Accurate sintering morphologies from Monte Carlo modelling

Transcription

1 Computtionl Methods nd Experiments in Mteril Chrcteristion II 259 Accurte sintering morphologies from Monte Crlo modelling S. Bordère 1, D. Gendron 2, J.-M. Heintz 1 & D. Bernrd 1 1 Institut de chimie de l Mtière Condensée de Bordeux, CNRS, Frnce 2 CERMEP, Frnce Astrct Through the two-dimensionl prolems of the sintering of n infinite row nd n infinite squre rrngement of mono-sized prticles, the relevnce of the Monte Crlo methodology sed on the non-discrete energeticl model is pointed out to clculte equilirium morphologies. In the sence of corsening, nd for the two cses of free nd constrined sintering, the clculted stle morphologies re shown to precisely superimpose tht otined from nlyticl resolution. Therefore, n ccurte drwing of surfce curvtures nd dihedrl ngles results from tht methodology which is one of the required conditions to del with more complex microstructures nd to e le to compre the clculted microstructures with experiments. Keywords: sintering, Monte Crlo simultion, interfces, microstructure. 1 Introduction Grin scle modelling of sintering processes is still the oject of mny numericl developments in order to del with complex microstructures close to rel systems. Two min numericl pproches re distinguishle depending on the stochstic or deterministic wy to solve the prolem. In the field of deterministic methods, the lst developments sed on the vritionl principle nd the finite element formultion re demonstrted to e efficient to simulte microstructurl evolution in polycrystlline mterils involving surfce nd grin oundry diffusion [1]. The phse field modelling is lso very powerful pproch to del with grin growth nd non-densifying sintering processes, producing simulted microstructure imges indistinguishle from rel microgrphs [2]. In contrst to the deterministic pproch, stochstic modelling minimises n energeticl

2 260 Computtionl Methods nd Experiments in Mteril Chrcteristion II potentil using Metropolis Monte Crlo lgorithm. Recent ppers hve shown the interest of tht pproch to del with complex microstructures [3], even in 3D [4]. In these pulictions irregulr interfce morphologies nd inccurte contct ngle re oserved. This is minly due to the comined ction of the dicretistion method nd the Potts energeticl model, which do tke into ccount unphysicl fluctutions of the system. In response to these limittions, different Monte Crlo methodology hs een developed [5]. In this pper, we im to demonstrte tht this methodology ccurtely simultes sintering microstructures for etter nlysis of experimentl microstructures. 2 Simultion method 2.1 Energetics nd discretistion The energetics of the non discrete Monte Crlo methodology is defined in the isotherml isoric system y the potentil φ, φ = γ s A s + γ g A g + E coh + PV (1) Here, γ s nd γ g re the surfce nd grin oundry energies respectively, A s nd A g their respective surfce res, E coh is the cohesive energy of the mteril, P the externl pressure nd V the totl volume of the system. In the present clcultions P is considered equl to zero. To model the sintering of different cylinder rrngements, the 2D version of the code ws used in which only the section of the cylinders is discretised. In fig. 1 n exmple of tringulr mesh used for clcultion is presented. The mesh is utomticlly redpted during clcultion to suppress element degenercy induced y morphologicl chnges. To model the sintering of n infinite row of prticles only two prticles were considered ssuming periodic oundry conditions t the order (fig. 1). The distnce L etween the oundries is chrcteristic length of the system, which ws fixed for constrined sintering nd ws vrile for free sintering. For the cse of n infinite squre rrngement of prticles, the clcultion unit hs een reduced to one prticle ssuming periodic oundry conditions in the two directions (Fig. 1). Only constrined sintering ws studied for the twodimensionl rrngement so tht the length L etween the oundries in the X,Y-directions ws fixed for equilirium morphology clcultion. 2.2 Monte Crlo minimistion The φ-potentil is minimised using the Metropolis lgorithm, which consists in rndom chnges of the prticle configurtion in order to induce potentil vritions φ of the system. The configurtion chnges re ccepted with the proility P( φ)=min(1,exp( φ/k B T), where k B is the Boltzmnn constnt nd T the sintering temperture. The physicl constnts considered for the clcultion nd the minimistion of the φ-potentil re presented in tle 1.

3 Computtionl Methods nd Experiments in Mteril Chrcteristion II 261 Periodic oundry conditions Visulistion unit for pore morphology L=4 Y X Figure 1: Tle 1: L=4 Clcultion unit L=4 Clcultion unit with periodic oundry conditions Initil configurtion unit for the clcultion of the sintering equilirium morphology of ) n infinite row of cylinders nd ) n infinite squred rrngement of cylinders. Visulistion within the cylinder section of the tringulr mesh considered for clcultion. Physicl constnts considered for the clcultion nd the Monte Crlo minimistion of the φ-potentil, eqn. (1). Physicl constnts vlue Young s Modulus, E Y (GP) 4 Poisson s coefficient, ν 0.26 Cylinder rdius, (µm) 0.5 Surfce tension, γ sv (J/m 2 ) 1.2 Grin oundry tension, γ g =R γ.γ s (J/m 2 ) R γ =1.73, 1.00, 0.52, 0.00 Externl pressure, P (P) 0 Temperture, T (K) 1500 Two types of rndom chnges re considered which re repeted numerous times for complete minimistion: - the first one concerns the configurtion chnges y rndom displcements of the mesh nodes within the section plne. In the peculir cse of constrined sintering, the nodes locted on the periodic oundry re constrined to move on it. The φ-vrition induced y the node displcements includes the cohesive energy vrition, the surfce energy vrition when the selected node is locted on the surfce nd the grin oundry energy vrition when the selected node is locted on the grin oundry. The cohesive energy vrition of the mesh tringles is clculted in the frmework of elstic strin energy vrition depending on the Young s modulus E Y nd Poisson s coefficient ν. The node displcement route enles the shpe chnge of the system nd the clcultion of the hydrosttic nd sher stress grdients, which re the elstic mteril response to tht shpe chnge.

4 262 Computtionl Methods nd Experiments in Mteril Chrcteristion II - the second one concerns the configurtion chnges y mss trnsports from tringle to one of its neighours. The induced φ-vrition only includes the cohesive energy vrition. As in the previous route the cohesive energy vrition is converted into elstic strin energy vrition. This second route enles the sttisticl clcultion of the mss trnsport in reltion to the stress grdient. Tht methodology for potentil minimistion hs een completely descried in [5]. The periodic oundry conditions imply tht the dimension of the system hs to e defined y chrcteristic length L. In the peculir cse treted here of free sintering of n infinite row of prticles, this length needs to vry in order to model prticle shrinkge. So, in tht cse third type of rndom chnges is considered. It concerns the rndom chnge of the system length L. Here rndom vrition D X of the system length in the X-direction is chosen. The new system length is L =L+D X which implies tht the new X-co-ordintes of the entire mesh nodes re x =x+x(l -L)/L. Tht system deformtion is quntified through the vrition of the elstic strin tensor ε of ll the mesh tringles (ε 11 =(L -L)/L), ε 12 =ε 21 =ε 22 =0) llowing the clcultion of the φ-vrition. 3 Clcultion of equilirium sintering morphologies for different prticles systems The minimistion of the φ-potentil eqn. (1) leds to morphologicl evolution of the prticles system until the stle end-stte. This evolution of the system cn e followed through tht of the surfce nd grin oundry energies (fig. 2). 1 E Energy / [γ sv (4π) ] 0.5 E g Monte Crlo Step Figure 2: Evolution for the infinite row of prticle system of the normlised surfce energy E s nd grin oundry energy E g ssuming grin oundry to surfce energy rtio R γ =0. The lrge plteu otined for oth curves with converging vlue lower thn 10-5 ensures tht the system is in its equilirium stte. The Monte Crlo equilirium sintering morphologies presented in fig. 3 nd fig. 4 respectively for

5 Computtionl Methods nd Experiments in Mteril Chrcteristion II 263 the infinite row of prticles nd the infinite squre rrngement of prticles hve een otined from tht criterion L=4 L=4 L=4 ψ e =62 ψ e =120.5 ψ e =150 L=1.8 L=3.6 L=2.5 ψ e =60.5 ψ e =120 ψ e =149 Figure 3: Visulistion of the end-stte morphologies clculted from Monte Crlo minimistion of n infinite row of cylindricl prticles for constrined sintering (line ) nd for free sintering (line ). Different grin oundry to surfce energy rtio hs een considered: 1) R γ =1.73, 2) R γ =1.00, 3) R γ =0.52. The vlue of the contct ngle ψ e is indicted on ech figure. ψ e =60.5 ψ e =120 ψ e =150 ψ e =180 c d Figure 4: Visulistion for constrined sintering of the pore morphology clculted for different grin oundry to surfce energy rtio: ) R γ =1.73, ) Rγ =1.00, c) R γ =0.52, d) R γ =0. The vlues of the clculted equilirium contct ngles ψ e re indicted on ech configurtion. Our im now is to demonstrte how these clculted morphologies ccurtely fit the exct sintering morphologies. First, we hve compred the vlues of

6 264 Computtionl Methods nd Experiments in Mteril Chrcteristion II dihedrl ngles ψ e t ech three-interfce junction with the exct ones, which cn e determined from the thermodynmic reltion eqn. 2, R γ g ψ e = 2cos( ) (2) γ 2 γ = s The result is tht the men vlues mesured from the Monte Crlo morphologies (fig. 3 nd fig. 4) re very close to the exct equilirium contct ngles: ψ e =60 for R γ =1.73, ψ e =120 for R γ =1.00, ψ e =150 for R γ =0.52 nd ψ e =180 for R γ =0.00. The smll discrepncy otined (less tht 3%) is minly due to the segment discretistion of the surfce. 4 Comprison of the numericl sintering morphologies with nlyticl morphologies First of ll, we hve to point out tht the minimistion is chieved t constnt pressure nd temperture vlues, which implies tht the volume of the system cn fluctute. These volume fluctutions re consistent with physics since rel physicl constnts re introduced to chrcterise the potentil (tle 1). They re not visile t the scle of the surfce energies plotted in fig. 2, nd consequently cnnot e discernile for prticle morphologies, s it cn e seen from fig. 3 nd fig. 4. A fine drwing of the prticle contour is then otined minly due to the tringle mesh discretistion. Moreover, since the elsticity of the prticles re quntified through the cohesive energy eqn. (1), the volume of the prticle slightly increses (less tht 0.05%) during the stilistion of the system in response to the elstic strin-energy relxtion induced y the modifiction of the surfce curvture. Physicl constnts different from those of tle 1 would led to different volume increses. Nevertheless, for physicl constnts representtive of rel mterils systems, this volume is lwys negligile, so tht the equilirium morphologies otined cn e considered to e only dependnt of the surfce to grin oundry rtion R γ. For tht reson, the Monte Crlo morphology cn e compred to the exct morphology, which cn e otined nlyticlly, ssuming constnt volume for the system. The solution otined within this hypothesis from the minimistion of the totl surfce nd grin oundry energy will e detiled in next pper. The resolution procedure is the sme thn tht developed y Cnnon nd Crter for the 3D system constituted of n infinite row of sphericl prticles [6]. The exct morphologies otined for different R γ rtio re drwn in fig. 5 in full line for constrined sintering nd in dshed line for free sintering. Only the qurter of prticle is represented since perfect symmetry is otined for the entire clcultion unit. The Monte Crlo morphologies re lso drwn in fig. 5. The full nd open circles re reltive to constrined nd free sintering respectively. These circles correspond to the nodes of the discretised prticle surfce. The cler superimposition of the numericl prticle drwing with the nlyticl one demonstrtes how ccurte is the present Monte Crlo methodology in otining equilirium sintering morphology. Tht

7 Computtionl Methods nd Experiments in Mteril Chrcteristion II 265 result llows us to consider tht the pore morphologies clculted for the squre rrngement of prticles (fig. 4) re certinly very close to exctness y/2 0.5 y/2 0.5 y/ Figure 5: x/ x/ c Comprison of the Monte Crlo stle morphologies of n infinite row of cylinders with exct morphologies, for different grin oundry to surfce energy rtio: ) R γ =1.73, ) R γ =1.00, c) R γ =0.52. Anlyticl solution for constrined sintering ( ), free sintering ( ), Monte Crlo solution for constrined sintering ( ), nd free sintering ( ). 5 Comprison of clculted sintering morphologies with experiments The interest of tht methodology is to del with more complex microstructures close to rel systems without ny dditionl numericl difficulties. To illustrte tht point, we hve modelled the free sintering of n hexgonl rrngement of cylinders with lrge pore inside (fig. 6) nd the sintering of finite squre rrngement of prticles deposited on n identicl mteril sustrte (fig. 7). The vlue R γ =0 considered for surfce to grin oundry rtio leds for constrined sintering to circulr pore t the equilirium stte fig. 4d. For free sintering, the equilirium corresponds to the disppernce of the pore leding to full densifiction. So the configurtions presented in fig. 6 nd fig. 7 correspond to intermedite stges of the sintering process. We cn nevertheless notice from fig. 7 tht some pores re circulr, showing locl qusi-equilirium stte. The simulted microstructures re compred to experimentl ones otined from the sintering of glss mterils for which R γ is close to zero. First, for the hexgonl rrngement system, sintering experiment hs een performed t 950 C on Pyrex glss cylinders (fig. 6). Thus, strict comprison cn e done etween numericl nd experimentl microstructures, which revels gret similrity for the pore nd interfce morphologies. Then, for the other system (fig. 7), experiment hs een performed on sod-lime glss prticles (Sovitec) of sphericl shpe nd glss sustrte. Even if strict comprison x/2

8 266 Computtionl Methods nd Experiments in Mteril Chrcteristion II cnnot e done etween the 2D simultion of sintering nd the 2D chrcteristion of 3D sintering, we cn nevertheless conclude tht the curvture of the prticle/sustrte interfce hs een well simulted s well s the shpe of the pore ner the sustrte. Figure 6: Comprison for n hexgonl cylinders rrngements with lrge pore of ) the Monte Crlo intermedite stge microstructure showing hydrosttic stress (σ h ) grdient from tensile vlue σ h. / γ s = 1 (light grey) to compressive vlue σ h. / γ s = 1 (drk grey) nd ) the experimentl microstructure of n equivlent glss cylinder system sintered à 950 C. 60 µm Figure 7: Free sintering of mny prticle system deposited on sustrte. ) Monte Crlo intermedite stge morphology exhiiting mss trnsport fluxes; ) Experimentl imge otined from scnning electron microscopy (ck-scttered electrons). The morphologies of the pore nd the prticle/sustrte interfce re very close from one figure to the other.

9 6 Conclusion Through the modelling of different 2D microstructures, the effectiveness of the here descried Monte Crlo methodology hs een demonstrted in otining ccurte sintering morphologies. Actully, the simulted equilirium morphologies strictly superimpose to the exct ones otined from nlyticl resolution. A regulr drwing of the interfces, ccurte contct ngles nd pore shpe re then simulted. This is n improvement compred to the Potts model modelling. It is minly due to the non-rigid discretistion nd to the physicl description of the interfce energy llowing good quntifiction of the fluctutions. The comprison of simulted microstructures with rel microstructures hs lso shown how relistic tht simulted results re. Acknowledgment We grtefully cknowledge computtionl fcilities provided y the intensive clcultion pole M3PEC-MESOCENTRE of the University of Bordeux I- DRIMM, prtly finnced y the regionl Council of Aquitine. References Computtionl Methods nd Experiments in Mteril Chrcteristion II 267 [1] Ch ng, H.N. &Pn, J., Modelling Microstructurl evolution of Porous Polycrystlline Mterils nd Numericl Study of Anisotropic Sintering. Journl of Computtionl Physics, 204(2), pp , [2] Wng, Y., Liu, Y., Cionu, C. & Ptton, B.R., Simulting Microstructurl Evolution nd Electricl Trnsport in Cermic Gs Sensors. Journl of Americn Cermic Society, 83(9), pp , [3] Brginsky, M., Tikre, V. & Olevsky, E., Numericl simultion of solid stte sintering. Interntionl Journl of Solid stte sintering, 42, pp , [4] Mtsur, H., Computer Simultion on Sintering Grin Growth. Journl of the Cermic Society of Jpn, 113(4), pp , [5] Bordère, S., Originl Monte Crlo Methodology devoted to the study of sintering processes.. Journl of Americn Cermic Society, 85(7), pp , [6] Cnnon, R.M. & Crter, W.C., Interply of sintering microstructures, driving forces, nd mss trnsport mechnism.. Journl of Americn Cermic Society, 72(8), pp , 1989.