Flow Behaviour of Semi-solid Metal Alloys

J. Non-Newtonian Fluid Mech. 112 (2003) 141–160

Flow behaviour of semi-solid metal alloys

J. Koke∗, M. ModigellInstitut fuer Verfahrenstechnik, Aachen University of Technology (RWTH), Turmstr. 46, 52056 Aachen, Germany

Received 8 November 2002; received in revised form 1 April 2003

Abstract

Semi-solid metal alloys, as used in industrial thixoforming, have a special microstructure of globular grainssuspended in a liquid metal matrix. The complex rheological properties are strongly influenced by the local solidfraction, particle shape, particle size and state of agglomeration. It was analysed how the microstructure develops independence of the shear rate and cooling rate during the solidification and it was observed that the average particlesize increased with increasing shear rate and decreasing cooling rate. In order to account for those phenomena, therate of crystal growth and the relationship between average particle diameter and viscosity was modelled by applyingthe Sherwood two-film model for the mass transport. The dependence of the viscosity from the particle size weremodelled with a modified Krieger–Dougherty model. Based on the rheological and microstructural observations anevaluation method was elaborated that allows for the construction of objective master curves that are independent ofthe particle growth during the experimentation. The isothermal experiments for the characterisation of the rheologicalbehaviour consisted of step-change of shear-rate and yield-stress experiments. From the experimental data thesteady-state flow curves could be determined as well as the time-dependent relaxation of the shear stress after a changeof shear rate. The steady-state rheological behaviour was found to be shear thinning. Nevertheless, immediately aftera shear-rate change an overshoot was observed that resulted from a short-time shear-thickening behaviour. The yieldstress was found to strongly depend on the microstructure and the degree of agglomeration of the solid phase. Withincreasing rest time the yield stress was increasing strongly, because of the agglomeration of the solid particles. Basedon the step-change of shear-rate experiments a single-phase flow has been developed that consists of a modifiedHerschel–Bulkley approach and accounts for the thixotropic as well as for the yield-stress behaviour of the alloys.© 2003 Elsevier B.V. All rights reserved.

Keywords:Semi-solid alloys; Suspension; Yield stress; Thixotropy

1. Introduction

Thixoforming has become an important forming technology during the last years. In contrast to theclassical die casting and forging processes, the alloy is processed in the semi-solid state, with globular

∗ Corresponding author. Tel.:+49-241-80-95159; fax:+49-241-8888-252.E-mail address:[email protected] (J. Koke).

0377-0257/$ – see front matter © 2003 Elsevier B.V. All rights reserved.doi:10.1016/S0377-0257(03)00080-6

142 J. Koke, M. Modigell / J. Non-Newtonian Fluid Mech. 112 (2003) 141–160

grains suspended in the liquid phase. Most commonly this process is used for the production of complexaluminium automotive parts. There is a high demand for models and numerical software tools allowingfor the accurate prediction of semi-solid casting processes. Although a lot of research has already beendone for the rheological characterisation of the semi-solid alloys (e.g.[4,13,19]), it is obvious that severalresults are in disagreement. In this paper, tin–lead alloys were investigated with a concentric-cylinderrheometer and it was analysed how the microstructure developed in dependence of the shear rate andisothermal shearing time. One motivation for this paper is to clear up some of the inhomogeneities beingfound in literature by systematically analysing the influence of microstructure and the type of measuringprocedure. In addition to the steady-state flow behaviour, the dynamic, isostructural and yield behaviourwere examined for different solid fractions and microstructures.

2. State of the art

Joly and Mehrabian[7] were the first to show that the representative viscosity decreases when shear-ing an alloy in the semi-solid state. It is commonly accepted that the steady-state viscosity is de-termined through the dynamic equilibrium of the formation and disruption of particle bonds. In theliterature similar observations on different alloys are reported, mainly tin–lead and aluminium-basedalloys [5,7,9,10,13,18]. When focusing on the often used tin–lead alloys it becomes obvious that manyresults do not compare very well. These inconsistencies can be detected when comparing the flow curvesfor Sn–15% Pb at similar solid fractions[7,9,13,15,21]. The direct comparison (Fig. 1) shows that someresults do not match or are even contradictory, as in the case of the data of McLelland et al.[13] andPeng and Wang[15]. While Peng found the shear stress to be an increasing function of the shear rate,Mclelland found the opposite at lower shear rates. In the paper of Joly and Mehrabian[7], the viscositiesare generally far higher than in the other cases.

Fig. 1. Comparison of steady-state flow curves of different authors, measured with concentric-cylinder rheometers on Sn–15%Pb.

J. Koke, M. Modigell / J. Non-Newtonian Fluid Mech. 112 (2003) 141–160 143

Similarly, the transient rheological analysis, which is necessary for the characterisation of the time-dependent or thixotropic behaviour, lead to different results. Often, step-change of shear-rate experimentswere carried out[11–16]. Especially the observations concerning the flow behaviour immediately afterthe shear-rate jump (isostructural flow behaviour) were divers. While the isostructural behaviour wasfound to be approximately Newtonian by Quaak et al.[16] and Mclelland et al.[13], Kumar et al.[9]observed a clearly shear-thickening flow behaviour.

Some reasons being responsible for the contradictions seem to be the different procedure of thesemi-solid alloy preparation and variations of the measuring procedures. Also the geometry of the mea-suring devices was different (grooved/smooth).

3. Apparatus

The experiments were performed with a Searle-type rheometer (Physica UDS200), which could beoperated in deformation and stress controlled modes. For the tests on semi-solid alloys the rheometerhad to meet the requirement of accurate temperature control and homogeneity. Therefore, in additionto the commercial heating unit (Physica TEZ400) an extra heater was installed in order to compensatefor the heat losses through the sample surface. By that the axial temperature difference could be keptwithin a range of dT = 0.2–0.5 ◦C. Besides, the sample was flooded continuously with hot nitrogento avoid oxidisation of the melted alloy. A schematic overview of the apparatus is given inFig. 2. Forthe design of the measuring system it was attempted to keep the gap as small as possible in order toapply rather homogeneous shear rates within the gap. However, the gap should be large enough to allowthe measurements of suspensions with particle diameters between approximately 250 and 650�m. Theused measuring system is a compromise between the maximum possible outer dimensions, the maximumtorque range and the rheometrical requirements for suitable concentric-cylinder device. The gap measurebetween inner and outer cylinder was 4 mm and the grooved inner cylinder had a diameter of 30 mm.

Fig. 2. Sketch of the experimental set up (1, computer; 2, rheometer electronic; 3, rheometer-measuring device; 4, thermostat;5, electrical heater for the cylinder TEZ400; 6, PT100; 7, controller for TEZ400; 8, electrical heater for nitrogen; 9, controllerfor nitrogen heater; 10, nitrogen).


Fig. 3. Viscosity and temperature during the material preparation within the rheometer (Sn–14.2% Pb;T = 198◦C; fs = 0.48;γ = 100 s−1, dT/dt = −4 ◦C/min).

The suitability of this specific measuring device was consolidated by comparison with geometricallydifferent devices. The concentric-cylinder devices used for similar alloys by other researchers consist ofgaps between 1.3 and 3 mm[7,9,13,15,21].

4. Standard procedure for the preparation of the globular microstructure

Prior to each measurement was the production of the globular solid phase. First of all the tin–leadalloy was melted within the shearing unit of the rheometer (T = 225◦C) and homogenised by shearingit. Afterwards the sample was cooled down with the desired cooling rate (usually dT/dt = −4 ◦C/min)under constant shear. Finally, the alloy was sheared at a constant shear rate for a certain time, so that therewere just little microstructural changes at the begin of the measurement. The temperature and resultingviscosity versus time are shown inFig. 3. Since the initial preparation of the alloy is of major importancefor the resulting microstructure, the pre-treatment parameters such as cooling rate, isothermal shearingtime, temperature and theoretical solid fraction are added to the figures.

5. Isothermal experiments

5.1. Step-change of shear-rate experiments

Step-change of shear-rate experiments were performed for Sn–14.2% Pb atT1 = 195◦C (fs = 0.53),T2 = 198◦C (fs = 0.48) andT3 = 200◦C (fs = 0.44). After cooling and 60 min of isothermal shearingwith γ = 100 s−1 the shear rate was lowered down toγ = 25 s−1, then step-wisely changed up toγ = 400 s−1 and finally decreased toγ = 100 s−1. In order to obtain rather stable values, the shear rateswere kept constant for 10 min before the measurement was taken.

In Fig. 4, the shear stresses are plotted versus time for three temperatures. After each step increasethe shear stress increased strongly before dropping to an almost steady-state value, where just minorchanges appear. It should be stated that in this complex two-phase material hardly ever a real steady


Fig. 4. Step-change of shear-rate experiments at different solid fractions: shear stress and shear rate vs. time (Sn–14.2% Pb;T = 195, 198, 200◦C; fs = 0.53/0.48/0.44;γpre = 100 s−1, dT/dt = −4 ◦C/min).

state exists, since the microstructure is characterised by continuous changes such as agglomeration anddeagglomeration (short-time scale), particle growth and dissolution (long-time scale). However, thesedata will be called steady-state in this paper in correspondence to other authors, meaning the equilibriumbetween agglomeration and deagglomeration. The effect of the immediate shear-stress increase is calledovershoot. Inversely an undershoot can be found when the shear rate is dropped. The corresponding“quasi” steady-state shear-stresses were obtained by averaging the last 45 s of each measuring step andare plotted inFig. 5. Especially the measurement at shear rate 400 s−1 exhibits step-changes in the shearstress (fs = 0.48 and 0.44), most likely due to the break-up of big agglomerates.

Fig. 5. Steady-state values from the step-change of shear-rate experiments for different solid fractions: shear stresses vs. shearrate (Sn–14.2% Pb;T = 195, 198, 200◦C; fs = 0.53/0.48/0.44;γpre = 100 s−1, dT/dt = −4 ◦C/min).


Fig. 6. Steady-state values from two experiments A and B on identically prepared alloys with shear-rate jumps in different order:experiment A [25, 50, 100, 200, 300, 400], experiment B [400, 300, 200, 100, 50, 25] (Sn–14.2% Pb;T = 198◦C; fs = 0.48;γpre = 100 s−1, dT/dt = −4 ◦C/min).

5.2. Influence of measuring procedure

The following experiment was carried out in order to emphasise the influence of the measuring param-eters on the results. Two step-change of shear-rate experiments were performed on identically preparedmaterial, one with ascending (A) and one with descending (B) shear-rate jumps. From the evaluated flowcurves it is noticeable that the two curves are different: while curve A has a negative slope, curve B is anincreasing function of shear rate (Fig. 6).

The reason for this discrepancy might be the continuous shear-rate dependent particle growth duringthe experiment. The contradicting literature results may also be explained by that phenomenon.

6. Microstructural analysis

The rheological properties of semi-solid alloys are basically influenced by the morphology of thesolid phase, which depends on the crystallisation conditions during the solidification and which changescontinuously due to Ostwald-ripening. Thus, the analysis of the microstructures in dependence of theshear rate, cooling rate and time are essential for a general rheological model.

In the following experiments, the parameters like shear rate, cooling rate and shearing time were variedsystematically, since they influence the crystallisation conditions and thus the particle number, size andshape.

6.1. Shearing-time influence

For the rheological characterisation of semi-solid alloys a stationary microstructure is desirable. Aqualitative investigation of this shearing-time influence was done with an experiment of continuousshearing at a constant shear rate. The resulting viscosities are plotted inFig. 7. During cooling the viscosity


Fig. 7. Viscosity vs. time at constant shear rate (110 s−1). Initial cooling rate is 1◦C/min. Steady-state temperature is 198◦C,corresponding to a solid fractionfs = 0.45. The metallographic pictures show samples quenched after 1 and 6 h of constantshearing (Sn–15% Pb).

increased due to the precipitation of solid phase. When shearing the sample isothermally the viscositydropped slowly until an apparent steady-state value has been reached after approximately 180 min. Fromthe course of the viscosity curve, it may be concluded that immediately after the cooling the microstructureis far from a steady state. It is likely that at the very beginning already dendritic particles are formed. Aftera longer shearing time under isothermal conditions the dendrites are shaped into globular grains due toabrasion, agglomeration, welding and ripening. The steady-state microstructure is characterised by thenumber, size and shape of the particles and their interactions. Furthermore, inFig. 7pictures of samplesfrom experiments with different shearing duration are printed. While the microstructure of the sample tothe left was obtained after 1 h of isothermal shearing, the second sample has been sheared for 6 h at thesame shear rate. It can be noticed that during shearing significant particle growth has occurred that alsocaused a continuous viscosity decrease within the first 240 min. Later on some instabilities occur and theviscosity increases slightly which is most likely due to the blocking of huge agglomerates in the measuringgap. Therefore, all subsequent relevant experiments were generally limited to a maximum duration of180 min in order to avoid those instabilities and in order to keep the particle size in a reasonable range.

A more detailed analysis with different shearing duration is presented inFig. 8. Samples after 1, 3and 6 h of constant shearing were analysed metallographically. The basic data obtained from the imageanalysis were the average particle diameterdcircle and the shape factorFcircle, which were defined asfollows:

dcircle = 2

√A

π, (1)

Fcircle = 4πA

U2. (2)

The particle cross-sectional area isA and the corresponding perimeter isU. The average particle diameterand the shape factor are plotted versus shearing time.


Fig. 8. Relation between shearing time and the average particle diameter (T = 198◦C; γ = 100 s−1; tvpre = 1–6 h;γpre = 100 s−1,dT/dt = −4 ◦C/min).

All experiments presented in this paper started after a pre-shearing time of 60 min. In the next section,a method is described that has been developed for the correction of the influence of the particle growth.

6.2. Shear-rate influence

Three different shear rates have been applied:γ= 100, 200, 400 s−1. The cooling rate was−4 ◦C min,and after reaching the process temperatureT = 198◦C, a 60 min interval of isothermal shearing followed.The particle diameters increased with higher shear rates, while the shape factors decreased slightly (Fig. 9).

Fig. 9. Relation between shear rate and average particle diameter (Sn–14.2% Pb;T = 198◦C; fs = 0.48; γpre = 100–400 s−1;tpre = 3 h; dT/dt = −4 ◦C/min).


The increase of particle size may be based on the improved convective mass transfer at higher Reynolds-numbers.

7. Correction method for measured data

From the microstructural analysis it became obvious that details of the experimental performance, likemeasuring time and the order of the shear-rate steps, have to be accounted for at the evaluation of theresults, since irreversible effects like particle growth are superimposed to the thixotropic properties. It isnecessary to correct for the influence of the time-dependent particle growth on the flow behaviour in orderto obtain representative and independent results, related to the size at the start of the experiment. For thatpurpose the particle growth in dependence of the shear rate and shear time as well as the influence of theparticle size on the viscosity had to be modelled.

7.1. Model for the particle growth

Wan and Sahm[22] suggested a model for the particle growth in dependence of the shear rate. Thisdid not satisfactorily predict experimental values as determined here[8]. That is why in this work theparticle growth will be modelled with a two-film model, as used in crystallisation research and chemicalengineering.

The mass transfer consists of two steps: the diffusion through the boundary layer to the surface andthe build-in process into the crystal matrix. With the assumption that the build-in process is a first-orderreaction, the differential equation is as follows[8]:

∂m

∂t= kdkr

kd + krA(c − csat), (3)

with the mass-transfer coefficients for the diffusionkd and the build-in processkr. While the saturationconcentrationcsat has to be determined from the alloy-phase diagram, the concentrationc of the su-persaturated melt is unknown and cannot be measured directly. The determination of the mass-transfercoefficients is difficult, because they cannot be measured separately.

If the reaction is limited by the build-in process, the particle growth would be independent of thetransport of the crystallising component, and there is no influence of the hydrodynamic conditions.In the contrary, if the reaction is limited by the diffusion process, an increase of the relative velocitybetween particle and fluid would lead to a reduction of the diffusion boundary-layer thickness and thusto a rise of the crystal-growth rate. Based on literature and the above presented experiments, it is likelythat the shear rate has a significant influence on the growth rate[1]. Assuming thatkr � kd Eq. (3)becomes

∂m

∂t= kdA(c − csat); kr � kd. (4)

The mass transfer can be described as a function of the relevant dimensionless numbers as Sherwood-,Reynolds- and Schmidt-number:

Sh= f(Re,Sc). (5)


The characteristic Reynolds-number for spherical particles with the diameterd in the shear field is

Re= ργd2

2η. (6)

The Sherwood-numberShis the dimensionless mass-transfer coefficient.

Sh= kdd

D. (7)

D is the diffusion coefficient in the liquid phase. The Schmidt-numberSccontains solely material con-stants.

Sc= η

Dρ. (8)

A well-known type for such a Sherwood relation is the Froessling equation:

Sh= 2 + ΦRep Scq. (9)

For 1< Re< 450 the parameter were determined as[6]:

Φ = 0.6; p = 12; q = 1

3.

With Eq. (9), the averaged surface mass-transfer coefficients for spherical particles can be calculated independence of the shear rate, the diameter and the viscosity of the liquid phase. With the choice of theparameters and when substitutingEqs. (6) and (8), Eq. (9)postulates that the mass-transfer coefficient isa linear function of the square root of the shear rate.

The basic mechanism for Ostwald-ripening is the minimisation of surface energy. Thus, during Ostwald-ripening big particles can grow while small particles dissolve. It has to be discussed, which effect is thetime limiting step: the particle growth or the particle dissolution. InsertingEq. (9)into Eq. (4)results in

1

V

∂m

∂t=

[12D

d2+ 3.6D

d

(ργ

2η

)1/2

Sc1/3](c − csat). (10)

FromEq. (10), it can be concluded that the dissolution of small particle has to be the faster step and thus itshould be sufficient continue with modelling the particle growth. For modelling purposes the particle-sizedistribution will be neglected and the average particle diameter will be taken as a parameter.Eq. (4)canbe transformed for the particle diameter:

∂d

∂t= 2

kd

ρ(c − csat). (11)

Since the driving concentration gradient cannot be measured separately, it was determined by adjustingthe growth equation to the experimental data.Fig. 10shows the calculated growth curve in addition tothe experimentally obtained data. The corresponding parameters are given inTable 1.

The model should be regarded as a reasonable engineering approach that is able to reproduce the growthprocess within the relevant time interval. With the help of this model the particle growth in dependenceof the shear rate can be predicted and used for the correction of measurement data.


Fig. 10. Average particle diameter vs. time. Experimental data compared to those obtained with the new crystal-growth model(Sherwood model).

7.2. Model for the particle-diameter influence on the viscosity

From the above described shearing experiment, it is already known that an increase in the averageparticle size causes a viscosity decrease. This may be explained by the following model: a thin layerof liquid phase is immobilised to the particle surface, which means that there is no relative motionbetween layer and particle. The immobilised layer leads to an increase of the effective solid volumefraction and thus to a viscosity increase. It becomes obvious that an increase in particle size leadsto a lower specific surface, which results into a smaller effective solid fraction and finally to a lowerviscosity.

The thicknessδ of such a layer depends on the chemical interactions between the particles and theliquid as well as on the hydrodynamics. For simplification, it is assumed that the hydrodynamic influ-ence is negligible. The ratio of the effective and the true particle volume shall be simply defined asthe effectiveness factorβ:

β = Φeff

Φ= (d + 2δ)3

d3. (12)

Table 1Parameters for the new crystal-growth model (Sherwood model)

Parameter Description Value

D (m2/s) Diffusion coefficient 1.97× 10−9

Sc Schmidt-number 148c − cs Difference of Pb-mass concentrations 0.87ρ (kg/m3) Average density of the liquid melt 7900


The viscosity of a spherical suspension in dependence of the solid fraction can be modelled with theapproach of Krieger–Dougherty:

ηrel =(

1 − Φeff

Φmax

)−kΦmax

. (13)

In the case of ideally shaped spherical suspensions the exponentkequals to 2.5 which is in correspondenceto the well-known Einstein model for weakly concentrated suspensions. In the other case of non-idealsuspensions this factor will be greater than 2.5, e.g.k = 4.5 [17]. The maximum solid fraction will beassumed to beΦmax = 0.68.

In order to investigate the influence of particle size experimentally, several experiments were performedwith differently prepared alloys by varying the cooling rate and shearing time at a shear rateγ = 100 s−1.The grain sizes were determined metallographically after quenching the material. Finally, the measuredviscosities could be plotted versus average grain size.

For the adjustment of the presented model the layer thicknessδ was set to a fixed value ofδ = 21�m,which is about 2–4% of the particle diameter. However, the assumption of a constant layer thicknessleads to a limitation of the model, because the effective volume might exceed the maximum volumefractionΦmax. Here, the determined parameters are valid for particle diametersd > 350�m. In Fig. 11,the viscosities for the model and the experiment are plotted versus the particle diameter at a constantshear rate.

It should be stated that the experimental data points can be predicted by the model reasonably well.

7.3. Correction of experimental data

With the help of the above derived model the shear stresses and viscosities can be corrected and relatedto the initial particle diameter d0, which corresponds to that diameter obtained immediately after thematerial preparation. By applyingEq. (13)the ratio of the corrected viscosityηkorr and actual viscosity

Fig. 11. Viscosity vs. particle diameter (fs = 0.48): experiment and modified Krieger–Dougherty model.


η can be written as

ηkorr

η= Skorr

S= [1 − (Φ0

eff/Φmax)]−kΦmax

[1 − (Φeff/Φmax)]−kΦmax. (14)

The effective volumetric fractionΦeff can be calculated byEq. (12)and the time-dependent diameterwith Eq. (11). Since the density difference of solid and liquid phase is very small it may be also writtenas

ηkorr

η= Skorr

S= [1 − (f eff0

s /fmaxs )]−kfmax

s

[1 − (f effs /fmax

s )]−kfmaxs

. (15)

First, the time-dependent particle diameters were calculated. The particle size prior to the first shear-ratejump is d0 = 534�m. The calculated time-dependent particle sizes for the previously described step-change of shear-rate experiments with ascending (A) and with descending (B) shear-rate jumps areplotted inFig. 12. Due to the shear-rate dependence of the growth rate both curves enclose a certainarea.

With the calculated particle diameters it is possible to correct the measurement data by applyingEq. (13)and the parameters fromTable 1. The data is related to the starting particle diameter. The original andcorrected flow curves are plotted inFig. 13.

Even if the curves are not congruent, the qualitative contradiction disappears, which proves the suitabil-ity of this method. Both shear-stress curves exhibit the same course with an apparent minimum at aboutγ = 100 s−1. It is not clarified yet if a flow-curve minimum is physically plausible[20], or if the datapoints at low shear rates are falsified by shear fracture. A shear-stress minimum would lead to an unstableflow behaviour, because two different shear rates could occur at the identical shear stress. However, somemacroscopic observations may not be explained other than by a minimum of the flow curve. Engelund

Fig. 12. Calculated particle growth for experiments A and B.


Fig. 13. Corrected and original shear stresses from two experiments A and B on identically prepared alloys with shear-rate jumpsin different order (Sn–14.2% Pb;T = 198◦C; fs = 0.48; γpre = 100 s−1, dT/dt = −4 ◦C/min).

and Wan[3] demonstrated that the existence of a flow-curve minimum is the only explanation for theoscillating flow of bentonite suspensions at low shear rates.

The correction method was applied on the following experimental results. After the data correction,the flow curves (Fig. 5) became increasing functions of the shear rate (Fig. 14) except for the curve atfs = 0.53 that still decreases at low shear rates.

Fig. 14. Corrected steady-state values from step-change of shear-rate experiments for different solid fractions: shear stresses vs.shear rate (Sn–14.2% Pb;T = 195, 198, 200◦C; fs = 0.53/0.48/0.44;γpre = 100 s−1, dT/dt = −4 ◦C/min).


8. Yield stress

An important issue for the rheological modelling is the question if there exists a yield stress. Whilethe determination of the yield stresses by extrapolation from the flow curve is usually inaccurate andproblematic[2], the employment of shear-stress controlled rheometers offers the possibility of the directyield-stress measurement. The extent of the yield stress depends on the inner structural forces, e.g. causedby particle bonds or dendrites.

The term yield stress ideally means that there is no irreversible deformation below the critical yieldstress. In the case of the semi-solid alloys a minimal creeping behaviour could be noticed, thus thefollowing presented yield values are apparent yield stresses.

In the following experiment, the applied shear stress was linearly increased from rest and also decreasedfrom continuous shear. InFig. 15, the shear stresses and the resulting shear rates are plotted for an Sn–15%Pb alloy that has been presheared with a cooling rate of dT/dt = −1 ◦C/min. It can be clearly seen thatthe material remains unsheared until a critical shear stress is reached. The reversed experiment withdecreasing shear stress shows again that there exists a critical shear stress below which the materialsolidifies and becomes unsheared. This phenomenon is commonly observed at viscoplastic substances[23] and makes it necessary to distinguish between two yield stresses:

(1) if the fluid was at rest prior to the application of the shear stresses, the yield point may be called astatic yield stress;

(2) if the yield stress was measured from a continuously sheared fluid, the yield point may be called adynamic yield stress.

The reason for the existence of two different yield stresses can be found in the dependency of the yieldstress from the microstructure. When taking an unsheared material the particle–particle bonds have tobroken (destroyed) and the adhesion forces have to be overcome. If the fluid was agitated, the particleshave to develop bonds during shear.

The time dependency of the static yield stress was investigated in an experiment, at which an identi-cally prepared sample was kept at rest for different duration. The shear stresses are plotted versus the

Fig. 15. Linear shear-stress ramps for the measurement of yield stresses. Left: increasing shear stress. Right: decreasing shearstress.


Fig. 16. Results from shear-stress ramp-experiments after different rest times. Left: shear stress vs. deformation angle. Right:shear stress vs. rest timetr (Sn–15% Pb;T = 195◦C; fs = 0.5; γpre = 100 s−1; dT/dt = −1 ◦C/min).

deformation angle inFig. 16(left) and the yield stress can be determined from they-axis. As can be seenin Fig. 16(right), the yield stress increases with time, which is attributed to the fact that the solid particlesagglomerate at rest and develop a spatial network that becomes more and more stable with increasingtime.

9. Isostructural flow curve

In the literature the term isostructural behaviour describes the spontaneous change of the flow behaviourafter a sudden change of shear rate. It is assumed that the inner structure or degree of agglomerationremains unchanged in the first instant. The so called isostructural flow curve can be determined with aspecial procedure of the step-change of shear-rate experiment as will be explained withFig. 17.

Here, the steady-state flow curve is given. When applying the shear rateγ2, the equilibrium shearstress of the flow curve (due to an equilibrium microstructure) will be approached after sufficient time.

Fig. 17. Scheme for the construction of the isostructural flow curve and the steady-state flow curve for thixotropic materials.


Fig. 18. Experimental procedure and isostructural flow curve for atγ = 100 s−1 (Sn–15.8% Pb;T = 198◦C; fs = 0.41;γpre = 100 s−1, dT/dt = −4 ◦C/min).

If the shear rate is immediately increased fromγ2 to γ3, the suspension will still have the equilibriummicrostructure ofγ2. Thus, the jump results into a shear-stress increase fromS∗(γ2) to S2(γ3), untildropping to the new equilibrium valueS∗(γ3). The shear-rate jump to a lower shear rateγ1 causes animmediate shear-stress decrease, until the new equilibrium valueS∗(γ1) is reached. The curve connectingthe spontaneous shear stresses with equal microstructure is defined as the “isostructural” flow curve. Foreach shear rate an isostructural curve exists, crossing the steady-state flow curve.

The experimental procedure is explained inFig. 18(left). After each shear-rate jump, the substanceis sheared with the shear rate fromγ0 in order to adjust the equilibrium before the next jump. Thisprocedure was applied on an alloy with a solid fraction offs = 0.41 and a shear rate ofγ = 100 s−1

as the equilibrium shear rate. After each shear rate change the material was sheared withγ = 100 s−1

until an almost steady state was achieved for the next jump. Inertia effects were excluded from the dataevaluation. The isostructural curve is plotted inFig. 18(right). A curve fit with a Herschel–Bulkley modelresults into the following function for the isostructural flow curve:

Siso (100 s−1) = 6.1 + 0.006γ2.07.

Obviously, the material exhibits elastic or plastic properties that cause the material to react as shear-thickening. Here, the authors model this complex behaviour as a shear-thickening “isostructural” flowbehaviour with the flow-exponentn = 2.07. This observation is of high importance for the technicalprocess and emphasises that the steady-state flow curve is not sufficient and appropriate for the simulationof thixocasting processes.

10. Model

From the above presented experimental studies, the following criteria for semi-solid alloys can bederived:

• the material consists of a yield stress;• the material exhibits a shear-thinning steady-state flow behaviour;


• the flow behaviour is time-dependent (thixotropic) (neglecting the irreversible structural changesthrough ripening);

• the isostructural flow behaviour is shear thickening.

Furthermore, all properties are influenced by the solid fraction, the microstructure and the degree ofagglomeration.

The derivation of a general flow model, describing the experimental results, consists of two steps: thecomposition of the steady-state flow curve and subsequently the extension to the full thixotropic model.

10.1. Steady-state flow curve

The semi-solid metal alloys consist of a Newtonian matrix fluid and a solid particle phase. If the solidfraction fs equals zero, the alloy has the Newtonian viscosityηl . At higher solid fractions the viscositybecomes shear-rate dependent, caused by the particle–particle interactions of the solid phase. Thus, theviscosityη shall be defined as the sum of a Newtonian and a non-Newtonian contribution. The shear-ratedependent term is modelled with a Herschel–Bulkley model that is commonly used for the descriptionof suspensions with yield stresses and shear-thinning properties:

η = ηl + S0(fs)

γ+ k(fs)γ

n−1, (16)

with the yield stressS0(fs), the solid-fraction-dependent consistency-factork(fs) and the flow indexn. Thesolid fraction may be calculated with the Scheil-equation.

10.2. Equation of state

In order to account for the thixotropic behaviour, an additional parameterκ has to be introduced,characterising the degree of agglomeration in dependence of the time and the shear rate. Thus, thegeneral equation of state becomes

η = f(γ, κ, fs). (17)

The kinetics of the structural parameter is described by a differential equation:

∂κ

∂t= f(κ, γ). (18)

For a time-dependent problem, even at constant shear rateγ, the viscosityη is a function of time throughthe variation of the structural parameterκ taken as the solution ofEq. (18)

η(t) = f(γ, κ(γ, t), fs). (19)

11. Conclusion

From the above presented experimental results, it was possible to get a broader understanding aboutthe rheological properties of semi-solid alloys. The experimentation with concentric-cylinder rheometersis very practical since a big variety of different experiments can be performed with the same apparatus.


Unfortunately, the common experimental procedure, as described in this paper, takes a rather long timewhich is accompanied by the disadvantage of particle growth during the experiment. Thus, an effectivemethod to correct for that phenomenon has been presented and applied successfully. Based on the experi-mental data, a thixotropic flow model could be derived. The detailed derivation of the full single-phase andtwo-phase model[8] and the definition and determination of the necessary parameters will be presentedin another paper.

Acknowledgements

The authors gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft(DFG) within the Collaborative Research Centre (SFB) 289 “Forming of metals in the semi-solid stateand their properties”.

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Flow Behaviour of Semi-solid Metal Alloys