The application of blending law on linear viscoelastic properties of linear?star and star?star polymer blends

The application of blending law on linear viscoelastic

properties of linear±star and star±star polymer blends

Wen-Bin Liau

_{*, Jeng-Shrong Uen}

_{, Wen-Yen Chiu}

a_{Institute of Materials Science and Engineering, National Taiwan University, Taipei, Taiwan} b_{Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan}

Received 20 July 1998; received in revised form 15 January 1999; accepted 29 January 1999

Abstract

A quadratic binary blending law of the terminal relaxation spectra is used to predict the viscoelastic properties of star±star and linear±star polymer blends. The results are compared with the experimental data reported by Struglinski et al. Star polymers with long arms are assumed to dominate the relaxation behavior of the star±star polymer blends. In the linear±star polymer blends, star polymers are assumed to dominate the relaxation behavior. The `iso-relaxation time' approach is proposed to represent the linear chain by an equivalent star chain for the linear±star polymer blends. The predicted viscoelastic properties agrees with the experimental observation very well. Also, the well known relation, …Jo

e†bA1=v2(v2is the volume fraction of long-arm star chains), can be deduced from the binary blending law for the concentrated long-arm star chains entangled with much shorter chains. Furthermore, it also predicts that it would systematically depart from the relation, …Jo

e†bA1=v2, when the molecular weight of shorter chain becomes closer to the molecular weight of long-arm chain. In the limit, M1=M2ˆ 1, the relation is switched to …Jo

e†bAv2. # 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction

The star polymers, because of the chain architecture of a central point and equal branches, exhibit quite di€erent viscoelastic behaviors from the linear poly-mers [1±14]. In the past years, extensive work has been done theoretically and experimentally on the viscoelas-tic properties of the monodispersed star polymers [2± 6]. Attention has primarily been focused on two steady shear ¯ow properties of polymer solutions, the zero-shear viscosity (Z0) and steady-state recoverable com-pliance (Jo

e). The Z0 relates to the relaxation with long relaxation time of the polymer solution and the Jo e

relates to the broadness of the relaxation time spec-trum [15].

In the regime of low and moderate concentration that the Rouse±Ham theory applies, the zero-shear vis-cosity (Z0) and steady-state recoverable compliance (Joe) of the star polymers, are comparable to those of the linear polymers with the same radius of gyration as the star polymers. In other words, the Z0 and Joe of the star polymers are much lower than those of the linear polymers, with the same molecular weight.

On the other hand, in the regime of high concen-tration and high molecular weight that the entangle-ments are formed among polymer chains (Mw> Mc, Mc is the characteristic molecular weight), the Z0 and Jo

e of the star polymers are much greater than those of the linear polymers. For the star polymers, the Z0 increases exponentially with the increasing molecular weight of the arm, Ma. However, the Z0 of the linear

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polymers has a power law relation with the molecular weight, i.e., Z0AM3:5. It is surprising that the recover-able compliance, (Jo

e), is found to still can be described by Rouse±Ham theory which is suitable for the none-ntangled systems. However, this does not mean that the Rouse±Ham mechanism is prevailing in the entangled star polymers [11]. From the terminal relax-ation spectra of the entangled star polymers, one can ®nd that it is di€erent from and much broader than that of the linear polymers [6,7].

Therefore, the relations of Z0and Joe relaxation spec-tra, are quite di€erent for the linear and star polymers in the entangled regime. From the recent theories based on the tube model [11±14], it has been suggested that there are two competing mechanisms of relaxation for the entangled linear polymers, reptation and con-straint-release (CR) by tube renewal, while they are path-breathing (arm retraction) and CR for the entangled star polymers. The relations between these mechanisms are complicated and still needed to be stu-died.

Another important factor for the viscoelastic beha-vior of polymers is the molecular weight distribution (MWD). To study the e€ect of MWD on the viscoelas-tic behavior, polymer blends containing two narrow MWD polymers are studied extensively. Yoshida et al. [10±12] and Struglinski et al. [9] have prepared several series of linear±star and star±star polymer blends to examine the e€ect of chain architecture on relaxation behaviors and viscoelastic properties. In the `dilute blend' [10], a dilute component (probe chain) with high molecular weight, M2, is added to the component

(test component) with low molecular weight, M1

(M2 M1> Mc). By investigating the motion of the probe chain, Yoshida et al. evaluated the mutual inter-action of two components. They suggested that the behavior of the probe chain is essentially identical with that in its own monodispersed dilute system and prac-tically controls the relaxation modes of the test com-ponent. In this research area, some studies have been done for linear±star and star±star polymer blends [13,14].

On the other hand, in the `concentrated blend', the component with high molecular weight is large enough to form chain entanglements, not only with the other component, but also among themselves. On the con-trary to the dilute blend, the studies are relatively few for the concentrated polymer blends.

Although, several mixing rules of linear viscoelastic behaviors for linear±star and star±star polymer blends have been proposed. However, a general and quanti-tative blending law for all blends is helpful to describe the viscoelastic behaviors. The purpose of this work is to propose a general blending law on relaxation spec-tra for the star±star and linear±star polymer blends. The compositional dependence of linear viscoelastic

properties is then derived and compared with the ex-perimental results reported by Struglinski et al. [9].

2. Theoretical treatment 2.1. Star±star polymer blends 2.1.1. Binary blending law

For lack of entanglements in the dilute or low mol-ecular weight systems, the relaxation spectrum H…t† for a blend of two monodispersed polymer chains with

molecular weight M1 and M2 can be simply

rep-resented by the Ninomiya blending law [10,15]: Hb…tb† ˆ v1H1 t_lb 1  ‡ …1 ÿ v1†H2 t_lb 2  …1† where the subscript `1', `2' and `b' denote the polymer 1, polymer 2 and polymer blend, respectively; H the relaxation spectrum; tb the relaxation time of the blend; l the shift factor; and v1the volume fraction of polymer 1.

In the concentrated and high molecular weight blends (M1,M2> Mc), the entanglements are formed among the same species as well as the di€erent species (i.e., `1-1', `1-2', `2-2' entanglements are formed). Thus, the linear blending law is not suitable in such a system. The quadratic blending law proposed by Montfort [16] and Kurata [17] is accepted favorably. It has the form: Hb…tb† ˆ v21H1 t_lb 1  ‡ÿ1 ÿ v2 1
H2 t_lb 2  …2† When the above blending law is constituted simply by the entanglements, it is found that the v2

1H1…t1† is pro-vided by the `1-1' chain contacts, v2

2H2…t2† by the `2-2' chain contacts and 2v1v2H12…t12† by the `1-2' chain contacts. In quadratic blending law, it is assumed that the shorter chain (polymer 2) is predominant. Therefore, the contribution of `1-2' entanglement is replaced by 2v1v2H2…t2†.

In an extreme interpretation, the relaxation of `1-2' entanglements should cooperate the two-sided e€ect, the relaxation of the short chain is retarded by the long chain and the relaxation of the long chain is accelerated by the short chain at the same time [11]. For the linear±linear polymer blends, the relaxation of the short chain is weakly retarded by the long chain and the relaxation of the long chain is strongly acceler-ated by the short chain. Thus, the short chain is predo-minant.

However, this e€ect of blending can possibly be di€erent for the star chains in the blend. The induced relaxation spectra for the polymer chains with di€erent chain architecture can therefore be very di€erent. For

lack of the study in this area, it is not known whether the short star chain or the long star chain predomi-nated the relaxation behavior in the star±star polymer blends. In this study, we assume that the long star chain is predominant. Thus, the quadratic blending law [16,17], that the long star chain dominate the relaxation in the entangled star±star polymer blends, is used to simulate the relaxation spectra of blends with respect to its individual components.

Because the star chain with long arms is the domi-nant component, the subscript `2' in Eq. (2) represent the long-arm star chain and `1' represent the short-arm star chain. The v1 is the volume fraction of the short-arm star chain.

2.1.2. Shift factors

The de®nitions of shift factors l1and l2 are the fac-tors by which the relaxation times t1,b and t2,b of the polymer `1' and `2' in the blend are altered relative to their values (t1 and t2) when the polymer is sur-rounded by others of its own kind [15].

l1ˆt_t1,b

1, l2ˆ

t2,b

t2 …3†

From the generalized proposed tube models of the monodispersed star polymer [2±6], the relaxation time of pure component is expressed as:

tiA…Z0†iMi …4† …Z0†iˆ A exp _vM i Me  , i ˆ 1,2 in this system …5†

where …Z0†i is the zero-shear viscosity of ith com-ponent, Me is the molecular weight corresponding to entanglement spacing Mi is the arm molecular weight of ith component, and v is the constant. Parameter, `A' depends on the molecular weight [3,18,19].

Whereas the relaxation time of each star component in the blend ti,bcan be expressed as [15]:

ti,bA…Z0†b M2

i 

Mw …6†

where Mwˆ v1M1‡ v2M2 (average arm molecular weight of blend) and …Z₀†_bis the zero-shear viscosity of the blend. For the ®rst trial to estimate the shift factor

of each component, it is assumed that …Z0†b is

expressed in the same form as in Eq. (5), i.e., …Z0†bˆ A exp v0_M w Me ! …7† Then the shift factors, l1 and l2 in Eq. (3) could be derived as below: l1ˆ t_t1,b 1 ˆ … Z0†b ÿ M2 1= Mw
…Z0†1 ÿ M2 1=M1
…8a† ˆ exp ÿ v0_M w=Me
M1 exp…v0_{M1=Me† Mw} …8b† l2ˆ t_t2,b 2 ˆ … Z0†b ÿ M2 2= Mw
…Z0†1 ÿ M2 2=M2
…9a† l2ˆ exp ÿ v0_M w=Me
M2 exp…v0_{M2=Me† Mw} …9b†

Since the parameter, A's in Eqs. (5) and (7) are depen-dent on molecular weight, Eqs. (8b) and (9b) are not quite accurate. However, Eqs. (8b) and (9b) are used to estimate the initial values of shift factors only. The re®nement of shift factors is performed according to Eqs. (8a) and (9a).

2.1.3. Zero-shear viscosity

All the linear viscoelastic functions are theoretically interrelated. The zero-shear viscosity …Z0†, can be expressed as moments of the relaxation spectrum H…t† [15]:

Z0ˆ …₁

0 H…t† dt …10†

Similarly, in the polymer blends: …Z0†bˆ

…1

0 Hb…tb† dt …11†

If Eq. (2) holds in the concentrated blend, the zero-shear viscosity of the blend in terms of those of the in-dividual pure components is given by

…Z0†bˆ …1 0 Hb…tb† dt ˆ …₁ 0  v2 1H1…tb=l1† ‡ÿ1 ÿ v21
H2…tb=l2†dt ˆ v2 1l1 …1 0 H1…tb=l1† d…tb=l1† ‡ ÿ 1 ÿ v2 1
l2 …₁ 0 H2…tb=l2† d…tb=l2† ˆ v2 1l1…Z0†1‡ ÿ 1 ÿ v2 1
l2…Z0†2 …12†

There are few words to say about the calculation of …Z0†b and the shift factors. (1) They are calculated by iteration. (2) The shift factors are calculated from Eqs.

(8b) and (9b) ®rst then the values of shift factors are substituted into Eq. (12) to obtain the …Z0†b. (3) Because of the assumption made in Eq. (7), we must correct the shift factors calcualted from Eqs. (8a) and (9a) by the new value of …Z0†b, and the procedure is repeated until the …Z₀†_b closed to the experimental value.

Once the shift factors and zero-shear viscosity …Z0†b of individual components are known, the compo-sitional dependence of zero-shear viscosity of blends can be obtained from Eq. (12).

2.1.4. Steady-state recoverable compliance The steady-state recoverable compliance (Jo

e) can be expressed as the moments of the terminal relaxation spectrum H…t† in the entangled regime (M > Mc), as well as the zero-shear viscosity [15]:

Jo e ˆ …₁ 0 H…t†t dt  …1 0 H…t† dt 2 ˆ 1 Z2 0 …1 0 H…t†t dt …13†

For the polymer blends, ÿ Jo e
bˆ 1 …Z0†2b …1 0 Hb…tb†tbdtb …14†

If Eq. (2) holds in the concentrated blend, the steady-state recoverable compliance of the blend in terms of the individual pure components is given by

ÿ Jo e
bˆ 1 …Z0†2b …1 0  v2 1H1 t_lb 1  tbdtb‡ÿ1 ÿ v21
H2 t_lb 2  tbdtb  ˆ 1 …Z0†2b  v2 1l21 …₁ 0 t1H1…t1† d…t1† ‡ ÿ 1 ÿ v2 1
l2 2 …1 0 t2H2…t2† d…t2†  ˆ 1 …Z0†2b h v2 1l21 ÿ Jo e
1…Z0†21‡ ÿ 1 ÿ v2 1
l2 2 ÿ Jo e
2…Z0†22 i ˆ v2 1 " …Z0†bM1 …Z0†₁Mw #2" …Z0†1 …Z0†b #2_ÿ Jo e
1‡ ÿ 1 ÿ v2 1
 " …Z0†bM2 …Z0†2Mw #2" …Z0†2 …Z0†b #2_ÿ Jo e
2 ˆ v2 1  M1 Mw 2_ÿ Jo e
1‡ ÿ 1 ÿ v2 1
 M_M2 w 2_ÿ Jo e
2 …15†

When v140…v241†, Eq. (15) can be simpli®ed as: ÿ Jo e
b1 ÿ 1 ÿ v2 1
 M_M2 w 2_ÿ Jo e
2 ˆÿ2v2ÿ v22
0 B @ 1 M1 M2‡  1 ÿM_M1 2  v2 1 C A 2_ÿ Jo e
2 ˆ 2 v2ÿ 1  M1 M2 1 v2‡  1 ÿM_M1 2 2 ÿ Jo e
2 ˆ 1 ‡ v1 v2  M1 M2 1 v2‡  1 ÿM_M1 2 2 ÿ Jo e
2 1 1 v2  M1 M2 1 v2‡  1 ÿM_M1 2 2 ÿ Jo e
2 …16†

If M1=M2  1 then …Joe†bAv12. Thus, the mixing rule

proposed here can predict the relation …Jo

e†bAv12 as it is

observed in the experiment [9,11]. 2.1.5. Dynamic moduli

In a similar way as the derivation of the recoverable compliance, the dynamic storage and loss moduli can also be expressed in terms of the moments of the term-inal relaxation spectrum H…t† in the entangled regime …M > Mc† [15]: G0_ˆ…1 ÿ1H…t† o2_t2 1 ‡ o2_t2d ln…t† and G00_ˆ…1 ÿ1H…t† ot 1 ‡ o2_t2d ln…t† …17†

Substitute Eq. (2) to Eq. (17), the dynamic storage and loss moduli of the blend in terms of the individual pure components are given by

…G0_† bˆ …1 ÿ1v 2 1H1…tb1l1† o 2_t2 b 1 ‡ o2_t2 bd ln tb‡ …1 ÿ1 ÿ 1 ÿ v2 1
H2…tb=l2† o 2_t2 b 1 ‡ o2_t2 b d ln…tb†

ˆ v2 1 …₁ ÿ1H1…p† l2 1o2p2 1 ‡ l2 1o2p2d ln…p† ‡ ÿ 1 ÿ v2 1
…1 ÿ1H2…q† l2 2o2q2 1 ‡ l2 2o2q2d ln…q† where p ˆ tb=l1and q ˆ tb=l2. Furthermore, …1 ÿ1H1…p† l2 1o2 1 ‡ l2 1o2p2 d ln…p† ˆ G 0 1…l1o† and …₁ ÿ1H2…q† l2 2o2q2 1 ‡ l2 2o2q2d ln…q† ˆ G 0 2…l2o† Thus, …G0_† bˆ v21G10…l1o† ‡ ÿ 1 ÿ v2 1
G0 2…l2o† …18†

Similarly, the loss modulus can be written as: …G00_† bˆ …1 ÿ1v 2 1H1…tb=l1†_{1 ‡ o}otb_2t2 bd ln…t†b‡ …1 ÿ1 ÿ 1 ÿ v2 1
H2…tb=l2†_{1 ‡ o}otb_2t2 bd ln…t†b ˆ v2 1 …₁ ÿ1H1…p† l1op 1 ‡ l2 1o2p2d ln…p† ‡ ÿ 1 ÿ v2 1
…₁ ÿ1H2…q† l2oq 1 ‡ l2 2o2q2d ln…q† …1 ÿ1H1…p† l1op 1 ‡ l2 1o2p2 d ln…p† ˆ G 00 1…l1o† and …1 ÿ1H2…q† l2oq 1 ‡ l2 2o2q2d ln…q† ˆ G 00 2…l2o† Thus, …G00_† bˆ v21G100…l1o† ‡ ÿ 1 ÿ v2 1
G00 2…l2o† …19†

2.2. Linear±star polymer blends 2.2.1. `Iso-relaxation time' approach

There are two ways to treat the linear±star polymer blends. One way is that the star chain is treated as a linear chain and a linear±linear binary blending law is used. The other way is that the linear chain is treated as a star chain and a star±star binary blending law is used. Since the linear viscoelastic properties derived above is based on the star±star polymer blends, the lat-ter will be used in this work. A linear chain can be treated as an equivalent star chain if the relaxation time of the linear chain is equal to that of the star-equivalent chain. Therefore, instead of the `iso-vis-cosity' approach [9], we initiated the `iso-relaxation time' approach to obtain the star-equivalent molecular weight M _{for the monodispersed linear chains of the} blend. Thus,

…Z0†linearM ˆ …Z0†star
Mˆ A exp 

v0_M fMe



M _20†

where …Z0†linear, M are the viscosity and molecular weight of monodispersed linear component and M_{, f} are the star-equivalent molecular weight and number of arms. Thus, a linear±star blend system can be theor-etically represented by a star±star blend system with the Eq. (20). The linear chains will behave like their corresponding star-equivalent chains on the relaxation spectrum.

2.2.2. Binary blending law

For the linear±star polymer blends, the binary blending law can similarly be represented by Eq. (2) with the `iso-relaxation time' approach. From the pub-lished experimental data, it is found that the volume fraction of the star polymer, predominantly determined the relaxation behavior [11±13]. Therefore, the sub-scripts `1' and `2' in the Eq. (2) represent the linear

Table 1

Viscoelastic properties of pure-component polybutadiene [9] Sample M  10ÿ4 _Jo e 107cm2dyneÿ1 Z0 10ÿ4(258C, P) (M)a10ÿ4 41L 3.68 1.78 1.35 4.07 105L 10.0 1.95 43.7 7.63 174L 16.8 1.82 295 9.65 75S 7.46 8.31 47.9 127S 12.7 13.5 2750

a_M_{obtained from `iso-relaxation time' approach in this work}

…Z0†M ˆ …Z0†star:Mˆ A exp…M=B †M

and star components, respectively. As the star±star blends, the shift factors, zero-shear viscosity, steady-state recoverable compliance, and dynamic moduli at very low frequency of the linear±star polymer blends can be obtained from Eqs. (8a), (9a), (12), (15), (18) and (19).

3. Results and discussion

To evaluate the applicability of quadratic binary blending law to the star±star and linear±star polymer blends, the predicted results are compared with the ex-perimental data reported by Struglinski et al. [9]. The data and discussion reported by Struglinski are detail enough to evaluate the quadratic binary blending law easily. The materials used in Ref. [9] are blends of nearly monodisperse linear and three-arm star polybu-tadienes. The viscoelastic properties of those materials are listed in Table 1 The ®fth column of Table 1 is cal-culated according to the `iso-relaxation time' approach. Five series of binary blends are prepared by Struglinski, four with linear and star components (the linear±star series, 41L/127S, 105L/127S, 174L/127S, 105L/75S) and one in which both components are stars (the star±star series 75S/127S). The notations have followed the Struglinski's work. All blends are well entangled.

It is found that the determination of constants, A and v0 _{of the Eq. (5) for the monodispersed star} poly-mer, is very important. Theoretically, constants A and v0 _{can be obtained from the data ®tting, if there are} enough data of the monodispersed star polymers. However, there are only two samples for the monodis-persed star polybutadiene in the Ref. [9]. If we proceed directly from the data ®tting, there could be many

errors. Therefore, instead of the data ®tting from only two samples, the proposed value, v0_{ˆ 0:6, for the} monodispersed star polybutadiene and polystyrene are adopted [3,5,9]. Then the constant A can be obtained from one of two monodispersed star samples. In this study, we determine the constant A from the monodis-persed star sample, 75S.

For the star±star polymer blend, the blend, 75S/ 127S, of polybutadiene is examined. The long-arm star chain component (127S) is assumed to dominate the relaxation behavior and the quadratic blending law is used to simulate the relaxation spectrum of the blend, with respect to its individual components. The results con®rm that the assumption is correct. Also, the

quad-Table 2

Shift factors for polymer blends

v2 l1 l2 75S/127S Series 0.05 1.16 0.034 0.1 1.35 0.040 0.2 1.82 0.054 0.3 2.47 0.073 0.5 4.69 0.14 0.7 11.40 0.34 0.85 19.07 0.57 41L/127S Series 0.05 1.36 0.0021 0.1 1.59 0.0024 0.2 2.69 0.0041 0.3 4.77 0.0073 0.5 16.8 0.026 0.75 96.4 0.15 105L/127S Series 0.05 1.16 0.031 0.1 1.36 0.036 0.15 1.59 0.042 0.2 1.86 0.049 0.3 2.56 0.068 0.5 5.04 0.133 0.75 12.81 0.34 174L/127S Series 0.05 1.16 0.057 0.1 1.35 0.066 0.2 1.82 0.089 0.3 2.46 0.12 0.5 2.29 0.323 105L/75S Series 0.05 1.02 0.911 0.1 1.04 0.928 0.2 1.07 0.957 0.3 1.10 0.981 0.5 1.13 1.01 0.7 1.15 1.02 0.8 1.14 1.02 0.9 1.13 1.01 0.95 1.13 1.01

Fig. 1. Zero-shear viscosity …Z0†b vs. volume fraction of 127S

star-component …v2† for the 75S/127S series of polybutadiene

system. The ®lled points denote the experimental data and the solid line represents the predicted values.

ratic blending law is examined with the assumption that the short-arm star chain is predominent. However, the results do not match the experimental data.

3.1. Shift factors

Recalled that in the evaluation of shift factors, we must correct the shift factors until the …Z0†b closed to the experimental value. The results for the star±star polymer blend are shown in Table 2 and Fig. 1. It was found that the di€erences between calculated values and experimental values are small enough, not more than four iterations. Therefore, correct shift factors can be obtained by iteration.

For the linear±star polymer blends, the `iso-relax-ation time' approach for the monodispersed linear chain component is adopted to obtain the correspond-ing star-equivalent chain. The results are shown in the ®fth column of Table 1 for polybutadiene. Then the linear chain is treated as an equivalent star chain and the binary blending law is applied. When the M _of the linear chain with the molecular weight of the star chain is compared, the star chain is the dominant com-ponent Table 2 shows the results of trials on shift fac-tors for linear±star polymer blends. As the star±star polymer blends, the di€erences between calculated and experimental values are no more smaller than four iter-ations (Figs. 2 and 3).

3.2. Zero-shear viscosity

The calculated values of …Z0†bof the 75S/127S blend from Eq. (12) and the experimental values are plotted in the Fig. 1. As the experimental data, it is found that the predicted zero-shear viscosity of the blend increased monotonically with the increasing of the volume fraction of 127S star components, v2. From Fig. 1, it is found that the predicted values can match the experimental values very well. Therefore, the assumption that the long-arm star chain is predomi-nant in the star±star polymer blends is suitable in the binary blending law. This result is contrary to the entangled linear chain system which the short linear chain predominantly determins the relaxation behavior as shown by Montfort et al. [16] and Kurata [17]. This may be due to the strong e€ect of chain architecture that results in the di€erent relaxation mechanisms

between entangled linear and star polymers.

Considering the tube model, [11±14] there are two competing mechanisms of relaxation for the entangled linear polymers, repetition and CR, while they are path-breathing (arm-retraction) and CR for the star polymers. These di€erent relaxation mechanisms may result in the di€erent binary blending laws for entangled linear±linear and star±star polymer blends.

Fig. 2 shows the comparative result of zero-shear viscosity between experimental data and theoretical prediction for the 105L/75S blend system. As pointed out by Graessley et al. [8,9], the component viscosity is nearly same for this blend and there is a weak maxi-mum around v2ˆ 0:5. The quadratic binary blending law also predicts that there is a weak maximum around v2ˆ 0:65. The predicted zero-shear viscosity of the blends changes less than the experimental data across the composition range. The di€erence between the predicted values and experimental data is less than 15%. Fig. 3 shows the comparative results of zero-shear viscosity between experimental data and theoreti-cal prediction for the 41L/127S, 105L/127S, and 174L/ 127S blends. As in the star±star blends, …Z0†b increases

Fig. 2. Zero-shear viscosity …Z0†b vs. volume fraction of 75S

star-component …v2† for the 105L/75S series of polybutadiene

system. The ®lled points denote the experimental data and the solid line represents the predicted values.

Fig. 3. Zero-shear viscosity …Z0†b vs. volume fraction of 127S

star-component (v2) for 41L/127S (*), 105L/127S (R), and

174L/127S (Q) series of polybutadiene system. The ®lled points denote the experimental data and the solid line rep-resents the predicted values.

monotonically with the increasing volume fraction of the star components and the predicted values ®t the ex-perimental data very well.

3.3. Steady-state recoverable compliance

The calculated and experimental results of the steady-state recoverable compliance, …Jo

e†bof the blends is showed in Fig. 4. For 75S/127S, 41L/127S, 105L/ 127S, and 174L/127S series, it is found that …Jo

e†b in-itially increases with the increasing volume fraction of long-arm star components, v2, then reaches a maxi-mum and ®nally it decreases with the increasing volume fraction, v2. However, the recoverable compli-ance of the blend, 105L/75S, increased monotonically with the increasing volume fraction of the star com-ponents. The predicted values can ®t the experimental data and predict the trend very well. It recon®rms that the quadratic binary blending law and the assumption that the long-arm star chain is predominant in the relaxation behavior of star±star polymer blends are reasonable.

Another interesting thing is the relation between the steady-state recoverable compliance, …Jo

e†b, and the volume fraction of long-arm star polymer, v2, near v2ˆ 1. Yoshida et al. [11] and Struglinski et al. [9] have pointed out that the steady-state recoverable compliance, …Jo

e†b, of the linear±star and star±star blends containing concentrated long-arm star chains entangled with much shorter chains, is inversely pro-portional to the volume fraction of the long-arm star chain. This relation can be predicted from the mixing rule proposed in Section 2. Furthermore, it can be easily deduced that …Jo

e†bAv2 with M1=M2ˆ 1 from Eq. (16). The relation agrees with the experimental

ob-servation very well. Struglinski et al. [9] pointed out that the steady-state recoverable compliance is pro-portional to the volume fraction of star chain for 105L/75S, where the star-equivalent molecular weight of 105L is nearly same as the molecular weight of star chain. Also, it can be proved from Eq. (16) that it will

systematically depart from the relationship,

…Jo

e†bA1=v2, when the molecular weight of short-arm chain becomes closer to the molecular weight of long-arm chain. In Fig. 4, it is shown that the prediction of relation between …Jo

e†b and v2 agrees with the exper-imental data very well.

Fig. 4. Recoverable compliance …Jo

e†b vs. the volume fraction

of star-component …v2† for 41L/127S (*), 105L/127S (Q),

174L/127S (R), 75S/127S (w), and 105L/75S(q) series of polybutadiene system. Points denote the experimental data and the solid line represents the predicted values.

Fig. 5. Dynamic storage modulus for selected compositions of the 105L/127S series. Points denote the experimental data and the solid line represents the predicted values. v2ˆ 0:75 (W,r),

0.5 (T,t), 0.3 (Q,q), 0.2 (R,r), and 0.1 (*,w)

Fig. 6. Dynamic loss modulus for selected compositions of the 105L/127S series. Points denote the experimental data and the solid line represents the predicted values. v2ˆ 0:75 (W,r),

3.4. Dynamic moduli

In Ref. [9], the published experimental observation on the frequency dependence of storage and loss moduli are limited only to the 105L/127S series. Thus, only the dynamic moduli of the 105L/127S blend are predicted here. The values of dynamic storage and loss moduli of blends can be predicted from Eqs. (18) and (19) with known values of dynamic moduli of pure components. The results are shown in Figs. 5 and 6. The lines represent the calculated results of the linear± star polymer blends. The ®lled and open points rep-resent the experimental data [9]. The experimental data at the high frequency side …log o > 1† is so crowded that it is dicult to extract from original ®gure of Ref. [9]. However, the di€erence of dynamic moduli of star and liner polymers is very small in the high frequency region [9]. Thus, the mixing rule should predict the dynamic moduli quite well in this region. From Figs. 5 and 6, it was found that the predicted values can ®t the experimental data very well.

4. Conclusion

In this work, a quadratic blending law on the relax-ation spectra for the star±star and linear±star polymer blends is proposed. On the contrary to linear±linear blends, for the linear±star polymer blends, the star component is dominant and the long-arm star com-ponent is dominant for the star±star polymer blends. In addition, the `iso-relaxation time' approach is pro-posed to obtain the `star-equivalent' molecular weight,

M_{, for the monodispersed linear component of}

blends.

Both the quadratic binary blending law and `Iso-relaxation time' approach can provide a generally excellent prediction on the compositional dependence of the zero-shear viscosity, steady-state recoverable compliance, and dynamic modulus for the entangled linear±star and star±star polybutadiene blends.

The trends that, …Jo

e†binitially increases with

increas-ing volume fraction of star component (v2) then

reaches a maximum, …G0

0†bAo2 and …G000†bAo, all can be predicted fairly well with the experimental obser-vations.

The quadratic binary blending law can predict the

well known relationship, …Jo

e†bA1=v2, near v2ˆ 1. Furthermore, it predicts that it will systematically depart from the relationship, …Jo

e†bA1=v2, when the molecular weight of short-arm chain (or star-equival-ent chain) becomes closer to the molecular weight of long-arm chain. In the limit, M1=M2ˆ 1, the relation-ship is switched to …Jo

e†bAv2.

Acknowledgements

We are indebted to the National Science Council for ®nancial support of this work through the grand num-ber: NSC-85-2216-E-002-013, NSC-86-2216-E-002-007, and NSC-87-2216-E-002-022.

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