NUCLEATION AND SPINODAL DECOMPOSITION IN TERNARY-COMPONENT ALLOYS

Transcription

1 NUCLEATION AND SPINODAL DECOMPOSITION IN TERNARY-COMPONENT ALLOYS COLLEEN ACKERMANN AND WILL HARDESTY Abstract. The Cahn-Morral System has often been sed to model the dynamics of phase separation in mlti-component alloys on large domains. In this paper e examine phase separation on small one-dimensional domains time independently. In particlar e se AUTO to create bifrcation diagrams of eqilibrim soltions for to different nonlinearities and se Matlab to obsere the strctre of the material at arios points on the diagrams. We compare the reslts to determine if sing different nonlinearities significantly affects the behaior of the Cahn-Morral System. Date: Jly 3, 29. Dr. Thomas Wanner and Dr. Eelyn Sander.

2 NUCLEATION IN ALLOYS Figre. The processes of ncleation and spinodal decomposition on to dimensional domains are depicted in these plots. The left plot is an example of a ncleation pattern and the right plot demonstrates spinodal decomposition. Introdction Alloys are composite materials hich are formed by mixing a nmber of pre metals together at a high temperatre and then rapidly qenching or cooling the mixtre to form a solid. Dring the process of qenching, the components ndergo a phase separation in hich they begin to form patterns. These pattern formations can be diided into to classes: ncleation and spinodal decomposition. Qalitatiely ncleation occrs hen the indiidal components begin to materialize from the homogeneos mixtre as isolated droplets or droplets. Spinodal decomposition occrs hen the components form connected snakelike patterns. This behaior is modelled mathematically by the Cahn-Morral system. 2. Backgrond and Research Methods 2.. Cahn-Morral system. The Cahn-Morral system is gien by, () t = (ε 2 + f( )) on Ω ν = = on Ω ν

3 2 C. ACKERMANN AND W. HARDESTY here R 3. The energy of the system is modelled by the Van der Waals free energy fnctional, E ε [ ] = Ω ( ) ε F ( ) dx here R 3. The term f is defined as the deriatie of the doble-ell potential, F from the free energy fnctional. The domain Ω R = [, ], has Nemann bondary conditions imposed on the edges here F : R 3 R and f : R 3 R Gibbs Simplex. The Gibbs simplex is defined as: G = {(,, ) R 3 : + + =,,, }. here G t. The Gibbs simplex also represents the set of all possible states or aerage mass concentrations of the ternary alloy, hich follos from the conseration of mass. G can be diided into regions corresponding to ncleation and spinodal decomposition, hich are determined throgh linearization analysis. Gien a state (ū,, ) G, the stability of that particlar state is gien by compting the eigenales of J f (ū,, ), here J f is the Jacobian of f() [4]. The state (ū,, ) is in the ncleation region if J f has no positie eigenales. Otherise it lies in the spinodal region [2]. These regions can be depicted graphically ith the Gibbs Triangle, here each color represents a different region The nonlinearities. In or research e sed to nonlinearities; a qadratic nonlinearity: and a logarithmic nonlinearity: F (,, ) = ( )( 2 ) 4 F (,, ) = 3.5( + + ) + ln + ln + ln. The nonlinearities ere chosen so that they ere triple ell potentials and symmetric. Symmetric means that each component contribtes to the nonlinearity in the same ay, making inestigations simpler.

4 NUCLEATION IN ALLOYS 3 Figre 2. The Gibbs triangle for the qadratic nonlinearity is shon on the left, and the triangle for the logarithmic nonlinearity is depicted on the right Both of the nonlinearities hae been inestigated preiosly in both the one and to dimensional cases [3],[4],[2]. Hoeer, most of the preios research as done ith time ariation. Or research differs in that e are looking at soltions at one moment in time and comparing reslts from different nonlinearities. Also in or case e are sing mch smaller domains so that e can stdy the behaior of indiidal droplets. This differs from past research hich dealt ith stdying the behaior of grops of droplets on mch larger domains. 3. Reslts and Discssion 3.. Gibbs Triangle. For each of or nonlinearities e created Matlab code hich shos hich regions on the Gibbs simplex hae zero, one or to positie eigenales. In the pictres belo the Gibbs simplex is projected onto the plane. The red area represents the ncleation region here there are no positie eigenales. The light ble and dark ble represent the regions here there are one and to positie eigenales respectiely, these to areas are considered part of the spinodal region Path folloing. Or ltimate goal in path folloing as to trace paths in the parameter space of the ncleation region. Hoeer, the nontriial branches of eqilibria in the ncleation region are not connected to the triial branch. To reach the non-triial ncleation branches e had to begin in the spinodal region.

5 4 C. ACKERMANN AND W. HARDESTY Bifrcation diagram for: cmdqadrn3e Second primary branch for log-nl 5 L2-norm L2-norm Mass Concentration Mass Concentration Parameter Parameter Figre 3. This figre displays the bifrcation diagrams for the λ rns along the second primary branch for both nonlinearities. The diagram for the qadratic nonlinearity is depicted on the left and the logarithmic nonlinearity is shon on the right Bifrcation diagram for: logcmd2a3bottomalpha33 Bifrcation diagram for: alpharn 5 L2-norm L2-norm 5 Mass Concentration Mass Concentration Parameter Parameter Figre 4. This figre compares the alpha rns from both nonlinearities; the plot on the left is from the qadratic nonlinearity and the plot on the right is from the logarithmic nonlinearity Bifrcation diagram for: cmdqadbeta6267c Bifrcation diagram for: logcmd2a3bottombeta L2-norm L2-norm Mass Concentration Mass Concentration Parameter Parameter Figre 5. This figre compares to sccessfl β rns from inside the ncleation region for both nonlinearities; the plot on the left depicts the β rn for the qadratic nonlinearity here λ = 8, α.9344 and the plot on the right shos the β rn for the logarithmic nonlinearity ith λ = 45 and α 2..7

6 NUCLEATION IN ALLOYS 5 UZ at Label 49 ith lambda =., alpha = 5 and beta =. UZ at Label 43 ith lambda = 2., alpha = 5 and beta = UZ at Label 443 ith lambda = 33., alpha = 5 and beta =. UZ at Label 479 ith lambda = 77., alpha = 5 and beta = Figre 6. This figre depicts the change in droplet formation for the qadratic nonlinearity, as λ increases along the secondary branch of figre 3. This is de to the fact that since the ncleation region is stable, it isn t possible to branch ot. First e aried λ = /ε 2. To get onto the second primary branch in the spinodal region. After folloing the second primary branch and its bifrcations e attempted to follo in α = (ū + )/2 back into the ncleation region. Finally e aried β = (ū )/2 in the ncleation region. We ere interested in the second primary branch since symmetry of the soltions along that branch ensred that e cold get centralized bbble formations. The primary branches occr at the points λ = (nπ) 2 /µ here n N and µ is the positie eigenale of J f (). The goal as to follo this branch to a sfficiently large ale of λ so that hen that ale is fixed and α is aried back from the primary branch and / or any secondary branch that it ill become small enogh to reach the ncleation region.

7 6 C. ACKERMANN AND W. HARDESTY UZ at Label 24 ith lambda = 5., alpha = 5 and beta =. EP at Label 62 ith lambda = 5, alpha = 5 and beta = UZ at Label 89 ith lambda = 5., alpha = 5 and beta =. UZ at Label 7 ith lambda = 5., alpha = 5 and beta = Figre 7. This figre depicts the change in droplet formation for the logarithmic nonlinearity, as λ increases along the secondary branch of figre 3. As shon in figre 3, the primary branch for the qadratic nonlinearity as folloed ot to λ 8 here only one secondary branch as detected by AUTO. To reach the ncleation region the λ ale as fixed at 8, here the α rn originating from the secondary branch reached sfficiently small ales. Unfortnately, the α rn originating from the primary branch as nable to reach a small enogh ale to enter the ncleation region. The α rn hich did enter the ncleation region prodced seeral points from hich a β arying simlation cold be rn; it trned ot that all bt one of these points reslted in sprios rns. The plot on the left in Figre 5 displays the bifrcation diagram prodced from the aforementioned β rn. For the logarithmic nonlinearity, the first nontriial branch as easy to find and there ere also seeral bifrcations originating from it. Hoeer, the droplet

8 NUCLEATION IN ALLOYS 7.9 LP at Label 6267 ith lambda = 8., alpha = and beta = Figre 8. This figre depicts the droplet formation occrring at the label from hich the β rn for the qadratic nonlinearity depicted in figre 5 originates. diagrams ere not ery interesting, becase the aerage concentration of the and components ere alays exactly the same. Changing β from to. yielded doble and single edge droplets, since the ū and concentrations ere different. The second branch as mch more difficlt to get onto than the first branch. We cold not get onto the second branch ith either β = or β =.. Therefore e decided to try forcing the symmetry by setting (x) = ( x), hich redced the complexity of the problem by eliminating a component from the ector. This alloed s to add more modes to the discrete cosine transform in or spectral code ithot sing significantly more compting poer, hich in theory shold hae increased the accracy of or reslts. Unfortnately hen sing the ne symmetric code the second branch still did not appear.

9 8 C. ACKERMANN AND W. HARDESTY.9 UZ at Label 59 ith lambda = 45., alpha = and beta = Figre 9. This figre depicts the droplet formation occrring at the label from hich the β rn for the logarithmic nonlinearity, depicted in figre 5 originates. We then changed the step size from 8 to 4 and the symmetric code as able to go onto the second branch. To determine if the symmetric code played any part in the transition to the second branch e tried sing or original code again ith the ne step size. Once again e sccessflly moed onto the second branch. This probably orked becase no AUTO as getting far enogh onto the second branch in the first step that it old not accidentally conerge to the original branch on the second step. Next e proceeded to start folloing the second branch and all of its bifrcations. We fond many other branches, bt the shape of or diagram appeared to indicate that e ere missing a pitchfork bifrcation. At the time e ere sing the nonsymmetric code ith β =. hich cold hae cased the branch that seemed to be missing to be slightly separated from the other branches. We decided to go

10 NUCLEATION IN ALLOYS 9 RG at Label 6393 ith lambda = 8., alpha = and beta = RG at Label 6396 ith lambda = 8., alpha = and beta = RG at Label 645 ith lambda = 8., alpha = and beta = MX at Label 6437 ith lambda = 8., alpha = and beta = Figre. This figre depicts the change in droplet formation for the qadratic nonlinearity, as β increases along the path in figre 5. back to sing the code ith β =, hich alloed s to find not only that branch, bt seeral other branches ot to λ = 5. Finally e started trying to follo in α to get into the ncleation region. Beginning ith λ = 45 e ere able to follo back to α = 2 from the branch ith the loest L 2 norm. This as in the ncleation region. We tried to trace back into the ncleation region at λ = 45 from a fe of the other branches, bt AUTO either started jmping beteen branches or simply intersected another branch and started moing aay from the ncleation region. On or one sccessfl rn into the ncleation region e obtained α = 2 at for different points. We tried to ary in β starting at each point. For the first three e only receied end points from AUTO. We ere able to follo the forth

11 C. ACKERMANN AND W. HARDESTY RG at Label 67 ith lambda = 45., alpha = and beta = e-6 RG at Label 69 ith lambda = 45., alpha = and beta = RG at Label 7 ith lambda = 45., alpha = and beta = BP at Label 75 ith lambda = 45., alpha = and beta = Figre. This figre depicts the change in droplet formation for the logarithmic nonlinearity, as β increases along the path depicted in figre 5. point ith an L 2 norm of 35 and β =.6265 briefly before AUTO started finding constant bifrcation points. Figres 3, 4 and 5 compare and contrast the bifrcation diagrams reslting from the λ, α and β rns respectiely, for both nonlinearities. Interestingly, the diagrams depicting the α rns for both non-linearities are ery different hile the diagrams prodced from the β rns are almost identical. This sggests, at least qalitatiely, that hen in the spinodal region, the to nonlinearities behae ery differently bt once inside the ncleation region they behae more similarly. If similar reslts are obtained from ftre research then this old imply that from a materials science perspectie, that the behaior of the Cahn-Morral system is not affected by the change in nonlinearity.

12 NUCLEATION IN ALLOYS 4. Ftre Work We old like to sccessflly trace α back into the ncleation region from seeral branches at the same λ ale and compare droplet formations. We hope to eentally find hich droplet formations are possible at each ale of λ for each nonlinearity. Or third partner James O Beirne ill be inestigating a sixth degree polynomial nonlinearity and comparing the reslts ith or inestigations. Eentally the research shold be expanded to to and three dimensions to see if reslts ary significantly. Also more than three components ill be sed. Both of these extensions, thogh reqiring more compting poer, ill make the model considerably more reliable. We hope that ftre research ill gather enogh data to either disproe or spport the assertion presented in this paper that the once inside the ncleation region, the system behaes identically, despite haing different nonlinearities. Perhaps een a rigoros mathematical argment that explains the impact of changing nonlinearities. 5. Acknoledgements The athors ish to thank Dr. Thomas Wanner, Dr. Eelyn Sander, James O Beirne, the Department of Defense, the National Science Fondation, and the Department of Mathematical Sciences at George Mason Uniersity. References [] Kathleen T. Alligood, Tim D. Saer, and James A. Yorke, Chaos, An Introdction to Dynamical Systems, Springer- Verlag Ne York, Inc., Ne York, Ne York, 997. [2] Jonathan P. Desi, Hanein H. Edrees, Joseph J. Price, Eelyn Sander, and Thomas Wanner, The Dynamics of Ncleation in Stochastic Cahn-Morral Systems, in preperation (29). [3] Jnseok Kim and Kyngken Kang, A nmerical method for the ternary Cahn-Hilliard system ith a degenerate mobility, Applied Nmerical Mathematics 59 (29), [4] Stanislas Maier-Paape, Barbara Stoth, and Thomas Wanner, Spinodal Decomposition for Mlticomponent Cahn-Hilliard Systems, Jornal of Statistical Physics 98 (999), [5] Rüdiger Seydel, From Eqilibrim to Chaos: Practical Bifrcation and Stability Analysis, Elseier Science Pblishing Co., Inc, 52 Vanderbilt Aene, Ne York, Ne York, 988.

13 2 C. ACKERMANN AND W. HARDESTY Department of Mathematics, Virgina Tech Department of Mathematics and Statistics, Uniersity of Maryland Baltimore Conty address: address: