Capillary rheometry of a binary mixture polymer/CO2 in a single screw extruder

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Capillary rheometry of a binary mixture polymer/CO2

in a single screw extruder

Audrey Common, Martial Sauceau, Élisabeth Rodier, Jean-jacques

Letourneau, Jacques Fages

To cite this version:

Audrey Common, Martial Sauceau, Élisabeth Rodier, Jean-jacques Letourneau, Jacques Fages.

Cap-illary rheometry of a binary mixture polymer/CO2 in a single screw extruder. Chemical Engineering

and Processing: Process Intensification, Elsevier, 2015, 93, pp.21-26. �10.1016/j.cep.2015.04.004�.

�hal-01611093�

Capillary

rheometry

of

a

binary

mixture

polymer/CO

2

in

a

single

screw

extruder

Audrey

Common

1

_,

_Martial

_Sauceau,

_Elisabeth

_Rodier,

_Jean-Jacques

_Letourneau,

Jacques

Fages

*

UniversitédeToulouse,EcoledesMinesd'Albi,CNRS,CentreRAPSODEE,81013Albi,France

Keywords: Hot-meltextrusion Supercriticalcarbondioxide Capillaryrheometry Polymerviscosity

ABSTRACT

Processingbio sourcedpolymerswithsupercriticalfluidsisapromising routetowardsnew green engineeringprocesses.Supercriticalcarbondioxide(sc CO2),issolubleinlargequantitiesinmolten

polymers,whereitactsasplasticizerandswellingagent.Itisused,inblendingorfoamingofpolymers, particleformationandpolymerisationprocess.

Theprocessofhot meltextrusionassistedbysc CO2allowedthedevelopmentofanon lineviscosity

measurementbasedoncapillaryrheometry.Appliedtoabio sourcedpolyamide,itwasvalidatedby comparisonwithaclassiccapillaryrheometer.Bothdatasetswereingoodagreement.

Apseudoplasticfluidbehaviourwasobservedwitha30%viscositydecreasefrom46to32Pasat 5000s 1_and220"_{C,uponadditionofCO}

2.However,viscositydecreasedtoaplateaubeforereachingthe

thermodynamicsolubility.Thecomparisonwithamodelcouplingsolubilityandflowallowedtoidentify themethodlimitations,whichwereattributedtothekineticsofdissolutionandmixing.Thehigherthe shearstress,thehighertheamountofCO2atwhichtheviscosityplateauisreached.Thesemeasurements

mayquantifytheimpactoftheCO2ontherheologyofthesystembutalsooftheefficiencyofthemixing

processinourexperimentalsetup.

1.Introduction

Most of the polymers used in industry nowadays have a petrochemicalorigin.However,duetothepredictedexhaustionof theworldpetroleumreserves,itisnecessarytoreplacethemwith bio sourced polymers with equivalent properties. It is also importanttofindnewwaysofprocessing them,whichcomply withtherequirementsofgreenchemistry.Theuseofsupercritical fluidsinpolymerprocessinghasconsiderablygrownupinthelast decades[1 3].Themostusedissupercriticalcarbondioxide(sc CO2).Itissolubleinlargequantitiesinmanymoltenpolymersand

canbeaddedinextrusionprocesseswhereitactsasaplasticizer andswellingagent[4].Itisknowntobeagreenprocessingagentof greatinterestinthepolymerfield,suchasblendingofpolymers, polymerfoaming,particleformationorpolymerizationprocess.

Extrusionisaprocessconvertingarawmaterialintoaproduct ofuniformshapeanddensitybyforcingitthroughadieunder

controlledconditions[5].Ithasextensivelybeenappliedinthe plastic and rubber industries, where it is the most important manufacturingprocess.

Coupling sc CO2 and extrusion modifies the rheological

properties ofthe polymer whileflowing through thebarrel of the extruder [5]. The reduction of viscosity decreases the mechanical constraints and the operating temperature within theextruder.Thus,thismayallowtheuseoffragileorthermolabile molecules, like active pharmaceuticalingredients. Furthermore, theabsenceofresiduesinthefinalmaterialisalsoanadvantagefor pharmaceuticalapplications[6,7].

Using anewsc CO2assistedextrusionprocess,microcellular

foamsofabiocompatibleamorphouspolymerwereelaborated[8]. However,understandingandimprovingsuchaprocess requires theknowledgeofphysicalproperties,likethesolubilityofCO2into

the polymer, the diffusion coefficient andthe viscosityof the mixture.Anextruderprovidesahighshearrate,particularlyinthe die.However,theviscosityofthe binarysystemunderprocess conditionsisverydifficulttoreachinconventionalrheometers. Viscosityunderpressurecanbemeasuredindifferentways[9]. Onegroupofmeasurementsisbasedonvibratingsurfaceorfalling ballrheometers[10].ThesetechniquesrequireNewtonianorlow viscosity polymers. The other main group of measurement

*Correspondingauthorat:EcoledesMinesd'Albi,CampusJarlard,F-81013Albi, France.Tel.:+33563493141;fax:+33563493025

E-mailaddress:Jacques.Fages@mines-albi.fr(J.Fages).

techniques is based on capillary rheometry. This technique is widely usedforviscouspolymersandimplementationsexistto measuretherheologyofmixturesofpolymerandsc CO2[11].

Inthiswork,wehaveimplementedthison linetechniqueona single screwextruderinordertoestimatetheviscosityofabio sourced semi crystalline polymer. Firstly, the validity of the methodwillbecheckedbycomparingtheobtainedresultswith measurementsmadeonaclassiccapillaryrheometer.Secondly,we will applythistechnique tothe binary systempolymer/carbon dioxide. Finally, the comparison with a flow model will be implementedtoidentifysomelimitationsofthetechnique. 2.Theoryofcapillaryrheometry

Theprincipleofcapillaryrheometryistoforcemeltpolymer throughadieoflengthLanddiameterD.Knowingthevolumeflow rateQandthepressureloss

D

P(P Patm)createdbythedie,the

shearrate _

g

_app,thestress

t

pandsubsequentlytheviscosity

h

can

be calculated (Fig. 1). To determine this viscosity, several assumptions need to be made. A Poiseuille flow in a tube is assumedwiththefollowingadditionalhypotheses:

# incompressiblefluidwithalaminar,isothermalandestablished flow,

# noendeffects, # nowallslip.

Ifthesehypothesesareverified,thenthefollowingformulas apply:

t

p¼D_4L

D

P (1) _

g

app¼ 32Q

p

D3 (2)

h

¼

_g

_{_}

t

p app (3)

Fornon newtonianfluids,theRabinowitchcorrectionshould apply:

_

g

_w¼ _

g

_app 3mþ1 4m ! "

withm¼dIn_dInQ

t

p (4)

Another arguable hypothesis is the absence of end effects, whichmayrequirecorrections.Onewayistousealongdieinorder tomaketheeffectsnegligible.Theothermethodistoimplement theBagleycorrection[12].Itconsistsincarrying out measure ments with dies of different lengths and to use the slope of

D

P f(L/D)tocalculate

t

p.Theproblemofthismethodisthehigh

numberofexperimentsrequired. 3.Materialsandmethods

Polyamide PA 11, commercial name Rilsan1 _{(BMFO grade,}

Arkema,France),isabio sourcedpolymeramino 11 undecanoic acidobtainedfromcastoroil.Itisalinearpolymerwithanumber averagemolarmassof8700gmol 1_{andaweightaveragemolar}

massof20,010gmol 1_.

Melting onset temperature of the PA 11 is in the range 180 200"_{Candtheglasstransitiontemperaturewithin40 60}"_C.

Thesoliddensity

r

P,determinedbyheliumpycnometry(Micro

meretics,AccuPYC1330)was1027&5kgm 3_{.Meltdensitywas}

found to be 979kgm 3 _{at 220"C with a magnetic suspension}

balance(Rubotherm,Germany).

SolubilitymeasurementsofCO2inthepolymerwerecarried

outonthe samemagneticsuspensionbalance at202,220and 231"_{Catdifferentpressures.}

Theexperimentalsetup(Fig.2)haspreviouslybeenusedwith several polymers including PA 11 [13 16]. The single screw extruderhasa30mm screwdiameterandalengthtodiameter ratio(L/D)of35(Rheoscam,SCAMEX,France).AgreatL/Dratio generallyindicatesagoodcapacityofmixingandmeltingbutan importantenergyconsumption.Thescrewisdividedintothree parts.Thefirstonehasalengthtodiameterratioof20whilethe othershavealengthtodiameterratioof7.5.Betweeneachpart,a restrictionringhasbeenfittedoutinordertoobtainadynamic gastight, which prevents sc CO2 from back flowing. The first

conicalpartallowsthetransportofsolidpolymersandthen,their melting and plasticizing. Then, the screw has a cylindrical geometryfromthefirstgastightringtothedie.Thetemperature insidethebarrelisregulatedatfivelocations:TaandTbbeforethe

CO2injection,TcandTdaftertheinjectionandTeinthedie.All

temperaturesweresetat220"_C.

Therearethreepressureandtwotemperaturesensors:P1after

theCO2injector,P2andT1beforethesecondgastightringandP3

andT2bythedie.Thisallowsmeasuringthetemperatureandthe

pressureofthepolymerinsidetheextruder.Errorsassociatedto pressureandtemperaturemeasurementswereabout0.2MPaand 1"_{C,respectively.}

Threedieshavebeenused.Theyallhaveadiameterof1mm andtheirlengthsare7,17and22mm.Thosedieswillbecalledhere L7,L17andL22,respectively.

CO2(N45,AirLiquide,France)ispumpedfromacylinderbya

syringepump(260D,ISCO,USA)andthenintroducedatconstant volumetricflowrateatalengthtodiameterratioof20fromthe feedhopper.ThepressureintheCO2pumpiskeptslightlyhigher

than the pressure P1. The pressure, the temperature and the

volumetricCO2flowratearemeasuredwithinthesyringepump.

CO2 density, obtained on NIST website by Span and Wagner

equationofstate[17,18],isusedtocalculatemassflowrateand thustheCO2massfractionwCO2.CO2flowratewasvariedandits influenceontheviscosityofthemixturewasstudied.Oncesteady stateconditions(testedonP andTmeasurements)arereached, massflowrateandpressure(P3)aremeasured.

Fortheimplementationofthetheoryofcapillaryrheometry, the melt density has been considered as independent of the pressure,andtakenequaltothevaluemeasuredbyamagnetic suspensionbalanceat220"_{Candatmosphericpressure.Moreover,}

inthecaseofCO2injection,wehavestatedthatthehypothesisof

incompressiblefluidandconstantdensityisacceptablesincethe solubilisationoflessthan5%ofdenseCO2(densityofwhichis

400kgm 3_{) with an increase involume of the same order of}

magnitudewillresultinadensityvariationofabout2 3%. Parallelmeasurementsonacapillaryrheometer(CR)(Instron model3211,USA)were madeattheLaboratoire desMatériaux Polymères etdes Biomatériaux(ISTIL EPUL, Lyon, France). This rheometerisequippedwithapistonof0.9525cm2_{andwithaforce}

sensor of 20kN with a precision of0.2%. One die of L/D 140 (L 70mmandD 0.5mm)isused.Measurementswerecarried outat220"_C.

4.Results

4.1.Comparisonofbothmethods

Measurementsontheextruderwerecarriedoutwiththethree dies(Table1).Experimentswerecomparedwithresultsobtainedon the Instron capillary rheometer. At first, end effects could be postulated as negligible with thelongest dieL22 and only the Rabinowitchcorrectionwasapplied(m 0.56).Nevertheless,Bagley correctionwasalsoimplementedontheextruderexperimentsin ordertoevaluateendeffects.Experimentalresultsobtainedwith thethreedies wereinterpolatedatthesameshearrate,which allows plotting the Bagley curves (Fig. 3). According to the procedure ofanalysis describedhereafter, thepressure dropis reportedonthexaxis,whereasthecapillaryaspectratioisreported ontheyaxis.Thenon linearshapeofthesecurvesiscommonly attributedtotheeffectofthepressureontheviscosity[19].

To estimatethe effectof pressureon viscosity, Pantani and SorrentinohaveproposedthefollowingexpressionfortheBagley curves[20]: L D ¼

a

D

P2þb

D

P c (5) with

t

op¼_4b1 e D ¼c

b

¼2a_b 8 > > > > > < > > > > > : (6)

t

opisthestressatzeroentrancepressure,eanadditionallength

whichtakesintoaccounttheentranceeffectonpressuredropand

b

aparameterdescribingthepressureeffectonviscosityinthe Baruslaw.AscanbeseeninFig.3,thedatawerewellrepresented byEq.(5).The

b

valuesobtaineddecreasedfrom55'10 9Pa 1_at

4000s 1_to35'10 9_Pa 1_at10000s 1_{(Fig.4).Thesevaluesare}

coherentwith theonesreportedat an averagetemperature of 210"_{Cforpolypropylene(PP)andpolystyrene(PS),buthigherthan}

those obtained for PA66 found to be around 7'10 9Pa 1 at temperatures below 300"_{C [21]. Finally, theentrance pressure}

drops were extrapolated fromFig. 3, allowingto calculatethe corrected stresses. After the application of the Rabinowitch correction(m 0.54),thecorrectedviscositycanbecalculated.

Fig.5showsthecomparisonofresultsobtainedontheextruder andonthecapillaryrheometer.Bothseriesofresultsareingood agreement. However, results on the extruder treated with the Bagley correction give significantly lower viscosity values. Therefore,thecorrectionseemstobeusefulforthisdie.Theset ofviscosityvaluesobtainedwiththecommercialrheometerare higher.

Table1

Dataobtainedontheextruderwiththethreedifferentdies.

L7 L17 L22

DP(MPa) Q(cm3_s 1_{) DP(MPa) Q(cm}3_s1_{) DP(MPa) Q(cm}3_s 1₎

5.8 0.50 8.3 0.22 14.5 0.31 7.6 0.65 10.3 0.35 18.0 0.45 9 0.82 12.5 0.50 20.0 0.58 10 1.01 14.9 0.65 23.0 0.71 11 1.20 17.7 0.82 25.5 0.86 11.6 1.38 18.2 0.97 28.0 1.00 19 1.14

Fig.3.Bagleycurvescalculatedatdifferentshearrates.

Thisdiscrepancymightbeascribedto:

# temperaturecontrolduringtheexperiment.Bothmethodsused temperaturesetat220"_{Cbuttheheatingtakesplacethrougha}

staticpolymer inthe CRduring around 1hwhereas there is mixing in the extruder with short residence times (around 3min).

# pressurelevelisquitedifferent.PressureintheCRisintherange 6 200 MPa and particularly within 60 200 MPa, which correspondstotherangeofshearratesencounteredinthedie of the extruder. This high level of pressure may induce an increaseinviscosity.Pressurelevelintheextruderisbetween7 and25MPa.

# samplestorage leading to a possiblechange in themoisture content.PA11isquitesensitivetohumidity.

Inordertofurtherinvestigatethesediscrepanciesandtouse thiscomparisonasavalidationofthemethod,wecouldusealess sensitive polymer andevaluate pressure correctionsto the CR results.Nevertheless,untreatedresultsstillgiveagoodorderof magnitudeoftheviscosity.

4.2.Evolutionofviscosityofthemixture

Experimentson the binary systemwere carried out on the extruderonlywiththeL22die,whichensuresagoodpressurefor solubilisationofCO2intheextruderandminimizesthenumberof

experiments.However,ithastobekeptinmindthateventhough thequantitativevalueisnotexact,duetothelackofcorrection,the qualitativeevolutioniscorrect.

SolubilityexperimentaldatawererepresentedbyaSanchez Lacombeequationofstate[22,23]byusingtheexpressionofthe fugacitycoefficientsofacomponentinthemixtureproposedby Neau[24].Thepurecharacteristicparametersofeachcomponent weretakenasindicatedinTable2.Moreover,thefollowingmixing ruleswereused,endowedwithonefittedbinaryparameterforthe characteristicenergy

e

*:

e

(_¼X i X j

f

_i

f

_j

e

( ijwith

e

(ij¼ð1 kijÞð

e

(i

e

(jÞ0:5 (7)

Thismodelwasthenusedtoevaluatethesolubility(weq)ateach

pointofthedie.Solubilityatthedieentranceisnotedweqi.

Viscosityofthebinarysystemdecreaseswiththeincreasein CO2contentandshearrate(Fig.6).Thebehaviourintermsofshear

rateisnotsurprisingsincepolymersareoftenshearthinning.The decreaseinviscosityrelatedtoCO2additionisalsoconsistentwith

literatureobservations.Thisdecreaseusuallyfollowsapowerlaw

[26].However,ourmeasurementssuggestaplateau,whichisnot consistentwithsuchalaw.Thismaybeduetothedesorptionofthe CO2inthedieandtheoccurrenceofatwo phasesystemthatcould

disturbmeasurements.

Infact,animportantissuewithbinarymixtureistoknowifthe systemisinasingle phase.Multiplepressuresensorsalongthedie wouldbenecessarytocheckthisassumption[27].However,our systemwassetwithonlyonesensorattheentranceofthedie. Thus,anevaluationoftheexactlocationofdesorptioninthedie was made. A linear depressurisation was assumed from the entrance to the exit [28]. This linear depressurisation was confirmedbymodellingthedieflowwithComsolMultiphysics1

byusingNavier Stokesequationsforanincompressiblefluid.A Carreaulaw(Fig.5)wasusedtorepresenttheviscosity

h

ofthe polymer(Table3):

Fig.5.Comparisonofdifferentrheologicalresults.

Table2

PurecharacteristicparametersforSanchez–Lacombeequationofstate. Compounds r*(g/cm3_{) T*(K) P*(MPa)}

PA11[25] 1.035 765.0 465.4 CO2a 1.2518 354.1 329.3 a_{CalculatedbyusingdataobtainedonNISTwebsitebySpanandWagnerequationof}

state[17,18].

Fig.6.EvolutionofthebinarysystemviscosityasafunctionofCO2contentand

shearrate.

Table3

ParametersfortheCarreaulaw.

h0(Pas) h1(Pas) l(s) n(!) a(!)

63.46 0 3.40'10 4 _{0.40 2}

h

¼

h

₁þð

h

0

h

1Þ 1þð

l

g

_Þa+ n 1=a ð Þ

h

(8)

h

0istheviscosityatzeroshearrate,

h

1theviscosityatinfinite

shear rate,

l

the relaxation time, n the power index and a a dimensionlessparameterdescribing thetransition betweenthe firstNewtonianplateauandthepowerlawregion.

An example of the resultsfound by the flow simulationis presentedonFig.7.Figs.8 10showthelocationofthepointof desorptionforeachexperimentalmeasurementatthethreeshear ratesstudiedandfordifferentCO2contents.Desorptionisassumed

tooccurwhenw weq.

AstheCO2contentincreases,desorptionoccursearlierinthe

die.Whendesorptionoccurswithinthefirstthreequartersofthe die length (16.5mm), viscosity measurements differ from the powerlaw,asshownbythelogarithmicchartinFig.11.Theindex

ofthepowerlaw,n,forthethreeshearratesarelistedinTable4. ViscosityresultscanthusbeassumedaccurateaslongastheCO2

contentisnottoohighanddesorptiondoesnotoccurtooearlyin thedie.Inthisstudy,thelimitappearstobew/weqi<0.27.

Finally,adecreaseoftheviscosityaround25%isobservedinthe single phasecriterionregiondefinedabove(w/weqi<0.27,corre

spondingtoamasspercentageupto2%ofCO2).Theappearanceof

theplateauathigherCO2contentisduetothepredominanceofthe

two phaseoccurrencealongthedie.Theviscositymeasuredinthis zonecanbeseenasanapparentviscosityofthemixtureinthe process.Itreflectscouplingofflowingwithphysicalphenomenaof two phase occurrence. Then, it can be used to apprehend phenomenological behaviour of the flowing mixture in the process.

5.Conclusion

Capillaryrheometryisanefficientmethodtomeasureviscosity ofpolymerathighshearrate.Weimplementedthistechniqueon line on an extruder in order to carry out measurement in processingconditions.Validityofthemeasurementswaschecked with a commercialcapillary rheometer. Measurements on the binary mixtureCO2/polymer were thencarriedout. Desorption

pointalongthediewaslocatedbymeansofasolubility model (Sanchez Lacombe) coupled with a flow modeland correlated withexperimentaldata.AminimaldistancebeforeCO2desorption

mustbeattainedtoensureaccuracyofthedata.Thoseexperiments andcalculationdemonstratetheefficiencyofcapillaryrheometry to measure viscosity of polymer and binary system polymer/ sc CO2.Theset upwithonlyonepressuresensorattheentranceof

the dieworksatlowratio ofCO2.Abovea givenCO2 content,

depending on the solubility, pressure and length of the die, depressurisation occurstooearlyinthedieandmeasurements giveaccesstoapparentviscosityinprocessconditions.Toaccess absoluteviscosityofthemixture,asystemwithmultiplepressure sensorswouldbenecessary.

Acknowledgements

This paper is dedicated to the memory of our esteemed colleagueElisabethRodierwholeftusmuchtooearly.

Fig.8.Desorptionpointat3640s 1_{at3differentinitialratios.}

Fig.9.Desorptionpointat5025s 1_{at3differentinitialratios.}

Fig.10.Desorptionpointat7707s 1_{at2differentinitialratios.}

Fig.11.Logarithmicchartoftheevolutionofviscosityofthebinaryasafunctionof CO2contentandshearrate.

Table4

Indexofthepowerlaw. _

gðs1Þ 3640 5025 7707

Thisworkwasmadepossiblethankstofinancialsupportofthe ANR (French National Agency for Research) and to Arkema companythatsuppliedthepolymer.

AuthorswouldliketothankPr.Jean PierrePuauxfromLMPB (Lyon,France)laboratorywhokindlyaccepttocarryoutexperi mentsonhiscapillaryrheometer.ThetechnicalsupportofOlivier Ezequelisalsogratefullyacknowledged.

References

[1]R.B.Yoganathan,R.Mammucari,N.R.Foster,Densegasprocessingofpolymers, Polym.Rev.50(2)(2010)144–177.

[2]H.P.Zhang,M.C.Chen,Polymerizationinsurpercriticalcarbondioxide,Prog. Chem.21(9)(2009)1869–1879.

[3]E.Reverchon,R.Adami,S.Cardea,G.DellaPorta,Supercriticalfluidsprocessing ofpolymersforpharmaceuticalandmedicalapplications,J.Supercrit.Fluids 47(3)(2009)484–492.

[4]M.Sauceau,J.Fages,A.Common,C.Nikitine,E.Rodier,Newchallengesin polymerfoaming:areviewofextrusionprocessesassistedbysupercritical carbondioxide,Prog.Polym.Sci.36(6)(2011)749–766.

[5]C.Rauwendaal,PolymerExtrusion,HanserPublishers,München,2001. [6]Z.K.Nagy,M.Sauceau,K.Nyúl,E.Rodier,B.Vajna,G.Marosi,J.Fages,Useof

supercriticalCO2aidedandconventionalmeltextrusionforenhancingthe dissolutionrateofanactivepharmaceuticalingredient,Polym.Adv.Technol. 23(2012)909–918.

[7]T.Vigh,M.Sauceau,J.Fages,E.Rodier,I.Wagner,P.L.Sóti,G.Marosi,Z.Nagy, EffectofsupercriticalCO2plasticizationonthedegradationand residual crystallinity of melt-extruded spironolactone – non-invasive testing of immediate-releasesoliddispersions,Polym.Adv.Technol.25(2014) 1135–1144.

[8]C.Nikitine,E.Rodier,M.Sauceau,J.-J.Letourneau,J.Fages,Controllingthe structureofaporouspolymerbycouplingsupercriticalCO2andsinglescrew extrusionprocess,J.Appl.Polym.Sci.115(2)(2010)981–990.

[9]A.Shine,Polymersandsupercriticalfluids,Phys.Prop.Polym.Handb.18(2007) 319.

[10]B. Calvignac,E.Rodier, J.-J. Letourneau,P. Vitoux,C. Aymonier,J. Fages, Development of an improved falling ball viscometer for high-pressure measurementswithsupercriticalCO2,J.Supercrit.Fluids55(1)(2010)96–106.

[11]M.Nobelen,S.Hoppe,C.Fonteix,F.Pla,M.Dupire,B.Jacques,Modelingofthe rheologicalbehaviorofpolyethylene/supercriticalCO2solutions,Chem.Eng. Sci.61(2006)5334–5345.

[12]E.Bagley,Endcorrectionsinthecapillaryflowofpolyethylene,J.Appl.Phys.28 (5)(1957)624–627.

[13]M.Sauceau,C.Nikitine,E.Rodier,J.Fages,Effectofsupercriticalcarbondioxide onpolystyreneextrusion,J.Supercrit.Fluids43(2007)367–373.

[14]C.Nikitine,E.Rodier,M.Sauceau,J.Fages,Residencetimedistributionofa pharmaceuticalgradepolymermeltinasinglescrewextrusionprocess,Chem. Eng.Res.Des.87(6)(2009)809–816.

[15]C.Nikitine,E.Rodier,M.Sauceau,J.-J.Letourneau,J.Fages,Controllingthe structureofaporouspolymerbycouplingsupercriticalCO2andsinglescrew extrusionprocess,J.Appl.Polym.Sci.115(2010)981–990.

[16]A.Common, E.Rodier,M. Sauceau, J.Fages,Flow and mixingefficiency characterisationinaCO2-assistedsingle-screwextrusionprocessbyresidence timedistributionusingRamanspectroscopy,Chem.Eng.Res.Des.92(2014) 1210–1218.

[17]Carbondioxide,NISTChemistryWebBook,http://webbook.nist.gov/cgi/cbook. cgi?ID=124-38-9.

[18]R.Span,W.Wagner,Anewequationofstateforcarbondioxidecoveringthe fluidregionfromthetriple-pointtemperatureto1100Katpressuresupto 800MPa,J.Phys.Chem.Ref.Data25(6)(1996)1509–1596.

[19]H.Laun,Polymermeltrheologywithaslitdie,Rheol.Acta22(2)(1983) 1435–1528.

[20]R. Pantani, A. Sorrentino, Pressure effect on viscosity for atactic and syndiotacticpolystyrene,Polym.Bull.5(4–5)(2005)365–376.

[21]S.Ceccia,C.Cocquet,L.Trouillet-Fonti,D.R.Long,Influenceofpressureon polyamide66shearviscosity:acasestudytowardspolarpolymersbehavior, Rheol.Acta53(2014)181–190.

[22]I.C.Sanchez,R.H.Lacombe,Anelementarymoleculartheoryofclassicalfluids. Purefluids,J.Phys.Chem.80(21)(1976)2352–2362.

[23]I.C.Sanchez,R.H.Lacombe,Statisticalthermodynamicsofpolymersolutions, Macromolecules11(6)(1978)1145–1156.

[24]E.Neau,Aconsistentmethodforphaseequilibriumcalculationusingthe Sanchez–Lacombelattice-fluidequation-of-state,FluidPhaseEquilib.2(2002) 133–140.

[25]P.A.Rodgers,Pressure–volume–temperaturerelationshipsforpoly(vinylidene fluoride)andpolyamide-11,J.Appl.Polym.Sci.50(12)(1993)2075–2083. [26]J.R. Royer, Y. Gay,J.M. Desimone,S.A. Khan, Highpressure rheologyof

polystyrene meltsplasticizedwith CO2: experimentalmeasurementand predictivescalingrelationships,J.Polym.Sci.PartB:Polym.Phys.38(2000) 3168–3180.

[27]A.Xue,C.Tzoganakis,Rheologicalpropertiesofpolystyrene/supercriticalCO2 solutionfromanextrusionslitdie,J.Polym.Eng.23(1)(2003)1–22. [28]X.Han,K.W.Koelling,D.L.Tomasko,L.J.Lee,Effectofdietemperatureonthe

morphologyofmicrocellularfoams,Polym.Eng.Sci.43(6)(2003) 1206–1220.