J. Rheol. revised paperEvaluation of Models combiningRheological Data with the MolecularWeight DistributionD. Maier 1 �A. Eckstein 1Chr. Friedrich 1 yJ. Honerkamp 1;21 Freiburger Materialforschungszentrum, Stefan-Meier-Stra�e 21,D-79104 Freiburg im Breisgau, Germany2 Universit�at Freiburg, Fakult�at f�ur Physik, Herrmann-Herder-Stra�e 3, D-79104 Freiburg im Breisgau, GermanySYNOPSISThe aim of this article is to test and to improve existing models combining rhe-ological data with the molecular weight distribution (MWD). This process oftesting and improving was separated into two distinct steps: First, the mixing be-havior of the relaxation modulus obtained from a polymer blend was investigatedand compared with the proposed mixing behavior derived from the linear and thequadratic mixing rules with a mixing parameter � equal to 1 and 2. Second, forthe �rst time, kernel functions weighted with the MWD are estimated directlyfrom experimental data and compared to the weighted kernel functions publishedin the literature. This procedure was performed with polystyrene blends com-�Electronic mail: dima@fmf.uni-freiburg.deyElectronic mail: chf@fmf.uni-freiburg.de 1

posed of two monodisperse polymers with narrow MWD determined using sizeexclusion chromotography (SEC). It was shown, that the existing mixing rulesare not able to adequately describe the mixing behavior of the polystyrene blends.An improved mixing rule was derived with � of 3:84. Furthermore, only for theimproved model, a kernel function exists which is able to describe the experimen-tal data over a wide molecular range. It is shown how improved estimates of theMWD of a polymer can be determined using the new model.I. INTRODUCTIONIn order to understand how the molecular weight distribution (MWD) a�ects theviscoelastic properties of polymers, \homopolymerblends" have been utilized asmodel systems. These model systems are composed mostly of two monodispersepolymers. These monodisperse components have the same chemical nature butdi�erent molecular weights. Agarwal (1979), Mills (1969), and Braun et al. (1996)show, that with these binary mixtures, it is possible to investigate the in uenceof polydispersity on various material parameters, such as the zero-shear-viscosity�0 or the terminal relaxation time �0.The dependence of these material parameters or such measures on the molec-ular weight structure of a mixture determines the mixing behavior which is for-mulated in terms of a mixing rule.From a molecular point of view, mixtures are characterized by their MWDw(M) whereas the material function is often taken to be the relaxation modulusG(t). The corresponding standard mixing rules de�ne the hypothesized relation-ship between them. Two such mixing rules have been proposed in literature. Doiand Edwards (1986) introduce the \linear mixing rule", whereas des Cloizeaux(1988) and Tsenoglou (1987) formulate the \quadratic mixing rule". For thelinear mixing rule, the relaxation modulus of the mixture is given by the linearsuperposition of the relaxation moduli of the components. For the quadratic2

mixing rule, one must sum up the squareroots of the weighted relaxation mod-uli of the components to obtain the squareroot of the relaxation modulus of themixture.These two mixing rules can be derived theoretically. The \reptation model"introduced by Doi and Edwards (1986) yields the linear mixing rule. Thequadratic mixing rule was obtained by des Cloizeaux (1988) and Tsenoglou (1987)and further discussed by Milner (1996). Des Cloizeaux derived it by utilizing the\double reptation" concept whereas Tsenoglou examined entanglements concept.The linear and the quadratic mixing rule can be formulated as special casesof the following more general parametric mixing rule:G(t)=G0N = 0B@ 1Zln(Me) F 1=�(t;M)w(M)d(lnM)1CA� : (1)In eq. (1), F (t;M) is the kernel function, describing the relaxation behaviorof a molecular weight fraction with a molecular weight M , and � is a parame-ter which characterizes the mixing behavior. Me is the entanglement molecularweight and G0N the plateau modulus.The following kernel functions have been proposed in literature: the Tuminellostep function [Tuminello (1986)], the single exponential [Tsenoglou (1991)], theDoi [Doi (1986)], the BSW (Baumgaertel, Schausberger, Winter) [Baumgaertelet al. (1990)] and the Des Cloizeaux [Des Cloizeaux (1990)] kernel. They dependtypically on parameters (e.g. the terminal relaxation time �0 or the zero-shear-viscosity �0) which must be estimated from appropriate data.The possible values of � known from the literature are � = 1 (linear mixingrule), � = 2 (quadratic mixing rule), or a value between 1 and 2 which describesa more general combination of these two mixing rules as introduced by Anderssenand Mead (1997). Mixing rules which correspond to � = 1 or 2 will be calledconventional mixing rules. If, additionally, one of these conventional mixing rulesis combined with one of the above kernel functions, the combination will be called3

a conventional model. In addition, Anderssen and Mead (1997) have shown, thatthe molecular weight scaling is independent of �, which in many ways justi�esthe parametric model of eq. (1).Considerations made by Malkin and Teishev (1991) as well as by Nobile etal. (1996) about the mixing behavior of the viscosity function �( _ ), in which _ is the shear rate, yield a similar rule for the determination of the MWD.For the model of eq. (1), the kernel function and the value of � must beknown before the MWD w(M) of a polymer can be calculated from its relaxationmodulus G(t). It is possible to measure the relaxation modulus G(t) directly ina step shear experiment. But, in this case, there are technical limitations forshort and long times. Therefore, one may calculate G(t) from the relaxationtime spectrum derived from the measurement of the dynamic moduli G0(!) andG00(!). This is the typical procedure nowadays. For the determination of theMWD from the relaxation modulus G(t), eq. (1) must be solved for w(M) whichis equivalent to an inversion of a �rst kind Fredholm integral equation.Up to now, there exists in principal three di�erent methods for the inversionof such an integral equation for the MWD: First, the non-parametric determina-tion of the MWD with a regularization method, as proposed by Wasserman andGraessley (1992) and by Wasserman (1995). A second, often used method, is aparametric one, in which a parametric form of the MWD must be assumed. Thiskind of inverse problem was examined by Nobile et al. (1996) as well as by Mead(1994). The parameters are determined by �tting this model to the experimentaldata. Finally, computational stabilization can be achieved by simply computingthe moments of the MWD with the so called cumulant method [Mead (1994)].This approach has been formalized by Anderssen et. al (1997).In general, the results of these procedures can be summarized as follows:unimodal MWDs can be reconstructed relatively easily, but in most cases bi-modal MWDs give problems. This may result from the use of an inappropriatemodel, which is not in quantitative agreement with the mixing behavior of the4

homopolymers.Therefore, it is the aim of this work to evaluate the model given by eq. (1).In order to do this, a more general parametric mixing rule, in which � in eq. (1)is introduced as a parameter to be determined from appropriate data: By �t-ting this model to the measured data, an optimal value of � can be estimated.The procedure presented below yields also, for the �rst time, an approximateestimate of the kernel function from the measured G(t) of bimodal mixtures.With this procedure, a generalized model is obtained, which is compared withthe conventional models.The MWDs obtained by the conventional model and the generalized modelbased on rheological experiments, are compared to direct measurements of theMWD obtained by size exclusion chromotography (SEC).In the following section, the experimental details are presented about thepreparation of the bimodale mixtures of PS. Then, the dynamic moduli of thesemixtures were measured. From these dynamic moduli, the relaxation time spectrawere estimated to calculate their relaxation moduli. The values of these relaxationmoduli were then utilized (in Sec. IV) to assess the conventional mixing rules, toobtain an improved mixing rule, and to estimate the kernel functions weightedwith the MWD. These estimated weighted kernel functions were compared withthe weighted kernel functions published in the literature (Sec. V). In Sec. VI,the coventional models (� = 1 and � = 2 with an appropriate kernel function)are compared with the improved model. In Sec. VII, the results are summarizedand conclusions are presented.II. EXPERIMENTSA. MaterialsThe narrow distributed polystyrene samples used in this study were pre-pared by anionic polymerization at low temperatures (�780C) using tetrahy-5

drofuran (THF) as the solvent, sec-buthylithium as the initiator, and 4,5-methylenephenantrene as the indicator. Details of the synthesis are given inS�anger et al. (1996). The samples were then precipitated, puri�ed with methanol,and dried in vacuo at 600 C till they had reached constant weight.B. CharacterizationThe MWD w(M), normed by 1Zln(Me) d(lnM)w(M) = 1; (2)the molecular mass Mw, and the polydispersity P = Mw=Mn, satisfyMw= 1R�1 d(lnM)M2w(M)1R�1 d(lnM)Mw(M)P = 1R�1 d(lnM)M2w(M)( 1R�1 d(lnM)Mw(M))2 : (3)They were determined by SEC, using toluene as the solvent. SEC was calibratedwith PS which has a narrow MWD. The ow rate of the solvent was determinedin the range 1:003 { 1:007ml=min before each SEC-measurement. The molecularmasses Mw of PS varied between 56000 g=mol and 643000 g=mol.Static light scattering measurements were performed at 200 C with a fullycomputerized and electronically modi�ed SOFICA photogoniometer (Baur, In-strumentenbau, Hausen, Germany) in the angular range from 300 to 1400 in stepof 50. A helium/neon laser (�0 = 632 nm) was used as the light sources. Arefractive index increment of 0:194 was used for the measurements in toluene.Additionally, the number average molecular weightsMn were determined fromosmotic pressure measurements by the gonotec membrane Osmomant 090 os-mometer in THF as the solvent at 250 C for the low molecular weight PS.6

The molecular weights Mw and polydispersity P of �ve narrowly distributedpolystyrene samples are listed in Table I. Furthermore, Fig. 1 shows the MWD ofthese samples determined by SEC. The samples are abbreviated in the followingwith the titles PS56, PS60, PS177, PS250, and PS643. For PS56, a small highmolecular component was detected.TABLE I. The values of Mw and P for the �ve narrowly distributed polystyrenesinvestigated. sample Mw=[g=mol] PPS56 56400 1:03PS60 60400 1:04PS177 176700 1:03PS250 250000 1:04PS643 642700 1:07C. MixturesThree series of bimodal polystyrene mixtures were made by solution blending.The low and high molecular weight polystyrene component were solved in THFat room temperature. After stirring for 2 hours, the polstyrene solutions wereprecipitated in methanol. The obtained polystyrene blends were dried in a vacuooven at 600 C till they had reached a constant weight. The mixtures were pro-duced, so that the molecular weight ratiosMwLM=MwHM di�ered among the threeseries. The indices LM and HM stand for \low molecular" and \high molecu-lar". The molecular weight ratios were approximately 1=3 (for the �rst series)1=5 (for the second series), and 1=11 (for the third series):1. In the �rst series, mixtures are prepared which consist of PS60 and PS177with 10%, 20%, 40%, 60%, and 80% of the high molecular polymer (PS177).7

2. In the second series, mixtures are prepared which consist of PS56 andPS250 with 10%, 30%, 50%, and 70% of the high molecular polymer(PS250).3. In the third series, mixtures are prepared which consist of PS60 and PS643with 20%, 40%, 60%, and 80% of the high molecular polymer (PS643).D. Rheological measurementsFor the rheological experiments, each sample of the PS mixtures was dried undervacuum at 600 C for several days. The samples were then cold compressed andannealed for 30 min at 1900 in vacuum and compression molded to form smalldisks of 25 mm diameter and thickness of about 1 mm.The rheological behavior was studied using dynamic oscillatory tests in aRheometrics Mechanical Spectrometer RMS 800 with parallel plates of 25 mmdiameter. The angular frequency ! varied from 10�2 to 102 rad=s and the tem-perature range was changed in steps of 100 C from 2400 C to 1300 C depending onthe molecular weight. All samples were investigated under nitrogen atmospherein order to prevent thermooxidative degradation. The isotherms were shifted toobtain mastercurves at a reference temperature of T0 = 1700 C using the programLSSHIFT generated by Honerkamp and Weese (1993). The mastercurves of the�rst mixture series (PS60=PS177) are shown in Fig. 2.E. Determination of Material ParametersIn the high frequency range (aT! � !r � 102rad=s), all mastercurves mergenearly into one curve independent of the composition. In this region the functionG0 reaches the plateau modulus G0N of 2 �105Pa, determined by the tan� criterion(see Fig. 2 horizontal line), a value which is well known [Ferry (1980)].The steady-state compliance J0e was estimated fromJ0e = lim!!0 G0(!)G02(!) +G002(!) ; (4)8

yielding the error range 0:9 � 10�5 Pa�1 � J0e � 1:2 � 10�5 Pa�1 for the studiedsamples. J0e di�ers a little among the investigated samples. The averaged valueover all samples was determined to 10�5 Pa�1.Furthermore, the entanglement molecular weight Me was estimated using[Ferry (1980)] Me = �RTG0N ; (5)in which R is the universal gas constant (8:314 J=molK), T the temperature atwhich G0N is measured, and � the density of the sample at this temperature T .For the PS-samples, a value of Me of 18000 g=mol was obtained.The terminal relaxation time �0 is a very important material parameter. Itcan be estimated from the relaxation time spectrum calculated with the programNLREG [Weese (1993)]. In order to determine �0, one must choose the largesttime for which the relaxation time spectrum possess a local maximum.The dependence of �0 on the molecular weight Mw obeys the well knownscaling law as can be seen in Fuchs et al. (1996):�0(Mw) = K � �MwM0 �a ; (6)in which M0 is the unique molecular weight of 1 g=mol. In order to determinethe parameters K and a, the terminal relaxation time of the �ve monodispersesamples listed in Table I was estimated. To obtain a better estimation of a andK, three additional samples which were not included in the mixing experimentswere investigated. The resulting values were a = 3:67 and K = 6:919 � 10�20 s(see Fig. 3).

9

III. DATA ANALYSIS OF BINARY MIX-TURESThe MWDs of the polymer mixtures under investigation are superpositions oftwo monodisperse polymers. Therefore, the MWD of the blend can expressed asw(M) = xwHM(M) + (1� x)wLM (M); (7)in which wHM is the MWD of the polymer with the high molecular weight, wLMis the MWD of the polymer with the low molecular weight, and x is the weightfraction of the high molecular polymer in the bimodal mixture.Substitution of eq. (7) into eq. (1) yields the following model for the �-th rootof the normalized relaxation modulus Gx(t)=G0N , in which Gx(t) makes explicitthe dependence of the relaxation modulus G(t) on the component x:y(x; �; t) =< F 1=� >wLM (�; t) + xf< F 1=� >wHM (�; t)� < F 1=� >wLM (�; t)g;(8)in which < F 1=� >w (�; t) = 1Zln(Me) d(lnM)w(M)F 1=�(t;M); (9)and y(x; �; t) = Gx(t)G0N !1=� : (10)The weighted kernel function < F 1=� >w (�; t) will depend on the choice ofthe kernel F (t;M), the parameter �, and the given MWD w(M).It follows from eq. (8) that, for a �xed � and a speci�c time t0, y(x; �; t)will be a linear function of the component x. This fact can be used to test andimprove the formulation of the mixing rules. As a test, the model of eq. (8), with� = 1 (linear mixing rule) and � = 2 (quadratic mixing rule), was �tted to thedata.In addition, an improved �t to the data was obtained by optimizing the choiceof � in eqs. (8,10). From this model, one obtains additionally direct experimental10

estimates of the weighted kernel functions < F 1=� >wLM (�; t) and < F 1=� >wHM(�; t). They can be compared with the corresponding values of eq. (9) for variouschoices of F (t;M) (e.g. one of the kernels known from literature), and the w(M)(e.g. obtained by SEC measurement). In this way, one can assess the extent towhich various models to date explain the measured data.IV. INVESTIGATION OF THE EXPONENT �A. Conventional ExponentsFrom the relaxation time spectra, the normalized relaxation moduli G(t)=G0Nwere calculated by direct integration. In Fig. 4 the relaxation moduli of the �rstmixture series are shown. It can be seen, that the relaxation moduli G(t) ofthe monodisperse sample PS60 (see dotted line in Fig. 4) have a shoulder atapproximately 1 s caused by an additional high molecular weight part. This partcan also be seen in the SEC curve (see Fig. 1).In order to determine the time region [tmin; tmax] in which a reliable investiga-tion of the relaxation moduli is possible, the following points must be considered:First, the lower bound tmin of this time region is that time, for which all relaxationmoduli merge into one curve. For smaller times, there exists no dependence onthe molecular weight so that these times possess no information about the MWD.The time tmin was determined to be 10�2 s and is indicated in Fig. 4 by the leftdashed line. The upper bound tmax depends on the error of the measured data.Since one can not measure dynamic moduli below 1 Pa with our equipment, thedetermined values of the normalized relaxation modulus which are below 1Pa=G0N� 10�4 are only extrapolated values (see horizontal line in Fig. 4). The errorsin these extrapolated values increase for very long times. This can be seen inFig. 4 in which two relaxation moduli nearly cross each other. This occurs be-cause of the increasing of the extrapolation errors. It is reasonable to conclude,that the largest time for which no such crossings occur is the largest time for a11

reasonable investigation and is therefore set identical to tmax. The time tmax wasdetermined to be 103 s and is indicated by the right dashed line in Fig. 4. Theseconsiderations lead to a time range to which all the following investigations wererestricted: t � [tmin; tmax];with tmin = 10�2 s and tmax = 103 s: (11)It should be noted, that no such crossing occurs for the third mixture series(PS60=PS643) in the time range of eq. (11). Nevertheless, 103 s is large enoughsince the upper bound tmax needs only to be larger than the terminal relaxationtime �0. For PS643, �0 is 126 s (see Fig. 3 and eq. (6)) and consequently smallerthan 103 s.Combining eq. (6) and eq. (11), one �nds that the maximum accessible molec-ular weight range is: M � [4:5 � 104; 106] g=mol: (12)One is now in a position to investigate the dependence of the relaxation mod-ulus G(t) on x. The model of eq. (8) is �tted to the data for � equal to 1 and2, and the weighted kernel functions < F 1=� >wLM (�; t) and < F 1=� >wHM (�; t)are estimated.The time interval is subdivided into 100 equally spaced points ti (i =1; : : : 100) on the logarithmic scale. For every point ti, the values< F 1=� >wLM (�; ti) and < F 1=� >wHM (�; ti) were determined by minimizing�2(�; ti) = LXj=18<:y�(xj ;�; ti)� h< F 1=� >wLM (�; ti) + xjf< F 1=� >wHM (�; ti)� < F 1=� >wLM (�; ti)gi(�j=�)y�(xj ;�; ti) 9=;2(13)12

with respect to the values of < F 1=� >wLM (�; ti) and < F 1=� >wHM (�; ti).In eq. (13), the xj (j = 1; : : : ; L) denote the di�erent mixtures tested, the �jsare the relative measurement errors and y�(xj; �; ti) are the measured values ofy(xj; �; ti). The minimization was performed with � = 1 and � = 2 for all threemixture series to test the quality of the conventional mixing rules.For the various choices of ti, the results did not di�er greatly. The �t witht0 = 18 s , which is a representative of all others, is shown in Fig. 5 for the �rstmixture series. Since, in Fig. 5, y(x; �; t) versus x for �= 1 and � = 2 is shown,all data values should lie on a straight line if either the linear or the quadraticmixing rule is able to describe the data. But the �gures indicate, that the datado not lie on a straight line.Together, these �ts yield the conclusion that the linear and the quadraticmixing rules do not adequately describe the experimental results.B. Optimization of the exponent �Since the conventional mixing rules are unable to describe the dependence ofthe investigated relaxation moduli on the high molecular component, the ideaemerges to generate a phenomenological mixing rule, in which the �t is optimizedwith respect to the choices of �.Hereby, � was chosen to minimize�2(�) = 100Xi=1�i�2(�; ti); (14)in which �2(�; ti) is the minimum value, at each time ti (see eq. (13)). The valuesti are the same as in eq. (13) and �i is given by ln(ti)� ln(ti�1).Figure 6 shows the dependence of �(�) on � for the three di�erent mix-ture series. The optimal values for these three series are �1 = 3:79 � 0:20 (forPS60=PS177), �2 = 3:82 � 0:20 (for PS56=PS250), and �3 = 3:92 � 0:20 (forPS60=PS643). These three values are compatible with each other, so it can beconcluded, that for the investigated polystyrene mixtures the value of � is �xed.13

The weighted mean value is � = 3:84� 0:10; (15)which is the best value for PS in the molecular range given by eq. (12). In Fig. 6,this value is indicated as a dashed line.Now, all investigated data are �tted with the optimal value of � = 3:84. Thevalues y(x; �; t)j�=3:84 for all three mixture series lie on a straight line in contrastto the behavior when � was chosen equal to 1 and 2 (see Fig. 5). Figure 7 showsone representative example (t0 = 18 s of the �rst mixture series PS60/PS177).Consequently, the PS data are represented best with a �-value of 3:84, verydi�erent from the �-values of the linear and the quadratic mixing rules.V. ESTIMATION OF THE KERNEL FUNC-TIONA. The conventional kernelsThe �ve kernel functions Fj(t;M) (j = 1; : : : 5) (see eq. (1)), which are typicallyused in literature, are (see also the article of Wasserman and Graessley (1992)):1. The Tuminello-kernel: F1(t;M) = 1 if t < �0(M)F1(t;M) = 0 else: (16)2. The Single-Exponential-kernel:F2(t;M) = e�t=�0(M): (17)3. The Doi-kernel: F3(t;M) = 8�2 1Xn�odd e�tn2=�0(M)n2 : (18)14

4. The BSW-kernel (Baumgaertel, Schausberger and Winter):F4(t;M) = � 1Z0 duu��1exp � t�0(M)u! (19)with � = vuut J0eG0NJ0eG0N � 1 � 1: (20)In Sec. II E G0N is determined to 2 � 105 Pa. J0e varies in the error range of(0:9 � 10�5� 1:2 � 10�5) Pa�1 for the di�erent PS samples with an averagedvalue of 10�5 Pa�1. This leads to the error range (0:31� 0:62) for � withan averaged value of 0:41. These deviations of � have nearly no in uenceon the BSW kernel, so that in the following � is �xed to 0:41.5. The Des Cloizeaux-kernel:F5(t;M) = " 8�2 1Xn�odd 1n2 exp��n2 � t�0(M) + M12:5Me g � tM12:5Me�0(M)���#2(21)with g(x) = 1Xm=0 1� exp(�m2x)m2 :The material parameters such as �0(M), Me, G0N and J0e are known (seeSec. II E), so that all �ve kernels can be numerically evaluated.B. Choice of the optimal kernel functionIn order to calculate the MWD from the relaxation modulus, a kernel functionmust be chosen (see eq. (1)). Since, in literature, �ve di�erent kernel functionsare suggested, we have to decide which one to take.For this purpose, we introduce a criterion based on the weighted kernelfunctions < F 1=�j >w (�; t) (see eq. (9)) of the di�erent kernels Fj(t;M).< F 1=� >w (�; t) can be calulated, if the kernel function F (t;M), the MWDw(M), and the mixing parameter � are given. The MWD was taken from theSEC-measurement (see Fig. 1), and � was chosen to be 1 (linear mixing rule), 2(quadratic mixing rule), and 3:84 (improved mixing rule). < F 1=�j >w (�; t) are15

then compared to the values of < F 1=�est >w (�; t) which are estimated directlyfrom the data (see eq. (8)).Estimates < F 1=�est >wi (�; t) of the weighted kernel functions < F 1=� >wi (�; t)for each of the 5 di�erent PS samples (PS56, PS60, PS177, PS250, and PS643)can be made according to eq. (8), in which the index i stands for the correspondingPS sample. For example, < F 1=� >w56 stands for the kernel function weightedwith the MWD of PS56. For the �rst and the third mixture series < F 1=� >wLM isidentical to < F 1=� >w60. For that reason, we have two estimates of < F 1=� >w60(�; t) for each �. For the second mixture series, < F 1=� >wLM corresponds toPS56. The value of < F 1=� >wHM corresponds in the �rst mixture series toPS177, in the second to PS250, and in the third to PS643.In Fig. 8 estimates < F 1=�est >wi (�; t) of w60 and w177 are represented by cross-marks. Furthermore, for each of the investigated PS samples, the weighted kernelfunctions < F 1=�j >wi (�; t) of the �ve di�erent kernels Fj(t;M) are calculated.For w60 and w177, these functions are represented in Fig. 8 by �ve di�erent lines.In the next step, the di�erence d (ln(< Fj >wi; �; t))d (ln(< Fj >wi); �; t) = ln �< F 1=�j >wi (�; t)�� ln �< F 1=�est >wi (�; t)� (22)between the logarithms of the calculated and the estimated weighted kernel func-tions is introduced. The smaller this distance the better the kernel functionFj(t;M) describes the experimental data for a given pair (�; t). In order to de-cide, which kernel function describes best the experimental data for a given mix-ing parameter �, independent of the time, the norm �wi(Fj; �) of the distance iscalculated: �wi(Fj; �) = vuuut tmaxZtmin d2 (ln(< Fj >wi; �; t)) d ln t: (23)The norm �wi(Fj; �) was calculated for every possible combination of the �veinvestigated PS samples, the �ve kernel functions Fj, and the three possiblemixing parameters �. �wi(Fj; �) describes quantitatively to what extent the data16

are described by the kernel function Fj(t;M). Additionally, the complete norm ofthe low molecular weights �wLM (Fj; �), the high molecular weights �wHM (Fj; �),and all molecular weights �(Fj; �) are calculated according to:�wLM (Fj; �)=�w601 (Fj; �) + �w56(Fj; �) + �w602 (Fj; �);�wHM (Fj; �)=�w177(Fj; �) + �w250(Fj; �) + �w643(Fj; �);�(Fj; �)=�wLM (Fj; �) + �wHM (Fj; �); (24)in which �w601 (Fj; �) is calculated from the estimates of the �rst mixture seriesPS60&PS177 and �w602 (Fj; �) is the corresponding norm of the third mixtureseries PS60&PS643.TABLE II.The values �(Fj; �), �wLM (Fj; �), and �wHM (Fj; �) (j = 1; : : : ; 5) determined byeq. (23) and eq. (24) for the �ve kernel functions Fj(t;M) known from literaturefor the three interesting values of �. The smallest values of �, implying that thecorresponding kernel function is the optimal one, are printed bold faced.� = 1 � = 2 � = 3:84kernel Fj(t;M) �(�wLM ;�wHM ) �(�wLM ;�wHM ) �(�wLM ;�wHM )F1 : Tuminello 411(234; 177) 670(458; 212) 961(565; 396)F2 : single-expon. 312(11; 301) 20:6(14:0; 6:6) 14:6(8:6; 6:0)F3 : Doi 289(11; 278) 20:5(15:3; 5:2) 15:7(8:3; 7:4)F4 : BSW 296(56; 240) 47:3(42:6; 4:7) 35:6(20:5; 15:1)F5 : des Cloizeaux 539(490; 49) 581(454; 127) 365(259; 106)In the second column, the norms for the mixing parameter � = 1 (linear mixingrule) are listed. The corresponding estimates < F 1=�est >wi (�; t)j�=1 as well as thecalculated values < F 1=�j >wi (�; t)j�=1 of the weighted kernel functions for w60and w177 are shown in Fig. 8a. Table II clearly shows, that a di�erence of the norm�wi(Fj; �)j�=1 exists between the low and the high molecular weight polymers.17

The norms of the Tuminello kernel and the Des Cloizeaux kernel are large for thelow molecular weight polymers (see Table II �wLM (Fj; �) in the second column)whereas the norms of the three other kernels in this case are small (see Table II�wLM (Fj; �) in the second column). In contrast, the situation is reversed for thehigh molecular weight polymers (see Table II �wHM (Fj; �) in the second column).Consequently, no kernel function known from literature is able to describe thedata investigated in this article assuming the linear mixing rule. Nevertheless,for the mixing parameter � = 1, we decide to take the kernel function with thesmallest norm �(Fj; �): the Doi kernel.In the next step, � was chosen equal to 2 (quadratic mixing rule). For w60and w177, the estimated < F 1=�est >wi (�; t)j�=2 and calculated values < F 1=�j >wi(�; t)j�=2 are shown in Fig. 8b. The fact, that a di�erence between �wLM and�wHM exists (see Table II third column) is comparable to the situation detectedfor the linear mixing rule. The smallest value for � (20:5) is clearly smaller thanthe one obtained with � = 1 (289). Therefore, for the quadratic mixing rule aswell it is not possible to obtain a consistent description with one of the knownkernel functions. However, compared to the linear mixing rule the situation isclearly better. The values of the weighted kernel function calculated with theDoi kernel describe the estimates best.Finally, the optimal value of � ( = 3:84) was chosen. For w60 and w177, theresults are shown in Fig. 8c, and in the forth column of Table II the values of �,�wLM , and �wHM are listed. In this case there is nearly no di�erence between�wLM and �wHM . For the single exponential kernel, the norm is very small,independent of the molecular weight. The result is, that this kernel functiondescribes very well all estimates made with � = 3:84.In summary, it was shown, that for the conventional mixing rules none of thetypically kernel functions presented in literature describe the estimates< F 1=�est >wi (�; t) which are derived from experimental data. The distancesbetween the calculated values of the weighted kernel functions assuming a con-18

ventional model and the estimates strongly depend on the molecular weight. Incontrast, this disadvantage does not exist for the improved mixing rule.Nevertheless, it is possible to choose an optimal kernel function for each valueof �. This is the kernel function with the smallest norm � (Fj; �). For theconventional mixing rules this is the Doi kernel, and for the improved mixingrule the single-exponential kernel.VI. COMPARISON OF DIFFERENT MODELSIn Sec. V, we have motivated three versions of the generalized model combiningrheological data with the MWD. Wasserman and Graessley (1992) favored a forthversion. These four models are special cases of the generalized model given byeq. (1). The �rst model was �xed with � = 1 and the Doi-Kernel. The secondmodel with � = 2 and the Doi kernel and the third one with � = 3:84 and thesingle exponential kernel. The forth model favored by Wasserman and Graessley(1992) holds to � = 2 and the des-Cloizeaux kernel.In order to compare these four models we decided to test them in two di�erentways. First, we calculate the dynamic moduliG0 and G00 from the MWD obtainedby SEC. Then, the calculated and the measured moduli are compared. Second,we solve the inverse problem and estimate the MWD from the measured dynamicmoduli. This problem must be solved with a special technique serving numericalstability: a regularization method. Then, the calculated MWD is compared withthe MWD obtained by SEC.A. Comparison with the direct problem: Calculation of the DynamicModuli from the MWDIn this part, the relaxation moduli G(t) are calculated for the four di�erentmodels directly from eq. (1). These relaxation moduli are converted into the cor-responding dynamic moduli via the relaxation time spectra. Then, the calculateddynamic moduli are compared with the measured ones.19

As a representative example, the dynamic moduli of 20% PS60 and 80%PS177 are shown in Fig. 9 (solid lines). The dynamic moduli calculated withthe �rst model (� = 1 and Doi kernel) are shown in Fig. 9a (dashed lines). Thedashed lines of Fig. 9b show the calculated dynamic moduli of the second model(� = 2 and Doi kernel) and the chain dashed lines show the dynamic moduliof the forth model (� = 2 and Des Cloizeaux kernel). In Fig. 9c the calculateddynamic moduli of the third model, the generalized model with � = 3:84 and thesingle exponential kernel are shown (dashed lines).This comparison clearly demonstrates, that the �rst, the second, and the forthmodel are not able to describe the measured dynamic moduli in a quantitativecorrect way. In contrast to this result, the third model, which is the improvedone, is able to describe the experimental data. It further shows, that for theinvestigated PS samples and for � = 2 the Doi kernel describes the experimentaldata better than the Des Cloizeaux kernel. This was a result of Sec. V and isherewith con�rmed.B. Comparison with the inverse problem: Determination of the MWDfrom the Dynamic ModuliIn order to determine the MWD w(M) from the relaxation modulus G(t)eq. (1) has to be inverted. This inversion problem is ill-posed (for a de�nitionof ill-posed problems see Morozov (1984)), so that a regularization method hasto be used. In this article, the Tikhonov regularization is applied, in which thevalueV (�) = NXi=1 24G�(ti)G0N � 1Rln(Me) d lnMF 1=�(t;M)w(M)!�352�2i + � 1Zln(Me) d lnMw002(M)(25)has to be minimized with respect to w under the condition that w(M) � 0 and1Rln(Me) d lnMw(M) = 1 holds. The summation forces the solution to be compatiblewith the data. The integral smoothes the solution and the value � weights it with20

the sum. For a good estimation of w an optimal value of � has to be chosen. Inthis article � is selected on the basis of the SC-method introduced by Honerkampand Weese (1990).The MWD w(M) is determined with this regularization procedure for the fourmodels investigated in Sec. VI A. One representative result for the �rst model(� = 1 and Doi kernel) is shown in Fig. 10a. In this �gure the experimental dataof the SEC measurement for 20% PS60 and 80% PS177 (solid line) are shown aswell as the estimated result with the regularization procedure (error bars). Bothpeaks obtained by the SEC measurement were reconstructed only poorly.The result for the second model (� = 2 and Doi kernel) obtained by theregularization procedure is shown in Fig. 10b. The estimation of the MWD withthis model is better than the estimation with the �rst model. This result wasexpected since the norm � for the �rst model is larger than that for the secondmodel (see Table II).In Fig. 10c the result for the model with � = 2 and the des-Cloizeaux kernel isshown. No peak is reconstructed. Obviously in this case, the mixing parameterand the kernel function are not optimally chosen.In the third model, representing the improved model, � was �xed to 3:84and the single exponential kernel was chosen. Figure 10d shows the estimatedMWD in comparison to the MWD obtained by SEC (solid line). Both peaksare very well reconstructed resulting from an optimal mixing parameter � witha corresponding optimal kernel function F (t;M).VII. CONCLUSIONSIn this article the relationship between rheological data and the molecular weightdistribution (MWD) was investigated. Up to now this relationship is typicallydescribed with the linear or the quadratic mixing rule. It was shown for the inves-tigated PS samples, that the conventional mixing rules are not able to describe21

the rheological data well.For that reason, a new phenomenological mixing rule described with a mixingparameter � was introduced. This parameter includes the linear mixing rule(� = 1) and the quadratic mixing rule (� = 2). The new mixing parameter wasobtained by �tting the experimental data of the PS samples to the componentsof the molecular weights. We found an optimal value of � equal to 3:84� 0:10.Furthermore, for the �rst time, the kernel function, weighted with the MWD,was directly estimated from experimental data. These estimates are comparedto the corresponding expressions of the kernel functions given in literature. Wefound that for the conventional mixing rules no kernel function is able to describethe experimental data well. Either they describe the low molecular weight part orthe high molecular weight part well, but no kernel function was able to describeboth parts. The improved mixing rule (� = 3:84 � 0:10) does not have thisdisadvantage. The result is, that only with the combination of the optimallychosen mixing parameter and the optimal kernel function it is possible to describethe experimental data.Comparison of the MWD obtained by SEC to the MWD estimated withthe di�erent models shows, that the investigated PS samples are much betterreconstructed with the improved model than with the conventional ones.It is also possible to apply the procedure described in this article to dynamicmoduli G0 and G00. Such a method has the advantage, that the conversion of thedynamic moduli to the relaxation modulus is not necessary anymore. This willbe done in future. Furthermore, we will check the validity of this generalizedmodel for other polymers.ACKNOWLEDGEMENTWe thank Professor R. Anderssen (Div. of Math. and Statistics, Canberra,Australia) for helpful discussions. 22

ReferencesAgarwal, P. K., \Relationship between steady-state shear compliance and molec-ular weight distribution," Macromolecules 12, 42-44 (1979).Anderssen R. S. and D. W. Mead \Theoretical derivation of molecular weightscaling for rheological parameters," J. Non-Newtonian Fluid Mech. (in press).Anderssen R. S., D. W. Mead, and Driscoll, \On the recovery of molecular weightfunctionals from the double reptation model," J. Non-Newtonian Fluid Mech. 68,291-301 (1997).Baumgaertel, M., A. Schausberger, and H. H. Winter, \Relaxation of polymerswith linear exible chains of uniform length," Rheol. Acta 28, 400-408 (1990).Braun, H., A. Eckstein, K. Fuchs, and Chr. Friedrich, \Rheological methods fordetermining molecular weight and molecular weight distribution," Appl. Rheol.6, 116-123 (1996).des Cloizeaux, J., \Double reptation vs simple reptation in polymer melts," Eu-rophys. Lett. 5, 437-442 (1988); 6, 475 (1988).des Cloizeaux, J., \Relaxation and viscosity anomaly of melts made of long en-tangled polymers. Time dependent reptation," Macromolecules 23, 4678-4687(1990).Doi, M. and S. F. Edwards, The Theory of Polymer Dynamics, (Clarendon, Ox-ford 1986), Chap. 7.23

Ferry, F. D., Viscoelastic Properties of Polymers, 3rd ed. (Wiley, New York1980).Fuchs, K., Chr. Friedrich, and J. Weese, \Viscoelastic properties of narrow-distribution poly(methyl methacrylates)," Macromolecules 29, 5893-5901 (1996).Groetsch C. W., The Theory of Tikhonov Regularization for Fredholm Equationsof the First Kind, (Pitman, London, 1984).Honerkamp, J. and J. Weese, \Tikhonovs regularization method for ill-posedproblems," Cont. Mech. Thermodyn. 2, 17-30 (1990).Honerkamp, J. and J. Weese, \A note on estimating mastercurves," Rheol. Acta32, 57-64 (1993).Honerkamp, J. and J. Weese, \A nonlinear regularization method for the calcu-lation of relaxation spectra," Rheol. Acta 32, 65 -73 (1993).Malkin, A. Ya. and A. E. Teishev, \Flow curve-molecular weight distribution: isthe solution of the inverse problem possible?" Polym. Eng. Sci. 31, 1590-1596(1991).Mead, D. W., \Determination of molecular weight distributions of linear exiblepolymers from linear viscoelastic material functions," J. Rheol. 38, 1797-1827(1994).Mills, N. J., \Rheological properties and molecular weight distributions of poly-dimethylsiloxane," Europ. Polym. J. 5, 675-681 (1969).24

Milner, S. T., \Relating the shear-thinning curve to the molecular weight distri-bution in linear polymer melts," J. Rheol. 40, 303-315 (1996).Morozov, V. A., Methods for Solving Incorrectly Posed Problems, (Springer,Berlin, 1984).Nobile, N. R., F. Cocchini, and J. V. Lawler \On the stability of molecular weightdistributions as computed from the ow curves of polymer melts," J. Rheol. 40,363-382 (1996).S�anger, J., C. Tefehne, R. Lay, and W. Gronski, \Simpli�ed procedure of anionicpolymerization of styrene and diene using 4,5 methylenephenanthrene," Polym.Bull. 36, 19-26 (1996).Tsenoglou, C., \Viscoelasticity of binary homopolymer blends," ACS Polym.Prepr. 28, 185-186 (1987).Tsenoglou, C., \Molecular weight polydispersity e�ects on the viscoelasticity ofentangled linear polymers," Macromolecules 24, 1762-1767 (1991).Tuminello, W. H., \Molecular weight and molecular weight distribution from dy-namic measurements of polymer melts," Polym. Eng. Sci. 26, 1339-1347 (1986).Wasserman, S. H., \Calculating the molecular weight distribution from linearviscoelastic response of polymer melts," J. Rheol. 39, 601-625 (1995).Wasserman, S. H. and W. W. Graessley, \E�ects of polydispersity on linear vis-coelasticity in entangled polymer melts," J. Rheol. 36, 543-572 (1992).25

Weese, J., \A regularization method for nonlinear ill-posed problems," Comp.Phys. Comm. 77, 429-440 (1993).

26

FIGURE CAPTIONSFIG. 1. The MWD of the homopolymers measured with SEC. The solid lineshows the distribution of PS56, the chain-dotted line of PS60, the dotted line ofPS177, the dashed line of PS250, and the dot-chained line of PS643.FIG. 2. The measured dynamic moduli G0 and G00 for the �rst mixtures series(PS60/PS177) with 0%, 10%, 20%, 40%, 60%, 80%, and 100% of the high molec-ular polymer (PS177). The dashed line characterizes the maximum frequency!max = 102 rad=s. For !r � !max all curves merge nearly into one curve. Thedotted line shows the plateau modulus G0N = 2 � 105 Pa for polystyrene.FIG. 3. The determined terminal relaxation times �0 in dependence on themolecular mass Mw for eight di�erent monodisperse polymerblends. The pointsof the �ve polymers investigated in this paper are showed with a cross whereasthe three points of the additional polymers are indicated with a star. The straightline show the �t with eq. (6) leading to the parameters K = 6:919 � 10�20 s anda = 3:67.FIG. 4. The determined relaxation moduli G(t) normalized with the plateaumodulus G0N for the �rst mixtures series (PS60/PS177) with 0%, 10%, 20%,40%, 60%, 80%, and 100% of the high molecular polymer (PS177). The dashedlines indicate the minimum and maximum time [tmin; tmax] in which useful inves-tigations of the relaxation modulus are possible. The horizontal line subdividethe graph into two regions. In the region below this line, the extrapolated valuesof the normalized relaxation modulus are shown. The dotted line represents thelow{ and the dashed line the high molecular monodisperse sample.FIG. 5. The values y(x; �; t0) with � = 1:0 and � = 2:0 at the time t0 = 18 sin dependence on the high molecular component x for the �rst mixture series27

(PS60/PS177). The straight lines characterize the �t on the data.FIG. 6. The value � in dependence on the phenomenological parame-ter � for the �rst mixture series (PS60&PS177), the second mixture series(PS56&PS250), and the third mixture series (PS60&PS643). The dashed lineshows the optimal value of � (= 3:84) for all mixture series.FIG. 7. The values y(x; �; t0) with � = 3:84 and t0 = 18 s in dependence onthe high molecular component x for the �rst mixture series (PS60/PS177). Thestraight line characterizes the �t on the data.FIG. 8. The estimates < F 1=�est >wi (�; t) of the weighted kernel functions (rep-resented by crossmarks) and the calculated values of < F 1=�j >wi (�; t) for thedi�erent kernel functions known by literature: Tuminello ( -�- ), single exponen-tial ( � � � ), Doi ( - - - ), BSW (-� � ��-), and Des-Cloizeaux ( -��- ). In (a) � is setequal to 1 (linear mixing rule), in (b) equal to 2 (quadratic mixing rule), and in(c) equal to 3:84 (improved mixing rule).FIG. 9. The measured dynamic moduli of 20% PS60 and 80% PS177 areshown (solid lines). The dashed lines represent the calculated dynamic modulifor (a) the �rst model (� = 1 and Doi kernel), (b) the second model (� = 2and Doi kernel), and (c) the third model (� = 3:84 and single exponential). Thechain dashed lines in (b) represent the forth model (� = 2 and Des Cloizeauxkernel proposed by Wasserman and Graessley (1992)). The dotted lines show themaximum frequency.FIG. 10. The MWD of the binary mixture with 20% of PS60 and 80% ofPS177 obtained by the SEC-measurement (black line) and the regularizationprocedure (error bars). The regularization procedure uses in (a) the model com-28

bining the linear mixing rule (� = 1) and the Doi kernel, in (b) the quadraticmixing rule (� = 2) and the Doi kernel, in (c) the quadratic mixing rule (� = 2)and the des Cloizeaux kernel, and in (d) the improved mixing rule (� = 3:84) andthe single exponential kernel. The dotted line shows the entanglement molecularweight Me of the PS sample (18000 g=mol).

29

1 5 10 50 100 500

*104

0.0

0.5

1.0

1.5

2.0

FIGURE 1Maier et. al.J. Rheol.

30

FIGURE 2Maier et. al.J. Rheol.

10−3 10−2 10−1 100 101 102 103

100

102

104

10−3 10−2 10−1 100 101 102 103

1

10

100

1000

*102

31

4 6 8 10 20 40 60 80 100 200

*104

10−3

10−2

10−1

100

101

102

103

FIGURE 3Maier et. al.J. Rheol.

32

FIGURE 4Maier et. al.J. Rheol.

10−3 10−2 10−1 100 101 102 103

10−18

10−16

10−14

10−12

10−10

10−8

10−6

10−4

10−2

100

33

FIGURE 5Maier et. al.J. Rheol.

mixture series 1: PS60&PS177(�;t0) = (1;18s)

(�;t0) = (2;18s)0.0 0.2 0.4 0.6 0.8 1.0

0*100

1*10−4

2*10−4

3*10−4

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.005

0.01

0.015

0.02

34

FIGURE 6Maier et. al.J. Rheol.

mixture series 1: PS60&PS177mixture series 2: PS56&PS250

mixture series 3: PS60&PS6431 2 3 4 5 6 7 8

0

5

10

15

20

25

30

35

�

�

35

FIGURE 7Maier et. al.J. Rheol.mixture series 1: PS60&PS177(�; t0) = (3:84;18 s)

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.05

0.1

36

FIGURE 8Maier et. al.J. Rheol.< F 1=�j >wi (�; t)

0.01 0.1 1.0

10−4

10−2

100

0.01 0.1 1.0

10−4

10−2

100

0.01 0.1 1.0

10−4

10−2

100

0.1 1.0 10.0

10−4

10−2

100

0.1 1.0 10.0

10−4

10−2

100

0.1 1.0 10.0

10−4

10−2

100

PS60 PS177(a)(b)(c)

37

FIGURE 9Maier et. al.J. Rheol.20% PS60 & 80% PS177

10−2 100 102

100

102

104

106

10−2 100 102

100

102

104

106

10−2 100 102

100

102

104

106

(a)

(b)

(c)

38

FIGURE 10Maier et. al.J. Rheol.20% PS60 & 80% PS177

1 2 4 7 10 20 40 70 100

*104

0.0

0.4

0.8

1.2

1.6

(a) � = 1Doi kernel

1 2 4 7 10 20 40 70 100

*104

0.0

0.4

0.8

1.2

1.6

(b) � = 2Doi kernel

1 2 4 7 10 20 40 70 100

*104

0.0

0.4

0.8

1.2

1.6

(c) � = 2des Cloizeaux kernel

1 2 4 7 10 20 40 70 100

*104

0.0

0.4

0.8

1.2

1.6

(d) � = 3:84single-expo. kernel

39