TacticityEffectsonPolymerBlendMiscibility.2.RateofPhaseSeparation r",,=M-(S"2",!)<3>

1993, 26,

Tacticity ^Effects

^on

Polymer Blend Miscibility.

^2.

^{Rate of}

Phase Separation

G. Beaucage*-1and R. S.Stein

Department of Polymer^ScienceandEngineering, University ofMassachusetts, Amherst,Massachusetts01003

ReceivedMay^{5, 1992}

ABSTRACT: Cloud-point^curves ofisotactic andatactic poly(vinyl methylether) (i- anda-PVME)^blended with atacticpolystyrene (PS)^were analyzed using theFlory-Huggins-Staverman(F-H-S)approachinthe first_paper1ofthisseries. Afunctional formforthecomposition-dependent interactionparameter,“g”, in these two polymer^{blendswas obtained.} In this_paper the “g” functionspreviously obtained ^{are used}to predict^{the rate}ofphaseseparationin termsofthe apparentdiffusionconstant. TheF-H-S _analysisof cloud-point^curves predicts^adrastically^slowerrateof_{phaseseparation}fori-PVMEinblendswithPSin comparisonwith a-PVMEblends. Therateof_{phaseseparation}isexperimentally^determinedforthesetwo polymer blends using lightscattering. A_comparison is madebetween the measured apparentdiffusion constant and that predicted from simple cloud-pointmeasurements using the F-H-S approach. The experimentalkineticsdata_agreewiththedrastically^slowedrateof_{phaseseparation}predicted for the isotactic blend. A comparisonofF-H-Sresults andthe measuredkinetics_yields^atranslationaldiffusionconstant which _agreeswith literature values. A briefdiscussionof^a novel analysis oflight scattering datafrom intermediate-stagephaseseparation^{is also}presented. This_analysisis basedon amodifiedCahn-Hilliard approach and^uses the“g” functions ^obtainedfrom cloud-point^curves using theF-H-S _approach.

Introduction

Equilibrium thermodynamics govern thedirection in which nonequilibrium phase separation

will

tend in a

binary^blend. In^a practical blend the domain^sizesand final morphology

will

be governed by phase-separation thermodynamicscoupledwiththetransport^andinterfacial properties ofthe blend components. It is convenientto divide phase-separation behavior in _polymer blends of criticalcompositionintothreestages. The earlystageof spinodalphaseseparation^isto some degreeof_accuracy described by^abalanceofthermodynamic^andtransport properties using Cahn-Hilliard _theory.2³ The interme- diatestageof_phaseseparation^{can beviewedas a}process displaying spinodal-like behavior with ^a modified ther- modynamic driving force based on a changing local composition. This hasbeen a matter ofrecent investi- gation^andwillbediscussedin thelatter_{part of}this paper.

Late-stage phase separation ^is generally ^described by Ostwald ripening.4·5 Late _stages depend ^on surface tensions between components, the viscosityofcomponents, andvariety ofother parameters such^asstressfieldsin the material. Thebulk of_{this paper}

will

dealwith_{the early} stageinterms ofCahn-Hilliard_theory.

In the

first

_paper of this series1 the Flory-Huggins- Staverman

(F-H-S)

equation ^{was used} to determine ^a functional form for the composition ^and temperature dependenceofthe interaction function, “g”, in isotactic and atactic poly(vinyl methyl ether) (i- and a-PVME)/

polystyrene (PS) blends basedon cloud-point^measure- ments. Usingthis_approachthe_parametersa,bo,bu and

c(thelatter_beingtermed the Staverman parameter)^are determined in the

F-H-S

_equation.

g(<t>2,T) ⁼ a + ^(b0+

bJT)

(1^- c02) (1)

rameteratthespinodal temperature. Inthis_paperit will

be shown

that

the _g(<£,T) functions predict drastically slowedkinetics of_phaseseparation fori-PVME/PSblends (incomparisonwith a-PVME/PSblends). Since theg{<t>,T)

functionsare determined from cloud-point^curves at the miscibility

limit,

^{we view} the predictions for _phase- separation kinetics well into the_{two-phase regime} as a

test of the usefulness of the

F-H-S

_{approach. Light} scattering measurements

will

be usedto determinethe rateof_{phaseseparation}intheisotactic andatacticPVME/

PS blends. The resultsofkinetics measurementsinthe immiscible _regime ^are _compared with the predictions derivedfrom cloud-point^curves atthemiscibility

limit.

A quantitativeagreement

will

be demonstrated.

Inaddition, intermediatestagesof_{phaseseparation}will bebrieflyexplored. Functionalformsfor thecomposition and temperature dependence of the thermodynamic driving^forceforphaseseparation^aresuccessfully usedto describeintermediate _stagesusing avariable localcom-

position.

Cahn-Hilliard Theory

ThelinearCahn-Hilliardtheory2·3for_{the earlystages} of phase separation ^{has been used} to _analyze light scattering^data6’7fromphase-separatingpolymer^blends.

During^the

initial

_stagesof_phaseseparation theintensity of light ^{scattered at} all _angles is expected to increase exponentially,

Kq,t) ⁼ I(q,t=Q) exp(2R(q)t) (2) where_q⁼

(4 / )

sin (0/2),^tisthetimeof_phaseseparation, and R(q)^istheamplification^{factor for}phaseseparation

or the growth^rate forasize scaledescribe by2v/q. Cahn- Hilliard _theorydefinesR(q)foran incompressible blend

as

The thermodynamic driving^forceforphaseseparation at earlystages, (g-gs)/gs^(seethe appendix),^{can be}calculated, where _{g„ is} the composition-dependent interaction pa-

fSupported inpartby^agrantfrom Polysar Inc. Present address:

SandiaNational Laboratories, Albuquerque,NM 87185.

r",,=M-(S"2",!)

^<3>

whereDc is thetranslationaldiffusion constant,

f

^isthe

free energydensityofthe system atcomposition %,and is the coefficientofthe composition gradient.

0024-9297/93/2226-1609$04.00/0 ^&copy; 1993 American Chemical Society Downloaded via UNIV OF CINCINNATI on December 1, 2019 at 04:17:49 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

TableI

Flory-Huggins-Staverman Analysis Parameters from^Fits totheCloud-PointCurves(Values Determined in the First

Paperof This Series1)

a(X103) feo(X102) fe, c

helero -1.43 1.36 -4.69 0.36“

iso -0.57 1.77 -4.08 0.425

Avalué

0.86 0.20 0.14 0.17

avgvalue

” FromBondi.10

For binary^blendsoflinear polymers under the^mean- fieldapproximation,eq³hasbeen givenby^(seerefs6and

8) Temperature(*C)

R(q) ⁼ Dcq2

(x^- x8) Rp2q2\

36

/

⁽⁴⁾

where istheFlory-Hugginsinteraction parameter, 9^is theinteraction parameter at the spinodal temperature, andR0isthe unperturbed chain dimension. Equation⁴ suggestsplotsof_R(q)/q2versus _q2inorder to determine the apparentdiffusion _{constant,Dapp,}

D.„

⁼

D.(T)(^)

⁽⁵⁾

and (DcRp2/36). Asis indicated byeq 5,_Dappisequalto

zero at the spinodaltemperature. The^valueofqfor which R(q)^{- 0 is}termedthecritical_q(qc). Composition fluctu- ationswithwavenumberslarger than thisqcvaluewillnot decomposeintophase-separatedstructures. Equation⁴ also predicts ^{a balance between} transport ^{and thermo-} dynamic^effectswhichleads toamaximum_{in R(q)}at“qm”.

“qm” is relatedto_“qc”by

q ²

//2

⁽⁶⁾

“qm” istheqvalueatthe

intitial

_stagesforthe scattering

“halo”from spinodal decompositionofblends discussed inaprevious paper.9

For blends displaying^acomposition-dependentinter- action parameter,g(<£), eq ^{4 can} be approximated by

R(q) ⁼ Dcq2

(

^(g~g9)Ss

Rp2q2\

36 / ⁽⁷⁾

and_eq 5 by

= 18)

asdiscussed in the appendix.

Predictions

_Concerning

Kinetics

of Phase Separation

from F-H-S Analysis of

Cloud-Point Curves

Using^{the values}for_o,b0,bx,andc(Table I) determined from cloud-point^curves inthe

first

_paperofthis series,1

it ^is _{possibleto predict}g (composition-dependent -pa- rameter)over arangeoftemperatures. ^Oneuse forsuch predictedg values^isinthe estimationofthegrowth^rate for _phasesin the immiscible regime using eqs ⁷ and 8.

Since the behavior of_{g is} predicted from data ^at the miscibility limit (for the ^case of cloud-point data), ^a successfulpredictionintothe immiscible_regimecouldbe considereda testoftheformalismsused in deriving the g function.

As discussed above,the thermodynamicdriving^force for _phase separation in the immiscible regime ^{can be}

Figure ^1. Predicted behaviorfor (g^- g,)/gsfromtheF-H-S analysis. This is the thermodynamic driving^force for _phase separation^abovethe coexistencecurve. Isotactic PVMEshows lowermiscibility^andmuch slowerkineticsabovethemiscibility limit (g^- gs)/gs⁼ 0.

approximated^as (appendix)6-8 DWP ^_ (g~g9)

DC(T) ^g9

W^econsiderblendsof critical composition (=75^%PVME by weight)^{so as} toavoidthe metastable binodal_region) and predict the thermodynamic driving^forcefor_phase separation for the isotactic and atactic blends (Figure^1).

In this _{analysis gs} was calculated _using the

F-H-S

parameters of Table

I

in _eq 1 and the cloud-point temperature^fora75%PVMEblend under the assumption that this is the critical composition ^and the critical temperature (Tcp,i80 ⁼ 109 °C, TcPiatactiC⁼ 121 °C). (g^- g8)/g9valuescalculatedin this_way^are much smaller for theisotactic blendforsimilarquenches abovethecritical temperature. The^rateofphaseseparation for thei-PVME blendis predicted to^be drastically^slowerthan the rate forthe a-PVME/PS blend(DC(T) isexpectedtohavevery similarvaluesfor thedifferenttacticitiessincethey^have essentially identical glass transition temperatures ^and similar molecular _weights). The spinodal interaction parameter^canalsobecalculatedfrom thesecondderivative ofthefree energywhich yields^acomparable valuefor_gs.

The experimentally^measuredcritical_pointwasused since

it was felt thatthiswas amore direct value.

Experimental

^Section

IsotacticPVME,i89(Mw⁼ 89 000, MN⁼ 49 100,MZ^{= 144}900), was prepared using cationicpolymerizationfollowed by solvent fractionationaspreviouslydescribed.9 Atactic PVME,a99(Mw

= 99 000,Mn ⁼ 46 500, M2 ^{= 151}300), ^was purchased from ScientificPolymer Products Inc. and fractionated.9 Thetriad tacticities ofthe two polymers ^were determined using proton NMR (a99, 31% isotactic and 69% heterotactic; i89, 55%

isotactic, 40% heterotactic, and ^5% syndiotactic). The glass transition temperature for the two polymers ^was essentially identical, -29 °C, in accord with the findings of ^{Karasz and} MacKnight11forothermonosubstituted vinylpolymers. Nom- inallymonodispersePS,Mw/Mn^{= 1.03,}ofmolecular weight Mw

= 120000 (Polymer Laboratories, Amherst,MA)^was blended witheachofthe twoPVME samples. Molecular_weightsof all polymers ^are reported^as measured by GPC in termsof_poly- styrene standards. A Mark-Houwink analysisoftheisotactic and atactic PVME indicated that this leads to errors in the molecular weightoflessthan5%, which^iswithinthe accuracy ofthe measuredvalues.

Blendsof PVME/PS^{were cast}fromtoluene solutions at30

°Candallowed to air dry. The films ^were further dried ina vacuum oven at70°Cforaweekandfinally^{ina vacuum oven}

at 100 °Cforat least6h before scatteringmeasurements were

Macromolecules, 26, Tacticity^Effects Separation

0 123456789

q <*"')

Figure^{2. Series}of_lightscattering measurementsforisotactic PVMEblendedwithPS (75% PVME) at130°C. (Thecritical temperature^{is 109}°C.) The patterns^are separatedby about¹ minandmonotonically^increasein intensitywithtime.

made. Kineticmeasurementswere madeonthinfilms_generally closeto 10µ thickheld betweentwo_glasscover slips. These thinsamples^arenecessaryin order toreducemultiplescattering.

Aone-dimensional solid-state detector(OMAImade byPrinceton Instruments)coupledwithaPCwas usedto perform the kinetics studies. A screen was used to produce ^a real image of the scatteringpattern,^anda macrozoom lenswas attachedtothe detector. The_detector,lens,andscreen were tiltedatan angle of30°fromthemainbeam. Thehot_{stageand sample}were set atanangleofabout25°inorder tocircumventinternal reflection fromthecover slip/airinterface. Usingthis_{apparatus,angles} from0°to75°could easily^beobserved. Adjustments^forasmaller angular range could^bemadein minuteswiththis_setup. The kineticsmeasurementsreported in thispaper ^were performed

on 75%byweight PVMEblends parepared^asdescribedabove.

Thiscomposition^hasbeendeterminedtobecloseto thecritical compositionforthese blends; thus,onlyspinodal decomposition isexpectedin _{the earlystages.} (Off-criticalcompositionswill

phaseseparatevia spinodal decomposition only indeepquenches).

Samplesforthekineticsmeasurements^were equilibrated at about 100 °C on a hot_stage near the apparatus and quickly transferred to the apparatus which was at the experimental temperature. Due to^thesmallmass ofthecover slips and samples, thermal equilibrium^{was reached}inmuchlessthanaminuteas measuredwithathermocouple attached to the^cover slip.

Kinetics

of Phase Separation

From the predictionsof_Figure ¹ we can make three qualitativestatements concerning the relative kinetics for phaseseparationinisotacticandatacticblendsof PVME/

PS. First, the driving ^force for _{phase separation} is

dramatically ^smallerforisotacticPVME inblendswith PS in comparison with atactic PVME, second, the de- pendenceof^thegrowth rate^on quenchdepth fori-PVME

is very ^weak (the _slope of(g ^- gs)/g6 in temperature ^is much smallerthanthe_slopeforatacticPVME),^andfinally the miscibility limit ^{is lower}forisotacticPVME blends (a direct result ofthe cloud-point measurement at Tc).

Figures²and3showaseriesof lightscatteringpatterns (reducedintensity^versus q)for isotacticandatacticPVME blended with polystyrene (75% by weight PVME). In both cases the intensity monotonically ^{increases as a} function of_{time, the} curves beingseparatedby about ¹ min inbothcases. The cloud points for theblends shown in_Figures2and 3occur at about109°Cforthe isotactic blend and at about 112 °C for the atactic blend. The patternsfor_Figure2(isotactic)were measuredat130°C (aquenchdepthof²¹°C). Figure^{3was measured}at123

°C (aquenchdepthof2 °C). Curvesinboth figures^are separatedby^about1min. Sincethekineticdatain_Figures

2 and 3 show comparable ^rates of_{phase separation} for drasticallydifferent_{quenches. They}dramatically^show

atacticity^{effecton the}phase-separationkinetics. When

0123456789

q(*"')

Figure^{3. Seriesof}lightscattering measurements for atactic PVMEblendedwithPS (75% PVME)at 123°C. (Thecritical temperature^is121°C.) Thepatterns^are separatedby about¹ minandmonotonically^increasein intensity with^time.

Time (min)

Figure^4. <7muversus timeforisotactic andatactic PVME/PS blendsofabout4°C quench.

Figure^5. Dependenceofthemaximum in_scatteringintensity

on timeforisotactic andatactic PVME/PSblends.

comparisons^are madebetweendifferent_quenchesofthe isotacticblends,only small changesinthe rate ofphase separation^{are observed.} Thus, thesmall_changesin_phase separation ratewith_quenchdepth^andtherelatively^slower rateof_{phaseseparation}forthe isotacticblendspredicted by the

F-H-S

_analysisare qualitatively^confirmed.

Plots of the q value for the maximum in scattering intensity,qmai,^versus timeandthecorresponding reduced intensity7max versus timeare givenin_Figures4and5for the isotacticandatacticblendsataquenchdepth of^about 4°C(120and126°C,respectively). Atthis_quenchdepth theisotactic materialshowsaregionofconstant_qmaxfor about 10 min, followed by movement ofqmazto smaller angles. The atactic materialshowslittle if _{any region}of constant<?mu,and_qmaxrapidlydecays tovery low^values.

Thus,for similar_quenchdepths the isotacticPVME/PS

Time (min)

Figure^{6. InI versus time fora 16}°C quenchoftheisotactic blend assuggested byeq^1.

Figure^7. R(q) ^vs qforthe isotactic blendof_Figure 5.

blenddisplays slowedkineticswhen comparedwithatactic PVME/PS ^{blends as} predicted by the

F-H-S

_analysis

discussed in the companionpaper.1

In_Figure5the isotactic blendshowsa large regionof linear growth^followedby^anonlinear_region(intermediate stagesof_phaseseparation). The atactic material^on the

same timescalerapidlydecomposesintoaphase-separated material. Clearly, the growth in volumeofthe_{phases is} much slower in the isotactic blend as predicted by the

F-H-S

analysis. Inmeasurements at equal quench depths thekineticsare effected by the temperaturedependence of Dc. This effect would increase the rate of _phase separation(throughDC(T))forthe isotactic blendsince

it

ismeasuredatahigher temperature. The^observedrate is,however, slower than that ofthe atacticPVME.

In_Figure6,

In/is

_{plottedagainst}timefora16°C quench oftheisotactic blend,^assuggestedbyeq^2. Aseriesof_q valuesare plotted.

At

_earlystages(t^{< 10}min)^alinear_regimeisobserved for mostofthe _{q range.} This correspondsto the early stagesofphaseseparation fortheisotactic blend. Devi- ation fromlinearCahn-Hilliardbehavioroccurs atlater times. The _slopes of the linear region ^of Figure ^{6 are} plotted^versus qin_Figure7. Amaximum in_R(q) occurs as indicated byCahn-Hilliard theroy.

Asnoted_above,eq 4 suggestsplotsof_R(q)/q2versus q2.

Figure^{8shows sucha} plot for^{the data}of_Figures6 and 7. A linear_regionisobserved, and^avaluefor theapparent diffusion constant of7.6 X 10~3/¿m2/s is obtained. For thiscase qc⁼ 15 Mm-1andavalueofabout_qm ⁼ _{10 µ}nr1 is calculated from _eq 6. This value is higher than ^the valueindicated in Figure⁷ (about⁷

µ 1).

^Thismay^be accountedfor _bythe distance ofextrapolation from the measured values used in determining ^{qc (an} order of magnitude) in the ^analysis of_R(q)/q2versus q2 plots.

Valuesfor£)app versus temperature^are plotted in Figure 9. An extrapolation ^to Z)app ⁼ 0 is made in order to

Figure^8. R(.q)/q2^versus q2forthe isotactic blendsof_Figures

5and6.

Figure^9. D,pp^versus quenchtemperature for⁷⁵wt%isotactic and heterotactic PVME/PS blends for extrapolation to ^the spinodalpoint^atDapp⁼ 0. LinesareF-H-Sfits_{using(~KcT)((g} -g»)/g«)(withKc^astheonlyfit_parameter)fortheisotactic and atacticblends.

TableII

SpinodalTemperatures for Isotactic andHeterotactic PVME/PS Blends of 75wt % PVMEAs Measured by

Dt„

Extrapolation,QuenchMeasurementsfor Thin Films,^and Cloud-PointMeasurementsforThick Films

isotacticblend atacticblend

T%CD«PP),°c 113 119

Tmm(thinfilms),^{°C 115 121}

Tcp(thick films),°C 109 121

determine the spinodal temperature (eq 5)whosevalues are given^as Tsin TableII. Three spinodal temperatures

are indicated in Table

II

forthe75%PVMEblends. Values obtained by the^aboveextrapolation differed by about²

°Cfromthose measured usingthe cloud-point technique.

Tmin (thin films)^is the minimumtemperature at which phase separation could ^be induced by the quenching technique ^used in thekineticsmeasurements. Thedif- ference betweenthelatterandthe cloud-point^measure- ment may ^be due to the difference in film _thickness, indicated in the Experimental Section,1·5’9 ^as has been noted by^CohenandReich.12 The smalldriving^forcefor phaseseparation in the isotacticblends discussed above may^beshifted by^{surfaceeffects}toalarger^extent inthe isotactic_blends,leading^toalarger discrepancyintheTm¡n

and _Tcpvalues in TableII.

Correlation

between

Flory-Huggins-Staverman Analysis

^and Kinetics Measurements k

In the first _paper of this series1 F-H-S _analysis was

usedtodetermineafunctional form for the composition- and temperature-dependent interaction parameter, “g”, from cloud-pointmeasurements. Thesefunctionalforms for “g” ^{were used to} predict the thermodynamicdriving forcefor_phaseseparation in the immiscibleregime, (g^- gs)/gs.

Macromolecules, 26,No. 7, Tacticity^{Effects the Rate} Separation From the discussion given above there is qualitative

agreement betweenthe predictionsofthe

F-H-S

_analysis1

andthemeasuredkineticsdata (Figures¹and9). In_Figure

9, ^a

fit

ofthe

F-H-S

functionalform for _(g^- _ga)/gahas beenapplied^to the measured _Z)app data. It ^{is assumed} that the translational diffusion constant has a linear temperaturedependenceinthe20°C_rangestudied,^such that

D_app ⁼ _DC(T)(g~ga)

ga

= KCT(g-g>)

g* (9)

The

F-H-S

_{equations for “g”} ^are used to _{generate (g}^- gs)/gs,andKc isusedto

fit

thekineticsdata.

It

isexpected that tacticity^wouldhave

little

effect^on thetranslational diffusion constant DC(T) since the two PVME’s have identical _glasstransition temperatures ^and similarmo-

lecular weights. The values for Dc(125 °C), i.e., KCT, obtained from the fit ofFigure^{9 are}

Dcieo(125°C) ⁼ -0.22gm2/s

Sc,atact¡c(125°C) ⁼ -0.24Mm2/s

These valuesare closeto other experimentally determined valuessuchasthose ofNishi, Wang,andKwei13forthe PS/PVMEblend (determinedusingNMR, Dc⁼ -0.28).

The agreement between the two Dc values, and the agreement with literature values, supports the

F-H-S

analysis ofthe CPC reported in the companion paper.1 Thus, ^{we have} quantitatively ^shown that the

F-H-S

analysis of cloud-point^{curves at the} miscibility

limit

is useful inpredictingphase-separationkinetics datainthe immiscible_regimefor_earlystagesof_phaseseparation.In the nextsection

F-H-S

functionalforms for “g” will ^be usedto perform^a preliminaryanalysisofintermediate- stage phaseseparation (i.e., thenonlinear_regimeof_Figures

5 and 6).

Intermediate

_Stages

of

Phase Separation

TheCahn-Hilliardanalysis(presented in the previous sections)^was basedon theinitial_stagesof_{phaseseparation} inwhichlinear_regionsof^InI^versus time_plotsareobserved.

This_regimeisdescribedtosome degreeofsatisfaction by linearCahn-Hilliard_theory. Ascan beseen from Figure 6this _regime (about 15 min in _Figure6)accounts fora

very smallportion of^thedata, andfor_{largerquenches}in blendswithsizable valuesofZ)appthis_{regionmay not} be observedatall. Many polymer blendsystems have been reportedintheliteraturefor whichalinear_regimecannot be measured,14and generally^alinear_regimeisnotobserved for metalurgical systems. Therefore, it ^is desirable to understand, at^leastinan empiricalmanner,therich data which follows the

initial

linear_region.

It

shouldbenoted thatthe behavior whichis observedin_Figure6 is

fairly

universal, occurring^over the_quenchdepth spectrumfor both isotactic and atacticPVME of criticalcomposition

as well ^as for other blend _systems reported in the literature.6·14 A discussion ofa novel approach ^to the analysis ofintermediate-stage phase separation ^is pre- sented below in which the variation of _(g ^- _g8)/g„ with composition (determined using theF-H-Sparameters)^is usedto fit lightscattering data from intermediatestages

of_phaseseparation at^asinglebulkcomposition. A similar approach^was suggestedbyLanger18with_regardtometallic alloyswhich showsimilarbehavior (Figure ^2,ref 18).A briefdiscussionof preliminary^resultswill^bepresented.

Afullpresentationoftheresultsofthis analysis^asapplied

to the isotactic/atactic PVME/PS blends and other polymer blends

will

bepresentedin the future.

For a molecularly miscible binary blend the bulk composition^describesthe composition atallsize scalesof interest. Similarly, ^{atthe earliest}stagesof_phasesepa- ration for a blend of critical composition the bulk composition^{can beused}to ascertainthe thermodynamic driving^forceforphaseseparation. Immediately following

aquenchintoatwo-phase regime,thebulkcomposition

no longer describesthe blend at certainsize scales. As phaseseparationproceedsthesize scaleofthecoexisting phasesgrows,resultinginadiscrepancy^betweenthebulk composition at larger ^and larger size scales. Thus, the phase-separationprocessinvolvestwo coexistingprocesses.

In a critical composition binary blend of A and B for instance,component A^isexcludedfromaphasepredom- inantlycomposedofBand component B^isexcludedfrom

a phase predominantly composed ofA. Eachof these phase-separation processes might ^be expected to be governedby^abalanceofthermodynamics^andtransport properties^asdescribedbyCahn-Hilliard_theory. In this

case the thermodynamic driving^forceis composedby^a weighted^sum oftwo terms,^one describing^eachofthese processes. Sincethe composition^isshifted towardamore

miscible_regimeasone leavesthecriticalcomposition, the averagethermodynamicdriving^forcefor_{phaseseparation} is diminished. (Reference shouldbe madeto _Langer,18 who discusses fine-grained and coarse-grained composi- tionsinmetallic_alloys.)

For the ^case of a binary polymer blend of critical compositionthe composition dependenceofthemiscibility

limit

is generally

flat

in the immediate

vicinity

^ofthe criticalpoint. Thus, ^one would_expectonly small shifts in the thermodynamic driving ^force with composition changesat theinitialstages. As_phaseseparationproceeds, changesintheaveragedepthofquench causedby shifts in composition of the coexisting phases reduce the thermodynamicdriving^force. Thisapproach when^cou- pledwithCahn-Hilliardtheory predicts^acollapsing halo in the lightscatteringpattern^asiscommonly^observed.

The collapsing halo^ispredicteddirectly^fromeqs^{4 or} 8 since,^as

(

^-

8)/ „

^or (g^- ga)/gebecomessmallerwitha

shiftin composition, the maximuminR(q) generatedby q2((g^- gB)/gB^~ fio2<?2/36)becomessmallerforafixedfio2.

Using this _approach, a functional form forg(T,4>) is ^a prerequisite for further analysis of intermediate-stage phaseseparation. The

F-H-S

functional formwillbeused in this _regard.

Recently,15 Hashimoto and co-workers have observed that _{the early} stages of _{spinodal phase separation} in polymer blends give way to an intermediate _stage of pseudospinodalphaseseparation characterizedby^aself- similar growthofcomposition fluctuations. Thatis,the size scale ofthe fluctuations _grow while the structure remains similarto that ofthe early stages of_spinodal decomposition, albeit^{on a}larger^sizescale(Figure11,top figures). This_maybeanindicationthatprocessesrelated to_spinodaldecomposition^areimportantbeyond the linear Cahn-Hilliard _regime.

Asshownschematically in Figure¹⁰thelinear theory

assumes that

(

^-

„)/ „

^or (g-ga)/ga^isconstantforafixed temperature of_quenchin the equation

R(q) ⁼ Dcq2

(

^(g~ga)ga

RoV)

36

/

⁽⁷⁾

Sinceig^- ga)/ga(g isusedfor^acomposition-dependent

)

depends ^on composition, the linear theory effectively

assumes that composition ^{is constant} throughout ^the

I ^- lo exp(2 R(q) t)

R(q)⁼ Dc q‘2 <( - 8)/ ^-Ro‘2 q‘2/36)

Figure ^{10. Schematic of linear}Cahn-Hilliard _theory. Left:

schematic structure for spinodal decomposition (dark lines indicateregionsof composition fluctuations). Right:dependence of_R(q)on q.

m

^{= f(time) =} f(time)

(X^- )/ 8 ^{= f(Ts andc)}

c

TIME

B => R(t)^-0

C =>R(t)^» l/t

D => X⁼ Xs TIME

Figure ^11. Self-similarity in intermediate stages^on polymer phase separation. (Dark ^lines in the top figures indicate, schematically,regionsof composition fluctuations. Magnification remains constant in thetop three figures.)

phase-separation process. Experimentally,this_assump- tion_appearstobevalidover alimited timerange due to the flatnessofthe spinodal^{curve near} thecritical point (Figure^11,points marked“A”)^{and dueto}the small local changeincompositionofthe phase-separatingspeciesat theinitial _stagesofphase separation.

Initially

^adecreasing(g^- g8)/gsin_eq4yields^asmaller value for_R(q) andconsequently^adecreasing slopeof^In I⁼ In7o+ 2R(q) t (eq2). At^some point

(

^- 8)/ 8^becomes equal to(f?o2q2)/36(eq4)and the_slope

ofln(I)

^versus time (eq2) goesto zero (B in Figure^11). Beyondthistime in the_phaseseparation, theslope (2R(q)) becomesnegative (since(Ro292)/36 islarger than(g^- gs)/gs). Thisisobserved in the data (Figure ^{6). As} timeprogressesfurther (C in Figure11),composition^becomesapproximately linear in time in^aregion of composition for which timethe spinodal temperature^islinearincomposition;thus,(g-g8)/g8,which dependsroughly^on the inverseofthespinodaltemperature foraconstant quench, becomeslinear in 1/timeandInI (=lnlo + 2R(q)t)^becomesindependentoftime. Atlater stages ^one would predict ^a linear decrease of intensity with time as (g ^- gs)/gs goes to ^zero. Actually, ^a slight increaseinintensity^isobserved at verylong time ^scales in the blends studied here (D in Figure 11), indicating thatthe pseudospinodal model^isno longerappropriate.

Thefunctionalform for “g”^obtainedusingtheF-H-S approach^{was used to} estimate^adecreasing(g-ga)/gswith

Figure ^12. Typicalfit ofthe scattering data using thepseu- dospinodal approach and theF-H-S_equationfor_(g^- _g,)/g,(eq 5.24). (123.5°C, heterotacticPVME/PS,q ⁼ 1.84µ 1.) timeof_quench. Inorder tosimplify^theanalysis,only^one of the two coexisting phase compositions ^{was used} to generate(g^- gs)/gs. Thisisthe equivalentof_assuming^a symmetricphasediagram about¿critical· “g”^valuesfrom the polystyrene-richphase^were usedinthe analysis. Figure 12isatypical fit of^thedatausingthisapproach andthe

F-H-S

equation for (g ^- gB)/ga (eq 1). An exponential decrementin composition

(

^{versus time in}Figure ¹¹⁾

was usedto modelthebehaviorshown schematicallyin Figure11,yielding^twounknownfittingparameters.. These parameters, however,^{can be}determinedfrom the dataif itisassumedthatthecompositionchangewithtime^isnot dependent^on the sizescale ofmeasurement (i.e.,the _q value) within the _range ofq values measured. (In this

case the previous assumption yields⁵⁰⁰plots (for^a500- channeldetector), similar to Figure 12,foreach quench.

Ineachoftheseplots^thesetwo_parametersmustremain constant.) The spinodal temperatures^andthea,bo, bi, andcvalues ascertainedin the

F-H-S

_analysiswere used to generate(g^- g8)/g8andR(q) ^data. Frompreliminary results _using this approach, values of_{R(q) have} been determinedforisotacticPVME/PSblendswhichshowa

significant linearCahn-Hilliard_regime. R(q)u™..values generally^{are close} to the R(q)values determined _using this_technique. ValuesofRo(the unperturbed radiusof gyrationforan averagepolymer chain in the blend) may beobtainedinthis_{way (one}ofthetwofittingparameters).

Generally the radiusof_gyrationwas in the 100-Á range which would_agreewithan average chainofthePVME/

PS polymers.

This _approach to analyzing the scattering datayield fitsto the data^{over a}wide_rangeofqandtime. Although notusedin the_{presentpaper due}to time constraints(each quench generated 5001^versus qdatafileseachwith time dependenciessimilar to Figure 11),it isbelievedtobea

very promising approach which can result in a Cahn- Hilliard_typeanalysisofscattering data^on systemswhich showaverysmall(or no) linear_regimein the In/ versus

time data. Simplified functional ^forms for _g{T,$) are

adequatefor this_analysis. For thequenches examined, preliminary^valuesforZ?appdidnot appear tosignificantly varyfrom thelinearCahn-Hilliardvaluesfor_quenchesin which the linearregime ^was well-defined. The use of^a much broader_rangeofqvalues andthelossofthelinear assumption^are hoped toyield^more accuratefitsand values for_Dapp. This_maybeparticularly^{useful inblendswhich} showonly^alimitedlinearCahn-Hilliard_regime. It would appearthatthisapproach tointermediatestagesof_phase separation, with minor modifications, ^is applicable ^to nonmacromolecularblends. Amore detailed presentation oftheresultsofthis analysisintheisotactic/atacticPVME/

Macromolecules, Tacticity ^{the Rate} Separation PSblendsandapplication to other polymer^blendswillbe

presented in the future.

Conclusions

The Flory-Huggins-Staverman theory using cloud- point ^{data at the} miscibility limit predicts drastically slowed phase-separationkineticsintheimmiscible_regime forthe isotacticPVMEblendswithPS. Quantitatively, measurementsofthe phase-separationkinetics surpris- inglysupport the prediction ofthe

F-H-S

analysis. As

a test of thisprediction the translational diffusion ^con- stants,Dc,ofthetwotacticities of PVMEwere calculated from fits ofthe apparentdiffusion constant versus the quenchdepth. The^valueofDccalculatedin this_wayfor isotactic^and atactic PVME/PS^{blends was} very similar and _agreed with literature values obtained _using an

unrelated technique.

(It

^isexpectedthatthesetwovalues should_agreefordifferent tacticitymonosubstitutedvinyl polymersofsimilar molecular weight which^donotshow a shift in _Tg.) Thus, ^a quantitative ^assessment of ^the agreement between the theoretical prediction ^and the measured kineticswas upheld.

An_analysisofintermediate_stagesofphaseseparation takingadvantageof^the

F-H-S

_predicted dependencies of the interaction parameter ^on composition ^and tem- perature and using ^a modified Cahn-Hilliard _approach

was presented. Scattering data^{over a}wide rangeofqand time (well beyond the linearregime)^{can be}

fit

_usingthis approach. Preliminary ^results qualitatively agree with the results of a linear Cahn-Hilliard _analysis. A

full

presentationofthe application of thistechnique to the PVME/PS ^{blends and} other blend _systems will be presentedin thefuture. This_analysistechnique may^be very usefulforblends whichshow only ^a limited linear Cahn-Hilliard_regime. ThemodifiedCahn-Hilliard_ap- proach appears to ^be applicable ^to nonmacromolecular blends.

given by

[ µ '/µ RT

^-

µ '/µ^ ]

⁼

S(AG'/NRT)/

(where

µ '/µ

^RTand

µ µ RT

^are analogousto _eq 10 and11ofthefirst_paperinthisseries1withtheaddition ofagradientenergytermandAG'/NRT^isgiven above).

The chemicalpotential^differenceis givenby 5(AG'/NRT) ^In

l ^In 2

0(p-y 77lj 7712 771-2

2 [~3^ 2( ^

^(A2)

The continuityequation

,

+ divJ⁼ 0 (A3)

isusedtorelate the chemicalpotentialdifference to the growth rate of composition fluctuations through the relation

J

⁼ -AV

'

µ/

µ2' I

,µ^ µ^ß ^{( 4)}

where is an OnsagercoefficientandJisthe localflux ofspecies ^1.

Equations^{A2 and}A4are usedto derive

J

_{(eq 3.4}ofref 8)

Using this _expression in _eq A3 and taking the Fourier transform using the derivative rule,^{we have}

Appendix:

Development of

Equation

⁸

Derivations_concerningnonequilibrium thermodynam- icsofpolymer blendsintheimmiscible_regimeusingthe Flory-Hugginstheory^were

first

_{presentedby Van}Aart-

sen.16 VanAartsen_developedan expressionfor the_phase sizeofmaximum growth rate during the early stagesof spinodal decomposition, ^{de Gennes8}andPincus17devel- oped^an expressiondescribing thegrowth rateR(q) using Flory-Hugginstheory. ^Referenceshouldalsobemadeto the workof_Langer18 inmetal_alloyswhichdiscussesthe issue ofcoarse-grain composition in ^a phase-separating system and his generalworkon first-order_phasetransitions whichhe hasrecentlyreviewed.19 Thederivationofeq 8

followsthe developmentofdeGennes8andHashimoto.6 The free enthalpy in the spinodal regime ^has been described by extending the Flory-Huggins equation to includeaterm _describingspatial variationsin composition.

Similarly, for ^a composition-dependent interaction pa- rameter,1g, and ignoringpolydispersity,^{we have}

AG' NRT

\

02

= ^— In0i ^{+ — In}02+ £0102 +

771^ 7712 360^02(V0,)2

(Al)

where“a” istheKuhn statistical_segmentlength,0, isthe volume fraction of component i, m, is the number of occupiedlattice sites formacromolecule i, ^andN is the total numberof latticesitesin moles.

The chemical potential ^{difference between the two} components of ^an early-stage nonhomogeneous binary system under the condition that the coarse-grain ^com- position ^{can be} represented by ^a single composition ^is

^=Wl',A¡[^+¿¡"2e+a"2*',í+

^]s,-36^'|lA

The growth rate inthe spinodalregime ^is givenby

R(q) ^{= - — = 1 5(50q)q - «2}

t-

.

^{6t = ~Q A}

f-^r-

⁺

^--2g

⁺

[

^{1202 m202}

(1-2#l)^

⁺

*l#!Ai

⁺

_36#A,

^{° q2 (A7)}

The interaction parameter ^at the spinodal point ^is defined by¿2(G/IVI?T)/50i2 ^{= 0} which yields

<A8>

Thus,eqA7 can be written as

R(q) ⁼ q2A(q)

((2g

- 2gs)^- (A9)

Under the assumptionthatthe entanglement properties andfriction coefficientsare thesame forthetwo “neat”

components andforthe_componentsin^ablend (i.e.,g ⁼ 0 but

6g/

^and 52g/ó0i2 ^are not necessarily zero), the Onsager coefficient at the small _q limit _(long diffusion