Diplomarbeit - Research Group Fidler

2ThesisParticle Dynamics Simulationof the Compaction of Hard Magnetic Powderperformed at the Institute ofAngewandte und Technische Physikder Technischen Universität Wienunder the supervision ofA.o. Univ.Prof. Dr. Josef FidlerDr. Thomas SchreflfromHeinz ZwickMat.Nr. 8526700Haymerlegasse 8/6A-1160 Wien

51 AbstractPowder assembly simulation provide the most detailed information for theoreticalinvestigations of the dynamical properties of the particle behavior. In a usual MDsimulation the particles are simulated explicitly considering force and torque using amolecular dynamics approach. The discrete element method (DEM) is a numericaltechnique that solves engineering problems that are modeled as a large system of distinctinteracting general shaped (deformable or rigid) bodies or particles that are subject to grossmotion. However, the available computer time imposes restrictions on modeling thesesystems: only a few particles can be considered during short simulation times. Theserestrictions can be decreased by applying a hierarchical technique for the calculation of themagnetic interactions and modeling of simple systems such as spheres.We used a real micromagnetic approach with minimization of the total magnetic freeenergy and time integration of the equations of motion alternately which is some kind ofelectrodynamically expanded DEM. The particles interact via linear spring force, magneticdipol forces and friction forces. The torque provided by the magnetic crystalline anisotropycauses the alignment of the particle. The influence of the solution (gas or fluid) is included.The main goal of this thesis is to describe the magnetization of the green compacts ofNd 2 Fe 14 B, SmCo 5 and barium ferrite as a result of processing, external parameters andintrinsic properties. Processing parameters are the applied magnetic field and the isostaticor uniaxial pressing route with wet or dry friction. Internal properties are the particledensity, anisotropy constants and the saturation magnetization of the particles. Externalparameters comprise the particle size, box size and box shape. The ratio of mechanicaltorque versus the torque originating from the magnetocrystalline anisotropy determines theresulting magnetization and compaction defects. This defects occur as a particle densitydefect or a misalignment of the c-axes.The compaction consists of rearrangement and pressure activated processes where theamount of rearrangement processes is greater. The pressure activated processes, where theparticles overcome a volume barrier, determine the amount of defects. This activationprocesses are more dominant in irregular shaped particles due to the coupling of themagnetocrystalline anisotropy constant to the translational degree of freedom.The dynamic behavior of the particle clusters depends on the imposed force (parallel ortransverese to the direction of applied field), but in all cases the chains are desintegrated insmall cluster which are more stable.Our main interest concerns the optimization of sintered Nd 2 Fe 14 B magnets. The pressing ofthe particles as one part of this processing route gives an improvement up to 10 % in theremanent mean magnetization of anisotropic magnets utilizing multiaxial compactioninstead of uniaxial compaction. This holds for spherical particles in contrast to platelets(barium ferrite) where uniaxial compaction is favored due to geometrical constraints.

62 Micromagnetism2.1 IntroductionMagnetic materials are an important group of special materials and are necessary in ourdaily business such as in computers, cars, motors and many other applications. This widerange of applications results in subsequent improvement of the performance in the lastdecade. Both experimental and theoretical investigations has been carried out to understandand optimize magnetic materials. Different manufacturing techniques has been investigatedincluding the sintering route. A variety of models has been used to simulate powdercompaction [1],[4]-[10] which is one part of this route. The mathematical calculation isexact within numerical accuracy in contrast to the prerequisites of the model and theintroduced data, especially surface parameters of the particles. Our micromagneticapproach takes both the electrodynamical and mechanical aspect into account. Theimplementation of the algorithm falls in the realm of scientific computing which is definedas 'the use of high-performance computer technology in an innovative and essential way toaddress and advance the state of knowledge in an applications discipline' [63].2.2 Micromagnetic equationsMicromagnetism includes several theories for the calculation of magnetization processes.In this approximation we consider small volume elements instead of single spins, thatmeans a transition from discrete to continuos values of the magnetization vector. Thevolume element includes at least several crystal cells. Additionally we suppose in oursimulations that the vector length of magnetization stays constant in the probe and rotatescoherently. For sake of simplicity we consider only homogenous, single phase crystals. Inrealitas considering Sm-Co alloys the powder is softened after milling because of workhardening (neutralization of previous cold forming).In the case of coherent rotation and homogenous, isotropic material, the maximumnucleation field is the anisotropy field [38].The magnetization of each particle aligns parallel to the effective field at the particlelocation and therefore minimizes total free magnetic energy . The effective field is the sumof stray fields (own and neighboring particles), crystal anisotropy field and Zeeman field.This process can be calculated either dynamically solving the Gilbert- or Landau-Lifschitzequationor statically with minimum search of the total free magnetic energy with respectto magnetization directions , according to the variational problem3δ∫ E E E E E d r(2.1)δE = { k+A+stat+streu+sp }where E k , E A , E stat , E streu and E sp denote magnetocrystalline anisotropy-, exchange-,magnetostatic-, stray field- and stress anisotropy energy density, respectively.

In the general case there are several minimas. The direction of magnetization adjusts to thenearest minimum (in the direction of negative gradient) in the static case and walks anintricate path (with spiral precession around the changing effective field) in the dynamiccase, but not necessarily to the nearest minimum. An advantage of variational formulationis the insensitivity of the energy minimum regarding the exact functions of energycontributions [45]. Hence one has to choose appropriate functions for the physical problem[54]. The numerical expense increases rapidly with increasing number of variationalparameters. On the other hand convergence decreases with increasing number ofparameters.Within framework of this theory the spontaneous magnetization is described with respectto a cartesian coordinate system.7Magnetocrystalline anisotropy energy densityFe and Co feature a large magnetization and high curie temperature, but low coercivitiesdue to their low magnetic anisotropies. Because of high anisotropies of rare earth,transition metal rich rare earth compound, especially light rare earth because of theirferromagnetic coupling with Fe and Co, are prominent candidates for hard magneticmaterials [65]The origin of the preference direction of magnetic moment in the crystal (cell) is the LScouplingof rare-earth metals (relativistic-quantummechanical description). An externalelectrostatic field or the crystal field of neighboring particles shift the energy levels of anatom in the following way: the higher the binding energy (core electrons) the smaller theshift. A coupling between the states within the atom is only possible in the case of externalinfluence (nonorthogonal eigenfunctions)[53]. Spin and orbit and therefore magneticmoment and quadropol charge distribution of 4-f electrons are coupled. So the rotation ofmagnetic moment causes a rotation of anisotropic charge cloud. Because the electrostaticenergy between 4-f charge distribution and charge distribution of neighboring atoms (onthe lattice sites) increases depending on symmetry, the magnetic 'easy' direction is preferred[42]. In other words energy must be supplied to rotate the magnetization in the lattice.We reiterate that the crystal field establishes two effects, a coupling of eigenfunctions andan spatial anisotropic energy distribution with respect to magnetization direction.The general form of magnetocrystalline anisotropy energy density in the hexagonal casehence follows2466E = K + K sin ( ϕ - θ) + K sin ( ϕ - θ ) + K sin ( ϕ - θ ) + K sin ( ϕ - θ ) cosφ (2.2)k u0 u1u2u3u4and simplified with respect to the leading termEK= − K ( u ⋅ α)2 J = J sα (2.3)uwhere u defines direction of easy c-axis. Angleϕ ,θ and φ describes magnetizationdirection, the direction of easy c-axis and the azimuth difference betweenmagnetization and any crystal axis in the basal plane respectively [38]. J, J s and α denote

9Magnetostatic energy density in the inner field (long range interaction)The magnetostatic energy tries to align the magnetization parallel to the inner field.Generally the inner field is the sum of external field and demagnetizing field (stray field ofall other particles) [40]H int = H ext - H d (2.7)This stray field considerably reduces the coercive field.The contribution of the inner field to the total energy density is given byE stat = -J s ⋅ H int (2.8)The contribution E=-J s H ext is the Zeeman energy. The spatial variation of the field alongthe cross-section is small as compared to the average field. Therefore every sphere can beapproximated with a dipole located in the center [43]EDN N31 µ 3mi ( mjrji ) r0ji= ∫ Edd r = − ∑ ∑5−2 4πi= 1,i≠j j=1 rjim mriji3j(2.9)where m and r denote the magnetic moment and the interdistance vecor between thedipoles. The factor 1/2 indicates no self-energy but originates from the summation.Stray field-, Shape anisotropy energy densityThe own stray field is the energy stored in the magnetic field of a magnetized volume (selfenergy) and is minimal for multidomain particles (net magnetization is zero). Thismagnetic field originates from the finite size of a magnetized body. An anisotropy ofgeometrical shape of a magnetized body can give a magnetic preferential direction (shapeanisotropy) due to the angular dependence of the own stray field [38]The stray field contribution isE s= − 1 s s2 H M (2.10)with the stray field H s =-∇ U and the magnetic scalar potential U obeys the poisson equation∆U = divM s(2.11)The stray field minimizes energy due to the Maxwell law div B=0, closed field lines yieldsminimal stray field energy. In some formulations [2] this field is named demagnetizingfield because interior it opposes the magnetization and the inner field is the sum of external

10and stray field, but the former one is the Zeeman energy and therefore no self energy term(factor 1/2). The stray field energy overcomes the magnetocrystalline anisotropy energyonly in the case of soft magnetic materials.In spherical particles this field is isotropic and constant, in the case of elliptical particleshomogen. The stray field energy depends on the direction cosinus α´x,y,z of magnetizationwith reference to axes of the ellipsoidµEd = 0 M 2 s( N α′ 2 x+ N α′ 2 α β y+ Nχα′2z)2(2.12)and Nα+ Nβ+ Nχ= 1. In the special case of spheres N α= N β= N χ= 1/3This equation is correct under the assumption, that a homogen external field yields ahomogen magnetization in the probe and this homogen magnetization results in a homogenstray field [2]The contribution of N spherical particles to the total energy isE = N2µ 0M s6(2.13)Because of the continuum Maxwell theory this value is exact for particles with cubicsymmetry only. The simple reason originates from the neighboring dipoles in the particleitself. Their magnetization do not necessarily add up to zero (see Lorentz field in thesection definitions).Two types of stray fields have been introduced. H s results from the particles itself and iscalculated from the magnetization M s within the particle. H d is the stray field originatingfrom all other particles [41]Stress anisotropy densityThe stress anisotropy is the energy applied to rotate the spontaneous magnetization in avolume element subjected to stress from the equilibrium to a given position. Thiscontribution is big, and substantial, in the case of soft magnetic materials as FeSi sheetmetal and is small as compared to the other contributions in hard magnetic material.

2.3 Physical Limits of our Simulation ModelSuperparamagnetismThe time for spontaneous reversal of magnetization due to thermal excitations is given byτ−1= τK V−− 1 kT0e11eff(2.14)−1where τ 0denotes the larmor frequency of the particle of volume V. The energy barrierK eff V corresponds to the maximum anisotropy energy. A single spin is able to overcomesmall potential energy barriers due to the thermal energy E th =kT, but the exchange energyfixes domains in relative low energetic positions [2].For spheres with uniaxial symmetry we findK eff =K u1 +K u2 +K u3 +K u4 cosφ (2.15)Thermally stable magnets are in the rangeK u1 V/kT >>1Considering a factor 100 in these energies we obtainkTd > 1200 th 3π K1lower limit of particle size(2.16)This formulae gives a radius of 3 nm for Nd 2 Fe 14 B and an average lifetime betweenfluctuations of 10 years.Multidomain particlesAn optimal sintered magnets consists of single domain particles embedded in anonmagnetic matrix The grains should be magnetically decoupled [12]. The shapeanisotropy energy will be easily minimized by means of a bloch wall, if energeticallypreferred, where the quadropol energy is the half of the dipol energyµ0 ⎛ 4π⎞µ π3120 ⎛ 4 ⎞31R M M = πR 4 K1A + R M M2 ⎝⎜3 ⎠⎟34 ⎝⎜3 ⎠⎟3B0 2s⇒ D < 18µ γcritJcrit s s crit crit s s(2.17)This gives for Nd 2 Fe 14 B with exchange constant A= 7.7 10 -12 A/m and K 1 =4.33 10 6 J/m 3 adiameter of D crit =100 nm [39] for spherical particles and up to 380 nm for polyhedralgrains.Coherent rotation is for small particles dominant (d th

12However after complete saturation (with a puls magnetization device) the particles aresingle domain. Our model takes only monocrystaline particles into account. Hence theupper limit of particle size is in the range of the grain size.Nucleation fieldThe nucleation field of one single particle in a homogeneous external field parallel to theeasy c-axis is governed by∂ET = = 2Ku1 sinϕ cosϕ − µ0MsHNsinϕ= 0∂ϕ(2.18)Hence for the magnetization antiparallel to external field direction followsHNK≅ 2 1 (2.19)JsThe torque exerted by the external field and the crystal anisotropy field balance at thispoint. The magnetization rotates reversibly up to an angle ϕ N where M s spontaneously tiltover to the opposite direction. The rotation takes place uniformly within the particle. Theminimum instability field is obtained for ψ0= π / 4 (angle of homogen external field withrespect to negative easy c-axis) in the case of K 1 >4K 2 [41]KHmin N≅ 1 (2.20)JsThe nucleation field for an ensemble of randomly orientated non interacting single domainspheres is given byHcK≅ 0 48 2 1. Stoner-Wohlfarth value [66]Js(2.21)This value represents the limit of external field for perfect spheres, but in the case ofcoercive field the magnetization of a single particle does not necessarily tilt over but the netmagnetization within the assembly is zero.The nucleation field of rough particles,, i.e. with realistic irregularities, is governed by theequation [51]Hα[( γ γ ) ]1=5− / ∆ − D M2µ0MSRCo O L S(2.22)where γ RCo5and γ Odenotes the domain wall energy for the bulk and around oxideprecipitation respectively, ∆ is the width of region where γ changes from γ RCo5to γ OandD L M S is the local stray field around an irregularity. Actually domain nucleation can occurbefore the internal field in the sphere is zero in agreement with measurements at extremely

13rough surfaces. According to Searle et al in this case the nucleation field is always zero.Experimentally the smoothest spheres yield a roughness depth of about 1 µm.However spheres that were ground in the air-blast grinder for a few days very often show ameasurable H N . In fact there are two concurrent processes involved in establishing thevalue of H N• the likelihood of defects in the particles decreases with decrement of particle size andan increase of H N is expected with decreasing particle size• the likelihood of oxide contamination increases with decreasing particle size due toprolonged milling sequenceLocal stray fields around irregularities (e.g. oxide precipitations) does not increasenecessarily with decreasing particle size ( assuming same roughness).Hence experimentally the total H C remains contant, but scatters considerably due todifferent milling treatments [51]

143 Permanent MagnetsHard magnetic materials base upon alloys of the ferromagnetic transition metals iron,cobalt and nickel as well as the rare-earth metalsThe macroscopic quality characteristics of a permanent magnet are given by• maximum energy density or maximum energy density product (BH) max• squareness• reversible permeability• magnetic ordering temperature T c (curie temperature)The maximum energy density product (BH) max is a measure for the maximummagnetostatic energy per volume unit which is kept up by the magnet in the air gap [38].Hence it defines the size and application of the magnetic system.The squareness of demagnetizing curve (hysteresis loop in 2 nd quadrant) is theoretically upto 1, technically the best values are in the range of 0.74 [6]The reversible permeability describes the system at variable working conditions (gapwidth). Depending on the geometry the system adopts a working point on thedemagnetizing curve.The curie temperature determines the use and employment.remanencehighRE magnetsAl-Ni-Colowhard ferritshighCu-Ni-Colowcoercive fieldFig. 3.1relative size of magnets consisting of different material with same effect: volume proportional (BH) max -1 ,area prop. B r -1 and the length prop. H c -1 [38]Additional requirements depend on the application• high remanence induction• high coercive field• high electrical resistance (eddy currents)• little temperature dependence of remanence and coercive field• chemical resistance• mechanical properties• processing, especially forming

15• availability, costThe microscopic quality characteristics of a permanent magnet (hard magnetic material)are given by• high magnetocrystalline anisotropy• make movement of domain wall most difficult (defects, composition, internalstress)Actually there are 3 microcrystalline mechanisms determining the coercive field, shapecontrolled (Fe-Cr-Co, Al-Ni-Co) , nucleation controlled (hard ferrites, SmCo 5 , Nd 2 Fe 14 B,Sm 2 Fe 17 N 2.7 ) and pinning controlled (Sm 2 Co 17 ) magnets. Fig. 3.2 compares microscopicand macroscopic characteristics during this century.1000K1BHmaxSm Co 2 17Sm Fe N 2 17 2.7SmCo 5RE-Co #18Nd-Fe-B10(BH) max[kJ/m 3 ]10010Fe-Cr-CoAl-Ni-Co#5Al-Ni-Co#9hard ferrit1K 1[MJ/m 3 ]36% Co-steel10.11920 1940 1960 1980 2000yearFig. 3.2The historical development of (BH) max as a macroscopic parameter andthe magnetocrystalline anisotropy as a microscopic parameter [7][12][15][42]3.1 SmCo 5 and Nd-Fe-B based sintered magnetsThe binary system SmCo 5 was developed in the late 1960s together with other RE-metals(Y, Ce, Pr) instead of Sm. Sm yields the most promising results when prepared with thepowder metallurgical process, but the ingots are expensive. However it is one of the mostimportant type of permanent magnets mainly due to the high Curie temperature.

16table 3.1physical properties of SmCo 5density 8200...8400 [kg/m 3 ]E-modulus, Youngs modulus 1,1 10 5 [N/mm 2 ]thermal expansion coefficient [K -1 ]parallel to easy axis +7.10 -6normal to easy axis +13.10 -6specific electric resistance (50 ... 60) 10 -6 [Ω.cm]application temperature 200 ... 250 [°C]diameter without thermal fluctuations d th 3 [nm]grain size 5 ... 10 [µm]remanence polarization J r 0,80 ... 1,15 [T]saturation polarization J S[T]anisotropy constant K 1 17,1 [MJ/m 3 ]coercive field J H c 500 ... 800, max. 3500 [kA/m]maximum energy density product (BH) max up to 200 [kJ/m 3 ]curie temperature T c 700 ... 720 [°C]mechanism of magnetic hardeningFine Particle Magnet,crystal anisotropymagnetizing field (for pulsing up the system) 2500 ... 3000 [kA/m]Nd 2 Fe 14 B magnets (Stadelmaier 1983 and Sagawa 1983,Fujimura 1984) features a highenergy density product up to 250 kJ/m 3 [7], an optimized Nd 15.1 Fe 78.4 B 6.1 Ga 0.1 Cu 0.3sintermagnet yields 360 kJ/m 3 with a remanence of 1.38 T [8]. Therefore this ternarysystem seems promising for applications as small electromotors, clocks, microphones andloudspeakers. Unfortunately it features the plain disadvantage of deterioration of coercivefield beginning at the range of 100°C and therefore are not appropriate for largerelectromotors (testing of power consumption at the temperature of 95°C). Other ingots asin the system Sm-Fe-N improve the temperature stability of this property but they arethermodynamically instable (diffusion of N at high temperatures). Other possibilities toovercome temperature difficulties are cooling of the system either with fans, a coolingsystem, dynamically cooling with the motion of the system itself as in the case ofsubwoofers (loudspeakers) or with ferrofluid vapor in small systems as tweeters(loudspeakers). In the case of ferrofluids the damping decreases the upper vibrations of theloudspeaker membran and cools due to high thermoconductance (Rockford-Fosgate Coop.)Cool particles are attracted from the magnetic system and press the 'hot' particles withlower J s ,due to higher temperature, to the outer, cool wall (self circulating mechanism).However this fascinating process opposites the price of up to 2000$ per liter.Thus the motor of subsequent intensive research is the availability and the low costmaterial for this permanent magnetic system.Intrinsic propertiesThe most important factor is intrinsic properties because they fix processing and externalparameters. The high saturation polarization and the anisotropy constant originates fromVan Hoove singularities (flat bands in the bandstructure yields high density of states)[40]and the spin-orbit coupling [42] respectively. There are 3 principal system available for

17Nd 2 Fe 14 B hard magnetic grains in a nonmagnetic matrix, exchange coupled nanocrystallinegrains and remanence enhanced magnets with hardmagnetic grains embedded in a softmagnetic matrix [42]. In the latter case the exchange coupling aligns the soft magneticferrite parallel to the hard magnetic easy c-axis.table 3.2physical properties of Nd 2 Fe 14 Bdensity 7200 ... 7600 [kg/m 3 ]E-modulus, Youngs modulus 1,4 10 5 [N/mm 2 ]thermal expansion coefficient [K -1 ]parallel to easy axis +3,4 10 -6normal to easy axis -4,8 10 -6specific electric resistance 150 10 -6 [Ω.cm]application temperature 80...180 [°C]diameter without thermal fluctuations d th 5 [nm]grain size 10 (...20) [µm]remanence polarization J r 1,05 ... 1,40 [T]saturation polarization J S 1,61 [T]anisotropy constant K 1 /K 2 4,33 / 0,65 [MJ/m 3 ]coercive field J H c 720 ... 2000 [kA/m]maximum energy density product (BH) max up to 400 [kJ/m 3 ]curie temperature T c 310 [°C]mechanism of magnetic hardeningFine Particle Magnet,magnetocrystallineanisotropyExternal parametersExternal parameters depend on particle size. This size influences the compaction in thefollowing way. Small, monocrystalline particles are pulsed up to a single domain particle inthe case of anisotropic magnets. Small particles features a small strayfield as compared tocrystal anisotropy and therefore the stray field disturbs the alignment during pressing onlyto a small extend. Particle surface factors such as shear forces and shape changesdepending on treatment as milling or hydrogen decripitation (HD).A calibration of the model based on experimental results is necessary to obtain accurateresults.Processing parameters (processing kinetics)Generally there are two processing routes for compaction of particle fractions- compaction with pressure- compaction without pressureThe former processing group comprises uniaxial, biaxial or multiaxial compaction withstatic pressure in matrices with upper and lower punch, isostatic pressing for all-roundcompaction, hot pressing, sinter forging, extrusion pressing and rolling.The latter group includes pouring (filters) , slurry casting (thermoelement tubes) andvibration compaction (pellets).An intensive mixing guarantees a homogen and statistical distribution of particles. Here thepouring density and and pressability should be optimized. The pouring density influencesthe stroke of the punch, the size of the box and therefore the manufactured green compacts

18per time unit. The optimization of pressability results in reduction of pressure for a givendensity of green compact. Hence a high pressure yields a high stability of the compactresulting in improved handling [6].An external field tries to align particles during pressing in the case of anisotropic magnets.Nd 2 Fe 14 B particles (1...5 µm) are aligned in a pulse field (4 10 6 A/m) and pressed (200MPa) parallel or transverse to a static magnetic field (10 6 A/m). Afterwards the greencompact is sintered at 1100°C and quenched (prevention of extensive grain growth). Thesubsequent annealing (630°C) maximizes the coercive field (Fig. 3.3).3.2 Barium FerriteThis oxide permanent system was developed in the 1960's parallel to the Al-Ni-Co system.Actually 3 compositions, BaFe 12 O 19 , SrFe 12 O 19 and PbFe 12 O 19 , can be produced. Only twoof them are fabricated due to special precautions in the Lead Ferrite production (toxicvapors). The important physical properties of the hexagonal Barium Ferrite is listed in thefollowing table.table 3.3physical properties of Ba-Ferrite [38][59]solid matter density4970 ... 5280[kg/m 3 ]raw densitybulk density44201900E-modulus, Youngs modulus [N/mm 2 ]thermal expansion coefficient 10 -7 [K -1 ]specific electric resistance 10 3 .. 10 4 [Ω.cm]application temperature 100 ... 150 [°C]diameter without thermal fluctuations d th >12 [nm]grain size 0,5 ... 100 [µm]remanence polarization J r0,22 ... 0,25 isotropic [T]0,35 ... 0,41 anisotropicsaturation polarization J S 0,43 [T]anisotropy constant K 1 0,320 [MJ/m 3 ]coercive field J H c 320 [kA/m]maximum energy density product (BH) max 8,5 ... 31 [kJ/m 3 ]curie temperature T c 450 [°C]mechanism of magnetic hardeningtexturecrystal anisotropyThe primary product consists either of aggregated raw material produced with a pelletingdisc or the spray dried and pressed raw material. The subsequent synthesis performedforms the desired hexagonal ferrites. The milling gives the desired single domain particles.Oxidation is not as crucial as in Nd 2 Fe 14 B or SmCo 5 fabrication.

19alloyingvacuum melting (NdFeB),inductive or ARC furnaceor(co) reduction NdFeB2Nd 2 O 3 + B 2 O 3 + 28Fe + 9Ca → 2Nd 2 Fe 14 B + 9CaOin hydrogen atmosphere at 1100 to 1200 °Creduction-diffusion SmCo5R 2 O 3 + 10 Co + 3 Ca → 2RCo 5 + 3CaOin hydrogen atmosphere at 1150°C, 3 hourscrushing and millingcrushing. mortar grinding and premilling lumpsjet milling in nitrogen or argon orattrition milling in an organic liquid under inert atmospherecomposition controlandadjustmentoverstoichiometry of RE orblending of RE-rich milled alloy with the milled alloydue to depletion of RE(contamination,particularly oxidation)alignment in magnetic fieldpulsed or static field or combination ofSmCo 5 H ext = 0,96 MA/m (hot pressing) staticNdFeB H ext = 5 ... 10 MA/m staticpressinguni-,bi-, multiaxial, isostatic, rubber isostaticsinteringSmCo 5NdFeBT s =1100°CT s =1100°C, 1 hannealingSmCo 5NdFeB1...2 h800...900°Cquenched after temperingto prevent desintegration fromSmCo 5 to Sm 2 Co 171...3 h900-600°Cmachiningmagnetizinggrinding, cutting, spark machining for complex shapesFig.3.3Processing of Nd 2 Fe 14 B and SmCo 5 sintered magnets [38][56][48]Alternatively the milling can be performed in one processing step by a electromechanicalprocess applying electromagnetic fields. Due to this space and time dependent fields theparticles (with different directions of magnetization due to movement in turbulent fluid or

20gas 1 ) collide and shatter into pieces [59][60]. This route reduces the costs about 90% andgives the fine-disperse ferrite representing the desired grain size distribution with smalldegree of dispersion (≤1.5 µm). Additionally an improvement in the magnetic andmechanical properties of the final product can be observed (and therefore recrystalizationannealing can be chanceled). No premilling steps and working media (steel balls) arenecessary. The wear of the working media yields a contamination of the magnetic powder.Thus considerable reduction of mechanical strength and magnetic properties results insubsequent processes (compaction,sintering). Therefore cost intensive cleaning can bechanceled. Additionally these steel balls and the accompanied stochastic mechanicaltreatment results in damage of crystal lattice and therefore again in deterioration ofmagnetic properties. In the case of conventional milling the powder must be annealed dueto cold working. In electromechanical milling the fracture is initialized at the grainboundaries related basically to the micro flaws existing in the stressed body. Thus aquenching starting from previous calcination results in an improvement of 2% in milledmaterial [59]We reiterate that the milling should result in a magnetic powder or suspension with greatcleanness and small grain size distribution.The subsequent two processing routes yields magnets with different properties. Theisotropic system exhibit a smaller remanence induction (40-50%) and a smaller fabricationcost as compared to the anisotropic systemA partially substitution of Fe 3+ ions (5µ B ) with nonmagnetic ions as Al 3+ ,La 3+ and Ga 3+results in a considerable increase in coercive field (500 kA/m) [38]Hc⎛ 2K⎞u1= α − NMs⎝⎜ µ M ⎠⎟0s(3.1)Unfortunately the remanence decreases. Hence there exist remanence enhanced andcoercive field enhanced type anisotropic ferrite.Finally a high magnetocrystalline anisotropy and a specific texture (grain size, -distribution, pores, -distribution, second phases) yields a high coercive field in connectionwith a high remanence (maximal energy density products)Actually a third processing route has been developed, the aerosol synthesis technique.Unfortunately the remanence (J r =32.5 mT) and the coercive field (H c =49.3 kA/m) are smalldue to noncollinear magnetic structure and surface magnetic dead layers [49]. The particlesexhibit poor crystal perfection (K u1 ≥2.6 kJ/m 3 ) and therefore shape anisotropy gives avalue comparable to crystal anisotropy.However the different forms of fabricated barium ferrite comprise rings, segments, plates,cylinders, discs and cords for refrigerator seals. The application includes speaker systems,electrical machines, measuring devices and so on.1 If the gas carrier stream strongly flows upwards into a granular bed from the bottom, the bed becomesfluidized. Actually speaking, this fluidied bed totally behaves like fluid. Even stronger air flow causesspontanious bubbling. It looks like nothing but Boiling water.

21storage of raw materials in silosFe 2 O 3 , SrCO 3 , BaCO 3 . Suprasil, Al 2 O 3weigh in : Fe 2 O 3 : BaO bzw. SrO = (5,4 ... 5,8) :1mixing :4000 l drum mill,wet; Eirich mixer,drydrying :spray dryer,filter press; pelletizationpresintering :formation of hexagonal ferritsBaCO + Fe O → BaOFe O + CO ↑3 2 3 2 3 2BaOFe O + 5 Fe O → BaO ⋅ 6 Fe O2 3 2 3 2 3isotropicanisotropic1000°C air 1350°C,airmilling: ball triturator (mill) dry attritor wet(Blaine surface:(Blaine surface:3000-4000 m 2 /g) 6000-7000 m 2 /g)forming: spray granulation slurry about 64% solid, rest waterwhirl sintering (powder painting) (0,5...1 µm single domain particlesin suspension)dry pressingwet pressing in magnetic field 5 kA/cmsintering: tunnel furnace 1300°C tunnel furnace 1300°Cgrinding,cutting:diamant grindingdiamant cuttingarming:glue, plast molding to manufacturecompound ceramic-metalmagnetizing:puls magnetizationquality control:quality verificationFig 3.4fabrication of Barium Ferrite [38][15]

4 Micromagnetic 2-D Particle Dynamics22What this model can do ...• Understand the role of individual magnetic systems in the dynamics of magnetically hardparticles with uniaxial preferred magnetic direction• Trace particle trajectories, gaining insight into the physical mechanism of particledynamics• Estimate trends in magnetic parameters such as magnetization• Estimate particle distributions within the box applying different processing parametersWhat this model does not do ...• Particle trajectories are restricted to two dimensions yielding only qualitative results, butdeviations should not be too large• Parameters has been estimated partially with macroscopic values due to lack of database.Therefore they are less reliable under conditions where only few observations exist.The magnetization vector of each particle in the ensemble will align parallel to theeffective inner field, which is the sum of magnetocrystalline anisotropy-, Zeeman- andstray field, and so localizes in the minimum energy state. The approach is to represent eachmajor energy system and then add up the individual contributions and minimize this totalenergy. The force balance gives the response of the system to external and intrinsicparameters. Starting from the total energy of the system subsequent minimization to obtainthe magnetization angles and time integration to derive the motion reflect the dynamics ofthe system (Fig.4.1).We consider the motion of spheres restricted to two dimensions (planar motion). The easyc-axes of the particles lie within the plain and therefore the magnetization is constrained tothis plane (within a deviation of up to 5 deg) We reiterate that owing to magnetostaticinteractions which are small as compared to the magnetocrystalline anisotropy themagnetization will deviate from this plane up to 5 deg.The limit of the packing factor in two dimensions (planar density PD) for equal sizedspheres is 0.907; the most frequent density is about 0.82 (experimentally with resincylinders [33]). The PD-limit of equal sized spheres in three dimensions is of course 0.74,the most frequent density is 0.57 (3D tiling simulation of isostatic pressing [34]). Thecomparison of a 2-D simulation with experimental or 3-D simulations should always takethe different limits of the particle density into account. Two-dimensional MD results canqualitatively only transformed to the 3-D case, when motion of particles is confined such asin our box , but not in the case of free surfaces [29].

23startread parameterssetup system(scaling, positions, angles)energy minimum search withconjugate gradient methodwith respect tomagnetization directions ϕ idynamic startcalculation of nearestneighbor tablewith respect to r < r ccalculation of magnetic forcesfor particles r>r cwithquadratic equal-sized meshintegration of system overcommunication intervall dt searchwith constant time step dtexact calculation of magneticforces for nearest neighborsderivativesnocompactiondensityorcompactionpressurereacheddynamic endyesoutput of final angle of c-axesandfinal coordinatesterminalendFig.4.1flow chart of simulation run

244.1 Setup of systemTable 4.1 lists the material parameters used for the Nd 2 Fe 14 B simulation runs. Our effectivefriction is one order of magnitude smaller than the value for the hard sintered particles. Inour simulations the field of 1.2 kOe and 8 kOe corresponds to an experimental field of12.kOe and the anisotropy field for Nd 2 Fe 14 B respectively. This values are in agreement toour small effective mechanical torque T contact =µF n (at least one order of magnitude smallerthan the experimental torque). The relative increase of the dipole stray field and shear forceas compared to the external field and normal force gives the correct ratio of mechanical tomagnetic torque due to the irregular shape of particles.Table 4.1material properties of Nd 2 Fe 14 B for calculationgeometrical factorsbox size V box4000 µm 2 (t=0)diameter d 5 µm, all particles equal sizednumber of particles N 100particle surface factorsstiffness k n 4.5 10 4 N/mdamping constant γ n1.6 10 -5 Ns/m [5]shear constant γ s0.8 10 -3 Ns/m [5]friction coefficient µ 52 interparticle50 particle-wallphysical factorsdensity 7400 kg/m 3 (Nd 2 Fe 14 B)magnetic polarization J 1.61 T [3]anisotropy constant K u1 4.3 10 6 J/m 3 (Nd 2 Fe 14 B) [3]processing factorsexternal field H ext 0; 10 5 A/m(1.2kOe); 0.637 10 6 A/m (8kOe); 5.4 10 6 A/mcompaction velocity5 m/ssolutionN 2 gas, standard conditionsnumerical factorsatol 10 -2The initial configuration is either random, tiled up in direction of external field or pulsedup to equilibrium (experimental). The difference between the tiling and pulsing consists inthe initial density distribution. In all cases the initial configuration of c-axes can be chooseneither in direction of external field (H→ ∞, Nd 2 Fe 14 B, SmCo 5 and Sm 2 Fe 17 N 2.7 ) orrandomly or the stationary c-axes directions can be calculated (no displacement, onlyrotation without friction). In the case of pulsing the initial configuration is an uniformdistribution (equidistant chains of particles) with friction and applied static field (thepulsing field can be considered to be static during the simulation run of a fewmicroseconds), but without pressing motion. This uniform distribution is calculated untilthe magnetic potential energy and the particle distribution reaches the stationary state (theparticles block together). The calculated postions and angles of c-axes are the input for thesubsequent compaction simulation.

254.2 Calculation of magnetization directionsThe minimization of the total magnetic free energyE = E K + E zeeman + E S (4.1)with respect to magnetization angles ϕ iis solved numerically with a conjugate gradientmethod employing a NAG library routine (evolution of magnetization of sphere describedby energy terms with infinite damping during equilibration 2 , it starts from an initial valueand converge towards an equilibrium solution. The choice of initial solution affectscomputation time and can affect final solution)The uniaxial anisotropy energy approximation with K u1 >0E K = V K u1 sin 2 ( ϕi− θi)(4.2)the Zeeman energyE zeeman = -µ 0 m.H ext cos( ϕi − ψ )(4.3)and the dipole approximation of the stray field energyN N1 µ 3mi ( mjrji ) rjimimj(4.4)0E S= − ∑ ∑5−32 4πi= 1,i≠j j=1 rjirji3πasm Ms=µ0= 4 3sM Jwhere angle of the c-axis θ iof monocrystalline particle is a mechanical variable , but angleof external field ψ and angles of magnetizationϕ i as a result of energy minimization areelectrodynamical ones (Fig 4.2); m i, r jiand V denote magnetic moment, interdistancevector and volume of the spheres respectively. Induced contributions (eddy currents) aredisregarded.The stray field energy of the particles can be neglected because this contribution only shiftsthe total energy but does not change position of minimum.The magnitudes of the energy contribution is E k >E Zeeman >E stray . We assume r ji =const duringthe short adjustment (10 -10 s = minimal time step for minimum search) of magnetizationdirections (vector length remains constant)An approximation of the mechanical rotation time results inIθ = −2VKsin( ϕ − θ ) cos( ϕ − θ )ω2VK≈I2 u1u1i i i i(small amplitudes)(4.5)where I and ω denote the mass momentum of inertia and angular velocity, repectively.2 hence a dynamical conjugate gradient technique seems to be approbriate for finite damping of magnetizationvector during equilibration: the magnetization do not get caught at a flat local minimum

The required maximum time step for minimum search, after which ϕ ihave to berecalculated isdtsearch26= 2 π I(4.6)10 2 VK1uDuring this time step in the order of 10 -8 s the change in E remains small. The subsequentupdate of ϕ iwill not change the torque significantly.zyHφ[0001]θϕxmP(t)x1x2Fig.4.2description of variables4.3 Dynamic motion of particlesThe Newton equations of motions2d xiM2dt2icontactimag= F + F + Fd θiiiI2= Tcontact+ Tmag+ Tdtisolisol(4.7)are decomposed in differential equations of 1.order

27ii i( contact mag sol )1xi= ∫ F + F + FMxii=∆t∆t∆tx dti∆t( contact)ii imag sol1θi= ∫ T + T + T dtIθ =∫∫θ dtidt(4.8)and integrated using a third-order GEAR predictor corrector sheme with constant time step.Here F, T, x and θ denote force, torque, center-of-mass particle position and angle of c-axis, respectively. The choice of anisotropy constant K 1 (high rotational acceleration) orstiffness constant leads to a time step of the order of 10 -13 s (atol denotes a numericaltolerance) [25]⎧5Md I M⎫−44π2π (4.9)dt = atol ⋅ 11210 , ⋅ min ⎨⎪⎬⎪2, ,−2⎩⎪ µ0mK1V5⋅10 k ⎭⎪The time step is governed by the maximum acceleration acting on a particle, resultingeither from the magnetostatic interactions, the magnetocrystalline anisotropy or the contactforces.Higher orders of predictor corrector sheme are recommended [29] because of the intrinicatemotion originating from different time scales in rotation and translation. In our case higherderivations than the acceleration would increase the numerical errors, because of a timestep close to the machine precision.For equations of translation we used magnetic dipole forcesF ( m ∇ ) B(4.10)magn =and linear contact forces originating from the harmonic oscillator potentialF = −∇V(4.11)2knrV ( rij) = ( d − rij)2 θBoth normal and shear components in contact forces are includedijcontact{ n(ij) / 2 γn(ij) / 2}F = k d − r − v ⋅ n nvrel+ − min( γs, µ Fstiff⋅ n ) sign( vrel) s⎧⎨⎩ 2⎫⎬⎭(4.12)where vr s θ θ and n and s the normal (interdistance) and shear unit vector= + d( + )rel ij i jrespectively (more details see [25][26][29]). In our case shear constant γ nis two orders of

28magnitude smaller than γ s. The roughness depth compared to particle diameter is big 3 . Thistwo values are approximately equal in the millimeter range of the spheres. In our modelsurface factors can be important. It was assumed further that the particles do not undergoplastic deformation which holds for the hard Nd 2 Fe 14 B.The influence of the solution is included withFTisolisol= −6πηrv= −ηi4πrVBox4πNθi(4.13)where η and r denote the viscosity coefficient and radius of sphere. The equations ofrotation include contact forces and the magnetic partTTijcontactimagd= − µstiff⋅ sign θi+ θj2 F n ( )E= − ∂ ∂θ =2 1sin( ϕ − θ ) cos( ϕ − θ )iVK u i i i i(4.14)The calculations are performed in usual reduced units. Length are measured in units ofdiameter of the particles, time in units of diameter/pressing velocity, masses in units ofparticle mass.The magnetic dipole forces are calculated exactly during time integration only for nearestneighbors due to time expensive trigonometric functions whereas the magnetic forces fromthe other particles were processed before the integration and introduced with a equal sizedquadratic mesh, with the particle in the center, containing up to 8 particles per cell. Hencefor every particle another mesh was generated. The distance between particle and center ofthe mesh cell is the averaged distance and the sum of magnetic dipole moments in the cellrepresents an effective magnetic moment [37]. This reduces the numerical expense from N 3to N 1.2 where N denotes the number of particles in the box.The simulation terminates at final compaction density of 0.82 or at final end pressurewhich are input parameters.The model was implemented in Fortran 77 code on a Digital Alpha RISC and MIPS64machine in an UNIX and IRIX environment respectively.4.4 ValidationThe total algorithm consists of energy minimization and time integration alternately. Forminimization a NAG routine was utilized, which features a build-in error and validationanalysis too. Several criteria has to be hit for successful minimum search. This criteria canbe controlled with several parameters, including numerical tolerance, supplied to theroutine by the user. See the NAG routine documentation for detail.3 see module documentaion of fcontactlinrol in appendix for details

29The discrete (distinct) element method (DEM) is a family of related techniques designed tosolve problems in applied mechanics, especially problems which exhibit gross motion ordeformation that maybe of discontinuous nature. A DEM-algorithm is a numericaltechnique which solves engineering problems that are modeled as a large system of distinctinteracting general shaped (deformable or rigid) bodies or particles that are subject to grossmotion. Engineering problems that exhibit such large scale discontinuous behavior cannotbe solved with a conventual continuum based procedure such as the Finite ElementMethod. The discrete element procedure is used to determine the dynamic contact topologyof the bodies. It accounts for complex non-linear interaction phenomena between bodiesand numerically solves the equations of motion. Since the DEM is a very computationallyintensive procedure, many existing computer codes are limited to modeling either twodimensionalor small three-dimensional problems that employ simple body geometries.The forces acting on a body in molecular dynamics can originate from all other bodies inthe system, even the bodies not in physical contact with the body being considered andpotential energy of particles is in the order of magnitude of thermal energy kT. In the DEMonly the forces resulting from the physical contact between bodies are used and potentialenergy of particles is orders of magnitude larger than thermal energy. Our problem involvesboth long range magnetostatic (global forces) and short range contact interactions and istherefore some kind of electrodynamically expanded DEM.The long range magnetostatic interaction features no discontinuities and the choose of timeintegration algorithm is not problematic in such a case.Lets consider the problem of contact forces. Several potentials were adapted for MDcalculations such as the soft Lenard Jones for molecule dynamics and a modification of theMoliere potential, the Buckingham potential. Both potentials, with appropriate scaling,feature no discontinuities. While the Lenard-Jones is a polynomial , the 'hard' Buckinghamwith an exponential decay seems appropriate for the ceramic sintered particles. But on thecontrary this advantage opposites the trivial disadvantage of expensive calculation (fourierrow in 64 bit precision) and a small integration time step in connection with implicitintegration routines (expensive calculation of Jacobi matrix) due to the stiffness of system.The potential of harmonic oscillator is appropriate for hard particles and features a linearforce response in agreement with elastic stress-strain-response of metals and ceramics.Unfortunately it features a singularity of order k (stiffness constant) in the secondderivative which yields either expensive calculation (implicit ADAMS routine) ortermination of calculation due to singularities (BDF-routines). A singularity occurs everytime a contact is broken or established. Our integration time step, taken from literature[25], is usable due to the fact of relative accuracy. Library routines such as the NAG libraryproposes an even smaller, but variable time step. To state the validity of a result one mustimpute to it an accuracy related to an understood criterion. This is important, especially inour case, because simulation itself is an approximate process, employing hypothetical orstatistically varying data, constructs and parameters. The process requires the use of anunderstood criterion of validity such as order of magnitude/polarity comparison andestimation (of solution) when subjecting the system to selected stresses [52]. In oursimulation the criteria are• energy propagation and• stability of integration

30In the case without applied torque the c-axes angles should be almost conserved duringtime integration, which reflects the experiment (2 walls hem the particle rotation).Environmental influences can produce small disturbations of the system [53]. But thisholds for molecules. The accuracy of calculated trajectories and stability of integrationrequires a time step far enough smaller than the time where forces on a particle changesignificantly [20] (factor of 1,12⋅10 -4 in eq 4.9) In this context simulation results with aspecified accuracy may be valid for one purpose but invalid for another. The numericaltolerance atol takes the user-desired accuracy into account, but one should be aware of theeffect that the numerical error increases with decreasing time step beyond machineprecision (our Alpha RISC: 10 -16 ). Finally the time step calculated results from the rapidestchanging force (Eq 4.9)Another aspect comprises the stiffness and friction constant. For rough particles the grossstray- field gradient and the gross shear force near the surface appears to be larger than foridealized spheres. This effect takes the choice of the friction coefficient into account. Anenhanced friction coefficient (µ>1) increases the shear constant and takes into account anincreased mechanical torque as compared to elastic repulsive force (dry particle).Additionally the ratio of stray field to normal force is enhanced.However the functionality of the program have to be checked by the user of the programand default values (atol=1) are recommended.Two particle interactions are included both in magnetic and mechanical treatment. Theerror should be small due to small accompanied shift in the magnetic energy and small timestep, respectively.4.5 ClustersParticles with lower symmetry can be treated assuming clustered spheres. Generally twopossibilities are proposed for the treatment of clusters, which differ in numerical expenseand application.4.5.1 Lagrange equations of motion 1 st kind (bonded coordinates in cartesian form)The system consists of N particles with L=L 1 +L 2 +L 3

H e l e n a A l m i r a t i - Mesa deartesanos de MontevideoBuenos días. Voy a hablar desde el sectorartesanal.En primer lugar aclarar que no todoslos artesanos practicamos la economíasolidaria; pero más allá de que el artesanoes un trabajador que muchas vecestrabaja solo, lo hace en general bajo laspautas del trabajo que define la economíasolidaria, y a su vez cuando se asocia,en general lo hace en un relacionamientolaboral con las bases de la economíasolidaria. Entonces a la hora de analizar larealidad del sector artesanal coincide en algunos aspectos con la realidad dela economía solidaria.Segundo, en relación a la economía informal, queremos aclarar que engeneral a lo que habitualmente se considera informalidad no lo llamamosasí, creemos que habría que analizar cuál trabajo es formal y cuál informal.Consideramos si que hay trabajadores que no están incluidos en el sistemade la seguridad social, pero más allá de cómo lo definimos, el tema es cuáles el problema que tenemos y cómo le encontramos una solución.El sector artesanal en su mayoría no está incluido en el sistema de seguridadsocial, y esto implica dos cosas: por un lado que el activo no está aportandopara sostener al pasivo, pero por otro lado que no recibe ningún beneficio.Esa es la realidad del sector, frente a la cual en los últimos años hemosintentado elaborar propuestas: propusimos crear organismos públicos yprivados, así como instancias de diálogo, que busquen encontrar solucionesde acuerdo a la realidad y características del sector artesanal , porque engeneral lo que vemos es que cuando pasa el tiempo y el sector artesanal noestá incluido por más que haya una herramienta, lo que creemos es que esaherramienta no fue la adecuada para la realidad del sector.El monotributo existe desde el año 2002, se modifica en la reforma tributariay luego a través de un decreto reglamentario. En las dos modificacionesque hubo fueron incorporados muchos de los reclamos que hicimos desdeel sector, y creemos que algunos que son muy importantes, no fueron incluidos.Para lo cual a través de los organismos en los cuales está representadoel sector, hicimos algunas propuestas, y actualmente se formó un grupo detrabajo en la órbita del BPS donde se están estudiando las soluciones. Par-KOLPING Uruguay31

32T 2nβ c θ cT 1x c ,y cFig 4.3 reduced coordinate system for a cluster with size 2The Lagrange equations of motion 2 nd kind with clusters up to size 2 have beenimplemented in order to simulate the behavior of a mixture of single spheres and dumbbells.4.6 Statistical evaluation2-D simulation is qualitative and therefore subjective [44], only 3-D simulation yieldsquantitative results. However, a qualitative comparison concerning trends is possible. Agraphical sampling is sufficient in our case. The investigation comprises only the coarsestructure, which parameter effects much to get a robust design. In our case the intrinsicparameters cannot be eliminated, but defused (Tagucchi). Design of Experiments is theproper instrumentPrinciples of Design of Experiments (DoE)• true repeats of experiments possible• randomization: random selection of samples is possible• blocking: uniform experimental conditions within blocks of experiments , the errororiginating from numerical calculation is fixed and cannot be reduced• appropriate arrangement of test points gives better evaluation and interpretation ofresults• all factor-effect interactions should be independent from each otherAdditionally all 'natural' variables as friction, strength and direction of external field,anisotropy constant and particle size can be transformed to overcome subjectiveinterpretation.With 7 factors we find 3×2 7 = 381 experiments (each experiment 3 times for statisticalevaluation). This enormous number have to be reduced to the interesting cases, which issubjective too, due to expensive time for a simulation run.In the evaluation we differ between effect and principal effect. An effect is the response ofthe system to one or more factors. A principal effect is the reaction of the system to onefactor only. The change of response function in connection with value change of one factorgives a numerical definition of effect. In fact there are two definition differing with factor 2

33Only the classical definition applies for qualitative evaluations 4 . Another interesting featureis the effect of interaction between different factors, which is a combined effect of factorson response function: the principal effects are not additive. The definition of the factorinteraction is the ' influence of effect of one factor by means of adjustment of one or moreother factors' [17]The essential results of a distribution is the mean value and the variance or for a smallnumber of particles the median and the interquartil distance. In our simulations thevariances are smaller as compared to 3-D simulations or irregular shaped particles due tothe smaller stray fields for spherical particles. For the evaluation of the results we appliedF-tests [31] to the fitted values comparing the variances of easy c-axes distribution. Thevariance is similar to the quality loss function of Tagucchi, only distributions withcoincidence of mean value with nominal value and minimal dispersion represent theoptimum.Another possibility for evaluation is the Signal-to-Noise ratio. The ratio of thedemagnetizing field to the external field influences this value in the case of anisotropicmagnets. A quantative definition is represented by LOG 10 (m max /m demag ) with the relativemaximum magnetization of the frequency plot (m max

34simple modelprinciple factors (5-20)full factoriell experimenteffects and factor interactionsmax. 4 factorsscatter diagramoptimizing and tolerating factorsprocess comparisoncomparison between oldandnew, improved processingFig. 4.4qualitative DoE (Shainin)The effectivity of DoE for 2, 3 and 4 factors is given by 75%, 88 % and 94% respectively.Hence the variation of more than 4 factors is situated within random error of result (5%).

355. Results and Discussion5.1 Nd 2 Fe 14 B, SmCo 5 and Sm 2 Fe 17 N 2.7 compactionThe main types of compaction processes simulated were isostatic and uniaxial, with orwithout applied field. The 'applied field' means pulsing up the particles in a high field tomagnetic saturation J S and then pressed with a static field whereas 'no field' means that theparticles were pulsed up and then compacted without static field. In the case of prepulsedparticles a big precompaction is desired. This was simulated by tiling up a precompactwhich gives a preferential orientation (chains of dipoles) in the tiling direction (Fig.2). Thisconfiguration reflects the state after pulsing up the system. The simulation time, of course,is shorter too.The processing route without applied static field needs careful handling of the precompactafter pulsing, because the particle system is in an unstable state with high magnetic freeenergy. The system will easily minimize its energy leading to an uniform distribution of thec-axes. Indeed the simulation sequence with no friction shows such a behavior leading touniformly distributed m vectors in the equilibrium. The contact forces blocks the system,so it remains in a high energy stateExtern energy or induced forces such as the pressing machine or even shaking causes thislabile state to move in phase space towards an energy minimum (Fig.5.1). The decrease ofmagnetic potential energy is less significant for isostatic pressing because shorter pressingtime which holds only for the shock pressing velocity.m agnetic potential energy (arb.units)0-200-400-600-800-1000-1200isostatic, no applied external fielduniaxial, no applied external field-14000.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85packing density (% )Fig.5.1energy propagation during pressing of isotropic magnets from transition H ext →∞ to H ext →0. The decrease oftotal free magnetic energy originates from alignment parallel to the inner field. The major contribution is themagnetocrystalline anisotropy energy.Dynamic cluster analysis shows that smaller clusters are more stable due to great innermobility of such chains. Similar results were obtained for nonmagnetic particles too [33] .

36Obviously dry friction would be better than wet friction, but the F-tests carried out on 5%and 1% niveau shows no significant difference. But of course wet friction causes a highergreen compaction density (experimentally about 10 %). This is realized with compactingslurry or with additives like Lithiumstearat [6]60 same time5040isostaticuniaxialfrequency3020100-90 -60 -30 0 30 60 90-10alignment-20-30-40-50-60Fig 5.2aComparison of c-axes distribution at the same time in isotropic magnets. The isostatic and uniaxial sampleexhibits the density of 84 % and 60% respectively1500isostaticuniaxial1000500frequency0-500-50 0 50c-axes-difference-1000-1500same timeFig 5.2bpair correlation function of easy c-axes, the c-axis of one single particle was correlated to all other c-axes dueto long range magnetostatic interaction

38c-axismisalignment= +45°refc-axisrefmisalignment= -45°Reference axis is direction of external field or pressing directionFig.5.3aFig.5.3bIsostatic pressingUniaxial pressingFinal particle distribution in the box. The color scale reflects the alignment of the particles referring to thepulsing field direction with no applied field (maximum ratio strayfield to external field) during pressing92%87%99%96%94%97%89%84%99%Fig.5.4aIsostatic pressingThe misalignment increase to the side with the lowpouring density (top side). Additionally there is agradient in the profile in both directions81%Fig. 5.4bUniaxial pressingThe misalignment exhibit a gradient to the side withthe lower initial density (here the right side) and agradient in the pouring direction (left to right side)The misalignment distribution within the box. The side bands are in the range up to two particle layers. Thegradient is steeper in the case of uniaxial compaction due to the longer stroke during compaction (longertrajectories). The chains at the opposite side disintegrate into small clusters which are slowly shifted into oneanother and thus remain stableAn applied external field during densification gives an other path in phase space.Here zero friction would give the best results because then we have the desired needledistribution instead of a gaussian with an uniform part. The contact forces cause theparticles to misalign during compaction and the anisotropy aligns the particle. Of course, ifthe anisotropy constant K 01 is small the misalignment increases (but within a decrease of10% in K 01 no significant difference occurs).

39Simulations were performed for three cases : H ext = 10 5 A/m (1.2 kOe), Hext= 6.4 10 5 A/m(8 kOe) and H ext = 5.4 10 6 A/m (2K 01 /J s [8]) with the uniaxial route. At high field thesystem shows approximately needle-distribution, but this effect may be described in asimulation using particles with lower rotational symmetry. In other words using spheresone can always find a field strength beyond which a rotation in field direction occur due tolack of geometrical constraints. Where the perfect alignment of c-axes for any kind ofparticles, within geometric constraints, is possible, a perfect alignment of magnetization isimpossible because the compact will easily minimize its magnetic strayfield energy with abloch wall through the pressed block.Comparing isostatic and uniaxial compaction (Fig.5.5) at the static field of 8 kOE nosignificant difference could be found.frequency60H=8kOe50403020100-10-20-30-40-50-60isostaticuniaxialalignmentFig 5.5Comparison of isostatic and uniaxial die pressing for an applied field of 8 kOe during compactionIn realitas this field of 8 kOe represents the anisotropy field due to our small surface (friction) and stiffnessparametersAt this field the misalignment occurs at the final compaction stage and here the pressuredistribution on particle is approximately isotropic in both cases for nonmagnetic particles[30]. In our case of magnetic particles the pressure dependence was found to beanisotropic. Inhomogeneous regions in the uniaxial case according to the preferentialorientation of the dipole chains. The pressure distribution in the box is homogen andisotropic in isostatic pressing. Pressure and stay fields are high at the corners, where ingeneral the mechanical forces are at least one order of magnitude higher. The high speed ofthe pressing punch does not significantly influence this kind of simulations due to strongfield in contrast to the case with no applied field. Here further simulations with 50 µmparticles revealed that the axes distribution is influenced by the characteristic time, theoscillation time of rotation for the free particle t osz =10 t search . If this oscillation time is inthe range of pressing time then the variance remains small: the particle cannot rotate muchin this time. If the pressing time is 10 times larger than the oscillation time the spectrumspreads and we can find a significant difference compared to the previous case. When weconsider an even smaller pressing time, let's say the factor 100, we find no significantdifference to the case with the factor 10. It seems that for the particle there is no difference

40between 10 or 100 oscillations during compaction because in both cases it hashypothetically enough possibilities to assume all possible angles before it stuck. A verylow speed of the punch(es) ensures that the ensemble remains close to the current energystate at all time [44]. The fact, that the particles can oscillate, questions the stability of oursystem or the time integration, because there can be an instantaneous angular velocity dueto small imbalances in torque and numerical errors [30]. But the integration step is fixedsmall enough resulting from the angular acceleration and a tolerance for numerical errors.A global damping factor in the angular equations (4.13) is included. The punch can inducesmall oscillations, in most cases on the punch side as mentioned above. This oscillationincrease with punch speed and anisotropy constant [44]. The substantial characteristic isthat the pressure increases enormously at the very last stage of the compaction [30] andhere the particles stuck.The comparison of direction of external field, parallel or transverse to the pressingdirection in the uniaxial case, shows no significant differences as reported by an early workof Kihara et al [4] in contrast to the newly published release [24]. Kihara et al uses nomicromagnetic model but the possible effect of magnetic coupling between particles iswithin channel width of his histogram; Tamura et al uses an anisotropy torque behaviordepending on rotational velocity. We should mention that the ideal magnetic dipol torquemxH is realized when the coupling between the c-axis and the direction of magnetizationdue to strayfields is infinite. In our model the coupling is realized via the anisotropyconstant K 1 (high K 1 gives strong 'hard' coupling between magnetization directions and c-axes) and the size of the external magnetic field is incorporated implicitly in the angledifference between c-axis and magnetization direction (4.14). In our case this coupling ofm due to dipole strayfields is in the range up to 4 deg per particle.Initial preferential orientation of the particles in the box influences both the axesdistribution and the resulting magnetization: when the applied field is parallel to thepreferential orientation (made up by chains of particles, not c-axes) the result inmagnetization is about 10% higher compared to the case, in which the applied field istransverse to this preferential orientation due to different total magnetic energy. This effectonly occurs at low fields (two orders of magnitude) compared to the anisotropy field. Foran external field in the range of the anisotropy field chains do not influence both themagnetization and axes-distribution, but affect the pressure distribution within the box.

411.000.95normalized magnetization0.900.850.800.750.70isostatic;1.2kOeuniaxial;1.2kOe0.650.5 0.6 0.7 0.8packing density (PD)Fig.5.6gives the magnetization as a function of packing density. The planar density beyond which totalmagnetization adopts a constant value is 0.60. This is 75% of maximum produced packing density. The totalmagnetic free energy increases for higher packing densities owing to friction. The random error of result wasconfirmed with 5 simulation runs in the isostatic case.There exists a critical density (about 0.60 for equal sized spheres) for the particles abovewhich stucking occurs (Fig.5.6). The particles cannot move anymore. Here the frictionshear energy is in the order of the magnetic potential energy and minimizing of this energydoes not monotonly decrease. Sagawa et al [35] reports experimentally from a criticalprecompaction density, above which magnetization (that means particle alignment)decreases, a PD of 0.38 for static isostatic and 0.59 with prepulsed isostatic pressing.Isostatic pressing leads to an improvement of about 10% in magnetization and 30% in(BH) max after sintering as compared to uniaxial pressing. The simulation of isostatic anduniaxial pressing (Fig.5.7) yields this behavior as reported by Sagawa et al.Transverse uniaxial pressing gives an improvement of about 7% in magnetization and 17%in (BH) max as compared to parallel uniaxial pressing 6 . The coercive field H c is not affected.In his experiments the particles with an average grain size of 3.8 µm was pulsed up in afield up to 45 kOe and afterward pressed in a static field of 12 kOe. In our simulation ofthis experiment we used a tiling routine as mentioned above to simulate the prepulsing anda static field of 1.2 kOe to keep simulation time small.6 Here parallelity corresponds to direction of punch and applied static field.

42Fig. 5.7aFig. 5.7bVector force plot for isostatic pressingVector force plot for uniaxial parallel pressingFinal force distribution in the box with applied field H ext =1.2 kOe. The color scale reflects the strain betweenthe particles. Preferential directions and inhomogeneous areas occur only in the case of uniaxial diecompactionfrequency60H=1.2kOe50403020100-10-20-30-40-50-60alignmentisostaticuniaxialFig. 5.8comparison of isostatic and uniaxial pressing at shock pressing conditions (vp=5m/s) with a 'realistic' appliedfield of H ext =1.2 kOeThe energy equilibrium changes between pulsing up the system and the compaction withapplied static field due to field intensity change (relative to the strayfields). This can causea reordering as in the case with no applied field due to different ratio of external field tostrayfield (depends on stability of chains). With no applied field the system minimizesenergy and c-axes align in random directions with normalized magnetization of 2/π andadditionally if the pressing time is in magnitude of typical rotational time the pressingpunch is faster than the minimization of energy and friction holds the system in thisnonequilibrium state. The result at high external field is in general different as compared tolow field. The difference originates from reordering of the system due to energyminimization and external forces. Fig. 5.9 shows how the chains of particles break intosmall clusters during pressing.

4315105total clustersinitialinitial1µs1µsfrequency0-510 20 30 40 50clustersize-10linear clusters-155µm NdFeB particles, transverse compaction, uniform tilingFig. 5.9The initial configuration were 10 chains with equidistant spacing and ideal oriented particles subjected totransverse uniaxial compaction with an applied field of 1.2 kOe. The chains are not pressed together as theyare but disintegrate into small clusters (Poisson distribution). Linearity is assigned for a minimum ratio of 3referring to length versus width. No round clusters with size three could be observed in either case.The mean improvement is about 10% with isostatic compaction compared to uniaxialcompaction, no significant difference between parallel and transverse uniaxial compaction.frequency60H=1.2kOe50403020100-10-20-30-40-50-60NdFeB isostaticFeSmN isostaticalignmentFig.5.10Comparison of Nd 2 Fe 14 B and Sm 2 Fe 17 N 2.7 (J s =0.71 T) with same initial configurationAnother aspect concerns the comparison of the Nd 2 Fe 14 B system to the more recentlydeveloped Fe 17 Sm 2 N 2.7 . The technical realizable Sm-Fe-N system reveals a higheranisotropy constant K 1 =8.9 MJ/m 3 and a smaller saturation polarization of J s =0.71 T [39]as compared to Nd 2 Fe 14 B which results in harder coupling of magnetization with respect toc-axis and smaller demagnetizing field (Fig.5.10). A simulation with the theoretical limit ofJ s =1.52 T results in a larger deviation of magnetization from the c-axes (enhanceddemagnetizing field) and therefore smaller relative magnetization after compaction. But the

44theoretical limit exhibit a polarization twice as for the technical material, but the reductionof relative (normalized) magnetization m/m max is only about 20 %. The original material isa powder with proper pressability and therefore friction is probably smaller than in the casesimulated and density distribution should be better. However Sm 2 Fe 14 N 2.7 cannot beannealed after compaction due to diffusion of nitrogen out of the interstitial lattice sites(problem of thermodynamics), so probably the system can only be compacted with coldhardening binder.60H=1.2kOeNdFeB isostaticSmCo 5isostatic4020frequency0-20alignment-40-60Fig 5.11Comparison of Nd 2 Fe 14 B and SmCo 5The third system considered is the SmCo 5 hard magnetic system (Fig.5.11). The intrinsicvalues for the anisotropy constant and saturation polarization are K 1 =17.1 MJ/m 3 andJ s =1.06 T. This values should result in an even better relative magnetization as compared toNd 2 Fe 14 B, but the deviation is within error of result (Fig.5.12)

45m=99 %isostatic 1.2 kOem=97 %isostatic 6 kOem=96 %uniaxial 1.2 kOem=91 %uniaxial 6 kOe-100 -50 0 50 100Fig.5.12box charts for SmCo 5 with same initial configuration and parameters, only varying external field and pressingroute. The markings indicate 0,1,5,25,50,75,95,99,100 percentiles(50 % of data values within the plotted box)Isostatic compaction results in higher relative magnetization than uniaxial compaction. Theinfluence of field strength is within random error of result.

5.2 Summary of simulation runs with spheres46Several parameters were varied due to processing. But some of the parameters does notchange the behavior significantly due to the simple model, 2-D calculation or insensitivityof variational routine to parameter variation. Hence a dynamic 3-D calculation shouldelucidate this problem. However a DoE with 2 parameters ( 75 % efficiency) can be carriedout (table 5.1; 5.2)preferentialdirectionof forcetable 5.1simulation results of uniaxial compactiontotal 17 runs, 11 out of these 17 are anisotropic compactioninhomogeneousareasof forcemisalignmentincreases towardsouter side of the boxtotal 8 7 8 6anisotropic 6 5 7 4gradientof alignmenttowards low initialpouring densityColumn 1 reads: 8 runs of 17 yield a preferential direction of force, 6 runs out of the 8 runs were anisotropic(with external field)We can derive the following results1) an external field yields an preferential direction (column 1) and inhomogeneous areas(column 2) concerning force. This inhomogeneous areas appear more frequently withincreasing shear force. In the case of rough particles (shear constant 100 times higher ascompared to smooth particles) holes in the compact appear due to a kind of self-hemmechanism . Therefore slurry yields better results. The shear force is a principal factor.2) in the anisotropic case the misalignment at the outer side is at the pressing punch, in theisotropic case the demagnetization (and hence misalignment) takes place uniformly withinthe box (column 3)preferentialdirectionof forcetable 5.2simulation results of isostatic compactiontotal 20 runs, 11 out of these 20 are anisotropic compactioninhomogeneousareasof forcemisalignmentincreases towardsouter side of the boxtotal 0 7 9 5anisotropic 0 6 4 3gradient ofalignmenttowards low intialpouring densityWe derive the new result in the isostatic case, that isostatic pressing yields in neither case apreferential direction in force (column 1)Comparing the uniaxial and isostatic case we find1) the influence of mechanical forces is bigger as compared to magnetic forces (column 1)Especially in the corners the enhanced mechanical forces [44] overcome the enhancedmagnetic strayfield forces. Experimentally the alignment degree (of c-axes) decreases withincreasing pressure [57].

472) less simulations exhibit a gradient in alignment in the isostatic case as compared to theuniaxial case.3) the misalignment does not increase toward the outer side of the box in all cases (50%).This could be due to the finite size effect. Assuming a linear probability for long-rangeinteraction (due to magnetostatic potential)P = d(5.1)rwhere d and r denote the distance between nearest neighbors and othermost particles (inboth directions). The size of the compact should be chosen so large that the energycontribution that would occur for distances larger than L/2 are negligible small to avoidfinite size effects [67]. The size of the box with negligible interaction (

48perfect crystal latticeJ s =380 mT, K 1 =320 kJ/m 3imperfect crystal latticeJs=50.3 mT, K1=2.583 kJ/m 3Fig. 5.13Pulsing of system with same initial configuration , H ext = 500 kA/m but different physical parametersThe left hand picture shows particle stacking in the right-upper cornerThese platelets features the following new characteristics• Rolling can occur (single spheres only jump at this high rotational velocities asmentioned above) but is limited during compaction due to high initial density. But therolling may influence the pulsing procedure.• Torque and therefore magnetocrystalline anisotropy is coupled to translation degree offreedom (equ.4.17) and therefore the forces are in order of magnitude of 'soft' elasticforces. Utilizing a stiffness constant of k n =10 7 N/m and platelets with perfect crystalstructure the strain ε is about 10% of the diameter in our simulation. Actually thestiffness constant of ceramics is one order of magnitude higher, but micro flaws reducesthe necessary force for intra- or intercrystalline fracture. In case of pellets the weakbonded particle agglomerations are easily shattered. This seems to be some kind ofnotch impact and the explanation of microprocesses in electromechanical milling. Themagnetic force results from the interaction between the magnetized ferrite pellets (2-7mm initial size) and the electromagnetic field, which varies with time and position(patented procedure [60]). This results in relative movement of the magnetic parts sothat mechanical stresses ( such as pressure, friction, blow impact, baffle) occur whichleads to a progressive crushing.The magnetic pull consists of 2 main contributions• eddy current contribution with approximately quadratic field strength dependency• magnetocrystalline contribution with approximately linear dependency in anisotropyconstant and applied field strength (varying in space and time)The former contribution is small in ferrites as compared to Nd 2 Fe 14 B due to highspecific electric resistance. However in experimental observation the total magnetic pullis related approximately quadratically to the local field strength and is proportional tothe size of the active interface of the body.• The low symmetry of 2 in contrast to infinite symmetry of spheres. The particle shapesdetermine the pressability and alignment with respect to pressing direction. Thepressability is given by [58]density of green compactpressability =applied pressure(5.2)

49Recrystalization annealing after work hardening and sliding additives increasepressability at a given size distribution of particles. In the case of anisotropic magnetsthe 'sliding' additive is water yielding a decrease of friction coefficient (and shear force).pressing velocity V p = 5 m/s76%V p = 0.5 m/s82%64%72%84%62%87%99%89%shock multiaxial compactiontotal 76 %99%multiaxial compactiontotal 87 %93%99%99%86%77%99%93%96%83%shock parallel uniaxial compactiontotal 87 %78%parallel uniaxial compactiontotal 93 %74%68%79%73%77%97%75%75%99%shock transverse uniaxial compactiontotal 78 %99%transverse uniaxial compactiontotal 80 %Fig. 5.14Boxplots of simulation runs with Ba-ferrite subjected to different conditions. Relative magnetization indifferent areas and total magnetization are depicted. Parallel pressing is more favorable than transversepressing due to particle size [15] Pressing takes place from the right hand side in uniaxial compaction. Themagnetization values correspond to the projection on the x-axis (m.x) Big density gradients in all cases aredue to wrong loading of the die (particle stacking as indicated in the pulsed precompact)

50The symmetry of the platelets influences the alignment in the following way- isostatic : c-axes for platelets are inclined with respect to applied field due to isotropicforce transfer- uniaxial parallel : the expectancy value of c-axis is near zero (more platelets in fielddirection)- uniaxial transverse : the expectancy value is >45° (more platelets transverse to fielddirection)Therefore parallel die pressing yields the best results for Ba-ferrite as depicted in Fig.5.14isostatic compactionuniaxial compactionFig. 5.15Different compaction conditions assuming imperfect or destroyed crystal structure in the anisotropic case(H ext = 50 kA/m) with compaction velocity v p = 0.5 m/sIn the processing route of the Ba-ferrite the particles are agglomerated to pellets (granules).Granules may be described as hard or soft, depending on the type of bonding. In our case itis assumed to be soft (stacking of particles due to friction and magnetic forces).The determination of appropriate conditions for die pressing is crucible due to theimportance in sintering. Minimization of density gradients within sample is critical [61]because distortion in the case of rubber die pressing and cracks both in uniaxial andisostatic pressing results. The influence of loading on density gradients and misalignmentgradients is depicted in the figure 5.15Traditionally the compaction process has been thought to occur in several stage: at lowpressures the granule rearrange with little or no fracture. At higher pressure deformation orfracture of individual grains occur, allowing further rearrangement of the remaininggranules. Finally the particles or granule fragments that comprise initially the granulesshows further fracture or plastic deformation. The evaluation is based on the Pressure-Activation model basing on physical arguments [61]. This model describes two processesthat occur simultaneously during compaction• An initial rearrangement, whose features are that, for small pressure values, the densitybears a linear relation to the applied pressure while it saturates after the pressure exceeds acertain value.• A pressure activated process that occurs when the probability of an appropriate barrierbeing overcome becomes sizable.For sake of simplification we consider the Single-Barrier Model , which can be applied ifthe granules have the same size, nature and environment as in our case. This modelassumes that volume, rather than energy, is been exchanged between systems 7 .7 The independent variable is the compactivity

51ρ =ρ0⎛ ρ − ρ ⎞⎧⎡ ⎛ ⎞⎤−⎢ − −⎝⎜ρ ⎠⎟⎣ ⎝⎜⎠⎟ ⎥ ⎦+ − ⎛⎝⎜ − ⎞⎫∞ 0FFu1 ⎨⎪c 1 exp ( 1 c) exp ⎬⎪⎩⎪⎠⎟∞FlF ⎭⎪(5.3)Here ρ ∞ is the maximum planar density (=0.90) only at infinite force F, and ρ 0 is the initialdensity of compaction at zero force 8 . The rearrangement process and the activation processis characterized by the parameter F l (linear force response) and F a (single barrier activationforce) respectively. F l is the saturation value of the force at which further increase inpressure has no effect during that process. Linearity occurs for FF l . But as F exceeds F a an abrupt increase in the probabilityof void filling occurs (crushing, chipping, fracture), a subsystem overcomes a volumebarrier. Little effect is observed for F

52table 5.3extracted fitting parameters , fitted with the total force utilizing eq. (5.3)compaction rearrangement amount linearity force activation force quality of fit χ 2CF L (arb. units) F a (arb. units)uniaxial v p =5 m/sperfect structure0.76 2.60 10 7 10 10 - 10 12 3.5 10 -4isostatic v p =5 m/sperfect structure0.86 2.85 10 8 10 10 - 10 12 7.9 10 -5isostatic v p =0.5 m/sperfect structure0.59 8.00 10 6 3.70 10 8 8.0 10 -4uniaxial v p =0.5 m/simperfect structure0.65 0.80 10 6 0.90 10 8 4.8 10 -4isostatic v p =0.5 m/simperfect structure0.62 5.20 10 6 2.16 10 8 2.6 10 -4isostaticuniaxialforce exerted on 2 wallsforce exerted on 2 walls0.65 0.70 0.75 0.80 0.85relative densityisostatic (multiaxial-) die pressing ;the force response exhibit a 'smoother' behaviorwhich corresponds to more rearrangements0.65 0.70 0.75 0.80 0.85relative densityuniaxial die pressing;the upper part corresponds to the overcoming of abarrier due to stacking of particlesisostaticuniaxialforce (LOG scaling)force (LOG scaling)0.65 0.70 0.75 0.80 0.85relative densitysame correlation as above, but in LOG 10 scaling forthe force(semi logarithmic plot) An exponentialfitting in one variable for the powder compressionequation seems appropriate as suggested byKonopicky or Nishihara0.65 0.70 0.75 0.80 0.85relative densitysame correlation as above, but in LOG 10 scaling forthe force. Subsequent rearrangement and activationoccurs (fitting in 2 variables). The activation forceincreases progressively. The voids beeing filled is anaddititive quantityFig.5.16pressing with imperfect or destroyed crystal structure (K 1 =2.583 kJ/m 3 , parameters taken from aerosolsynthesized Ba-Ferrite) and v p =0.5 m/s; the scaling of the plots to each other reflects the size of force exertedon two wallsThe big insensitivity of F a corresponds to a great dispersion of this the activation force.Generally this dispersion is smaller for soft particles as compared to hard particles.

Hence a better description is derived, when the activation force is replaced by hisexpectancy value53ρ =ρ0⎛ ρ − ρ ⎞⎧⎡ ⎛ ⎞⎤F max∞ 0F−⎢ − −⎝⎜ρ ⎠⎟⎣ ⎝⎜⎠⎟ ⎥ ⎦+ − ⎛⎝⎜ − Fu⎞1 ⎨⎪c 1 exp ( 1 c) ∫ g( F ) exp⎩⎪⎠⎟∞FFdFal0a⎫⎬⎪⎭⎪(5.4)In the case of shifted-gaussian distribution we deriveg( F ) =aπ1/21⎡ ⎛0Fa⎢1+erf⎣ ⎝⎜2Fσ⎞⎤⎥F⎠⎟⎦σ⎡0( Fa− Fa) ⎤exp−2⎣⎢4F⎦⎥σ(5.5)where F a 0 , F σand erf (x) are the mean value, the spread of the force barrier and theerrorfunction, respectively. The distribution is restricted to F a >0. This case can be treatedanalyticallyρ =ρ0⎛ ρ − ρ ⎞⎧⎡ ⎛ ⎞⎤−⎢ − −β⎝⎜ρ ⎠⎟⎣ ⎝⎜⎠⎟ ⎥ ⎦+ − ⎛⎝⎜ − ⎞⎫∞ 0FFu1 ⎨⎪c 1 exp ( 1 c) exp ( F)⎬⎪⎩⎪⎠⎟∞FlF ⎭⎪⎛σerfc F 0F ⎞a−⎝⎜F 2Fσ⎠⎟⎛ Fβ ( F) =exp⎛0F ⎞ Fa⎝⎜1+erf⎝⎜2F⎠⎟σ2σ2⎞⎠⎟(5.6)(5.7)where erfc (x)=1-erf (x) denotes the complementary errorfunction. The additional featureof this approach is the existence of multiple barriers expressed by an distribution of F a orthe multiplicative factor β ( F ) in agreement with Fig.5.16. This overall pressure-densityrelation does not address the inhomogenities of pressure within the compact, so importantdetails such as the presence of density gradients and the propensity for compacts to containdefects are hidden or lost.isotropic caseuniaxial caseFig. 5.17vector force plot of Ba-Ferrite (perfect lattice) compaction before reaching the final density (-4%).

Latter on we will see that other contribution such as coupling between easy c-axes andtranslational forces and kinetic energy contribute to the problem too. These inhomogenitiesare more important in the case of perfect crystal structure due to enhanced crystalanisotropy constant. Inhomogeneous density regions within the box are filled well in thecase of isotropic magnets due to isotropic pressure as depicted in Fig.5.17 The particlestacking of the dumb bells (coupling of magnetocrystalline torque with translational force)is indicated in the right upper corner as a result of pulsing. Hence the geometricalconstraints inhibit a homogeneous density distribution in the case of uniaxial compaction(additionally the platelets are at the punch side because smaller particles can accumulate atthe bottom, which is a kind of segregation effect)545.4 Isotropic Barium Ferrite MagnetsFor the initial configuration the particle trajectories were calculated until total freemagnetic energy does not change significantly (equilibration of system). This simulationrepresents the die filling of the particles.An additional advantage of isotropic magnet manufacturing consists of the higher greendensity of the compact as compared to the uniaxial case. In the latter case the mechanicalforce increases misalignment approximately quadratically when the green densityconverges to a saturation value [57]. Rubber isostatic case is unfavorable for very highpressures, isostatic compaction in an autoclave is possible. However our calculationterminated at the same final density because our simple model disregards plasticdeformation and crushing.isostatic casedie filling from the top side, at the outer sides thesmall single spheres, in the center the stacked upplateletsuniaxial casedie filling from the right hand side, the single particlein the large pore is surrounded by plateletsFig. 5.18isotropic magnets with different compaction techniques, perfect lattice, pressing velocity v p =5m/sin both cases the expectancy angle of c-axes of the platelets is 45° approximately. Coupling of magnetizationdirection with respect to easy c-axis is up to 14 deg.

555.5 Compaction DefectsCompaction defects can be minimized with simple die shapes, low-aspect-ratio (i.e., heightto diameter) and small frictional effects, but cannot be eliminated. They are present in allpressed powder compacts.A general and coarse inspection comprises• heterogeneous densification• homogeneous densificationThe defects appear• as a particle density defect (PDD)• as a misalignment of easy c-axes (MoC)Heterogeneities are undesirable because they lead to defects after compaction andwarping/cracking after sintering due to different shrinkage. The following list of defectsrefer to Glass and Ewsuk [68]. For a more accurate description of the origins of this defectsrefer to their article. Here a short description and example is provided for each defect withemphasis on the misalignment of easy c-axes. In a compact a single kind or anycombination of this defects are present.Shape distortion is due to density gradients in the compact. Friction yields the particledensity defect, punch impact the misalignment of easy c-axes as mentioned in the chapter5.1density gradientFinal density of 5 µm Nd 2 Fe 14 Bspheres with friction coefficientµ=50, stiffness k n =4.5 10 4 N/m,pressing velocity v p =5 m/s andanisotropic (H ext =1.2kOe)isostatic compaction. Inspite oflow-aspect ratio of the diewrong die filling (initial densitygradient) in connection with therelative high friction (particlejam) yields a density gradientRing Capping is an outer ring separation that forms at the outer edge of the pressing punchface. Ring capping of c-axes is due to the enhanced stray field at the corner, but asmentioned above the final mechanical forces are enhanced too at the corners. A ringcapping of MoC is indicated in Fig.5.3b for an isotropic Nd 2 Fe 14 B magnet (maximum ratiostrayfield to external static field).End capping is a central cone shaped separation. The PDD and MoC forms at any side ofthe compact and at the punch side, respectively.Laminations are periodic circumferential cracks in a compact that originate on the diesurface perpendicular to the pressing direction. The PDD forms when pressing at excessivehigh or low pressures or when die wall friction is high. Laminations of MoC are due toparticle stacking as depicted in the chapter isotropic magnets.

end capping56pore with inclusionpunchsurface defectFinal density of a mixture of Ba-Ferrite particles. (5 µm spheres,dumb-bells with length to widthratio of 2) with frictioncoefficient µ=0.5 , stiffnessk n =4.5 10 7 N/m and pressingvelocity of v p =5 m/s inconnection with uniaxial diepressing (isotropic magnet). Thepore with the inclusion issurrounded by dumb-bells. Thelamination starting from thispore to the bottom of thecompact (not indicated) wouldbe closed at a higher pressure.Die wall sticking could be thereason for such a behavior.Vertical cracks are elongated cracks that form parallel to the pressing direction in theinterior region of a compact. The PDD forms in bodies with an excessively high compactratio (i.e., compact density to die-fill density). High pressing rate as in our case can alsocontribute to the problem.punchcracksSame initial condition as above,but another final arrangement.In the case of isotropicmaterials, especially irregularshaped particles, the particlesperform intrinsic motions andthe surface parameters are notwell-known. Even the simpleexample of a rod in one pointcontact (initially rolling) with atabletop can have multiplesolutions for its impedingmotion (one with continuingrolling, another with theinitiation of sliding)[69]. Theresult may simply reflect thestochastic nature of thecompaction process [68]Large pores in a compact are produced from large and irregularly shaped granules and/orhard (nondeformable ) granules that do not deform sufficiently during compaction. Themagnetic moment near the pore is different as compared to a homogeneous region and canseverely degrade performance of a sintered hard magnet.

57porepunchSame initial condition as above,but another, improved finalarrangement. This is due to alack of spikes in the kineticenergy propagation. The defectsincrease with increasing numberand higth of the spikes. Thenumber and higth of spikes isrelated to the ratio of activationprocesses and the strength of theactivation forces duringcompaction, respectively.end cappingSurface defects including large pores and a cobblestonelike finish are produced from large, irregularly shaped , hard granules and from rough surface dies with high wall friction.Surface defects are critical for hard magnetic materials because they decrease the effectivedemagnetization factor, as mentioned in chapter Physical Limits of our Simulation Model,and deteriorate the demagnetization curve.multiaxialFinal density of 5 µm Nd 2 Fe 14 Bspheres with friction coefficientµ=50, stiffness k n =4.5 10 4 N/m,pressing velocity v p =5 m/s andanisotropic (H ext =1.2kOe)isostatic compaction. Thecobblestonelike finish resultsfrom the huge frictioncoefficient.cobblestonelikefinishSame as above, but with frictioncoefficient µ=5. The compacthas a smoother surface. In bothcases the kinetic energy hassimilar behavior.multiaxialsmooth finish

586 SummaryA simple model for powder compaction of nucleation controlled hard magnetic materialsfeaturing single crystals has been developed and validated. The Coulomb friction law hasbeen adopted for µm particles. Isostatic and uniaxial pressing were compared in theprepulsed and static case. Both single spheres and a mixture of single spheres and dumbbellshas been utilized to calculate the systems Nd 2 Fe 14 B, Sm 2 Fe 17 N 2.7 , SmCo 5 and Ba-Ferrite.Alignment of c-axes, gradients of alignment, particle distribution and arrangement, forcedistribution and force gradients has been investigated both in the isostatic (multiaxial) anduniaxial pressing route. A theoretical model for rearrangement and activation forces hasbeen used to evaluate pore filling processes during compaction.The milling sequence influences sensitively the physical ( including intrinsic) properties ofthe final powder.For the single spheres an improvement of 10% in magnetization has been calculated forisostatic compaction as compared to uniaxial parallel pressing. In the case of the mixture(easy c-axes normal to plane of 'platelets') the uniaxial parallel pressing route is mostfavorable.In the manufacturing of isotropic magnets the misalignment is not significantly influencedby the applied final pressure. On the other hand in the case of anisotropic magnets theparticle misalignment decreases quadratically with the applied pressure (above 100MPa)[57]. Therefore isostatic pressing is favorable for anisotropic magnets.Further improvements of this program can include different particle sizes and shapes,different anisotropy constants for the particles, different cluster geometries and differentdie shapes. Pellets and their crushing can be taken into account with variable constraintconditions.

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61[29] S.Dippel,G.G.Batrouni and D.E.WolfCollision-induced friction in the motion of a single particle on a bumpy inclinedlinePhys.Rev.E, Vol 54, number 6, Dec. 1996.[30] Jianmin Lian and Susumu ShimaPowder assembly simulation method by particle dynamics methodInternational Journal for Numerical methods in Engineering, Vol.37, 1994.[31] DIN 53804 T1Statistical evaluation,measurable (continuos) characteristicsDIN Deutsches Institut für Normung e.V. 1989.[32] A.S. Lemak and N.K. BalabaevMolecular Dynamics Simulation of a Polymer Chain in Solution by CollisionalDynamics MethodJ.Comput.Chem., Vol.17, No.15,1685-95 (1996).[33] Takayasu IkegamiTheoretical description of a Two dimensional Compaction Process of CylindersJ.Am.Ceram.Soc. 790, 153-60 (1996).[34] G.Q.Lu, X.ShiComputer simulation of isostatic powder compaction by random packing ofmonosized particlesJ.Mat.Sci.Letters 13, p.1709-11, 1994.[35] M. Sagawa and H. NagataNovel Processing Technology for Permanent MagnetsIEEE Transaction on Magnetics, Vol.29, No.6, Nov. 1993.[36] J.D.Park,T.S. Jang,W.Y.Jeung,C.S. KwakThe effect of stress on the magnetic alignment of hot-pressed FeNdB magnetsderived by computer simulationIEEE Transactions on Magnetics, Vol.29, No6, Nov.1993.[37] Miles,J.J. and Middleton,B.K.A hierarchical micromagnetic model of longitudinal thin film recording mediaJ.Magnetism Magn.Mat. 95 (99-108) 1991.[38] Michalowsky L., Heinecke U., Schneider J., Wich H.Magnettechnik, Grundlagen und AnwendungenFachbuchverlag Leipzig-Köln 1995, ISBN 3-343-00897-4.[39] Katter M.New Rare-Earth-Iron based Hard-magnetic MaterialsDiss. 1991, TU-Wien.[40] Mohn P.Magnetism in the Solid StateVorlesungsskriptum 1997.[41] Kronmüller H.Mikromagnetism And magnetization processes in modern magnetic materialsScience and Technology of Nanostructured Magnetic Materials,Plenum Press, N.Y. 1991.

63[57] O.B.G. Assis, V.Sinka, M.FerranteThe influence of pressing pressure on particle alignment in Nd-Fe-B greencompacts, J.Mat.Sci.Letters 13 (1994) 1141-1143.[58] Fritz, Schulze, Haage, Knipfelberg, Kühn, RohdeFertigungstechnikVDI Verlag, 2. Auflage 1990.[59] Halbedel, KillatElektromechanische Autogenmahlung von HartferritenAufbereitungstechnik 34 (1993), Nr.10.[60] Halbedel, KarstenVerfahren zur autogenen Zerkleinerung von HartferritenOffenlegungsschrift DE 41 29 360 A1, Deutsches Patentamt 1993.[61] Kenkre, Endicott, Glass, HurdA Theoretical Model for Compaction of Granular MaterialsJ.Am.Ceram.Soc. 79 (12) p.3045 f. (1996).[62] H.M.Jaeger and S.R.NagelPhysics of the granular stateScience, Vol.255,1523-1531, 1992.[63] A. I. HustrulidParallel Implementation of the Discrete Element MethodEngineering Division, Colorado School of Mines.[64] D.J.BensonThe calculation of the shock velocity - particle velocity relationship for a copperpowder by direct numerical calculationWave Motion 21 (1995) 85-99, Elsevier Science B.V.[65] R.F.Sabiryanov, S.S.JaswalAb Initio Calculations of the Curie Temperature of Complex Permanent-MagnetMaterials, Phys.Rev.Lett. Vol 79, Nr.1, 1997.[66] E.C.Stoner, E.P.WohlfarthPhil.Trans.R.Soc. 240 599 (1948).[67] D. W. HeermannComputer Simulation Methods in Theoretical PhysicsSpringer Verlag 1986.[68] S.J. Glass and K.G. EwsukCeramic Powder CompactionMRS Bulletin, December 1997.[69] Trinkle J.C.; Pang J.-S.; Sudarsky s.; Lo G.On Dynamic Multi-Rigid-Body Contact Problems with Coulomb FrictionZAMM, Zeitschrift Angewandte Math. Mech. 77 (1997) 4, 267-279.

648 Appendixinformation-QQ-requirements of customermachine performance-Qservice-Qevaluation-Qdesign -Qqualificationofpersonneltesting-Qdata-Qcompiler-Qnumerical-Qvalidation-Qquality-loop of model development8.1 Module documentation (design and testing)readfile, simuldata,writefilea. purposeread parameters from input files and write data on output fileb. specificationsubroutine readfile (user,title,ctitle,number)implicit nonecharacter*40 ctitle(100),titleinteger numberreal*8 user(900)C-------------------------------------------------------------------subroutine simuldata (ctitle,tanf,tend,tstep,integrator,& n,h0,hmin,hmax,jceval,rtol,atol,petzld,maxstp,& norm,itask,itrace,sens,u,eta,mode,mult)implicit nonecharacter*1 integrator,jceval,norm

65real*8 h0,hmin,hmax,rtol(1200),atol(1)integer n ,maxstp,itask,itrace,modereal*8 tanf,tend,tstep,sens,u,eta,multcharacter*40 ctitle(100)logical petzldC---------------------------------------------------------------------------subroutine writefile (user,title,ctitle,number,tanf,tend,tstep,& integrator,h0,hmin,hmax,jceval,rtol,atol,petzld,& maxstp,norm,itask,itrace,sens,u,eta,mode,mult )implicit nonec. description//sintdata.f// reads data from input file SINTDATA.D , transforms the data to SI-systemand calculates some constants as volume,viscosity, etc. from input data//simuldata.f// reads data from input file SIMULDATA.D//writefile.f// writes both the data from //sintdata.f// and //simuldata.f// and, additionally,the initial coordinates, c-axes and other calculated data as the smallest single sphereparticle for given intrinsic parameters, etc. on the file SINTOUT.Dd. referencesnonef. parameters1: user - real arrayarray containing the packed data, see//*convert*.f// module documentations3: ctitle - character arraycontain the name of parameters5: tanf - realstarting time for integration7: tstep - realtime step for output of data (coordinates, c-axes, clusters)9: atol - real arrayatol(1) contains the user desired accuracy ofthe integration time step11: mult - realmultiplicator for total energy and gradientsin the minimum search routine; should bemult=1 with appropriate scaling2: title - charactercontains title of input file4: number - integernumber of particles6: tend - realtermination time of integration8: integrator - characterfixes the integration routine10: mode - integerfixes the input mode for c-axes, see//caxis.f// module documentation12: petzld - logicalpetzld=.TRUE. : performs convergence testin GEAR integration routinepetzld=.FALSE. : no convergence test,otherwise for NAG integration routinethe other inputvaraibles in //simuldata.f// are for NAG integration routines only and has notbeen used in our simulation yet.g. error indicators and warnings'error opening file SINTDATA''error reading data from SINTDATA''unexpected EOF in SINTDATA''error closing file SINTDATA'h. further comments//sintdata.f// packs the data in the user - array onlythe data from //simuldata.f// must not be packed

66the implicit none fortran statement is recommendend, because in this case the user mostexplicitly declare all variable and constant types and no implicit type assignment, whichcan be an error source, is performed.caxis,readkoord,xyposa. purposeassigns initial coordinates and angles of c-axesb. specificationsubroutine caxis (theta,psi,n,mode)implicit noneinteger n,modereal *8 theta(n),psireal*8 g05dafC-----------------------------------------------------------------C read coordinates from file 'xykoord.d'subroutine readkoord (xkoord,ykoord,user,n,nmax)implicit nonereal*8 xkoord(200),ykoord(200),user(900)integer n,nmaxC---------------------------------------------------subroutine xypos(xkoord,ykoord,user,n)implicit noneinteger nreal *8 user(900), xkoord(n),ykoord(n)real*8 g05dafc. description//caxis.f// initializes the c-axes of particles with the following optionsMODE=0 totally random from 0...2*PI (uniform distribution)MODE=1 random along or opposite external fieldMODE=2 read from file CCAXIS.D//readkoord.d// reads data from the standard coordinate file XYKOORD.D and positionsthe punch at the outermost right side of the particles//xypos.f// positions the particles within the box either randomly (box very large) orordered randomly (medium box), that means the box is divided into cells and these cellsare occupied randomly. For a small box this routine is terminated with an error indicator.d. referencesnonef. parameters1: theta - real arrayangles of c-axes3: n - integeron exit: number of particles5: xkoord - real arrayx-coordinates of particles7: user - real arraypacked data, see //*convert*.f// modules2: psi - real arrayangle of external field at particle position(for inhomogeneous external field), not used4: mode - integerinitialization mode for c-axes6: ykoord - real arrayy-coordinates of particles8: nmax - integeron entry: maximum number of particles=200

67g. error indicators and warnings//readkoord.f// 'error opening KOORD.D''error reading data from KOORD.D'//xypos.f// 'warning: pressing area may be too small for random distribution'h. further comments//readkoord.f// is called prior to //caxis.f// in the main program, therefore the number ofcoordinates found in the input file XYKOORD.D determines the number of angles of c-axes.scalinga. purposesubroutine for scaling SI data to natural unitsb. specificationsubroutine scaling (user,xkoord,ykoord,tanf,tend,tstep,& h0,hmin,hmax,n)implicit noneinteger nreal*8 user(900),xkoord(n),ykoord(n),tanf,tend,tstepreal*8 h0,hmin,hmaxc. descriptionscaling on diameter (length),mass (mass),vpress/diameter (time), thus for micrometerparticles the time is microseconds and for millimeter particles the time is milliseconds; ifvpress is approximately zero time is scaled with 1/diameter.d. references[32] Molecular Dynamics Simulation of a Polymer Chain in Solution by CollisionalDynamics Methodf. parameters1: h0 - realfirst time step taken in integration routinefor NAG routine only, not used in GEARroutines2: hmin - realminimum time step allowed duringintegration, for NAG routines only3: hmax - realmaximum time step allowed duringintegration, for NAG routines onlyother parameters see //readkoord.f// and //simuldata.f// module documentationg. error indicators and warningsexchange energy and boltzmann factor are not convertedh. further commentsevery new physical parameter must be scaled accordingly (and eventually packed into theUSER array afterwards), so the user is recommended to test the array he submits to anyroutine if it contains the up-to-date data.

68eminimuma. purposecalling routine for minimum searchb. specificationsubroutine eminimum (n,x,iuser,user,objf,ifail,check,check1)implicit nonelogical check,check1integer nreal*8 X(n)real*8 OBJF,USER(900)integer IUSER(1),IFAILreal*8 X02AJFreal*8 g05dafexternal OBJFUNc. descriptionthis routine performs the following tasks: it checks, if the gradient is too small forminimum search and shifts randomly the initial position (cheap test,optionally); it checks ,if the gradient is too small for minimum search and assumes a minimum (cheaptest,optionally); it calls the minimum search routine //E04DGF// from the NAG library.The new version of the NAG-library includes minimum search routines with the secondderivative and therefore this subroutine, when changing to the new version, is unnecessary.d. referencesNAG library documentationf. parameters1: n - integernumber of particles , n≤2002: x - real arraymagnetization directions (angles) supplied in units of radians3: iuser - integer arraynot used, confer NAG library documentation for //E04DGF//4: user - real arraythis array includes all physical parameters, angles of c-axes, coordinates. In the routine//objfun// this array is unpacked and assigned more 'conceivable' or 'readable' variables asxkoord(), ykoord(). This solution was used to avoid COMMON blocks and errors ondifferent machines.5: objf - realthe value of the objective function at x6: ifail - integeron exit: IFAIL = 0 until the NAG routine detects an error or gives a warning7: check - logicalcheck = .TRUE. : testing if minimum or maximum point and shift position, if necessary8: check1 -l ogicalcheck1 = .TRUE. : for subsequent calls in integration routine, only if a minimum point isassumed9: objfun - externalsubroutine, which evaluates the objective function and gradients for minimu search routineg. error indicators and warningsbecause the values of the output parameters of the NAG routine are even useful if IFAIL≠0on exit, this value is set to IFAIL=-1 before entry. Therefore the NAG routine does not

terminate execution of the whole program but writes an error message on standard outputunit. These SOFT errors have not appeared in the usual use of the package.69objfuna. purposesubroutine //objfun.f// for NAG routine //E04DGF// (calculation of energy minimum)b. specificationsubroutine OBJFUN (MODE,N,X,OBJF,OBJGRD,NSTATE,IUSER,USER)implicit noneinteger MODE,N,NSTATE,IUSER(1)real*8 x(N),OBJF,objgrd(N),user(900)c. descriptioncalculates the objective function F(x) and its gradient g(x)=Nabla f(x) for a specified nelement vector xd. references[1] NAG F77 manualf. parameters1: objf - realthe objective function2: N - integerthe number of variables, N

7024∑{ 0 1 sin ( ϕi θi ) u2sin ( ϕi θi) }E = V K + K − + K −kristt u ui=1∂E∂ϕkristiiNN⎡ ∂ϕj∂θ3j= V∑{ 2Ku 1sin( ϕj− θj) cos( ϕj− θj) + 4Ku2sin ( ϕj− θj) cos( ϕj− θj)}−⎣⎢∂ϕ ∂ϕj=1∂θj= 0 (independent variable)∂ϕ1 1EForm= − ∑µ 0mi( − M)= const2 i 3(sphere)∂E= 0£∂ϕ¢i¡N¢N1 µ 3mi ( mjrji ) rjimim0jEStreu= − ∑ ∑5−32 4πi= 1,i≠j j=1 r rmrEijii= ⎛ cosϕ⎞m⎝ ⎜ ⎟sin ϕ ⎠⎛ x= ⎜⎝ yStreu3−∂Eiii− x− yjj⎞⎟⎠1 µ0m=2 4π2i= 1,i≠j j=1r[ cos ϕj( xi − xj) + sin ϕj( yi − yj)]5[ cos ϕi ( xi − xj) + sin ϕi ( yi− yj ]∂ϕ1 µ0m=2 4πrNjiN∑ ∑jicosϕ cosϕ + sinϕ sinϕ+ − cosϕ + jsinϕiδik sin ϕjcosϕiδik3−r3−3−streuk2∑ ∑i= 1,i≠j j=1jij i j iji3[ − sin ϕjδjk( xi − xj) + cos ϕjδjk( yi − yj)] 5[ cos ϕi ( xi xj) sin ϕi ( yi yj) ]r[ cos ϕj( xi − xj) + sin ϕj( yi − yj)]5[ − sin ϕiδik ( xi − xj) + cosϕiδik ( y i− yj)]rjiNjiNji− sinϕ cosϕ δ + cosϕ sinϕ δr−j i jk j i jkji3+− + − −j. Testing of Algorithm• case of anisotropy field and external field: particle magnetization and therefore c-axesrotates in direction of applied field: location of minimum• case of anisotropy field and stray field of fixed other particles: same as aboveii⎤⎦⎥

71convert, backconvert, userconvert, userbackconverta. purposeconverting data for minimizing routine,pack and unpack the user array into 'conceivable' and readable arraysb. specificationsubroutine convert (user,higth,width,dens,diameter,stiff,& friction,frictionw,hext,psi0,pres,k0,k1,mass,imass,mag,vol,& my,pi,daempf,k2,vpress,etai,& a,b,e,sigma)implicit nonereal*8 user (900),higth,width,dens,diameter,stiff,pireal*8 friction,frictionw,hext,psi0,k0,k1,mass,imassreal*8 mag,vol,my,pres,k2,vpress,etai,a,breal*8 daempf,e,sigmaC-------------------------------------------------------subroutine backconvert (user,higth,width,dens,diameter,stiff,& friction,frictionw,hext,psi0,pres,k0,k1,mass,imass,mag,vol,& my,pi,daempf,k2,vpress,etai,& theta,xkoord,ykoord,psi)implicit nonereal*8 user (900),higth,width,dens,diameter,stiff,pireal*8 friction,frictionw,hext,psi0,k0,k1,mass,imassreal*8 mag,vol,my,pres,k2,vpress,etaireal*8 theta (200),psi(200),xkoord(200),ykoord(200),daempfC------------------------------------------------------------------subroutine userconvert (n,xkoord,ykoord,theta,user,psi)implicit noneinteger nreal*8 user(900)real*8 xkoord(n),ykoord(n),theta(n),psi(n)C-------------------------------------------------------------------subroutine userbackconvert (n,xkoord,ykoord,theta,user,psi)implicit nonec. descriptionParameters are submitted to the minimum search routine //E04DGF// via the USER -arrays(to avoid COMMON blocks)//convert.f// unpacks the USER array with respect to parameters//backconvert.f// packs the parameters into the USER array, counterpart of //convert.f////userconvert.f// packs the coordinates,c-axes, angle of external field into the USER array//userbackconvert.f// unpacks the USER array, counterpart of //userconvert.f//d. referencesnonef. parameters1: user - real array2: friction - realarray of packed variables3: frictionw - realfriction particle - wallinterparticle friction4: hext - realstrength of external magnetic field

725: higth - real6: width -realhigth of boxwith of box7: psi0 - real8: dens - realangle of external field in radiansdensity of particle ρ9: diameter - real10: pres - realdiameter of particlesmaximum (termination) pressing force11: stiff - real12: k0 - realstiffness constantmagnetocrystalline anisotropy constant K u013: k1 - real14: mass - realmagnetocrystalline anisotropy constant K u1 mass of oneparticle15: imass - real16: mag - realmass moment of inertia of one particle magnetic moment of one particle17: vol - real18: my - realvolume of one particlemy=4π 10 -719: pi - real20: daempf - realpi=3.141...damping constant γ n21: k2 - real22: vpress - realmagnetocrystalline anisotropy constant K u2 pressing velocity23: etai - real24: a - realviscosity coefficientnot used25: b - real26: e - realnot usednot used27: sigma - real28: n - integernot usednumber of particles29: xkoord - real array30: ykoord - real arrayx- coordinates of particlesy- coordinates of particles31: theta - real array32: psi - real arrayangles of c-axesnot usedg. error indicators and warningsnoneh. commentswhen a parameter or coordinate is changed, the cooresponding USER entry has to bechanged too !!xyconvert, xybackconverta. purposepacks and unpacks coordinates and velocities for use with an integratorb. specificationsubroutine xyconvert (n,xkoord,ykoord,theta,y)implicit noneinteger nreal*8 xkoord(n),ykoord(n),y(6*n),theta(n)C -----------------------------------------------------------subroutine xybackconvert (n,xkoord,ykoord,theta,y)implicit nonec. descriptionAccording to NAG routines the integration of equations of motions needs two sets ofvariables: the independent and dependent variables. The vector of functions correspond tothe vector of variables in the first order differential equations system. Therefore the

73independent variables (coordinates, angles of c-axes, velocities and angular velocities) arepacked into one array y(). Therefore NAG integration routines and routines from othersource (NIST) can be used.//xybackconvert.f// packs set of independent variables into independent variable y()// xyconvert.f// unpacks y()d. referencesNAG manualf. parameters1: n - integer2: xkoord - real arraynumber of particlesx-coordinates of particles3: ykoord - real array4: theta - real arrayy-coordinates of particlesangles of c-axes5: y - real arraycontains angles of c-axes, angular velocities, x-coordinates, velocities in x-direction, y-coordinates, velocities in y-directiong. error indicators and warningsnoneneighblist, mesha. purposepreprocessing for hierarchical techniqueb. specificationsubroutine neighblist (y,molij,neighb,ra,n)implicit noneinteger nreal*8 y(6*n), raC -------------------------------------------------------------------------------subroutine mesh (y,x,fmagxmesh,fmagymesh,mag,my,pi,rt,higth,& actwidth,n)implicit noneinteger nreal*8 mag,my,pi,rt,higth,actwidthreal*8 y(6*n),x(n),fmagxmesh(n),fmagymesh(n)c. description//neighb.f// is a subroutine for calculating the nearest-neighbor-list for all particles,maximum 50 neighbors per particle//mesh.f// is a subroutine for calculating mesh in box and calculates the forces with anoverall value of magnetization in mesh elements with exception of nearest neighbors,which are calculated exactly with nearest-neighbor-listd. references[37] A hierarchical micromagnetic model of longitudinal thin film recording media[20] Molekulardynamik, Grundlagen und Anwendungenf. parameters1: pi - realpi=2*asin(1.)=3.14159...2: my - realmy=4π 10 -7 (not scaled)3: N - integernumber of particles, N

744: x - real arraydirections of magnetizations (angles with respect to polar coordinates)5: y - real arraycontains angles of c-axes, angular velocities, x-coordinates, velocities in x-direction, y-coordinates, velocities in y-directiony(i) theta y(i+n) theta'y(i+2n) x(i) y(i+3n) x'y(i+4n) y y(i+5n) y'6: mag - realmagnetization moment of one sphere =M s V7: neighb - integer arrayelement i contains the quantity of nearest neighbors for particle i8: molij - integer arraymolij (i,j) contains the explicit numbers of the particles , which are nearest neighbors(j=0,...,neighb(i)) for the particle i. For example, for particle 1 the particles 4 and 11 arenearest neighbors: neighb(1)=2, molij(1,1)=4, molij(1,2)=119: fmagxmesh - real array outputx-component of magnetic forces acting upon particle i, i=1,...,N calculated prior to timeintegration with hierarchical technique10: fmagymesh - real array outputy-component of magnetic forces, confer 9:11: higth - realactual higth of box12: actwidth - realactual width of box13: rt,ra - realsame parameter, indicates the size of one mesh element = ( rt × rt )g. error indicators and warningsnonei. Algorithmic detailsa mesh is created around every particle, that means there is not one absolute meshdistribution, but n meshs with respect to actual particle coordinates. The nearest neighborsconsists of particles in the innermost mesh element for a specified particle.fbody, fbodyneigh, fbodymesha. purposecalculates the magnetic forces exterted on particle i, i=1,...,N either including all particles//fbody.f // or with nearest neighbors table //fbodyneigh.f// or in connection with the meshroutine //fbodymesh.f//b. specificationsubroutine fbody (n,x,xforce,yforce,mag,my,pi,y)implicit nonereal*8 mag,my, piinteger nreal *8 x(200) ,y(1200), xforce (200),yforce (200)C--------------------------------------------------------------------C bodyforces with neighborlistsubroutine fbodyneigh (y,x,xforce,yforce,molij,neighb,mag,my,pi,n)implicit none

integer ninteger neighb(n),molij(n,50)real*8 mag,my, pireal*8 y(6*n)real *8 x(n), xforce (n),yforce (n)C--------------------------------------------------------------------C bodyforces for meshroutinesubroutine fbodymesh (xzell,yzell,xlimu,xlimo,ylimu,ylimo,& my,pi,fx,fy,xi,yi,angi)implicit noneinteger nreal*8 pi, angireal*8 my,xzell,yzell,xlimu,xlimo,ylimu,ylimo,fx,fy,xi,yireal *8 x(200), y(1200)real*8 xforce (200),yforce (200)75c. descriptionthis 3 routines are designed to calculate magnetic forces in the 2-D case. The first routineuses all particles in the box to calculate the magn. forces at each call and is therefore timeexpensive (N 3 ). //fbodymesh.f// is a routine in connection with the meshroutine tocalculate the magnetic force exerted on particle i in a box with a specified distance andspecified magnetic moment. The complementary routine //fbodyneigh.f// calculates themagnetic forces originating from nearest neighbors only and is not critical with respect totime (N 1.2 )d. references[37] A hierarchical micromagnetic model of longitudinal thin film recording media[20] Molekulardynamik, Grundlagen und Anwendungenf. parameters1: - 8: see mesh module documentation9: xforce - real arraymagnetic force in x-direction exerted on particle i, i=1,...,N10: yforce - real arraymagnetic force in y-direction exerted on particle i, i=1,...,N11: xzell - realx-component of the magnetic moment in the cell element referred to12: yzell - realy-component of the magnetic moment in the cell element referred to13: angi - realangle of magnetic moment of particle i (on which the magnetic force if exerted to)14: xlimu - realx- coordinate of lower boundary of mesh element referred to15: xlimo - realx- coordinate of upper boundary of mesh element referred to16: ylimu - realy- coordinate of lower boundary of mesh element referred to17: ylimo - realy- coordinate of upper boundary of mesh element referred to18: xi -realx-coordinate of particle i19: yi -real

y-coordinate of particle i20: fx - realmagnetic force (x-component) from the mesh element exerted on particle i21: fy - realmagnetic force (y-component) from the mesh element exerted on particle ig. error indicators and warningsif (r.LT.(1.d-30).OR.r.GT.(1.d+30)) thenwrite (*,*) r,' range r exceeded !'stopendifprogram stops if the interparticle distance exceeds the limitsh. further commentsfbodyneigh is always used in connection with the mesh routinefbody may not be used with the mesh routinei. Algorithmic detailsthe magnetic forces⎧Bmij⎛ µ0H= ⎜⎝ µ0Hextxextyi= ⎛ cosϕ⎞m⎝⎜sinϕ⎠⎟i¡⎞⎟ +⎠76[ j i j j i j ]⎫Nµ m 3 cos ϕ ( x − x ) + sin ϕ ( y − y ) ⎛ xi− xj⎞ ⎛ ϕj⎞01 cos∑ ⎨⎪−⎬⎪5 34πy yj i jr⎝⎜i−= ≠j ⎠⎟ijr ⎝⎜sinϕ1,j ⎠⎟⎩⎪ij ⎭⎪magnetic force exerted on particle iF ij= ( mi ∇i) Bj= m(cosϕ∂i+ sin ϕ∂i) B∂x∂yiij

Fix3−3−Fiy3−3−77[ j i−j+j i−j ]2Nµ m 3 cos ϕ 2( x x ) sin ϕ ( y y ) cosϕ0j= + (cosϕi∑ {5+53( xi− xj) −4πj= 1,i≠jrr2[ cos ϕj ( xi − xj ) + sin ϕj ( yi − yj )( xi − xj )]rij2[ cosj( xi − xj) + sinj( yi − yj)( xi − xj)]µ= +5( x − x ) + sinϕ ϕ cosϕ75( yi− yj) +5rrijijij} ϕ ∑ {[ sin ϕj ( xi − xj )]7 i j i5j= 1,i≠j rij[ ϕj i−j ]2Nm 3 cos ( y y ) sinϕ0j(cosϕi∑ {5 54πj= 1,i≠j rijrij2[ cos ϕj( xi − xj)( yi − yj) + sin ϕj( yi − yj) ]+ sinϕN∑ij= 1,i≠j{rij[ cos ϕj( xi − xj) + sin ϕj( yi − yj)]2[ cos ϕj( xi − xj)( yi − yj) + sin ϕj( yi − yj) ]rij73 27rij5+ 3( x − x ) −iji}ij5( x − x ) +−jjNi3j}3( y − y ) )sinϕj5( yi− yj) +53( yi− yj) })rij−i. Testing of Algorithm• if the dipole directions are normal to each other, no energy is necessary to approach . Theforce is normal to the interdistance vector• F 12 =F 21 (3 rd Newtonian law)c) parallel oriented dipoles attract, antiparallel repulse each other• F=-grad E p (5)mbodya. purposecalculates the magnetic torque on particle i, 1=1,...,Nb. specificationsubroutine mbody (n,x,moment,k1,k2,vol,y)implicit nonereal*8 k1,k2, volreal*8 y(1200) ,moment (200), x(200)integer nc. descriptionthis routine calculates the torque originating from the magnetocrystalline anisotropyd. references[41] Mikromagnetism And magnetization processes in modern magnetic materialsf. parameters1: k1 - real

78anisotropy constant K u12: k2 - realanisotropy constant K u23: vol - realvolume of sphere4: n - integernumber of particles, n

79& xforce,yforce,stiff,daempf,pi,f,mass,neq,actwidth,& acthigth,mcontact,& fstempel,a,b,bottom,left)implicit nonereal*8 acthigth,left,bottomc. descriptioncalculation of contact forces using a repulsive linear stiffness force from harmonicoszillator potential, rolling and gliding coulomb frictiond. references[29] Collision-induced friction in the motion of a single particle on a bumpy inclinedline[25] Granular Flow: Friction and the Dilatancy Transitionf. parameters1: n - integernumber of particles3: ycontact - real arrayy-component of contact force5: frictionw - realparticle-wall friction coefficient7: diameter - realdiameter of particles9: yforce - real arraynot used11: daempf - realdamping contant (used in connection withstiffness constant)13: f - real arraynot used15: neq - integernot used17: higth - realhigth of box2: xcontact - real arrayx-component of calculated contact forceacting on particle i, i=1,...,n4: friction - realinterparticle friction coefficient6: y - realcontains angles of c-axes, angular velocities,x-coordinates, velocities in x-direction, y-coordinates, velocities in y-direction8: xforce - real arraynot used10: stiff - realstiffness constant12: pi - realpi=3.14159...19: mcontact - real arraytorque originating from contact forces21: a - realnot used23: bottom - realbottom coordinate of box (isostatic pressing)25: e - realnot usedg. error indicators and warningscorr=1.d0corrw=1.d0C correctur factor for rolling, if friction coeffizient greater than 1.14: mass - realmass of one particle16: actwidth - realactual width of box (due to pressing punchin uniaxial pressing)18: acthigth - realactual higth of box (due to pressing punch inisostatic pressing)20: fstempel - realforce on pressing punch22: b - realnot used24: left - realleft coordinate of box (isostatic pressing)26: sigma - realnot used

80if (friction.GE.(1.d0)) corr=1.d1if (friction.GE.(1.d1)) corr=1.d2if (friction.GE.(1.d2)) corr=1.d3if (friction.GE.(1.d3)) thenwrite (*,*) 'interparticle friction too large !'stopendifendifif (frictionw.GE.(1.d0)) corrw=1.d1if (frictionw.GE.(1.d1)) corrw=1.d2if (frictionw.GE.(1.d2)) corrw=1.d3if (frictionw.GE.(1.d3)) thenwrite (*,*) 'friction particle-wall too large !'stopendifh. further commentsnonei. Algorithmic detailsIn all cases the shear constant is assigned withγγ = n corrcf. setupg module2sThis values holds for the simulation of the impuls magnetization where only magneticforces act upon the system (no pressing) and decrease simulation time enormously in thiscase. Alternatively the values for k n and γ ncan be read in from the input file.In both cases the parameter corr corrects the Coulomb friction in the following wayµ

81real*8 fxstroem (200),fystroem(200),y(1200)real*8 etaiinteger nC--------------------------------------------------------------------subroutine momstroem (n,y,mstroem,higth,width,diameter,pi,etai)implicit nonereal*8 higth,width,diameter,pireal*8 mstroem(200),y(1200)real*8 etaiinteger nc. descriptionthe friction resulting from the solution (viscosity) generate both a force and torque actingupon the particle. The default value (viscosity=zero in the input file SINTDATA.D) is a N 2gas with standard conditions.d. references[10] Statistische Physik, Berkeley Physik Kurs Bd.5f. parameters1: y - real arraycontains angles of c-axes, angular velocities, x-coordinates, velocities in x-direction, y-coordinates, velocities in y-direction2: fxstroem - real array3: fystroem - real arrayx-component of force acting upon the y-component of force acting upon theparticleparticle4: pi - real5: diameter - realpi=3.1415926...particle diameter6: etai - real7: n - integerviscosity parameternumber of particles8: mstroem - real array9: higth - realtorque acting upon the particlehigth of box10: width - realwidth of boxg. error indicators and warningsnoneh. further commentsnonei. Algorithmic DetailsThe force and torque in a fluid consists of a pressure contribution and a friction viscositycontribution. The Newtonian theorem of viscosity is given byτxyη ∂ vxdF= − =∂ydfπη = m. v. l . n where n =8xyNVπ⇒ η = ρ . v.l8This formulae is correct up to high-vacuum (the free mean path has to be smaller the boxdimensions)Nitrogen N 2 at standard conditions (STP) yields

¡ρ=1.251 g/lv =420 m/sMaxwell velocity distributionl =3.2 10 -7 mA linear velocity gradient is taken into account (in fact near the surface it nonlinearprogressive) with a'radius' from the velocity on the surface to zero velocity of82R =B.HB, H box dimensions, N number of particlesπNThis distance should be a good approximation for uniform distributed particles in the box.The approximation of the fluid viscosity resistance of rotation is4rMströmung = −η 4πθBHπ NThe fluid viscosity resistance of a sphere is given byF = −6πηrvStokes [5]Strömungclusterchecka. purposechecks the coordinate arrays on correct structure for time integrationb. specificationsubroutine clustercheck (y,yc,xkoord,ykoord,diameter,nclust,n1,n)implicit noneinteger n,nclust,n1real*8 xkoord(n),ykoord(n),y(6*n),yc(3*n)c. descriptionthe following structure of coordinate files is assumed: from the top of array (and thereforeinput file XYKOORD.D) two subsequent values correspond to the two spheres for a dumbbell.At the very beginning of the coordinate arrays the single paricles are situated. Forexample: n=5 particles ⇒ nclust=1, n1=3 assuming (x,y)koord(1...3) the single spherecoordinates, (x,y)koord(4...5)= the coordinates of the two spheres for the dumb bell with andistance difference of maximum diameter+eps, where eps=5 10 -6Finally the routine calculates the initial values of the independent coordinates (center ofmass coodinates and velocities) for the clustersd. referencesnonef. parameters1: yc- real array outputpacked reduced coordinates, center of mass coordinates of the dumb-bells2: n1 - integer outputnumber of single spheres3: nclust - integer outputnumber of dumb-bells, e.g. for n=100 ⇒ n1=50, nclust=25 possible4: xkoord,ykoord - real arrays inputx,y coordinates of the particles, respectivelyconfer //gearclust.f// module documentation for the other parameters

g. error indicators and warnings'clustercheck failed !!''cluster number from top of file:',nclust-i+1This fatal error message indicates the position of the inconsistent coordinates in number ofclusters (one cluster consists of 2 particles)83setupg,setupia. purposesetup routines for GEAR predictor-corrector integrator called prior to time integrationb. specificationsubroutine setupg (user,xkoord,ykoord,atol,stiff,daempf,& diameter,mass,imass,dt,rt& my,pi,mag,k1,vol,nsearch,n,tvgl)implicit noneinteger nreal*8 user(900),nsearch,tvgl,xkoord(n),ykoord(n)real *8 stiff,daempf,diameter,mass,imass,dt,rt,atolreal *8 my,pi,mag,force ,k1,volC---------------------------------------------------------------------------------subroutine setupi (user,xkoord,ykoord,atol,stiff,daempf,diameter,& mass,imass,dt,rt& my,pi,mag,k1,vol,nsearch,n,tvgl)implicit nonec. descriptionthe routine choose the correct time step both for the time integration and minimum search,the mesh size for hierarchical technique and shifts the compact, if necessary, to the originof the coordinate system. This shift can be necessary after a pulsing simulationd. references[25] Granular Flow: Friction and the Dilatancy Transitionf. parametersconfer //convert//, //xyconvert//1: tvgl - realtime step for output of files (SIMULDATA.D)2: nsearch - realnumber of time steps dt of the integrator , after which a new minimum search is performed3: rt - realelement size of quadratic equal-sized mesh4: dt - realtime step for integrator5: atol - realtolerance parameter for the time step supplied by the userg. error indicators and warningsnoneh. further commentsnonei. Algorithmic detailselastic constants, automatic build-in values

84The appropriate values for stiffness and damping can be derived from the maximum forcein the systemkγnn510 µ0m=52πd= 50M2205µ m4πd Mor the values can be user supplied.The value nsearch*dt, determining the integration time between the minimum searches,should not be greater than the time between the output of data files and should not besmaller than the equilibration time for the magnetic moment (10 -10 s).gearb, gearbc, gearclusta. purposenumerical solver for system of ordinary DGL 1.order, explicit gear predictor-corrector3.order, initial value problemb. specificationsubroutine gearb (y,f,fmagxmesh,fmagymesh,molij,neighb,&t,tend,dt,n,neq,ncorr,trace,reset)implicit noneinteger n,neq,ncorrlogical trace,resetreal*8 y(neq),f(neq)real*8 fmagxmesh(n),fmagymesh(n)integer molij (n,50),neighb(n)real*8 t,tend,dtC-------------------------------------------------------------------------subroutine gearbc (y,f,fmagxmesh,fmagymesh,molij,neighb,&t,tend,dt,n,neq,ncorr,trace,reset)implicit noneinteger n,neq,ncorrlogical trace,resetreal*8 y(neq),f(neq)real*8 fmagxmesh(n),fmagymesh(n)integer molij (n,50),neighb(n)real*8 t,tend,dtC---------------------------------------------------------------------------------subroutine gearclust (y,f,yc,fc,fmagxmesh,fmagymesh,molij,neighb,& t,tend,dt,n,neq,ncorr,trace,reset,& n1,nclust,isocompaction,diameter)implicit noneinteger n,neq,ncorr,n1,nclustlogical trace,resetreal*8 y(neq),f(neq)real*8 fmagxmesh(n),fmagymesh(n)integer molij (n,50),neighb(n)real*8 t,tend,dt

85real*8 yc(6*nclust),fc(6*nclust)logical isocompactionc. descriptionmodule //gearb.f// performes the same task as //gearbc.f//, but the former is for uniaxialcompaction and the later is for isostatic compaction in connection with hierarchicaltechnique. The system of equations refered to by this two routine are different.module //gearclust.f// performs a time integration of a mixture of single spheres and dumbbells,the systems of equation for the mixture according to Lagrange 2 technique must besuppliedd. references[20] Molekulardynamik, Grundlagen und Anwendungen p.67f. parameters1: y - real arrayvector of dependent variables, contains angles of c-axes, angular velocities, x-coordinates,velocities in x-direction, y-coordinates, velocities in y-direction2: f -real arrayvector of functions according to the differential equations y' = f (t,y)3: fmagxmesh - real array inputx-component of magnetic forces acting upon particle i, i=1,...,N calculated prior to timeintegration with hierarchical technique4: fmagymesh - real array inputy-component of magnetic forces, confer 3:5: molij - integer arraymolij (i,j) contains the explicit numbers of the particles , which are nearest neighbors(j=0,...,neighb(i)) for the particle i. For example, for particle 1 the particles 4 and 11 arenearest neighbors: neighb(1)=2, molij(1,1)=4, molij(1,2)=116: neighb - integer arrayelement i contains the quantity of nearest neighbors for particle i7: t - realstarting value for time integration of independent variable time8: tend - realend value for time integration of independent variable time9: dt - realfixed time step for time integration10: n - integernumber of particles11: neq - integernumber of differential equations of 1. order = 6*n12: ncorr - integernumber of corrector steps, 1 ≤ ncorr ≤ 313: trace - logicaltrace =.TRUE. : convergence test control , routine writes on standard output device whichvariables do not converge performing the corrector steps; ncorr must be greater than 114: reset - logicalreset=.TRUE. : the first derivation of accelerations b i (t) is set to zero15: isocompaction - logical.TRUE. isostatic compaction; uniaxial otherwise16: diameter - realdiameter of particles17: fc - real array

86vector of functions for dumb-bellsg. error indicators and warningsdelta... and converg... denote the present and prior difference of accelerations, confer eq.(2) in section Algorithmic DetailsC tracing = convergence test controlif (trace) thenif (dabs(deltaw(i)).GT.dabs(convergw(i)))& write (*,500) 'THETAi',i,tif (dabs(deltax(i)).GT.dabs(convergx(i)))& write (*,500) 'Xi',i,t&if (dabs(deltay(i)).GT.dabs(convergy(i)))write (*,500) 'Yi',i,tconvergw(i)=deltaw(i)convergx(i)=deltax(i)convergy(i)=deltay(i)500 format (1X,A6,' konvergiert nicht:Teilchen',1X,I4,2X,';Zeit:',E10.5)endifh. further commentsif the differential equations have singularities, the sheme does not converge. Try an evensmaller time step, or if at limit of computer performance, take only one corrector step andcheck if the solution is appropriatei. Algorithmic detailsGEAR predictor corrector algorithm 3 rd order [20]This variant includes the third derivation of timeh hri , p( t + h) = ri ( t) + vi ( t) h + ai ( t) + bi( t)2 6hvi,p( t + h) = vi ( t) + hai ( t) + bi( t)2a ( t + h) = a ( t) + hb( t)i,p i i2 32(1)In the predictor step b i (t) does not change. Then the forces are recalculated with the newcoordinates and the resulting accelerations are the corrected values a i,c (t+h)∆ai = ai, c( t + h) − ai,p( t + h) (2)Finally the corrected values are calculatedhri , c( t + h) = ri , p( t + h)+ ∆ai125hv,( t + h) = v,( t + h)+ ∆a121bi , c( t + h) = bi ( t)+ ∆aihi c i p i2(3)Now the corrector step can be reiterated (up to 3 times) or the corrected values from thefirst corrector step can be accepted as in our simulations. If another order of GEARpredictor-corrector is employed all coefficients in the corrector step change !The local discretization error is of third order

87ε local = O(dt 3 ) (4)Additionally the algorithm for Lagrange 2 //gearclust.f// includes:• correction of nearest-neighbor-list to avoid inner forces between the particles in thedumb- bell• subsequent time integration of single spheres, clusters and transformation from reducedsystem to particle system for dumb-bellsThe functional system //fclust.f// calculates the desired functional vector with the commonsingle particle coordinates, so the routines for the calculation of forces and torque remainunchanged.j. testing of algorithmtesting with two interacting hard magnetic particles, but can be tested with any system ofdifferential equations and compared to analytical solution. The cluster routine was testedregarding conservation of distance and angle difference within the cluster.fb, fbc,fclusta. purposeevaluates the functions f i (y,t), that is the first derivatives of the differential equations forgiven argumentsb. specificationsubroutine fb (t,y,f,fmagxmesh,fmagymesh,molij,neighb,number)implicit nonecommon /var/ user,x,etai,fstempel,ninteger number,nreal*8 t,y(6*number),f(6*number)real*8 user(900),x(200)real*8 fstempel,etaireal*8 fmagxmesh(number),fmagymesh(number)integer molij(number,50),neighb(number)C--------------------------------------------------------------------------subroutine fbc (t,y,f,fmagxmesh,fmagymesh,molij,neighb,number)implicit nonecommon /var/ user,x,etai,fstempel,nC--------------------------------------------------------------------------subroutine fclust (t,y,yc,f,fc,fmagxmesh,fmagymesh,molij,& neighb,number,n1,nclust,isocompaction)implicit none integer ninteger n1,nclustreal*8 yc(6*nclust),fc(6*nclust)logical isocompactionc. description//fb.f// contributes the function system for uniaxial pressing (integrator B inSIMULDATA.D), //fbc.f// for isostatic pressing (integrator C in SIMULDATA.D); there isno difference in the calling sequence, but in the calculation of the box dimensionsboth routines contribute the velocities and accelerations to the vector of functions f iadditionally the routine //fclust.f// contributes the vector of functions fc i for dumb-bells forboth the uniaxial and isostatic case

d. references[5] Mechanik, Berkeley Physik Kurs Band 1f. parametersconfer //gearb.f//, //gearbc.f//,//gearclust.f// module documentation1: t - realindependent variable (time)2: y - real array3: f - real array4: fmagxmesh - real array5: fmagymesh - real array6: molij - integer array7: neighb - integer array8: number - integernumber of particles number≤200g. error indicators and warningsnoneh. further commentsnonei. Algorithmic detailssee section dynamic motion of particles for details88analysislin, kinenergy, potenergylin, clusteranalysea. purposeperform analyses of the particle systemb. specificationsubroutine analysislin (user,y,tout,tstep,fstempel,mass,imass,stiff,& sigma,rshift,rc,& diameter,objf,clustsize,clustformlin,clustsizemax,n)implicit noneinteger n,clustsizemaxinteger clustsize(clustsizemax),clustformlin(clustsizemax)real*8 user(900),y(6*n),tout,fstempel,mass,imass,stiff,sigmareal*8 diameter,rc,tstep, objf ,rshiftreal*8 kinenergy,potenergylinC--------------------------------------------------------------------------------real*8 function potenergylin (y,stiff,sigma,rshift,diameter,objfun,&mult,&higth,width,n)implicit noneinteger nreal*8 y(6*n),stiff,sigma,width,higth,diameter,objfun,mult, rshiftC--------------------------------------------------------------------------------real*8 function kinenergy (y,mass,imass,n)implicit noneinteger nreal*8 y(6*n),mass,imassC----------------------------------------------------------------------subroutine clusteranalyse (y,clustsize,clustformlin,rc,diameter,& clustsizemax,n)

89implicit nonereal*8 y(6*n),rc, diameterinteger clustsize(clustsizemax),clustformlin(clustsizemax)c. description//analysislin.f// writes data during simulation run on output file in SINGLE PRECISION .For this purpose the following 3 routines are refered to by this module//kinenergy.f// kinetic energy//potenergylin.f// total magnetic potential energy, elastic potential energy for harmonicoszillator potential//clusteranalysis.f// clusteranalysis, analyses clustersizes and shapesAdditionally the pressing force is written on output file 'FORCEvsTIME.d'Actually there are functions for other potentials than the harmonic oszillator potential, buthave not be used in connection with GEAR predictor corrector; see the README.TXT filein the source directory for detailsd. references[22] Normenblatt Stahlgußstücke für hydraulische Maschinen, Annahmekriterienf. parameters1: clustsize - integer arraynumber of total clusters with size of index i3: clustsizemax - integerthe size of the biggest cluster allowed (=50)5: objf - realtotal magnetic potential energy derived fromthe routine //objfun// originating fromminimization7: user- real arraypacked array with parameters2: clustformlin - integer arraynumber of linear clusters with size of index i4: rc - realwithin this maximum distance in units ofparticle radius the particle is counted to be inthe cluster6: stiff - realstiffness constant between particles8: y- real arraycontains actual (angular) velocities andcoordinates9: mass - real10: imass - realmass of a single spheremass moment of inertia11: diameter - real12: tout - realdiameter of sphereactual time of output13: tstep - realtime span between outputsall other parameters are not refered to for harmonic oszillator potentialg. error indicators and warningsnoneh. further commentsclustsize(clustsizemax) and clustformlin(clustsizemax) contains clusters with sizeclustsizemax and biggermaximum 1000 particles : size of arrays cluster and welementthe elastic potential energy does not include the wall-particle interactioni. Algorithmic Detailsclusterform: linear, if length : width = 3 : 1 (more accurately: linear arranged)

90presstermination, pressterminationisoa. purposechecks termination conditions for programb. specificationsubroutine presstermination (user,y,x,fstempel,pres,t,n)implicit noneinteger nreal*8 user(900)real*8 y(6*n),x(n),treal*8 fstempel,presC this routine for isostatic pressing onlysubroutine pressterminationiso (user,y,x,fstempel,pres,t,n)implicit nonec. descriptionThis modules check whether 2 out of 3 termination conditions are fulfilled: endpressure ordensity beyond which no further simulation is desired. The critical pressure is supplied bythe user in the input files, the critical density for equal sized sphaeres is 0,82. Here thecritical density value has been chosen bigger because of fluctuations in pressure. An output(time, density) on standard output device is generated for density >0,65 . When terminationis detected, the final values of magnetization, c-axes and coordinates are written oncorresponding files . The maximum value of coupling of magnetization with x-axes for oneparticle is written on standard output device.d. references[33] Theoretical description of a Two dimensional Compaction Process of CylindersJ.Am.Ceram.Soc. 790, 153-60 (1996)[34] Computer simulation of isostatic powder compaction by random packing ofmonosized particles, J.Mat.Sci.Letters 13, p.1709-11, 1994f. parameters1: fstempel - realactual force exerted on pressing machine3: t - realactual simulation time5: user - real arraypacked data7: x- real arraymagnetization directionsg. error indicators and warningsnone2: pres - realuser-supplied force from input fileSINTDATA.D4: n - integernumber of particles6: y - real arraypacked integration variables

918.2 Input dataThe program package consist of*.f Fortran-77 source code*.txt text files with additional informationwith the following 3 contributionsTILERSINTMAGEVALUATIONThe package is available on ATP330 /usr3/users/zwick/simulation/source (date ofpublishing) or on 3.5" disc alternatively. The package is delivered as it is, no warranty anduse at one's own risk !!The program TILER tiles up a precompact. The options are the direction (X or Y) and thedistribution (RANDOMLY or UNIFORMLY). The random distributions is for spheresonly, the uniform distribution is recommended in the case of clusters, because it creates thedesired input file for the pulsing or compaction simulation.The program SINTMAG carries out the trajectory calculation of the particles. It requires atleast two input filesSINTDATA.DSIMULDATA.DXYKOORD.D(optionally)CCAXIS.D(optionally)These two input files for this program, SINTDATA.D and SIMULDATA.D , containingthe physical parameters and simulation data respectively. The input parameters forSINTDATA.D are common SI units. Some options arestiffness constant = 0.0 the program calculates the default stiffness and dampingparametersas mentioned above in the section elastic constantsviscosity of the solution= 0.0 the program calculates the viscosity of N 2 STP. Thecompaction is performed under 'inert' gas conditionsnumber of particles = 0 the coordinates of the particles (max.200) will be read infrom the input file XYKOORD.DThe box dimensions should be set properly, the program shifts, if necessary, the position ofthe particles to the left-bottom corner. The uniaxial die pressing is performed from the rightside of the box in negative x direction, the punch is positioned automatically on theoutermost side (see Fig. xy). In this case the width of the box is adjusted automatically,where the higth remains at the input value. Almost the same procedure applies formultiaxial pressing (pressing with 4 punches), but higth and width are adjustedautomatically.ATTENTION: viscosity and damping should be restricted to the range that no STIFFdifferential equation system appear.The input file SIMULDATA.D contains the input data for the simulation run. the datacomprise the starting time (in secs), the output time of some files ( CAXIS#.D,KOORD#.D,CLUSTER#.D) and the end time of the simulation. But the program can stop

at final end pressure or final density (0.82) too. Hence it stops when one of this conditionshas been satisfied.The next ASCII character fixes the integrator92RIBCPQGEAR Predictor and Corrector 3 rd order with rolling and gliding friction,constant time step and linear stiffness function, spheres and uniaxial pressingsame as R , but for isostatic compactionsame as R , but with mesh and nearest neighbor tablesame as B , but isostatic compactionsame as B , but with spheres (N/2) and clusters with size 2 (N/4)same as P , but isostatic pressingwhere N denotes the number of particles. In the case of clusters the number of spheres and'platlets' should reflect the realistic frequency [33]The most simulations were carried out with B, C, P and Q integrators due to smallersimulation time.The parameter ATOL denotes the relative accuracy referring to the time step taken fromliterature. In most cases the default value of 1 is recommended. Only at high values of thefriction coefficient one must be careful and check the origin of the time step. If the timestep originates from the stiffness the parameter ATOL must be assigned withatol=1/corr (see module documentation of fcontactlinrol)Additionally this value can be used to introduce a stochastic element into the simulationruns. A variation of atol can result in different results as demonstrated in the case ofisotropic magnets. In this case the particles perform intrinsic motions and a variation ofatol and solution (damping) results in different particle distribution and alignment.total magnetic free energy (arb.units)0.0-0.5-1.0-1.5-2.0-2.5atol=100, N2-solutionatol=10 , N2-solutionatol=2 , N2-solutionatol=20 , high-viscous-solution (1000xN2)0 1 2 3 4 5time (arb.units)Mixture of Ba-Ferrite particles with k n =4,5 10 7 N/m and different global damping factor (solution). Thecalculation time with atol=2 was 6 days on a MIPS 450 MHz machine.

93Warning: The result must be validated by the user, so an even smaller time step than theminimum recommended time step is suggested.The next parameter fixes the angle distribution of the c-axes- randomly- in or contrary to field direction- taken from input file CCAXIS.DThe other parameters are for STIFF integrators.PETZLD fixes the convergence test for this integrator.FALSE. no convergence test.TRUE. convergence test during runIn the latter case the integrator writes the convergence failures during the corrector steps onthe standard output file, but only in the case of at least 2 corrector steps !! Problems withconvergence appear at singularities at mentioned , but the error in our simulation should bewithin evaluation error. The parameter ATOL controls this convergence behavior andshould be set smaller if desired.8.3 Output dataThe output data comprise the following filesSINTOUT.D double precision, original datacomprise the input file SINTDATA.D , some output data as mass of particle, etc. , theangles of c-axes, the coordinates and the used data from SIMULDATA.D. This data is notscaled yet !!This is the 'de facto' input data for the program and should be compared carefully with theoriginal data !!!MDIST.D (angle, number of particles)distribution of initial easy c-axes of particle as histogramMDIST1.D (angle,number of particles)distribution of magnetic moment of particles after the first minimum search as histogramCAXIS#.D (c) single precision ,scaled unitswhere # denotes a number between 0001 and 9999 (max. output files !!). This file(s)contain the c-axes of the particle (raw data in SINGLE PRECISION) The file CDISTF.Dcontains the corrected c-axes angles with respect to the direction of the applied externalfield at program termination.KOORD#.D (x,y) single precision ,scaled unitscoordinates of particles . File KOORDF.D contains the coordinates at program termination.CLUSTER#.D (cluster size, total number of clusters, linear number of clusters)single precision, integerdistribution of (dynamic) clusters during simulation run . The criterion for clustering isgiven by the distance of particles (within 1% of diameter)A length vs. width ratio of 3 vs. 1 results in a linear cluster. Clusters with a size greaterthan 50 are counted as a cluster with size 50.kENERGYvsTIME.D(time, kinetic energy) single precision, scaled units

contains the kinetic energy of the system at the specified output times.pENERGYvsTIME.D(time, magnetic energy, elastic energy) single precision, scaled unitscontains the total free magnetic and elastic energy of the system at the specified outputtimes. The elastic energy is the bulk elastic energy only (without wall interaction)FORCEvsTIME(time, force) single precision, scaled unitscontains the force versus time during simulation runWARNING: the variable tstep in SIMULDATA.D contains the output time for thenumbered files above, so care should be taken in assigning this variable. Otherwise thehard disc is filled up with data !errorsConsult the NAG manual for errors in the NAG minimum search routine !The errors during integration originates either from improper input data or from animproper time step controlled by the parameter ATOL.The coordinates from the tiling program TILER are in appropriate SI units due to inputdata from the SIMULDATA.D file.Note 1: When a pulsing simulation is performed the data must be converted (back scaled)before starting a pressing simulation (only the coordinate file)Note 2: the CAXIS#.D from the pulsing simulation must be copied to the CCAXIS.D fileNote 3: number of particles N=0 in the SINTDATA.D fileNote 4: c-axes distribution =2 (from file) in the SIMULDATA.D fileThe program EVALUATION performs a simple evaluation of one or more file(s).- normalized magnetization- magnetization in the four sections top, bottom, left, right and center of the box- simple evaluation of the c-axes distribution(s) assuming gaussian distribution- generates histograms of the c-axes distributions on the file(s) CAXISHISTO#.D- generates ASCII output files, e.g. for AVS graphical software94

959 DefinitionsAnisotropie (anisotropy) having different properties in different directions [materialsscience and engineering lectures]Attritor eine Maschine, in der Materialien (wie Körner oder Gewürze) zwischen zweigezahnten, gegeneinander rotierenden Metallscheiben pulverisiert werden [Webster]Ausbauchungsfaktor (squareness) (BH) max wird wesentlich durch den Verlauf derEntmagnetisierungskurve, d.h. durch die Ausbauchung bestimmt [6]-H23BH c1BB rUntere Grenzkurve 1: χ=0,25Obere Grenzkurve 2: χ=1,0Technischer Höchstwert 3: χ=0,74Basisebene (basal plane) the special name given to the closed packed plane in hexagonalclosed-packed unit cells [materials science and engineering lectures]Brown´sche Paradoxon Das Brown´sche Paradoxon besagt, daß die Koerzitivfeldstärkenicht kleiner als die Anisotropiefeldstärke werden kann: H c,theoret. =H A (=2K 1 /J s ) . DieAussage des Brown´schen Paradoxons wird verständlich, wenn man die kohärente Rotationder Magnetisierung in der ganzen Probe mit der Drehung der magnetischen Momente innur einem Teilbereich vergleicht. In dem letzteren Fall ist wegen der unterschiedlichenOrientierung benachbarter Spins noch der durch die Austauschwechselwirkung bedingteEnergieanteil zu berücksichtigen, so das damit H c nicht kleiner als H A werden sollte.Größer als H A kann die Koerzitivfeldstärke aber auch nicht werden. Das Brown´scheParadoxon gilt für eine ideale unendlich ausgedehnte Probe. [38] Lösung: Inhomogenitäteninnerhalb des Kristalls [41].Nach dem Brown´schen Paradoxon ist die Keimbildung nur an besonderen Stellen derProbe möglich. Besondere bevorzugte Gebiete für die Keimbildung sind Schwankungenoder Unstetigkeiten der Werte A, K und J s , also Oberflächen , scharfkantige Ecken, Poren,Kratzern, Ausscheidungen, nicht unterbrochene Diffusionsvorgänge und andereInhomogenitäten.Cluster (engl. Klumpen, Traube, Nest) Physik: als mehr oder weniger einheitlich Ganzeszu betrachtendes Gebilde zusammenhängender (gebundener ) bzw. in ihrer Bewegungkorrelierter Teilchen, insbesondere in Vielteilchensystemen (z.B. Atomkernen) [MeyerLexikon]DEM , distinct (discrete) element method A discrete element algorithm is a numericaltechnique which solves engineering problems that are modeled as a large system of distinctinteracting general shaped (deformable or rigid) bodies or particles that are subject to grossmotion. Engineering problems that exhibit such large scale discontinuous behavior cannotbe solved with a conventual continuum based procedure such as the Finite ElementMethod. The discrete element procedure is used to determine the dynamic contact topologyof the bodies. It accounts for complex non-linear interaction phenomena between bodiesand numerically solves the equations of motion. Since the DEM is a very computationally

96intensive procedure, many existing computer codes are limited to modeling either twodimensionalor small three-dimensional problems that employ simple body geometries.Domänenstruktur,Ursache (cause of domain structure): Ein unendlich ausgedehnterKristall würde eine einheitliche spontane Magnetisierung und keine Domänen aufweisen.Ein realer Kristall hat Grenzflächen, die an unmagnetische Bereiche anschließen. Durch dieDomänenaufteilung wird die Streufeldenergie stärker minimiert, als andere Energietermezunehmen (z.B. Energie in Blochwand). [3]Exzeß : gibt die Abweichung der Wölbung der Verteilung von einer Normalverteilung anflachgipfelig: negativ Normalverteilt: null spitzgipfelig: positivEin positiver Exzeß zeigt einen Werteüberschuß in der Nähe des Mittelwertes und an denVerteilungsenden an. Der Exzeß ist die um 3 verminderte Kurtosis (exzeß= kurtosis-3),wird aber trotzdem manchmal als Kurtosis bezeichnet.Fluidisierung (fluidization) If air strongly flows upwards into granular bed from thebottom, bed becomes fluidized. If you are a good swimmer and the bed is large enough,you may be able to swim in it! Actually speaking, this fluidied bed totally behaves likefluid. Air bubble blown into bed goes up just like that in ordinary fluid. Even stronger airflow causes spontanious bubbling. It looks like nothing but Boiling water.(http://granular.com)Granulare Materialien (granular materials) they are large conglomerations of discretemacroscopic particles. If they are non-cohesive, then the forces between them areessentially only repulsive so that the shape of the material is determined by externalboundaries and gravity. If they are dry then any interstitial fluid, such as air, can often beneglected in determining many, but as we will see below, not all of the flow and staticproperties of the system. They feature three important aspects: the existence of staticfriction, the fact that temperature is effectively zero and, for moving grains, the inelasticnature of their collisions.They play an important role in our industries, such as mining, agriculture, civil engineering,geological processes and pharmaceutical manufacturing.Granulieren (granieren) eine Substanz in körnige Form oder in die Form kleinerTröpfchen, Linsen, Kügelchen u.ä. bringen (z.B. zur leichteren Weiterverarbeitung)[Brockhaus]Härte (hardness) the resistance of a material to penetration by a sharp object [materialsscience and engineering lectures]Haufwerke (material) Allgemein werden Haufwerke von Teilchen nach ihrem ungefährenDurchmesser in DIN 30900 wie folgt bezeichnet- Granulate, größer 1mm Durchmesser- Pulver, kleiner 1 mm Durchmesser- Kolloide, kleiner 1 µm DurchmesserHydrodynamisches Paradoxon: d´ Alembert: Ein Körper erfährt in einer stationären,ebenen, inkompressiblen , reibungsfreien Parallelströmung keinen Strömungswiderstand,also keine Kraft in Strömungsrichtung [11]Jetmühle (jet mill): a kind of air-blast mill (with N 2 instead of air)Kaltverfestigung (strain hardening, wear hardening, work hardening): bei derKaltformung metallischer Werkstoffe eintretende Verfestigung, die sich durch Anstieg derHärte, Streckgrenze und Zugfestigkeit /.../ bemerkbar macht /.../ Aufhebung derKaltverfestigung durch Glühen /.../ [Meyers Lexikon ]

¡ ¡97Kanalglühofen (Tunnelofen) /.../ in denen die mit Blechen gefüllte Kiste auf einemWagen, dessen Plattform aus feuerfestem Material gemauert ist, zunächst eineVorwärmzone, dann eine Glühzone, etc. und schließlich eine Abkühlzone durchfährt;wärmetechnisch wirtschaftlich [Lueger Lexikon der Technik]Keramiken (ceramics) Materials consisting of compounds of metallic and nonmetallicelements [materials science and engineering lectures]kohärent: hier: gleichartige und zusammenhängende Drehung der Magnetisierung [2]Korn (grain) a portion of a solid material within which the lattice is identical and orientedin only a single direction [materials science and engineering lectures]Kornwachstum (grain growth) movement of grain boundaries by diffusion in order toreduce the amount of grain boundary area. As a result, small grains shrink and disappearwhile other grains become large [materials science and engineering lectures]Korrelationsanalyse (correlation analysis) Die Untersuchung der Zusammenhängezufälliger, nicht berechenbarer Signale oder periodischer Zeitfunktionen unter Verwendungvon Korrelationsfunktionen. Bei der Auto-K. wird die innere Struktur einer einzelnen,zufällig verlaufenden Zeitfunktion betrachtet. Mit Hilfe der Auto-K. lassen sich dabei vonStörungen überlagerte periodische Vorgänge identifizieren. Bei der Kreuz-K. werden mitHilfe der Kreuzkorrelationsfunktion Aussagen über die strukturelle Ähnlichkeit zweier miteinem festen zeitlichen Abstand ablaufender zufälliger Vorgänge geliefert. Die K. geht vonder Voraussetzung aus, daß die beobachteten Prozeße stationär und ergodisch sind, d.h. daßdie gewonnenen Verteilungs- und Dichtefunktionen nicht vom Anfangszeitpunkt derMessung abhängen und daß die aus einem Ensemble gleichartiger Messungen gebildetenErwartungswerte bzw. Scharmittel mit den zeitlichen Mittelwerten jeder Einzelmessungübereinstimmen. Die Korrelationsfunktion dient der Charakterisierung desZusammenhangs von 2 gleichzeitig verlaufenden Schwankungsvorgängen. Sie hängt vondem Zeitparameter τ ab und ist ein Maß dafür, wie stark die Werte von x (Anm.Zeitfunktion 1) mit den den um das Zeitintervall später auftretenden Werten von y (Anm.Zeitfunktion 2) im Mittel über den ganzen Zeitverlauf zusammenhängen.Der Korrelator ist ein Gerät zur Ermittelung von Korrelationsfunktionen, mit dessen Hilfez.B. Informationssignale aus einem überlagerten Störfeld (Rauschen) herausgelöst undsowohl ihr Zeitpunkt als auch ihr Effektivwert bestimmt werden. [Brockhaus]Kurtosis (curtosis) ein Maß der Abweichung einer Verteilung gegenüber einerNormalverteilung mit demselben Erwartungswert und Mittelwert (siehe Exzeß)Lorentzfeld das elektrische Feld E Lin einem fiktiven Hohlraum eines außerhalb desHohlraums homogen polarisierten Dielektrikums. Dieses Feld wurde zuerst von Lorentzberechnet. Es rührt von den Oberflächenladungen des Hohlraums her, derenFlächenladungsdichte P cosθ ist. Dabei sind P die Polarisation und θ die polareKugelkoordinate zwischen Polarisationsrichtung und betrachteter Raumrichtung. Aus derBerechnung des elektr. Feldes im Mittelpunkt einer Hohlkugel mit dem Radius r ergibt sich1 1die Beziehung EL= P = E( εrel− 1). Dabei ist ε3ε reldie relative Dielektrizitätskonstante.03In Wirklichkeit ist aber ein bestimmtes Molekül nicht von einem Hohlraum, sondern vonbenachbarten Dipolen umgeben. Es kann angenommen werden, daß sich die Wirkungbenachbarter Dipole gerade zu Null ergänzt. Das ist insbesondere bei Kristallen mitnichtkubischer Symmetrie eine zu weit gehende Näherung. Die von E Lund dem äußeren

981Feld herrührende Feldstärke am Ort des betreffenden Dipols ist EI = EL+ E = E( εrel+ 2).3[ABC Physik]Magnetische Flüssigkeiten (ferrofluids) Bezeichnung für stabile kolloidale Suspensionenferromagnetischer Teilchen mit den makroskopischen Eigenschaften einer echtenFlüssigkeit. Sie besitzen einen Sättigungsmagnetismus in der Größenordung von 50 kA/m(vgl. paramagnetische Salze: 2 kA/m, Eisen: 1700 kA/m)Herstellung: a) Magnetit Pulver wird in Anwesenheit einer Polymerlösung etwa 40 Tagelang in Kugelmühlen gemahlen und zwischendurch mehrfach zentrifugiert bis diegewünschte Teilchengröße erreicht ist. b) Ausfällung von Magnetit Fe 2 O 3 aus einerSalzlösung in Anwesenheit von oberflächenaktiven Polymeren und einer Trägerflüssigkeit.Der Preis pro Liter liegt je nach Qualität bis zu 2000 $.Die magnetische Flüssigkeit besteht aus Teilchen von ca. 10 nm Größe, die von einer 2 nmdicken Polymerhülle umgeben sind. Die Bezeichnung Ferrofluid bezieht sich nur auf dasInnere eines Teilchens, die Eigenschaft der Flüssigkeit als Ganzes ist paramagnetischerNatur. [Römpp Chemie Lexikon]Die Suspension ist dann stabil (keine Agglomeration der Teilchen), wenn die thermischeBewegungsenergie der Teilchen (Brown´sche Bewegung) groß gegenüber der potentiellenEnergie infolge der magnetische Kräfte zwischen den Teilchen und der van-der-WaalsKräfte ist [38]Magnetostriktion: Erscheinung, daß es bei Änderung der spontanen Magnetisierung (inBetrag und Richtung) zu einer Verzerrung des Kristalls kommt. DieMagnetostriktionskonstante liegt in der Größenordnung vom thermischenAusdehnungskoeffizienten. [3]Matrize (matrice): Preßform für den Grünling (gepreßter Sintermagnet), im Unterschiedzur Patrize, mit der die Matrize hergestellt wird.Maximales Energieprodukt (maximum energy product) (BH) max Auf derEntmagnetisierungskurve (siehe Ausbauchungsfaktor) stellt sich in Abhängigkeit von derGeometrie des Magnetsystems der sogenannte Arbeitspunkt ein. Die dort gespeichertebzw. zur Verfügung stehende Energie ist die Energiedichte oder das maximaleEnergieprodukt (BH) max [6]Merkmal ist eine Eigenschaft, die das Unterscheiden und Beurteilen von Einheitenermöglicht [21]Mischen (mixing) verfahrenstechnische Grundoperation, bei der zwei oder mehrere Stoffemiteinander vermengt werden, daß sich die Einzelkomponenten möglichst auf die gesamteMischung verteilen. [Meyer]Monte-Carlo-Verfahren (monte-carlo-method): 1) Numerische Verfahren,bei denen inder Berechnung zufällig erzeugte Zahlen eine Rolle spielen. Es wird ein demmathematischen oder physikalischen Problem entsprechendes Wahrscheinlichkeitsmodellgeschaffen und innerhalb des Modells werden Stichproben vorgenommen. Durchumfangreichere Stichproben wird ein genauerer Schätzwert des Ergebnisses erzielt[SYBEX-Computerlexikon]2) The Monte-Carlo method is defined as representing the solution of a problem as aparameter of a hypothetical population, and using a random sequence of numbers to

99construct a sample of the population, from which statistical estimates of the parameters canbe obtained [14]Mühle (mill) Maschine zum Mittel- und Feinmahlen, bei der das Mahlgut durch Druck- ,Schlag- , Prall- und Scherbeanspruchung zerkleinert wird. Speziell Trommelmühle (drummill): zerkleinern das Mahlgut in rotierenden (auch schwingenden) Mahltrommeln, diesesind mit losen Mahlkörpern aus verschleißfestem Material zu 20-40% gefüllt, sodaß dasMahlgut durch die Mahlkörper zerrieben oder zerschlagen wird [Meyer]Nucleation controlled Magnete Domänenwände bewegen sich leicht innerhalb derKörner, aber die Korngrenzen behindern ihre Bewegung von Korn zu Korn. ImSättigungsbereich, wo die Domänenwande außerhalb der Körner liegen, könnenmagnetische Keimbildungen nur noch durch ein starkes Koerzitivfeld hervorgerufenwerden. Beispiele für nucl.-contr.-Magnete sind Nd 2 Fe 14 B oder SmCo 5 .Die leichte Verschiebbarkeit der Domänenwände resultiert in einer großenAnfangssuszebtibilität nach dem thermischen Entmagnetisieren. Die maximale Remanenzist mit einer geringen Feldstärke H s,max im Vergleich zur Koerzitivfeldstärke verbunden.Pelletieren (pellet) ein Verfahren zum Stückigmachen (Agglomerieren) feinkörnigerStoffe für nachfolgendes /.../ Sintern. Es beruht darauf, daß sich bei der Bewegung vonfeinem Gut, das hinsichtlich Körnung, Kornverteilung, Oberflächenform der einzelnenTeilchen, Benetzbarkeit und Quellfähigkeit bestimmte Forderungen erfüllt und mit einemAnfeuchte- bzw. Bindemittel versetzt ist, in einer Trommel oder einem schräg stehendentotierenden Teller (Pelletierteller) Kugeln (pellets) schalenförmig aufbauen. Gut geeignetfür Korngrößen unter 0.1 mm. Pellets von 3-40 mm. Im allgemeinen muß die Kornklasseunter 0.06 mm 40% betragen, um genügend druckfeste Pellets zu erhalten. [Lueger]Pelletieren gehört wie das Schütten zur Formgebung ohne Druckanwendung(Vibrationsverdichten) [6]Permeabilität,reversible: bei sehr kleinen Feldstärkeänderungen und gleichzeitigerVormagnetisierung (Überlagerung eines Gleichfeldes durch kleine Wechselfelder,z.B.Tonbandaufnahmen) hat man innerhalb der Grenzschleife auf irgendeiner Hysteresiskurveeinen Punkt B-H erreicht und kehrt die Änderungsrichtung der Feldstärke H um, so besitztdie nun entspringende Hysteresiskurve eine kleinere Steigung als die Einlaufsteigung deralten Kurve,wenn ∆H sehr klein wird, dann erreicht man wieder den Ausgangspunkt, d.h.ein umkehrbarer Prozeß.∆BHµ rev= lim∆H→0 B∆HDie rev. Permeabilität ist am größten bei B=0 und ist dort mit der Anfangspermeabilitätidentisch.Sie hängt nur von der Induktion, nicht von der Feldstärke ab. [Lueger][7]Physik (lat. physica = Naturlehre < griech physike(theoria) = Naturforschung(griech.theorein = schauen) < griech.physikos = die Natur betreffend) Naturwissenschaft,die bes. durch experimentelle Erforschung und messende Erfassung die Erscheinungen undVorgänge, die Grundgesetze der Natur, vor allem die Strukturen, Eigenschaften undBewegungen, die Erscheinungs- und Zustandsformen der unbelebten Materie sowie dieEigenschaften der Strahlungen und der Kraftfelder untersucht [Duden]Pinning controlled Magnete Die Domänenwände werden stark entweder an ebenenAusscheidungen (magnetische Inhomogenitäten) innerhalb der Körner oder an den

100Korngrenzen festgehalten. Für pinning controlled Magnete (z.B.Sm 2 Co 17 ) ist ein weithöheres Sättigungsfeld nach dem Entmagnetisieren nötig als für nucleation controlledMagnete.Qualitätskreis (Q-loop, Q-spiral): conceptual model of interacting activities thatinfluence the quality of a product or service in the various stages ranging fromidentification of needs to the assessment of whether these needs have been satified [DINISO 8402]consists of planing, realization and usenote1: the q-loop does not necessarily reflect the time evolution of activitiesnote 2: the model q-loop does not apply in the cases of activities and processes with repectto quality considerations because of the lack of the element ' identification, whether theseneeds have been satisfied' [DIN 55350 T11]robust: insensitive to disturbations [55]Schiefe (skewness) einer Verteilung ist die Abweichung von der symmetrischen Form derNormalverteilung: Verteilung = Linkssteil ⇒ Schiefe = positivSelbstenergie (self energy): definiert als jene Energie die aufgewendet werden muß, umden Körper aus infinitesimalen Bausteinen zusammenzufügen, die anfangs unendlich weitvoneinander entfernt waren. [Berkeley Physik Kurs Bd.1]Shape controlled Magnete: Die Ummagnetisierung wird hauptsächlich durch die meistnadelförmige Form der magnetischen Partikel hervorgerufen (z.B. Al-Ni-Co).Simulation: building a modell + making experiments with the modell [19] The process ofsimulation comprises the steps of experiment definition, modelling, computerimplementation, validation and data gathering [Encyclopedia of Computer Science]spontane Magnetisierung (spontaneous magnetization): unter spontaner Magnetisierungverstehen wir also die Parallelstellung der magnetischen Dipole auf den Gitterplätzen übereinen größeren Bereich des Kristalls.Anm.: wenn der ferromagn. Kristall über die Curietemperatur erwärmt wird und dannabkühlt, tritt spontane Magnetisierung auf, die Bereiche weisen jedoch regellos verteilteRichtungen zueinander auf; erst durch äußeres Feld Wachstum der ´günstig´ orientiertenBezirke auf Kosten ungünstig orientierter Bezirke) [1]Sprühtrocknung (spraydrying) Verfahren zum Trocknen von flüssigen oder breiigenSubstanzen, bei dem diese in speziellen Trockentürmen fein verteilt einem erwärmtenTrockengas gegengeführt werden [Meyer]Statistische Versuchsplanung (design of experiments) befaßt sich mit dersystematischen Planung und Auswertung der Versuche, sie verbindet dabei fachlogischeund statistische Aspekte [17]Steifheit (stiffness) a qualitative measure of the elastic deformation produced in a material.A stiff material has a high modulus of elasticity [materials science and engineeringlectures]Suprasil aus Siliziumchloriden hergestelltes synthetisches Quarzglas mit hoher UVDurchlässigkeit [Römpp Chemie Lexikon]Ummagnetisierung (magnetization reversal): Umkehrung der Magnetisierungsrichtungdes Materialsweichmagnetisch: Verschieben der Blochwändehartmagnetisch: Drehung, Umklappprozesse der Weiß´schen Bezirke

101Validierung (validation) - validation is the most perplexing aspect of the simulationprocess. In the simulation community there is a wide range of opinion as to the meaning,necessity, and techniques of validation. Validation , in general, refers to estimating thedegree of validity of the simulation results and is a property somewhat comparative toaccuracy. However while accuracy has an 'absolute' connotation, validity has a 'relative'connotation. Validity, therefore , assumes the nature of relative accuracy [Encyclopedia ofComputer Science]- Die Validierung (Feststellung der Gültigkeit) geschieht aufgrund der Übereinstimmungder Testergebnisse mit einem Kriterium außerhalb von Testwerten (z.B. über einSchätzurteil gewonnen wird : Kriteriums-V.) oder aufgrund des Zutreffens einerVorhersage (predictive-V.) oder logisch-inhaltlich (context-V.) oder im Kontext belegbarerTheorien (Konstrukt-V.) [Meyer]Validität (validity) (spätlat. validitas=Stärke) die Gültigkeit eines wissenschaftlichenVersuchs oder eines Meßverfahrens, insbesondere eines Tests. Die Validität gibt den Gradder Genauigkeit an, mit der ein Verfahren das mißt, was es messen soll [Meyer]Versuchsplanung (Design of Experiments DoE): Versuchsplanung in diesem Sinne isteine Methode, um die Parameter eines Produktes oder Prozesses vor Beginn derSerienfertigung zu optimieren. Dabei wird davon ausgegangen, daß auf ein Produkt odereinen Prozeß mehrere Einflußgrößen wirken, die wiederum ein oder mehrereQualitätsmerkmale (Ausgangsgrößen) beeinflussen. Bei den Einflußgrößen werden dieSteuergrößen (Parameter) von den Störgrößen unterschieden. Die Steuergrößen werdeneinmalig bestimmt und während der Entwicklung festgelegt, sodaß eine Veränderung durchden Bediener oder Benutzer nicht mehr möglich ist. Die Optimierung der Steuergrößenerfolgt durch die Versuchsplanung. Die Störgrößen sind gar nicht und nur sehr aufwendigund kostenintensiv zu kontrollieren. Sie können sich ändern und sind nur statistisch mitMittelwert und Standarabweichung zu erfassen. Die Störgrößen sind die Ursachen für dieunerwünschten und unkontrollierbaren Abweichungen eines Qualitätsmerkmals (sieheMerkmal) von seinem Zielwert [55]Vibrationssegregation (segregation induced by vibrations) When we shake a granularmedium, large grains go up. Unlike in ordinary gas, however, kT plays no role in a granularmaterial. Instead the relevant energy is the potential energy, mgd, of a grain of mass mraised by its own diameter, d, in the gravity of earth, g. For typical sand this potentialenergy is at least 10 12 times kT at room temperature. Because kT is evectively zero,ordinary thermodynamics become useless. For example many studies have shown thatvibrations or rotations of a granular material will induce particles of different sizes toseparate into different regions of the container. Since there are no attractive forces betweenthe particles, this separation would at first appear to violate the increase of entropyprinciple, which normally favors mixing. In a granular material , on the other hand, kT=0implies that entropy considerations can be outweighted by dynamical effects that nowbecome of paramount importance. Unless perturbed by external disturbances , eachmetastable configuration of the material will last infinitely. Because each configurationshas its unique properties, the reproducibility of granular behavior , even on large scales andcertainly near the static limit, can only be defined in terms of ensemble averages. A secondrole of temperature in ordinary gases or fluids is to provide a microscopic velocity scale.

102Again, in granular materials this role is completely supressed, and the only velocity scale isthe one imposed by any macroscopic flow itself [H.M.Jaeger, University of Chicago] [62]Weichglühen (softening) Wärmebehandlung zum Vermindern der Härte einesWerkstoffes auf einen vorgegeben Wert [Böhler Edelstahl]Wirbelsintern (whirl sintering) Tauchschmelzverfahren, es wird durch Tauchen in eineWirbelschicht aus Kunststoffpulver auf ein heißes Werkstück ein fester Kunststoffüberzugaufgebracht [Meyer]

10310 Symbolsα´x,y,z direction cosinus of magnetization with reference to axes of the ellipsoidα direction cosinus of the saturation polarizationβ c direction cosinus of θ cγ n shear constant in normal (interdistance) directionγ s shear constant in tangential directionγ RCo5 domain wall energy for the bulkγ O domain wall energy around oxide precipitation∆ width of region where γ changes from γ RCo5to γ Oη viscosity coefficientϕ magnetization directionψ angle of external fieldθ , θ direction of easy c-axis ρ ∞ is the maximum planar density (=0.90)ρ 0 initial planar densityφ azimuth difference between magnetization and any crystal axis in the basal plane−1τ 0larmor frequencyω angular velocityc the amount of compaction because of rearrangementD L M S local stray field around an irregularityE A exchange energy densityE k magnetocrystalline anisotropy energy densityE sp stress anisotropy energy densityE stat magnetostatic energy densityE streu stray field energy densityF forceF a single barrier activation force0F a the mean value of the force barrierF σ the spread of the force barrierF l linear force responseH d demagnetizing field, stray field originating from all other particlesH s stray field originating from the particle itselfI mass momentum of inertiaJ ex exchange integralJ magnetic polarization vectorJ s saturation polarizationK ui anisotropy constantsM mass of particlem , m imagnetic momentNα , Nβ , Nχdemagnetizing factors along the axes of an ellipsoidn normal (interdistance) unit vectorr radius of spherer1 × F1torque, r 1 denote the vector from center of mass of dumbbell to center of sphere 1r , r jiinterdistance vector between the dipoless shear unit vectorspin vector on a lattice siteS i

104TuUVxtorquedirection of easy c-axismagnetic scalar potentialvolume of particle, volume of the spherescenter-of-mass particle position

105DanksagungMein Dank gilt allen, die bei der Erstellung dieser Arbeit mitgewirkt haben, vor allem abermeinen Eltern, die mich das ganze Studium hindurch unterstützt haben.Weiterhin bedanke ich mich bei der Kärntner Landesregierung für den Leistungszuschußund bei dem BM für Unterricht und Forschung für das staatliche Stipendium.Weiters gilt meine gebührende Anerkennung Herrn Prof. Fidler und Herrn Dr. Schrefl fürdie Idee und dem Vorschlag zu dieser Arbeit, viele Tips und das Bereitstellen derHochleistungsrechner.LebenslaufName:ZWICK HeinzAdresse:Haymerlegasse 8/6, A-1160 WienGeburtsdatum: 22-04-1967Geburtsort: VillachEltern:ZWICK WalterZWICK CarlaGeschwister: ZWICK Jürgen GilbertZWICK Walter HolgerSchulbildung: 1973 bis 1977 Volksschule in St.Georgen/Kärnten1977 bis 1981 Hauptschule in Nötsch1981 bis 1985 BORG in Hermagor1985 mit Auszeichnung maturiertAb 1985 TU-Wien1988 1.Diplomprüfung bestanden1993/94 Präsenzdienst