W is miscible with Nb [4], but has a psitive heat mixing with u [5], and (2) W start t be mbile abve 600 in u, we assume that the adding W will act as traps r Nb, which reduces the Nb precipitate size in u matrix, and the mbility Tungsten will enhance this reductin. In rder t prve ur assumptin, we use KM simulatin t study the dynamic evlutin the precipitates in a simpliied ternary ally system. First we will ind ut the prper parameters s that ur simpliied KM cde can qualitatively present real materials; then we will cmpare ur 2-step annealing simulatin results with the XRD spectrums _2hr 600_2hr_700_2hr 700_2hr Nb W u cunts θ Fig. 1. XRD results r dierent annealing cnditins. 2. KM apprach 2.1 KM intrductin Kinetic Mnte arl (KM) is widely used t study diusin-cntrlled phenmena, and its main advantage is that it has a gd cmprmise between the atmic scale mechanism and the macrscpic time scale. Given a system with initial cniguratin, and a set transitin rate between dierent cniguratins, the evlutin the system can be studied [6]. 2

3 2.2 Flwchart 2.3 KM mdel The gal this prject is t build a system that can best describe the u-nb-w ternary ally system. The simulatin bx is set t have 64*64*64= atms in ttal, with a cc crystal structure. There is ne and nly ne vacancy in the bx. In the binary system, we chse 90 at% type A atms and 10 at% type B atms, and in the ternary system, we replace 1 at% A t atms, thus having a A0.89B system. Only irst nearest neighbr interactin is cnsidered. The exchanging rate between vacancy and its neighbrs is determined by the lcal envirnment and the pint energy that atm, i.e. when exchanging happens, the atm and vacancy need irst t break all the bnds they have with the lcal envirnment and then the atm jumps t the pint psitin, then t the rmer vacancy psitin. The energies envlved are energies the bnds ( ε AA, εbb, ε, εab, ε A, s s s ε B, εav, εbv, ε V ) and the pint energies A, B, (it is assumed the same type atms have the same pint energy). The pint energies are set t be input, based n the value r pure A system with just ne vacancy ( eV). The rest the energies are calculated in the cde, based n the chesive energy ch ch ch pure A, B, systems (,, ), the rdering energy between AB, A and B A B 3

4 rd rd rd rm rm rm (,, ), and vacancy rmatin energy in pure A, B, (,, ). AB A B AV BV V ch ch They are set t be input and their values are chsen t be: = = 4.34eV ch (experimental data r pure Ni), = 4.7eV (cnsidering the act that represents rd tungsten, whse bnds are strnger than A and B), = eV [ 7 ] (typical rd rd experimental data r u- system), A = 3 AB = eV, rd B = 0 (cnsidering the act that u and Nb nly have a slightly psitive heat mixing but u and W have very large psitive heat mixing, and that Nb and W is generally rm rm rm negligible since Nb and W are miscible.), AV = BV = V (experimental data r pure u [8], but this value has inally been adapted r the reasn related t the time scale, see sectin 3.1.). All parameter needed in ur cde are listed in Table.1. AB A B rder ch V Binary system A 90 B 10 rder AB = eV ch A ch B = 4.34eV = 4.34eV AV BV S A S B = eV = eV Ternary system A 89 B 10 1 rder A rder AB = eV = eV rder B = 0eV ch A ch B ch = 4.34eV = 4.34eV = 4.70eV AV BV V S A S B S = eV = eV = eV 9.8eV 9.5eV 9.2eV Table. 1. All the energy parameters r KM mdel. 3. Results and discussin 3.1 Real time scale: determinatin V The vacancy rmatin energy determines the relatinship between KM simulatin step and real time. Fr real material, the vacancy cncentratin at thermal equilibrium is abut ~ In ur mdel, we increase it t KM 3 6 X V = 1/ 64 = Accrdingly, mre atm-vacancy exchanges ccur in the mdel, which means ur KM simulatin is much aster than real time experiments. The rati between real time and KM time is a unctin as temperature: real KM t ( T ) X V A = = KM eq t X ( T ) V 4

5 Where X eq V is the vacancy cncentratin at thermal equilibrium: eq ΔGV X V = exp( ) RT Table 2 lists several A values at dierent temperatures, i we assume all the rm rm rm vacancy rmatin energy are the same, e.g. AV = BV = V (The reasn r this assumptin will be discussed belw). Based n these parameters, ur annealing simulatins were run r 4*10 10 KM steps in a temperature range rm 100 t 500. By scaling t the real time scale, the changes the average particle sizes with time r bth binary and ternary systems are pltted in Fig. 2. It is shwn that the ternary system has much smaller precipitates (~10 3 atms) than the binary system (~10 4 atms). The relatin between adding tungsten and the decrease particle size will be discussed mre later. Fig. 2 als illustrated a act that the lwer the annealing temperature, the mre time it will take t rm cmparable size particles as higher temperature, which is just the case in experiment. I we chse a slightly dierent vacancy rmatin energy (e.g. 1.6eV), the new A values will yield much lnger nucleatin time: r 500 t~30h, and t~2700h r 400. mpared with real annealing experiments, rm rm rm = = is a mre reasnable chice r ur mdel. AV BV V T( ) A Table 2. Values the rati A at dierent temperatures (a) (b) <n> Binary A 90 B Ternary A 89 B Real time [s] Real time [s] <n> 5

6 T=300 (c) (d) Fig. 2. (a) and (b) shw average particle size as a unctin real time at dierent temperatures: (a) is r binary system, (b) is r ternary system. (c) and (d) are visualizatin pictures the ally structures r bth systems at T=300. The red dts are Nb atms, the blue nes are W atms. Bth (c) and (d) are pltted rm the inal time step (~10 8 s), when <n> changes with time very slwly. 3.2 Precipitate size vs. time As shwn in Fig. 2, the average precipitate size the ternary ally is much smaller than that binary ally, apprximately ne rder magnitude lwer when the rest the parameters are set t be the same. It is als und that the tw systems llw dierent grwth law the slp Fig. 2(a) is higher than that Fig. 2(b). This phenmenn agrees quite well with what we und in experiment, i.e. the third element behaves like a cntrller t the precipitate grwth. 3.3 :determinatin the mbility atm T prvide a quantitative descriptin the mbility atms, the ive-requency mdel [7] is adpted. The schematic diagram Figure.3 shws the ive dierent requencies r vacancy jump in an ininite dilute slutin i nly irst n. n. is cnsidered. T be simple, w0 is the unperturbed hst atm-vacancy exchange requency, w1 is the requency that vacancy jumping arund the slute atm, w2 is the slute atm-vacancy exchange requency, w3 is the dissciatin requency the slute atm and the vacancy, and w4 is the assciatin requency the slute atm and the vacancy. By calculating the relative energy r each type jump, thse ive requencies are determined by the llwing equatin: wi = υ e 14 Where υ 0 is the base requency which is set t be s i ij bnding. Figure shws the relatin between w2 and = i, j 0 i kt by input, and at 300, assuming nly ne atm in a pure A matrix. Since w2 is the direct -V exchange requency, the lwer the value, the less the mbility. 6

7 Figure. 3. Five-requency mdel r vacancy jumps in the presence a reign atm, shwing (111) plane. [9] w Figure. 4. The requency the exchange between the slute atm and the vacancy as a unctin the pint energy atm. Frm Figure.4 we can see that when = 9.2eV, w2 is s lw that atms are indeed immbile at 300. This trend can als be seen rm Figure.5, which is the visualizatins micrstructures at the same real time r the ur dierent. 7

8 (a) (b) (c) Figure.5. Visualizatin micrstructure precipitates at ur dierent pint energy atm : (a) = eV ; (b) = 9.8eV ; (c) = 9.5eV ; (d) = 9.2eV. In ur experiment, when we irst annealed the sample at 600, tungsten shuld be immbile based n the experimental data rm u-w binary system, and when we annealed the sample at 700, tungsten shuld starts t be mbile. Because ur simulatin des nt mean t reprduce everything in experiment, we just chse = 9.2eV and chse T = 300 t be the cnditin similar t T = 600 in experiment. Further we und ut that at 500 in ur simulatin, atms starts t be mbile, which is kind the case that happened in experiment at 700. These are the simulatin cnditin we use in the llwing part. 3.4 Mdeling experiments Based n the analysis abve, we chse three cnditins as a relectin experimental cnditins: anneal at 300, anneal at 300 then anneal at 500, anneal at 500. Any the temperatures has been maintained r relatively lng enugh time t have distinguishable cniguratins and reach a quasi-steady state, since in the annealing situatin, i waiting lng enugh, the inal equilibrium state is just ne big B particle in A matrix, and this is bviusly nt what we want. Figure.6 shws the visualizatins micrstructures in the three cnditins. Dierent atm distributins and particle size distributins are bserved. (d) 8

9 (a) (b) (c) Figure.6. Visualizatin micrstructures precipitates under three dierent cnditins: (a) directly annealing at 300 ; (b) irst annealing at 300, then annealing at 500 ; (c) directly annealing at 500. annealing temp () Amunt B in matrix Amunt in matrix % 69% <13% <50% <11% <41% Table.3. Distributin B and atms under three dierent cnditins. Annealing temp () n tt <n atm > Table.4. The ttal number precipitates and the average size (the average number atms in particles) under three dierent cnditins. Table.3 shws the atm distributin in each cnditin, and Table.4 shws the ttal number particles and their average size (average number atms) in each cnditin. These data shw us a picture similar t what happened in experiment. At 300, 69% atms are in the matrix, which is an indicatin the immbility, s the A and B atms behave quite like in AB binary system, which can be seen rm the small value ntt and large value <natm>. I urther annealing at 500, there is a signiicant reductin (mre than 19%) number atms in the matrix, while a simultaneus increase B atms in the matrix. Therere, the rati number B atms in particles and number atms in particles decreases, i.e. mre in particles, which gives the shit the XRD ally particle peak t tungsten side in experiment. And since there are mre B atms in the matrix, the number B 9

10 atms that can g t surace increases, which gives the increase in intensity Nb peak in XRD prile. I ging directly t 500, since atms start t be mbile, they behave as strng traps r B atms, thus giving a sharp reduce in particle size, which is als shwn bviusly in XRD prile, i.e. the hal-maximum width ally particle peak annealing at 700 is much smaller than that either annealing at 600 r annealing irst at 600 then nclusins Althugh ur KM mdel is quite simple, the results we have are qualitatively in gd agreement with experimental bservatins, prviding us with sme insights the atmic scale mechanism. The precipitatin size in binary allys AB will increase drastically withut limitatin. Just by adding 1 at% a third element, which is much less mbile at all temperatures cmpared with the ther slute element B and has a very high heat mixing with the matrix element A, the precipitatin size has been reduced signiicantly and under cntrl. The mbility is the key pint t the precipitatin prcess. In general the mre immbile is, the smaller the average size is. But i at a temperature when is indeed immbile, the material will behave mre like binary material, and large precipitates will be rmed. The mbility is mstly determined by its pint energy. One particular pint energy is chsen t make a representatin W in the u-nb-w ally, and the experiments were re-perrmed in simulatin. Again the mbility shwed its imprtance. T make ur explanatin mre cnvincing, mre details need t be cnsidered, and mre supprtive data are required. Reerence [1]. Hrnbgen, Jurnal Light Metals, 1 (2001) [2] [3]. Btcharva, Jurnal Allys and mpunds, 365 (2004) [4] Y. V. Lakhtkin, J. Phys. IV France 05 (1995) [5]. S. Xing, Nanstructured materials, 5 (1995) [6] P. Belln: Kinetic Mnte arl Simulatins in rystalline Allys: Principles and Selected Applicatins [7] J. Russel, P. Belln, Phys. Rev. B., 63, (2001) [8] W. Tritshauser, Appl. Phys., 6 (1975) [9] J. Philibert, Atm mvements: diusin and mass transprt in slids, Mngraph de Physique F-91944, France