Comparison of spheresize distributions obtained from rheology andtransmission electron microscopy in PMMA/PS blendsChr. Friedrich, W. Gleinser, E. Korat, D. Maier, and J. Weese Citation: J. Rheol. 39, 1411 (1995); doi: 10.1122/1.550720 View online: http://dx.doi.org/10.1122/1.550720 View Table of Contents: http://www.journalofrheology.org/resource/1/JORHD2/v39/i6 Published by the The Society of Rheology Related ArticlesLetter to the Editor: Comments on “Thermodynamic Admissibility of the Reptation Model” [J. Rheol. 48, 53(2004)] J. Rheol. 48, 705 (2004) Computer simulation of the film blowing process incorporating crystallization and viscoelasticity J. Rheol. 48, 525 (2004) Constitutive equation to describe the nonlinear elastic response of aqueous foams and concentrated emulsions J. Rheol. 48, 679 (2004) Prediction of bubble growth and size distribution in polymer foaming based on a new heterogeneous nucleationmodel J. Rheol. 48, 439 (2004) Theory for drop deformation in viscoelastic systems J. Rheol. 48, 417 (2004) Additional information on J. Rheol.Journal Homepage: http://www.journalofrheology.org/ Journal Information: http://www.journalofrheology.org/about Top downloads: http://www.journalofrheology.org/most_downloaded Information for Authors: http://www.journalofrheology.org/author_information

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Comparison of sphere-size distributions obtainedfrom rheology and transmission electron microscopy

in PMMAlPS blends

Chr. Friedrich.V W. Gleinser, E. Korat, D. Maier, and J. Weese

Freiburg Materials Research Center (FMF) and Institut furMakromolekulare Chemie, Hermann-Staudinger-Haus, Albert-Ludwigs­

Universitiit, Stefan-Meier-Straf3e 21 and 31, D-79104 Freiburg,Germany

(Received 5 May 1995; accepted 24 July 1995)

Synopsis

Often, blends of two immiscible polymers have a morphology with one component building amatrix in which spherical inclusions of the other component are embedded. The rheologicalresponse of such blends contains an elastic contribution which can be attributed to the formrelaxation of the inclusions. This process has a characteristic relaxation time which is proportionalto the radius of the inclusions divided by the interfacial tension between the blends' components.Thus a distribution of radii leads to a distribution of relaxation times. It is shown that rheologicaldata together with an emulsion model can be used to determine the volume weighted sphere-sizedistribution up to a scaling depending on the interfacial tension. The procedure is applied to data offour PMMAJPS blends and the results are compared with the corresponding distributions obtainedfrom transmission electron microscopy (TEM). If the concentration of the spherical inclusions issmall, both results are in excellent agreement. For larger concentrations, deviations between theresults from rheology and TEM are observed. © 1995 Society of Rheology.

I. INTRODUCTION

One possibility to design new polymeric materials with advanced properties is theblending of commercially available polymers. Most of the common polymers are incom­patible, so that they form multiphase systems. In the case of two components, the minorphase is usually dispersed in form of spherical inclusions of different sizes in the majorphase. The processing and ultimate properties of the blends depend on the size distribu­tion of the inclusions and the interfacial tension between the components. Therefore, it isof great interest to get model predictions for the rheological behavior and methods for thecharacterization of the morphology.

The starting point for models describing the rheological behavior of such polymerblends are emulsion models as presented, for example, by Oldroyd (l953, 1955) and Choiand Schowalter (1975). These models deal with emulsions and suspensions of two New­tonian ftuids and predict an elastic contribution in the rheological response which can beattributed to the form relaxation of the inclusions. The characteristic time of this processdepends on the radii of the inclusions and on the interfacial properties between theblends' components.

a)Corresponding author.

© 1995 by The Society of Rheology. Inc.J. Rheol. 39(6), November/December 1995 0l48-6055/95/39(6)/1411/15f$7.00 1411

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1412 FRIEDRICH ET AL.

The influence of the form relaxation can also be observed in polymer blends. Emul­sion models describing the rheological behavior of such a blend must, however, accountfor the viscoelastic properties of the components. In 1990, Palieme published such amodel which is the most widely used rheological model of polymer blends with a matrixand spherical inclusions since its appearance. This model leads to an equation relating thelinear viscoelastic material functions of the blend and its components to the interfacialproperties and the sphere-size distribution of the inclusions.

Given results from rheology and information about the size of the inclusions whichcan, for example, be obtained from microscopy, the rheological models can be verifiedand, additionally, the interfacial tension between the blends' components can be deter­mined. This has been shown by experimental studies of Graebling and Muller (1990) andGramespacher and Meissner (1992). Gramespacher and Meissner used an extension ofthe model of Choi and Schowalter in combination with results from light microscopywithin their study. The study of Greabling et at. (1993) is based on the model of Palieme.Though the influence of a polydisperse sphere-size distribution is discussed by them, theydo not make an attempt to compare experimental results for the entire sphere-size distri­bution with rheological data. All investigations published on this topic up to now (see,e.g., the papers of Muller's group and Grarnespacher and Meissner) deal only with amean value of the radii and do not take into account the entire sphere-size distribution.

In this contribution it is shown how the Palieme model can be used to determine thesphere-size distribution of an incompatible polymer blend from rheological data, and it isstudied to what extent the sphere-size distribution derived from rheology is in agreementwith the corresponding distribution obtained from transmission electron microscopy(TEM). For the investigations presented, PMMAIPS blends with 2, 5, 10, and 20 wt % PShave been used.

In the following section, the determination of the sphere-size distribution from rheol­ogy using the model of Palieme is described. As TEM does not lead to direct informationabout the sphere-size distribution [see, e.g., Gleinser et al. (l994a and b)], the reconstruc­tion of this distribution from TEM data is also described in Sec. II. Section III containsthe experimental details concerning the materials, the rheological measurements, and thecharacterization of the morphology with TEM. In Sec. IV the results are presented whichare discussed in Sec. V. Section VI contains the conclusions derived from our investiga­tions.

II. THEORY

A. Estimation of sphere-size distributions using the model of Palierne

The model of Palieme (1990) is intended for describing the rheological behavior of adispersion of viscoelastic spherical inclusions in a viscoelastic matrix. Under the assump­tion that the deformation of the inclusions remains small, Palieme derived the linearviscoelastic properties of such a blend. The model accounts for the fact that the spheresmay have different radii, and uses an electrodynarnical analogy in order to take interac­tions between different inclusions into account. If the interfacial tension a is the soleparameter describing the interfacial properties between the components, the result for thecomplex shear modulus is given by

with

1+3'2'i'/)#(w,R;la)

G:(w) = G~(w) 1-22.icPiH(w,R;la)'(I)

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RHEOLOGY AND STRUCTURE OF BLENDS 1413

4a1Ri[2G:(w)+ 5Gj(w)]+[Gj(w)-G~(w)][16G:(w)+19Gj(w)]H(wR-Ia) = (2)

• I 4a1Rj[G:(w)+Gj(w)]+[2Gj(w)+3G:(w)][16G~(w)+ 19Gd'(w)]'

where G~ (w). Gd(w), and G: (w) are the complex shear moduli of the matrix, thedispersed phase. and the blend. respectively. ¢i denotes the volume fraction of the inclu­sions with radius Ri . The entire volume fraction of the dispersed phase is ¢J = 2:¢Ji . Inthe case of a continuous distribution Eq. (1) must be slightly modified. Introducing thevolume weighted sphere-size distribution vCR) where vCR) dR is the volume fraction ofparticles with radii between Rand R +dR, Eq. (1) can be rewritten as

1+3J;H(w.RJa)v(R)dR

Gt(w) = G:(w) I -2J;H(w,RJa)v(R)dR' (3)

For the analysis of the rheological data two further modifications are necessary. First,RIa is substituted by R' and the scaled distribution

u(R') = av(aR') (4)

is introduced. in order to get an equation which does not depend on the interfacial tensiona. That means that the distribution vCR) can only be determined up to a scaling definedby Eq. (4). Second. the differential dR / is replaced by R' dOn R') in order to indicatethat the distribution u CR ') is calculated on a logarithmic scale. This modification ismotivated by the proportionality of the particle radius and the corresponding form relax­ation time in the relaxation-time spectrum which is usually calculated and plotted on alogarithmic scale. too. With both modifications Eq. (3) can be rewritten as

1+3J:Jf(w,R')u(R')R'dOn R')G*(w) = G*(w) ----==-----------,-

b m 1-2J':J[(w,R')u(R')R'dOnR')'

Equation (5) can be split into two nonlinear integral equations.

(1+3J':J[(w,R')u(R')R'dOn R'»)

G;(w) = K' (u)ew) = Re G~(w) 1-2j':J[(w.R')u(R')R'dOn R') ,

(I +3j':J[(w.R')u(R')R' dOn R

I »)G~(w) = K"(u)(w) = 1m G~(w) 1-2j':J[(w,R')u(R')R'd(ln R') ,

(5)

(6a)

(6b)

relating the distribution uCR') to the shear storage modulus G~(w) and the shear loss

modulus G~(w) of the blend. The integral operators K' and K" depend on the complex

shear moduli of the matrix and the inclusions, G~(w) and Gd(w), respectively. Thedetermination of the distribution u(R') from experimental data for the dynamic modulirequires the inversion of these integral equations. As this is an ill-posed problem. aregularization method must be used to perform the inversion [see Morozov (1984) for adefinition of ill-posed problems and Groetsch (1984). Bertero (1986), Louis (1989), andEngl (1993) for a review of the basic principles of regularization].

With Tikhonov regularization, an estimate for the distribution u(R') is obtained from.. 'rI'rI n tr tt a f' I

nOiSY experimental data G b.1 ... ·'Gb.n ' Gb,I •... 'Gb,n or Gb(Wl}, ... ,Gb(wn ) ./I /I . , I II "( h .Gb(wtl .... ,Gb(wn), With errors 0"\ .... ,an • a\ .... ,an for t e expenmental data a

relative error was assumed) by minimizing

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1414 FRIEDRICH ET AL.

1(C li fT II 2

(Til 2 b,i- K (u)(wi»,

nn1

V(A) = L ----ri:(C~~-K'(u)(w)2+ Li=\(Ti' i=\

-rxJ~eo (u"(R'»2 d(lnR'). (7)

In this equation the so-called regularization parameter A has been introduced. With anappropriate value for this parameter, the first term on the right-hand side of Eq. (7) forcesthe result to be compatible with the data. The second term leads to a smooth estimate forthe distribution u (R').

The calculations in the following sections have been performed with the programNLREG [Weese (1993)] which can also be used for the determination of relaxation-timespectra [Honerkamp and Weese (l993b); Weese and Friedrich (1994); Friedrich et at.(1995)]. Given noisy experimental data and the corresponding data errors, the programNLREG computes the approximate solution of nonlinear integral equations defined byTikhonov regularization. In addition NLREG offers several features such as the ability toestimate the optimal value of the regularization parameter with the SC method[Honerkamp and Weese (1990)]. This feature is very important, because it dependsmainly on this parameter, whether the result obtained by a regularization method is goodor not. Furthermore, the ability to calculate a positive solution of Eq. (6) was used.

B. Reconstruction of sphere-size distributions from TEM data

In order to study a polymer blend with TEM, thin slices of 50-200 nm have to beprepared, whereas typical particle radii range from about 50 nm up to several microme­ters. Obviously, only a two-dimensional structure can be observed with TEM, and imageanalysis of thin slices yields the radii of profiles of the particles rather than the radii of thespherical particles themselves.

A relation between the actual sphere-size distribution and the so-called profile sizedistribution can be derived under the following assumptions [Goldsmith (1967)]:

(1) The particles are distributed randomly and homogeneously.(2) The particles are opaque.(3) Overlapping particles can be neglected.(4) Effects due to rim of the slice can be neglected.

Denoting the number weighted sphere-size distribution by q(R)(f q(R)dR = 1), thecorresponding profile-size distribution by per), and introducing the average particle ra­dius

Rn = LeoRq(R)dR (8)

and the thickness d of the slice, this relation can be written as

d 2r Leo 1per) = --- q(r)+--- ~ q(R)dR.

d+2Rn d+2Rn 0 "R--r-(9)

The first term in this equation is caused by particles with the center inside the slice. Forthese particles, the true particle radii are obtained. The second term is due to particles cutby the slice, but with the center outside the slice. For these particles, radii smaller thanthe true radii are measured.

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RHEOLOGY AND STRUCTURE OF BLENDS 1415

TABLE I. Molar mass M w • polydispersity index M.JM n • and zero shear rate viscosity l'Jo at 190 °C of theblends' components.

PS PMMA

M w (kg/mol)MwlM n

l'Jo (Pa s) (Tref = 190 ·C)

1001.039450

311.19

92000

The sphere-size distribution can be reconstructed with a method published recently byGleinser et al. (I 994b). The first step of the reconstruction procedure is the constructionof a histogram from the measured profile radii. Then Eq. (9) is solved by regularizationusing the program FTIKREG [Weese (992)]. Finally, the resulting distribution is normal­ized.

For the investigations presented here, the reconstruction method is modified. First, theprofile radii are sampled in such a way that a histogram with logarithmically distributedbins is obtained. Second, the relation

vCR) = cR3q(R) (0)

with

(I 1)f/J

~R3q(R)dRc = -_

is used to replace the number weighted sphere-size distribution q(R) by the volumeweighted sphere-size distribution vCR) and the differential dR is changed into Rd(ln R):

(2)d vCr) 2r fro 1 vCR)

per) = ----~ +---- ~ d( In R).d+2Rn cr d+2Rn -co~ cR

This equation is inverted instead of Eq. (9). Third, the resulting distribution is normalizedin such a way that

f:v(R)dR = f/J, (3)

where f/J denotes the volume fraction of the inclusions. All modifications are necessarybecause the comparison with results from rheology requires the determination of thevolume weighted sphere-size distribution on a logarithmic scale instead of the numberweighted sphere-size distribution on a linear scale.

III. EXPERIMENT

A. Materials

As blend components PMMA and PS samples donated by BASF AG (Ludwigshafen,Germany) were used. Both polymers were synthesized by anionic polymerization. Themolecular weights were chosen in such a way that the influence of the form relaxation iscompletely in the measuring window of the rheometer. The molecular parameters and thezero shear rate viscosities are summarized in Table 1. It should be noted that the PMMAhas a tacticity of 80% syndiotactic triades (from I H-NMR spectrum) and a glass tem­perature of 125 -c.

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1416 FRIEDRICH ET AL.

The blends were prepared by common precipitation of a solution in tetrahydrofuran(10 wt %) into methanol. After drying for at least 3 days at 70°C in vacuum, the powderwas pressed at 180°C in vacuum to disks with a radius of 12.5 mm and a thickness of 1mm. These samples were used to estimate the rheological and morphological character­istics. The investigations presented in this paper refer to compositions of 2, 5, 10, and 20wt % PS in a PMMA matrix. The abbreviations for the blends are B02, B05, B10, andB20, respectively.

B. Rheology

The rheological measurements were performed on a Rheometries Mechanical Spec­trometer RMS-800 with parallel-plate geometry and a plate radius of 12.5 mm. Isother­mal frequency sweeps (100-0.1 rad/s) were recorded between 150 and 240°C in steps of10°C. Strain was kept below 0.3 to stay within the region of linear viscoelasticity. Allmeasurements were done under N2 atmosphere to avoid thermal decomposition of thesamples. Using the procedure of Honerkamp and Weese (l993a), the frequency sweeps ata given temperature were shifted to master curves referring to a temperature of 190°C.The shift factors showed WLF behavior with parameter values which are in the range ofthose for PMMA.

In this context it must be noted that the emulsion model of Palierne is not necessarilyin agreement with the time-temperature superposition principle. The situation is evenmore complicated if a temperature dependence of the interfacial tension a is taken intoaccount. With aa/aT = - I .25 x 10 - 5 N K - 1 m- 1 according to Wu (1970), the inter­facial tension's variation is about 0.6 mN/m within the temperature range the isothermalfrequency sweeps have been measured in. For that reason the time-temperature super­position principle may be violated and the isothermal frequency sweeps may haveslightly different shapes. On the other hand, the shift procedure led to excellent mastercurves for all blends. It can therefore be assumed that possible inaccuracies caused by aviolation of the time-temperature superposition principle are small.

C. Transmission electron microscopy (TEM)

After the rheological measurements the samples were taken out of the rheometerwithout destruction. Ultrathin cuts were obtained by a Leica Ultracut-E microtome witha diamond knife. The cuts were made in the center of the disks. The thickness of thesections was about 60 nm indicated by interference color. From these samples, TEMelastic bright field images were taken on a Zeiss CEM 902 (operated at 80 kV) withmonoenergetic electrons (ESI mode). A 90 mm objective diaphragm was used corre­sponding to a 17.3 mrad aperture in back focal plane. The morphology of the blends wasanalyzed using the image processing system IBAS 2.0 (firmware by Kontron).

IV. RESULTS

A. Rheology of components and blends

Figure 1 shows the dynamic moduli of the pure components PMMA and PS. Theexperimental data are indicated by symbols. Both samples show the typical behavior ofnarrow distributed polymers with G I ( w) - w2 and Gil (w) - w in the flow region[Ferry (1980)]. The solid lines represent the moduli obtained after recalculations from thecorresponding relaxation-time spectrum. This kind of interpolation allows the evaluationof the moduli at frequencies where the moduli of the blends are given. This is necessaryin connection with the computation of the volume weighted sphere-size distribution ac­cording to Sec. II A.

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RHEOLOGY AND STRUCTURE OF BLENDS 1417

7

~ 6 f 0_....rl' '" q

mQ:.

b5Ol.Q

4

(ij'3Q:.

'5,2.Q

. 1

o f

,( PMMA, I ! " .I d , " "I

-3 -2 -1 0 1 2 3 4

log (0 [radls]

7

(ij'6D-

bsl ~9 •• ,":

a 0_ 0

Ol.Q

4

~3m~

C> 2Ol

.Q

J L PSI , "I , "",,1 • " ,.1 "' ,I

-3 -2 -1 0 1 2 3 4

log (0 [radls1

FIG. 1. Shear storage modulus G I (w) and shear loss modulus G" (w) of the PMMA matrix and the dispersedPS phase. The data refer to a temperature of 190°C. The symbols mark the experimental data. The solid linesrepresent the moduli recalculated from the corresponding relaxation-time spectra. This kind of interpolation isnecessary for the estimation of the volume weighted sphere-size distribution from rheological data described inSec. II A.

In Fig. 2 the shear storage and loss moduli of the four blends B02, B05, B10, and B20are plotted. The experimental data are represented by symbols. The shoulder at smallrelaxation times can be attributed to the form relaxation of the inclusions. With increasingPS content this contribution is more pronounced. Such a behavior has also been reportedfor PMMAIPS blends by other authors such as Valenza et al. (1991), Wippler (1991),Gramespacher and Meissner (1992), and Greabling et al. (1993, 1994).

Given the data of both components (Fig. 1) and the blends (Fig. 2), the volumeweighted sphere-size distributions were estimated up to a scaling with the procedure

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1418 FRIEDRICH ET AL.

presented in Sec. II A. The result is shown in Fig. 3. It can be observed that, withincreasing the amount of the PS sample, the distribution becomes broader and is shiftedto larger radii. For each distribution the volume fraction was calculated according to Eqs.(4) and (13) and designated as <l>rheol' The resulting values are listed in Table II togetherwith the volume fraction used within the preparation of the blends. These values are ingood agreement with each other. Finally, the shear storage and loss moduli have beenrecalculated (Fig. 2, solid line). The fit is in excellent agreement with the experimentaldata, too. It should be noted that the range of the fit is restricted by the range in which themoduli of both components can be evaluated (Fig. 1, solid line).

7 E i I Iii i I iii I I 11"'3

~6

~

t9 5~

4

~ 3mQ:.t9 2~

"o •

B02

432o 1

log (j) [rad/s]

-1-2

o ! , , ",! , "Il'! t! ,,' "".t ! t, I , ",,' I '!,,' !

-3

7 Ii' I I I !3

'iV 6Q:.

t9 5~

4

~ 3tils,t9 2E

.~

BOS

432o 1

log co (rad/s I-1-2

a! I ,

-3

FlG. 2. Shear storage modulus G I (w) and shear loss modulus G"( w) of the blends B02, B05, B 10, and B20.The data refer to a temperature of 190 "C. The symbols mark the ellperimental data. The solid lines representthe moduli recalculated from the volume weighted sphere-size distributions in Fig. 3.

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RHEOLOGY AND STRUCTURE OF BLENDS 1419

7

lU 6 [ ___ 0 i

e:.b 5Cl..2

4

~ 3IIIeo.C> 2Cl..2 ~ ~

~

BI0

o t " .. ! , ! , ,,, II'! , , , I , 1 III ,I , '" ,,I , ,

-3 -2 -1 0 1 2 3 4

log m [rad/s]

7

~ el ___ Q itlI~

b 5Cl..2

4

_ 3III

~

b 2Cl.Q

1

o [B20

" , I ! """,1 , '''If,,1 , ",I . , '!II "II'!

-3 -2 -1 0 1 2 3 4

log eo [rad/sj

FIG. 2. (Continued.)

B. Morphology of blends by TEM

In Fig. 4 representative TEM images of the blends are shown. The dispersed phase(PS) appears dark. As a result of the cutting procedure the particles have ellipsoidalshapes. This was confirmed by rotating the cutting direction by 90°. Within digital imageanalysis, the ellipses were treated as circles with the same area. Altogether about 2000profiles were measured for each blend. In Fig. 5 the results are plotted in the form, ofhistograms on a logarithmic scale.

Starting with the histograms, the volume weighted sphere-size distributions were cal­culated. The distributions were normalized using the volume fractions from rheology(Table 11). These curves are shown in Fig, 6. With increasing PS content the distributions

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1420 FRIEDRICH ET AL.

1000 I , i 'I i

-2.00

-- B02...... B05

-- 610

620

'.~

V~ \

/-\ \\ .\.,,\ .

A

\ , ,,,"

-4.00 -3.00logR' =log(Rla) [m2/N]

"r'. /.: /

,//-5.00

..,./

v··--OLI_~~

-6.00

800

200

E 600Zii:::'5" 400

FIG. 3. Scaled volume weighted sphere-size distribution u(R') of the four PMMNPS blends obtained fromrheology.

become broader and are shifted to larger radii. It is also remarkable that all distributionsare unimodal except that of the blend B20 which is bimodal. From the volume weightedsphere-size distributions the profile-size distributions were recalculated. These curves(Fig. 5, solid line) are in good agreement with the corresponding histograms.

C. Estimation of interfacial tension

By comparing moments of the distributions obtained from rheology and TEM, theinterfacial tension can be estimated. The corresponding values (Table II) were calculatedaccording to

I~Rv(R)dRa=

I~R'u(R')dR'(14)

which has been derived from Eq. (4). These values were used to rescale the distributionsobtained from rheology. The resulting curves are displayed in Fig. 7 (dashed lines)together with the distributions obtained from TEM (solid lines).

TABLE II. Volume fraction <f; used within the preparation of the sample, volume fraction ¢mool derived fromthe rheological data [Eqs, (4) and (13)], and estimates of the interfacial tension a for the four PMMNPS blends.

¢ <;Drheol 1J/ (10-3 Nfm)

B02 0.024 O.oI8 2.52B05 0.057 0.047 3.10BIO 0.112 0.103 2.64B20 0.220 0.205 2.08

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v. DISCUSSION

RHEOLOGY AND STRUCTURE OF BLENDS

FIG. 4. TEM images of the blends B02. 805 . Bl0, and B20.

1421

In the preceding section results for the volume weighted sphere-size distribution of thePMMAIPS blends determined in two independent ways have been presented: From rhe­ology the distribution has been estimated up to a scaling depending on the interfacialtension. By analysis of the TEM data, this distribution has been determined up to anormalization factor which can be obtained from the volume fraction of the inclusions.Both results have been compared in order to calculate a value for the interfacial tensionand to get the final result for the volume weighted sphere-size distribution from rheology.

Both results (Fig. 7) show that the average particle size as well as the width of thedistribution increase with increasing concentration of the inclusions. For a small amountof spheres in the blend (samples B02 and B05), the results from rheology and TEM arein excellent agreement. If the concentration of the inclusions is increased, differences can

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1422 FRIEDRICH ET AL.

3 I I 'I

802 B05

2

1'0.,-

...0::

o1,0 I ' " , I

B10

0,8

i 0,6

'0.....'2 0,40::

0,2

0,0-8 -7 -6 -5-8 -7 -6

820

-5

log r jrn] log r [m)

FIG. 5. Histograms of the profile radii for the blends B02, B05, B10, and B20. The solid lines represent theprofile-size distributions recalculated from the volume weighted sphere-size distributions in Fig. 6.

be observed. The deviations are especially large for the blend B20: Rheology leads to aunimodal distribution with a considerable amount of small spheres. There is, however, noindication of these small particles in the TEM data. In addition, TEM leads to a bimodaland not to a unimodal distribution.

The excellent agreement for the blends B02 and B05 shows that the volume weightedsphere-size distribution can be determined up to a scaling from rheological data. Thedeviations for the samples B10 and B20 indicate that the interactions between the differ­ent inclusions are not adequately taken into account. On one hand, in the derivation ofEq. (9) it is assumed that the inclusions are distributed independently. Clearly, this is notpossible for blends with a high concentration of inclusions because of a kind of excludedvolume effect: If there is a sphere at one point, there cannot be another one in a distanceless than the radius of this sphere. On the other hand, the velocity field around oneinclusion is distorted by the presence of other inclusions in the matrix. This effect mayonly approximately be taken into account by the Palierne model.

The values of the interfacial tensions (Table II) vary between 2.52 and 3.10 mN/m(Tref = 190°C). Because the distributions for the blend B20 differ considerably, the

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RHEOLOGY AND STRUCTURE OF BLENDS 1423

-8

4

I- 802

I 3 r... ···805

-- 810

_. 820

'0 2......

~">

o I " ,,!

-9

/.. \

. j \:0.. "'---' o~. 1\.1 I \ \j /: \ .

./ I \~.o-7 -6

logR [rn]

-5

FIG. 6. Volume weighted sphere-size distribution v(R) of the four PMMAJPS blends obtained from TEM.

corresponding value for the interfacial tension is expected to have a large error. Neglect­ing this value, a mean value of 2.75:!:::0.30 mN/m is obtained. A comparison with valuesfrom literature [Wu (1982); Joseph et al. (1992); Gramespacher and Meissner (1992)]shows that this result is realistic. The differences can be explained by a temperaturedependence of the interfacial tension [Wu (1970)] and different molar mass and molarmass distributions of the polymer samples. In addition, the tacticity of the PMMA mayhave an influence on the interfacial tension.

4 I I i

-5

820

810

-6-7-80' ,Id,I,<j,! >' ' ,d ':s> I

-9

3

:s"b 2......

ci:">

log RIm)

FIG. 7. Volume weighted sphere-size distribution v(R) of the four PMMA/PS blends obtained from TEM(solid lines) in comparison with the results from rheology (dashed lines).

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1424

VI. CONCLUSIONS

FRIEDRICH ET AL.

It has been shown that rheological data can be used to determine the volume weightedsphere-size distribution in an immiscible polymer blend up to a scaling depending on theinterfacial tension. The procedure is based on the emulsion model of Palieme and anonlinear regularization method for the numerical calculations. It has been applied to datafor the shear storage and loss moduli for four PMMAIPS blends. The results have beencompared with TEM data in order to estimate the interfacial tension and to get the finalresult for the volume weighted sphere-size distribution from rheology.

Both results show that, for the PMMAIPS blends studied, the average sphere radiusand the width of the distribution increase with increasing the concentration of the inclu­sions. For small concentrations (2 and 5 wt % PS), the results from rheology and TEMare in excellent agreement. Deviations at larger concentrations (IO and 20 wt % PS)indicate that the interactions between different inclusions are not adequately taken intoaccount in the rheological model of Palierne, in the stereological model, or in both. As aconsequence, further studies with the aim to refine the models are necessary.

ACKNOWLEDGMENTS

We thank S. Heinzmann for performing the rheological measurements and BASF AG,Ludwigshafen, Germany for the donation of the PMMA and PS samples.

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