Linear rheology of multiarm star polymers diluted with short linear chains a)

Linear rheology of multiarm star polymers diluted with short linear chains a) A. Miros and D. Vlassopoulos b) FORTH, Institute of Electronic Structure and Laser and Department of Materials Science and Technology, University of Crete, 7111 Heraklion, Crete, Greece A. E. Likhtman Physics Department, University of Leeds, Leeds LS2 9JT, United Kingdom J. Roovers NRC, Institute for Chemical Process and Environmental Technology, Ottawa, Ontario K1A R6, Canada (Received 8 July 22; final version received 3 October 22) Synopsis We present experimental results on the linear rheology of multiarm star/linear polymer mixtures, the latter having molecular weight much smaller than the star arm molecular weight. In such a case the linear chains act as ideal macromolecular solvents, which dilute entanglements of the arms. Using different star polymers we show that it is possible to account for this dilution and describe the linear rheology of the mixtures using the Milner McLeish theory for arm relaxation, complemented by the longitudinal modes of stress relaxation and high frequency Rouse modes. A universal description of the isofrictional arm relaxation time as a function of the number of entanglements is obtained for stars of any functionality and degree of dilution. The slow structural mode, related to the diluted star s colloidal core, also depends on the number of entanglements, but in a more complex way. 23 The Society of Rheology. DOI: 1.1122/1.1529172 I. INTRODUCTION It is now widely accepted that the tube model of entanglements can describe the complex rheological properties of polymer melts Doi and Edwards 1986 ; Marrucci et al. 1999 ; McLeish 1997, 22 ; McLeish and Milner 1999 ; Watanabe 1999 ; these range from linear polymers, where the dominant reptation mechanism is complemented by additional modes, the most notable of which is the contour length fluctuations Doi 1981, to branched polymers. The latter include star polymers, where arm relaxation takes place via an activated diffusion, because of the presence of the center branching point McLeish and Milner 1999 ; Fetters et al. 1993 and polymers that combine linear and star behavior, such as combs Daniels et al. 21a ; Roovers and Graessley 1981, H polymers McLeish et al. 1999 ; Roovers 1984, and pom-pom polymers a Dedicated to Professor G. Marrucci on the occasion of his 65th birthday. b Author to whom all correspondence should be addressed; electronic mail: dvlasso@iesl.forth.gr 23 by The Society of Rheology, Inc. J. Rheol. 47 1, 163-176 January/February 23 148-655/23/47 1 /163/14/$25. 163

164 MIROS ET AL. McLeish and Larson 1998 ; Houli et al. 22. Branched polymers are of considerable interest because of their practical implications in the understanding and characterization of the commercial long-chain branching effects, and also because they serve as systems for elucidating the influence of macromolecular architecture on the polymer rheology Larson 21 ; Wood-Adams and Costeux 21 ; Hatzikiriakos 2. Inherent to the description of the dynamics of branches is the concept of dynamic dilution at different time scales Ball and McLeish 1989. In general, the effect of dilution of the entanglement network on the rheological properties of well-defined branched polymers is a subject of considerable interest. Linear polymers in concentrated solutions exhibit a weight or volume fraction,, dependence of the entanglement molecular weight M e 5/4 which corresponds to plateau modulus of G N 9/4 ) under good solvent conditions Adam and Delsanti 1977 ; Colby and Rubinstein 199 and M e 4/3 which corresponds to G N 7/3 ) under theta conditions Adam and Delsanti 1984 ; Colby and Rubinstein 199. On the other hand, a recent systematic study with concentrated hydrogenated polybutadiene solutions in n-alkane solvents Tao et al. 1999 seems to support M e 1 scaling with G N 2 ). Nevertheless, despite the fact that the exact scaling between 4/3 and 1 is still unresolved, it is possible for the same polymer to control its rheological properties by selectively tuning the number of entanglements via the addition of solvent. A recent investigation Daniels et al. 21b examined entangled solutions of linear, three-arm star and H-shaped polyisoprenes in oligomeric theta-like solvent squalene, and demonstrated that by taking into account the dilution effects as well as some polydispersity and high frequency Rouse modes the tube models can describe the entire frequency spectrum well. In this work we consider blends of multiarm star and linear polymers within the limit of very small linear molecular weight (M linear ) compared to the star arm molecular weight (M a ). Such a case is viewed as star polymer solutions in a macromolecular solvent. For this case we demonstrate the validity of entanglement dilution and describe the linear rheology of these systems over the entire frequency range excluding the glass for a variety of blend compositions and arm molecular weights. In this respect the present investigation represents an extension of an earlier dilution study Daniels et al. 21b in several ways: i different chemistry here 1,4-polybutadienes are employed, confirming the universality of the findings. ii Very high number of arms, resembling the behavior of ultrasoft colloids Vlassopoulos et al. 21 ; Grest et al. 1996 ; Likos 21, which in addition to star arm relaxation, exhibit the structural relaxation mode as well, the latter also depending on the number of entanglements. iii Macromolecular solvent linear polymer instead of a molecular or oligomeric one. iv In a theoretical description of the dynamics, the longitudinal modes of stress relaxation are considered and their necessity is demonstrated. This approach adds two important additional parameters for future consideration, namely, the ratio of the linear polymer to the arm molecular weight and the architecture of the solvent e.g., the star. Section II describes the materials and techniques used. The main findings are presented in Sec. III, and discussed in view of the theories based on the tube model available, and the dilution of the network entanglements. Finally, a summary of the conclusions is presented in Sec. IV.

LINEAR RHEOLOGY OF MULTIARM STAR POLYMERS 165 TABLE I. Molecular characteristics of the 1,4-polybutadiene stars. code f M a g/mol T g ( C) R g a nm PB1 2 5 b 96 1.1 estimated 6415 6 12 1 92 12.7 643 56 23 9 92 18.5 646 61 47 5 92 28 a From light scattering measurements in dilute cyclohexane good solvent solution. b The effective arm molecular weight of the linear polymer is considered to be half its total molecular weight. II. EXPERIMENT A. Materials The 1,4-polybutadienes used in this study are listed in Table I, along with their molecular characteristics. The star polymers, with nominal functionality f 64 and varying arm molecular weight, were synthesized using a dendrimer scaffold and chlorosilane chemistry Roovers et al. 1993, whereas the linear one was purchased from Polymer Source, Canada. Star-linear polymer mixtures of different compositions were prepared by dissolution of the polymers in a good solvent cyclohexane about 5wt % total polymer concentration under gentle stirring for about 2 days, followed by solvent evaporation in a vacuum oven at room temperature for another 24 h. All samples used in this work were optically transparent. To reduce the risk of degradation, a small amount of antioxidant 2,6-di-tert-butyl-p-cresol.1% was added to the solution. B. Methods The dynamic response of the mixtures was studied with small amplitude oscillatory shear measurements under nitrogen atmosphere over a wide range of temperatures from 1 to 6 C). A Rheometric Scientific ARES strain controlled rheometer was employed with a dual range force rebalance transducer 2KFRTN1 and temperature control of.1 C achieved via a nitrogen convection oven. The sample was placed between two parallel plates of 8 mm diameter, reaching a gap of about 1.5 mm. Dynamic measurements consisted of strain sweeps to obtain the strain range that corresponded to the linear response for different frequencies, time sweeps to ensure stable conditions, and frequency sweeps in the range of 1.1 rad/s to obtain linear viscoelastic spectra of the storage (G ) and loss (G ) moduli. III. RESULTS AND DISCUSSION Figure 1 depicts master curves of G and G for the 646/PB1 mixture at different compositions, including the pure components. They were obtained by horizontal shifting of the individual frequency sweeps at different temperatures along the frequency axis, according to the time temperature superposition principle Ferry 198. Except for the PB1 linear component, all other samples exhibited entangled polymer behavior. As can be noted in Fig. 1 a, however, the plateau modulus G N drops significantly upon addition of the small linear chains, which effectively dilute the entanglements created by the star arms; the corresponding entanglement molecular weight increases. In this respect the short linear chains act as ideal macromolecular solvents. In fact, as long as the relation degennes 1979 M linear M a holds see Table I, the linear chains should penetrate the stars, although this is quite rough since the role of the functionality needs

166 MIROS ET AL. FIG. 1. Typical master curves of storage, G a, and loss, G b, moduli for a 646/PB1 mixture at different compositions from the top: 1/, ; 8/2, ; 5/5, ; 3/7, ; /1,, with a reference temperature of 19 K. further consideration. The data in Fig. 1 conform to the picture of short linear chains nearly uniformly penetrating the star molecules; this was experimentally confirmed using dynamic light scattering measurements Vlassopoulos et al. 1999. Furthermore, from Fig. 1 b one can appreciate the effects of the decreasing number of entanglements by adding solvent in reducing the G minimum. To quantify the dilution of the linear chains, we determined the plateau modulus and checked it against the Colby Rubinstein 199 prediction. Note that the G N determination from the G ( ) curve in Fig. 1 a is somewhat ambiguous, because G ( ) exhibits a weak power law rather than a true plateau, a feature expected for branched polymers Graessley and Roovers 1979 ; Roovers 1985 ; Vlassopoulos et al. 21 ; McLeish and Milner 1999 ; Islam et al. 21. Consequently, G N was also estimated from

LINEAR RHEOLOGY OF MULTIARM STAR POLYMERS 167 FIG. 2. Dependence of the plateau modulus G N a and entanglement molecular weight M e b on the volume fraction of the star in star-linear polymer PB1 blends: 646: ; 643: ; 6415:. Dotted and solid lines represent the scaling predictions for G N 7/3 and 2, respectively and M e ( 4/3 and 1, respectively, discussed in the text. integration of the G ( ) curve around the terminal region, according to G N 2/ G Gs d ln Ferry 198 ; Roovers 1985, 1986 ; the subscript s refers to the contribution from the high frequency Rouse-like transition region, which is practically identical to the linear case. However, this procedure involves some uncertainty, since additional mechanisms of relaxation such as contour length fluctuations and longitudinal modes, which also relate to the cut-off value of G s, were not considered. Nevertheless, the plateau values from this integration were comparable to those estimated from G ( ) directly. Figure 2 depicts the dependence of G N and M e on the star volume fraction in the three star/linear polymer mixtures investigated. The entanglement molecular weight was determined from M e ( RT)/G N with being the density; the prefactor 4/5 was consistently omitted throughout this work, since the theoretical star relaxation model does

168 MIROS ET AL. FIG. 3. Temperature dependence of the frequency shift factors for various stars, linear polymers, and star-linear blends, at a reference temperature of T ref 19 K. not include it either Milner and McLeish 1997, 1998. In addition, we note that the M e value of the pure stars was 1815 g/mol, conforming well to values reported in the literature Ferry 198 ; Milner and McLeish 1998. In this consistent manner, the comparison between model and experimental data is satisfactory for star polymers, as will be further discussed below. It is evident that, within experimental error, the data lie between the two slopes Colby and Rubinstein 199 ;Taoet al. 1999, namely, 7/3 and 2 and 4/3 and 1. Therefore, determination of the exact scaling of G N and M e with remains an unresolved problem that requires carefully designed experiments that involve a large variety of polymer systems and concentrations Fetters et al. 1994, 1999 ; Colby 1997, and this is beyond the scope of the present work. For the purposes of this study it is sufficient that the dilution works, and we choose the Colby Rubinstein 199 scaling exponents 7/3 and 4/3. Daniels et al. 21b have reported a similar observation of dilution for three-arm star and H-polyisoprenes with the oligomeric solvent squalene; note however, that squalene is probably a better solvent for polyisoprene than PB1 for polybutadiene Brandrup and Immergut 1989. Depsite the exact value of the exponent, in the present case it is interesting that given the topology of the multiarm stars the macromolecular dilution still holds; moreover, the macromolecular solvent here, PB1, having 22 repeat units, barely qualifies as an oligomeric solvent. There is a small part of the arm near the center in close contact to its neighbors that forms an effective core and may be partially stretched, and it is not known how much the short linear chains can penetrate. However, the measured G N of the stars is found to be very similar to that of linear chains, within experimental error Kapnistos et al. 1999 ; Pakula et al. 1998 ; Roovers 1986, neglecting any possible small temperature dependence Graessley 1982 ; this suggests that a small amount of stretching of the arms does not contribute appreciably to the plateau modulus. In addition, the deviations of the data in Fig. 2 from the theoretical slopes here taken for theta solvent conditions, the difference from good solvent being very small are really small. It is thus safe to conclude that the dilution effect is universal and that the multiarm star topology essentially does not affect it quantitatively. All mixtures exhibited the same thermorheologically simple behavior, as seen in Fig. 3, which depicts the temperature dependence of the frequency shift factor T. The well-known WLF expression Ferry 198, log T c 1 (T T ref )/(c 2 T T ref ) represented all data for various stars, linear polybutadiene and their mixtures well, with T ref 19 K and values of c 1 and c 2 being about 1 and 6 K, respectively.

LINEAR RHEOLOGY OF MULTIARM STAR POLYMERS 169 The terminal region is characterized by two-step relaxation. The faster of the two relaxation processes is well established as corresponding to star arm relaxation. A theoretical account of this process was recently presented by Milner and McLeish, who used the concept of dynamic dilution with appropriate scaling of the entanglement length Milner and McLeish 1997 ; Vega et al. 22. This theory was developed within the framework of the tube model and was proven successful in describing the arm relaxation of stars of any functionality and of different chemistry, without adjustable parameters Kapnistos et al. 1999 ; Milner and McLeish 1998. The only ones used, namely, the entanglement molecular weight M e, respective plateau modulus G N, and the Rouse relaxation time of an entanglement segment e N e 2 b 2 /3 2 k B T, with the monomeric friction coefficient and b the Kuhn segment entanglements set at 1/ e ) can be obtained from the data. The relaxation modulus G MM (t) is obtained from work by Milner and McLeish 1997 G MM t x 1 G N 1 ds 1 s x exp t/ s, 1 where s is the relaxed fraction of the arm and x 4/3 is the dilution exponent ( MM stands for Milner and McLeish. The total arm relaxation time, (s) (e Ueff(s) )/ early (s) 1/ activated (s) 1, where U eff is the effective potential, incorporates early fast diffusion of the free end of the arm and activated arm retraction, and depends on e and the number of entanglements per arm; the latter is reduced by the dynamic dilution effect. The analysis of the multiarm star data using this theory considers that a small fraction of the arm near the center is included in the core and does not contribute to this process. The high frequency region ( 1/ e ), in which tube constraints do not significantly affect the relaxation modes of the stars, proceed via Rouse dynamics as follows Doi and Edwards 1986 : G R t G N N a N a /N e exp 2n 2 t/ R, 2 n N a /N e with the Rouse time being R (N a /N e ) 2 e Likhtman and McLeish 22. Calculation of the contribution to the dynamic response is obtained through the appropriate Fourier transform. It is now straightforward to calculate the dynamic response of the mixtures in frequency space that encompasses the frequency range from arm relaxation to the Rouse modes using Eqs. 1 and 2. Typical results for melt 646 and the diluted star 646/ PB1 mixture are presented in Figs. 4 a and 4 b, respectively dotted lines. The necessary parameters of the theory used in the fitting procedure were obtained selfconsistently from the experimental data, discussed in detail by Kapnistos et al. 1999. Both the plateau modulus and entanglement molecular weight were determined from the frequency spectra Fig. 4 described above and shown in Fig. 2 as well. We did not obtain the friction coefficient by fitting; instead, we determined the Rouse time of an entanglement segment e from the high frequency limit of the rubber plateau Kapnistos et al. 1999 ; Ferry 198, and then as a check we determined the friction coefficient at 3 K using b.7 nm Fetters et al. 1994 and the shift factors in Fig. 3; we found that for the star systems considered here, varies from 1 1 1 to 3.1 1 1 N m/s, which is in good agreement with values reported in the literature Ferry 198 ; Milner and McLeish 1998 ; Vega et al. 22. The agreement between theory

17 MIROS ET AL. FIG. 4. Comparison of experimental data and theoretical predictions for G and G for the 646 star melt a and two star-linear 646/PB1 blends: 7/3 and 8/2 at T ref 19 K; the latter data were shifted vertically by a factor of 1 to facilitate a comparison. Solid lines represent Eq. 4 where the longitudinal relaxation was accounted for; dotted curves represent theory without the longitudinal modes Eqs. 1 and 2 only. The values of e used were 2 s 646 star, 4s 7/3, and 2.8 s 8/2. and experimental data is very good over the whole frequency range except for onset to the Rouse-like transition zone and the slow terminal relaxation mode, which will be discussed later and confirms that the Milner McLeish approach captures the basic physics of star arm relaxation for any functionality when the dilution effects of macromolecular solvents are properly accounted for. Furthermore, these results seem to indicate that dynamic dilution does not discriminate between intra- and intermolecular entanglements Vlassopoulos et al. 21 ; Grest et al. 1996. The former would be expected to be more prevalent in 64-arm stars than in 3-arm stars and linear polymers. The greatest disagreement between theory and experimental data observed in Fig. 4 that lies in the area of minimum of G needs some further consideration; to this end, we have added the longitudinal stress relaxation to the original Milner McLeish expressions, that was recently calculated for the case of linear chains Likhtman and McLeish 22.

LINEAR RHEOLOGY OF MULTIARM STAR POLYMERS 171 To describe the physical origin of this mode, let us consider small step deformation of the isotropic entangled melt. Because different tube segments are oriented differently before deformation, some of them will stretch and some of them will contract. The stress after the Rouse time of one entanglement segment e will be G N RT/M e. However, after time e, chain segments can become redistributed along the tube as a result of new segment lengths, i.e., some chain segments will move from compressed segments to stretched segments. After this relaxation, Doi Edwards theory predicts the stress to be equal to 4 5G N, i.e., 1/5 of the stress stored in the tube can relax after time t e as a result of the longitudinal mechanism described. This longitudinal relaxation must not be confused with contour length fluctuations CLFs. Whereas both mechanisms are derived from the bead-and-spring model of the chain inside a tube, the CLF mechanism describes escape from the original tube by fluctuations; on the other hand, longitudinal relaxation is due to motion inside the original tube. As mentioned earlier, Likhtman and McLeish 22 calculated the dynamics of this process for linear chains. Repeating the same derivation for the case of a star we get G long t 1 5 G N N a /N e (N a /N e ) 1 p exp p 1/2 2 t R. 3 The complete equation now reads G t G MM t G R t G long t. 4 It should be noted that although Eq. 4 is adequate for the present discussion, it is still not the final quantitative prediction. The calculation of both the early and the late times in the Milner McLeish approach is somewhat approximate, and an exact calculation is beyond the scope of the present work Figure 4 a shows the experimental linear viscoelastic data for the 646 star melt, along with the predictions of Eq. 4, by the solid curve. The effect of the longitudinal modes is remarkable indeed. Based on this comparison, it can be stated that Eq. 4 describes the full spectrum of star relaxation except for the segmental dynamics and the ultraslow dynamics of colloidal nature well and that longitudinal relaxation should be accounted for. In similar manner, Fig. 4 b demonstrates the success of this approach for the two diluted 646 stars at different linear chain concentrations and thus a different number of arm entanglements. The longitudinal mechanism again captures the data around the G minimum well. Note that the 7/3 data exhibit more noise, but, on the other hand, no vertical shifting was utilized. Despite the satisfactory description of the experimental data using Eq. 4, it should be kept in mind that the Milner McLeish 1997 model has limitations, which have been recently discussed in the literature; in particular, dynamic dilution apparently breaks down a few entanglement segments near the branch point which probably follow constraint release dynamics Watanabe et al. 22 ; Shanbhag et al. 22. However, it still remains the most complete and accurate model for star arm relaxation at the moment and as such it was employed in this comparison. Figure 5 presents the dependence of the arm relaxation time under isofrictional conditions, normalized to the segmental time and scaled with the number of arm entanglements ( a / s )(M a /M e ) 5/2, on the number of entanglements per arm M a /M e for a variety of star polymer melts with functionality ranging from 4 to 128, all being 1,4- polybutadienes Vlassopoulos et al. 21, as well as the present star polymer blends; in the latter case solvent-mediated dilution of entanglements Fig. 2 is taken into account in the horizontal axis (M a /M e ) in Fig. 5. The arm relaxation time was determined

172 MIROS ET AL. FIG. 5. Semilogarithmic representation of the normalized isofrictional arm relaxation time ( a / s ) (M a /M e ) 5/2 vs the number of entanglements per arm (M a /M e ) for various star polymers ( f 128, ; 64, ; 32, ; 18, ; 4, and star/linear 1 mixtures 6415, ; 646, *; 643, ). consistently for all samples from the inverse crossover frequency to the terminal region, whereas the segmental time from the inverse crossover frequency to the Rouse-like transition Pakula et al. 1998. This type of representation stems from development of the tube theory for arm relaxation see, e.g., Milner and McLeish 1997, that predicts a (M a /M e ) 5/2 exp ( /2 M a /M e ) with being the spring constant of the quadratic potential. This plot suggests rather universal behavior and differs from that reported by Watanabe and Kotaka 1983 and Watanabe et al. 1996a, who studied the viscoelastic relaxation of a mixture of a styrene core butadiene arms diblock copolymer with low molecular weight polybutadiene. They found that the micelle arm relaxation times were much longer as much as two orders of magnitude compared to the corresponding star arm relaxation, and concluded that this fast relaxation mechanism is similar but not completely the same in the two systems. Apart from the fact that the ratio of linear to star or micelle arm molecular weights, M linear /M a, is not the same in the two systems although it conforms to the above mentioned penetration criteria in both cases, an important difference relates to the larger core/shell ratio of the micelle Watanabe et al. 1996a compared to that of the star Vlassopoulos et al. 21. Therefore, this discrepancy provides further evidence of the difference between multiarm stars and block copolymer micelles Halperin 1987 ; Vlassopoulos et al. 1999. Another important difference that possibly affects the above results relates to the fact that whereas the polystyrene core of the micelles Watanabe et al. 1996b is glassy and thus rigid, that of the stars is rather fuzzy, i.e., soft and can deform Vlassopoulos et al. 21. The present well-defined star systems provide a clear physical picture of the diluted arm relaxation mechanisms. However, a few additional remarks are in order. Within the uncertainty due to scattering of the data, we find a value of the effective spring constant of the quadratic potential that is about.7 for M a /M e 2, which is smaller than the extracted value of.96 from lower functionality polyisoprenes Fetters et al. 1993 ; Rubinstein and Colby 22. For more than 2 25 entanglements per arm, the arm relaxation times in Fig. 5 apparently level off; this is probably not physical, but due rather to the procedure of extracting the relaxation times as well as to the 5/2 power which

LINEAR RHEOLOGY OF MULTIARM STAR POLYMERS 173 FIG. 6. Double logarithmic representation of the isofrictional structural relaxation time ( slow / s ) vs f 2.5 (M a /M e ) 5 for stars ( f 128, ; 64, and star/linear mixtures 646, ; 6415,. The line is to guide the eye. maybe too large. One can also observe small deviation of the blend data from the single star data, which however does not alter the conclusions drawn here. This could also relate partly to the extraction of the relaxation times and the number of diluted entanglements. The slow relaxation process, detected at the lowest frequencies in Fig. 1, is established as related to the cooperative structural rearrangements of the weakly ordered liquid-like stars Kapnistos et al. 1999. This type of ordering has been documented in the literature for stars with functionality f 128 or 64 arms, based on small angle x-ray scattering measurements Pakula et al. 1998. However, in the case of the blends, the star star distances increased due to the presence of linear chains that shift the ordering peak to outside of the detectable wave vector range. Despite the dilution effect, the slow mode can still be detected in a high-star content blend; as the linear chain concentration increases, this mode becomes weaker and eventually disappears at intermediate concentrations it fuses with arm relaxation. The mean-field scaling approach developed to account for star melt structural relaxation is applicable to the present diluted case as well. This mode is considered an activated process that involves partial disentanglement of the interpenetrating stars followed by displacement of the star into a neighboring cell, separated by a distance of its size, a process controlled by the free energy of corona elastic deformation arm stretching. The net result of this analysis suggests the following scaling relation Kapnistos et al. 1999 : slow 1/3 f 11/9 26/9 1 N X 1 a Ne exp 1/3 s N X 2 2 N a 3 a N e f 4/3, f 5/3 11/3 5 where (a 2 2/3 ) and a is the monomer size, and X 1 and X 2 are unspecified numerical constants. The main outcome is the strong dependence of structural relaxation on both the functionality and arm molecular weight, supporting the experimental findings. Based on the data from star melts with 64 and 128 arms, a plot of the isofrictional slow time slow / s vs f 2.5 (M a /M e ) 5 has been proposed as representing the structural mode of all stars Vlassopoulos et al. 21. For the case of mixtures, since arm disentanglement participates in structural relaxation, dilution by the macromolecular solvent should be considered. The is shown in Fig. 6 which is an attempt to describe the structural mode of

174 MIROS ET AL. all diluted stars. Given the complex nature of the scaling in Eq. 5, as well as the difficulty in accurate experimental determination of slow from the intersection of the terminal G 2 and G lines; see also Pakula et al. 1998, especially when low enough frequencies were not reached, the clear message in Fig. 6 is a universal trend of the experimental data, in qualitative agreement with the prediction. For completeness, we note again the difference of the present results from block copolymer micelles in a nonentangled matrix, where the slow mode was assigned to the Stokes Einstein diffusion of micelles Watanabe et al. 1996a, 1998 ; Gohr and Schärtl 2. A issue that remains is the role of the size ratio M linear /M a, as already mentioned. Whereas the present results support the penetration of small chains into the stars, at higher linear chain molecular weights the entropic cost of penetration is too high Raphaël et al. 1993 ; Halperin and Alexander 1988 ; Leibler and Pincus 1984 and the conformation of the mixture as well as its properties is different; in such a case it should be treated as a star-linear mixture in which both arm relaxation of the star and reptation of the linear chain participate and should be accounted for Milner et al. 1998 ; Roovers 1987 ; Struglinski et al. 1988. At the same time, the importance of star functionality should not be underestimated, since it can lead to a non-negligible core size and eventually the mixture can exhibit many similarities to micelle/linear polymer systems the slow relaxation mechanism being the most notable one Watanabe et al. 1996b ; Watanabe and Kotaka 1984 ; Gohr et al. 1999 ; Gohr and Schärtl 2. Naturally, the crossover of M linear from macromolecular solvent behavior to a star-linear blend is of particular interest, and it will be addressed in the future. IV. CONCLUDING REMARKS The linear rheology of mixtures of multiarm star and linear polymers having molecular weight much smaller than the star arm molecular weight as investigated. These systems were considered solutions of stars in macromolecular solvents which dilute entanglements of the arms. Using a variety of mixtures different star arm molecular weights and compositions we were able to describe the response of diluted stars over the entire frequency spectrum excluding the glass, based on the the Milner McLeish theory for arm relaxation, the longitudinal stress relaxation which was introduced and calculated for the first time for star polymers, and the high frequency Rouse modes. A virtually universal description of isofrictional arm relaxation time as a function of the number of entanglements was obtained for stars of any functionality and degree of dilution. The slow structural mode, related to the diluted star s colloidal core, also depends on the number of entanglements, as was indicated by a recent mean field scaling approach, but in a rather complex way; nevertheless, good qualitative agreement with the data was attained. ACKNOWLEDGMENTS This work was carried out at the Institute for Theoretical Physics, University of California, Santa Barbara; two of the authors A.E.L. and D.V. would like to acknowledge support by the National Science Foundation under Grant No. PHY99-7949. Additional support was received from the European Union Grant No. HPRN-CT-2-17.

LINEAR RHEOLOGY OF MULTIARM STAR POLYMERS 175 References Adam, M., and M. Delsanti, Dynamical properties of polymer solutions in good solvent by Rayleigh scattering experiments, Macromolecules 1, 1229 1237 1977. Adam, M., and M. Delsanti, Viscosity and longest relaxation time of semi-dilute polymer solutions. II. Theta solvent, J. Phys. France 45, 1513 1521 1984. Ball, R. C., and T. C. B. McLeish, Dynamic dilution and the viscosity of star-polymer melts, Macromolecules 22, 1911 1913 1989. Brandrup, J., and E. H. Immergut, Polymer Handbook, 3rd ed. Wiley, New York, 1989. Colby, R. H., Dynamics in blends of long polymers with unentangled short chains, J. Phys. II 7, 93 15 1997. Colby, R. H., and M. Rubinstein, Two-parameter scaling for polymers in solvents, Macromolecules 23, 2753 2757 199. Daniels, D. R., T. C. B. McLeish, B. J. Crosby, R. N. Young, and C. M. Fernyhough, Molecular rheology of comb polymer melts. 1. Linear viscoelastic response, Macromolecules 34, 725 733 21a. Daniels, D. R., T. C. B. McLeish, R. Kant, B. J. Crosby, R. N. Young, A. Pryke, J. Allgaier, D. J. Groves, and R. J. Hawkins, Linear rheology of diluted linear, star and model long chain branched polymer melts, Rheol. Acta 4, 43 415 21b. de Gennes, P. G., Scaling Concepts in Polymer Physics Cornell University Press, Ithaca, NY, 1979. Doi, M., Explanation for the 3.4 power law of viscosity of polymeric liquids on the basis of the tube model, J. Polym. Sci., Polym. Lett. Ed. 19, 265 273 1981. Doi, M., and S. F Edwards, The Theory of Polymer Dynamics Oxford, New York, 1986. Ferry, J. D., Viscoelastic Properties of Polymers, 3rd ed. Wiley, New York, 198. Fetters, L. J., D. J. Lohse, S. T. Milner, and W. W. Graessley, Packing length influence in linear polymer melts on the entanglement, critical and reptation molecular weights, Macromolecules 32, 6847 6851 1999. Fetters, L. J., A. D. Kiss, D. S. Pearson, G. F. Quack, and F. J. Vitus, Rheological behavior of star-shaped polymers, Macromolecules 26, 647 654 1993. Fetters, L. J., D. J. Lohse, D. Richter, T. A. Witten, and A. Zirkel, Connection between polymer molecular weight, density, chains dimensions and melt viscoelastic properties, Macromolecules 27, 4639 4647 1994. Gohr, K., and W. Schärtl, Dynamics of copolymer micelles in a homopolymer melt: Influence of the matrix molecular weight, Macromolecules 33, 2129 2135 2. Gohr, K., T. Pakula, T. Kiyoharu, and W. Schärtl, Dynamics of copolymer micelles in an entangled homopolymer matrix, Macromolecules 32, 7156 7165 1999. Graessley, W. W., Effect of long branches on the temperature dependence of viscoelastic properties in polymer melts, Macromolecules 15, 1164 1167 1982. Graessley, W. W., and J. Roovers, Melt rheology of four-arm and six-arm star polystyrenes, Macromolecules 12, 959 965 1979. Grest, G. S., L. J. Fetters, J. S. Huang, and D. Richter, Star polymers: Experiment, theory and simulation, Adv. Chem. Phys. XCIV, 65 163 1996. Halperin, A., Polymeric micelles: A star model, Macromolecules 2, 2943 2946 1987. Halperin, A., and S. Alexander, On the dynamics of densely grafted layers. The effect of stretched configurations, Europhys. Lett. 6, 329 334 1988. Hatzikiriakos, S. G., Long chain branching and polydispersity effects on the rheological properties of polyethylenes, Polym. Eng. Sci. 4, 2279 2287 2. Houli, S., H. Iatrou, N. Hadjichristidis, and D. Vlassopoulos, Synthesis and viscoelastic properties of model dumbbell copolymers consisting of a polystyrene connector and two 32-arm star polybutadienes, Macromolecules 35, 6592 6597 22. Islam, M. T., L. A. Archer, Juliani, and S. K. Varshney, Linear rheology of entangled six-arm and eight-arm star polybutadienes, Macromolecules 34, 6438 6449 21. Kapnistos, M., A. N. Semenov, D. Vlassopoulos, and J. Roovers, Viscoelastic response of hyperstar polymers in the linear regime, J. Chem. Phys. 111, 1753 1759 1999. Larson, R. G., Combinatorial rheology of branched polymer melts, Macromolecules 34, 4556 4571 21. Leibler, L., and P. A. Pincus, Ordering transition of copolymer micelles, Macromolecules 17, 2922 2924 1984. Likhtman, A. E., and T. C. B. McLeish, Quantitative theory for linear dynamics of linear entangled polymers, Macromolecules 35, 6332 6343 22. Likos, C. N., Effective interactions in soft condensed matter physics, Phys. Rep. 348, 267 439 21. Marrucci, G., F. Greco, and G. Ianniruberto, Rheology of polymer melts and concentrated solutions, Curr. Opin. Colloid Interface Sci. 4, 283 287 1999. McLeish, T. C. B., ed., Theoretical Challenges in the Dynamics of Complex Fluids, NATO ASI Vol. 339 Kluwer, London, 1997. McLeish, T. C. B., Tube theory of entangled polymer dynamics, Adv. Phys. 51, 1379 1527 22.

176 MIROS ET AL. McLeish, T. C. B., and R. G. Larson, Molecular constitutive equations for a class of branched polymers: The pom-pom polymer, J. Rheol. 42, 81 11 1998. McLeish, T. C. B., and S. T. Milner, Entangled dynamics and melt flow of branched polymers, Adv. Polym. Sci. 143, 195 256 1999. McLeish, T. C. B., et al., Dynamics of entangled H-polymers: Theory, rheology and neutron-scattering, Macromolecules 32, 6734 6758 1999. Milner, S. T., and T. C. B. McLeish, Parameter-free theory for stress relaxation in star polymer melts, Macromolecules 3, 2159 2166 1997. Milner, S. T., and T. C. B. McLeish, Arm-length dependence of stress relaxation in star polymer melts, Macromolecules 31, 7479 7482 1998. Milner, S. T., T. C. B. McLeish, R. N. Young, A. Hakiki, and J. M. Johnson, Dynamic dilution, constraintrelease, and star-linear blends, Macromolecules 31, 9345 9353 1998. Pakula, T., D. Vlassopoulos, G. Fytas, and J. Roovers, Structure and dynamics of melts of multiarm polymer stars, Macromolecules 31, 8931 894 1998. Raphaël, E., P. Pincus, and G. H. Fredrickson, Conformation of star polymers in high molecular weight solvents, Macromolecules 26, 1996 26 1993. Roovers, J., Melt rheology of H-shaped polystyrenes, Macromolecules 17, 1196 12 1984. Roovers, J., Properties of the plateau zone of star-branched polybutadienes and polystyrenes, Polymer 26, 191 195 1985. Roovers, J., Linear viscoelastic properties of polybutadiene: A comparison with molecular theories, Polym. J. Tokyo 18, 153 162 1986. Roovers, J., Tube renewal in the relaxation of 4-arm-star polybutadiens and linear polybutadienes, Macromolecules 2, 148 152 1987. Roovers, J., and W. W. Graessley, Melt rheology of some model comb polystyrenes, Macromolecules 14, 766 773 1981. Roovers, J., L. L. Zhou, P. M. Toporowski, M. van der Zwan, H. Iatrou, and N. Hadjichristidis, Regular star polymers with 64 and 128 arms. Models for polymeric micelles, Macromolecules 26, 4324 4331 1993. Rubinstein, M., and R. H. Colby, Polymer Physics Oxford, New York, 22. Shanbhag, S., R. G. Larson, J. Takimoto, and M. Doi, Deviations from dynamic dilution in the terminal relaxation of star polymers, Phys. Rev. Lett. 87, 19552 22. Struglinski, M. J., W. W. Graessley, and L. J. Fetters, Experimental observations on binary mixtures of linear and star polybutadienes, Macromolecules 21, 783 789 1988. Tao, H., C. Huang, and T. P. Lodge, Correlation length and entanglement spacing in concentrated hydrogenated polybutadiene solutions, Macromolecules 32, 1212 1217 1999. Vega, D. A., J. M. Sebastian, W. B. Russel, and R. A. Register, Viscoelastic properties of entangled star polymer melts: Comparison of theory and experiment, Macromolecules 35, 169 177 22. Vlassopoulos, D., G. Fytas, T. Pakula, and J. Roovers, Multiarm star polymer dynamics, J. Phys.: Condens. Matter 13, R855 R876 21. Vlassopoulos, D., G. Fytas, G. Fleischer, T. Pakula, and J. Roovers, Ordering and dynamics of soft spheres in melt and solution, Faraday Discuss. 112, 225 235 1999. Watanabe, H., Viscoelasticity and dynamics of entangled polymers, Prog. Polym. Sci. 24, 1253 143 1999. Watanabe, H., and T. Kotaka, Viscoelastic properties of blends of styrene-butadiene diblock copolymer and low molecular weight homopolybutadiene, Macromolecules 16, 769 774 1983. Watanabe, H., and T. Kotaka, Viscoelastic properties of blends of styrene-butadiene diblock copolymer and high molecular weight homopolybutadiene, Macromolecules 17, 342 348 1984. Watanabe, H., Y. Matsumiya, and T. Inoue, Dielectric and viscoelastic relaxation of highly entangled star polyisoprene: Quantitative test of tube dilation model, Macromolecules 35, 2339 2357 22. Watanabe, H., T. Sato, and K. Osaki, Viscoelastic properties of styrene-butadiene diblock copolymer micellar systems. 1. Behavior in nonentangling, short polybutadiene matrix, Macromolecules 29, 14 112 1996a. Watanabe, H., T. Sato, and K. Osaki, Viscoelastic properties of styrene-butadiene diblock copolymer micellar systems. 2. Behavior in entangling, long polybutadiene matrices, Macromolecules 29, 113 118 1996b. Watanabe, H., T. Sato, K. Osaki, M. W. Hamersky, B. R. Chapman, and T. P. Lodge, Diffusion and viscoelasticity of copolymer micelles in a homopolymer matrix, Macromolecules 31, 374 3742 1998. Wood-Adams, P., and S. Costeux, Thermorheological behavior of polyethylene: Effects of microstructure and long chain branching, Macromolecules 34, 6281 629 21.