Shear Rate Thickening Flow Behavior of Semisolid Slurries

PRATYUSH KUMAR, CHRISTOPHE L. MARTIN, and STUART BROWN

The constitutive f low behavior o f semisolid slurries is a strong function o f temperature and pr iorprocessing history. The history dependence o f this constitutive behavior lends itself to modelingvia an internal variable framework that separates the evolution o f structure from the flow be-havior at a constant structure. We have assembled a computer-controlled, high-temperature Couetterheometer and have characterized the constant structure constitutive behavior o f a tin-15 pctlead system from 0.3 to 0.45 fractions solid. The flow behavior at a given material structureis shear rate thickening and not shear rate thinning, as is commonly assumed.

I. I N T R O D U C T I O N

INCREASING attention is being given to net shapeprocessing o f components using technologies involvingsemisolid precursors. Semisolid processing involves per-mitting a melt to partially solidify before shape-makingoperations,tuzl The degree o f solidification may vary from20 to over 90 pct solid, and the constitutive behavior o fthe resulting slurry has a strong dependence on percent-age solid. The semisolid is usually stirred while coolinginto the two-phase condition, producing a slurry o frounded solid particles that are smoother than normaldendrites. At solid fractions above 0.05 to 0.1, the slurrybehaves as a non-Newtonian, history-dependent fluid.At higher solid concentrations (above 0.5 to 0.6), theslurry may act as a nonlinear viscoplastic solid whichmay be handled and processed via high-temperatureforming methods used in traditional metal forming, suchas forging, extrusion, injection, and rolling.

Semisolids frequently exhibit what has been charac-terized as thixotropy, where the effective shear resis-tance o r apparent viscosity decreases as deformationoccurs. This thixotropic behavior is generally assumedto result from the breaking up o f particle agglomeratesdue to continued shear deformations. [1,2.3] These agglom-erates are characteristic o f stirred semisolids. An arresto f flow or decrease in flow rate is accompanied by anincrease in deformation resistance due to the reformationo f agglomerates. The material consequently exhibits avery strong coupling between flow behavior and themicrostrnctural state o f the material.

The experimental characterization o f semisolids hasnot normally corresponded to actual processing condi-tions, however. T o our knowledge, most o f the experi-mental characterization has involved "steady state"behavior, where the flow response is measured after tenso f minutes or hours o f constant shearing, tim The mea-sured flow behavior from steady state experiments is shearrate thinning, pseudoplastic, where the apparent viscos-ity decreases with shear rate. Actual processing, such asdie filling or forging, lasts seconds. Since the kineticso f structure evolution tightly couple to flow behavior,there is no reason to expect that steady state shear rate

P R A T Y U S H KUMAR and CHRISTOPHE L. MARTIN, ResearchAssistants, and STUART BROWN, Associate Professor, are with theDepartment of Materials Science and Engineering, MassachusettsInstitute of Technology, Cambridge, MA 02139.

Manuscript submitted May 12, 1992.

thinning behavior represents constitutive behavior duringdynamic processing conditions. As we will demonstratein Section IV, the short-term flow response is very dif-ferent from the steady state response.

The inherent constitutive behavior o f semisolid slur-ries depends on a number o f macroscopic factors, in-cluding temperature, shear rate, and characteristics o f theslurry. These factors translate to a set o f state variablesthat govern the rheology o f the material. A subset o fthese state variables is cal led internal variables o r struc-ture variables and can include a measure o f the degreeo f agglomeration among solid particles, fluid phase vis-cosity, particle s ize , particle morphology, and distribu-tion o f particle sizes. Internal variables are distinct fromexternal variables, such as temperature o r shear rate, inthat internal variables cannot b e directly imposed on theslurry but instead evolve according to specific kineticprocesses. Given these internal variables at any point,however, the state o f the material can be characterizedirrespective o f its pr ior history.

There exists a number o f models to characterize thephenomenology exhibited by metal semisolid sys-tems. H'5-91 The constitutive behavior o f these materialsis rather complex; the interaction among the solid par-ticles coupled with the hydrodynamic flow makes theresponse exlremely nonlinear. None o f the existing modelscaptures the complete constitutive response.

II. I N T E R N A L V A R I A B L EM O D E L F R A M E W O R K

An internal variable framework effectively combinesthe microstructure level structural kinetics with the mac-roscopic f low behavior. Internal variable models involvetwo sets o f relations, one set that represents the f lowbehavior o f the material at a given structure defined bythe internal variables

^"Fij = f i j ( ' Y m n , T, Sl...k) [1]

The second set o f relations represents the evolution o fthe internal variables as a function o f the flow conditionsand the internal variables themselves:

as,dt Sp = ~p(Ym,, T, s l ..k) 1 <--p <-- k [2]

where Y,,n = shear rate;^f0('/m,, T, Sl..D = flow equation;

METALLURGICAL TRANSACTIONS A VOLUME 24A, MAY t993--1107

T = temperature;rq = shear stress;

SL..k = set of k internal variables that char-acterize the state of the material; and

~p('~,,,,, T, sL k) = evolution equation for internal vari-able sp.

The preceding equations can equivalently be written as

~ m , ~- ~]mn(T i j , T , S l . " .k) [3]

and

dsp de ~p('cij, T, s~...k) 1 <-- p <- k [41dt

The total model therefore consists of a set of coupled,nonlinear, ordinary differential equations that can be in-tegrated throughout a flow history.

An appropriate flow equation for semisolid materialsmust represent certain fundamental phenomena. It hasbeen observed that within the liquid phase, individualsolid particles can agglomerate, tl°~ This observation istrue even for moderate solid fractions. These agglom-erates hydrodynamically interact with other agglomer-ates. Each agglomerate also undergoes plasticdeformation. We define an internal variable s (0 -< s1) representing the degree of agglomeration. In a fullyagglomerated state, when all particles are connected ina network fashion, s = 1, and in fully disagglomeratedstate, when none of the particles are connected together,S ~--- 0 .

The flow resistance of the fluid is believed to comefrom two different mechanisms. The first mechanism isthe energy dissipation due to disruption of particle bonds.The second mechanism is associated with the hydro-dynamics of flow describing the interaction a m o n g par-ticles and agglomerates suspended within the fluid. Weassume that the rate of energy dissipation can be takenas a dissipation potential, fijJ For simplicity, we furtherassume that the deformation in the solid phase occursdue t o shear flow and neglect the normal stresses. Therate of energy dissipation due t o plastic deformation ofsolid particle bonds during deformation, Ed, can be ex-pressed as

L J V b

where ~'so~d = shear stress in the deforming solid volume;7so~d = Shear rate in the deforming solid particle-

particle bond region;Cb = volumetric concentration of bonds; andVb = volume of the deforming solid.

Assuming that the shear stress, %~d, and shear rate,4Aol~d, in the particle-particle bond reg ion (we will referto it as "weld") can be taken as average values, we have

E d = Tsolid'Ysolid(~ d V b l Cb [6]L J Vb -.I

Depending on the exact particle morphology and three-dimensional connectivity, the total plastically deformingsolid volume fraction can be written as

[71

where al = geometric factor; andf~ = volume fraction solid.

Variations in the microstructure mean that deforma-tion is not occurring uniformly but instead takes placein local weld sites. We therefore determine the forcetransmitted t o the agglomerates due to an imposed mac-roscopic deformation rate and calculate the average shearstress acting aross an individual weld. Following Albersand Overbeek, ~:J who developed an expression for theforce transmitted across a spherical particle pair, we ex-tend this result t o a general expression for force trans-mitted to a particle agglomerate as

F ~ ~2~fT(hlAagg) [8]

where a2 = proportionality constant;/xy = fluid viscosity;3~ = macroscopic shear rate;

Aagg = cross-sectional area of the agglomerate; andAI = geometric parameter.

Thus, the stress in the particle "weld":

tangential component of force F across the weldTsolid

t ~ /.EFT AiAaggTsolid --

A2(r)Aagg

where

weld area

[9l

&(T) [10]

is a geometric parameter relating the agglomerate areat o weld areas and a~ is a constant that scales the totalhydrodynamic force acting on the agglomerate to thetangential force acting across a weld. The term Az(T) hasa temperature dependence of diffusional mass transfer,because we believe that the particle weld area grows ata rate governed by diffusive processes. Simplifying,

a;;tl"l'solid .= ~2---~[~f"y [ 11]

Imposing a power-law dependence of strain rate on shearstress in the solid,

"Ysolid = /~(T)Tsnolid [ 12]

where

n > l

and

fi,(T) = A0exp l~l-aVl [131k, R T /

is the temperature-dependent power-law coefficient. Theterm Q~ is a deformation associated activation energy,on the order of that for the lattice diffusion. SubstitutingEq. [11 ] for "fsoud,

1108--VOLUME 24A, MAY 1993 METALLURGICAL T R A N S A C T I O N S A

Ed

or

~solid ~- ,4(T) ,~n [14]

Inserting Eqs. [11], [14], and [7] into Eq. [6] results in

a;A, ] A a '2A, " n . n

where

E d = C(T)s f s txT+l ~"+1 [16]

r- t ~n+l

C ( T ) = a~[ a~A, / A(T) [17]LA2(T)J

Equation [ 16] is therefore the final expression fo r en-ergy dissipation rate due to a particle-particle welddeformation.

The second mechanism for the energy dissipation, thehydrodynamic effect due to relative particle mot ion , isgiven by an expression derived by Frankel and Acrivos.[13]We have incorporated it with slight modification to takeinto account the agglomeration factor:

(C[[Cmax)1/3 1E,. = A ( s ) 1 - (C/Cm.x) u3 " 2 /x:~2 [18]

where A(s) = hydrodynamic coefficient whichdepends on particle s ize . mor-phology, and geometricarrangement;

c = f~ (1 + 0. Is) = effective fraction solid accom-modating entrapped fluid withinagglomerates; and

Cmax = 0.625 -- 0. Is = maximum effective volumepacking fraction solid at a givenlevel o f agglomeration.

The total rate o f energy dissipation can therefore bewritten as

(C[ /Cmax) 1 / 3 le = A ( s )

1 - - (C/ /Cmax) 1/3 --2/[£f~/2

+ C(T)s fAx~+lg/n+l [19]

Given the assumption that e is a dissipation potential.we have

0e~" = - - [ 2 0 1

aS,Therefore,

(C/[Cmax) 1/3"r = A(s) " lx/~

1 - - (C/ /Cmax) 1/3

+ (n + 1)C(T)Sfsl~;+l~: [21]

We have obtained the value o f n --~ 4 from the highhomologous temperature 0.99 Tm rate-dependent f lowbehavior o f pure lead. This exponent was obtained froman earl ier set o f compression t e s t s[141 and also corre-sponds to Frost and AshbytlSj deformation maps.

The single internal variable representing the degree o f

agglomeration will evolve according to competing ki-netics for agglomeration and disagglomeration. A rea-sonable first-order functional representation for theevolution o f s is

ds

dt

where H ( T ,

= ~ = H(T,f~) (1 - s) - R ( T , f s ) s ~ " [22]

fs) = agglomeration parameter; andR ( T , f~) = disagglomeration parameter.

Although we have proposed a specific functional formfor the evolution equation, more experimental investi-gation is necessary to either validate those forms o r mo-tivate other kinetic equations. Relatively simplesimulations o f semisolid processing, however, havedemonstrated that such an internal variable model canrepresent important constitutive features, including shearrate thickening, thixotropy, and hysteresis.

Although other structural characteristics can be in-cluded as internal variables, our experience is that at agiven temperature, the fastest evolving characteristic isthe degree o f agglomeration. The fraction solid, to f i r s torder, can be evaluated as a function o f temperature.Particle size and morphology also affect the flow resis-tance o f the semisolid slurries at these solid fractions andare reasonable next candidates fo r inclusion as internalvariables.

Phenomenologically, semisolids exhibit time-dependentthixotropy. This implies that they exhibit hysteresis whenshear stress is plotted vs shear rate fo r a cyclic loadinghistory. In addition, they exhibit shear rate thinning be-havior, p s e u d o p l a s t i c i t y , at steady state, where the shearresistance increases at a decreasing rate as the shear rateis increased. Classically, this translates to a decreasingapparent viscosity with increasing shear rate. However ,the inherent behavior o f semisolids is nonlinear and ap-parent viscosity provides limited insight into constitutivebehavior. W e have experimentally observed that semi-solids also show an increasing rate o f change o f flowresistance with shear rate, dilatancy,* at constant structure.

*Dilatancy has also been referred to the volumetric expansion of anonlinearfluid in shear flow. Here, we use the association with shearrate thickening.

Figure 1 illustrates the combination o f these consti-tutive behaviors for semisolids. The dashed lines rep-resent the shear stress/shear rate response of the materialat constant structures (or states). For ease o f illustration,we assume that there is only one state variable, s. If theshear rate is changed quickly enough so that the structuredoes not have enough time to evolve, then the shear stressvs shear rate curve will fol low these constant structurelines. The constant structure response is shear rate thick-ening. However, if sufficient time is spent at each shearrate so that the structure achieves steady state conditions,then every point on the curve corresponds to a differentstructure. This response is shear rate thinning and is rep-resented by the heavy dashed line in Figure 1. The pro-posed model adequately captures the major features o fsemisolid flow behavior under both steady state and tran-sient conditions.

This article primarily addresses the constitutive be-havior represented by Eq. [1], that is, the flow behavior

METALLURGICAL TRANSACTIONS A VOLUME 24A, MAY 1993--1109

• , '

/ /

¢is/~jtlf; ,' . , '

/ / . . / " ~

:,' (Steady ,ate Respo~),.'"

;' ,,," .,'" ~(Constant Struel~re K~ponse).-"'"'"'

Shear r~ate. "~

Fig. 1--Schematic illustration of variation of flow behavior as afunction of shear rate and different material structures or states.

at a given structure. We limit our attention to the mod-erate volume fraction solid range where the constitutivebehavior is fluid-like. Existing steady state experimentaldata are not appropriate fo r decoupling flow from struc-ture kinetics, because both flow and structure are chang-ing simultaneously. We have developed an experimentalapparatus that achieves this decoupling and find that shorttime flow behavior is quite different from steady stateflow. The implication is that the assumption o f shear ratethinning behavior yields incomplete understanding o fsemisolid flow, especially under expected processingconditions.

Section III provides a description o f our experimentalapparatus and the important problems addressed beforeproper experiments could be performed. Section IV thendescribes our experimental procedure and provides theresults o f experiments using a tin-15 pct lead alloy. Theexperiments cover a range o f slurries solid concentra-tions from 30 to 45 pct volume fraction solid. We alsorelate these results to the internal variable model, andwe conclude in Section V with a discussion o f our resultsand proposals fo r continued work.

III. EXPERIMENTAL A P P A R A T U S

Figure 2 provides a schematic o f a high-temperatureCouette rheometer that we have fabricated to evaluatethe constitutive behavior o f semisolid slurries. The rhe-ometer is a Couette geometry, with an outer rotating cupwith a stationary inne r bob. The cup rotates to inhibitthe onset o f Taylor vortices that lead to incorrectly hightorque measurements. The frame o f the rheometer is

r,. ! I~

I ¢ :m Coupling

Control ~ . . Brushless

To Data ServomotorAt

Controller

Fig. 2--Schematic of high-temperature rheometer.

composed o f four solid posts to provide very high tor-sional stiffness. The cup and bob are located within aradiant tube furnace capable o f rapid heatup to temper-atures exceeding 1200 °C. The graphite cup and bob re-main in an argon atmosphere throughout an experiment.

Figure 3 provides a schematic o f the cup and bob ge-ometries. The b o t t o m surface o f the bob is tapered toprovide a more uniform shear rate as a function o f radialposition. The inner surface o f the cup and outer surfaceof the bob are grooved longitudinally with a groove widthand depth o f 1 mm and a periodic spacing o f 2 ram. W ehave also employed a cup and bob with a 0.5 mm d e p t hand spacing and have found little difference in the mea-sured response. Temperature uniformity is measured alongthe outer surface o f the bob using a l inear array o f four ,calibrated, chromel-alumel thermocouples, with loca-tions as indicated in Figure 3. Axial and radial temper-ature uniformity within the cup and bob can be maintainedwithin a 0.5 °C range. Accurate measurement and con-trol o f temperature gradients and variation o f tempera-ture with time are critical, since the fraction solid is avery strong function o f temperature, particularly in thehigh fraction solid regime.

It is critical to control shear rate and provide p rec i seshear rate histories due to the evolution o f slurry struc-ture and the strong dependence o f the constitutive b e -havior on shear rate histories. The Couette ar rangementprovides a homogeneous, well-understood f low field,

1110--VOLUME 24A, MAY 1993 METALLURGICAL TRANSACTIONS A

Stationary bob

T I

T 2 Control T.C.

T 3

Rotating cup

Fig. 3--Schematic of graphi te rotating cup and stationary bob.

where the shear rate can be determined explicitly. Mea-surements o f flow using an impeller are very difficult,if not impossible, to calibrate, particularly if the materialis nonlinear and history dependent. Cone and plate rhe-ometers are also problematic, since the flow field is againvery difficult to characterize for nonlinear materials andsince pressure gradients resulting from the cone and plategeometry can cause separation between the liquid andsolid phases. A servomotor with tachometer feedback isused to rotate the cup, permitting precise control o f an-gular velocity histories. Angular velocities up to ap-proximately 2000 rpm are possible. W e can thereforespecify arbitrary shear rate histories ranging from 0 toapproximately 1500 per second. This is, o f course, afunction o f the gap between the cup and bob which , inthe case o f the data presented here, is 3 mm. This rhe-ometer can change between the limits o f its angular ve-locity within tens o f milliseconds.

A torque transducer is attached to the bob to measurethe torque transmitted through the fluid in the annulartest volume from the cup to the bob, permitting mea-surement o f the true constitutive response o f fully de-veloped flow under both steady state and transientconditions. The torque transducer stiffness is very highto improve its response to transient tests. The shear rateis calculated through an expression derived byKreiger, 06,~71

where rc = inner radius o f the rotating cup,rb = outer radius o f the stationary bob,

~'b = stress at the outer surface o f bob, and12~ = angular rotation speed ( rad / s ) o f cup,

and defining

r~

rb

and

[23]

d lOge ~cm = - - [24]

d IOge Zb

Kreiger 's expression with 1 pct convergence is given by

lqc- - [ 1 +mlogee] if mlogee<0.2 [25]log~ e

and

fie 1 + m loge e + ~ m 2 d mloge e 3 d IOge ~'b

if 1 . 0 > m l o g ~ e > 0 . 2

[26]

If the gap between the cylinders is small enough, theabove expression can be approximated by a mean rateo f shear,

Equations [25] and [26] provide an accurate calculationo f shear rate independent o f the nonlinear constitutivebehavior o f the slurry.

A 386 microcomputer is used both to control theservomotor and to sample dc signals from the motortachometer, rheometer thermocouples, and torque sen-sors. A Keithley data acquisition system serves as theinterface fo r both digital-to-analog control and analog-to-digital signal processing. Sampling rates exceeding2 kHz are possible using this system, which permits res-olution o f very rapid transients.

We have performed spectral analyses on the torqueoutput signal and have used the resulting information tolimit contamination o f the torque signal due to cup andbob misalignment. W e expect flow development to bevery rapid across the cup and bob gap. With a minimumapparent viscosity o f 0.5 Pa-s (for a Sn-15 wt pet Pbsystem at 0.3 fraction solid), the characteristic time toobtain fully developed flow conditions has been esti-mated to be within milliseconds fo r a 3 mm gap. Ourservomoter has been tuned to provide a jump durationtime o f the order o f tens o f milliseconds to ensure de-veloped flow conditions. For the range o f Reynoldsnumbers observed during our experiments, inertial ef-fects can be safely assumed to be negligible.

IV. EXPERIMENTALP R O G R A M AND R E S U L T S

Semisolid slurries that exhibit a flow behavior depen-dent on their structural state require particular experi-mental techniques to separately determine f low behavior

METALLURGICAL TRANSACTIONS A VOLUME 24A, MAY 1993-- 1111

and structural kinetics. Steady state constitutive behaviorrepresented by Eqs. [1] and [21 can be expressed as

r,, = )~(5~,s, T, ss,) [281

where the ss subscript represents steady state conditionsand s** represents the steady state structure. This struc-ture, however, is a strong function o f temperature andshear rate. Steady state structure can be found by solvingthe following set o f nonlinear algebraic equations whichresult from setting the internal variable rates o f changeto zero.

dsi--- 0 = gi(5%, T, s]...k) 1 --< i -- k [29]

dt

.Unambiguous evaluation o f the flow equationf ( 5 ' , T, s) requires experiments where the shear rate israpidly changed. If the structure does not change overthe period o f the rate change, then the torque responseis a function o f t h e inherent f low behavior alone.

r = a~(5', T, s = constant) (30)

A set o f rate change experiments is illustrated inFigure 4, where a set o f rate changes from the same stateprovides a measure o f the flow behavior at that state.Each experiment provides an additional data point for agiven structural state's shear stress/shear rate response.If the structure evolution rate is sufficiently slow, thena single ramp in shear rate can efficiently characterizethe entire range o f shear stress/shear rate response at agiven structure. This experiment is also illustrated inFigure 4. Shea r rate ramp experiments therefore providea very efficient means o f evaluating inherent flow be-havior, because a single experiment can characterize theflow behavior at a given structural state. These transientexperiments however require very high data acquisitionrates to capture the transient response. One must also beconfident of negligible structural change and fully de-veloped flow in the rheometer.

The results o f our experiments on the tin-15 pct leadsystem follow. We determined the volume fraction solidusing previous work by Joly,14} correlating the fractionsolid at a given temperature. Quenched slurry micro-structures indicated that the particle size was on the ordero f 50 to 100 /xm. Experiments were performed at202 °C, 198.6 °C, and 196 °C, corresponding to 0.30,0.40, and 0.45 volume fraction solid. Higher fractionsolids produced large "spikes" in torque that we attributeto the formation o f particulate "bridges" between the cupand bob. This phenomenon has been observed by Turngand Wangt9] in a semisolid slurry and by Brady andBossis tlS] in hard sphere suspensions.* Torque, angular

*We expect this phenomenon to appear when the vo lume fractionsolid plus the vo lume of liquid entrapped in the agglomerates ap-proaches a maximum volume fraction (~r/6 for a simple cubicpackingapproximation). This vo lume fraction l imi t appears to be shear ratedependent: higher shear rates permit a higher fraction solid, whereaslower shear rates reach this percolation threshold earlier. W e attributethis, in part , to the greater degree of agglomeration at lower shearrates, forming a larger effective vo lume fraction so l id due to en-trapped liquid. Other factors affecting this threshold include hydro-dynamic effects that inhibit agglomeration or appropriate alignmentof particles/agglomerates.

(a)

(b)

(c)

I I~ u , , s ,~ c o n s t a n t

I' Y O , ~ I

I• I

I II I! I

Fig, 4--Schematic of shear rate change and shear rate ramp exper-iments: (a) shear rate increase indicating rapid decrease in shear stressafter change; (b) shear rate decrease with slow subsequent change inshear stress; and (c) shear rate ramp sampling a range of shear rates.

rotation rate, and thermocouple temperatures were col-lected throughout the experiment. The data acquisitionrate during transients was 100 Hz. All data were smoothedusing a LOWESS procedure implemented throughRS/1 data analysis software.tl9]

Figure 5 presents shear rate jump and drop experi-ments. We define shear rate drop and jump experimentsas those where the change in shear rate is very fast(order o f 10 ms). The changes in shear rate were made

1112--VOLUME 24A, MAY 1993 METALLURGICAL TRANSACTIONS A

1750

1500

1250

1000

750 Shear RateDrop

500

Shear RateJuanp

250

Sn-15wt%Pb at 196 C(0.45 Solid Fraction)Shear Rate Jump from300 to 436 i/sec andSheer Rate Drop from764 to 0.77 1/sec.

• Rate

Shear RateDrop

i Dr°Pl I

-I0 0 10 20 30

Timm (see.)

Fig. 5--Shear rate jump from 300 to 436 l/s and drop from 764 to0.77 1/s experiment on Sn-15 pct Pb at 196 °C.

200 Sn-lSwt%Pb at 202 C10.30 Solid Fraction)Shear Rate Ramp Downfrom Steady States

150" . /':~:'./y/////......y"/""100 .7' /"' .....""

50 ." / / , . - " /

o i;o ,oo 5;0 6;0 8oo

Shear Rate (1/sec.)

- - ~xperiment............. Model

Fig. 6--Shear rate ramp decrement experiments and model on Sn-15pct Pb at 202 °C, corresponding to 0.30 pct fraction solid.

from steady state conditions• We define steady state con-ditions in the following manner. We lower the temper-ature from the fully liquid state to the desired fractionof solid in approximately 5 minutes whi l e shearing at theinitial shear rate. We continue shearing the slurry at thesame shear rate for 25 minutes to reach a minimal tem-perature gradient of less than 0.5 °C and a temperaturestability over time of less than 0.2 deg. Notice that thetorque increases rapidly and then immediately decreasesin the shear rate increment, indicating a rapid evolutionof slurry structure. This is consistent with theagglomeration/disagglomeration interpretation ofstructure-flow coupling, where increasing the shear rateproduces a rapid breakup of particle agglomerates. Theshear rate decrement experiment, however, produces amuch s lower transient, indicating a s lower change instructure.

The slow change in shear stress immediately after arate decrement indicates that it may be possible to de -crease the shear rate more s lowly without changing thematerial structure and thereby characterize the constantstructure flow behavior in a single experiment. Shear rateramp experiments represent a s lower rate of change inshear rate (on the order of seconds). The duration of theramp was sufficiently short to ensure constant structurebut long enough to capture a complete f low curve be-tween the two shear rate extrema.

Figures 6 through 8 present the shear rate ramp dec-rement experimental data and simulated result obtainedfrom the internal variable model for the tin-15 pct leadsystem at three different fractions solid. The curves

5001[ Sn-15wt%Pb et 198.6 C(0.40 Solid Fraction)

450 Shear Rate Ramp Downfrom Steady States

400]

350-

300-

250-

200-

150-

100-

50-

100 2;0 , 0 0 , ;0 5 ;0 600 7 ;0 6 ° 0

Shear Rate (1/sec.)

- - Experiment............. Model

Fig. 7--Shear rate ramp decrement experiments and model on Sn-15pct Pb at 198.6 °C, corresponding to 0.40 pct fraction solid.

M E T A L L U R G I C A L T R A N S A C T I O N S A V O L U M E 24A, M A Y 1 9 9 3 - - l 113

70C

S00-

500~g

~ 4 0 0 -

300-

200

100

100 200 300 400 500 600 700 800

Sn-15wt%Pb at 196 C(0.45 Solid Fraction)Shear Rate Ramp Downfrom Steady States

800"

Shear Rate (I/eee.)

- - Experiment. . . . . . . . . . . . Model

Fig. 8--Shear rate decrement, ramp decrement experiments, andmodel on Sn-15 pet Pb at 196 °C, correspondingto 0.45 pet fractionsolid. Shear rate drop from 763 to 581, 281, and 0.77 1/s.

represent the variation in shear stress as a function ofshear rate for different shear rate ramp decrement ex-periments. Each curve therefore represents one rampdecrement experiment, starting initially from steady stateconditions, where the angular velocity of the cup de-creases linearly with time. We performed three ramp ex-periments from each initial steady state, and each shearstress/shear rate curve is representative of the threeexperiments.

We validated the ramp experiments with two addi-tional experiments. First , we performed shear rate dropexperiments from the same initial shear rate (and there-fore material state) to different intermediate shear rates.The resulting shear stress immediately after the drop ap-proached the shear stress curve obtained from a rampdecrement. The data points in Figure 8 represent mini-mum shear stresses due to shear rate drop experiments,each data point representing a single experiment. Noticethat the shear rate ramp decrement experiments approx-imate the abrupt rate decrement data, indicating that rampdecrements are adequate to evaluate constant structure(or state) flow behavior. Second, we performed a de-creasing shear rate ramp followed immediately by a rapidincreasing ramp back t o the original shear rate. Figure 9presents this experiment, where the shear rate was de-creased linearly t o 0.77 I/s, from 764 i / s in 3.47 sec-onds, and then immediately increased back to 764 1/sin 1 second. The curves are very close, indicating shearrate thickening behavior whether the shear rate is de-creased or increased. The final shear stress coincides well

800

7 0 0

600

500

400

300

200

Sn-ISwt%Pb at 196 C(0.45 Solid Fraction}Shear Rate ~ Downfrom Steady Stateat 764 I/seethen Ramp Up

fti

iRamp

- - 7 - t i i i i i ~ i i ~ I ix 0 0 2 0 0 3 0 0 4 0 0 5 0 0 s o o 7 0 0 8 0 0

Shear Rate |I/see.)

Fig. 9--Shear rate ramp decrement and increment experiment onSn-15 pct Pb at 196 °C. The shear rate ramp decrement occurred in3.47 s, while the ramp return to the original shear rate occurred inIs.

with the initial shear stress, indicating no change in slurrystate.

The constant structure data presented for the tin-15 petlead system indicate that the inherent flow behavior at ag iven structure is not shear rate thinning. Instead, it fol-lows a behavior characteristic of shear rate thickeningflow. This response is predicted by the m o d e l as well.Transient flow conditions, such as those encounteredduring mold filling, therefore cannot be simulated as-s u m i n g shear rate thinning behavior. Our data addition-ally indicate that the flow resistance does not reduce t ozero at very low shear rates, potentially indicating a shearrate dependent yield stress. We have not examined thisphenomenon in detail.

The steady state behavior of the tin-15 pet lead slurryresembles the results of other investigators. Figure 10indicates the steady state flow response of the Sn-15 petPb semisolid with fraction solid. If interpreted u s i n g ap-parent viscosity, this would correspond t o a shear ratethinning flow behavior. The model captures this steadystate shear rate thinning response as well.

The large dependence on the relative fraction solid im-plies a very strong dependence of flow resistance ontemperature. We have noticed this during our experi-ments, a variation of 0.5 °C can correspond t o very largevariations in measured shear stress. This is also impor-tant in the interpretation of transient behavior after abruptchanges in shear rate. Deformation heating effects canchange the temperature of the slurry and therefore the

1 1 1 4 - - V O L U M E 2 4 A , M A Y 1 9 9 3 M E T A L L U R G I C A L T R A N S A C T I O N S A

iii Sn-15wt%Pb at0.30, 0.40 and 0.45Solid FractionSteady State Pointsand Modal.

0,45

300

Model

0.40

2 0 0 t / / 0 . 3 0

I I I P2 0 0 4 0 0 600 800

Shear Rate (1/sac.)

Fig. 10--Steady s ta te flow response of Sn-15 pct Pb as a functionof vo lume fraction solid.

measured shear stresses. We have consistently observedsmall but significant (on the o rde r o f 1 °C) delayed de-creases in temperature during shear rate change experi-ments. These decreases are larger if the test furnace isprogrammed to give a constant output p o w e r during theexperiment. Smeulders e t al.t2°J have suggested that thesefluctuations in temperature, particularly loca l tempera-ture gradients in the region o f the agglomerate weld dueto local deformation, can affect the weld between ag-glomerated particles. This temperature effect will there-fore accentuate the disagglomeration and agglomerationprocesses during changes in shear rate. We intend toevaluate the kinetics o f agglomeration and disagglom-,eration from the transients occuring af te r shear rate in-crements and decrements. For our initial modeling efforts,we have taken functions H(T, f~) and R(T, fs) (Eq. [22])as parameters, determining them from ramp down andsteady state experiments, and considered only isothermalcases.

The hydrodynamic coefficient, A(s) (Eq. [21]), has beenretained as a parameter in our current model. W e intendto investigate the functional dependence o f A with re-spect to s.

V. CONCLUSIONS AND F U T U R E E F F O R T S

The transient constitutive behavior o f semisolid slur-des at constant structure and moderate fractions solid ischaracterized by an increasing rate o f flow resistance withshear rate (dilatancy or shear rate thickening). The steady

state response over relatively long time periods is shearrate thinning. Semisolid flow can be represented by aninternal variable constitutive formulation, although com-plete model formulation at intermediate fractions solidrequires additional characterization o f the structural vari-ables and their kinetics that affect flow, particularly thedegree o f agglomeration.

The actual mechanisms that govern agglomeration anddisagglomeration are not completely understood. The solidparticles combine into agglomerates through a weldingprocess that depends on particle and fluid compositions,temperature, shear ra te , particle s ize , and potentially onsuch subtle features as particles' relative crystallographicorientations. The disagglomeration process similarly in-corporates these influences through a different kineticmechanism. More theoretical and experimental effort isrequired to describe these kinetics before they can bereliably included within an internal variable model.

Shea r rate thickening behavior over short deformationperiods is desirable, since it encourages stable flow. Theinstabilities encountered during semisolid flow thereforemay not be a consequence o f the inherent f low behavior.Instead, the flow irregularities may reflect temperaturegradients and associated variations in fraction solid. Shearrate thickening constitutive models are also computa-tionally fortuitous, since shear rate thickening behaviorinhibits flow localization. Difficulties associated with thedegree o f discretization in a numerical simulation maybe less serious than with materials that are flow softening.

The internal variable model given by Eqs. [21] and[22] captures the flow behavior o f semisolid materialsadequately. This model can be easily used fo r simulatingreal industrial applications, like die filling. History-dependent response o f semisolid metal alloy slurries andassociated material instabilities can be predicted by thismodel.

R E F E R E N C E S

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