Unified application of the coupling model to segmental, Rouse, and terminal dynamics of entangled polymers

Journal of Non-Crystalline Solids 352 (2006) 342–348 www.elsevier.com/locate/jnoncrysol

Unified application of the coupling model to segmental, Rouse, and terminal dynamics of entangled polymers Christopher G. Robertson

a,*

, Liviu Iulian Palade

b,c,*

a

b

Bridgestone Americas, Center for Research and Technology, 1200 Firestone Parkway, Akron, OH 44317-0001, United States Centre de Mathe´matiques, Institut National des Sciences Applique´es de Lyon, Baˆtiment Leonard de Vinci, 21 Avenue Jean Cappelle, 69621 Villeurbanne cedex, France c Institut Camille Jordan, CNRS UMR 5208, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France Received 17 August 2005; received in revised form 19 December 2005 Available online 9 February 2006

Abstract The temperature dependence and relaxation function breadth of segmental dynamics (a-relaxation) for 1,2-polybutadiene and 1,4polybutadiene are used to predict their respective temperature-dependent terminal relaxation times by unified application of the Ngai coupling model. Literature results for the terminal flow of near-monodisperse linear polybutadienes having widely varying molecular weights are successfully represented using the coupling model by variation of only a single parameter, C, which is the proportionality constant between the longest Rouse relaxation time and the primitive relaxation time which underlies the cooperative segmental process. The value of C varies with molecular weight (M) according to C / Mb where b is found to range from 1.8 to 2.1, in close agreement with the expected exponent of 2. Contrary to experimental data and coupling model predictions, reptation theory predicts identical influences of temperature on Rouse and terminal relaxation processes; we suggest that invoking a temperature dependence for contour length fluctuations, and hence number of effective entanglements per chain, may resolve this deficiency of the tube model.
2006 Elsevier B.V. All rights reserved. PACS: 83.60.Bc; 83.10.
y; 81.05.Lg; 81.05.Kf Keywords: Dielectric properties; Relaxation; Electric modulus; Glass formation; Polymers and organics; Glass transition; Rheology; Structural relaxation; Viscoelasticity

1. Introduction

1=ð1
na Þ

sa ðT Þ / ½n0 ðT Þ

; sR ðT Þ / n0 ðT Þ;

1=ð1
nF Þ

It is common for the segmental relaxation (a-relaxation) time, sa, and terminal flow relaxation time, sF, to exhibit distinct temperature dependences for polymers [1–7]. The coupling model of Ngai [8–11] is a useful tool to interpret such thermorheological complexity. According to the coupling model, the segmental, Rouse, and terminal relaxation processes all depend uniquely on the temperature-dependent monomeric friction coefficient, n0(T): * Corresponding authors. Tel.: +1 330 379 7512; fax: +1 330 379 7530 (C.G. Robertson), tel.: +33 472438916; fax: +33 472438529 (L.I. Palade). E-mail addresses: [email protected] (C.G. Robertson), [email protected] (L.I. Palade).

0022-3093/$ - see front matter
2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.01.001

sF ðT Þ / ½n0 ðT Þ

.

ð1Þ

In the above, the longest Rouse relaxation time is represented by sR and the coupling parameters for a-relaxation and terminal flow are given by na and nF, respectively. At very high temperatures, segmental cooperativity ceases according to the coupling model which is in agreement with the experimentally noted change in dynamics well-above the glass transition temperature of glass-forming liquids [12]. For the very special case where both segmental and entanglement couplings are no longer active, relaxation response should be thermorheologically simple: sF ðT Þ / sR ðT Þ / sa ðT Þ / n0 ðT Þ.

ð2Þ

C.G. Robertson, L.I. Palade / Journal of Non-Crystalline Solids 352 (2006) 342–348

343

For practical temperature and frequency/time ranges, however, relaxation couplings play major roles; the disparities among the temperature dependences of various motions therefore remain, although they can become less noticeable at high temperatures where n0(T) becomes less sensitive to temperature. The coupling model relationships given in Eq. (1) suggest that thermorheological complexity is always expected when measurements are conducted across a wide enough expanse of frequency/time and temperature, even for the simplest polymers with amorphous, linear structures. The isothermal breadth of the segmental relaxation, measured by dielectric or dynamic mechanical spectroscopies, is simply related to the segmental coupling parameter by na = 1
b, where b is the exponent of the well-known Kohlrausch–Williams–Watts [13,14] (KWW) function. The experimental value of na depends on polymer chemistry, ranging from 0.45 for polyisobutylene to 0.76 for polyvinylchloride [15]. In contrast, the coupling model relationship for the zero shear viscosity in the terminal flow region [9–11]:

herein referred to as 1,4-polybutadiene. The segmental relaxation times from the mechanical data were assigned according to sa = (xmax)
1 where xmax is the oscillatory frequency, in units of rad/s, where the loss modulus displays a maximum. These results are plotted in Fig. 1. The VFTH parameters which capture these data are Aa = 6.607 · 10
15 s, Ba = 1214 K, and T1,a = 139.1 K. In a similar manner, dielectric results from Hofmann et al. [31] and Colmenero et al. [32] were combined with dynamic shear data [26] for 1,2-polybutadiene (1,2-content > 93%) as shown in Fig. 2. These sa data for 1,2-polybutadiene can be represented by the following VFTH parameters: Aa = 1.023 · 10
18 s, Ba = 1601 K, and T1,a = 225.6 K. The Tg-normalized temperature dependence of segmental relaxation time, or fragility, can depend on molecular weight [33–35]. Evidence suggests that the concentration of chain ends is the dominant influence in the molecular weight dependence of both Tg and fragility [36]. Indeed, both Tg and fragility reach plateau values at a similar molecular weight [33–35]. The M-dependence of Tg for 1,4-polybutadiene was reported by Colby et al. [23] to be

g0 / M 2=ð1
nF Þ

T g ðKÞ ¼ 174
12:1=M ðkg=molÞ.

indicates that the value of nF should be approximately constant at a value near 0.41 for all entangled linear polymers [9,16,17] based on the usual approximately 3.4 power law exponent observed for the molecular weight (M) dependence of viscosity [18–23]. Even for the case of polyisobutylene where na  nF and the effects of temperature on sa and sF are similar, thermorheological complexity is still noted in the softening zone due to the weaker temperature dependence of sR [24,25]. The widespread failure of time–temperature superposition is anticipated by the coupling model as illustrated in Eq. (1). Robertson and Rademacher [26] recently demonstrated the unified application of the coupling model to mathematically link the segmental and terminal dynamics, thus allowing one to predict sF(T) from sa(T) with the use of only a single adjustable parameter. In this study, we further test the utility of this approach by considering literature terminal relaxation data for linear 1,2-polybutadienes and 1,4-polybutadienes as a function of molecular weight. 2. Results and discussion Temperature-dependent sa data are commonly represented using the Vogel–Fulcher–Tammann–Hesse (VFTH) expression [27–29]:   Ba sa ðT Þ ¼ Aa exp ; ð4Þ T
T 1;a where Aa, Ba, and T1,a are fitting parameters. The sa(T) data from the dynamic oscillatory shear measurements of Robertson and Rademacher [26] were combined with dielectric data of Arbe et al. [30] for standard anionicallypolymerized polybutadienes with low 1,2-content (<10%),

ð5Þ

Analysis of the Tg data of Roovers and Toporowski [37] for 1,2-polybutadiene gives: T g ðKÞ ¼ 274
12:9=M ðkg=molÞ.

ð6Þ

Therefore, the sa(T) data for the 38.7 kg/mol 1,4-polybutadiene and the 80.1 kg/mol 1,2-polybutadiene studied 105 1,4-polybutadiene

relaxation time, τ (s)

ð3Þ

100

τF

10-5

τα

10-10

10-15 0.6

0.8 Tg / T

1.0

Fig. 1. Relaxation times for a-relaxation (mechanical: [26] n; dielectric: [30] h) and terminal flow of 1,4-polybutadiene. The solid circles are sF data assigned from the frequency location of the terminal peak in G00 as previously reported [26], and the open circles are sF determined from superposition of new data acquired on the same 1,4-polybutadiene at other temperatures. The solid line through sa(T) is the VFTH fit. The solid line through sF(T) is the coupling model fit to the terminal relaxation data by variation of C only (C = 6.1 · 106), and the dashed line is this coupling model fit shifted downward to allow comparison with the temperature dependence of the a-relaxation. The dielectric a-relaxation results of Arbe et al. [30] for T < 235 K are vertically scaled by a factor of 10
0.1 to overlap the mechanical results of Robertson and Rademacher [26]. The reference Tg is the temperature at which sa = 100 s.

344

C.G. Robertson, L.I. Palade / Journal of Non-Crystalline Solids 352 (2006) 342–348 105

relaxation time, τ (s)

1,2-polybutadiene

100

τF

10-5

τα

10-10

10-15 0.7

0.8

0.9 Tg / T

1.0

Fig. 2. Relaxation times for a-relaxation (mechanical: [26] n; dielectric: [31,32] h) and terminal flow of 1,2-polybutadiene. The solid circles are sF data assigned from the frequency location of the terminal peak in G00 as previously reported [26], and the open circles are sF determined from superposition of new data acquired on the same 1,2-polybutadiene at other temperatures. The solid line through sa(T) is the VFTH fit. The solid line through sF(T) is the coupling model fit to the terminal relaxation data by variation of C only (C = 4.6 · 107), and the dashed line is this coupling model fit shifted downward to allow comparison with the temperature dependence of the a-relaxation. The dielectric data of Hofmann et al. [31] and Colmenero et al. [32] are vertically scaled by factors of 10
3.3 and 10
2.95, respectively, to overlap the mechanical of Robertson and Rademacher [26]. The reference Tg is the temperature at which sa = 100 s.

previously by Robertson and Rademacher [26] should be representative of the limiting high molecular weight behavior, and we can use these results for all samples of interest herein. The shape of the loss modulus, G00 , versus frequency response in the segmental region is largely invariant to temperature for 1,4-polybutadiene (Fig. 3) and 1,2-poly-

butadiene (not shown here; see Ref. [26]). However, the corresponding peaks in tan d occur two decades lower in frequency (Fig. 4) and therefore reflect appreciable contribution from Rouse relaxation. It is clear that the superposition principle is violated for tan d in this region, and this specific experimental manifestation of thermorheological complexity is quite common [1–3,38]. The softening zone is where segmental and terminal relaxation functions, often with different temperature dependences, overlap. While this can certainly cause failure of time–temperature superposition, this is also the region where Rouse dynamics play a key role. We believe that the weaker influence of temperature on the Rouse process, as predicted by the coupling model in Eq. (1), is the dominant feature which leads to the extreme thermorheological complexity of tan d in the transition/softening zone. The KWW function [13,14] is typically used to describe the breadth of segmental relaxation response: b

GðtÞ ¼ Gg exp½
ðt=sK Þ .

ð7Þ

In the above, G(t) is the time-dependent relaxation modulus, Gg is the glassy modulus, sK is the KWW relaxation time, and b is related to the breadth of relaxation response. The relaxation is exponential (Debye) when b = 1, and the relaxation function is ‘stretched’ or broadened as b decreases from this value. The shape of the segmental peak in the loss modulus can be evaluated by transforming the KWW function to the frequency domain. Values of b = 0.47 (na = 0.53) and b = 0.40 (na = 0.60) were determined [26], respectively, for 1,4-polybutadiene and 1,2-polybutadiene from fitting the KWW function to the segmental loss modulus peaks as illustrated in Fig. 3. The primitive process, s0,a(T), of the coupling model is the independent local relaxation of molecular segments. For times greater than a characteristic time, tc,a,

4 171.3 K 172.7 K 174.3 K 175.6 K 176.8 K 178.3 K 179.8 K 187.5 K

3 1.0x108

5.0x107

loss tangent, tan δ

loss modulus, G" (Pa)

1.5x108

171.3 K 172.7 K 174.3 K 175.6 K 176.8 K 178.3 K 179.8 K 187.5 K

2

1 KWW Fit β = 0.47

0.0 10-4

10-3

10-2

10-1

100

101

102

103

frequency, ω (rad/s) Fig. 3. Dynamic mechanical loss modulus data for 1,4-polybutadiene from Robertson and Rademacher [26] are replotted in this figure. The thick solid line is the KWW fit to the data at 175.6 K with b = 1
na = 0.47.

0 10-4

10-3

10-2

10-1

100

frequency, ω (rad/s)

101

102

103

Fig. 4. Dynamic mechanical tan d data for 1,4-polybutadiene from Robertson and Rademacher [26] are replotted in this figure.

C.G. Robertson, L.I. Palade / Journal of Non-Crystalline Solids 352 (2006) 342–348

cooperativity causes the primitive motions to become correlated, and the coupling parameter, na, represents the strength of this interaction. The observed cooperative arelaxation time can then be derived from these concepts, leading to the following expression for sa(T) [8–11]: sa ðT Þ ¼

1=ð1
na Þ a ðt
n . c;a s0;a ðT ÞÞ

ð8Þ

The value of tc,a is in the picosecond range [39], and we assume a constant value of tc,a = 2 · 10
12 s for our calculations. As mentioned previously, the coupling parameter for segmental dynamics is simply related to the a-relaxation breadth, na = 1
b. Therefore, a noteworthy feature of the coupling model is that it predicts a connection between temperature dependence and breadth of the a-relaxation which agrees with data on numerous glass-forming polymers and small molecules [15]. For our purposes, it is useful to extract the temperature dependence of the primitive relaxation time from the measured cooperative relaxation time data by rearrangement of Eq. (8): 1
na

s0;a ðT Þ ¼ tnc;aa ½sa ðT Þ

.

ð9Þ

The s0,a(T) can be expressed in terms of VFTH parameters by incorporating Eq. (4):   1
na Ba na . ð10Þ s0;a ðT Þ ¼ tc;a Aa exp T
T 1;a A coupling model expression which is parallel to Eq. (8) can be written for the terminal flow of entangled polymers [9–11]: 1=ð1
nF Þ F sF ðT Þ ¼ ðt
n . c;F s0;F ðT ÞÞ

ð11Þ

The link between the coupling model equations for segmental and terminal flow is made via the Rouse relaxation process. The Rouse model can be considered to be valid for the flow of entangled polymers at very short times (t < tc,F) since intermolecular entanglement couplings have not yet influenced chain motion. A value of tc,F  4 · 10
9 s is selected [10] based on neutron spin echo data results which reveal such a timescale associated with the crossover from Rouse modes to slowed-down dynamics due to entanglement constraints [40]. According to the coupling model, the Rouse process is considered to be the primitive relaxation (s0,F) which underlies the terminal flow relaxation of entangled polymers: s0;F ðT Þ ¼ sR ðT Þ.

ð12Þ

The monomeric friction coefficient, n0(T), connects the Rouse relaxation process to the segmental dynamics. The longest Rouse relaxation time, sR, is given by [18,41] 2 2

sR ðT Þ ¼

M a n0 ðT Þ ; 6p2 M 20 kT

ð13Þ

where a2 is the mean square end-to-end length per monomer repeat unit, M0 is the molecular weight of the repeat unit, and M is the total polymer molecular weight. The

345

monomeric friction coefficient is considered to be proportional to the primitive segmental process (s0,a) [9]: n0 ðT Þ / s0;a ðT Þ.

ð14Þ

Compared to the variation of n0 with temperature, the influence of temperature on chain coil dimensions (a2) and the kT contribution in Eq. (13) are insignificant. The Rouse relaxation time is therefore approximately related to s0,a through a proportionality parameter, C: sR ðT Þ  Cs0;a ðT Þ.

ð15Þ

A direct link between the segmental and terminal dynamics is developed by combining Eqs. (10)–(12) and (15) in order to construct the following [26]:
n =ð1
nF Þ na =ð1
nF Þ tc;a

sF ðT Þ ¼ C 1=ð1
nF Þ tc;FF    Aa exp

Ba T
T 1;a

ð1
na Þ=ð1
nF Þ .

ð16Þ

Experimental data for segmental dynamics yield information about the relaxation function breadth and temperature dependence of that process; hence na, Aa, Ba, and T1,a are all known quantities for both 1,4-polybutadiene and 1,2polybutadiene. Also, we discussed earlier that the following parameters are constant: tc,a = 2 · 10
12 s, tc,F = 4 · 10
9 s, and nF = 0.41. It is clear that only one variable, C, is required to predict the temperature dependence of terminal flow relaxation times using the unified coupling model (Eq. (16)). With C = 6.1 · 106 for 1,4-polybutadiene and C = 4.6 · 107 for 1,2-polybutadiene, the terminal relaxation data for our polymers can be well-predicted from the breadth and temperature dependence of sa. The coupling model fits in Figs. 1 and 2 are shifted downward to illustrate the ability of the model to account for thermorheological complexity in both 1,4-polybutadiene and 1,2polybutadiene. We can test Eq. (16) by varying the C parameter in an attempt to fit literature terminal relaxation data for 1,4polybutadienes and 1,2-polybutadienes which vary extensively in molecular weight. For 1,4 polybutadiene, we consider the data of Palade et al. [42] and the results of Colby et al. [23]. The studies of Carella et al. [43] and Roovers and Toporowski [37] provide results for 1,2-polybutadiene which we can also analyze. In considering these studies, the reported values of recoverable creep compliance, J 0e , and zero shear viscosity, g0, were used to assign values to the terminal relaxation times at a reference temperature according to sF ¼ g0 J 0e .

ð17Þ

Shift factor (aT) data were then employed to evaluate sF at the other temperatures. Coupling model fits, generated by simply adjusting the parameter C, were able to capture well the temperature dependence for sF throughout a wide range of molecular weights as shown in Figs. 5–7. The validity of this coupling model approach for predicting terminal dynamics from segmental relaxation behavior

346

C.G. Robertson, L.I. Palade / Journal of Non-Crystalline Solids 352 (2006) 342–348

105

105

Palade, Verney, and Attané

1,2-polybutadiene

relaxation time, τ (s)

relaxation time, τ (s)

1,4-polybutadiene

100

τF τα

-5

10

51 kg/mol 71 kg/mol 87 kg/mol 229 kg/mol 464 kg/mol

10-10 0.4

CGF: Carella, Graessley, and Fetters RT: Roovers and Toporowski

100

τF 10-5

10-10

0.6

0.8

106 kg/mol (CGF) 346 kg/mol (CGF) 33 kg/mol (RT) 84 kg/mol (RT) 204 kg/mol (RT)

0.6

1.0

0.8 Tg / T

Tg / T Fig. 5. Plotted are relaxation times for a-relaxation (open symbols) and terminal flow (solid symbols) of 1,4-polybutadiene, the latter from the study by Palade et al. [42] at the indicated molecular weights. See caption to Fig. 1 for details about the a-relaxation data. The Tg data which were used to normalize the terminal relaxation results are the reported [42] differential scanning calorimetry values. The line through the a-relaxation results is the VFTH fit (Eq. (3)), and the lines through the terminal data are the coupling model fits (Eq. (15)) obtained by variation of C.

105

relaxation time, τ (s)

1,4-polybutadiene

100

τF

10-10 0.4

τα

46 kg/mol 76 kg/mol 130 kg/mol 355 kg/mol 925 kg/mol

Fig. 7. Plotted are relaxation times for a-relaxation (open symbols) and terminal flow (solid symbols) of 1,2-polybutadienes, the latter from the study by Carella et al. [43] and Roovers and Toporowski [37] at the indicated molecular weights (solid symbols). See caption to Fig. 2 for details about a-relaxation data. The Tg data which were used to normalize the terminal relaxation results are the reported [37,43] differential scanning calorimetry values. The line through the a-relaxation results is the VFTH fit (Eq. (3)), and the lines through the terminal data are the coupling model fits (Eq. (15)) obtained by variation of C.

1010

0.6

0.8

Colby, Fetters, and Graessley Palade, Verney, and Attané

1.0

Tg / T

can be tested by considering the molecular weight dependence of C. From Eqs. (13)–(15) the parameter C is predicted to vary with the square of molecular weight: ð18Þ

The role of molecular weight on the value of C was evaluated for 1,4-polybutadiene from fitting the literature results

C parameter

109

Fig. 6. Plotted are relaxation times for a-relaxation (open symbols) and terminal flow (solid symbols) of 1,4-polybutadienes, the latter from the study by Colby et al. [23] at the indicated molecular weights (solid symbols). See caption to Fig. 1 for details about the a-relaxation data. The Tg data which were used to normalize the terminal relaxation results are the reported [23] differential scanning calorimetry values. The line through the a-relaxation results is the VFTH fit (Eq. (3)), and the lines through the terminal data are the coupling model fits (Eq. (15)) obtained by variation of C.

C / M 2.

1.0

of Palade et al. [42] and Colby et al. [23] These results are presented in Fig. 8. The modest offset between the log(C) versus log(M) results for the two data sets in Fig. 8 may be a consequence of the slight difference in microstructure for the polybutadienes. The 1,2-content (vinyl content) was 11% for the 1,4-polybutadienes investigated by Palade, Verney, and Attane´ while the results of Colby, Fetters, and Graessley were obtained for polybutadienes with approximately 9% vinyl content. Because the number of samples was more limited for 1,2-polybutadiene, we com-

Colby, Fetters, and Graessley

10-5

τα

1,4-polybutadiene

108 1.8 ± 0.1 2.0 ± 0.1

107

106 102 molecular weight, M (kg/mol)

103

Fig. 8. This plot illustrates the weight-average molecular weight dependence of the C parameter determined by unified application of the coupling model to the terminal relaxation results of Palade et al. [42] and Colby et al. [23] for 1,4-polybutadiene. The slopes of linear fits to the data (log–log representation) are indicated.

C.G. Robertson, L.I. Palade / Journal of Non-Crystalline Solids 352 (2006) 342–348

nal flow relaxation time which is proportional to its Rouse counterpart; hence their temperature dependences are predicted to be the same. This relationship is given below:

Carella, Graessley, and Fetters 9

10

Roovers and Toporowski

sF ðT Þ / ZsR ðT Þ.

C parameter

1,2-polybutadiene

108

2.1 ± 0.2

107

106 101

10 2 molecular weight, M (kg/mol)

347

103

Fig. 9. This plot illustrates the weight-average molecular weight dependence of the C parameter determined by unified application of the coupling model to the terminal relaxation results of Carella et al. [43] and Roovers and Toporowski [37] for 1,2-polybutadiene. The slope of the linear fit to the combined data from these two literature sources (log–log representation) is indicated.

bined the data of Carella et al. [43] and Roovers and Toporowski [37] to create a log(C) versus log(M) plot in Fig. 9. We find that the molecular weight exponents for the proportionality constant C range from 1.8 to 2.1 (Figs. 8 and 9). These scaling exponents are in close agreement with the predicted value of 2 in Eq. (18). We would not expect such good agreement between the experimental and predicted scaling of C with respect to M if key assertions such as those represented by Eqs. (12) and (14) were flawed or if the basic coupling model relationships (Eqs. (8) and (11)) were unsound. We emphasize that no relaxation time data for the Rouse process were included in this analysis, hence it is not a circular finding that the use of a 3.4 exponent for the molecular weight dependence of terminal flow leads to an exponent of two for the molecular weight dependence of C. A key assumption of this coupling model approach is that the Rouse relaxation is the primitive process for terminal flow of entangled polymers and is also connected to the segmental process via sR(T)  Cs0,a(T). Our study provides experimental evidence for the validity of this coupling model bridge between sF(T) and sa(T). 3. Final comments We demonstrated that only one adjustable parameter is required in order to predict sF(T) from sa(T) using the coupling model. Except for the coupling model, no other theoretical treatments have been offered to explain the widespread reality of thermorheological complexity which results from distinct influences of temperature on segmental, Rouse, and terminal flow relaxations of entangled polymers. Reptation theory does yield insights into the relationship between sR(T) and sF(T). The reptation tube model of Doi and Edwards [44] gives an expression for the termi-

ð19Þ

In the above, Z is the number of entanglements per chain which is commonly represented as M/Me where Me is the molecular weight between entanglements. The only way for Eq. (19) to agree with the experimentally weaker temperature dependence for sR compared to sF is for Z to be a function of temperature. This is not unreasonable if we consider the process of contour length fluctuation whereby the effective number of entanglements is lower than M/Me due to motion near the polymer termini. Theoretical work by Milner and McLeish [45] employs an arm retraction model, originally developed for stars, to describe contour length fluctuations of linear polymers before those motions are eventually cut-off by reptation. As the chain ends fluctuate down the tube, the backbone segments involved in the motion must undergo higher energy states (e.g. gauche versus trans rotational conformations for many polymers) during this arm retraction [46], hence contour length fluctuation should be an activated relaxation process. This implies a temperature dependence for contour length fluctuations which causes the number of effective entanglements per chain to also depend on temperature, Z = f(T). Based on this argument, the proportionality between sF and sR should depend on temperature, and thermorheological complexity is thus predicted. In this paper, we illustrated the efficacy of the coupling model in accounting for thermorheological complexity. The coupling model provides a means to unify segmental, Rouse, and terminal dynamics which can all have different temperature dependences. We also suggested a modification to the reptation tube model to resolve the limitation concerning its implicit prediction of thermorheological simplicity. We hope that this work will spark new experimental and theoretical interest in the physics of polymer thermorheological complexity. Acknowledgments CGR expresses gratitude to Bridgestone Americas for granting permission to publish this research. We thank Professor Shi-Qing Wang at the University of Akron for suggesting that we consider the molecular weight dependence of C. References [1] M. Beiner, S. Reissig, K. Schroter, E.-J. Donth, Rheol. Acta 36 (1997) 187. [2] D. Ferri, L. Castellani, Macromolecules 34 (2001) 3973. [3] R. Zorn, G.B. McKenna, L. Willner, D. Richter, Macromolecules 28 (1995) 8552. [4] P.G. Santangelo, C.M. Roland, Macromolecules 31 (1998) 3715. [5] D.J. Plazek, V.M. O’Rourke, J. Polym. Sci.: Polym. Phys. Ed. 9 (1971) 209.

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