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\documentstyle[12pt]{article} \setlength{\textheight}{9in} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{0in} \setlength{\evensidemargin}{0in} \setlength{\headheight}{-0.30in} \setlength{\topmargin}{-0.30in} \renewcommand{\baselinestretch}{1.55} \begin{document} \vspace{\fill} \begin{center} \begin{Huge} Temporal Scaling Analysis:\\ Viscoelastic Properties of\\ Star Polymers$^{1}$\\ \end{Huge} \vspace{\fill} \begin{Large} George D. J. Phillies$^{*}$ Department of Physics Worcester Polytechnic Institute Worcester MA 01605 \end{Large} \end{center} %\vspace{\fill} \vfill \noindent $^{*}$ To whom communications may be addressed. EMail: phillies@wpi.wpi.edu (Internet). \vspace{1ex} \noindent $^{1}$ The partial support of this work by the National Science Foundation under Grant DMR94-23702 is gratefully acknowledged. \centerline{\tiny\today} \pagebreak \centerline{\Large\bf Abstract} A generalized temporal scaling ansatz for the frequency dependence of the loss and storage moduli and the shear dependence of the viscosity is tested against studies on entangled solutions of star polymers in good and theta solvents. At lower frequencies and shear rates, the ansatz calls for an exponential or stretched-exponential (e.g., $G_{0} \exp (- \alpha \omega^{\nu}))$ form for $G''(\omega)/\omega$, $G'(\omega)/\omega^{2}$, and $\eta(\kappa)$; at higher frequencies, the ansatz indicates that each of these quantities has a power-law dependence on its primary variable. Excellent agreement is found between the ansatz and literature data on solutions of poly-$\alpha$-methylstyrene, polybutadiene, polystyrene, and polyisoprene stars is found. A power-law correlation $\alpha \sim G_{0}^{2/3}$ is observed betweem the zero-frequency, zero-shear modulus and the low-frequency or low-shear decay constant of the stretched exponential. \pagebreak \centerline{\Large\bf Introduction} Recently, an ansatz\cite{Phillies99} has been advanced for describing the frequency dependence of the loss and storage moduli and the shear rate dependence of the viscosity of polymer solutions. The ansatz is based on a two-variable extension of the renormalization-group derivation\cite{RGderive} of the hydrodynamic scaling model\cite{HSM} of polymer dynamics, including the phenomenological\cite{Quinlan} analytic structure of the concentration dependence of the viscosity at elevated concentration. Preliminary tests\cite{Phillies99} of the ansatz against measured loss moduli find uniformly excellent agreement of the ansatz with experiment. It is not immediately transparent that the same ansatz would be applicable without modification to polymer melts. For the loss modulus, the ansatz predicts\cite{Phillies99} \begin{equation} \frac{G''(\omega)}{\omega} = G_{20} \exp(- \alpha_{2} \omega^{\nu}) \label{eq:G2s} \end{equation} for lower frequencies and \begin{equation} \frac{G''(\omega)}{\omega} = \bar{G_{2}} \omega^{x} \label{eq:G2p} \end{equation} at higher frequencies. For the storage modulus the corresponding forms are\cite{utspolystyrene} \begin{equation} \frac{G'(\omega)}{\omega^{2}} = G_{10} \exp(- \alpha_{1} \omega^{\nu}) \label{eq:G1s} \end{equation} for lower frequencies, and \begin{equation} \frac{G'(\omega)}{\omega^{2}} = \bar{G_{1}} \omega^{x} \label{eq:G1p} \end{equation} at higher frequencies. In these forms, $\nu$, $x$ and the $\alpha_{i}$ are scaling variables, $G_{20}$ and $G_{10}$ are zero-shear limits of the loss and storage moduli, and $\bar{G}_{2}$ and $\bar{G}_{1}$ are the $\omega = 1$ intercepts of the high-frequency forms for $G''/\omega$ and $G'/\omega^{2}$, respectively. At very high frequencies, an additional relaxation and a small constant term corresponding to the solvent must be incorporated in the above. Systematic comparison of the above equations with measured viscoelastic moduli\cite{utspolystyrene} of linear polystyrenes and polybutadienes in solution found good agreement with experiment. Because the functional forms above are in good agreement with the data to which they have thusfar been applied, conjectured relationships between $G'(\omega)$ and $G''(\omega)$ either must be satisfied within the fitting errors both by the forms and the data, or must be satisfied neither by the forms nor the data. The similarity in form between the frequency dependence of the loss modulus and the shear rate dependence of the low-shear viscosity implies for the dependence of $\eta$ on shear rate $\kappa$ that \begin{equation} \eta(\kappa) = \eta_{0} \exp(- \alpha_{\kappa} \kappa^{\nu}) \label{eq:etakdot1} \end{equation} at lower shear rates and \begin{equation} \eta(\kappa) = \bar{\eta} \kappa^{x} \label{eq:etakdot2} \end{equation} at more elevated shear rates. Equations \ref{eq:etakdot1} and \ref{eq:etakdot2} are also found \cite{thinning} to be in excellent agreement with results on linear polystyrenes in solution. In some models of polymer dynamics\cite{deGennes}, the available modes of polymer motion and hence concentrated-solution transport properties are strongly sensitive to chain architecture. The predicted differences arise because in entangled solutions some modes of polymer motion -- notable reptation -- are said to be available to linear chains but not to star polymers. Changes in viscoelastic properties of entangled polymer solutions attendant to changes in chain architecture therefore potentially supply sensitive tests of the accuracy and validity of those models. Furthermore, tests to date \cite{utspolystyrene,thinning} of the conjecture refer to linear polymers. It is therefore of significant interest to examine whether eqs \ref{eq:G2s}-\ref{eq:etakdot2} are only accurate when applied to linear chains, or whether they effectively describe viscoelastic properties of solutions of star polymers. In this paper, the temporal scaling ansatz is tested against literature data on the viscoelasticity of star polymers. The effect of chain architecture is studied by comparing these results with results on linear polymers that are chemically similar to the stars. The next section describes the actual fits of data to the conjectured equations. The final section treats fitting parameters, their concentration and molecular weight dependences, effects of chain architecture, and relationships to other published viscoelastic models. \vspace{1ex} \centerline{\Large\bf Comparison with Conjectured Forms} \vspace{1ex} In this section, eqs \ref{eq:G2s}-\ref{eq:etakdot2} are compared with published experiments on solutions of star polymers and their linear homologs, including results of Kajiura et al.\cite{Kajiura} and Isono, et al.\cite{Isono} on poly-$\alpha$-methylstyrene 3-arm stars and linear chains, Raju et al.\cite{Raju} on 4-arm polybutadiene stars and other polymers, Masuda et al.\cite{Masuda} on 6-arm polystyrene stars, and Graessley et al.\cite{Grpip} on linear and four-and six-arm polyisoprenes, as well as data\cite{Colby}-\cite{Nemoto} analyzed previously\cite{utspolystyrene,thinning} on linear polystyrenes and polybutadienes. Kajiura et al.\cite{Kajiura} report shear thinning of a molecular weight $3.57 \times 10^{6}$ three-arm star poly-$\alpha$-methylstyrene in $\alpha$-chloronaphthalene at concentrations 57-116 g/L, using a Weissenberg rheogoniometer cone-and-plate apparatus. Kajiura et al.'s data identify these measurements as referring to entangled stars, in the viscometric sense that the steady-state compliance satisfies $J_{e} \sim c^{-2}$ except perhaps at the lowest concentration studied. As seen in Fig 1 with fit parameters and fractional errors in Table I, eq \ref{eq:etakdot1} describes $\eta(\kappa)$ over the full observed range of shear rates and concentrations. The fitting parameters $\eta_{o}$ and $\alpha_{\kappa}$ both increase markedly with increasing polymer concentration. The range of shear rates examined in this work did not extend to the power-law regime of eq \ref{eq:etakdot2}, assuming that such a region exists in this system. The data is not sufficient to distinguish with certainty between a near-exponential $(\nu \approx 1)$ and a pure-exponential $(\nu = 1)$ dependence of $\eta$ on $\kappa$. Shear thinning in linear poly-$\alpha$-methylstyrenes was studied by Isono and Nagasawa\cite{Isono}, who observed a $1.65 \times 10^{6}$ molecular weight sample in $\alpha$-chloronaphthalene at concentrations 167-221 g/L using a Weisenberg rheogoniometer with cone-and-plate system. Based on Fig 6 of ref \cite{Isono}, these solutions were all entangled in the viscometric sense $J_{e} \sim c^{-2}$. As seen in Figure 2, eq \ref{eq:etakdot1} with $\nu = 1$ again describes $\eta(\kappa)$ to within measurement accuracy at all concentrations and shear rates. Fitting parameters (Table I) show the same qualitative trends for this data as for the data of Kajiura et al.\cite{Kajiura} on star polymers. Raju, et al.\cite{Raju} studied the dynamic moduli $G'(\omega)$ and $G''(\omega)$ of 3-arm (molecular weight $M=99,000$) and 4-arm($M =192,000$) polybutadiene stars in Flexon 391 (Exxon Chemical Company), and of linear polybutadienes, polystyrenes, polyisoprenes and hydrogenated linear polybutadienes in various solvents, using a mechanical spectrometer with an eccentric rotating disk geometry. Polymer volume fractions ranged from 0.109 up to 1.00. Based on the criterion $J_{e} \sim c^{-2}$, these solutions are all viscometrically entangled. Figures 3a and 3b and Table II show these measurements for the four-arm stars, together with fitting parameters based on egs \ref{eq:G2s}-\ref{eq:G1p}. With increasing $c$, $G_{10}$, $G_{20}$, $\bar{G}_{1}$, $\bar{G}_{2}$, and the $\alpha_{i}$ all increase markedly, while $\nu$ decreases modestly. At the same concentrations, $x$ has reached or soon reaches a fixed plateau value, different for the loss than for the storage modulus. As seen Figs 3a and 3b, agreement between each of the moduli and the temporal scaling forms is excellent over the entire range of observed frequencies ($0.001-300$ s$^{-1}$). Comparison is made below with the corresponding temporal scaling analysis\cite{utspolystyrene} results of Colby et al.\cite{Colby} on a 925 000 molecular weight linear polybutadiene in dioctyl phthalate and in phenyloctane at volume fractions 0.02-1. Colby et al.\cite{Colby}'s solutions all were at least marginally ($c [\eta] \geq 10$) entangled. Masuda et al.\cite{Masuda} examined 50 wt\% solutions of six-arm star polystyrenes in Kaneclor 500, a partially chlorinated biphenyl, determining $G'(\omega)$ and $G''(\omega)$ for polymers having $1.9 \times 10^{5} \leq M_{w} \leq 1.02 \times 10^{6}$. These measurements covered 6-7 orders of magnitude in frequency and 4-5 orders of magnitude in the value of the modulus. Based on $c [\eta]$ and the presence of a rubbery plateau in $G'(\omega)$, all solutions involved entangled stars. Masuda {\em et al.}'s\cite{Masuda}'s results and comparisons with the temporal scaling equations appear in Fig 4a and 4b, with fitting parameters in Table III. Over most of the frequency range examined, eqs \ref{eq:G2s}-\ref{eq:G1p} describe well the frequency dependences of the loss and storage moduli. In the low-frequency region, the $G_{i0}$ and $\alpha_{i}$ increase with increasing $M$, while $\nu$ falls from 1.0 to nearly 1/3. At high frequency, the power law forms are not quite adequate, for example because solvent contributions eventually become apparent. Consistent with earlier cases\cite{utspolystyrene} in which similar deviations were seen, addition of a constant $\eta_{1}$ to eq \ref{eq:G1p} was adequate to restore agreement with experiment. Similarly, addition of a terminal exponential (but not a terminal constant) $\eta_{1} \exp(- a \omega)$ to eq \ref{eq:G2p} gave agreement with experiment at all frequencies studied. The high-frequency parameters $\bar{G}_{1}$ and $\bar{G}_{2}$ decrease markedly with increasing $M$, while $x$ appears to saturate at larger $M$. Graessley et al.\cite{Grpip} report extensive measurements of viscosity and first normal stress of polyisoprene: tetradecane as a function of shear rate, including linear chains and 4- and 6-arm stars. Molecular weights extended up to $1.95 \times 10^{6}$; concentrations were as large as 340 g/L, including systematic variations of $c$ at fixed $M$ and $M$ at fixed $c$. Measurements were reported as superposition plots, one per value of arm number $f$; fits of single systems to eqs \ref{eq:G2s}-\ref{eq:etakdot2} are therefore precluded. However, Graessley et al. did report $\eta_{o}$ of eq \ref{eq:etakdot1}; they also reported the shear rate $\kappa_{o}$ at which $\eta(\kappa)$ had fallen to 80\% of $\eta_{o}$. Assuming $\nu \approx 1$ for shear thinning in this system, up to constants $\kappa_{o} \sim \alpha_{\kappa}^{-1}$ for $\alpha$ of eq \ref{eq:etakdot1}. These measurements are not readily used to validate the conjectured forms, but do shed light on certain systematicities of the fitting coefficients evident in the parameters of Tables I-III. In addition to these studies on star polymers in simple solvents, one notes results of Roovers\cite{Roovers} on polybutadiene stars in low- molecular-weight (but entangled) linear polybutadiene. Measurements in Ref \cite{Roovers} of the loss and storage moduli find results qualitatively similar to results treated above, but the additional complexities attendant to the solvent also being viscoelastic discourage a more detailed analysis here. There are also substantial studies on melts of star polymers, but these are beyond the scopy of this paper. \vspace{1ex} \centerline{\Large\bf Analysis} \vspace{1ex} For shear thinning in poly-$\alpha$-methylstyrene star and linear chain solutions, results of Kajiura et al.\cite{Kajiura} and Isono and Nagasawa\cite{Isono} agree that $\eta(\kappa)$ is an exponential $\eta_{0} \exp(- \alpha \kappa)$ in $\kappa$ out to the largest shear rates examined. $\eta_{o}$ and $\alpha$ (Table I) increase with increasing polymer concentration. Figure 5 compares these parameters for all poly-$\alpha$-methylstyrene solutions. The decay constant $\alpha$ and the zero shear viscosity are not independent. The line in the Figure is $\alpha \approx \alpha_{0} \eta_{0}^{2/3}$ for $\alpha_{0} = 7.9 \times 10^{-4}$. For poly-$\alpha$-methylstyrene solutions, the same function describes shear thinning of linear chain and star polymers. Furthermore, $\alpha$ and $\eta_{o}$ of linear and star polymers are linked by a power law having not only the same exponent but also at least very nearly the same proportionality constant. In many modern models \cite{deGennes} of polymer dynamics, linear chains and stars move through solution in dramatically different ways. If chain architecture actually has a dramatic effect on solution polymer dynamics, that effect is not apparent in shear thinning, for which the analysis here now shows \begin{equation} \eta(\kappa) = \eta_{0} \exp(- \kappa b \eta_{0}^{2/3}), \label{eq:shearthin} \end{equation} $b$ having the same value for linear and star polymers. Eq \ref{eq:shearthin} appears to be a novel quantitative result. Findings of Graessley et al.\cite{Grpip} on shear thinning in polyisoprene solutions are consistent with results on poly-$\alpha$- methylstyrene solutions. To describe $\eta(\kappa)$, Graessley et al. report $\eta_{0}$ and also a shear rate $\kappa_{o}$ such that $\eta(\kappa_{o})/\eta_{0} = 0.8$. For an exponentially decaying $\eta(\kappa)$, in terms of eq \ref{eq:etakdot1} up to constants $\kappa_{o} \sim \alpha^{-1}$. $\eta_{0}$ and $\kappa_{o}$ both increase markedly with increasing $c$ and $M$. Fig 6 plots $\kappa_{o}$ against $\eta_{o}$ for Graessley et al.'s\cite{Grpip} data. Writing $\kappa_{o} = \kappa_{o0} \eta_{o}^{-z}$, for $f = 2, 4, 6$ one has correspondingly $z = 0.61, 0.74, 0.76$ as best fits. These values of $z$ match well the 2/3-power exponent seen in Fig 5 for poly-$\alpha$-methylstyrenes, with perhaps a systematic increase in $z$ with increasing $f$. Graessley\cite{Graessley1,Graessley2} proposed a model for shear thinning in non-dilute solutions based on the dynamics of entanglement formation and on the depression of entanglement density by shear. The model employs a self-similarity assumption, namely that the characteristic time $\tau$ for formation of an entanglement in a solution at shear rate $\kappa$ is determined by the viscosity $\eta(\kappa)$ at the same shear rate, and not by the zero-shear limiting viscosity $\eta(0)$. How does this model compare with the analysis here? It is critical to reiterate that the {\em ansatz} leading to eqs \ref{eq:G2s}-\ref{eq:etakdot2} makes no assumption as to the nature of the dominant interpolymer interactions in solution. While the author has elsewhere\cite{RGderive,HSM} proposed that polymer solution dynamics under conditions above are in fact dominated by hydrodynamic rather than topological interchain interactions, the validity of that proposal\cite{RGderive,HSM} is entirely independent of the {\em ansatz} of ref \cite{Phillies99} being examined here. There is therefore no logical inconsistency in comparing the above analysis with Graessley's model\cite{Graessley1,Graessley2}. Graessley's model\cite{Graessley1,Graessley2} predicts that $\eta(\kappa)/\eta(0)$ is a specified, implicit, universal function of $\kappa \tau_{0}$, $\tau_{0}$ being the zero-shear-rate limit of $\tau$. Based on ref \cite{Graessley1}, $\tau_{0}$ is of the same magnitude as the longest viscoelastic relaxation time $\tau_{l}$ of the system of interest, so $\eta(\kappa)$ may equally be said to be a universal function of $\kappa \tau_{l}$. In this paper, from eq \ref{eq:etakdot1} $\eta$ is a function of $\alpha \kappa$, implying $\alpha \sim \tau_{l}$. In most entanglement based models\cite{deGennes}, $\tau_{l} \sim \eta_{0}$, implying $\alpha \sim (\eta_{0})^{1}$. The exponent of unity implied by Graessley's models\cite{Graessley1,Graessley2} is clearly larger than the observed exponent in $\alpha \sim \eta_{0}^{2/3}$. It is not apparent that this discrepancy would have been apparent if $\eta(\kappa)$ had here been analyzed by making superposition plots, as done in refs \cite{Graessley1,Graessley2}, as opposed to having a high-accuracy functional form to which individual sets of data were fit, as has been done here. The functional forms found here for the shear and loss moduli of star polymer solutions are precisely the same as those found earlier to describe well the same moduli of solutions of linear polymers\cite{utspolystyrene}. The fitting parameters of eqs \ref{eq:G2s}-\ref{eq:G1p} show systematic dependences on $c$ and $M$, the dependences seen here for star polymers being substantially the same as those seen earlier for linear polystyrenes\cite{utspolystyrene}. In specific: in the lower-frequency regime, the zero-frequency moduli $G_{0i}$ increase dramatically with increasing $c$ and $M$. The exponent $\nu$ is $\approx 1$ at lower molecular weights and concentrations, but decreases as $G_{0i}$ increases. $\alpha$ increases substantially with increasing $c$ at fixed $M$, but rather more weakly with increasing $M$ at fixed $c$. The $\bar{G}_{i}$ increase with increasing $c$, but at 50 wt\% $\bar{G}_{i}$ decreases with increasing $M$. The power-law slope $x$ increases with increasing $c$ and $M$; results here do not show that a large $c,M$ limit has been attained. There are well-known empirical correlations, such as the Cox-Merz rule, between shear thinning and the frequency dependence of the complex moduli. Motivated by these empirical rules, figures 7 and 8 examine if the correlation of figs 5, 6 is seen for the loss and storage moduli. Figure 7 treats polybutadiene solutions. Polybutadiene star polymer parameters are from Table II; parameters for linear polybutadienes are from ref \cite{utspolystyrene}, based on Colby et al.\cite{Colby}. In each case, $\alpha$ is plotted against $G_{10}$ or $G_{20}$ of eqs \ref{eq:G2s} or \ref{eq:G1s}. The parameters characterizing linear and star polybutadienes in good solvents are overlapped along the abscissa and lie on a common power-law line, the exponent $z$ in $\alpha = \alpha_{o} G_{i0}^{z}$ being 0.30 for the storage modulus ($i=1$) and 0.47 for the loss modulus($i=2$). At fixed $G_{i0}$, reducing the solvent quality generally increases $\alpha$ of linear chains (theta solvent points, plotted as open squares). Figure 8 treats linear and star polystyrenes in solution. Star polymer parameters are in Table III; linear polystyrene solutions were analyzed in ref \cite{utspolystyrene} using studies of Isono et al.\cite{Isono2}, Wolkowicz and Forsman\cite{Wolkowicz}, and Nemoto et al.\cite{Nemoto}. Unfortunately, there is very little overlap between values of $G_{i0}$ for linear and star polystyrenes. For fig 8a and $G''$, a single power law line with slope 1/2 characterises all of the somewhat scattered data. For fig 8b and $G'$, the displayed line of slope 0.3 describes linear chain and star polymer solutions having $G_{10} \leq 10^{11}$. Figs 7 and 8 exhibit a scaling relationship, common to both polymers, relating the zero-shear moduli and the corresponding frequency decay coefficients $\alpha$. The scaling exponent is approximately 0.5 for the loss modulus, and 0.3 for the storage modulus. The scaling relationship found in figs 5,6 for shear thinning is thus echoed, but not replicated, for the viscoelastic moduli. Each modulus corresponds to a different slope $z$. There are interesting similarities between the forms seen here for $G''(\omega)/\omega$, especially Fig 3b, and the theoretical curves of Morse\cite{Morse} for $G''(\omega)$. Treating semiflexible polymers, Morse's model describes a lower frequency regime in which $G''$ is dominated by curvature contributions to the stress, and a higher- frequency regime in which $G''$ is dominated by tension contributions to the stress. Tension contributions arise when short-wavelength bending modes both contribute significantly to the stress and are impeded by polymer entanglements. The tension contribution to $G''(\omega)$ is a higher-frequency tilted plateau describable (Figs 1 and 2 of ref \cite{Morse}) as an $\omega^{1/2}$ regime with crossovers at the ends. Morse's model specifically refers to semi-flexible polymers in which the persistance length of the polymer is much larger than the tube diameter. However, as noted by Morse, even a few cross-links between chains could lead to a large tension-dominated plateau; in the context of the model, it would appear that a star junction might serve the same purpose of inhibiting chain motions. In Fig 3b, plotting the loss modulus as $G''(\omega)/\omega$, a feature qualitatively like Morse's tension plateau appears as an additive exponential $\eta_{1} \exp(- a \omega)$ superposed on the background power-law decay. The exponential describes the plateau-like feature from beginning to end; a power-law (now $\omega^{-1/2}$) is tangent to part of the tilted plateau feature. The original model does not refer to star polymers, so these similarities in forms for $G''(\omega)$ cannot be said to confirm the model; the similarities are nonetheless striking. \pagebreak \begin{thebibliography}{66.} \bibitem{Phillies99} Phillies, G. D. J. {\em J. Chem.\ Phys.}, in press. \bibitem{RGderive} Phillies, G. D. J. {\em Macromolecules}, {\bf 1998}, 31, 2317-2327. \bibitem{HSM} Phillies, G. D. J., {\em J. Phys.\ Chem.} {\bf 1989}, 93, 5029-5039. \bibitem{Quinlan} Phillies, G. D. J., Quinlan, C. A. {\em Macromolecules}, {\bf 1995}, 28, 160-164. \bibitem{utspolystyrene} Phillies, G. D. J.; ``Temporal Scaling Analysis of Polymer Viscoelasticity: Linear Polystyrenes'', {\em submitted for publication}. \bibitem{thinning} Phillies, G. D. J., ``Temporal Scaling Analysis of Shear Thinning Effects: Linear Polystyrenes'', {\em submitted for publication} \bibitem{deGennes} deGennes, P.G. {\em Scaling Concepts in Polymer Physics}, Cornell U. P., Ithaca (1979). \bibitem{Kajiura} Kajiura, H.; Ushiyama, Y.; Fujimoto, T.; Nagasawa, M. {\em Macromolecules} {\bf 1979}, 11, 894. \bibitem{Isono} Isono, Y.; Nagasawa, M. {\em Macromolecules} {\bf 13}, 862-867 (1980). \bibitem{Raju} Raju, V. R.; Menenez, E. V.; Marin, G.; and Graessley, W.W. {\em Macromolecules} {\bf 1981}, 14, 1668-1676. \bibitem{Masuda} Masuda, T.; Ohta, Y.; Kitamura, M.; Saito, Y.; Kato, K.; and Onogi, S. {\em Macromolecules} {\bf 1981}, 14, 354-260. \bibitem{Grpip} Graessley, W. W.; Masuda, T.; Roovers, J. E. L.; Hadjichristidis, N. {\em Macromolecules} {\bf 1976}, 9, 127-141, especially Table II, samples series labelled P, S, and H. \bibitem{Colby} Colby, R.H.; Fetters, L. J.; Funk, W. G.; Graessley, W. W. {\em Macromolecules} {\bf 1991}, 24, 3873. \bibitem{Isono2} Isono, Y.; Fujimoto, T.; Takeno, N.; Kijura, H.; Nagasawa, M. {\em Macromolecules} {\bf 1978}, 11, 888. \bibitem{Wolkowicz} Wolowicz, R.I.; Forsman, W.C. {\em Macromolecules} {\bf 1985}, 18, 525. \bibitem{Nemoto} Nemoto, N.; Kishine, M.; Inoue, T.; Osaki, K. {\em Macromolecules} {\bf 1991}, 24, 1648. \bibitem{Roovers} Roovers, J. {\em Macromolecules} {\bf 1987}, 20, 148-152. \bibitem{Graessley1} Graessley, W. W. {\em J. Chem.\ Phys.} {\bf 1965}, 43, 2696. \bibitem{Graessley2} Graessley, W. W. {\em J. Chem.\ Phys.} {\bf 1967}, 47, 1942. \bibitem{Morse} Morse, D. C. {\em Macromolecules} {\bf 1998}, 31, 7044. \end{thebibliography} \pagebreak \centerline{\Large\bf Table I} \noindent Fit to eq \ref{eq:etakdot1} of viscosity of poly-$\alpha$-methylstyrenes as a function of shear rate, and percent fractional error RMS in each fit. Samples: a) molecular weight $3.57 \times 10^{6}$ three-arm star in $\alpha$-chloronaphthalene, from Kajiura et al\cite{Kajiura}; b) molecular weight $1.65 \times 10^{6}$ linear chain in $\alpha$-chloronaphthalene, after Isono and Nagasawa\cite{Isono}; [\ \ \ \ ] indicates a parameter frozen at an indicated value and not varied during the fitting process. \vspace{2ex} \noindent \begin{tabular}{|l|r|r|r|r|r|} \hline sample &$c$ (g/L) & $\eta_{0}$ & $\alpha$ & $\nu$ & RMS \\ \hline a & 56.9 & 28.2 & $6.40 \times 10^{-3}$ & [1] & 1.1 \\ a & 68.0 & 60.7 & $1.27 \times 10^{-2}$ & [1] & 1.3 \\ a & 72.1 & 79.9 & $1.38 \times 10^{-2}$ & [1] & 1.8 \\ a & 88.4 & 218 & $3.22 \times 10^{-2}$ & [1] & 1.8 \\ a & 101 & 429 & $5.00 \times 10^{-2}$ & [1] & 1.0 \\ a & 116 & 993 & 0.101 & [1] & 1.4\\ b & 167 & $5.06 \times 10^{3}$ & 0.229 & [1] & 3.2\\ b & 183 & $7.40 \times 10^{3}$ & 0.294 & [1] & 2.2\\ b & 201 & $1.31 \times 10^{4}$ & 0.411 & [1] & 2.3\\ b & 221 & $2.11 \times 10^{4}$ & 0.579 & [1] & 2.1\\ \hline \end{tabular} \pagebreak \centerline{\Large\bf Table II} \noindent Fit of the dynamic storage and loss moduli of 4-arm polybutadiene stars, from data of Raju {\em et al.}\cite{Raju} to temporal scaling forms $G_{0} \exp(-\alpha \omega^{\nu})$ and $\bar{G_{0}} \omega^{x}$, and root-mean-square percent fractional error RMS in each fit. [\ \ \ \ ] indicates a parameter frozen at an indicated value and not varied during the fitting process. \vspace{2ex} \noindent \begin{tabular}{|l|l|r|r|r|r|r|r|r|} \hline $\phi$ & $G$ & $G_{i0}$ & $\alpha$ & $\nu$ & RMS & $\bar{G_{i}}$ & $x$ & RME \\ \hline 0.108 & $G''$& 975 & 0.0080 & [1] & 5.2 & --& -- & -- \\ 0.206&$G'$ & $5.09 \times 10^{3}$& 0.207& 0.613 & 8.1 & -- & -- & -- \\ & $G''$ & $7.4 \times 10^{3}$& 0.064& 0.77& 3.5& $5.55 \times 10^{4}$ & 0.859& -- \\ 0.264& $G'$& $3.74 \times 10^{3}$ &0.609& 0.449& 8.2& $4.45 \times 10^{4}$ & 1.601& 6.2 \\ &$G''$ & $1.84 \times 10^{4}$& 0.075 & 0.86&3.7& $9.8 \times 10^{4}$ & 0.873& 1.0 \\ 0.403 & $G'$& $6.55 \times 10^{4}$& 0.77& 0.606& 7.5& $1.32 \times 10^{5}$ & 1.601& 6.2 \\ &$G''$&$1.16 \times 10^{5}$& 0.29& 0.724&5.4& $2.17 \times 10^{5}$ & 0.895& 5.1 \\ 0.609& $G'$& $7.68 \times 10^{6}$&2.87 &0.543&9.1& $5.19\times 10^{5}$ & 1.679& 4.1 \\ &$G''$& $1.59 \times 10^{6}$& 1.41& 0.588& 3.4& $4.14 \times 10^{5}$ & 0.869& 4.2 \\ 0.812& $G'$& $6.44 \times 10^{8}$& 7.95& 0.48&6.8& $1.22\times 10^{6}$ & 1.765& 5.5 \\ &$G''$&$1.7 \times 10^{7}$& 3.78& 0.44&5.4& $6.83\times 10^{5}$ & 0.850&2.2 \\ 1.00& $G'$& $2.78 \times 10^{10}$& 18.1& 0.46& 12& $2.3 \times 10^{6}$ &1.745& 7.1 \\ &$G''$& $1.68 \times 10^{8}$& 8.0& 0.38& 3.3& $1.03 \times 10^{6}$ & 0.868& 2.6 \\ \hline \end{tabular} \pagebreak \centerline{\Large\bf Table III} Fits of $G''(\omega)/\omega$ and $G'(\omega)/\omega^{2}$ 6-arm star polystyrenes in Kaneclor 500, based on Masuda, {\em et al.}\cite{Masuda} to eqs \ref{eq:G1s} and \ref{eq:G2s} at lower frequencies and to $\bar{G} \omega^{-x} + \eta_{1} \exp( - a \omega)$ at elevated frequencies, and percent root mean square fractional error (RMS) in the fit. \begin{small} \vspace{2ex} \noindent \begin{tabular}{|l|l|r|r|r|r|r|r|r|r|r|} \hline $M_{w}$ & $G$ & $G_{0}$ & $\alpha$ & $\nu$ & RMS & $\bar{G_{i}}$ & $x$ & $\eta_{1}$ & a & RMS \\ \hline 190 & $G''$ & $6.69 \times 10^{6}$& 4.48 & [1] & 7.3 & $8.21 \times 10^{5}$ & 0.67 & $3.08 \times 10^{5}$& $1.04 \times 10^{-2}$ & 8.1 \\ & $G'$ & $6.33 \times 10^{7}$& 9.4 & [1] & 17 & $9.1 \times 10^{5}$ & 1.59 & 620 & 0 & 6.9 \\ 316 & $G''$ & $2.45 \times 10^{7}$& 19.5 & [1] & 5.1 & $6.59 \times 10^{5}$ & 0.852 & $2.08 \times 10^{5}$& $8.5 \times 10^{-3}$ & 6.6 \\ & $G'$ & $8.26 \times 10^{8}$& 41 & [1] & 22 & $5.95 \times 10^{5}$ & 1.69 & 115 & 0 & 8.6 \\ 426 & $G''$ & $2.22 \times 10^{7}$& 15.1 & 0.76 & 6.3 & $2.49 \times 10^{5}$ & 0.946 & $8.1 \times 10^{4}$& $3.5 \times 10^{-3}$ & 6.5 \\ & $G'$ & $1.10 \times 10^{9}$& 34.7 & [1] & 14 & $1.43 \times 10^{6}$ & 1.68 & 2200 & 0 & 7.3 \\ 811 & $G''$ & $1.5 \times 10^{8}$& 49.9& 0.66 & 8.0 & $1.55 \times 10^{5}$ & 0.99 & $1.46 \times 10^{5}$& $1.5 \times 10^{-2}$ & 8.1 \\ & $G'$ & $1.14 \times 10^{11}$& 50.1 & 0.54 & 14 & $6.35 \times 10^{5}$ & 1.78 & 671 & 0 & 6.9 \\ 885 & $G''$ & $3.44 \times 10^{8}$& 29.7& 0.49 & 5.1& $1.16 \times 10^{5}$ & 1.08 & $1.53 \times 10^{5}$& $4.2 \times 10^{-2}$ & 3.7 \\ & $G'$ & $1.29 \times 10^{12}$& 27 & 0.33 & 7.8 & $7.29 \times 10^{5}$ & 1.848 & 1100 & 0 & 5.7 \\ 1020 & $G''$ & $7.8 \times 10^{8}$& 21.6& 0.36 & 6.6& $9.3 \times 10^{4}$ & 1.10 & $1.46 \times 10^{5}$& $6.6 \times 10^{-2}$ & 4.5 \\ & $G'$ & $3.19 \times 10^{12}$& 43.4& 0.37 & 8.1 & $6.1 \times 10^{5}$ & 1.86 & 0 & 0 & 4.5 \\ \hline \end{tabular} \end{small} \pagebreak \centerline{\Large\bf Figure Captions} \noindent Figure 1. Shear thinning of solutions of molecular weight $3.57 \times 10^{6}$ three-arm star poly-$\alpha$-methylstyrenes in $\alpha$-chloronaphthalene at concentrations (top to bottom) 116, 101, 88.4, 72.1, 68, and 56.9 g/L. Eq \ref{eq:etakdot1} with $\nu = 1$ describes $\eta(\kappa)$ well at all concentrations and shear rates studied. \noindent Figure 2. Shear thinning of solutions of molecular weight $1.65 \times 10^{6}$ linear poly-$\alpha$-methylstyrene in $\alpha$-chloronaphthalene at concentrations 221-167 g/L (top to bottom). Eq \ref{eq:etakdot1} with $\nu =1$ again describes $\eta(\kappa)$ well. \noindent Figure 3. a) Storage and b) loss modulus of 4-arm polybutadiene stars in Flexon-391 solutions at volume fractions (top-to-bottom at left margin) 1.00, 0.812, 0.609, 0.403, 0.264, 0.206, and 0.103, from data of Raju, {\em et al.}\cite{Raju} together with fits to eqs \ref{eq:G2s}-\ref{eq:G1p}, using fit parameters from Table II. Temporal scaling forms describe both moduli well over the full range of observed frequencies. \noindent Figure 4) a) Storage and b) loss moduli of 6-arm star polystyrenes in Kaneclor 500, based on results of Masuda, {\em et al.}\cite{Masuda} and fits to eqs \ref{eq:G2s} and \ref{eq:G1s} at lower frequencies and to $\bar{G} \omega^{-x} + \eta_{1} \exp( - a \omega)$ at elevated frequencies. Fitting parameters are in Table III. \noindent Figure 5) $\alpha_{\kappa}$ for shear thinning of 3-arm star(open circles) and linear (filled circles) poly-$\alpha$-methylstyrenes, plotted against $\eta_{o}$, based on Table I. Straight line is a best fit: $\alpha_{\kappa} = 7.89 \times 10^{-4} \eta_{o}^{0.667}$. \noindent Figure 6) $\kappa_{o}$ against $\eta_{o}$ for $f$-arm polyisoprene stars in tetradecalin, with $f$ of 2 (filled circles), 4 (open circles), and 6 (open squares), based on Graessley et al.\cite{Grpip}. Six high outlier points, not included in the fits, refer to samples of lower molecular weight. Solid lines are power laws. \noindent Figure 7) Scaling relation for polybutadiene 4-arm stars (Table II, filled points, data of Raju\cite{Raju}) and linear chains (open points, from ref \cite{utspolystyrene}, based on Colby et al.\cite{Colby}) in good (circles) and theta (squares) solvents, plotting $\alpha$ (eqs \ref{eq:G2s}, \ref{eq:G1s}) of the (a) loss modulus or (b) storage modulus against $G_{i0}$. A single power-law line describes entangled solutions of linear and star polymers in good solvents. \noindent Figure 8) Comparison of $\alpha_{i}$ and the corresponding a) $G_{20}$ (eq \ref{eq:G2s}) or b) $G_{10}$ (eq \ref{eq:G1s}) for solutions of polystyrene 6-arm stars (filled circles: Table III, data of Masuda et al.\cite{Masuda}) or polystyrene linear chains(open points: ref \cite{utspolystyrene}, based on Isono et al.\cite{Isono2} (circles), Wolkowicz and Forsman\cite{Wolkowicz} (starred circles), and Nemoto et al.\cite{Nemoto} (triangles). Solid line is a power law. \end{document}