Investigation of the Effect of Filler Concentration on the Flow ...

G. A. Campbell1*, M. D. Wetzel2

1Castle Associates, Jonesport, Maine, USA2Florence, Oregon, USA

Investigation of the Effect of Filler Concentrationon the Flow Characteristics of Filled PolyethyleneMelts

All polymeric slurries that have a high concentration of filler

are shear thinning. Shear thinning is an important character-

istic of polymers, filled and unfilled, because it enables an in-

crease in the throughput, shear rate in a die or an injection

molding system without having to use substantially more

power to increase the flow rate. Newtonian fluid-based slur-

ries show an increase in shear thinning as the concentration

of \filler" increases above the percolation threshold. As parti-

cle maximum packing concentration is approached the slur-

ries begin to approach a perfect pseudoplastic fluid. In some

cases, the shear thinning characteristics of a filled polymer

do not increase substantially as the filler loading is increased.

This is a quite different response than in Newtonian fluid-

based slurry. Therefore, it is important to understand the

materials engineering interactions that control shear thinning

so that process flow models can better predict the perfor-

mance of filled polymer systems. Highly filled polymers can

have processing issues, including high screw shaft torque, en-

ergy consumption, die pressure and melt temperature rise.

Previous theoretical developments and experimental evalua-

tions of highly filled polymer melts showed that the rheology

can be effectively described with a percolation model. In this

work, capillary rheometer measurements using several low-

density polyethylene resins, calcium carbonate and titanium

dioxide fillers are reported using percolation theory concepts.

The theoretical treatment of the rheology as a particulate per-

colating system with power-law behavior is used to analyze

capillary rheometer data. The observed effects of resin mole-

cular weight, filler type and size on rheology are described.

Engineers that design and debottleneck polymer processes

need to utilize the polymer viscosity at the minimum process

shear rate to determine the smallest motor that will allow the

process to run; in addition, the shear thinning characteristics

of the polymer are used to indicate how much increased pro-

duction may be possible with a given motor size. Thus, some

examples of expected effects on melt processing are also pre-

sented.

1 Introduction

Over the years, it has been found that a fundamental under-standing of the parameters that affect the viscosity and shearthinning, power law, behavior of polymer melts and compo-sites is necessary in the design and optimization of shear rateor shear stress dominated polymer processes. Recently, a theo-ry was developed that demonstrated that the value of the powerlaw exponent, n, was a measure of the fraction of the flowingNewtonian fluid-based composite that dissipates energy(Campbell et al., 2016a, b). The theory was based on the perco-lation concept that a structure is formed at a critical volumetricconcentration of the second phase and that there is cluster for-mation. A model based on macroscopic fluid mechanics wasdeveloped that had no adjustable parameters. This model fitavailable data. The viscosity characteristics, particularly shearthinning and the power law n of a polymer melt, can stronglyaffect process efficiency. In the late 1970’s and early 1980’s,UNIPOL polymers were introduced by Union Carbide, Piscat-away, NJ, USA as an alternative to conventional high pressurePolyethylene (HPPE). The new polymer transitioned to shearthinning at a higher shear rate than HDPE. The plastics indus-try found that productivity was diminished because extrudersdid not have enough power or torque to get to the same screwrotation rate and throughput as with HDPE polymers with thesame melt index. In order to utilize the new polymers to takeadvantage of its properties, industry spent substantial capitalon newly designed screws. This is a classic example of whatis found routinely in industrial production; anything that influ-ences the viscosity – shear rate relationship of a neat resin orparticle-filled polymer will affect the production rate. Two ma-jor areas of concern regarding the effects of polymer rheologyon processing are: 1) the polymer viscosity at low shear ratesand 2) the shear thinning characteristics of the polymer. Thus,the shear rate response and power law characteristics of New-tonian fluid-based slurries and filled and unfilled polymers areof practical importance to industry today.This investigation was the outgrowth of previous observa-

tions and experiments. When investigating and evaluating a\capillary rheometer" at General Motors Research (GMR),Warren, USA, in the 1970’s, researchers installed a capillarydie on the end of the injection unit of a molding machine. The

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Intern. Polymer Processing XXXIII (2018) 5 � Carl Hanser Verlag, Munich 619

* Mail address: Gregory A. Campbell, Castle Associates, Jonesport,Maine 04649, USAE-mail:[email protected]

G. A. Campbell, M. D. Wetzel: Effect of Filler Concentration on the Flow of Filled PE Melts

GMR investigation was focused on what material variableswould affect the potential decrease in fill time for a mold or in-crease in throughput through an extrusion die. Increased diethroughput and decreased mold fill time can result in a produc-tion increase and a higher financial return. Two resin responseswere typical. When using a Polystyrene, PS, with power lawconstant, n, of about 0.6, the maximum shear rate that couldbe achieved for the PS was about 50000 s–1; this limit in injec-tion rate would constrain any increase in potential productionrate. The inability to increase the shear rate was thought to bethe onset of the second Newtonian plateau. For a second poly-mer, a Polypropylene, with n of about 0.06, the apparent sec-ond Newtonian plateau was reached at about 200000 s–1, thisresult could lead to a much higher potential to increase produc-tion rate. These results suggest that the power law of a fluidcould be used to predict the shear rate where the fluid reachesthe second Newtonian plateau, the limit for potential produc-tion rate increase. At the time, there were no data available thatcould be used to test this hypothesis.From these observations and analyses, it follows that both

the low shear rate viscosity and the power law are importantwhen evaluating a resin regarding is potential utility in produc-tion. For the materials presented in this paper, the effect of aparticulate filler on the low shear rate viscosity is discussed inanother paper recently accepted for publication. When operat-ing a process in the power law region, the shear stress increasesmore slowly than the shear rate, reducing the process energydissipation per shear rate increment more than when in theNewtonian regime. When the shear rate increases to the secondNewtonian plateau of the fluid, the shear stress increases line-arly again with the shear rate. Increasing the shear rate nowleads to a proportional increase in the shear stress and conse-quently a proportional increase in process power consumption.These observations and hypotheses led to the issues investi-gated and presented in this paper.The primary issue to be investigated here is to compare the

change in the power law for three polyethylene polymers, PE,compounded with several particulate fillers including titaniumdioxide, TiO2 (Chemours, Wilmington, DE, USA) and withtwo sizes of calcium carbonate, CaCO3 (Omya Inc., Cincinnati,OH, USA), and to compare the results with the macroscopicchanges in power law for low viscosity carrier fluids filledcomposites previously reported. The objectives of the workpresented in this paper and issues to be examined are:1. As a foundation for understanding the current research into

CaCO3 filled PE, the filler concentration effects on thetransition into the power law region will be reviewed forlow viscosity Newtonian carrier fluid composites. Thesecomposites relate to polymers such as epoxies or poly-urethanes. A previously reported PE polymer filled withTiO2 and observations relative to how polystyrene meltscorrelated with percolation theory predictions will also bereviewed.

2. This paper will focus on developing an understanding ofhow PE molecular weight as indicated by melt index andCaCO3 size affect the shear thinning, power law of thepolymer-particle composite.

3. The response of the power law to the addition of filler willbe compared for the polyethylene systems and low viscos-ity carrier fluid systems.

4. Since most processing systems do not have enough powerto increase the shear rate after the initiation of the secondNewtonian plateau, this paper will review a theory that re-lates the power law to the transition into the second Newto-nian plateau. A functional relationship between the powerlaw and the ratio of the second to first Newtonian plateauwill be explored. This concept will be developed so that ifone knows the power law then the shear rate at the begin-ning of second plateau can be determined.

5. The importance of using filler volume fraction instead offiller weight fraction will be discussed relative to the in-crease in viscosity and the potential reduction in the powerlaw constant. This issue is often the cause of problems indeveloping alternative production systems.

When isotropic, or \spherical", fillers are added to fluids, con-ventional fluid mechanics-based models generally do not workwell after the filler concentration exceeds about 16 vol.%. Atthis point, the filler in the system tends to have a substantial ef-fect on the composite rheology by exponentially increasing theNewtonian viscosity, producing micro-structure in the com-posite, and decreasing the power law, n (Campbell et al.,2016a, b). For all polymer systems, the dissipated energy in-crease with the shear rate is related to an increase in shearstress. For a Newtonian response, the shear stress in linearlyproportional to the shear rate and the slope is the Newtonianviscosity. This correlates with a linear increase in process en-ergy dissipation with the flow rate. With power law fluids, theshear stress increases at a slower rate with an increase in shearrate than in the Newtonian regions. With a perfect pseudoplas-tic fluid, the power law index, n, is zero, so there is no increasein energy dissipation as the flow rate is increased; a violation ofthe second law of thermodynamics. Essentially all high mole-cular weight polymers fall between these two energy extremesat practical processing conditions, the power law constant, n,is less than 1.0 and greater than 0.0. The research reported herewas undertaken to evaluate the potential application of percola-tion concepts to help understand the effect of the power lawand viscosity change on the processing of concentrated disper-sions and slurries, both with low viscosity carriers often used inprocessing materials such as epoxies and polyurethane andhigh viscosity carriers such as polyethylene.The rheology of highly filled fluids and polymer melts has

been studied for many years and is reviewed extensively in pre-vious refereed publications (Campbell et al., 2016a, b; Camp-bell and Wetzel, 2016). Two original theories are often cited(Einstein, 1906; Batchelor, 1977). Others were developed bothbefore and after these two founding theoretical models, includ-ing a representative group (Maron and Pierce, 1956; Kriegerand Dougherty, 1959; Kamal and Mutel, 1985; Khan andPrud’homme, 1987; Tsenoglou, 1989; Kalyon, Yaras et al.,1993; Dealy and Wissbrun, 1995; Gupta, 2000; Kalyon, 2005;Kalyon et al., 2005; Kalyon and Aktas, 2014). These modelswere developed to predict composite viscosity as a function ofparticulate solid concentration. Many use particle maximumpacking concentration, um, as a critical parameter. Most ofthese models are empirical or quasi-empirical relationships,such as the Krieger-Dougherty and Maron-Pierce equations.Several researchers developed viscosity models based on parti-cle percolation using an approach analogous to electrical con-ductivity percolation theory and particle network formation

620 Intern. Polymer Processing XXXIII (2018) 5

(Zohrevand et al., 2014). Others utilized a percolation thresh-old, uc, and um (Zupancic et al., 1998). Campbell and Forgacs(1990) used percolation theory to model the Newtonian viscos-ity of concentrated suspensions. Other researchers investigatedthe effects of particles on viscosity where uc ranged from 16 to52 vol.%, depending on the fluid characteristics, particle typeand size (Wetzel et al., 2016; Campbell et al., 2016a; Campbellet al., 2016b). Another approach, particle element analysis, canlead to a robust solution of solids dominated flow (Moysey andThompson, 2004). However, the size of the computer and thecomputational time for a single solution makes the techniquea challenge for industrial utility. The literature models, eitherderived from conventional fluid mechanics or empiricallybased, often do not adequately represent the combined effectsof the particulate solids on the melt viscosity and the shearthinning of highly filled systems. As pointed out previously, apaper has been recently accepted that addresses the viscosityand related processing issues for the polymeric materials re-ported here. This paper focuses on the power law and the useof percolation theory to determine if the relationship betweenthe polymer, filler content and the power law can be better un-derstood and applied to practical industrial applications.

Percolation theory uses a statistical approach that predictsstructure formation in particulate slurries at much lower concen-trations than one would normally expect. Percolation effects areusually relegated to the onset of the existence of the percolating\infinite cluster." The particles are considered to be randomlydistributed throughout the slurry. Often cited to demonstratethe development of the infinite cluster is the conduction of elec-tricity through a particulate filled beaker. As a thought experi-ment, this has actually been carried out experimentally. Consid-er a beaker half filled with glass beads and with electrodes onopposite sides of the beaker. The electrodes are connected to acircuit with a battery and light bulb in series. The light will notlight with only the glass beads in the beaker since glass is an ex-cellent resistor and the glass filled beaker acts as an off switch.Now consider that gold beads are added slowly to the beaker,and the beads are mixed so that the gold beads are randomly dis-tributed. Gold is used because it does not corrode and thus hasexcellent contact conduction. As the beads are added, say in2% increments, the light will illuminate until the concentrationof gold beads is typically between 15 to 18% by volume. Ex-perimentally, because of the randomness of the distributed goldparticles there is not a single concentration for the conductionto occur. At the point where the light turns on, the beaker ofbeads is now an on switch. There is a path with some non-con-tinuous branches of the gold beads from one electrode to theother and the beads in the continuous path are considered to bethe percolation infinite cluster. Once the system has \perco-lated", it is often necessary to use this fact and other conceptsto quantitatively evaluate and predict many phenomena asso-ciated with \random" structure formation. This approach wasused in the investigation presented here.The general relationship for the viscosity of a Newtonian

fluid or polymeric based slurry as a function of shear rate is il-lustrated in Fig. 1. At very low shear rates the viscosity acts asa Newtonian fluid at the zero-shear viscosity, g0. As the shearrate is increased, the apparent viscosity decreases in the powerlaw region. At very high shear rates, a second Newtonian pla-teau is reached, the \infinite" shear viscosity, g?. For simple,

one-velocity component flow, the viscosity-power law rela-tionship is:

g _cð Þ ¼ k _cj j n�1ð Þ; ð1Þwhere g is the viscosity, _c is the one component shear rate (theabsolute value brackets indicate g is positive definite), n is thepower law constant, and k is the consistency index. The consis-tency is a necessary constant to anchor the power law line onthe viscosity axis through the power law region. However, ithas no theoretical basis for the evaluation of the effects of visc-osity in process simulations and analysis. When using powerlaw models in simulations, most researchers calculate the visc-osity for both the Newtonian and power law functions of thepolymer at a given process shear rate and use the lower valueas the proper system viscosity at that shear rate. This conceptis apparent if the power law function is extrapolated back tothe axis and the Newtonian viscosity is extrapolated to thehighest shear rate of interest per Fig. 1. If n = 1, then k is theshear rate independent viscosity and the function representsthe Newtonian viscosity.

2 Power Law Theory Development

2.1 Power-Law Index, n

Suspensions at moderate concentrations exhibit Newtonian be-havior at low shear rates and then shear-thinning behavior athigher shear rates (Campbell et al., 2016a, b; Campbell andZak, 2001; Zak, 1994; Radhakrishnan, 2002; Altobelli et al.,1991; Karnis et al., 1966a; Karnis et al., 1966b). As the shearrate increases within the flow field, it is proposed that large frac-tions of a cluster must cooperatively rearrange (Campbell et al.,2016a, b). At some point, the stress required for the particles ina cluster to locate and move into the existing holes as a group islarger than the locally applied stress on the cluster. When thisoccurs, that part of the cluster, where the particles cannot moveinto holes, flows without rearranging. Eventually, the wholecluster cannot deform. In the flow regions where there is no rear-


Intern. Polymer Processing XXXIII (2018) 5 621

Fig. 1. Polymer viscosity vs. shear rate response


rangement of clusters, individual particle surface area in theclusters is excluded from the dissipation rate in the slurry. Thefluid cannot perform work on those particulate surfaces in thenon-deforming cluster. This leads to an analysis of the thermo-dynamics of the viscous energy dissipation (VED) in suspen-sions (Campbell et al., 2016a, b).

No relative motion within a cluster suggests that there is noVED within that cluster. Thus, all of the energy dissipation musttake place in the regions surrounding the non-dissipative clus-ters. These regions can consist of pure suspending fluid, individ-ual particles in the suspending fluid; and smaller sized particledeforming clusters are the only surfaces upon which the fluidcan act. The shear stress of the slurry is thus related to the sur-faces that are available for viscous work: the container surface,the individual particles, the surface of the particles within thedeformable clusters, and the outer surfaces of the macroscopicnon-deforming clusters. It is the decrease of the surface areaupon which the fluid can perform work that is responsible forthe apparent decrease in the viscosity and thus the power law re-gion of flow (Campbell et al., 2016a, b).The hypothesis is that the power law, n, is the ratio of the

cluster filled slurry dissipation to the dissipation that would oc-cur if the flow characteristics were Newtonian under the sameflow conditions. A problem in general for evaluation of the ef-fect of cluster volume on the dissipation is how to determinethe fraction of non-deforming clusters in the shear flow forthe composite fluids, often characterized with a bob and cubor a cone and plate rheometer or similar rheometers. A techni-que was developed using tube flow to isolate the non-deform-ing cluster size (Campbell et al., 2016a, b; Zak, 1994; Radhak-rishnan, 2002) and a theoretical expression for the power lawexponent n based on the ratio of the dissipation in the compos-ite to that in the equivalent Newtonian fluid was developed fortube flow:

n ¼ 1� 4

3eþ 1

3e4: ð2Þ

Now define the volume fraction of the fluid in the infinite clus-ter(s) as

t ¼ e2; ð2aÞso that Eq. 2 becomes

n ¼ 1� 4

3

ffiffiffi

tp

þ 1

3t2; ð2bÞ

with

e ¼ Rc

R¼ sy

srzðr ¼ RÞ ; ð3Þ

where R is the tube radius and Rc is the radius of the non-de-forming, non-dissipating core region. It follows that Eq. 2 canbe cast in terms of this volume fraction with Eq. 2b. The upperand lower limits of n with Eq. 2 were found to be mathemati-cally correct. When there is no plug present, e = 0. Newtonianflow is predicted when n = 1. As the core size approachesthe size of the tube, e?1, the volume for dissipation ap-proaches zero and n approaches zero as well. The validity ofEq. 2 for n is substantiated by the fact that e = 1 is the only realroot. Historically, the most common and widely used non-Newtonian model for pseudoplastic fluids is the Ostwald and

de Waele’s Edisonian power law (1929). Since the late1920’s, this has been considered to represent an apparent or ef-fective viscosity as a function of shear rate for many types ofshear thinning systems. In contrast, what is reviewed here is afundamental engineering-based theory incorporating the ther-modynamics of shear thinning for at least Newtonian fluid-based slurries (Campbell et al., 2016a, b; Zak, 1994; Radhak-rishnan, 2002).

3 Experimental Procedures

Previously published investigations are reviewed here to pro-vide a foundation for the new results presented on CaCO3 filledPolyethylene melts. The initial work used an Instron capillaryrheometer to evaluate the rheological characteristics of CaCO3

slurries suspended in polyol to determine the effect of fillerconcentration on the power law (Campbell et al., 2016a, b).The Rabinowitsch correction (Rabinowitsch, 1929) was usedto correct the apparent shear rate. The additional pressure dropdue to the rearrangement of the fluid velocity profile at theends of the capillary can be quantified with a Bagley correction(Bagley, 1957). In the Bagley correction, capillaries with sev-eral different lengths, but of equivalent diameter, were used.A finite element scheme to investigate entry and exit lossesfor a power-law fluid flowing into a capillary was reported inthe literature (Boger et al., 1978). They found that the entranceregion that causes the additional pressure drop was less than 3capillary diameters. In the CaCO3 rheology part of this experi-mental project, most of the data were taken with a capillaryhaving an inside diameter of 0.78 mm and a length of101.79 mm. This gave an Lc/Dc ratio equal to 133.1. Anentrance region length of three diameters was 2.294 mm. or2.253% of the capillary length. So, the Lc/Dc ratio was suffi-ciently large to make the end effects negligible. Lastly, the datawere analyzed for the presence of wall slip. The wall slip ana-lysis of Mooney (1931) and Jastrzebski (1967) were used alongwith the observation reported by Yilmazer and Kalyon (1989),that with ammonium sulfate particles suspended in a Newto-nian poly(butadiene acrylonitrile acrylic acid) terpolymer, thecompound exhibited a dependence on capillary radius, whichindicated wall slip. The CaCO3 data reported here showed nodependence on capillary diameter, and thus no measurable slip.It is well known that a shear gradient can cause a concentrationchange in a flowing slurry system.The effect of the diameter of the flow tube, capillary, relative

to the particle diameter was investigated by Seshadri and Su-tera (1968; 1970). Their data showed that for Dc/a > 60, withDc = capillary diameter and a = particle diameter, the reduc-tion in concentration in the flowing slurry or dispersion wasnegligible for Newtonian fluid-based slurries (Seshadri and Su-tera, 1970).The rheometer was operated at pre-selected shear rates

(Campbell et al., 2016a, b; Campbell and Zak, 2001; Zak,1994). Four grades of CaCO3 listed in Table 1 were used in aNewtonian carrier fluid.Model fluid experiments using the four grades of CaCO3 re-

sulted in the percolation and maximum concentrations listed inTable 1. The percolation threshold concentration changed withparticle size. The largest two particle sizes showed essentially


the same percolation threshold of 17 vol.%. This value waswithin the theoretical range of 16 to 18%. For the 3.0 lmCamel-WITE particles, and then again for the 0.7 lm CAL-ST particles, uC shifted to a higher concentration with a de-crease in particle size. The size of the Omyacarb 10 and 21 par-ticles was such that hydro-dynamic interactions dominated col-loidal forces. However, the Camel-WITE and CAL-STparticles were small enough to appear to be influenced by col-loidal forces. Brownian motion tends to randomize the locationof particles in a suspension of small particles; in this case it wasproposed that this was the greatest colloidal interaction sincethe other colloidal forces had been minimized. The stearic acidtreatment of the CAL-ST particles probably played an addi-tional role in delaying the start of shear-thinning behavior. Thisincrease in threshold concentration as the particle size went be-low one micron could also be modeled by directed percolation.In directed percolation, the lattice is sensitive to particle prop-erties and the probability of connectedness is adjusted. Also,the Dc/a calculations indicate that the 21 lm particles mayhave a tendency to migrate while the other three particles haveDc/a values well above the 60 threshold reported by Seshadriand Sutera (1970).

A typical result for the power law function of CaCO3 slurryin a polyol is found as function of concentration, using a con-centration ratio, (u/um)

–2/3, plotted in Fig. 2, the blue diamonds(Zak, 1994). The intersection of the Newtonian curve at low

particle concentrations with the power law best fit is interpretedas the percolation concentration, uc, and the x-axis interceptleads to the evaluation of the maximum concentration. The dataindicated that there was a linear relationship between n and(u/um)

–2/3. Some deviation from the line is observed near thepercolation concentration. The n and (u/um)

–2/3 relationshipsmay imply that the power law is a function of some surface areameasure related to the particle size and shape distributions inthe dispersion state. These results led to work on the develop-ment of a theory that would relate the power law to the concen-tration of the filler in the slurry as reviewed above.The equation used to represent the power-law region after

the critical percolation concentration in Fig. 2 was developedby Zak (1994) and is written as:

n ¼uum

� ��2=3� 1

ucum

� ��2=3� 1

: ð4Þ

3.1 Application of the Power Law Theory

to Newtonian Fluid-Based Slurries

PMMA beads were used to evaluate the utility of the percola-tion-based theory described above. These power law data setswere analyzed using a similar method applied to the calciumcarbonate data per Fig. 2 (Zak, 1994; Radhakrishnan, 2002),and showed the same power law concentration relationshipwhen plotted in the same manner as the CaCO3 in Fig. 2. Asthe concentration increases, the value of n decreases (Campbellet al., 2016a, b).The PMMA data in Fig. 2 were then compared to the ther-

modynamic prediction of the power law theory from Eq. 2and are plotted in Fig. 3, showing that the data are in excellentagreement with the percolation-based theory. The experimen-tal value of n was determined using classical tube flow pressuredrop and flow rate analysis (Campbell et al., 2016a, b). Theplug size of non-deforming clusters was determined experi-



Fig. 2. Power law constants from three data sets (Zak, 1994; (JSR)Radhakrishnan, 2002)

Fig. 3. Power law data compared to cluster formation theory perEq. 2; n from pressure drop in tube from Zak, 1994 and Radhakrish-nan, 2002

CaCO3

Particle typeNominal avg.particle size

lm

uc um Dc/a

Omyacarb 21 21,0 0,17 0,59 36Omyacarb 10 10,0 0,17 0,60 76Camel-WITE 3,0 0,20 0,53 254CAL-ST 0,7 0,23 0,67 1092

Table 1. Calcium carbonate characteristics in a mode polyol fluidslurry experiment


mentally by visually tracking tracer particles in the flowingfluid (Zak, 1994; Radhakrishnan, 2002). Three sets of literaturedata (Altobelli et al., 1991; Karnis et al., 1966a; Karnis et al.,1966b) are included in the graph with n calculated from theflow velocities and structure reported by the investigators(Zak, 1994). It was found that all five data sets fit the theoryas shown in Fig. 3 (Zak, 1994; Radhakrishnan, 2002; Altobelliet al., 1991; Karnis et al., 1966a; Karnis et al., 1966b).

4 Application of Theory to Filled Polymer Melts

The experimental and model prediction results for filled powerlaw Newtonian fluid-based composites led to an investigationto determine if the proposed dissipation mechanism could be

used to interpret the development of the power law for polyethy-lene composites. The initial polymer-based evaluation of a filledpolymer melt system was previously reported and it focused on aTiO2 filled Low Density PE, LDPE (Campbell and Wetzel,2016; Wetzel, et al., 2016). The investigation was expandedother PE materials. The polymeric materials used in this andthe previous investigation are listed in Table 2. Four PE resins,one low density PE, LDPE, and three linear low density poly-ethylene, LLDPE, and three fillers, one pigmentary TiO2 andtwo CaCO3 particulate solids, were used in the polymer meltprocessing experiments. These materials had a wide molecularweight range as indicated by the melt index and confirmed forthree of the resins using GPC as listed in Table 3.

Equistar LDPE Petrothene NA-206 (Lyondell ChemicalCompany, Houston, Texas, USA) was characterized regarding


Apparent molecular weight averages from GPC

PS equivalent MW averageskg/mol

Polymer Mn Mw Mz Mw/Mn Peak area/conc

DNDA-1082 14,40 84,80 335,0 5,89 7298GRSN-9820 NT-7 19,60 346,00 2348,0 17,65 6609GRSN-2070 NT 28,10 152,00 487,0 5,41 8112

PE equivalent MW averages (kg/mol), converted from PS equivalentMW, M(PE) = A*M(PS), and A*0.43

Polymer Mn Mw Mz Mw/Mn MI

DNDA-1082 6,19 36,46 144,1 5,89 160,0GRSN-9820 NT-7 8,43 148,78 1009,6 17,65 20,0GRSN-2070 NT 12,08 65,36 209,4 5,41 0,7

Caution with the reported data: GPC condition may not be optimal for these samples. However, relative comparison should be still valid.

Table 3. Molecular characteristics of three of the LLDPE polymers

Polymer qg/cm3

MIg/10 min

(190 8C, 2.16 kg)

TMP

8CPower law n(> 100 s–1)

Equistar LDPE Petrothene NA-206 0,918 13,5 *110 0,341Dow LLDPE DNDA-1082 0,933 160,0 125 0,700

Dow LLDPE GRSN-9820 NT-7 0,924 20,0 123 0,598Dow LLDPE GRSN-2070 NT 0,920 0,7 121 0,393

Filler qg/m3

d50lm

Surf. Mod. Dc/a

Chemours Ti-Pure R-104 TiO2 3,900 0.20 to 0.25 Yes 4000,0OMYACARB 40-UL CaCO3 2,700 45,00 No 22,2OMYACARB 5-FL CaCO3 2,700 5,00 Yes 200,0

Table 2. Polyethylene polymers and fillers used

its viscosity as a function of shear rate at three temperatures. TheCross-WLF constitutive equations were used to fit the data andare plotted in Fig. 4. Using the last 8 data points, shear ratesfrom 223 s–1 to 4999 s–1, the power law index, n, was estimatedto be 0.361 for this resin. The NA –206 base resin was then filledwith TiO2 with solid fraction from 2.65 to 49.5 vol.%. Exam-ining Fig. 5, the shear thinning response to increased filler con-centration was dramatically different for these LLDPE compo-sites than for the filled Newtonian fluids. The power lawrapidly decreased as filler was added above the percolation con-centration in Fig. 3, but changed very slowly when filler wasadded to the LLDPE per Fig. 5. When the filler was added to aNewtonian carrier, Fig. 6A, the power law decreased rapidly ina sigmoid shaped function. The power law was found to be al-most constant when the raw data was examined for the TiO2 inLDPE as shown in Fig. 6B. A possible explanation for this dif-ference in power law response to filler concentration was devel-oped previously. It is well known that at or below the critical

molecular weight, Mc, PE has essentially no shear thinning. Atheoretical proposal was previously published for TiO2 filledPE (Campbell and Wetzel, 2016) that may help to explain theunexpected observation of the effect of filler on the shear thin-ning characteristics of high molecular weight polymers.

Under shear flow, polymer chains are thought to disentanglefrom one another (Campbell and Wetzel, 2016). It was proposedthat the low molecular fraction of the polymer, that fraction withmolecular weight belowMc,might be considered to be a theta sol-vent at all temperatures for the higher molecular weight polymermolecules. As the molecular weight increases, the fraction of thissolvent decreases such that the large molecules, *10 nm, be-come insoluble or swollen since they have collapsed into a\Gaussian sphere" per the literature. The large molecules are pro-posed here to act in the molten polymer as deformable \fillers"with the critical percolation concentration reached at or near Mc.When the molten polymer is sheared, the large molecules

decrease the volumetric dissipation in a similar manner that



A)

B)

Fig. 6. Power law as function of filler concentration for Newtoniancarrier and LDPE, A) CaCO3 effect on power law, B) power law vs. fil-ler concentration for filled PE

Fig. 4. Melt viscosity data and cross and WLF models vs. shear ratefor LDPE from 125 8C (389 K) to 145 8C

Fig. 5. Effect of CaCO3 filler on power law n


was observed for the Newtonian fluid slurries. That is, the largemolecules interact as infinite clusters. It is further suggestedthat as a particulate solid is added to the melt, it displaces someof the deformable filler, and because in the TiO2 case, of itsmolecular size, the polymeric \filler" can fill the gaps betweenthe solid filler or filler clusters causing the different responsefrom the Newtonian carrier composites shown in Fig. 5. Thus,the power law does not decrease as rapidly as it does when sol-id filler is added to a Newtonian fluid per Fig. 2. Returning tothe previous investigation of TiO2, in the PE polymer case,the dissipation of the base polymer and of the slurry at low fil-ler concentrations are thought to be dominated by the proposedcluster formed by the large polymer molecules in the base resinover this filler concentration range.

The relatively constant power law for this TiO2 filled PE fromthe lowest concentration of 0.0265 volume fraction to the high-est concentration of 0.495, suggests that the increase in the resis-tance to cluster deformation resulting from the increase in solidparticle concentration is in some manner offset by what occursin the resin plotted in Fig. 6B in the previously reported investi-

gation, the \polymer molecular filler" concentration of the basePE resin was estimated using the function n for PMMA spheresfrom Fig. 7 and the experimental power law for the base resin.For the LDPE polymer system shown in Fig. 9, the dissipa-

tion of the base polymer and of the TiO2 slurry at low fillerconcentrations is thought to be dominated by the proposedcluster formed by the large polymer molecules in the base re-sin. The \polymer molecular filler" concentration of the basePE resin was estimated with the function from Fig. 7. The rela-tionship between the power law and the filler concentration forthe PMMA experiments (Campbell, Zak and Radhakrishnan,2016) and the LDPE polymer/TiO2 system was found to havean effective volume fraction of 0.375 of the \polymeric filler"leading to percolating clusters. Consistent with this assump-tion, as the TiO2 filler was added the components of the slurry,polymer fluid, polymer \filler" and TiO2 solid behave as seenin Fig. 7. The detailed discussion of this proposal can be foundelsewhere (Campbell and Wetzel, 2016). The concentration ofthe PE fluid decreases in volume as does the fraction of the\PE filler" as TiO2 is added to the system. Using the percola-tion hypothesis relating to the high molecular weight polymer,the overall filler (or total solids concentration) increases at aslower rate than the TiO2 concentration.Using the estimated \total solids concentration for PMMA"

data from Fig. 7 and the measured neat LDPE power-law expo-nent n, the TiO2 filled polymer values for n show a remarkablefit to the percolation power law model as a function of concen-tration, per Fig. 8 and Eq. 4 (Campbell and Wetzel, 2016). Thedata indicate that the percolation concentration may be higherthan for three-dimensional percolation as filler is added to thebase polymer matrix. The power law n does not decrease sub-stantially as the filler loading is increased to 49.5 vol.%. It hasbeen reported that for 2-D percolation, the percolation concen-tration can lie between 42 and 50 vol.% for \spherical" fillers.As TiO2 is added to the system, the percolating structure maytransition from 3-D to 2-D percolation. These intriguing resultsfor TiO2 in a PE resin led to further research being conducted


A)

B)

Fig. 7. Filler concentration as function of power law and volume frac-tions of the components of the PE filled polymer, A) PMMA bead vol-ume fraction vs. n, B) interaction of solid polymer filler

Fig. 8. LDPE/TiO2 blends power law n plotted with percolation theoryand Newtonian fluids

on three other PE polymeric fluids and two CaCO3 gradeslisted in Table 2 and Table 3.

Given the background information and concepts detailedabove, three new LLDPE resins compounded with two gradesof CaCO3 are used in the remainder of the paper. If these con-cepts are to be potentially applicable to other polymer systems,it is necessary to determine if the new proposal that correlatesthe power law n with a filler-like response can be applied toother systems. The hypothesis is examined that polymers aboveMc act as an effective \filler" in a way that correlates with sys-tems previously discussed per Fig. 3. Several investigators havedeveloped relationships between molecular properties of poly-mers and the power law (Adams, 1987; Campbell and Adams,1990; Nichetti and Manas-Zloczower, 1998; Wood-Adams andDealy, 1996). A set of Polystyrene, PS, power law and molecu-lar weight data was available (Adams, 1987; Campbell andAdams, 1990) in addition to the power law n values calculatedfor the four PE resins used in their work.Table 4 lists the power law data for four PS materials with

different molecular weights. These data and the PE data forthe 4 unfilled polymers are plotted in Fig. 9. The \polymericfiller concentration" for all eight polymers was determinedusing the fitting equation in Fig. 7A. The percolation concen-tration function, Eq. 4, was used in this case. All data for boththe PE and PS resins fall along the same curve. This is an unex-pectedly good correlation for the combined results for all eightpolymers with two completely different molecular structures.

The data from both polymer sets fall on the percolation pre-diction as a function of polymeric \filler concentration". Thisrelationship shown in Fig. 7 for PMMA spheres was not corre-lated or developed with percolation theory. The PE data wereobtained from the as received resins (Wetzel et al., 2017). It isinteresting that polymer melt power law n can be related tothe effects observed for shear thinning slurries based on filledNewtonian liquids. This unexpected observation suggests thatthe underlying processes that relate to power law behaviormay be independent of the materials being evaluated andshould be further investigated.

The melt viscosity of the LLDPE resins, the CaCO3 master-batches and batch mixer letdown samples were measured in ac-cordance with ASTM D 3835 using a capillary rheometer (mod-el Dynisco LCR 7001, Dynisco, Franklin, MA, USA), (Wetzelet al., 2017). The viscosity of the three LLDPE resins weretested at 170, 190 and 210 8C and at shear rates from at least250 down to 1 s–1 using a 1.0 mm diameter die with length to di-ameter ratio L/D = 30 and a 1808 inlet angle. From the discus-sion below, this would potentially produce a 10% maximum er-ror in the viscosity. However, there would be essentially aconsistent error for all or the shear rates. Thus, the error shouldnot substantially affect the power law slope calculations. Dupli-cate tests with fresh samples were run at all temperatures. Thetest-to-test repeatability, expressed in the terms of the repeat-ability coefficient of variation, was less than 3% at all shearrates. Since only one capillary diameter was used, the wall slipcould not be evaluated in a traditional manner per the analysisof Mooney (1931) and Jastrzebski (1967). It has been proposedthat the onset of wall slip may occur at wall shear stress valuesof 0.2 to 0.3 MPa (Hatzikiriakos and Dealy, 1992). The onsetof wall slip is also indicated by a discontinuity or substantialchange in the shear rate versus shear stress slope. This slopechange and or discontinuity were reported as an increase inslope when the shear rate is plotted as a function of shear stress(Denn, 2001). In general, the wall shear stress was less than0.3 MPa, for many of the samples listed in Table 3. No shearstress-shear rate deviation from a smooth curve was noted inany of these samples. Also, there were no apparent surface de-fects in the rheometer extrudates. This does not preclude the ex-istence of some minor slip at the higher shear rates.

Another issue is the migration of fillers in shear flow. A sub-stantial body of work has been published recently and a represen-tative sampling dealing with particle migration are for theoretical(Hong et al., 2011), for experimental observation (Elias et al.,2008), and for reviews of recent work on suspensions (Normanand Wagner, 2009; Isayev and Palsule, 2016). One of the authorsco-published an experimental paper examining the migration ofrigid and rubbery fillers in laminar flow in injection molded parts(Dontula et al., 1996). Their data for filled polymer systems wereconsistent with the findings of Seshadri and Sutera (1970) wherethe largest particles tended to migrate. Since the goal of the pres-ent work was to determine what macroscopic effects the fillerhad on the power law behavior of polyethylene composites, andif these changes were similar to the observations from previouswork on Newtonian carrier composites, no investigation of themicroscopic migration was undertaken on the extrudates. At thetime of this work, resources did not allow the additional effortto examine the micromechanics. However, future investigationsshould probably evaluate migration given the results reported la-



Fig. 9. Percolation correlation for the power law for both PS and PEpolymer melts

n Concentration(vol. fraction)

Molecularweight

0,59 0,26 48.0000,39 0,33 117.0000,23 0,39 179.0000,19 0,40 217.000

Table 4. Power law estimates for a series of Polystyrenes (Campbelland Adams, 1990), and estimated concentration of polymer filler


ter in this paper. Dc/a calculations for the polymer composites in-dicate that the*40 lm particles may have a tendency to migratewhen it is the dominant particle while the other two particleshave Dc/a well above the 60 threshold value (Seshadri and Su-tera, 1970) per Table 2. It would be useful in a future investiga-tion to section and polish rheometer extrudates and determine ifit is particle migration (Dontula et al., 1996). Also, available re-sources did not allow for additional measurements using coneand plate or parallel plate rheometers.It was previously proposed that at low shear rates, particles

in a cluster have time to rearrange, allowing the cluster to de-form as a result of the imposed flow field (Campbell and For-gacs, 1990). The particles rearrange themselves by moving intoexisting holes within a cluster or past neighbors. This allowsfor stress to be transferred throughout the suspension and forthe fluid to perform work on all of the particle surfaces as wellas the container surface, thus yielding a Newtonian viscosity.Percolation theory was used to develop the following equa-

tion (Campbell and Forgacs, 1990) to predict the effect of fillerconcentration on the slurry Newtonian viscosity:

gr ¼ eum�ucum�u

� �

� 1; ð5Þwhere gr is the Newtonian viscosity ratio, filled vs. fluid, um isthe maximum packing concentration, uc is the percolation con-centration, and u is the particulate solid concentration. Equa-tion 5 has been previously shown to fit data for slurries usedto determine flow dynamics of slurries in tube flow (Campbellet al. 2016).

The shear stress, viscosity, and estimates of the yield stressfor the highest filler concentrations in two of the polymers usedin this current investigation may be found in Table 5 at twoshear rates. A typical rheological response, as represented bythe viscosity ratio, gcompound/gfluid, for the three PE resins andtwo fillers is plotted in Fig. 10. The percolation model of Eq. 5was used to fit for the two CaCO3 grades and a 70%*45 lm/30%*5 lm blend pre-compounded in LLDPE as a master-batch and then let down with the LLDPE in a batch mixer (Wet-zel et al., 2017). MB-7 was the response of the 0.7 MI resin andit showed a relatively small increase in viscosity as the fillerwas added and a marginal fit as indicated by the sum of squares

error, SEE1/2. A preliminary analysis of the viscosity data usedin this power law and the zero to infinite shear viscosity analysiscan be found elsewhere (Campbell et al., 2017; Wetzel et al.,2017). This paper extends the previous analysis and is focusedon the effect of fillers on the power law and the onset of the sec-ond Newtonian plateau for filled PE systems, both of whichhave ramifications for the processing of filled systems.For the compounded PE/filler dispersions, it is important to

evaluate the effect of the compounding, residence time andshear in the twin screw extruder on the estimated power lawconstant of the evaluated resins. It was found that in all cases,n increased as a result of the compounding experience. This isconsistent with an expectation of reduced molecular weight ofthe PE as a result of the time, melt temperature and stress his-tories. Because of time and resource constraints in an industrialenvironment the changes in molecular weights were not evalu-ated. The CaCO3 dispersion data were analyzed in the same


MB Sample MB1LD1

MB2LD7

MB3LD7

MB2LD7

MB3LD7

MB4LD1

MB5LD7

MB4LD7

MB6LD7

MB5LD7

MB4LD7

MB6LD7

LLDPE 1082 (160 MI) 9820 (20 MI)

CaCO3 – 5-FL Blend 5-FL Blend – 5-FL 40-UL Blend 5-FL 40-UL Blend

Vol. % 0% 60% 60% 60% 60% 0% 60% 60% 60% 60% 60% 60%cA (s–1) 75 75 75 20 20 75 75 75 75 20 20 20gA (Pa s) 66,2 1559,8 1597,3 3100,6 3266,7 389,1 1376,2 3150,2 3606,7 2476,8 5599,9 7210,2s (Pa) 4965 116985 119798 62043 65367 29183 103215 236265 270503 49561 112054 144276n’ 0,776 0,540 0,503 0,408 0,379 0,656 0,560 0,561 0,482 0,555 0,553 0,452

cW (s–1) 80,4 90,9 93,5 27,3 28,2 84,8 89,7 89,7 95,2 24,0 24,1 26,1s0 extrap. (Pa) 0,140 77,8 133,7 430,1 651,1 4,10 53,0 119,3 398,5 56,2 130,9 582,6s0 model (Pa) 0,005 0,110 0,110 0,110 0,110 0,036 0,214 0,214 0,214 0,214 0,214 0,214

Table 5. Rheological property estimates of several of the base polymers and particle masterbatches

Fig. 10. A relative viscosity plot of data and the percolation model ofEq. 5 with }c and }m as fitting parameters

manner as was developed and utilized in Fig. 7. As with TiO2

in LDPE, the effect of CaCO3 concentration on the change inn from 0 to 60 vol.% was not substantial as shown in Fig. 5.Overall, n decreased for all systems with increasing CaCO3

loading, and the maximum change in n was only about 0.2 unitsover the entire filler concentration range. This is much lessthan has been seen in Newtonian fluid-based dispersions perFig. 2 and 3.The effect of particle size, concentration and resin melt in-

dex of 20 were analyzed and are plotted in Fig. 11A and B.The effect of CaCO3 of two different sizes in 20 MI LLDPEon n is shown in Fig. 11A. The percolation point is consideredto be the onset of three-dimensional percolation, 16 vol.% fillerto the maximum concentration. This is the same value that cor-relates the melt data for both PS and PE per Fig. 9. The perco-lation point at 50% found in Fig. 11A can be considered asthe onset of two-dimensional percolation and is consistent withthe upper concentration limit proposed in the literature for 2-Dpercolation. The remaining data acted in a similar manner toFig. 11A with the data listed in Table 6. Previously in this pa-per, it was observed that all of the melt data for the four PE re-sins, MI 0.7, 13.5, 20, and 160, fall on the 3-D percolation lineper Fig. 9. Given that the resin \apparent filler" concentrationwas determined from a correlation of PMMA beads suspendedin a Newtonian fluid, the results suggest that one may wish tostart looking at a high molecular weight polymer as acting likefilled system with the polymer above the critical molecularweight acting as deformable filler. It is recommended thatmore detailed investigation of these observations should beconducted.The power law exponent for these systems initially fit a 3-D

percolation model. The resin falls on the 3-D percolation curve.All of the filler systems, single size 5 and 45 lm and a 70/30%(45/5 lm) mixture, show a similar reduction in power law as afunction of total concentration and show a strong tendency tomigrate toward 2-D percolation as the CaCO3 concentrationis increased. This was unexpected, particularly for the 70/



CaCO3 Typeand vol %

160 MI Polyethylene 20 MI Polyethylene 0.7 MI Polyethylene

Powerlaw n

Fillervol %

(u/um)–2/3 Fraction

ClusterPowerlaw n

Fillervol %


ClusterPowerlaw n

Fillervol %


Cluster

0 0.8430 0.1820 3.0720 0.0138 0.7350 0.2210 2.6991 0.0397 0.4190 0.3280 2.0740 0.198610% (45 lm) 0.8420 0.2638 2.3980 0.0139 0.7330 0.2989 2.2069 0.0403 0.3840 0.3952 1.8321 0.225330% (45 lm) 0.8790 0.4274 1.7388 0.0083 0.6730 0.4547 1.6685 0.0606 0.3640 0.5296 1.5073 0.241760% (45 lm) 0.5520 0.7311 1.2156 0.1152 0.4760 0.6884 1.2654 0.159510% (5 lm) 0.8720 0.2638 2.3986 0.0092 0.6790 0.2989 2.2069 0.0584 0.3560 0.3952 1.8320 0.248430% (5 lm) 0.6980 0.4274 1.7388 0.0516 0.7030 0.4547 1.6685 0.0499 0.3090 0.5296 1.5072 0.291060% (5 lm) 0.6080 0.6728 1.2849 0.0876 0.5420 0.6884 1.2654 0.120510% (70%45 lm)

0.8420 0.2638 2.3986 0.0140 0.6420 0.2989 2.2069 0.0728 0.3410 0.3952 1.8320 0.2618

30% (70%45 lm)

0.6220 0.4547 1.6685 0.0813 0.3410 0.5296 1.5072 0.2618

60% (70%45 lm)

0.5340 0.6728 1.2849 0.1251 0.4610 0.6884 1.2654 0.1693

Table 6. Rheological data for LLDPE and CaCO3 composites

A)

B)

Fig. 11. Effect of CaCO3 filler in Polyethylene on the composite powerlaw, A) CaCO3 fillers at 0, 30, 60% in MI 20: polymer filler concentra-tion from particulate n vs. }, B) n for filles PE vs. cluster volume frac-tion


30 mixed filler, as the 5 lm filler would normally have beenexpected to fill the voids between the larger 45 lm particlesand thus the total concentration would have a smaller effectthan would the single size particles alone as the total concentra-tion was increased. This may suggest that the particles organizein macroscopic clusters in the rheometer and slide by one an-other with a resin lubrication layer between the cluster layers.This observation should be investigated in future research. Thiseffect was not observed in the PMMA or CaCO3 slurries dis-cussed earlier where the suspending fluid was Newtonian. Atall concentrations, the power law followed the 3-D percolationfunction for the Newtonian fluid base with shear thinning slur-ries. It was also found that the bimodal filler concentrationsproduced results as expected for the PMMA filled slurries. Atthe same total concentration, the bimodal system had a lowerpower law (Radhakrishnan, 2002). This unexpected observa-tion for filled polymers may help explain some of the anoma-lies found when extruding highly filled resins, such as jettingin certain regions of an extrusion die.The change in power law as a function of cluster volume is

plotted in Fig. 11B and the data are listed in Table 6. The func-tion for all three resins and the two fillers follows the powerlaw function related to cluster volume per Eq. 3, just as theother Newtonian fluid-based composites did in Fig. 3. Eventhough the power law relationship appears to change from3-D to 2-D percolation as a function of filler content, Fig. 11A,the ratio of the dissipations for the slurry as represented by n,follows the same trend as the low viscosity slurries.

4.1 Implications for Extrusion and Molding Processes

The change in power law as a function of filler content or poly-mer molecular weight has a strong implication regarding theability to increase productivity by increasing the throughput inextrusion dies or injection molding flow passages. The end ofthe power law region is where the Newtonian infinite viscosityplateau begins. To understand the transition from power law tothe second Newtonian plateau, data from the 11 literaturesources were used as shown in Fig. 12 (Campbell and Zak,2017). A review of the development of the correlating Eq. 12is presented here to provide a basis for using the correlation.To correlate this data theoretically, the approach was to de-

velop a dissipation model based on a Bingham fluid viscosity(Campbell and Zak, 2017). The analysis started with the meth-od developed by Brenner (1958) to define the relative viscosityin Eq. 6. This is a continuation of approaching the analysis fo-cusing on dissipation as in the development of the power lawtheory per Eq. 2:

gr ¼

R

S0þP

Si

pniuidS

R

S0

p0niu0i dS

; ð6Þ

Here, pn is the outer normal traction on the surface, ui is the ve-locity, S is the surface area that is dissipating energy and Si isthe surface area of the particles in the slurry. The expressiondeveloped by Brenner (1958) for the relative zero-shear viscos-ity, Eq. 6, provided a way to write the relative infinite-shear

viscosity, gr,?. The suspending fluid can perform work on thecontainer surfaces, So, the surfaces of the non-dissipative clus-ters, ~Sc, and the surfaces of the single particles and thoseclusters that were able to deform,~Sd. Relative to the suspend-ing fluid, the infinite-shear viscosity is modeled as:

gr;1 ¼

R

S0þP

SiþP

Sd

pniuidS

R

S0

p0niu0i dS

: ð7Þ

Combining Eqs. 6 and 7, the infinite shear/zero shear ratio is:

gr;1gr;0

¼

R

S0þP

SiþP

Sd

pniuidS

R

S0þP

Si

p�niu�i dS

; ð8Þ

where? represents the fluid stresses in the infinite-shear case,and * represents the fluid stresses for the zero-shear flow case.Equation 8 has similar form as the definition for the power-law index, Eq. 2. gr,?/gr,o is the ratio of the limits of the energydissipation. The power-law index maps how the energy dissi-pation changes from the upper limit, expressed by gr,o, to thelower limit, expressed by gr,?. The key difference is that thestructure and dissipation in the suspension changes as a func-tion of shear rate as the shear rate progresses from the lowerzero-shear region to the upper infinite-shear region; the centralplug radius grows (Campbell et al., 2016), Radhakrishnan,(2002)). However, since the ratio gr,?/gr,o represents thosetwo regions of energy dissipation and the limiting power-lawindex represents the maximum amount of change betweenthese two extremes, a relationship between n and gr,?/gr,o isproposed. The energy ratio is defined as:

gr;1gr;0

¼

R

R

Rc

l DP0

2lBLr� Rcð Þ

h i2rdr

R

R

0

l DP02lBL

rð Þh i2

rdr

EV;B

EV;N

� �

; ð9Þ

for flow through a capillary, where EV,B/EV,N is an energy cor-rection to account for Bingham flow with subscripts V indicat-


Fig. 12. Effect of power law value on transition to second Newtonianplateau (Campbell and Zak, 2017)

ing the tube wall region, N for Newtonian and B for Bingham.The second assumption is that the shear stress is the same inboth cases. Therefore, the capillary geometries and the pres-sure drops are the same. The ratio of these two energy dissipa-tions can be expected to scale as:

EV;N

EV;B/ _cN

_cB

2

; ð10Þ

with

_cN_cB

¼ R

R� Rc: ð11Þ

After integration of the numerator and denominator in Eq. 9and canceling terms, the resulting function is combined withEqs. 10 and 11 to better energetically represent the Binghamflow that was observed in experiments, resulting in:

gr;1gr;0

¼ 1� eð Þ2 1� 8

3eþ 2e2 � 1

3e4

: ð12Þ

Using Eqs. 12 and 2 together, the theory line correlates wellwith all of the data representing the g?/go for slurries from 11literature sources, as shown in Fig. 12 (Campbell and Zak,2017). To our knowledge, this is the first theoretically basedanalysis that relates the power law to the ratio of the Newtonianviscosity to the second Newtonian plateau. This relationshipdemonstrates that if the power law and the Newtonian viscosityare known, the \infinite shear viscosity", g?, can be deter-mined with Eq. 16 when combined with Eq. 2 through the pa-rameter e.As illustrated in Fig. 13, processing through the power law

region requires little increase in equipment power because theshear stress increases much less than the shear rate. When thesecond Newtonian plateau is reached, the power needed to in-crease the velocity through a die or mold runners rises linearlywith flow rate since shear stress and shear rate are linearly re-lated. As described in the introduction, the onset of g? to theGMR researchers was indicated by the observation that settingthe injection speed control to a higher value resulted in essen-tially no increase in injection rate.From a process engineering perspective, several industrial

systems are now examined that can be affected by the increasein viscosity and decrease in power law. It is recognized thatmany polymer processes are dominated by infinite cluster for-mation above the percolation concentration as discussed in pre-vious sections. As the power law decreases toward zero, thetransition to the Infinite Shear Newtonian plateau occurs at alower apparent viscosity and at a higher shear rate, as shownqualitatively in Fig. 13. With both materials in Fig. 13 havingthe same g0, the \higher power law" resin reached the secondNewtonian plateau at about 20000 s–1. Beyond this shear rate,it would be very difficult to increase the production rate by in-creasing the flow rate in a die or mold because one would rap-idly exceed the equipment’s available power. The higher shearthinning resin, smaller n, did not reach the second Newtonianplateau until about 90000 s–1. Since the shear stress increasesmuch more slowly in the pseudo plastic viscosity region, theresin with the greater power law value cannot be processed ata high material velocity or shear rate, and thus would have apotential lower productivity than the one with the smaller n.

The data used in Fig. 13 were taken from the correlation inFig. 12.Highly filled polymers are important in a number of applica-

tions today and include heat conduction in electronic devices,sound deadening in automotive and other industries, and manymore industrial applications. In all of these applications, the fil-lers of interest are compounded into a polymer matrix, and thedispersion is cast, extruded or molded into the required shape.One real industrial problem that the authors are aware of is re-viewed here; the production of sound deadening material to beused in auto manufacture. Since sound deadening is a functionof the mass of the material between the origination of the soundwaves and the area to be dampened, it is natural to start with ahigh-density filler such as BaSO4 (q = 4.5 g/cm3). However,BaSO4 is costly, so there is an economic incentive to useCaCO3. At a major polymer materials supplier, the managerof the group charged with the development of this compositematerial instructed the group members to substitute CaCO3

for the BaSO4 in the formulation. Sound deadening is a linearfunction of the mass of the filler in the composite, so it takesmore CaCO3 by volume to achieve the same mass in the com-posite. As pointed out above, the low shear rate melt viscosityis an exponential function of volume of the filler in the com-posite per Eq. 5. Extrusion dies for this process are operated ata relatively low shear rate because of the high composite visc-osity. Figure 14 plots melt viscosity and the associated powerlaw, n, as a function of filler weight fraction. As the weightfraction approaches that required for effective sound deaden-ing, the viscosity of the CaCO3 system is about five times high-er than the BaSO4 system. Also, the power law approaches thevalue of 1.0 at a lower concentration indicating that the struc-ture in the composite is almost all cluster(s). In all likelihood,the compounding extruder would over-torque or over-pressurewith these highly-filled CaCO3 compounds. Furthermore, thecritical wall stress can be easily exceeded.The wall stress for TiO2 filled LDPE as a function of TiO2

concentration is plotted in Fig. 15 using the percolation con-vention, (u/um)

–2/3; a lower number on the x-axis infers higherfiller concentration. As the filler concentration is increased,



Fig. 13. Effect of power law on transition to second Newtonian pla-teau


both the wall stress and the plug stress at the center of the chan-nel increase. The power law rapidly approaches about 0.05 in-dicating again that the composite has a very high cluster con-tent and that the shear rate would be very high at the dieboundary. This can lead to jetting in flat dies and in gates in in-jection molds. For the case of the sound deadening project, thecritical wall stress was exceeded and the CaCO3 filled compos-ite jetted from the die so much so that no useful product couldbe produced.

5 Summary

The effect of filler concentration on the power law, n, in bothNewtonian fluids and three polyethylene polymers has beenpresented and compared. It was demonstrated that n can be re-lated to the volume fraction of the infinite clusters in the sys-tem and that the volume in these clusters does not dissipate en-

ergy. This leads to the realization that n can be interpreted asthe dissipation of the flowing material with percolating clusterformation divided by the dissipation if there were no clusterformation for both Newtonian fluid-based composites and forpolymer-based composites.Furthermore, it has been shown that if a polymer melt is ana-

lyzed as a combination of a low molecular weight fluid and ahigher molecular weight deformable filler the power law rela-tionship as a function of filler content exhibits the same rela-tionship per Eq. 4, as the fraction of filler in a Newtonianfluid-based slurry shown in Fig. 2 and the fraction of filler inthe neat polymer melt per Fig. 9.The measured power law in four filled PE polymer compo-

sites with filler concentrations up to about 60 vol.%, did notshow much of a reduction in n when compared to the effect offiller loading on n for Newtonian fluid-based dispersions. Itwas demonstrated that the behavior of polymer compositesdoes resemble the Newtonian fluid-based dispersions if thefilled polymeric power law is related to the combination ofthe deformable polymer filler (high molecular weight mole-cules), and the rigid filler added to the polymer melt.Although the Newtonian carrier-based fluid composites and

the unfilled polymers power law all fell on a curve of 3-dimen-sional percolation as a function of volume fraction, the filledpolyethylene composites tended to approach a 2-dimensionalpercolation function as the filler was increased using the samevolume fraction independent variable. This suggests that asthe filler is added to a high polymer, there is a change in me-chanistic flow characteristics.The power law constant of the polymers can be used to pre-

dict the shear rate at which the melt transitions from shear thin-ning to the \Infinite Shear Newtonian Plateau". Increases inproduction rate after this transition will be difficult because itwill require much more machine power per shear rate incre-ment.

References

Adams, M. E., \A Predictive Model for the Steady Shear Viscosity ofPolymer Melts," M. S. Thesis, Clarkson University, Postdam(1987)

Altobelli, S. A., Givler, R. C. and Fukushima, E., \Velocity and Con-centration Measurements of Suspensions by Nuclear Magnetic Re-sonance Imaging", J. Rheol., 35, 721–734 (1991),DOI:10.1122/1.550156

Bagley, E. B., \End Corrections in the Capillary Flow of Polyethy-lene", J. Appl. Phys., 28, 193–209 (1957)

Batchelor, G. K., \The Effect of Brownian Motion on the Bulk Stressin a Suspension of Spherical Particles", J. Fluid Mech., 83, 97–117 (1977)

Boger, V., Gupta, R. and Tanner, R. I. \The End Correction for Power-Law Fluids in the Capillary Rheometer", J. Non-Newtonian FluidMech., 4, 239–248 (1978)

Brenner, H., \Dissipation of Energy due to Solid Particles Suspendedin a Viscous Liquid", Phys. Fluids, 1, 338–346 (1958)

Campbell, G. A., Adams, M. E., \A Modified Power Law Model forSteady Shear Viscosity of Polystyrene Melts", Polym. Eng. Sci.,30, 587–595 (1990)

Campbell, G. A., Forgacs, G., \Viscosity of Concentrated Suspen-sions: An Approach Based on Percolation Theory", Phys. Rev. A:At. Mol. Opt. Phys., 41, 4570–4573 (1990)

Campbell, G. A., Wetzel, M. D., Howe, J. C. and Sterling, M. T, \In-vestigation of the Effect of Filler Concentration on the Flow Char-


Fig. 14. Effect of filler content on relative viscosity and the power lawn

Fig. 15. Power law, wall stress and plug stress relating to filler vol-ume fraction

acteristics of Filled Polyethylene Melt", SPE ANTEC Tech. Papers,1097–1106 (2017)

Campbell, G. A., Wetzel, M. D, \Characterizing the Flow of SlurriesUsing Percolation Theory Based Functions", Polym. Eng. Sci., 57,403–416 (2016), DOI:10.1002/pen.24435

Campbell, G. A., Zak, M. E. and Radhakrishnan, J. S., \Developmentof a Predictive Power Law Relationship for Polymer CompositesBased on Newtonian Carrier Concentrated Slurries", Polym. Com-pos, 28, 1172-1191 (2016a), DOI:10.1002/pc.24047

Campbell, G. A., Zak, M. E, \Characterizing the Flow of ConcentratedSlurries and Polymer Melts using Percolation Theory Based Func-tion", SPE ANTEC Tech. Papers, 186–198 (2017)

Campbell, G. A., Zak, M. E, \High Shear Rheology of Calcium Carbo-nate Slurries", SPE ANTEC Tech. Papers, (2001)

Campbell, G. A., Zak, M. E, Radhakrishnan, J. S. and Wetzel, M. D.,\Development of a Predictive Power Law Relationship for Con-centrated Slurries, Part 1: Theory", SPE ANTEC Tech. Papers,692–701 (2016b)

Dealy, J. M., Wissbrun, K. F.: Melt Rheology and its Role in PlasticsProcessing, Chapman & Hall, New York, p. 390–409 (1995)

Denn, M. M.," Extrusion Instabilities andWall Slip", Annu. Rev. FluidMech., 33, 265–87 (2001), DOI:10.1146/annurev.fluid.33.1.265

Dontula, N, Ramesh N. S., Campbell, G. A., Small, J. D. and Fricke,A. L." An Experimental Study of Polymer-Filler Redistribu-tion in Injection Molded Parts", J. Reinf. Plast. Compos., 13(2),98–110 (1994)

Einstein, A., \Eine neue Bestimmung der Moleküldimensionen\, Ann.Phys., 19, 289–306 (1906)

Elias, L., Fenouillot, F. Majesté, J.-C., Martin, G. and Cassagnau, P.,\Migration of Nanosilica Particles in Polymer Blends", J. Polym.Sci., Part B: Polym. Phys., 46, 1976–1983 (2008),DOI:10.1002/polb.21534

Gupta, R. K.: Polymer and Composite Rheology, 2nd Edition, MarcelDekker, New York, p. 224–265 (2000)

Hatzikiriakos, S. G., Dealy, J. M., \Wall Slip of Molten High DensityPolyethylene. I. Sliding Plate Rheometer Studies", J. Rheol., 35,497–523 (1991)

Hatzikiriakos, S. G., Dealy, J. M., \Wall Slip of Molten High DensityPolyethylenes. II. Capillary Rheometer Studies," J. Rheol., 36,703–740 (1992)

Hong, L.V, Sheng-Li, T. andWen-Ping, Z., \Direct Numerical Simula-tion of Particle Migration in a Simple Shear Flow", Chin. Phys.Lett. Vol., 28, 084708-1-084708-4 (2011)

Isayev, A. I., Palsule, S.(Eds.): Encyclopedia of Polymer Blends, Vol-ume 2: Processing, Wiley-VCH, Hoboken, p. 381 (2016)

Jastrzebski, Z. D., \Entrance Effects and Wall Effects in an ExtrusionRheometer during Flow of Concentrated Suspensions", I&ECFund., 6, 436–445 (1967)

Kamal, M. R., Mutel, A., \Rheological Properties of Suspensions inNewtonian and Non-Newtonian Fluids", J. Polym. Eng., 5, 293–382 (1985), DOI:10.1515/POLYENG.1985.5.4.293

Kalyon, D. M., Yaras, P., Aral, B. and Yilmazer, U., \Rheological Be-havior of a Concentrated Suspension: A Solid Rocket Fuel Simu-lant", J. Rheol., 37, 35–53 (1993), DOI:10.1122/1.550435

Kalyon, D. M., \Apparent Slip and Viscoplasticity of ConcentratedSuspensions", J. Rheol., 49, 621–640 (2005),DOI:10.1122/1.1879043

Kalyon, D. M., Dalwadi, D., Erol, M., Birinci, E. and Tsenoglu, C.,\Rheological Behavior of Concentrated Suspensions as Affectedby the Dynamics of the Mixing Process", Rheol. Acta, 45, 641–658 (2006), DOI:10.1007/s00397-005-0022-x

Kalyon, D. M., Aktas, S., \Factors Affecting the Rheology and Proces-sability of Highly Filled Suspensions", Annu. Rev. Chem. Biomol.Eng., 5, 229–254 (2014), PMid:24910916;DOI:10.1146/annurev-chembioeng-060713-040211

Karnis, A., Goldsmith, H. L. and Mason, S. G., \The Kinetics of Flow-ing Dispersions, 1. Concentrated Suspensions of Rigid Particles", J.Colloid Interface Sci., 22, 531–553 (1966a)

Karnis, A., Goldsmith, H. L. and Mason, S. G., \The Flow of Suspen-sions through Tubes. V. Inertial Effects", Can. J. Chem. Eng., 44,44181–44193 (1966b)

Krieger, I. M., Dougherty, T. J., \A Mechanism for Non-NewtonianFlow in Suspensions of Rigid Spheres", Trans. Soc. Rheol., III,137–152 (1959), DOI:10.1122/1.548848

Maron, S. H., Pierce, P. E., \Application of Ree-Eyring GeneralizedFlow Theory to Suspensions of Spherical Particles", J. ColloidSci., 11, 80–95 (1956), DOI:10.1016/0095-8522(56)90023-X

Mooney, M., \Explicit Formulas for Slip and Fluidity", J. Rheol., 2,210–222 (1931)

Moysey, P. A., Thompson, M. R., \Investigation of Solids Transport ina Single-Screw Extruder Using a 3-D Discrete Particle Simulation",Polym. Eng. Sci., 44, 2203–2215 (2004)

Nichetti, D., Manas-Zloczower, I., \Viscosity Model for PolydispersePolymer Melts", J. Rheol., 42, 951–969 (1998)

Norman J. M., Wagner, J., \Current Trends in Suspension Rheology",J. Non-Newtonian Fluid Mech., 157, 147–150 (2009),DOI:10.1016/j.jnnfm.2008.11.004

Ostwald, W., \The de Waele-Ostwald Equation", Kolloid Z., 47, 176–187 (1929), DOI:10.1007/BF01496959

Seshadri, V., Sutera, S. P. \Concentration Changes of Suspension ofRigid Spheres Flowing through Tubes", J. Colloid Interface Sci.,27, 1–143 (1968)

Seshadri, V., Sutera, S. P., \Apparent Viscosity of Course, Concen-trated Suspensions in Tube Flow", Trans. Soc. Rheol., 14, 351–373 (1970)

Rabinowitsch, B., \Über die Viskosität und Elastizität von Solen", Z.Phys. Chem., 145, 1–26 (1929)

Radhakrishnan, J. S., \Investigation of Poiseuille Flow of Concen-trated Monodisperse and Bidisperse Suspensions", PhD Disserta-tion, Clarkson University, Postdam (2002)

Tsenoglou, C., \Viscoplasticity of Agglomerated Suspensions",Rheol. Acta., 28, 311–314 (1989), DOI:10.1007/BF01329340

Wetzel, M. D., Howe, J. C., Sterling, M. T. and Campbell, G. A., \TheRheology of Concentrated Slurries: Experimental Evaluation andthe Effects on Polymer Processing", SPE ANTEC Tech. Papers,1144–1154 (2017)

Wetzel, M. D., Pettitt, D. R., Jr. and Campbell, G. A., \Development ofa Predictive Power Law Relationship for Concentrated Slurries,Part 2: Experiment and Processing Implications", SPE ANTECTech. Papers, 702–709 (2016)

Wood-Adams, P. M., Dealy, J. M., \Use of Rheological Measurementsto Estimate the Molecular Weight Distribution of Linear Polyethy-lene", J. Rheol., 40, 761–778 (1996)

Yilmazer, U., Kalyon, D. M., \Slip Effects in Capillary and ParallelDisk Torsional Flows of Highly Filled Suspensions", J. Rheol., 33,1197–1212 (1989)

Zak, M. E., \An Investigation of the Shear-Thinning Behavior andPower-Law Modeling of Concentrated, Non-Colloidal Suspen-sions," PhD Dissertation, Clarkson University, Postdam (1994)

Zohrevand, A., Ajji, A. and Mighri, F., \Relationship between Rheolo-gical and Electrical Percolation in a Polymer Nanocomposite withSemiconductor Inclusions", Rheol. Acta, 53, 1–20 (2014),DOI:10.1007/s00397-013-0750-2

Zupancic, A., Romano, L. and Kristoffersson, A., \Influence of Parti-cle Concentration on Rheological Properties of Aqueous a-Al2O3Suspensions," J. Eur. Ceram. Soc., 18, 467–477 (1998),DOI:10.1016/S0955-2219(97)00168-4

Date received: August 18, 2017

Date accepted: December 21, 2017

BibliographyDOI 10.3139/217.3571Intern. Polymer ProcessingXXXIII (2018) 5; page 619–633ª Carl Hanser Verlag GmbH & Co. KGISSN 0930-777X



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