Ferroelectrics 342:73-82, 2006 Computational Modeling of Ferromagnetic Shape Memory Thin Films

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1 Frrolctrics 4:7-8 6 Computational Modling of Frromagntic Shap Mmory Thin Films J. Liakhova M. Luskin and T. Zhang School of Mathmatics 6 Church St. SE Univrsity of Minnsota Minnapolis MN USA luskin@umn.du School of Mathmatics 6 Church St. SE Univrsity of Minnsota Minnapolis MN USA W propos a mathmatical modl for th dflction of singl crystal films of Ni MnGa frromagntic shap mmory alloys in rspons to th application of a magntic fild. W thn prsnt th rsults of numrical simulations obtaind from th finit lmnt approximation of this modl to study th dflction of th film du to th application of a magntic fild. Kywords: frromagntic shap mmory activ thin film computational modling INTRODUCTION Th Ni MnGa frromagntic shap mmory (FSM) alloy undrgos a cubic to ttragonal structural phas transformation that xhibits th shap mmory ffct and it is also frromagntic at tmpraturs blow its structural martnsitic transformation tmpratur []. Bcaus th thr ttragonal variants of martnsit hav orthogonal asy axs of magntization and th typical rotations of variants ar small in a compatibl microstructur a rvrsibl shap chang can b producd by applying appropriat magntic filds to rarrang th variants []. Thory indicats that singl crystal films will hav significant advantags ovr polycrystallin films for producing motion [] and singl crystal films of Ni MnGa that xhibit th frromagntic shap mmory ffct hav rcntly bn grown []. It has also bn dmonstratd that th transformation tmpratur of singl crystal films of Ni MnGa can b raisd abov room tmpratur by slightly varying th composition of Ni MnGa from its stoichiomtric composition []. W propos a modl for th dformation of singl crystal frromagntic shap mmory films in rspons to th application of a magntic fild. W us this gnral modl to dvlop a modl for singl crystal Ni MnGa films at tmpraturs blow its structural transformation tmpratur and w thn prsnt th rsults of numrical simulations obtaind from th finit lmnt approximation of this modl to study th dflction of th film by th application of a magntic fild.

2 BULK MODEL FOR FERROMAGNETIC SHAPE MEMORY CRYSTALS W dnot th rfrnc crystal domain for a frromagntic shap mmory crystal with thicknss << by Ω = {( x x x):( x x) S x } whr S R is a two dimnsional rgion (s Figur ). Dformations of th crystal will b dnotd by ux ( ): Ω R th dformd crystal by u( Ω ) th magntization of th crystal by mz ( ): u( Ω ) R th scalar potntial of th associatd magntic fild by ζ ( z):r R th applid magntic fild by hz ( ):R R and th tmpratur by θ. W not that th dformation uxis ( ) naturally dfind in matrial coordinats; whil th magntization mz ( ) scalar potntial ζ ( z) and applid magntic fild hzar ( ) naturally dfind in spatial coordinats. Figur. Rfrnc crystal domain Ω and dformd crystal domain u ( ) Ω. Th fr nrgy of a frromagntic shap mmory crystal with dformation ux ( ): Ω R and magntization mz ( ): u( Ω ) R can b modld by [] (.) { } ( ) = κ ( ) + ϕ( ( ) ( ( )) θ) ( ) um Dux ux mux dx Ω Ω u( Ω ) { µ z } ( ) mag + mz ( ) hz ( ) mz ( ) dz+ ( um ) { κ Dux ( ) ϕ( ux ( ) mux ( ( )) θ) } dx { µ z ( ) } Ω = + ( ) mag + mux ( ) hux ( ( )) mux ( ( )) dt uxdx ( ) + ( um ) whr κ is th surfac nrgy cofficint ϕ is th sum of th lastic and anisotropic fr nrgy dnsitis µ is th xchang nrgy cofficint and by ( ) mag ( u m) = ζ ( z) dz R with ( ζ + m) = for z R ( ) ( u m) is th magntostatic nrgy givn mag

3 whr th magntization m has bn xtndd by zro outsid of th crystal domain u( Ω ). Th magntization satisfis th magntic saturation condition mux ( ( )) dt ux ( ) = ms ( θ ) for x Ω whr th saturation magntization ms ( θ ) > for tmpraturs θ blow th structural transformation tmpratur. Howvr w will us th simplr condition (.) mux ( ( )) = ms ( θ ) for x Ω without significant loss of accuracy. Th trms in th fr nrgy (.) rprsnt from lft to right th surfac nrgy th combind lastic and magntic anisotropy nrgy th xchang nrgy th intraction nrgy du to th applid magntic fild and th magntostatic nrgy. W will assum that th film is rlasd from th substrat and fr on its top and bottom S but that it is attachd to th substrat on th dgs of th film latral surfacs { } S ( ) whr S dnots th boundary of th rfrnc domain rgion S. Mor gnral boundary conditions can also b tratd by our modl. Th surfac nrgy trm abov is dfind by ux ( ) κ Dux ( ) dx= κ dx. Ω Ω i j= xi xj This strain gradint surfac nrgy givs an infinit nrgy if th dformation gradint is discontinuous across an intrfac. W hav introducd an altrnativ total variation surfac nrgy in [45] that givs a finit surfac nrgy that is proportional to th surfac ara of intrfacs across which th dformation gradint is discontinuous. Th total variation of a dformation gradint u that is discontinuous across th picwis smooth surfacs σ j for j = J sparating th opn sts ω in th disjoint union L Ω = = ω is givn by J L u (.) D( u) = [[ u]] σ j ds dx Ω + σ j ω j= = m n= xm xn whr [[ u]] σ j dnots th jump of th dformation gradint across th intrfac σ j. W will show latr in this papr that th formula (.) givs th simpl xprssion (4.) whn applid to continuous picwis linar finit lmnt functions. Not only is th total variation surfac nrgy a mor accurat modl for th nrgy of martnsitic crystals with microstructur but it allows th us of continuous picwis linar finit lmnt approximation to th dformation rathr than th mor complx finit lmnt mthods ndd to approximat a strain gradint surfac nrgy [45]. W usd this mor accurat total variation sharp intrfac surfac nrgy in th computations prsntd in this papr but w will us th strain gradint surfac nrgy in som of th discussion of this papr for simplicity of xposition. W will modl th lastic and anisotropic fr nrgy dnsity for a frromagntic shap mmory crystal by a continuous and fram-indiffrnt functionϕ with th symmtry of th high tmpratur austnitic phas [78]. Th fram indiffrnc proprty [79] is that (.4) ϕ( RF Rm θ) = ϕ( F m θ) for all R SO()

4 whr SO () is th group of propr rotations and th crystal symmtry proprty [789] is that (.5) ϕ( FQ m θ) = ϕ( F m θ) for all Q G whr G is th symmtry group of th austnitic phas. For Ni MnGa th symmtry group G is th cubic symmtry group th group of propr rotations laving th cub invariant. Sinc th nrgy dnsity is invariant with rspct to th dirction of th magntization [] w also hav that (.6) ϕ( Fm θ) = ϕ( F m θ). At a fixd tmpratur blow th martnsitic transformation tmpratur th nrgy dnsity ϕ is minimizd at (.7) SO() [ U( θ )± m ( θ) ] SO() [ Un( θ) ± mn( θ) ] whr th U i s ar th symmtry-rlatd transformation matrics for th martnsit and th m i 's ar th corrsponding prfrrd dirctions of magntization (th asy axs). Crystallin Ni MnGa transforms from a high tmpratur phas with cubic symmtry to a low tmpratur phas with ttragonal symmtry []. If th rfrnc stat is takn to b th Ni MnGa crystal in its cubic phas at th martnsitic transformation tmpratur thn th martnsitic transformation matrics with thir associatd asy axs of magntization can b givn by [] β α U = α m = ms U = β m = ms α α α U = α m = m s β whr α =.6 and β =.9555 at th martnsitic transformation tmpratur and whr ms is th saturation magntization. W modl th sum of th lastic and anisotropic nrgy dnsity ϕ( Fm θ ) for Ni MnGa by (.8) ϕ( Fm θ) = ϕ( F θ) + ϕ( Fm θ) whr ϕ ( F ) θ is an lastic nrgy dnsity for a martnsitic crystal that undrgos a cubic to ttragonal transformation (s [7]) and κu m Bm ϕ( Fm θ) = β α β mm T whr B = FF is th lft Cauchy-Grn strain and κu is th magntic anisotropic constant []. W not that for θ blow th martnsitic transformation tmpratur ϕ ( F ) θ is constructd to b minimizd by th dformation gradints SO() U( θ ) SO() U( θ) SO() U( θ) and that ϕ ( Ui m θ ) is minimizd at m= ± msi. Thus th nrgy dnsity ϕ( Fm θ ) is minimizd on (.7). 4

5 A THIN FILM MODEL FOR FERROMAGNETIC SHAPE MEMORY CRYSTALS Our thin film modl for frromagntic shap mmory crystals combins th rigorously drivd thin film modl for martnsitic crystals of [4] with th rigorously drivd thin film modl for frromagntic crystals of []. Our modl is that local minima (stabl quilibria) ( ) um u m ar dformations satisfying ( ) of th bulk nrgy ( ) u( x x x) = y( x x) + b( x x) x+ o() for ( x x) S and x ( ) and magntizations satisfying m( u( x x x) ) = M ( x x) + o() for ( x x) S and x ( ) whr th dformations yx ( x): S R and bx ( x): S R and th magntizations M( x x): S R ar local minima (stabl quilibria) ( ybm ) of th nrgy dnsity pr unit thicknss E( y b M) = κ D y( x x ) + b( x x ) dx dx S ( ) + ϕ([ y ] ) S y b M θ dxdx (.) + { µ y( S) M M ( x x) h[ y( x x)] } dt y S y b dxdx + M ndt y y b dxdx. S W dnotd abov by [ y y b] th dformation gradint matrix constructd from th column vctors y y and b. W also dnotd by M y( S) th projction of th gradint M onto th tangnt plan of th surfac ys ( ). W not that th magntization nrgy ( ) mag ( u m) has bn modld as in [] by ndt M y y b dx dx S whr th componnt of th magntization normal to th film surfac is givn by M ( n x x ) = M ( x ) [ ( )] x n y x x with n[ y( x x) ] bing th normal to th film surfac y( S) at yx ( x ). W will prsnt computational rsults for a film attachd to th substrat at two opposit dgs and fr on th othr two dgs by assuming that Ω = {( x x x):< x < < x < < x < }. W furthr assum that th applid magntic fild is in th z - z plan hz ( ) = ( h( z z) h( z z) ) for z R and w sk nrgy-minimizing dformations and magntizations of th form ux ( ) = u( x x) α x u( x x) and mux ( ( )) = m( ux ( )) m( ux ( )). ( ) ( ) 5

6 Undr this constraint our modl is that local minima (stabl quilibria) ( um ) of th bulk nrgy ( ) ( u m) ar dformations of th form ux ( x x) = yx ( x) + bx ( x) x+ o() for < x < < x < < x < for (.) yx ( x) = ( y( x) α x y( x) ) and bx ( x) = ( b( x) b( x) ) < x < < x < and magntizations of th form (.) m( u( x x x) ) = M ( x x) + o() for < x < < x < < x < for (.4) M( x x) = ( M( x) M( x) ) for < x < < x < whr ( ybm ) ar local minima (stabl quilibria) of th nrgy dnsity pr unit thicknss givn by (.). W can formulat this problm as a on-dimnsional problm by introducing yx ( ) = ( y( x) y( x) ) bx ( ) = ( b( x) b( x) ) Mx ( ) = ( M( x) M( x) ) and th fr nrgy E ( y b M ) = κ{ y + b } + ϕ( y α b ) M θ dx (.5) + µ y( S) M M ( x) h( y ( x) ) dt y α b dx { } ( y M y M) ( ) + ( y ) + dt y α b dx. y W can s that E ( ybm ) = E( ybm ) by chcking that all of th trms in th intgrand of (.) dpnd only on x for dformations and magntizations of th form (.) and (.). W not that y b (.6) y y b [ y α b ] α = =. y b W rcall that th magntization m is constraind to satisfy th saturation condition (.) so (.7) M ( x) = ms for < x <. For our computations with th thin film nrgy (.5) w us th fram-indiffrnt fr nrgy dnsity ϕ( Fm θ) = ϕ( F θ) + ϕ( Fm θ) with th lastic nrgy dnsity blow th transformation tmpratur bing givn for dformation gradints (.6) by ϕ( F θ) = c ( FF -FF αβ) + c ( C + C ( α + β )) + cc + c4 ( CC α β ) whr c c and c 4 ar positiv lastic moduli c is a positiv constant chosn to nsur that T th dformation is orintation prsrving [9] and C = F F is th right Cauchy-Grn strain. Th anisotropic fr nrgy dnsity blow th transformation tmpratur is givn as abov by 6

7 κu m Bm ϕ ( Fm θ) = β. α β m m W prsnt th rsults of our computational invstigation into th control of th dformation of th thin film by th application of a magntic fild. W start our computation with th film in th variant U with magntization m= m and attachd to th substrat at th dgs x = and x =. Mor prcisly th initial dformation satisfis (.8) yx ( x) = ( β x αx) and bx ( x) = ( α) for < x < and < x < and th initial magntization satisfis (.9) M( x x) = ms () for < x < and < x <. Th film is attachd to th substrat at th dgs x = and x = so th dformation yx ( x) is constraind to satisfy th boundary condition (.) y( x) = ( α x) and y( x) = ( βα x) for < x <. W not that th initial dformation (.8) and magntization (.9) minimiz th fr nrgy (.) among all dformations and magntizations satisfying th boundary condition (.) and th saturation condition (.7). W prsnt computational rsults for th minimization of th fr nrgy (.) as th applid magntic fild H for (.) h( z) = ( H) for z R is varid. Our computational rsults simulat th fild inducd tunnl proposd in []. Figur. Fild inducd tunnl. NUMERICAL APPROXIMATION W start th numrical approximation with th applid magntic fild of th form (.) for fild strngth H = and th dformation (.8) and magntization (.9). W thn incras th fild strngth H incrmntally and at ach nw valu of H w minimiz th fr nrgy (.5) by a nonlinar vrsion of th Polak-Ribir conjugat gradint mthod [] with th initial itrat ( ybm ) bing th local minima obtaind at th prvious valu of H. Aftr stting 7

8 µ = and th applid magntic fild of th form (.) th nrgy dnsity E ( y b M ) taks th form E ( y b M ) = κ{ y + b } + ϕ( y α b ) M θ dx (4.) M( x) Hdt y α b dx ( y M y M) + dt y α b dx. y + y ( ) ( ) W approximat yx ( ) = ( y( x) y( x) ) by continuous picwis linar finit lmnt functions on a uniform msh that satisfy th boundary conditions y() = y() = β y() = y() =. W approximat bx ( ) = ( b( x) b( x) ) by picwis constant finit lmnt functions on th sam uniform msh of dimnsion N and w approximat M ( x) = ( M( x) M( x) ) by picwis constant finit lmnt functions on th sam uniform msh that also satisfy th saturation condition (.7). κ y + b dx is not finit for gnral Th strain gradint surfac nrgy { } continuous picwis linar yx ( ) and picwis constant bx ( ). In our computations w us th total variation surfac nrgy (.) which givs for this modl (4.) ( [[ y ]] + [[ b ]] ) whr dnots th nods of th msh and whr [[ b across th msh nod. y ]] and [[ b ]] dnot th jump of y and RESULTS AND DISCUSSION As an initial tst that our modl can prdict qualitativ bhavior of a frromagntic shap mmory alloy w usd in our computations non-physical dimnsionlss cofficints. W st th lastic moduli c = c = c = c 4 = th magntic anisotropic constant κ u = 4 th surfac nrgy cofficint κ = and th saturation magntization m s =. W us th transformation matrics givn arlir for Ni MnGa with α =.6 and β = W prsnt rsults in Figur for th dflction of th film by incrasing th dimnsionlss magntic fild from H = to H = with incrmnts H =.. Rsults ar prsntd for a uniform msh of siz 6. W can obsrv in Figur that as th applid magntic fild is incrasd from H = th film rmains stationary but th magntization starts to tilt in th dirction of th applid fild so as to rduc th intraction nrgy. At a critical applid magntic fild th film bgins to dflct. 8

9 Figur. Dflction of th film by incrasing th applid magntic fild. 9

10 To nsur that th film dflct upwards rathr than downward (th fr nrgy is symmtric with rspct to a rflction of th dformation about th initial dformation) w addd th pnalty trm ν y ( ) to th fr nrgy whr y ( ) = if y ( ) > and y( ) = y( ) if ( ) is zro onc th film has bn dflctd upward. y <. This trm Figur 4. First ordr convrgnc of th rror in th fr nrgy as th msh siz N. CONCLUSIONS W hav proposd a modl for th dformation of singl crystal frromagntic shap mmory thin film and w hav dvlopd this modl to th fit th transformation matrics of th Ni MnGa alloy. Our finit lmnt numrical approximation and computation dmonstrats that th modl prdicts th dflction of a singl crystal frromagntic shap mmory thin film by th application of a magntic fild. ACKNOWLEDGEMENTS This work was supportd in part by DMS-46 th Institut for Mathmatics and Its Applications and by th Minnsota Suprcomputr Institut. REFERENCES [] R.D. Jams and M. Wuttig Philos. Mag. A 77 7 (998).

11 [] J.W. Dong J.Q. Xi J. Lu C. Adlmann C.J. Palmstrøm J. Cui Q. Pan T.W. Shild R.D. Jams and S. McKrnan J. Appl. Phys (4). [] R.D. Jams and D. Kindrlhrr Philos. Mag. A77 7 (99). [4] P. Blik and Luskin Intrfacs Fr Bound. 4 7 (). [5] P. Blik and M. Luskin Math. Modls Mthods Appl. Sci (4). [6] P. Blik and M. Luskin Multiscal Modl. Simul. 764 (4). [7] M. Luskin Acta Numr. 5 9 (996). [8] R.D. Jams and K.F. Han Acta. Matr (). [9] M. Gurtin: Topics in Finit Elasticty. Philadlphia: SIAM; 98. [] K. Bhattacharya and R. D. Jams J. Mch. Phys. Solids 47 5 (999). [] G. Gioia and R. D. Jams Proc. R. Soc. Lond. 45 (997). [] K. Bhattacharya A. DSimon K.F. Han R.D. Jams and C.J. Palmstrøm Matr. Sci. Eng. A (999). [] P. Blik T. Brul and M. Luskin Math. Modl. Numr. Anal ().