勇敢的男人
A Pha Field Model for Grain Growth and Thermal Grooving in Thin
Films with Orientation Dependent Surface Energy
Nele Moelans 1,a ,Bart Blanpain 1,b ,Patrick Wollants 1,c
1Department of Metallurgy and Materials Engineering,Katholieke Universiteit Leuven,Leuven,Belgium
lans@mtm.kuleuven.be,
b bart.blanpain@mtm.kuleuven.be,
c patrick.wollants@mtm.kuleuven.be
Keywords:thin films,pha field simulation,grooving,grain growth,anisotropy
Abstract.A pha field model for simulating grain growth and thermal grooving in thin films is prented.Orientation dependence of the surface free energy and misorientation dependence of the grain boundary free energy are included in the model.Moreover,the model can treat different mechanisms for groove formation,namely through volume diffusion,surface diffusion,evaporation-condensation,or a combination of the mechanisms.The evolution of a groove between two grains has been simulated for different surface and grain boundary energies and different groove formation mechanisms.
Introduction
The grain size,grain size distribution and grain orientation strongly influence the strength,electronic properties and durability of polycrystalline films.Due to the high ratio of surface to bulk material,surface energy has an important effect on grain growth in thin films [1].In order to balance surface tension with grain boundary tension,grooves are formed where grain boundaries interct the film surface [2,3].It has been obrved experimentally that the grooves exert a drag force on moving grain boundaries and can stop grain growth [4].Furthermore,the surface energy of the grains may depend on their orientation.Favorably oriented grains with low surface energy may have a high driving force for grain growth and break free from the grooves,resulting in condary recrystallization.
This mechanism is often applied to obtain highly textured films with a huge grain size for micro-electronic devices [4–6].
Mullins [7,8]and Zhang [9]distinguish three different mechanisms for groove formation,namely through volume diffusion,surface diffusion or evaporation-condensation.The groove morphology depends on which is the dominant mechanism.
In this article we prent a pha field model for simulating the effect of thermal grooving and orientation dependent surface energy on grain growth in thin films.The three mechanisms of groove formation are considered in the model.
Model description
In order to describe the effect of surface tension on the evolution of the grains,the environment of the fihe atmosphere or substrate,is included in the simulations.A schematic rep-rentation of the simulation system for the ca of a film with two grains is shown in figure 1.The model assumes that groove formation is controlled by diffusion process,namely bulk or surface diffusion in the film,or diffusion of film material in the environment after evaporation or dissolution.
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Fig.1:System geometry and pha field variables ud in the simulations.马勋
The model is bad on the grain growth model of Chen and Yang for normal grain growth [10,11],in which different grains are reprented by different pha field variables ηi (−→r ,t )that are continuous functions in space and time.In each grain one of the ηi ’s equals 1and all the other equal 0.An extra pha field variable ψ(−→r ,t )is introduced into the model to distinguish between environment and film.It equals 1in the environment and 0in the film.ψ(−→r ,t )is a conrved variable.It can be interpreted as a scaled composition variable.For example in the ca of an Al-film in an argon atmosphere
ψ=c −c film
c env −c film (1)
where c is the local concentration of Ar,c film the solubility of Ar in Al and c env the equilibrium concentration of argon in Ar-gas above an Al-film.
The total free energy is formulated as a volume integral over the film and the environment
F = V
f 0(η1,η2,...,ηp ,ψ)+κ2p i =1(−→∇ηi )2+κ2(−→∇ψ)2 d V (2)with
f 0=m p i =1 η4i 4−η2i 2 + ψ44−ψ22 +p j<i γi,j η2i η2j +p i =1γi,ψη2i ψ2 ,(3)
the specific free energy as function of the pha field variables.The minima of the specific free energy correspond with the values of the pha field variables in the different domains in the simulation system.Within the film,f 0has minima at ψ=0,one of the ηi =±1and the other ηi =0.In the environment,f 0is minimal for ψ=±1and all ηi =0.In the prented simulations,only the minima with ηi =1or ψ=1are considered.m is a parameter related to the depth of the free energy well.p is the number of grain orientations considered in the simulation.
Orientation dependence of the surface free energy is introduced through the parameters γi,j and γi,ψand by giving κthe following dependence on the pha field variables [12,13]
κ= p j<i κi,j η2i η2j + p i =1κi,ψη2i ψ2
p j<i η2i η2j + p i =1η2i
ψ2.(4)
This formulation also allows to let the grain boundary energy depend on the misorientation between neighboring grains.At each interface,only one of the terms in the summation differs from 0.Accordingly,κ=κi,j at the interface between grains i and j .The κi,j and γi,j are related to the free energy of the grain boundary between the grains with orientations i and j and the κi,ψand γi,ψto the surface free energy of grain i .The model assumes that the interfacial free energies do not depend on the orientation of the interface with respect to the crystal structure of the grains.
Bad on the theory of Cahn and Hilliard for systems with diffu interfaces [14],it can be calculated that for a system with a free energy of the form (2)and (3)the widths of the interfaces are proportional to
l i,j ∝ κi,j (∆f 0)max = 32κi,j m (1+2γi,j )
,(5)with (∆f 0)max =f 0−f 0,min and the interfacial free energies equal
σi,j =1√3
κi,j m (1+2γi,j ).All types of interfaces are given the same thickness in the simulations for numerical convenience.Hence,for each interface,the parameters κi,j and γi,j must satisfy the relation入户申请书怎么写
κi,j m (1+2γi,j )=cte .(6)
The evolution of the non-conrved pha field variables ηi (−→r ,t )is given by a generalized Ginzburg-Landau equation
∂ηi ∂t =−L ∂f 0∂ηi +12 ∂κ∂ηi p j =1
(−→∇ηj )2+(−→∇ψ)2 −κ∇2ηi −−→∇κ·−→∇ηi .(7)The kinetic coefficient L is related to the mobility of the grain boundaries.The temporal evolution of the conrved pha field variable ψ(−→r ,t )governs a generalized Cahn-Hilliard equation
∂ψ∂t =−→∇·M −→∇ ∂f 0∂ψ+12 ∂κ∂ψ p i =1
(−→∇ηi )2+(−→∇ψ)2 −κ∇2ψ−−→∇κ·−→∇ψ .(8)To distinguish between bulk and interface diffusion M has the following spatial dependence
M =M η( p i =1η2i )+M ψψ2 p i =1η2i +ψ2for p j<i η2i η2j +p i =1η2i ψ2<10e −6(9)M =M gb p j<i η2i η2j +M s ( p i =1η2i ψ2) p j<i η2i η2j + p i =1η2i ψ2for p j<i η2i η2j +p i =1
η2i ψ2>10e −6(10)(11)
M ηand M ψare related to the bulk
diffusion coefficients of respectively the film and the envi-ronment.M gb and M s describe respectively grain boundary and surface diffusion.
Simulations
The evolution of a groove between two grains was simulated using the system geometry depicted in figure 1.The system was discretized using 256×256grid points.The thickness of the film was 128grid points.The lattice spacing ∆x was taken equal to 0.5×10−8m.For all simulations,L =1s −1,m =1J/m 3,∆t =0.01s.Simulations were performed for different values of the parameters κ,γand M .
A central finite difference scheme bad on 3grid points in each dimension was ud to approximate the laplacian and a central scheme bad on the 2neighboring grid points for the gradients.An explicit Euler technique was ud for the time stepping.Periodic boundary conditions were applied.Since equation (4)is not defined within the grains where η2i η2j =0,the parameter κwas only calculated at the interfaces,namely where p j<i η2i η2j + p i =1η2i ψ2>10e −6.
怎么获取root权限To prevent ψfrom taking values outside the interval [(0−c film )/(c env −c film ),(1−c film )/(c env −c film )],the term ∂f 0/∂ψwas implemented as ∂f 0∂ψfor 0−c film c env −c film ≤ψ≤1−c film c env −c film
−1for ψ<0−c film c env −c film and 1for ψ>1−c film c env −c film
Furthermore,for one iteration step (∆t =0.01s)for the Ginzburg-Landau equations,200iterations using a time step ∆t/200are performed
威风的反义词for the
Cahn-Hilliard equation.王者荣耀歌
a)b)
Fig.2:Simulation images for 2different ratios of the surface and grain boundary free energy.Both grains have the same surface free energy.a)σs =σgb =√2J/m 2:κ=2J/m 3,γ=1.b)σgb =√2J/m 3,σs =1.25√2J/m 3=(5/4)σgb :κ1,2=2J/m 3,κψ,1=κψ,2=2.5J/m 3,γ1,2=1,γψ,1=γψ,2=11/8.
Equilibrium angle When the surface free energies of both grains are equal,the equilibrium angle of the groove 1depends on the ratio of the surface free energy σs and the grain boundary free energy σgb according to Young’s law
σgb =2σs cos(β).(12)The simulation images for two different ratios of σs and σgb in figure 2
show that the equilibrium angle is well reproduced in the simulations.
1
Since it is assumed that groove formation is controlled by diffusion process,one may expect that the equilibrium angle is established during growth of the groove.
a)
b)
c)
Fig.3:Evolution of a system with two grains with different surface free energy:σs ,1=√2
J/m 2,σs ,2=1.25√2J/m 2=(5/4)σs ,1.Simulation images at time a)t =5s,b)t =60s,c)t =200s.Model parameters:κ1,2=2J/m 3,κ1,ψ=2J/m 3,κ1,ψ=2.5J/m 3,γ1,2=1,γ1,ψ=1,γ2,ψ=11/8,M =1×10−16m 2/s.
Orientation dependence of the surface free energy Figure 3shows a concution of simulation images for a system where the grains have a different surface free energy.The grain boundary obtains a curvature outside the plane of the film to balance the interfacial tensions at the root of the groove.Since grain boundaries move towards their center of curvature,the grain with low surface energy grows at the expen of the other grain.The equilibrium angles between the grain boundary and
the
grain surfaces
continuously adapt to the new position of the grain boundary.As a conquence,the grain boundary keeps moving until the grain with high surface energy has disappeared.
大学军训感言a)b)c)
Fig.4:Groove morphology for different groove formation mechanisms.a)Surface diffusion is dominant:M η=M ψ=M gb =0,M s =1×10−16m 2/s,b)Bulk diffusion is dominant:M =cte =1×10−16m 2/s c)Groove is formed by evaporation and condensation of film material:M η=M gb =0,M s =M ψ=1×10−16m 2/s,solubility of film material in environment is 0.001(c env =0.999).For the three simulations κ=2and γ=1.
Different mechanisms for groove formation In figure 4simulation images for the three different groove formation mechanisms distinguished by Mullins and Zhang [7,9]are compared.When bulk or surface diffusion is dominant,there is a maximum in the surface profile near the groove.The hill is wider for surface diffusion.In the ca
of groove formation through evaporation-condensation,there is no maximum in the surface profile.The steady-state groove profiles are in qualitative agreement with the analytical calculations of Mullins and Zhang,
except that their calculations predict an extra minimum next to the hill when surface diffusion is dominant.More simulations for a larger grain size and with a better resolution should be performed to analyze the groove profiles further.Furthermore,since the width of the interfaces in phafield simulations usually differs from the real interfacial width,it must be studied how to relate the diffusion mobilities to diffusion coefficients for interface diffusion.
Conclusions and outlook
A phafield model is prented for simulating grain growth and thermal grooving in polycrys-tallinefilms.To incorporate surface tension and to describe the shape of the grooves,both the film and it’s environment are considered in the simulations.The model can treat orientation dependent surface energy and misorientation dependent grain boundary energy.Furthermore, it allows to distinguish between bulk and surface diffusion.To validate the model,2D simula-tions were performed of the evolution of a groove between two grains.Results are in qualitative agreement with analytical theories,however further quantitative validation using a better res-olution is required.In the future,the model will be applied to simulate in3D grain growth and thermal grooving for thinfilms containing many grains.Then the effect of thermal pitting at grain boundary vertices can be studied as well.
Acknowledgments
The authors wish to acknowledge the Institute for the Promotion of Innovation through Science and Technology in Flanders(IWT-Vlaanderen)for funding this rearch.
References
[1]C.V.Thompson:Annu.Rev.Mater.Sci.Vol.20(1990),p.245
[2]W.W.Mullins:Acta Metall.Vol.6(1958),p.414
[3]M.J.Rost,D.A.Quist and J.W.M.Frenken:Phys.Rev.Lett.Vol.91(2003),p.026101
[4]J.E.Palmer,C.V.Thompson and H.I.Smith:J.Appl.Phys.Vol.62(1987),p.2492
中国网络经纪人登录平台[5]C.V.Thompson and H.I.Smith:Appl.Phys.Lett.Vol.44,(1984)p.603
[6]J.Greir,D.M¨u llner,C.V.Thompson and E.Arzt:Scripta Mater.Vol.41(1999),p.709
[7]W.W.Mullins:J.Appl.Phys.Vol.28(1957),p.333
[8]W.W.Mullins:Trans.Metall.Soc.AIME Vol.218(1960),p.354
[9]W.Zhang,P.Sachenko and I.Gladwell:Acta Mater.Vol.52(2004),p.107
[10]L.-Q.Chen and W.Yang:Phys.Rev.B Vol.50(1994),p.15752
[11]D.Fan and L.-Q.Chen:Acta Mater.Vol.45(1997),p.611
[12]A.Kazaryan,Y.Wang,S.A.Dregia and B.R.Patton:Phys.Rev.B Vol.61(2000),p.14275
[13]N.Ma,A.Kazaryan,S.A.Dregia and Y.Wang:Acta Mater.Vol.52(2004),p.3869
[14]J.W.Cahn and J.E.Hilliard:J.Chem.Phys.Vol.28(1958),p.258
