Binary Blends of Entangled Star and Linear Poly(hydroxybutyrate

Article pubs.acs.org/Macromolecules

Binary Blends of Entangled Star and Linear Poly(hydroxybutyrate): Effect of Constraint Release and Dynamic Tube Dilation Tannaz Ebrahimi,†,‡ Hamid Taghipour,§ Desiree Grießl,†,∥ Parisa Mehrkhodavandi,‡ Savvas G. Hatzikiriakos,*,† and Evelyne van Ruymbeke*,§ †

Department of Chemical and Biological Engineering, The University of British Columbia, 2360 East Mall Vancouver, Vancouver, British Columbia Canada ‡ Department of Chemistry, The University of British Columbia, 2036 Main Mall, Vancouver, British Columbia Canada § The Division of Bio- and Soft Matter (BSMA), Institute of Condensed Matter and Nanosciences (IMCN), Université catholique de Louvain (UCL), Louvain-la-Neuve, Belgium ∥ Chemical Technology, Technical University of Munich, Munich, Germany ABSTRACT: In order to further understand the relaxation behavior of binary blends of star and linear chains, new polymer blends consisting of linear poly(hydroxybutyrate) (PHB) matrix and PHB star molecules are designed, and their dynamics is investigated by varying the star concentrations and the molar mass of the linear matrix, while keeping few or no star−star entanglements in the blends. By studying the constraint release Rouse (CRR) relaxation of the star polymer diluted in the linear matrix at concentrations low enough to neglect star−star entanglements, we first point out the importance of the number of short linear chain entanglements on the CRR time of the long chains. For the blends composed of a larger proportion of star molecules, we then use this new definition of CRR time to determine the necessary time of a star−star entanglement segment to relax by CRR and explore its dilated tube, at the rhythm of the disentanglement/re-entanglements of the short chains. By considering this as the new reference time for describing the contour length fluctuations of the arm in its “fat” tube, i.e., the tube which only involves star−star entanglements, we propose a simple and consistent way to take into account two opposite effects resulting from the short linear matrix. On the one hand, fast relaxation leads to a large dilution effect, which is reflected by the dilation of the tube in which the long chains are moving. On the other hand, the long chains can only move in their fat tube at the rhythm of the motion of the short chains, which can slow down their relaxation compared to the motion of the same stars diluted in a real solvent.

1. INTRODUCTION

blends theoretical curves strongly depend on the assumptions used to describe these CR effects. This requires accurate description of the sharp transition between short times, at which the short chains are constraining long chains motion, and long times, at which short chains behave similarly as solvent molecules.22 Because of this, binary blends of monodispersed polymer chains are good candidates to enhance our understanding of constraint release mechanisms in polydispersed samples. In the past few years, many studies have been reported investigating the linear viscoelastic properties of binary blends, both from an experimental and from a theoretical point of view.20,22−26,28−35,37,39,41 As a first class of binary blends, polymer blends composed of two different linear polymers have been extensively studied.26−28,30−34,36,37,39 In particular, Stru-

1−3

Since its development in early 1970s, tube molecular theory has been the center of attention to improve our understanding of polymer dynamics.4,5 Although mathematical models based on this theory have been developed on the behavior of welldefined, narrow-dispersed polymers such as linear,6 star,7,8 H,9 combs,10−13 and other model branched polymers,14−18 modeling the relaxation behavior of polydispersed samples is still a challenge.19−21 Studying the linear viscoelastic behavior of binary blends composed of two different monodispersed components is an important step in this direction.22−39 To this aim, it is important to take into account different constraint release (CR) mechanisms according to which the dynamics of the slow relaxing chains are accelerated by the fast motions of the quickly relaxing components, which act as a solvent for the long chains.40 Since in binary blends all the short chains relax exactly at the same time, the influence of the CR processes is more pronounced and, in most cases, is clearly detectable in the viscoelastic results. This also means that with such binary © XXXX American Chemical Society

Received: December 8, 2016 Revised: March 3, 2017

A

DOI: 10.1021/acs.macromol.6b02653 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules ⎧ ⎡ t ⎤1/2 ⎫ ⎪ ⎪ ⎨ ϕ(tk + 1) = max φ′(tk + 1), ϕ(tk)⎢ k ⎥ ⎬ ⎪ ⎣ tk + 1 ⎦ ⎪ ⎩ ⎭

glinsky and Graessley22 used this class of blends to demonstrate that short linear chains could be considered as solvent molecules for the reptation of the long linear chains when their respective initial reptation times are very well separated. Indeed, the authors showed that short chains must be short enough to ensure that the relaxation time of the whole long chains through constraint release Rouse process (CRR), τCRR(M1,M2), is shorter than their initial reptation time: Gr =

where tk+1 is the time considered just after the time tk. This condition ensures that the derivative, dϕ(t)/dt, is always larger than or equal to −1/2, i.e., that the long chain never occupies its dilated tube faster than what is allowed by Rouse motion (for more details, see Appendix B of ref 34). In the case of a binary blend composed of υ1 wt % of short chains and υ2 wt % of long chains, the necessary time τlong−long ent for a long chain to occupy (not move in) its fat tube, i.e., the time after which ϕ(t) = υ2, can be easily determined: it is a constraint release Rouse time, which involves the relaxation of the υ1/υ2 short−long entanglements segments located between two consecutive long−long entanglements (i.e., entanglements created by two long chains).29 Therefore, if each of the short−long entanglements is characterized by a lifetime τobs, we find

3τeZ 2 3 3τeZ 2 3 3τeZ 2 3 Z = = = 23 > 1 2 τCRR (M1, M 2) τ1(M1)Z 2 (3τeZ13)Z 2 2 Z1 (1)

with Z2 the number of (initial) entanglement segments per long chain, τ1 (∝ M13) the reptation time of the short chain, and τe the Rouse time of an entanglement segment. Under these conditions, the long chains can be considered as moving in a fat tube, which is only composed by the entanglements between the long chains. While this criterion seems to work in general, it has been shown that the exact critical value of the parameter Gr, which fixes the limit between dilated or undilated tube, may differ from 1. For example, Park and Larson found a value of 0.064 in ref 27. Furthermore, in eq 1, the CRR time is determined for a long chain only entangled with short chains, i.e., neglecting the entanglements between two long chains. It is also considered that the whole chain is relaxing by CRR, while in reality, this process should be limited to subchains shorter than the length of the molecular segments between two long− long entanglements. The solvent effect coming from the short chains can be included in the models through the concept of dynamic tube dilation (DTD),40 according to which only the entanglements between unrelaxed chain segments must be considered in order to describe the motion of these unrelaxed chains. Therefore, the (effective) tube parameters at a specific time t must be rescaled such as36,40 Me = Me,0{ϕ(t )}−α

(2)

a(t ) = a0{ϕ(t )}−α /2

(3)

Leq (t ) = Leq,0{ϕ(t )}α /2

(4)

⎛ υ ⎞2 τlong − long ent = max(τobs , τe)⎜ 1 ⎟ ⎝ υ2 ⎠

(7)

It must be noted that the condition in eq 7, according to which the average lifetime of a short−long entanglement, τobs, cannot be shorter than its intrinsic Rouse time, τe, ensures that the long−long entanglement segments will never relax faster than by their intrinsic Rouse process. Before the time τlong−long ent, the long chains do not fully occupy their fat tube. On the other hand, just after this time, the long chains are locally relaxed (i.e., at the level of one long−long entanglement segments) but not yet globally (i.e., at the level of the whole chain). As a consequence of this local relaxation, after τlong−long ent but before the relaxation of the long chains, the tube parameters can be approximated by considering ϕ(t) = υ2 in eqs 2−5.36,40 This state corresponds to the frequency region where a secondlow frequencyplateau is observed in the storage modulus curve for binary blends with well-separated relaxation times. However, the local relaxation of the long chains does not mean that the long chains can freely move by reptation and contour length fluctuations in their fat tube, as if the short chains were equivalent to a solvent. Indeed, the relaxation of the long chains in their fat tube can only take place at the rhythm of the disentanglement/re-entanglement of the short chains, which can be quite slow, depending on the molar mass of the short chains. In fact, several studies have shown that the reptation and contour length fluctuations times of the long chains in their fat tube are a function of the relaxation time of a long−long entanglement, τlong−long ent (see eq 7), rather than a function of its intrinsic Rouse time, τRouse,long−long ent = τe(υ1/ υ2)2, as it would have been the case with a real solvent.34,36 Consequently, since the time τlong−long ent can be long, much longer than its intrinsic Rouse time, the motion of the long chains in their fat tube can be relatively slow. Therefore, the tube in which the long chains are relaxing will depend on the delicate competition between relaxing fast in a long (undilated) tube (being governed by the Rouse time of an entanglement segment, τe) and relaxing slowly in a short (dilated) tube (being governed by τlong−long ent), which strongly depends on the relaxation time of the short chains.36 Thus, the fast motions of the short chains have two major consequences on the viscoelastic properties at long time: first, a strong decrease of the level of G(t) due to local rearrangement

with Me and Me,0 being the effective and initial average molar mass between two entanglements, a and a0 being the effective and initial tube diameter, and Leq and Leq,0 being the effective and initial length of the primitive path of the chain. These effective values can evolve all along the relaxation of the sample, which is taken into account by the dilation factor ϕ(t) and the dynamic dilution exponent α. While the value of this last parameter varies between 1 and 4/3 in the literature,39,42,43 here we fix it as 1, in accordance with van Ruymbeke et al.36 A direct consequence of eq 2 is the rescaling of the relaxation modulus, G(t), which must depend on both the unrelaxed fraction of initial tube segments, φ′(t), and the dilation factor, ϕ(t): Gd(t ) = G N0μ(t ) = G N0φ′(t ){ϕ(t )}α

(6)

(5)

The dilation factor ϕ(t), which defines the diameter of the dilated tube, is a priori equal to the unrelaxed fraction of initial tube segments, φ′(t). However, it cannot decrease faster than according to a (constraint release) Rouse process:23,44 B


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Macromolecules (at the level of Me) induced by a constraint release mechanism (see eqs 5 and 6) and, second, if the whole chains are able to freely move in a fat tube, a possible speeding up of the reptation time of the long chains, which takes place at the rhythm of the disentanglement/re-entanglement of the short component. Recently, van Ruymbeke et al. have shown that the experimental viscoelastic data of a large set of samples could be well described with tube models, based on eqs 2−5 and by accounting for this possible relaxation of the long chains in a fat tube at the rhythm of the short chains motion.36 They attributed the origin of these slow motions to monomeric tension re-equilibration along the long chains34,35 and described it as a CR-dominated fluctuations process (CR-CLF). However, they could not reach the level of a fully predictive model, keeping the average lifetime of a short−long entanglement, τobs, as a free (fitting) parameter. The main objective of the present work is to extend this work to binary blends composed of a long, starlike component, blended to a short (linear or star) component. In such a case, it is expected that the long star molecules only relax through constraint release mechanisms and through contour length fluctuations process. Therefore, modeling the behavior of such blends requires to accurately taking into account the influence of these CR processes in the fluctuations of the arms on a very broad range of times, from their early fluctuations process to their terminal regime. In the case of monodisperse star polymers, this is usually achieved by extending the dynamic tube dilation concept to the arm retraction process,4,7,45 assuming that the relaxation time of the different molecular segments along an arm are very well separated. In such a case, we can consider that at the time at which a segment x is relaxing (with x being the location of the segment, from 0 at the extremity of an arm to 1 at the branching point), all the segments located between the arm end (x = 0) and this specific segment x are acting as a solvent. Indeed, at this specific time, these outer segments have had already the time to entangle and disentangle several times and therefore should not constraint the retraction of the deeper segments. Milner and McLeish23 extended, for the first time, this concept to the prediction of the relaxation of a specific blend of linear and star polymers presented by Struglinsky et al.46 By using this approach as well as by modeling the transition zone, from the star molecules moving in an undilated tube toward their motion in a fat tube, with a constraint release Rouse (CRR) process, they could correctly reproduce the experimental data. Today, most of the tube models use this concept to account for the faster relaxation of the long star in an environment of short chains. However, as described in the recent work of Desai et al.,41 many ambiguities remain about the way this tube dilation effect should evolve through time and, in particular, during the CRR regime.47 Furthermore, in this last work, the authors showed that the predictions obtained with the hierarchical model from Larson and co-workers or with the branch-on-branch (BoB) model15 developed by McLeish, Das, Read, and co-workers were not in good agreement with new experimental data of star/linear blends. They attributed this failure to the incapacity of the tube models to accurately describe constraint release events in situations where rather abrupt relaxation of a significant fraction of the polymer occurs, contrary to slip-links or slip-spring models.32,37,41

Besides this modeling issue, the failure of the DTD concept in the case of binary blends composed of star molecules as slow components was also experimentally demonstrated by Watanabe et al.,30,48 who compared the viscoelastic and the dielectric data of poly isoprene star/star blends. In their work, the authors showed that partial DTD must be used in order to obtain a good description of the data. Thus, being far from trivial, analyzing the viscoelastic properties of star/linear blends offers great opportunities to better understand the arm retraction process and to fine-tune the tube molecular theory. To this end, new star/linear blends have been designed and synthesized in order to produce a consistent set of data, allowing us to explore the influence of CR processes under different conditions. In particular, these blends are composed of long, well- entangled poly(hydroxy butyrate) (PHB) star molecules (containing 18.5 entanglement per arm) blended to different matrices of linear chains (containing either 1.8, 4.5, or 15.8 entanglements per chain).49 The proportion of star polymer has been varied from 2 wt % (in order to avoid star− star entanglement) to 10 and 20 wt %. With this new set of data, we are now able to further test the validity of the DTD concept. In addition to these new sets of data, we analyze the data presented by Shivokhin et al.,37 who worked on the stochastic slip spring model in order to predict the stress relaxation of binary blends composed of linear (Mw = 7.5 kg/mol) and star (Mw = 76.0 kg/mol) PBD chains. These data are interesting since they have also been analyzed by Desai et al.,41 who showed that tube models could not predict their relaxation behavior. Experimental data from Watanabe et al.30 for long PI star polymer (Mw,arm = 81 or 59 kg/mol) diluted in a matrix of short PI star polymer (Mw,arm = 9, 16, or 24 kg/mol) are also used in order to discuss the CCR time of the long stars in the case these ones do not contain any self-entanglement. The paper is outlined as follows. Section 2 describes the synthesis and characterization of the star and linear PHB molecules. Section3 describes the model proposed to analyze these blends. Section 4 presents and discusses the results. We first focus on the binary blends in which the long stars are not self-entangled. Then, we extend our analysis to blends containing a larger proportion of star polymer. Conclusions are drawn in section5.

2. EXPERIMENTAL SECTION 2.1. Materials. Unless otherwise specified, all the catalyst and polymer syntheses were performed under inert atmosphere (N2) using standard Schlenk, vacuum line, and glovebox techniques. A Bruker Avance 400dir MHz spectrometer was used to record the 1H NMR of the polymers to measure the polymerization conversions. Tetrahydrofuran (THF) was collected from a solvent purification system, further dried over Na/benzophenone, vacuum-transferred to a Strauss flask, and then degassed through a series of freeze−pump− thaw cycles. CDCl3 was dried over CaH2 and vacuum-transferred to a Strauss flask and then degassed through a series of freeze−pump−thaw cycles. Chiral diaminophenolate ligands, H(NNOtBu) dichloride indium complexes, (NNOtBu)InCl2, and [(NNHO)InCl]2(μ-Cl)(μOEt) (1), [(NNHO)InCl]2(μ-Cl)(μ-OBn) (2), and [(NNHO)InCl]2(μ-Cl)(μ-OTHMB) (3) were synthesized according to literature procedures.49,50 Monodispersed linear and three arm star poly(hydroxybutyrate) (PHB) of controlled molecular weight were synthesized via the ringopening polymerization of β-butyrolactone (BBL) using complexes 2 and 3 as the initiator and benzyl alcohol (BnOH) and tris(hydroxymethyl)benzene (THMB) as chain transfer agents. Polymerization results are summarized in Table 1. C


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scanning calorimetry (DSC), using a heating rate of 10 °C/min from −25 to 25 °C. As expected, the Tg values of the linear samples increase with molecular weight, and in blends Tg increases with the concentration of the star polymer. For the last set of blends with the matrix of highest molecular weight (Mw,L = 66.4 kg/mol), Tg is not affected by the star concentration. 2.3. Rheological Measurements. All the rheological measurements were performed under a nitrogen atmosphere to minimize degradation of the polymer samples during testing. The linear viscoelastic properties of the polymer samples have been determined by a rotational rheometer (Anton-Paar, MCR 501) equipped with parallel plate geometry (diameter of 8 mm). The gap was set to 0.5 mm. Small amplitude frequency sweep tests were performed at frequencies in the range from 0.01 to 100 Hz, with a constant strain of 1% and several temperatures ranging from 25 to 80 °C. Time− temperature superposition was applied to obtain the master curves presented in section 4.

Table 1. Polymerization of BBL in the Presence of BnOH (To Form Linear Chains) and THMB (To Form Three Arm Star Molecules)

1 2 3 4

ROH

[M]:[ROH]:[I]

conva (%)

Mn,theob (g/mol)

Mw,GPCc (g/mol)

Đc

BnOH BnOH BnOH THMB

5000/50/1 5000/23/1 5000/5/1 20000/6/1

>99 >98 >98 >95

8690 18300 84300 272000

7020 17800 66400 209000

1.01 1.02 1.01 1.02

a

Monomer conversion, determined by 1H NMR spectroscopy. Calculated from ([BBL]0/([ROH]/[I]) × monomer conversion × MBBL) + MROH (MBBL = 86.09 g/mol, MBnOH = 108.14 g/mol, MTHMB = 168.19 g/mol). cDetermined by GPC-MALS using dn/dc = 0.068 for PHB in THF, Đ = dispersity index. b

In a typical polymerization of BBL to form star PHBs, a 20 mL scintillation vial was charged with complex 1 in THF (0.0037 mmol). THMB (31 mg, 0.185 mmol) was dissolved in 2 mL of THF and added to the catalyst solution. After 1 h stirring, the solution was dried under vacuum for a few hours to remove resulted ethanol and all the solvent and generate complex 3 as a white powder. Complex 3 was then dissolved in 2 mL of THF, and BBL (1.50 mL, 18.3 mmol) was added dropwise to the stirring solution. The reaction mixture was stirred overnight and then quenched with 0.5 mL of HCl (1.5 M HCl in Et2O). A sample of the mixture was dissolved in wet CDCl3 to be analyzed by 1H NMR spectroscopy to determine conversion. The residue was quenched in cold methanol (0 °C); the precipitated polymer was solidified by immersing the vial in liquid nitrogen, and subsequently, the supernatant was decanted off. To remove the trace amounts of the catalyst, the last three steps were repeated three times, and the isolated polymer was dried under high vacuum for overnight prior to analysis 2.2. Blends Preparation. The symmetric three-armed star PHB (Table 1, entry 4) with a molecular weight of 209 kg/mol was used as the dispersed phase in matrixes of linear PHBs of different molecular weights to make the blends with different concentrations ϕs = 2, 10, and 20 wt % of the star polymer. In order to prepare the blends, a certain amount of the linear matrix and the star PHB were dissolved in dichloromethane (DCM) and stirred together overnight to form a homogeneous solution. Subsequently, the solvent was removed under vacuum and further dried in a vacuum oven for 4 days at 45 °C. The composition of the blends and the molecular weight of the linear matrix are listed in Table 2. The glass transition temperatures (Tg) of the samples are also given. These were determined using dynamic

3. THEORETICAL SECTION 3.1. Classical TMA Model. In order to analyze the viscoelastic properties of these samples, we first use the Time Marching Algorithm (TMA) developed by van Ruymbeke et al.9,44 Since this model has been described in detail previously, we only show here how the dilution effect coming from the short chains is taken into account in the fluctuation process of the star molecules (referred to by the index 2). This is achieved through the DTD picture, assuming that the part of the polymer already relaxed acts immediately as real solvent in the retraction process of the remaining oriented polymer fraction, as modeled by the time τactivated.9 On the other hand, following Milner and McLeish,4,23 it is assumed that early fluctuations of the arm extremities are not affected by CR effects.36,44 4 9π 3 ⎛ Marm ⎞ 4 τe⎜ τearly,arm(xi) = ⎟ xi 16 ⎝ Me ⎠

⎛ Marm,2 ⎞ ln τactivated,arm(xi) = ln τactivated,arm(xi − 1) + 3⎜ ⎟xi ⎝ Me ⎠ × ϕDTD,arm(xi)α (xi − xi − 1)

ϕsa

Mw,Lb (g/mol)

Tgc (°C)

L66 S209-L66-2% S209-L66-10% S209-L66-20% L17 S209-L17-2% S209-L17-10% S209-L17-20% L7 S209-L7-2% S209-L7-10% S209-L7-20% S209

0 2 10 20 0 2 10 20 0 2 10 20 0

66400 66400 66400 66400 17800 17800 17800 17800 7020 7020 7020 7020 209000

2.43 2.54 2.37 2.38 −3.63 −2.08 1.47 3.20 n.d.d −2.31 −0.58 0.52 3.46

(8c)

In eq 8c, the position xtr represents the position of the arm segment at which the potential is equal to kT, i.e., at which the transition between early fluctuations and deep retraction is taking place. In eq 8b, ϕDTD,arm(x) determines the polymer fraction which is not relaxed at the time the segment xi of the arm is relaxing: ϕDTD,arm(xi) = υ1(1 − x1) + υ2(1 − xi) if τfluc,arm(xi − 1) < τrept(M1) ϕDTD,arm(xi) = υ2(1 − xi)

(9a)

if τfluc,arm(xi − 1) > τrept(M1) (9b)

with x1 being the deeper molecular segment of the short chain relaxed by retraction at the time the arm segment xi of the star is relaxing and τrept(M1) being the reptation time of the short linear chains. Knowing the fluctuations times of the star segments and the reptation/CLF time of the short chains matrix (as detailed in

a c

(8b)

⎛ τactivated,arm(xi) ⎞ ⎟⎟ τfluc,arm(xi) = τearly,arm(xtr) exp⎜⎜ ⎝ τactivated,arm(xtr) ⎠

Table 2. Linear and Three-Armed Star PHBs Blends and Their Molecular Characteristics sample name

(8a)

Weight fraction of the star. bMolecular weight of the linear PHBs. Determined using DSC. dNot determined. D


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is noted that in the model τCRR is first considered as a fit parameter, which is further investigated and discussed in section 4. It is expected that eq 12 will be valid only if the fraction of the star polymer which is relaxed at the relaxation time of the short chains, τ1, is negligible, i.e. for star/linear blends with very well-separated relaxation time. Furthermore, in eq 12, we assume that the fluctuations process cannot take place simultaneously with the CRR process. These two assumptions are quite important, and will be validated in section4. B. Relaxation of Star Arms Containing Self-Entanglement. For the binary blends containing self-entangled star molecules, we assume that the star arms are moving in a fat tube, but at the rhythm of the entanglement/disentanglement of the short chains rather than as they were diluted in a real solvent.36 This is achieved by considering that the relaxation time of a long− long entanglement segment is well described by τlong−long ent (see eq 7) rather than by its intrinsic Rouse time. Consequently, the relaxation of the long star arms takes place through CRactivated CLF process.36

previous works36,51), the survival fraction of initial tube segments, φ′(t), is then determined and used in order to calculate the disentanglement modulus Gd(t) based on eqs 5 and 6. In addition to the disentanglement modulus Gd(t), the general relaxation function must also take into account the high-frequency Rouse process, GR(t): G(t ) = G R (t ) + Gd(t )

(10)

with G R (t ) =

∑ k

⎧ υk ρRT ⎪ 1 ⎨ Mk ⎪ ⎩4

Zk

p2 t ⎞ ⎟ ⎝ τR (Mk) ⎠ ⎛

∑ exp⎜− p=1

⎫ ⎛ p 2 t ⎞⎪ ⎟⎬ exp⎜ − ⎪ ⎝ τR (Mk) ⎠⎭ p = Zk + 1 n

+

∑

(11)

In eq 11, υk and τR(Mk) represent the weight fraction and the Rouse time of the chain k, respectively. 3.2. Star Relaxation through CR-CLF Process. While the approach proposed in section 3.1 will be first tested and commented, we would like to apply, in a second step, a new model, in which the long (star) components can relax in their fat tube but only at the rhythm of the (slow) motion of the short (linear) components. This approach is similar to the one proposed for binary blends of linear chains36 and is described here below in the specific case of long star component blended to a short (linear or star) matrix. Within this framework, two different cases must be considered: either the proportion of long star polymer is low enough to ensure that there is no selfentanglement of the long stars (i.e., no star−star entanglement) or the long star molecules are entangled with both the short polymer matrix and the other long star segments. A. Relaxation of Star Molecules Containing No SelfEntanglement. If the long star molecules are only entangled with the short chain matrix and if their relaxation times are strongly separated, it is expected that the star arms fully relax by constraint release Rouse process, while the short chains will relax by reptation/CLF in a thin tube.22 Therefore, we describe the corresponding relaxation function, Gd(t), as ⎡ ⎛ − t ⎞⎤ Gd(t ) = GN0 ⎢υ1φ′1(t )ϕ(t ) + υ2ϕ(t ) exp⎜ ⎟⎥ ⎢⎣ ⎝ τCRR ⎠⎥⎦

τCR − CLF(x fat tube) = τlong − long ent

9π 3 (υ2Zarm,2)4 xfat tube 4 16 (13)

where Zarm,2 is the number of initial entanglements along each arm of the star. It must be noted that with these systems containing only few long−long entanglement segments, we did not have to consider activated fluctuations for the relaxation of deeper segments in the fat tube since the corresponding retraction potential was lower than kT. In order to ensure that the star motions in a fat tube only takes place at time larger than the relaxation time of a long− long entanglement segment, τlong−long ent, we follow Shchetnikava et al.8 and use the reference system xarm, according to which (xarm = 0) at the arm segment (xfat tube= xe) for which τCR−CLF(xe) = τlong−long ent, while (xarm = 1) corresponds to the branching point of the star. In addition to these CR-activated motions, the star arms can also fluctuate in their thin tube, following eq 8a. However, the contribution of this relaxation process is negligible for the different samples considered here. Therefore, the survival probability p[2](xarm,i,t) of a star segment xarm,i can be simplified and described as [2] pCR (x , t ) = exp{−t /τCR − CLF(xarm, i)} − CLF arm, i

(12)

In this equation, φ′1(t) represents the survival fraction of tube segments belonging to the short chains 1 (thus φ′1(t) evolves from 1 to 0), and the dilation factor ϕ(t) is described by eq 6, considering that the survival fraction of tube segments φ′(t) = (υ1φ′1(t) + υ2). Thus, as described in eq 12, the dilution process is the only possible relaxation process for the long star component since the survival fraction of its tube segments, φ′2(t), is fixed to 1. This is consistent with the CRR picture according to which the long chains do not relax by reptation/ fluctuations but only by constraint release. Furthermore, right after the relaxation of the short chains, φ′1(t) will abruptly decrease, in such a way that the CRR condition of eq 6 will become active. This will ensure that the long chains will relax by CRR, as expected. Once the longest CR Rouse mode of the stars is relaxed, which corresponds to a time equal to τCRR, these chains should not contribute to the relaxation modulus anymore. This is ensured by the time decreasing exponential function in eq 12. It

φ′2 (t ) =

∫0

1

[2] pCR (x , t ) d x − CLF arm, i

(14)

(15)

On the other hand, the relaxation of the short component is modeled with the usual TMA, and the modulus Gd(t) is described as Gd(t ) = GN0 [υ1φ′1(t ) + υ2φ′2 (t )]ϕ(t )

(16)

where the dilation factor ϕ(t) is determined by eq 6. As described here above, this simple model contains two unknown parameters, τCRR (see eq 12) and τobs (see eq 7), which are discussed in the next section.

4. RESULTS AND DISCUSSION 4.1. Experimental Viscoelastic Properties of the Blends. The experimental linear viscoelastic properties of the PHB homopolymers and their blends are plotted in Figure 1, in E


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Figure 2. Logarithm of the zero shear viscosity of the different polymer blends as a function of the star concentration at the reference temperature of 50 °C. For the linear matrix L7, the zero shear viscosity has been calculated based on the results of Ebrahimi et al.48

for the polymer blends with the lowest matrix molecular weight, and it increases upon the addition of star polymer to the linear matrix. However, it can also be observed that the impact of the star polymer on the zero shear viscosity is much higher for polymer blends with lower molecular weight matrices. In particular, for the blends with a linear matrix of 7 kg/mol, there is a significant difference between the value of the zero shear viscosity of the pure matrix and the viscosity of the different blends. Even if we assume that the star molecules in the blends composed of 2 wt % of stars fully relax by CRR, the fact that there is a large viscosity increase for the blend in a short linear matrix against longer linear matrix is nontrivial and will be further investigated below. A last point to mention about Figure 2 is that the zero-shear viscosity values of blends with the high molecular weight matrix (Mw,L = 66.4 kg/mol) increase linearly in a logarithmic scale, i.e., exponentially with star concentration, while a positive deviation is visible for the blends with lower molecular weight matrixes. This is in contrast to the Milner and McLeish23 report which showed a negative deviation of the experimental results for blends of monodisperse star polybutadiene (PBD) and relatively long short linear matrix (Mw,L = 105 kg/mol and Mw,S = 127 kg/mol). This difference, which is most probably due to the solvent role played by the short chain matrix, demonstrates the nontrivial interpretation of the viscosity−concentration relationship of the binary blends of different polymers. 4.2. Predicted LVE Curves Based on the Usual TMA Model. As a first step, we used the classical version of the TMA (see section 3.1) in order to predict the linear viscoelastic properties of the monodisperse samples as well as of the blends. The material parameters of the PHB samples, i.e., the plateau modulus, the average molar mass between two entanglements, and the Rouse time of an entanglement segment, were determined based on the monodisperse samples. Their values are 650 kPa, 3800 g/mol, and 3 × 10−4 s, respectively. Compared to the relationship G0N = (4/5)(ρRT/Me), the value we consider for G0N is 7% too low. Comparisons between experimental and theoretical data are shown in Figure 3 for the monodisperse samples and in Figure 4 for the blends. As expected, the linear viscoelastic data of the monodisperse samples are well described by the model. It is also observed that the relaxation of the linear chains L17 (and, in consequence, the linear chains L7) is much faster than the relaxation of the star polymer, these last ones being still in their plateau region when the linear chains relax. This means that the assumptions behind eq 12, according to which we can neglect the fraction of star arms relaxed before the relaxation of the linear chains,

Figure 1. Master curves of the storage and loss moduli of the blends composed of (a) 2, (b) 10, or (c) 20 wt % of star polymer (Mw,s = 209 kg/mol) in different molecular linear matrices L7 (gray triangles), L17 (blue cycles), and L66 (black squared). The reference temperature for the master curves is Tref = 50 °C for blends composed of L66 and is such that (Tref − Tg) = constant for the other blends.

the form of master curves obtained by applying the time− temperature superposition principle, and at the reference temperature of 50 °C for the blends containing the longer linear chains, L60. For the other samples, slightly different reference temperatures were used in order to compensate for their difference in Tg (see Table 2): by working at the same “distance” from Tg (i.e., such as (Tref − Tg) is a constant), we ensure that the data superimpose in the high-frequency Rouse regime,53 as it would have been the case for samples with same Tg. As shown in Figure 1a, and in comparison to the viscoelastic properties of the monodisperse linear matrix (see Figure 3), it is also observed that adding 2 wt % of star in the linear matrix strongly affects the terminal regime, even if the star concentration is below the limit of self-entanglements. Since the average molar mass between two entanglement is equal to 3800 g/mol in the melt state (see below), the effective molar mass between two long−long entanglement of a molecule diluted at 2 wt % is equal to 190 kg/mol, i.e., is larger than the end-to-end molecular weight of the star molecules. In such a case, if the linear chains relax much faster than the star molecules in their undilated tube, the star molecules are expected to relax by the constraint release Rouse (CRR) process.22,28−30 This process, which is clearly observed at intermediate frequency for the blends involving the linear matrixes L7 and L17, takes time since the star arms are long and well entangled (each arm has around 18.5 entanglements). Interestingly, despite the fact that the relaxation of the star molecules is fully governed by the motion of the short chain matrix (see section 4.3), it is seen that the delay observed in the terminal relaxation time of these blends compared to the relaxation time of the linear matrix alone is decreasing with increasing the molar mass of the short linear matrix. This trend is also observed by looking at the corresponding zero shear viscosities of the different blends as a function of the star concentration, as depicted in Figure 2: As expected, at a fixed star concentration, the zero shear viscosity has the lowest value F


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Figure 3. Experimental (○) and theoretical (using the classical TMA model) () storage and loss moduli and of the monodisperse linear samples L17 (△) and L66 (□) as well as of the star polymer S209 (○).

Figure 5. Experimental (○) and predicted () storage (a) and loss (b) moduli and of a PBD star with Mstar = 76 kg/mol blended to linear PBD chains with M = 7.9 kg/mol, in different star proportions, equal to 100 (light blue), 50 (blue), 20 (red), 10 (black), 2 (orange), or 0 wt % (cyan). Data are from ref 37.

way the transition between early and activated fluctuations is implemented (see eqs 8). However, to further confirm it, a detailed comparison between the two models is required. However, despite this good agreement, a deeper analysis of the fluctuations times of the star arms raises important questions about the validity of this approach in this specific case of binary blends with star molecules containing only few star−star entanglements. Indeed, as shown in Figure 6 (by the Figure 4. Experimental (○) and predicted (using the classical TMA model) () storage and loss moduli and of the star S209 blended to the linear matrix L7 (a), L17 (b), and L66 (c), in different proportions of star polymer (black: 2 wt %; blue: 10 wt %; red: 20 wt %).

should be valid with the matrixes L7 and L17. On the other hand, we observe that the fraction of star arms relaxed at the time the linear chains L66 are relaxing is already more important. Therefore, in this last case, we expect that the assumption behind eqs 12 and 15, according to which we can neglect the relaxation of the long star molecules in their thin tube, should reach their validity limit. Furthermore, as it can be seen in Figure 4, the agreement between the experimental data and the one predicted by the usual TMA is quite good for the blends of the star molecules with the matrix L17 or the matrix L66. These results differ from those obtained by Desai et al.,41 according to which classical tube models are not able to correctly predict the viscoelastic properties of such blends. In order to allow direct comparison with their work, we compare in Figure 5 the experimental and theoretical rheological data of the set of PBD samples from Shivokhin et al.,37 since these samples have also been analyzed by Desai et al.41 These blends are composed of PBD star polymer with a total molar mass of 76 kg/mol, blended to short linear chains with a molar mass of 7.5 kg/mol, in different proportions. The material parameters for these PBD samples have been taken as GN = 1.2 MPa, Me = 1650 g/mol, and τe = 2.3 × 10−7 s.9,34,36 Again, a good agreement is found between experimental and predicted curves, contrary to the recent results reported by Desai et al.41 We believe that the large difference observed between the predictions obtained with the two models in the case of star/linear blends comes from the

Figure 6. Fluctuations times (red - - -) and early fluctuations times (black ) of the arm segments of the star, from the arm extremity (x = 0) to the branching point (x = 1), corresponding to the blend composed of 20 wt % PBD star with Mstar = 76 kg/mol and 80 wt % of linear PBD chains with M = 7.9 kg/mol.

red dashed curve), after the early fluctuations process of the arm extremities (from x = 0 to x = 0.28), the retraction times of the deeper arm segments (see eq 8b) are predicted to be nearly constant along 3/4 of the arm and do not increase with the depth of the molecular segment. This is due to the large proportion of linear chains, which are considered as a solvent during this retraction process (see eqs 9). In consequence, the total fluctuations times of the arms (see eq 8c) are becoming very short, and even shorter than their early fluctuations times (see eq 8a and the black continuous curve in Figure 6), which does not seem physical. Thus, for such extreme blends, the star relaxation is completely dominated by the time at which the transition between early and activated fluctuations takes place, i.e., at time τearly(x = xtr). Another limitation of this approach is noticeable in Figure 4a as for the PHB stars S209 blended with the linear L7 matrix, linear chains relax at very short times. This causes a pronounced solvent effect at the beginning of the retraction G


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12 and that the star molecules do not need any extra relaxation process, which would speed up their relaxation, in order to capture their relaxation moduli at intermediate frequencies. This good agreement suggests that (1) the fraction of star polymer relaxed before the relaxation of the linear matrix is negligible, as it was already proposed based on Figure 3, and (2) the relaxation of the chains by fluctuations is negligible during their CRR process. A good agreement was also found for the blends based on the linear matrix L60; however in this case, the CRR regime cannot be observed. In order to discuss the value of the CRR time based on a large set of data, other blends with no self-entanglement of the long stars are investigated, and the results are shown in Figures 8 and 9. These blends are a PBD star/linear blend Shivokhin et

process of the stars. As a result of this, the corresponding retraction potential of stars never reaches a value of kT. Therefore, the activated fluctuations of the stars never take place, and their fluctuation times are fully dominated by the early fluctuations until their branching points. This prevents a fast relaxation of the arms since the solvent effect from the linear matrix is not taken into account in the early stage of the fluctuations process. The two cases described above show that our tube model reaches its limits with these star/linear blends, with a too strong effect of the DTD in the retraction process and a too small effect of the DTD in the early fluctuations process. Therefore, in the next sections, we would like to consider another point of view, according to which the early fluctuations of the star molecules can take place in a fat tube, but at the rhythm of the disentanglement/re-entanglement of the short linear chains, as it has been described in section 3.2 and in ref 36 in the case of binary blends of linear chains. 4.3. Relaxation of the Binary Blends without Long Star Self-Entanglements. As a first step, we consider binary blends in which the long star component is not self-entangled, with the objective to test if the relaxation of such blends is well described by eqs 10−12, i.e., assuming that the long stars as fully relaxing by constraint release Rouse relaxation (CRR). Furthermore, we would like to discuss the value of the corresponding CRR time, τCRR, and check if it follows the scaling law proposed by Struglinski et al.,22 i.e., if this CRR time depends on the relaxation time of the linear matrix, τobs = τ1, and on the number of entanglement segments between two extremities of the star molecules, Z2 (= 2Zarm,2): τCRR = τobsZ 2 2 = τ1Z 2 2

Figure 8. Experimental (○) and theoretical () storage and loss moduli of a PBD blend composed of 2 wt % of star polymer with a molar mass of 76 kg/mol and 98 wt % of linear chains with a molar mass of 7.9 kg/mol (○) and compared to the monodisperse linear matrix (□).

(17)

To this end, we first consider this CRR time as a free (fitting) parameter, and then check the validity of eq 17. Results obtained for the PHB blends containing 2 wt % of star S209 in different linear matrices are shown in Figure 7. As

Figure 7. Experimental (○) and predicted (- - -, ) storage and loss moduli of PHB blends composed of 2 wt % of S209 in the different linear matrices, by considering (thick curves) or not (thin curves) that the star are fully relaxed at time τCRR.

Figure 9. Experimental (○) and theoretical () storage and loss moduli of (a) 3 wt % of PI star with Marm = 81 kg/mol diluted in different star matrix with Marm = 9, 16, and 23 kg/mol and (b) 4 wt % of PI star with Marm = 59 kg/mol diluted in the same star matrices.25

already mentioned, the proportion of star molecules is low enough to avoid self-entanglement of the star. If we assume the star molecules never reach their full relaxation (and thus their diffusive mode), we see that the model (i.e., eq 12 without the exponential factor) very well predicts the intermediate frequency data; however, the sample will never flow (see the thin curves). In order to allow the molecules to reach their terminal regime, we need to consider a time τCRR after which the stars are fully relaxed (as described by eq 12). In Figure 7, we see that the CRR regime of the stars diluted in the linear matrices L17 and L7 is very well described by eq

al. report37 as well as two series of samples composed of a long polyisoprene (PI) star in different matrices of short PI stars, coming from the work of Watanabe et al.30 In the last case, the short component is starlike rather than linear. However, these stars are short enough to relax well before the relaxation of the long star and therefore the CRR assumption stays valid. Again, for these samples, eqs 10−12 correctly capture the viscoelastic properties of the long component, which only relaxes by CRR process. It must be noted, however, that in order to correctly capture the relaxation of the short, poorly H


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oxide).53 Deviation from the simple expression (τCRR = τ1Z22) was also pointed out by Watanabe and co-workers (see Figure 37 in ref 5 and refs 54 and 55). However, they are in contradiction with other experimental works on polystyrene blends,56−58 which showed a power-law dependence on Z1.2−5 In order to explain this different scaling, Wang et al. suggested that other molecular factors, such as the chain stiffness or the free volume changes with molecular weight, could also influence this dependence. While no difference was observed here with blends of three different natures (PHB, PI, and PBD), we should extend this study to other polymers, and in particular to PS blends, in order to discuss the universality of eq 18. In Figure 10b., it is seen that the ratio between CRR time and (τ1/Z12)Z22 is around 3. The fact that the prefactor is not equal to 1 may be due to the way the value of τ1 has been determined, based on the experimental data, i.e., including CR effect on the short chains. It can also be due to the fact that assuming a single disengagement time (τ1) of the short−long entanglements is oversimplified since in reality there is a broad distribution of these short−long disentanglement times. This new definition of the lifetime of an entanglement can now be used in eq 7 for defining the relaxation time of a long− long entanglement segments in binary blends containing selfentangled long star polymers (see section 4.4), thus removing the unkwnown parameter of eq 13:

entangled component, one needs to avoid a too strong influence of the solvent in their fluctuations process, which can lead to a situation similar to the one described in Figure 6. To this end, we imposed that the fluctuations times of the short star matrices are at least equal to their early fluctuations time. We can now look at the value of the CRR time, τCRR, that we have used for describing the relaxation moduli of these different samples. According to eq 17, the ratio τCRR/Z22 should be equal to the relaxation time of the short matrix, τ1. The validity of this relationship is examined in Figure 10a. In this figure, the matrix

Figure 10. (a) Ratio τCRR/Z22 versus the terminal relaxation time of the short matrix, for the different blends presented in Figures 7−9. (b) τCRR/(Z1/Z2)2 versus the terminal relaxation time of the short matrix for the same blends.

relaxation time has been estimated as the inverse of the frequency at which lines superimposing experimental storage and loss moduli at low frequency with slopes of 1 and 2 crossover each other. Despite a clear trend observed in Figure 10a, it is obvious that the different data do not fall into a single line of slope 1. Thus, the relation proposed in eq 17 cannot be validated. Only the data for the PI blends, which are based on the same short chain matrices, superimpose, meaning that most probably the number of entanglements of the short chains, Z1, should also affect the value of τCRR. In fact, as illustrated in Figure 10b, it is observed that the CRR time also depends on the number of entanglement segments of the short chains and is well described as ⎛ τ ⎞ τCRR = τobsZ 2 2 ∝ ⎜ 12 ⎟Z 2 2 ⎝ Z1 ⎠

⎞⎛ υ ⎞ 2 ⎛ τ τlong − long ent = max⎜ 12 , τe⎟⎜ 1 ⎟ ⎠⎝ υ2 ⎠ ⎝ Z1

(19)

4.4. Relaxation of the Binary Blends Containing SelfEntangled Long Star Molecules. On the basis of eqs 13−16 as well as on the new definition of the relaxation time of a long entanglement segment eq 19, we can now predict the linear viscoelastic properties of the binary blends containing few selfentangled long star molecules. As a first set of samples, we look at the moduli of the long stars entangled with very short, unentangled linear chains. In this specific case, the relaxation time of the long entanglement segment, τlong−long ent, used in eq 13 is simply equal to its intrinsic Rouse time (see eq 19). As shown in Figure 11, the agreement between experimental and

(18)

A possible way to explain this observed scaling is to consider that after the relaxation time of a short chain, τ1, there are Z1 entanglements, which are lost (and not only one entanglement, as it is assumed in eq 17). Using this relationship allows us to explain the fact that the delay observed in Figures 7−9 between the relaxation of the long star molecules and the relaxation of the short matrix (i.e., τCRR/τ1) increases with decreasing the molar mass of the matrix, as mentioned in section 2 (see Figure 2). Furthermore, it allows explaining why in ref 36 the value of τobs determined based on eq 17 was found to be much shorter than the relaxation time of the short chains. In fact, this scaling was already observed in the literature.52,53 In particular, in ref 52, Wang et al. studied the dynamics of a large range of entangled binary blends, based on molecular dynamics simulation, and found that the diffusion coefficient of the long (probe) chain diluted in short linear matrix (i.e., short enough to ensure Rouse-like diffusion of the long chains) was showing a power-law dependence on Z1 which was close to power 1, i.e., in agreement with the results found here. This dependence was also observed by Smith et al. with tracer diffusion experiments on blends of linear poly(propylene

Figure 11. Experimental (○) and predicted () storage and loss moduli of the binary blends S209-L7-20% (blue symbols and black curve) and S209-L7-10% (red symbols and gray curve).

predicted curves is very good, suggesting that the CR-CLF process (see eq 13) well describes the motion of the star arms: these ones fluctuates in a fat tube and are not delayed by the motion of the very short linear chains. On the other hand, if the stars are blended to a longer linear matrix, the motions of the linear chains slow down the motions of the star arms, which are moving in their fat tube, and the relaxation time of a long entangled segment, τlong−long ent, I


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Figure 13. (a) Experimental (○) and predicted () storage and loss moduli and of a PBD star with Mstar = 76 kg/mol blended to linear PBD chains with M = 7.9 kg/mol, and containing 100 (magenta), 50 (blue), 20 (red), 10 (black), 2 (green), or 0 wt % of star polymer. While the value of τ1 (= 1.8 × 10−5 s) could be used for the 50% and 20% blends, a value of 1.5τ1 was used for the blends with 10 and 2 wt % of star polymer. Data are from ref 37.

Figure 12. Experimental (○) and predicted () storage and loss moduli of the binary blends S209-L17 (a) and S209-L66 (b), composed of 20% (blue symbols and black curve) or 10% (red symbols and gray curve) of star polymer. While with 20 wt % of star polymer, the value of τ1 is used, a value of (1.5τ1) was used for the blends with 10 wt % of star polymer. Results obtained based on τ1 are shown by the thin curves.

relaxation in a thin or fat tube, one should further play with the molar mass of the linear matrix: if the linear chains are becoming very long, the necessary time for the star arms to explore their fat tube (at the rhythm of the “short” chain motions) will become very long, and the relaxation of these last ones in a thin tube will be favored again. Thus, the way a long star polymer will relax in a matrix of linear chains is a delicate balance between CLF and CR-CLF, which depends on their concentration, their molar mass, and the molar mass of the short matrix. This picture differs from the proposition of Doi et al.59 to describe CR mechanisms in binary blends, which associate the relaxed short chains to a solvent. On the other hand, it rather agrees with the proposition of Viovy et al.,60 who define two different types of motions, either of the chain along the undiluted tube or of the skinny tube along the fat tube, at the rhythm of the short chains motions. However, the way we determine the equilibration time τlong−long ent and the influence of contour length fluctuations is different.

experimental and predicted curves is also found for the blends with 20 wt % of star polymer. It must be noted, however, that for the blends including only 10 wt % of star polymer one needs to consider a slightly higher value of τ1 (higher by a factor 1.5) in order to obtain an accurate description of the data. The origin of this discrepancy (observed by comparing the thin curves to the experimental data in Figure 12), which is exactly the same for the blends in the two different matrices, L17 and L66, is not clear and should be further investigated, based on a larger set of data. A possible explanation is the approximation we use for describing the survival probability of the dilated tube segment (see eq 14), presently described as an exponential distribution. This model is then used for predicting the viscoelastic properties of the PBD star/linear blends from Shivokhin et al. Results are shown in Figure 13, and the same conclusions as for the PHB blends are found: again, the data are well predicted by using eq 19; however, with a small amount of star chains (10 wt % or less), the relaxation time associated with the motions of the short chains is again, found to be 1.5 larger than τ1. We also note that with the blend composed of 50 wt % of star polymer the predicted contribution of the star is larger than in the experimental data. In order to correctly describe the experimental data, one should consider 46 wt % of star polymers rather than 50 wt % (not shown). Despite these small variations found in the value of τ1, it can be concluded that this simple approach allows obtaining a very good description of the experimental data, while keeping a clear physical meaning: the long star molecules are moving and relaxing in their fat tube, at the rhythm of the motions of the short chain matrix, which is taken into account through the time τlong−long ent. The star arms could also move and relax in their initial, undilated tube; however, in the blends considered here, their arms are very long and thus their motions in a thin tube are too slow in order to compete with their motions in their fat tube (the blends based on the linear matrix L66 being close to this limit). In order to modify the balance between star

5. CONCLUSION In order to further understand the relaxation of binary blends composed of star and linear chains, new polymer blends, consisting of a linear poly(hydroxy butyrate) (PHB) matrix and PHB star molecules, have been synthesized and studied, varying the star concentrations and the molar mass of the linear matrix, while keeping no or few star−star entanglements in the blends as well as considering components with well-separated relaxation times. This has allowed us to study the constraint release Rouse (CRR) relaxation of the star molecules diluted in the linear matrix at concentrations low enough such that star−star entanglements are avoided. On the basis of these data as well as other experimental data taken from the literature, we could show that the terminal relaxation time of the star depends on the relaxation time of the short chain matrix τ1 and on the number of entanglement segments in the stars Z2, as expected. However, we showed that it also depends on the number of the entanglement segments of the short chains, and discussed this dependence, compared to previous results from the literature. J


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Rheology Reviews; Binding, D. M., Hudson, N. E., Keunings, R., Eds.; The British Society of Rheology: 2007; pp 53−134. (7) Milner, S. T.; McLeish, T. C. B. Parameter-Free Theory for Stress Relaxation in Star Polymer Melts. Macromolecules 1997, 30 (7), 2159− 2166. (8) Shchetnikava, V.; Slot, J. J. M.; van Ruymbeke, E. A Comparison of Tube Model Predictions of the Linear Viscoelastic Behavior of Symmetric Star Polymer Melts. Macromolecules 2014, 47 (10), 3350− 3361. (9) van Ruymbeke, E.; Bailly, C.; Keunings, R.; Vlassopoulos, D. A General Methodology to Predict the Linear Rheology of Branched Polymers. Macromolecules 2006, 39 (18), 6248−6259. (10) Kapnistos, M.; Koutalas, G.; Hadjichristidis, N.; Roovers, J.; Lohse, D. J.; Vlassopoulos, D. Linear rheology of comb polymers with star-like backbones: melts and solutions. Rheol. Acta 2006, 46 (2), 273−286. (11) Inkson, N. J.; Graham, R. S.; McLeish, T. C. B.; Groves, D. J.; Fernyhough, C. M. Viscoelasticity of Monodisperse Comb Polymer Melts. Macromolecules 2006, 39 (12), 4217−4227. (12) Snijkers, F.; van Ruymbeke, E.; Kim, P.; Lee, H.; Nikopoulou, A.; Chang, T.; Hadjichristidis, N.; Pathak, J.; Vlassopoulos, D. Architectural Dispersity in Model Branched Polymers: Analysis and Rheological Consequences. Macromolecules 2011, 44 (21), 8631− 8643. (13) Ahmadi, M.; Bailly, C.; Keunings, R.; Nekoomanesh, M.; Arabi, H.; van Ruymbeke, E. Time Marching Algorithm for Predicting the Linear Rheology of Monodisperse Comb Polymer Melts. Macromolecules 2011, 44 (3), 647−659. (14) Larson, R. G. Combinatorial Rheology of Branched Polymer Melts. Macromolecules 2001, 34 (13), 4556−4571. (15) Das, C.; Inkson, N. J.; Read, D. J.; Kelmanson, M. A.; McLeish, T. C. B. Computational linear rheology of general branch-on-branch polymers. J. Rheol. 2006, 50, 207. (16) van Ruymbeke, E.; Orfanou, K.; Kapnistos, M.; Iatrou, H.; Pitsikalis, M.; Hadjichristidis, N.; Lohse, D. J.; Vlassopoulos, D. Entangled Dendritic Polymers and Beyond: Rheology of Symmetric Cayley-Tree Polymers and Macromolecular Self-Assemblies. Macromolecules 2007, 40 (16), 5941−5952. (17) Matsumiya, Y.; Watanabe, H.; van Ruymbeke, E.; Vlassopoulos, D.; Hadjichristidis, N. Viscoelastic and Dielectric Relaxation of a Cayley-Tree-Type Polyisoprene: Test of Molecular Picture of Dynamic Tube Dilation. Macromolecules 2008, 41 (16), 6110−6124. (18) van Ruymbeke, E.; Muliawan, E. B.; Hatzikiriakos, G.; Watanabe, T.; Hirao, A.; Vlassopoulos, D. Viscoelasticity and extensional rheology of model Cayley-tree polymers of different generations. J. Rheol. 2010, 54, 643. (19) van Ruymbeke, E.; Slot, J. J. M.; Kapnistos, M.; Steeman, P. A. M. Structure and rheology of branched polyamide 6 polymers from their reaction recipe. Soft Matter 2013, 9, 6921. (20) van Ruymbeke, E.; Lee, H.; Chang, T.; Nikopoulou, A.; Hadjichristidis, N.; Snijkers, F.; Vlassopoulos, D. Molecular rheology of branched polymers: decoding and exploring the role of architectural dispersity through a synergy of anionic synthesis, interaction chromatography, rheometry and modeling. Soft Matter 2014, 10, 4762−4777. (21) Read, D. J. From Reactor to Rheology in Industrial Polymers. J. Polym. Sci., Part B: Polym. Phys. 2015, 53, 123−141. (22) Struglinski, M. J.; Graessley, W. W.; Fetters, L. J. Effects of polydispersity on the linear viscoelastic properties of entangled polymers. 3. Experimental observations on binary mixtures of linear and star polybutadienes. Macromolecules 1988, 21 (3), 783−789. (23) Milner, S. T.; McLeish, T. C. B.; Young, R. N.; Hakiki, A.; Johnson, M. Dynamic Dilution, Constraint-Release, and Star-Linear Blends. Macromolecules 1998, 31 (26), 9345−9353. (24) Lee, J. H.; Archer, L. A. Entanglement friction and dynamics in blends of starlike and linear polymer molecules. J. Polym. Sci., Part B: Polym. Phys. 2001, 39, 2501−2518. (25) Rubinstein, M.; Colby, R. H. Polymer Physics; Oxford University Press: 2003.

For the blends composed of a larger proportion of star polymer, we then used this new definition of CRR time in order to determine the necessary time of a star−star entanglement segment to relax by CRR and explore its fat tube, at the rhythm of the disentanglement/re-entanglements of the short chains. We propose to consider this time as the new reference time in order to describe the contour length fluctuations of the arm in its fat tube. In such a way, we could consistently take into account the two opposite effects coming from the short linear matrix: on one hand, its fast relaxation leads to a large dilution effect which is reflected by the dilation of the tube in which the long chains are moving. On the other hand, however, the long chains can only move in their fat tube at the rhythm of the motion of the short chains, which can strongly slow down their relaxation, compared to the motion of the same stars diluted in a real solvent. This molecular picture has been implemented in our tube model, and the good agreement found between experimental and predicted viscoelastic data is encouraging. In the near future, we would like to test it on other binary blends, with not so well-separated relaxation times. To conclude, the way a long star polymer will relax in a matrix of linear chains is a delicate balance between their (fast) relaxation by fluctuations in a (long) thin tube and their (slow) relaxation by fluctuations in a (short) fat tube. This balance depends on the concentration of the star and also the molar mass of the star and the low molecular weight matrix.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail [email protected] (S.G.H.). *E-mail [email protected] (E.v.R.). ORCID

Hamid Taghipour: 0000-0002-1103-4586 Parisa Mehrkhodavandi: 0000-0002-3879-5131 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors are very grateful to Dimitris Vlassopoulos and Hiroshi Watanabe for helpful discussions and to Maksim Shivokhin for the experimental data of the PBD blends. This work was partly supported by the Fonds National de la Recherche Scientifique (E.V.R. and H.T.). Additional financial support for an NSERC Strategic Grant program is greatly acknowledged. T.E. acknowledges the scholarship graduate program (4YF) at the University of British Columbia.

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Binary Blends of Entangled Star and Linear Poly(hydroxybutyrate

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