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Prediction of density variation in powder injection moulding fi lling process by using granular modelling with interstitial power law fl uid Hao He a Yimin Lia Jia Loua Dapeng Lia Chen Liub a State Key Laboratory of Powder Metallurgy Central South University 932 Lushannan Road Yuelu District Changsha 410083 Hunan Province China b Guangxi Nonferrous Metals Group Co Ltd 9th Floor Mingdu Building 22 Jinpu Road Nanning 530022 Guangxi Province China a b s t r a c ta r t i c l ei n f o Article history Received 22 August 2015 Received in revised form 12 November 2015 Accepted 7 December 2015 Available online 8 December 2015 Inthepresentstudy granular modelling with aninterstitial power law fl uidwas used tosimulatethefi lling pro cess in powder injection moulding PIM The granular model enables direct prediction of the density distribu tion The feedstock consisting of stainless steel powder and a multicomponent binder system was injected in two moulds with designed features to verify the simulation results The effects of the moulding parameters on the homogeneity of the powder particle distribution were investigated The experimental results agreed well with the predictions Theresults indicatedthatthe granular model presented hereincould be used for mouldde sign and the determination of injection parameters 2015 Elsevier B V All rights reserved Keywords Powder injection moulding Discrete model Density variation Granular model Interstitial power law fl uid 1 Introduction Density variation is an inherent problem in the injection moulding processofmetallicandceramicpowders 1 2 Itisdesirabletomaintain a uniform powder packing density in the feedstock to eliminate any sources of distortion in the subsequent sintering process 3 5 To save the cost of materials numerical simulation is employed to predict the density variation in the mould fi lling process 6 11 There are three types of models used continuum model bi phase model and granular model The continuum model assumes that the feedstock maintains a constant density during moulding and thus it cannot pre dict the density variation To remedy this problem some researchers recommend using the mixture theory for modelling the PIM where the bi phase model is the most frequently used theory 12 15 In the literature the bi phase model is usually used to simulate the powder binder phase sep aration Barriere et al achieved good results simulating the powder binder separation 16 Greiner et al established a bi phase model to explain the yield stress and shear induced powder migration in the micro PIM process 17 Wang et al used the bi phase model to explain the boundary layer effect 18 However the bi phase model treats the fl ow of feedstock as the addition of two distinct but coupled fl ows characterised by the viscosities of the binder and powder As well as the binder the viscous behaviour of the rigid powder phase must be provided to simulate the fl ow of feedstock however the viscosity of powder in the binder cannot be measured directly Hence additional work and more simulation time are required to obtain the viscous be haviour of the powder Moreover powder particles are theoretically in capable of being treated as a continuous phase Thus the bi phase model suffers both practical and theoretical fl aws Asanalternativetotheabovecontinuummechanicsapproach adis crete model mightbe a better choice for describingmutual interparticle interactions even locally 19 Aizawa and Iwai developed a granular model to simulate the mould fi lling process of electropackaging mate rials 20 21 Themodelconsidersthemovementandtraceofeachgran ular element which is composed of a powder and a thin surrounding binder layer The prediction of density variations allows for direct pro cess simulations for investigating the formation of defects such as pores and density variations However their model treats the internal contact force between two granular elements as the sum of the elastic and damping forces This assumption fails to consider the fact that the actual contact force during moulding is a viscous force for most wax based binder systems Furtherwork needsbecarriedoutonmodellingthecontactforce The thermoplastic binder is power law fl uid The powder injection moulding process can be regarded as a process of squeeze fl owing of a power law fl uid between rigid spheres an idea that has been employed in the simulation of many real industrial problems including the classical lubrication of engineering bearings the press moulding of composite materials and powder granulation 22 26 However there is no research focusing on the simulation of powder injection moulding using this idea Powder Technology 291 2016 52 59 Corresponding authors E mail addresses he hao555 H He liyimin333 Y Li lou3166 J Lou 357003413 D Li liuchen0771 C Liu http dx doi org 10 1016 j powtec 2015 12 009 0032 5910 2015 Elsevier B V All rights reserved Contents lists available at ScienceDirect Powder Technology journal homepage In this work a granular model with an interstitial power law fl uid is proposed to simulate the powder injection moulding process The den sity variation is predicted and validated by comparison with experi ments This model can provide not only alternative solutions in understanding die fi lling and powder transfer mechanisms it also es tablishesanimprovednumericaltoolformoulddesignandinjectionpa rameter optimising 2 Theoretical model In this work we use the derivation of the leading order lubrication termforthe axisymmetricalsqueezefl owof apower lawfl uid between two rigid spherical particles under a no slip boundary condition The particles are assumed to be completely immersed in the fl uid of binder and to have the same radii R The effects of particle shape and particle size are ignored It is considered that every particle approaches every other particle alongtheir common axis As shown in Fig 1 it is assumed that one particle is stationary and the other has a velocity v thus the problem is formulated using cylindrical coordinates The governing equations of the granular model are expressed as fol lows 22 26 1 Under the lubrication assumption the momentum equation re duces to the following expression p r rz z 1 where p is the pressure and rzis the shear stress 2 Thecontinuityequationforthesymmetricradial fl owofthefl uid between the particles can be expressed as 1 r rvr r 1 r v vz z 0 2 where vrand vzare the radial and axial velocities respectively 3 Theconstitutive equation for a power lawfl uid can beexpressed as Fig 1 Schematic of normal motion of two particles Fig 2 Schematics of moulds for simulation and verifi cation work a Obstacle mould and b step mould unit m Fig 3 ln vs ln rate at different temperatures for the binder Table 1 Moulding parameters for simulation and experimental works ParametersValues Injection temperature C 150 160 170 Injection pressure MPa 90 180 Injection velocity m s 6 9 12 Holding pressure MPa 40 Cooling time s 40 53H He et al Powder Technology 291 2016 52 59 rz K vr z n sign vr z 3 where K is the fl ow consistency and n is the fl ow index The above rela tionships allow the radial pressure distribution P to be deduced thus P r 2K 2n 1 n n vn Z B r rn s0 r2 2R 2n 1 dr 4 whereR istheharmonicradius whichiscalculatedby1 R 1 R1 1 R2 Further integration of Eq 4 leads to an expression for the viscous force F The equation of motion can be expressed as Fsum V dt m 5 where Fsumis the resultant force of the nearby particles and m is the mass of the particles The above relationshipsallow thetracingof particle movementdur ing the fi lling process in powder injection moulding To evaluate the temperature effect on particle motion thermal conduction between particles is expressed as cVdQ dt K T2 rdr S 6 where is the density c is the specifi c heat V is the volume of the par ticle Q is the heat exchanged between particles and K is the thermal conductivity The calculation of S can be found in Iwai s research 21 3 Experimental procedure 3 1 Simulation The simulation was conducted using the Matlab software in three dimensional 3D cases Twoisoscelestrianglemoulds whosegeometries are shown in Fig 2 were designed for the simulation and verifi cation work The fi rst mould had an obstacle in the middle and the other had a thin step in the front The feedstock was injected though the gate located at the base of the triangle An obstacle or a step would generate a shear force on the feedstock which would lead to a density variation in the areas in front of and behind the obstacle or step In order to measure the density profi le the obstacle mould was divided into seven zones along the injection direction as illustrated in Fig 2 a The step mould was divided into two zones which represented the thin part and thick part asillustratedinFig 2 b Theeffectsofvariousparametersweresim ulated such as injection velocity injection temperature particle size and powder loading To evaluate the level of the density inhomogeneity the standard deviation of the density values in all zones was calculated In some conditions the injection process failed to fi ll the last zones of the mould For such cases the density value was regarded as zero 3 2 Materials and processing Forverifi cation gas atomised316Lstainlesssteelpowder provided by Osprey Metals Ltd d50 12 m and a multicomponent binder sys tem composed of 55 wt paraffi n wax 35 wt polypropylene and 10 wt stearic acid were used The powder and binder were initially blended in a three dimensional shaker for 30min atroom temperature andthencompoundedinanXSM1 20 80rubbermixerat165 Cfor2h The apparent viscosity and specifi c heat of the binder were measured using an Instron3211 capillary rheometer at temperatures ranging from 150 to 180 C A cylindrical die 1 2 mm in diameter and 51 1 mm high was used The shear rate was varied between 40 and 1200 s Viscosity shearratecurvesforthebinderattherelevantmouldingtem peraturesareshowninFig 3 Thetestingmethodsanddataforthepres sure volume temperature andthermalconductivityofthebinderwere taken from Zhang s work 27 Ignoring the temperature effect the spe cifi cheatandthermalconductivityofthebinderwere1900J kg 1 K 1 and 0 2 w m 1 K 1 respectively The injection moulding process was conducted on a JPH80c injection moulding machine manufactured by Guangdong Only Machinery Co Ltd China The moulding parameters are shown in Table 1 3 3 Rheological properties of the binder The dependence of the binder viscosity on the shear rate can be expressed by a power law model 28 as follows k n 1 8 where is the apparent viscosity of the feedstock k is a constant that represents the viscosity at a shear rate equal to 1 and is the shear Table 2 Values for k and n of the binder at different temperatures Temperature C knR2 1403 214 1040 45050 9930 1463 093 1040 45140 9933 1522 827 1040 45060 9943 1582 579 1040 45110 9961 1642 325 1040 45190 9973 1701 775 1040 47620 9915 Fig 4 Curve of n vs T for binder 54H He et al Powder Technology 291 2016 52 59 rate It is also noted that the fl uid may not strictly behave as a non Newtonianfl uidatveryloworveryhighshearrates buttheNewtonian behaviour is not considered in this work due to the simplicity of the simulation process The ln vs ln curvesare shown in Fig 3 Thevalues of n and k for the binder at different temperatures can be obtained by using linear fi t as shown in Table 2 The correlation coeffi cient for stan dards was above 0 99 The values of k and n at temperatures not included in the experi ments were also needed for the simulation work The dependence of the binder viscosity on the temperature can be expressed by the Arrhe nius equation k k0exp Q RT 9 where Qis thefl ow activationenergy Ris thegasconstant T is thetem perature and k0is a reference viscosity The ln k vs T 1curve is shown inFig 4 a Thevaluesobtainedwere k0 9 278965andQ 28 221 47 kJ mol average value In order to simplify the calculation a linear relationship between n and temperature was used in this study n aT b 10 Then vs T curveis shown in Fig 4 b The valuesobtained were a 6 21429 10 4andb 0 18922 Byusingtheseparameters theviscos ity of the binder at different temperatures can be calculated 4 Results and discussion 4 1 Simulation for obstacle mould For a certain feedstock the fi lling patterns of powder injection moulding mainly depend on the injection velocity injection tempera ture and powder loading Particle size is another possible factor The Fig 5 Simulated particle distributions for different parameters a injection velocity b injection temperature c particle size and d powder loading Unless otherwise noted the default parameters were 12 m s 160 C 40 m and 50 respectively Fig 6 Injection completeness of sample with injection velocity of 6 m s a simulated and b experimental 55H He et al Powder Technology 291 2016 52 59 effects of these parameters on the density distribution are shown in Fig 5 All the curves show similar trends the maximum values of all conditions are located in zone A they then sharply drop approximately 6 to the minimum values in zone B After that in most conditions the densities improve somewhat in zones C and D but then drop slightly in zonesEtoF AsshowninFig 5 somecurveshavenodotsinzonesDtoF which means that the feedstock was frozen before these zones and failed to fi ll the mould At the end of zone A the width of the path for feedstock fl ow de creases dramatically Particles are trapped before the triangle obstacle leading to the highest density Zone B exhibits a much lower density because far fewer particles can diffuse into it The feedstock continues to fl ow yet the temperature drops and the velocity of the frontier grad ually decreases If the injection parameters are not set correctly the frontier cools down too fast and the fl owability of the feedstock deteri orates resulting in insuffi cient injection into the mould In all conditions the feedstock in zone B is separated by the trian gle obstacle and then joins together The mould region near the bot tom of the obstacle triangle is diffi cult to fi ll Therefore zone B exhibits lower density and thus is more likely to form a wielding line Subsequent zones such as C and D have a somewhat higher den sity because the join process is more complete However at the tip Fig 7 Comparison between simulated and experimental particle distribution for different parameters a injection velocity of 6 m s b injection velocity of 12 m s c injection tem perature of 170 C and d powder loading of 55 Fig 8 BSE photographs for the sample with injection velocity of 12 m s for a zone A and b zone B 56H He et al Powder Technology 291 2016 52 59 such as in zones E and F the particles are subjected to a high pushing force from the fast moving particles behind them The fl ow of the frontier particles is accelerated and thus they are separated from the particles behind them FromFig 5 itcanbeconcludedthattheinjectionparametersstrong ly affect the density values and density distribution From the calcula tion of Fig 5 a the standard deviation values for 6 9 and 12 m s are 60 68 57 04 and 45 respectively The best injection homogeneity is achievedat12m s because anaccelerated injectionrateincreasesthe shear rate and reduces the viscosity of the feedstock As a result more zones can be fi lled However in as fi lled zones A B and C 6 m s shows the lowest density variation Future work will be focus on this phenomenon a possible reason is that the higher velocity traps more particlesbeforetheobstacle AsshowninFig 5 b highertemperatures increase the density and homogeneity of the density distribution The values for 150 160 and 170 C are 56 45 45 and 4 6 respectively 170 C shows a much lower value because all zones are fi lled As shown in Fig 5 c 50 m can fl ow into zone F with 40 density but 40 m fails thus 50 m shows better density homogeneity 10 12 than 40 m 45 As shown in Fig 5 d 55 powder loadinggivesahigherdensityvariation 62 79 than55 45 and fails to fi ll zone E Inordertosuggestagroupofoptimisedinjectionparameters thein jection completeness needs to be taken into consideration Thus al though 6 m s provides a lower density variation than 9 and 12 m s in the fi lled zones it exhibits the worst fl owability For better fl owability the simulated results recommend a higher injection velocity tempera ture coarser particle size and a lower powder loading From the rheo logical point of view the change of these parameters leads to a decreaseintheviscosityofthefeedstock whichgivesbetterfl owability Amongthem higher injection temperatures obviously promote density homogeneity lower powder loading and coaser particle size do not promote better homogeneity in the as fi lled zones but more zones are fi lled Higher velocities may trap more particles before the obstacle leading to a higher density variation in the as fi lled zones 4 2 Verifi cation for the obstacle mould Experiments were carried out to verify the simulation Some typical conditions in Fig 5 were chosen for the verifi cation as shown in Figs 6 and7 Forthepredictionofinjectioncompleteness theexperimentalre sults showed good agreement with the simulated results As shown in Fig 6 the feedstock shows an injection distance similar to that of the simulated sample The effects of injection velocity injection parameters and pow der loading are shown in Fig 7 Fig 8 shows the back scattered elec tronic BSE images of zones A and F of the sample shown in Fig 7 a The area of the white powder phase was calculated by an Image Proplus software to determine the density of the zone This is be cause the samples in present work were too small to do an accurate cut and density test As shown in Fig 7 the experimental density was approximately 5 6 lower than the simulated density This dif ference is mainly attributed to the pixel calculation at the edge of the white phase The polishing effect may be another reason Fig 8 indi cates that the different values from A to B 7 05 are similar to the simulated value 6 02 Although the values