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0.1 Phason elasticity and atomic dynamics of quasicrystals 1 0.1 Phason elasticity and atom

0.1 Phason elasticity and atomic dynamics of quasicrystals 1 0.1 Phason elasticity and atom
0.1 Phason elasticity and atomic dynamics of quasicrystals 1 0.1 Phason elasticity and atom

0.1Phason elasticity and atomic dynamics of quasicrystals

F.G¨a hler,S.Hocker,U.Koschella,J.Roth,and H.-R.Trebin

Institut f¨u r Theoretische und Angewandte Physik,

Universit¨a t Stuttgart,D-70550Stuttgart,Germany

0.1.1Introduction

As the order of a quasicrystal is quasiperiodic,it can be described as an irrational cut through a periodic structure in a higher-dimensional space.This mathematical trick has important consequences for the low energy excitations that can occur.Translating the cut space to a different position is a symmetry operation,which changes the quasicrystal structure,but not its energy.A small breaking of this symmetry,by chosing a cut of small and slowly varying slope(with respect to the ideal orientation)therefore leads to low-energy Goldstone modes, called phasons.In many respects,phasons are analogous to phonons,which are small and slowly varying distortions of a structure in physical space.The distortions related to phasons rest in the complementary,internal space needed for the embedding of the quasicrystal in the higher-dimensional crystal.

In much the same way as there is an elastic energy for phonon type distortions of a solid, there is an effective elasticity theory for the phason degrees of freedom,which moreover is coupled to the phonon elasticity.A non-zero phason strain,i.e.,a non-zero slope of the cut space,will cost energy.At higher temperatures,however,phason excitations,which cor-respond to a?uctuating phason strain,will become possible.This is analogous to phonons. There is one important difference,however.Whereas phonons are usually propagating modes, phasons are believed to be diffusive.

Phason strain in a quasicrystal is connected with a rearrangement of certain local atomic con?gurations.In an elementary form these rearrangements are called phason?ips.Under-standing the dynamics of phason?ips and other atomic rearrangements of the structure of a quasicrystal is essential for the understanding of the formation and stability of quasicrystals, and also for many of their physical properties.Perfect quasicrystals are usually obtained by high temperature annealing after solidi?cation.During this process,many defects initially present are eliminated.It is therefore necessary that phason?ips,atomic diffusion,and other dynamical processes are possible and effective at these temperatures.

Atomic diffusion and phason mobility are also important for the mobility of dislocations. Unlike in a crystal,in a quasicrystal a moving dislocation leaves a phason wall in its wake. This phason wall must be smoothed out and?nally eliminated by phason?ips and diffusion processes,for otherwise the material would harden very quickly.Mobile phason?ips are therefore necessary for the ductility of quasicrystals observed at high temperatures.

There are several other interesting consequences of phason?ips.Kalugin and Katz have suggested that phason?ips provide a mechanism for atomic diffusion[1].Even though all experimental evidence indicates that vacancy mediated diffusion is dominant in quasicrystals at high temperatures[2],a phason?ip mediated diffusion mechanism cannot be ruled out at lower temperatures.Bl¨u her et al.[3]found deviations from the Arrhenius law,indicating that another diffusion mechanism becomes more ef?cient there.Due to phason-phonon coupling,

it should also be possible to excite phason?ips by applying a mechanical stress.It is conceiv-able that a periodically varying mechanical stress induces phason?ips,which likely lead to internal friction.Weller and Damson[4]have measured internal friction in quasicrystals,but it remains unclear whether the underlying atomic processes are related to phason?ips.

Our goal here is to study structure changing dynamical processes in quasicrystals by means of computer simulations.Such processes are,in particular,phason?ips and other jumps of atoms.We are not concerned with the long wavelength phonon part of the dynamics, which may be obtained by dynamical matrix methods[5,6].Our study is divided in two main parts.A?rst one is concerned with elasticity theory.Assuming a given atomic interaction,the elastic constants for phonons,phasons,and the phonon-phason coupling are determined for a simple model structure.It turns out that for simple,purely attractive pair potentials the ma-trix of elastic constants is not positive de?nite,i.e.,the model quasicrystal is only metastable for such interactions[7].A closer analysis suggests,however,how the potentials have to be modi?ed to improve the situation.With the modi?ed potentials the matrix of elastic constants becomes positive de?nite,at least at constant stoichiometry[8].Unfortunately,this does not yet imply that the quasicrystal structure is the ground state.The reason is that applying a lin-ear phason strain is not the most general deformation one can apply to a quasicrystal.Still,by Monte-Carlo simulations we?nd that the ground state of the modi?ed potentials is a supertile random tiling structure very close to the perfect quasicrystal,which is a marked improvement over the original potentials.Such an analysis is an important?rst step in selecting interaction potentials having a quasicrystalline ground state,with which one then can perform further simulations.

On a more fundamental level,dynamical processes can be studied atomistically,by using molecular dynamics simulations.This is done in a second part.There are several ways to extract information.In principle,one can compute the Van-Hove correlation function or its Fourier transform,the dynamical structure factor,which contains all information about the dy-namics.This has been attempted in[9],but it is a formidable task to do simulations providing suf?ciently good statistics.This is even so if,instead of computing the whole time-averaged Van-Hove correlation function,the computation is limited to radially averaged displacement histograms at certain time intervals[9].Often,it is therefore more promising to measure directly the quantities one is interested in.

The search for phason?ips is not easy,however.One problem is that atomic jumps and ?ips overlap with thermal vibrations,so that special?ip detectors have to be used[10].They monitor certain local con?gurations and report changes.With this method it was possible to derive jump frequencies and jump distances[10].To a certain degree it was also possible to ?gure out the correlation of the jumps of nearby atoms.But a full investigation would require additional tools or the computation of the correlation functions.

Another dif?culty is that realistic quasicrystal models require very sophisticated potentials for their stabilization.One way to avoid this problem is to use simpli?ed models which are stable with simple model potentials.From such simple models one can expect only qualita-tively correct results,but this is still interesting.With such simple models we have studied phason?ips and atomic diffusion[11],atomic jump processes[12],and even shock waves in quasicrystals[13].

Eventually,the simulation of realistic models with realistic potentials has to be mastered, however.First attempts with such potentials for decagonal Al-Cu-Co had been made in[10],

but it turned out that they were good enough only at very low temperatures.At more elevated temperatures,the Al substructure was quickly decaying,indicating that the relative energy scales of Al and transition metals is not correct.Still,it was possible to observe the expected phason?ips.In this article,we report on simulations of decagonal Al-Ni-Co with newer potentials due to Moriarty and Widom[14,15],which turn out to be much better.With these potentials,the quasicrystal is now essentially stable up to the melting point.At higher temperatures,a large part of the aluminium atoms become very mobile,so that aluminium diffusion could directly be measured.Despite this aluminium mobility,the structure remains essentially unchanged.

All the dynamical processes discussed here(phason?ips,atom jumps)are candidates for the microscopic mechanisms leading to internal friction.If mechanical stress is applied to a material,the resulting deformation need not be instantaneous,because the induced atomic rearrangements need some time to complete.In the case of a periodic external stress,with a period comparable to the relevant relaxation times,this leads to a phase shift between the ap-plied stress and the deformation,which in turn leads to dissipation.This internal friction can be observed by mechanical spectroscopy experiments[4,16].Direct simulation of internal friction,i.e.,inducing phason?ips and atom jumps by applying periodically varying mechan-ical stresses,has proved to be too ambitious for the moment,but understanding the candidate microscopic processes is an important step towards this goal.

0.1.2Phason elasticity

In this section we?rst review the general form of linear elasiticity theory for quasicrystals, and then compute all elastic constants for a simple model quasicrystal using Lennard-Jones potentials.Since not all phason elastic constants turn out to be positive[7],the potentials are then modi?ed in a systematic way in order to make the model stable against phason strains,at least at constant stoichiometry[8].The ground state structures of the modi?ed potentials are determined by Monte-Carlo simulations.They turn out to be supertile random tiling structures similar to the perfect quasicrystal.

Generalized linear elasticity theory

Elasticity theory for quasicrystals describes both phonon and phason degrees of freedom. The pure phonon part is the same as for other solids.Phonon type elastic deformations are described by the Green deformation tensor

εij=1

?ξj

+

?u j

reads

f=

1

?ξ j

.(3)

Notice that,unlike the phonon strain,the phason strain is not symmetric in its two indices. Since the two indices refer to directions in two different spaces(internal space and physical space,respectively),the phason strain must not be symmetrized.In the general case,the free elastic energy density of a quasicrystal is

f=1

2

C⊥,⊥

ijkl

χijχkl+C ,⊥

ijkl

εijχkl+O (εij,χij)3 ,(4)

where the generalized Hooke tensor now contains6+10+12=28independent elastic constants for a two-dimensional quasicrystal with a two-dimensional internal space.

In the presence of a non-trivial point symmetry,the number of independent elastic con-stants is reduced.For a two-dimensional decagonal quasicrystal,as it is considered here, the phonon strain has two symmetry invariant modes,the one-dimensional bulk deformation ε(1)and the two-dimensional shear deformationε(6)i.The phason strain splits into two two-dimensional invariant modes,χ(6)i andχ(8)i.As a consequence,a two-dimensional decagonal quasicrystal has?ve generalized elastic constants,the bulk modulusλ3/2,the shear modulus λ5/2,two phason elastic constantsλ7andλ9,and a phason-phonon coupling constantλ6. The elastic free energy density as a function of the invariant strain modes then reads f=f phonon+f phason+f coupling(5)

f phonon=1

2

λ5 ε(6)12+ε(6)22

f phason=1

2

λ9 χ(8)12+χ(8)22

f coupling=λ6 ε(6)1χ(6)1+ε(6)2χ(6)2 .

The quasicrystal model

For the atomic con?guration of our two-dimensional decagonal model quasicrystal we choose the binary tiling quasicrystal of Mikulla and Roth(Fig.1),?rst mentioned in[17].Its con-struction based on the T¨u bingen triangle tiling[18]is described in[19].This particular model has the advantage that there are unambiguous atomic con?gurations without defects for a wide range of phason strains,which is not the case for most other models.

For the atomic interactions we choose the Lennard-Jones potentials usually used for these binary tiling structures,

φij(r)=?ij σij r 6 ,(6)

where r is the distance of the two atoms and i,j are the atom types.The parameters?ij and σij are chosen such that the potential minima are at ideal nearest neighbor distances,and the minimum of the potential for different atom types is twice as deep as the other ones:?AA=?BB=1,?AB=2,σAA=1.176,σAB=1,σBB=0.6180.(7) The purpose of the deeper minimum for different atom types is to avoid phase separation.For the simulations as well as for the analytical calculations we need to cut off the potential at some?nite radius R.Furthermore,the potentials are shifted,so that they vanish at the cutoff radius R:

(r)=φij(r)?φij(R)(r

-3.51898-3.51894

-3.51890

-3.5194

-3.5190

-3.5186

-0.0030.000

0.003

-3.5188

-3.5186

-3.5184

-0.003

0.000

0.003

a big atom(R=1.92).An analo-

atoms.(b)Local environments

(R=1.92).No.40does

phason strain. system:

phonon:λ3=250,λ5=90.2,

phason:λ7=?2.70,λ9=0.8,

coupling:λ6=?1.14.(9) Analytic computation of the phason elastic energy

In the last paragraph we have seen that the phason elastic constantλ7is negative,which violates the stabilization criteria and implies that the quasicrystal is metastable at temperature zero.We therefore want to analyse how the phason elastic energy depends on the potentials, in order to learn how to modify them so that they can stabilize our model quasicrystal.

For this purpose,we consider a quasicrystal with an arbitrary phason strain,but without phonon strain or local relaxation of atom positions.If we consider a cutoff radius R for the atomic interactions,the potential energy E i of a single atom depends only on the potentials

φand the atomic con?guration inside the ball B R,x

i with radius R around the position x i of

the atom.Therefore,we can calculate the zero temperature elastic free energy by adding the potential energies of the central atoms of each local environment,multiplied by the number of occurrences N i of the environment.In the same manner we can express the total volume by the V oronoi volumes of the central atoms of the occuring environments.We can easily rewrite this in terms of the frequencies n i of the environments.These,on the other hand,are proportional to the areas of the acceptance domains A i of the environments,shown in Fig.5:

f=E pot

i N i V i= i n i E i i A i V i.(10)

The energy E i of an atom with a given environment depends only on the potentialsφ,the corresponding acceptance domain A i depends on the phason strainχ,and the V oronoi vol-umes V i are?xed and known,so that we have derived the zero temperature free energy density

r1

1.00

r3

1.54

r5

1.90

2λ7χ26+

1

while the molecular dynamics relaxation simulations resulted in

λ7=?2.70,λ9=0.8.(13) These results agree only qualitatively.One source of discrepancy is that the cutoff radius used for the simulations(R=7)was bigger than the one used for the analytic expression. However,molecular dynamics simulations with the short range potential show that the differ-ence between the two results is mainly due to atomic relaxations,which are neglected in the analytic calculations.

Modi?cation of the potentials

Let us now have a closer look at the analytic expressions Eq.11.The nearest neighbor interac-tion terms for the two phason elastic constantsλ7andλ9have the same weights,but opposite sign(?rst column in Eq.11).This is due to the fact,that all binary tilings with the same sto-ichiometry have the same potential energy,if only nearest neighbor interactions are assumed. For all plausible potentials the nearest neighbor part ofλ7is negative,while the one ofλ9is positive.Since the phason strain modeχ6increases the relative number of small atoms while χ8decreases it,this describes the dependence of the potential energy density on bond density.

Stoichiometry preserving phason strains satisfyχ26=χ28,and thus have the phason elastic constant

1

2

1

-1

-2

00.51 1.52

Figure6:Original Lennard-Jones potentials and modi?ed potentialφAB,a mixed Lennard-Jones and Dzugutov potential.

expressions phason elastic constants which are all positive,but the simulations lead to com-pletely different results.So,if the repulsive part is too strong,atomic relaxations overshadow completely the effect of the energy dependence on the frequencies of local environments,and the analytic calculation is no longer realistic.

Ground states by Monte-Carlo simulations

A negative phason elastic constant at temperature zero means that there exist con?gurations with lower potential energy,and thus the quasicrystal is not the ground state.For binary tiling quasicrystals with Lennard-Jones potentials a crystalline ground state was detected already by Lee et al.[23].With Monte-Carlo simulations we could essentially reproduce their results. The ground state consists of an assembly of periodic nanocrystallites.We also performed Monte-Carlo simulations with the modi?ed potentials.Here,the ground state is a supertile random tiling,with supertiles that are legal local environments from the perfect quasicrystal. However,a long range quasicrystalline order is not present,and the fat rhomb supertile seems clearly favoured(see Fig.7).Since a tiling of only fat rhomb supertiles would lead to a different stoichiometry,some supertile defects occur to adjust the ratio of the atom types.In contrast to the simulations with Lennard-Jones potentials,the perfect quasicrystal now is a low energy con?guration,even though the potential energy of the ground state as found by the Monte-Carlo simulations is clearly lower than that of the quasicrystal and its approximants. Discussion

Our results show that with the original Lennard-Jones potentials the binary tiling quasicrys-tal is not stable,only metastable.For molecular dynamics simulations,this is not really a problem.Since the structure is close-packed,there are enormous energy barriers between the quasicrystal and other competing structures,which have only slightly lower energy.With

Figure7:Ground state of the model quasicrystal with the modi?ed potentials.

modi?ed potentials it is possible to obtain a positive phason elastic constant for stoichiom-etry preserving phason strains,whereas other phason strains can still have a negative elastic constant.In fact,thermodynamic stability does not require that stoichiometry non-preserving phason strains have positive elastic constants.Rather,for stability it is required that in an energy-composition diagram the quasicrystal is an extremal state,i.e.,it is on the boundary of the convex hull of the data points of all existing states(see[23]).An extremal state cannot decompose into two or more phases of lower energy,but the same average composition.So far,such an analysis has not been performed for the modi?ed potentials.

The Monte-Carlo simulations suggest that the quasicrystal is still not the ground state,but the lowest energy states are much closer to the quasicrystal than for the Lennard-Jones poten-tials.The situation can certainly be improved by including interactions of longer range,but the effort for the analytical computations soon becomes overwhelming,because the number of different local environments grows rapidly with their size.We also note that the frequencies of all local environments considered vary quadratically with the phason strain,which implies that the elastic energy has also a quadratic form.A non-analytic elastic energy,as it is as-sumed to be necessary for an energetically stabilized quasicrystal[24],can only be obtained if the cutoff radius is larger than the matching rule radius of the tiling,which is rather big in this case[19].

The molecular dynamics simulations show that local relaxation effects are very important.

Figure8:Top left:Drawing of phason?ip in decagonal Al-Cu-Co;circles are Al,squares Cu/Co.The remaining pictures show the potential landscapes of different stages of such a?ip as observed in a molecular dynamics simulation(from left to right,top to bottom).Cu/Co have deep potential minima,the?ipping Al atoms only shallow,elongated ones.

They can completely overshadow the contributions from the frequencies of the(perfect)local environments.This makes an extension of the analytical computations to much larger radii doubtful.Relaxation effects also strongly modify the results from continuum elasticity theory. Atomistic considerations are therefore essential to understand the stability of quasicrystals.It is not enough to know the continuum phason elasticity.

0.1.3Atomistic simulations

Molecular dynamics(MD)is an excellent tool for the study of dynamical processes at an atomistic level.In a MD simulation,the equations of motion of the particles in the system are integrated numerically over a suf?ciently long time interval.An accurate representation of the interactions between the atoms is therefore needed.Due to the aperiodic nature of qua-sicrystals,any reasonable model structure requires so many atoms,that a quantum mechanical treatment of these interactions is not feasible.For this reason,classical,effective potentials have to be used.The quality of the results of MD simulations crucially depends on the quality of these potentials,and the construction of reliable potentials is a very dif?cult task.

MD simulations of decagonal Al-Cu-Co quasicrystals using realistic pair potentials have already been presented in[10],but the the potentials were able to stabilize the model structures investigated only at very low temperatures.At higher temperatures,the Al subsystem quickly melted away,while the transition metal matrix remained stable.Hence the relative energy scales of these two subsystems were probably not correct.Still,a number of interesting results could be obtained in these simulations,among them the direct observation of a phason?ip. In Fig.8,different stages of this?ip are shown,each with the potential landscape seen by the atmos involved.Cu and Co atoms are in located in deep and sharp potential minima,whereas

the potential minima of the Al atoms are much shallower.Morover,the minima of the?ipping Al atoms are elongated in the direction into which the?ip takes place.

In the present article,we shall concentrate on simulations carried out with new potentials derived in the framework of Moriarty’s generalized pseudopotential theory(GPT)[14,15]. These GPT potentials consist of a systematic expansion of the total energy into n-body terms, which is derived from?rst principles.For computational ef?ciency it is desirable to truncate the series as early as possible.Unfortunately,if transition metals(TM)are involved it is not always possible to terminate already after the two-body term,but Al-Lehyani et al.[25]have found that it can be done if a correction term is added to the TM-TM interactions.The correc-tion term is?tted empirically to an ab-initio calculation of a simple quasicrystal approximant [25].For computational ef?ciency,we have cut off the corrected pair potentials shortly after the third minimum,guiding the potential functions smoothly to zero.All MD simulations were carried out with the code IMD[20]developed at our institute.

Using the same corrected pair potentials,Mihalkoviˇc et al.[26]have developed an opti-mized structure model of decagonal Al-Ni-Co.In Monte Carlo simulations the model was varied until a structure of minimal energy was reached.We use this model structure(with slight variations)also for our simulations.Since the model was optimized with respect to the potentials,it can be expected that they?t well together,which is important for our simulations. Stability of the model structure

The structure model of Mihalkoviˇc et al.[26]essentially consists of an alternating stacking of two different layers,which are decorations of the same hexagon-boat-star(HBS)tiling.The resulting periodicity is about4?A.Like in[26],we have worked with two different decorations which differ in stoichiometry.The simpler Ni rich decoration,shown in Fig.9,has a composi-tion of Al70Ni21Co9,belonging to the the”basic Ni rich”phase[27].In the other decoration, of composition Al72Ni17Co11,some of the Ni atoms are replaced by Co(see[26]).This decoration will be termed the Co rich decoration,even though it also belongs to the Ni rich corner of the existence domain of decagonal Al-Ni-Co[27].In the follwing,we shall mainly concentrate on the Ni-rich decoration.

In order to check the correctness of the energy scale,we?rst determined the melting temperature T m,by slowly heating the sample at constant pressure.The Ni rich sample melted at about1250K,whereas the Co rich sample melted at1170K.This is surprisingly close to the experimental melting temperature of the basic Ni rich phase of about1200K[27],given that the potentials were originally derived for zero temperature.

Next,to check the stability of the model structures we performed simulations at constant energy,at about T=0.5T m.While both structures are essentially stable,there are a few Al atoms which change their positions during the?rst steps of the simulation.The relaxation occurs primarily inside the stars tiles:the Al atoms there move to different positions in the quasiperiodic plane,and they moreover seek alternating positions from layer to layer,breaking the4?A periodicity.The relaxed equilibrium positions are shown in Fig.9.Inside the star tile, they depend on the number of boats around the star,which in turn is determined by the type of supertile in which the star is located(Fig.9).Stars with one or three adjacent boats show an exact8?A periodicity in equilibrium,whereas stars with?ve adjacent boats show an8?A periodicity only up to about0.5T m.At higher temperatures all of the?ve equivalent positions

Figure9:Ni rich decoration after relaxation.The positions of marked atoms differ from the original decoration[26].Small(large)dots indicate atoms in upper(lower)layer.Al:dark grey;

Ni:black;Co:light grey.Dashed lines mark supertiles contaning the three characteristic local neighbourhoods of the star tiles.Encircled atoms occur only in every second double layer,those with solid circles in one half of the double layers,those with dashed circles in the other half. inside the star are occupied with equal probability,and the periodicity is completely broken locally.At very high temperatures,the decoration of the hexagon supertile is further modi?ed and becomes completely symmetric.For the following simulations,we have directly used the modi?ed decoration corresponding the real equilibrium structure.The same modi?cations to the original model of Mihalkoviˇc et al.[26]have also been observed in[28].

Aluminium diffusion

At temperatures above0.6T m,signi?cant Al diffusion is observed(Fig.10).Only Al atoms neighboring Co atoms are involved in the diffusion,but these form a sizeable fraction of all Al atoms.Positions with mobile Al atoms are suf?ciently close to each other that long range diffusion is possible.Other Al positions can remain very stable even at high temperature, particularly those in rings with?ve Al and?ve Ni atoms.The stability of these rings can also be seen in(time averaged)atom density maps(Fig.11),where such Al positions are very sharp,whereas those near Co atoms are completely smeared out.

As noted earlier,at temperatures close to T m the decoration of the supertile hexagon(a boat-star pair)becomes symmetric.In the atom density maps of these supertile hexagons,one can see smeared-out parts of the Al distribution,namely pairs of banana-shaped regions near each of the Co atoms(Fig.11),and continuous,zig-zag shaped distributions along the z-axis (Fig.12),arising from the central dot in the supertile hexagon.These and other smeard-out parts of the Al distribution open diffusion channels.They are suf?ciently close to each other

Figure10:Al motion at T=0.9T m.Dark grey,large:Ni;light grey,large:Co;dark grey, small:Al initial positions;light grey,small:Al positions after1ns.Initial and?nal Al positions are connected.

Figure11:Atom density map,projected on xy plane.Co positions are marked with circles;Ni positions appear as sharp,almost black dots,whereas Al positions are grayish.

that long-range Al diffusion is possible.Continuous channels in the atom density maps have also been observed by Henley et al.[28].Fig.12suggests that a number of aluminium atoms leave the layers,and spend considerable time between the layers.This kind of disorder could explain part of the diffuse diffraction intensity observed in x-ray scattering at high tempera-tures[29].

Due to the high Al diffusivities,it has been possible to measure the Al diffusion constant

Figure 12:Atom density map similar to Fig.11.Here,the same slab of material is shown in projections on the xy-and the xz-plane.The central dot in a supertile hexagon consists of a continuous density along a zig-zag in the z-direction,which provides a diffusion channel.05

10152025024681012

14161820

m e a n s q u a r e d i s p l a c e m e n t [?2]time [ns]T=1010K T=1044K T=1079K T=1114K T=1148K T=1183K T=1218K T=1253K

010

20

30

40

50

60

70

0246810

1214161820

m e a n s q u a r e d i s p l a c e m e n t [?2]time [ns]T=1010K T=1044K T=1079K T=1114K T=1148K T=1183K T=1218K T=1253K Figure 13:Mean quare displacement of Al as a function of time,for different temperatures.Shown are the displacements in the x-direction (top)and the z-direction (bottom).

as a function of temperature and pressure.As expected,it follows the usual Arrhenius law:

D =D 0e ? ?H +p ?V

1e?131e?12

1e?110.80.91 1.1 1.2 1.3

1.4 1.5 1.6D [m 2/s ]1000/T [1/K]z Co?rich x Co?rich z Ni?rich x Ni?rich 1e?12

1e?11

1e?10

0510152025303540D [m 2s ?1]pressure [MPa]

z?direction y?direction x?direction

Figure 14:Left:Arrhenius plot for Al diffusion for the Ni rich and the Co rich decoration.Right:Pressure dependence of diffusion constant.

At ?rst we chose the volume of the system corresponding to a pressure p =0,and made a series of simulations with varying temperature.The resulting mean square displacements are shown in Fig.13.From these curves (and similar ones for the Co rich decoration)the activation enthalpies H and preexponential factors D 0could be determined,separately for the x-,the y-,and the z-direction.The resulting Arrhenius plot (Fig.14,left)shows that the activation enthalpies are almost isotropic in the Co rich case,but somewhat anisotropic in the Ni rich case,and that both the activation enthalpies and the preexponential factors for the Ni rich composition exceed the values for the Co rich composition.Furthermore,the diffusion is isotropic in the xy-plane,as it is imposed by decagonal symmetry.The numerical values are given in Table 2.

Table 2:Activation enthalpies and pre-exponential factors for two decorations.

Decoration

Co rich

0.89

0.64

1.8×10?8

6.3×10?9

high temperatures.A sizable fraction of the Al atoms,namely those near Co atoms,become very mobile at high temperature,which leads to a high Al diffusivity,comparable to that of vacancy diffusion in fcc Al[30].The diffusion we observe cannot be vacancy mediated,how-ever.The activation volume we determined is too small for vacancy diffusion[31].Moreover, there are no vacancies present in the structure,and during the short simulation time vacancies cannot form.Our results do not exclude,however,that there is also vacancy diffusion,once vacancies are present.Mehrer and Galler[31]have found that in decagonal Al-Ni-Co the diffusion of many transition metals,but also that of Zn,is predominantly vacancy mediated, as it is the case for other quasicrystals.The diffusion mechanism we observe must be due to the particular(local)structure of the quasicrystal.

Acknowledgements

We would like to thank Mike Widom and Marek Mihalkoviˇc,who kindly provided the po-tentials and the initial model structures used for the atomistic simulations.This work was supported by Deutsche Forschungsgemeinschaft under Project Ro924/4.

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实用计算器程序 1.基本功能描述 (1)可以计算基本的运算:加法、减法、乘法、除法。 (2)可以进行任意加减乘除混合运算。 (3)可以进行带任意括号的任意混合运算。 (4)可以进行单目科学运算:1/x、+/-、sqrt、x^2、e^2等。 (5)可以对显示进行退格或清除操作。 (6)可以对计算结果自动进行存储,并在用户需要的时候查看,并且可在其基础上进行再运算操作。 (7)界面为科学型和普通型,可在两界面间通过按钮转换。 2.设计思路 计算器属于桌面小程序,适合使用基于对话框的MFC应用程序设计实现。首先要思考的问题是:我的程序需要实现什么样的功能?需要哪些控件?需要哪些变量?需要哪些响应? 我们知道基于对话框的MFC应用程序的执行过程是:初始化、显示对话框,然后就开始跑消息循环列表,当我们在消息循环列表中获取到一个消息后,由相应的消息响应函数执行相应的操作。根据这个流程我们制定出计算器程序的程序框架主流程图,如下页图1所示。 根据程序主流程图可以看出,我们需要一些能响应用户操作的响应函数来实现我们的计算器相应按键的功能。

图1 程序主流程图 说明:所以流程图由深圳市亿图软件有限公司的流程图绘制软件(试用版)绘制,转 存PDF后导出为图片加入到word中的,所以可能会打印效果不好,但确实为本人绘制。

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