Dielectric Relaxation as an Independent Examination of Relaxation

Article pubs.acs.org/Macromolecules

Dielectric Relaxation as an Independent Examination of Relaxation Mechanisms in Entangled Polymers Using the Discrete Slip-Link Model Ekaterina Pilyugina, Marat Andreev, and Jay D. Schieber* Department of Chemical and Biological Engineering, and Center for Molecular Study of Condensed Soft Matter, Illinois Institute of Technology, 3440 South Dearborn Street, Chicago, Illinois, United States ABSTRACT: Dielectric spectroscopy is often used as a tool complementary to rheology for investigation of relaxation processes of a viscoelastic medium. In particular, dielectric relaxation of type-A polymers reveals the autocorrelation function of the end-to-end vector of a chain, whereas the relaxation modulus is the autocorrelation function for stress, which depends on more-local structural moments of the chain conformation. Here we examine published data for monodisperse linear, bidisperse linear and star-branched polyisoprene using the discrete slip-link model (DSM). Although the DSM makes predictions very similar to tube theory for linear viscoelasticity, there are some noticeable differences in the predicted contributions of different processes. Also, the DSM uses a single mathematical object for blends and varying chain architectures, meaning that the two adjustable parameters should be independent of molecular weight, blending, architecture or flow field. Here we use one set of parameters (aside from the temperature dependence of friction) to predict both the rheology and dielectric relaxation for all these systems as a strong test of the theory. We find that all circumstances save one are well described. Namely, dilute long chains in a sea of short chains can be predicted rheologically, but dielectric relaxation data show a reduction in the relaxation time of long chains greater than that predicted by either the DSM or the expected Rouse motion. We find that a modified Likhtman−McLeish model makes nearly identical predictions. The dilute long-chain contribution to dielectric relaxation remains a challenge for all entanglement theories.

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INTRODUCTION

effects of stretching, or constraint release could then be considered. Changing chain architecture also required modification to the mathematical tube models. Star-branched chains,4,5 H-shaped chains, or pom-pon-shaped chains6 required their own levels of description, as did bidisperse blends of linear chains.7 Sometimes one mathematical model is used for linear viscoelasticity, and another for flow.8 As a consequence, parameters that are similar in two different mathematical tube models might require some amount of modification in their values to describe the same chemistry. In their original papers, Doi and Edwards sketched an alternative to tubes for modeling entanglements. Instead of tubes, they illustrated rings through which the chains were constrained to pass. These were drawn to motivate the use of a certain expression from temporary network models to relate stress to chain conformations. Otherwise, they made no mathematical development of this idea. A similar picture was later used by Zimm9 to describe electrophoresis of DNA in a gel, which he called the “lakes−straights model”. The gel is pictured as a connected network of “lakes”, each of identical

The dominant picture motivating mathematical or simulation models for entangled polymers for nearly the past 35 years has been that of tubes.1,2 Underlying the picture is the idea that chain uncrossability is the physics dominating chain relaxation, and this is best captured by envisioning the chain as a random walk trapped in a tube. The tube represents the mean field preventing chain crossings. Within this picture, there have been several levels of description describing the chains. The original model of Doi and Edwards considered a single step of the primitive path (the tube centerline), and was restricted to flow rates that did not stretch the length of the primitive path and neglected fluctuations in the length of the primitive path, as well as motions of the tube (constraint release) that are expected to result from the relaxation of the surrounding matrix of chains. Subsequent models added more details of the primitive path in order to include additional physics that were found to be necessary to describe data. For example, this sometimes meant using one mechanical model for the tube (a 3D Rouse dynamics for the tube, with varying mobilities of the segments), and another (a 1D Rouse chain on a wire) for the primitive path chain within the tube.3 Then one assumes that the relaxation of the chain is the product of these two different processes. Within these assumptions and approximations, the © 2012 American Chemical Society

Received: December 7, 2011 Revised: June 18, 2012 Published: July 11, 2012 5728

dx.doi.org/10.1021/ma202658h | Macromolecules 2012, 45, 5728−5743

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tube diameter a, connected by “straights”. Free-energy tension differences, Brownian forces, and electric field forces drove dynamics of the chain. Some years later a similar simulation was developed independently by Hua and Schieber10 for polymer melts. This simulation used a one-dimensional Rouse chain that was forced to pass through the slip-links of the Doi−Edwards picture. Entanglements were paired to determine the entanglement destruction processes in constraint release. However, not all of the dynamics were written as mathematical expressions, detailed balance was not preserved in the constraint dynamics algorithm, and excluded volume of the Rouse beads gave somewhat artificial scaling of the primitive path length. Nonetheless, agreement with data was good, and the simulation gave rise to several new stochastic simulations11−13 and mathematical models.14−17 Of greatest interest here is the discrete slip-link model (DSM) that superseded the slip-link simulations.18 Unlike its predecessor, the DSM is a complete mathematical model of a mean-field chain rather than a computer simulation. It satisfies detailed balance in entanglement creation and destruction at chain ends, detailed balance in constraint dynamics, is compliant with the nonequilibrium thermodynamics formalism called GENERIC,19 and has a single mathematical model for arbitrary chain architecture in arbitrary deformation. It requires two adjustable parameters, one for a Kuhn-step-length time scale, and another for entanglement density, whose values are independent across chain architecture, cross-linking or blending. The latter might even be predicted from atomistic simulations.20,21 Some mathematical simplifications were made originally, such as not treating the number of Kuhn steps in a strand as integer,22 that have subsequently been examined. Because the model naturally includes fluctuations in primitivepath length, entanglement spacing, number of entanglements and monomer density, as well as constraint dynamics, it is a useful tool for examining assumptions typically made in tube models. In particular, the factorization assumption mentioned above for relaxation processes was recently examined for both monodisperse23 and bidisperse24 linear chains. While many tube-model assumptions were justified by the comparison with the DSM, some differences were seen in predictions of various contributions to relaxation. These differences were then seen in predictions of linear rheology for bidisperse blends with widely separated molecular weights, such as those used in “molecular probe rheology”.25 Liu et al.25 studied the dynamic modulus of blends of short and very long chains to elucidate the role of constraint dynamics on short chains. Their analysis suggested that constraint release contributes to the 3.5 scaling exponent of the zero-shear-rate viscosity with molecular weight for monodisperse chains. However, DSM predictions showed that these molecularprobe data could be predicted, and that no such conclusion about constraint dynamics was necessary. In other words, the scaling can be described completely by sliding dynamics using DSM, but not tube models. Here we consider another independent check that is provided by dielectric relaxation.26−29 In particular, for polymers such as polyisoprene, where the dipoles align along the backbone, one can follow the relaxation of the end-to-end vector of the chain, which is slower than the entanglement dynamics followed by the dynamic modulus.30−32 Two particular challenges are the simultaneous prediction of both the mechanical and the dielectric relaxations of both star-

branched polymers and bidisperse blends. In tube models, it is typically believed that “dynamic tube dilation” is necessary to describe dynamical relaxation in star-branched chains. For starbranched melts one postulates that the constraint dynamics felt by the star-arms speeds up the relaxation by fluctuations in the primitive path length (a result of sliding dynamics). While this postulate can explain star-branched chain relaxation, two different values for the time constant are necessary to explain both mechanical and dielectric relaxation data. We next give brief descriptions of the dielectric relaxation experiments and the discrete slip-link model that we use to analyze it. In the following sections, we examine the ability of the theory to describe both relaxations simultaneously without adjusting parameters. Dielectric Relaxation Spectroscopy. The basic idea of dielectric spectroscopy is to measure the relaxation of dielectric permittivity ε(t) as a response of the system to the application of a small external electric field E(t). Under the applied electric field, dipoles of the molecules interact with the field which is reflected in the polarization density P(t) and can be observed in the autocorrelation function ⟨P(0)·P(t)⟩eq. Because the perturbation is small it allows us to use linear response theory33 and connect the relaxation function ε to the autocorrelation function ⟨P(0)·P(t)⟩eq Δε*(ω) ≡ ε*(0) − ε*(ω) ∼ iω

∫0

∞

⟨P(0) ·P(t )⟩eq e−iωt dt

(1)

In the experiment, one measures the complex dielectric permittivity ε*(ω); therefore, knowing how P depends on structure allows extraction of some useful information about relaxation of the material. In general, the polarization density is the sum of all dipoles μi in a unit volume

P=

1 V

∑ μi

(2)

i 28

Stockmayer divided all polymers into three types depending on how the dipoles are situated and oriented on the chain. Type-A polymers have dipoles connected to the backbone of the chain and parallel to the chain contour; for type-B polymers dipoles are also connected to the backbone of the chain but oriented perpendicular to the chain contour; and in type-C polymers dipoles are attached to the side groups. We will discuss only the most well-studied casetype-A polymers without dipole inversion. For this type of polymer the dipole moment of a segment is aligned with the segment’s end-to-end vector and, therefore, the sum of dipoles in the chain is proportional to the end-to-end vector Ree of the chain. For polarization density we can write P=

∑ μ ∑ riα = μ ∑ R eeα α

i

α

(3)

rαi

where μ is the magnitude of dipole moment per segment, is the orientation vector of the ith segment on chain α, the sum over i counts the segments in the chain and the sum over α counts the number of chains in the unit volume. Here, we assume that all segments have the same dipole moment magnitude μ. We neglect any solvent contribution, which is reasonable because these interactions occur on faster timescales and are not seen in frequency ranges for relaxation of type-A dipoles. For the same reason we do not account for the segmental relaxation of type-B dipoles, which are also often 5729


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As was mentioned above, in the DSM the chain is represented as a random walk of (constant) Kuhn steps with point-like entanglements that are randomly distributed on the chain. Two processes represent the dynamics of the chain sliding dynamics (SD) and constraint dynamics (CD). Sliding dynamics is a shifting of the Kuhn steps through entanglements, determined by chemical potential differences and Brownian forces. This process allows the chain to “slide” through entanglements and also accounts for destruction and creation of entanglements at the ends of the chain: an entanglement is destroyed when the last Kuhn step in a dangling end shifts through this entanglement, and creation of entanglements is determined from detailed balance. Constraint dynamics allows entanglements to be destroyed and created anywhere on the chain. This is accomplished by assigning a characteristic lifetime (determined self-consistently from sliding dynamics) to each entanglement upon its creation. This approach allows us to mimic a Doi−Takimoto pairing algorithm47 within a single-chain model. The characteristic lifetime of an entanglement represents the process of entanglement destruction by sliding dynamics of the paired imaginary chain. The conformation of the chain is described as Ω ≡ (Z, {Ni}, {Qi}, {τCD i }), where Z is the number of strands in the chain, Ni is the number of Kuhn steps between two neighboring entanglements (i − 1 and i), Qi is the vector between two neighboring entanglements (i − 1 and i) and τCD is the i characteristic lifetime of the i-th entanglement. There are three molecular-weight independent parameters in the model: MK is the molecular weight of a Kuhn step, which depends on polymer chemistry; β is related to the entanglement density (β = exp (−μE/kBT), where μE is the entanglement chemical potential), depends on polymer chemistry and solvent concentration, and is assumed to be independent of temperature over moderate ranges; τK is the time constant of the model and is related to the friction coefficient of a single step in the chain. Note that the definition here differs from previous definitions by a factor of β + 1. It depends on temperature, polymer chemistry and solvent concentration. M K is determined independently, so is not adjustable. The stress tensor τ(t) for the model is determined by thermodynamics18,19 to be

present in type-A polymers. The cross-correlations between different chains are also negligible when the type-A dipole moment is small, which is accepted for cis-polyisoprene.30,34 Therefore, dielectric permittivity can give us information about the autocorrelation function of the end-to-end vector of the polymer chain Δε*(ω) ∼ iω

∫0

∞

R(t )e−iωt dt

(4)

where R(t): = ⟨Ree(0)·Ree(t)⟩, with some fitted prefactor. Relation of Dielectric Spectroscopy to Rheology. It was previously observed that the dynamic modulus G(t) measured in rheological experiments and the relaxation of dielectric permittivity ε(t) measured by dielectric spectroscopy have relaxation times close to each other,35,36 which suggests that there might be a relationship between the two relaxation functions. One simple entanglement polymer theorythe tube theory of double reptation37,38predicts that stress relaxation is proportional to the end-to-end vector relaxation R(t) squared, namely G(t ) ∼ R(t )2

(5)

Watanabe39 derived a similar relation (eq 6), applying the concept of dynamic tube dilation,40 which is widely used to describe star-branched polymers,4,41,42 to linear chains and showed that it works for linear monodisperse polymers. But, the proposed expression failed to explain results for bidisperse linear blends39,43,44 G(t ) ∼ R(t )1 + d ,

1 ≲ d ≲ 4/3

(6)

Dynamic tube dilation was originally used to predict the rheology of star-branched systems,4,41,42 but DTD (relation 6) is unable to predict the dielectric relaxation of star-branched monodisperse melts.45 In 2003, Schieber et al.18 proposed a model more detailed than tube modelsthe discrete slip-link model (DSM). The model is explained briefly in the next subsection, but we mention here that it is straightforward to extract R(t) predictions from DSM calculations. Therefore, we have an opportunity to obtain dielectric relaxation predictions from DSM and compare whether this model can predict rheological and dielectric experiments simultaneously. Discrete Slip-Link Model. The discrete slip-link model takes a different approach to describe polymer melt dynamics compared to tube models. There is no tube around the chain. Rather, it is a single-chain mean-field stochastic model based on thermodynamics. Therefore, some quantities which are treated as constants in tube models, become stochastic variables in the DSMthey fluctuate with time. For example, the DSM includes detailed fluctuations in the number of entanglements, monomeric density, entanglement spacing, and therefore provides a more-detailed level of description. Unlike tube theory, the DSM is a single mathematical object, with only two adjustable parameters, and can be used to predict rheological experiments for monodisperse and bidisperse linear chains as well as for monodisperse star-branched melts.23,24,46 By being more strict, DSM lacks the simplicity of tube models and also pays a significant computational cost for the detailed level of description. Here we describe the mathematical model briefly, focusing on the assumptions that are relevant in the analysis of results.

⎛ ⎞ ∂F(Ω) ⎟ ⎜ Q ∑ j⎜ ⎟ ⎝ ∂Q j ⎠T ,{N },{Q j=2

Z−1

τ (t ) = − n

j

i ≠ j}

(7)

where n is the number density of polymer chains, F(Ω) is the Helmholtz free energy of a chain, and ⟨...⟩ is an ensemble average or integral over the distribution function. In this paper we assume a Gaussian free energy, which is a good approximation for equilibrium dynamics of a random walk chain without flow F(Ω) = kBT

Z−1

⎛ 3Q 2 i

∑ ⎜⎜ i=2

⎝ 2Na i K

2

+

2 ⎤⎞ 3 ⎡ 2πNa i K ⎟ ⎥ ln⎢ 2 ⎣ 3 ⎦⎟⎠

(8)

The relaxation modulus G(t) is determined using the Green−Kubo expression33 G (t ) = 5730

1 ⟨τxy(0)τxy(t )⟩eq nkBT

(9)


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The autocorrelation function of the end-to-end vector of the chain is also straightforward to estimate Z(0) Z(t )

R (t ) =

∑ ∑ Q i(0)·Q j(t ) i=1 j=1

The plateau modulus G N0 =

(10)

eq

G0N

is found from G(t = 0) to be

NK − 1 ⎡ N −β− ρRT ⎢⎛ β ⎞ + K ⎜ ⎟ β+1 M w ⎢⎣⎝ β + 1 ⎠

2 ⎤⎥ ⎥⎦

(11)

where ρ is polymer density, R is the ideal gas constant, Mw is the molecular weight of the polymer chain, and NK = Mw/MK is the number of Kuhn steps in the chain. Note that this prediction is larger than what is usually estimated from dynamic modulus data. For a more detailed description of the model, see ref 23. In this work, we apply the DSM to predict dielectric relaxation spectroscopy experiments. Predictions of the theory are compared to published data. We show that DSM is able to predict simultaneously G* and ε* experiments for linear monodisperse, linear bidisperse and monodisperse starbranched systems, where all parameters of the modelβ and τKare determined from linear monodisperse G* data. For monodisperse linear chains we also compare our results to the Likhtman−McLeish tube model and find that it can describe rheological and dielectric data with the same quality as DSM. For bidisperse linear blends, the DSM predicts the viscoelastic relaxation very well for the entire range of volume fractions, but shows a discrepancy between predictions and the data of ε* for low concentration of long chains, while no discrepancy exists for the long-chain volume fraction greater than 10%. We construct a LM-like tube model for bidisperse blend of dilute long chains in a sea of short chains and find that it gives results nearly identical to the DSM predictions. For monodisperse star-branched melts we compare predictions from the DSM to the Milner−McLeish tube model and show that the MM model fails to describe dielectric relaxation experiments. We use the ability of the DSM to look at the relaxation by constraint and sliding dynamics separately and prove that dielectric relaxation of linear chainsmonodisperse and bidispersecan be described by sliding dynamics only, and constraint dynamics has no influence on the relaxation of the end-to-end vector autocorrelation function R(t). This is not true for star-branched polymers, where constraint dynamics significantly affects prediction results for R(t).

Figure 1. G*(ω) and ε*(ω) for linear polyisoprene 180 kDa (Tr = 40 °C). Symbols are the data by Watanabe and lines are prediction by DSM (β = 5, NK = 307, τK = 0.45 μs).

rheological data for polyisoprene taken in this lab.39,51 Ideally, the value for β should be universal for specific chemistry and should not depend on sample. However, the sensitivity of apparent G*(ω) to polydispersity, tacticity, and how the sample is loaded, may result in some discrepancy for data taken from different laboratories with different equipment. Therefore, sometimes the fitted value for β may vary somewhat from lab to lab, as we see below. The value of β = 5.0 was chosen for Watanabe’s data and used for all data from this lab. This gives G0N = 670 kPa according to eq 11. From the theory we obtain the relaxation functions G(t/τK) and R(t/τK). These are fit to a BSW spectrum52,53 and then converted to the frequency domain analytically. The details of the procedure can be found in.23 The time scale parameter τK is determined by a horizontal shift of the predictions to match experimental data. We performed calculations for five different molecular weights, using the same value of 5.0 for β and found τK for each molecular weight. Then, the average value of 0.45 μs was chosen as the universal τK and used for all predictions. We emphasize here that β and τK were completely determined from comparison to monodisperse rheological data; no parameters were adjusted to get dielectric relaxation predictions. Figure 2 shows results for all monodisperse linear chain data by Watanabe. We see that with the same set of parameters we are able to predict within uncertainty all the data quite well for both rheology and dielectric relaxation, although there is a noticeable discrepancy between data and DSM predictions of G* for the smallest molecular weight PI21. The τK value would need to be smaller in order to match these short-chain data. We mention here that PI21 has only 4−5 entanglements per chain. Smaller τK for PI21 is expected and has been seen before for other lightly entangled chains.23 DSM is a coarse-grained model and assumes that the dynamics is completely determined by entanglements, which may be a bad assumption here. It is interesting to note that DSM does predict the qualitative change in the dielectric relaxation from the 21 kDa chains to 49 kDa. The higher-molecular-weight chains show a change of

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RESULTS AND DISCUSSION We now use several experimental data sets taken from the literature. All data are for polyisoprenethe most popular type-A polymer. Unfortunately, the amount of data where dielectric spectroscopy and rheology were measured simultaneously and at the same temperature is small. Most were done by Watanabe and co-workers,48 but for linear monodisperse chains we also found data from Adachi et al.49 and, more recently from Glomann et al.50 Monodisperse Linear Melts. Linear monodisperse polymer melts are probably the most well-understood system in rheology. In Figure 1, we compare experimental data for linear polyisoprene with molecular weight 180 kDa taken by Watanabe51 with the predictions of DSM. The parameter β was found by matching the plateau modulus for different sets of 5731


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Figure 2. G*(ω) and ε*(ω) for linear polyisoprenes (Tr = 40 °C). Numbers indicate the molecular weight in kDa. Symbols are the data by Watanabe and lines are prediction by DSM (β = 5, τK = 0.45 μs). Figure 4. G″(ω) and ε″(ω) for linear polyisoprenes (Tr = 25 °C). Numbers indicate the molecular weight in kDa. Symbols are the data Glomann and lines are prediction by DSM (β = 6, τK = 1.2 μs).

slope in Δε″ for frequencies higher than the maximum, whereas the lowest molecular weight does not, and these features are predicted by the theory. Indeed, the better agreement between predictions and data for the shortest-chain dielectric relaxation might suggest that the higher-frequency Rouse motions missing from the theory, and presumably not affecting the dielectric data, are to blame. In Figure 3 and Figure 4 comparisons with data from other laboratories (Adachi et al.49 and Glomann et al.50) are shown. For the data from Adachi et al. the value for β is the same as that used for Watanabe’s data, but for Glomann’s data it is about 20% higher, which is acceptable, given the possible reasons for the difference discussed above. The plateau

modulus G0N given by eq 11 is estimated to be 550 kPa for Glomann’s data. The values for τK are also different, because the reference temperature for these experiments is different. All the τK values can be converted to one temperature by the time−temperature−superposition frequency shift factor43 and are consistent with each other. We mention that Glomann et al. compared their results to the Likhtman−McLeish tube model3 under the assumptions that constraint release does not contribute to ε* and that R(t) follows the single-chain relaxation function, and found that the model can predict these data as well. We also check the influence of entanglement complexity on the DSM predictions. It is common to treat entanglements as binary events. However, it is possible that an entanglement consists of three or more chains. Some tube models argue for the possible presence of nonbinary entanglements to explain the data.8 For ternary entanglements, when one of three chains exits, the entanglement is destroyed for all three chains. This would make the chain relaxation by constraint dynamics more probable, decreasing the influence of sliding dynamics and, therefore, increasing τK. We also calculated relaxation assuming a somewhat extreme case where all the entanglements were assumed to be ternary events. The results are presented in Figure 5. τK does become larger, as expected, but the shape of G* is inconsistent with the data (see the inset on the left of Figure 5, which blows up the crossover region of G*). This suggests that G* is sensitive to entanglement complexity and the assumption of binary entanglements is the only one consistent with the data. We also compare our results with the Likhtman−McLeish tube model.3 In that model, the relaxation modulus G(t) is calculated by

Figure 3. G″(ω) and ε″(ω) for linear polyisoprene 140 kDa (Tr = 0 °C). Symbols are the data by Adachi and lines are prediction by DSM (β = 5, τK = 12 μs). 5732


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Figure 5. G*(ω) and ε*(ω) for linear polyisoprene 180 kDa. Symbols are the data by Watanabe and blue lines are predictions by DSM with entanglement complexity 2, red dashed lines are prediction by DSM with entanglement complexity 3 (β = 5, NK = 307, τK = 0.6 μs).

Figure 6. G*(ω) and ε*(ω) for linear polyisoprene 180 kDa. Symbols are the data by Watanabe and blue lines are predictions by DSM and red dashed lines are prediction by the Likhtman−McLeish tube model (ZLM = 49, τe = 5 μs).

⎛ 4 1 G(t ) = Ge⎜⎜ μ(t )RLM(t , cν) + 5 5 Z LM ⎝

Z LM − 1

∑ p=1

⎡ p 2 t ⎤⎞ exp⎢ ⎥⎟⎟ ⎣ τR ⎦⎠ (12)

where Ge is the entanglement modulus, μ(t) is the single-chain relaxation function, τR is the Rouse time, RLM(t, cν) is the relaxation function of the 3-D Rouse tube, where the parameter cν goes from 0 to 1 and reflects the “strength” of the constraint release. For calculation of the end-to-end autocorrelation function R(t) we used just the single-chain relaxation function μ(t), similarly to what was done by Glomann et al.50 The justification of this choice will be given below (see subsection Examination of Sliding Dynamics Contribution to Dielectric Relaxation). Therefore, R(t) ∼ μ(t). Only two parameters were used: the (constant) number of entanglements ZLM and the time scale τe. The latter parameter is expected to be related to ours by τe ≈ N2e τK. In that model the number of Kuhn steps per strand is also constant and Ne = NK/ZLM. Note that in the DSM, Z is the number of strands, which equals the number of entanglements minus one, and is stochastic. Results for PI180 are plotted in Figure 6 and for other data sets by Watanabe, summarized in Figure 8. From this plot we can see that both theories give results of similar quality. Glomann et al. also compared their data to the Likhtman− McLeish theory,50 but they allowed the plateau modulus Ge, entanglement molecular weight Me, and cν parameters to be independent. We fixed them at cν = 1, Me = Mw/ZLM, Ge = ρRTZLM/Mw. Hence, our comparison is a little bit more strict. We mention here that a typical value for cν is around 0.4,50 also Nair and Schieber22 theoretically estimated the value for cν to be 1/12. In our calculations, we first tried to vary cν to find a better agreement with data. It turns out that with decreasing cν the time-constant τe for viscoelastic and dielectric relaxation

Figure 7. ε*(ω) and G*(ω) for linear polyisoprene 180 kDa. Symbols are the data by Watanabe and lines are predictions by the Likhtman− McLeish tube model for different cν values (ZLM = 49, τe = 5 μs).

requires different values in order to describe the data. Only with c ν = 1 were we able to describe both experiments simultaneously with a single value for τe. In Figure 7, predictions by the Likhtman and McLeish theory for three different values of cν are presented as an example. Similar behavior was observed for other molecular weights. Since ε* is not influenced by cν, we find τe by matching the dielectric relaxation data. Then we use this τe value to predict viscoelastic data. There is almost no influence of the cν parameter on the plateau height; therefore, cν and ZLM are independent. We vary ZLM to match the plateau modulus with data. It turns out that 5733


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parameters are now used to predict bidisperse blends. A series of experimental data for polyisoprene blends were taken from Watanabe et al.44 This series consists of PI blends with molecular weights 94 and 308 kDa, and the volume fraction of PI308 was varied from 0.05 to 0.5. There is another series of experiments from the same lab for blends of polyisoprenes with 21.4 and 308 kDa molecular weights.43 But as discussed in the previous subsection, PI21 is lightly entangled, so we do not consider it here. Results are shown in Figure 9 with comparison to experimental data. From Figure 9, one can see that DSM is able to predict G* well for blends at all volume fractions, as was previously shown in24 for a polystyrene blend. Predictions for dielectric relaxation in Figure 9 show good comparison for volume fractions of long chains down to 0.1. At 10% volume fraction there is a small deviation between DSM predictions and the data, but for the blends with very low concentration of long chains, where the volume fraction of long chains is 0.05 and 0.03, there is a significant discrepancy between data and DSM predictions. Similar discrepancies are seen for the blend with even shorter chains (see Figure 21 in the Appendix). Our predictions show a shoulder at low frequencies that correspond to the long chains, whereas data show that this shoulder disappears at low long-chain concentrations. (By “shoulder” we mean the presence of an inflection point. The presence of long chains in the blend does not vanish completelythere is still a small difference between data for the monodisperse case and the diluted blends.) Disappearance of this shoulder implies that long chains relax much faster than predicted, since we see no such effect in the theory: DSM predicts a shoulder indicating the relaxation of long chains at low frequencies. Recalling that dielectric relaxation is not strongly influenced by constraint dynamics, and also the fact that the dynamic modulus prediction of the model for this blend is consistent with the data, disagreement with the dielectric relaxation data is puzzling. We suspect that there is no theory for entangled polymers that can predict this rapid relaxation of long chains. Watanabe et al. described their data for blends PI94 with PI308 and PI21 with PI308 in terms of the dynamic tube dilation model. He reported that dynamic tube dilation is unable to predict blends with low concentrations of long chains in both

Figure 8. G*(ω) and ε*(ω) for linear polyisoprenes. Numbers indicate the molecular weight in kDa. Symbols are the data by Watanabe and lines are prediction by the Likhtman−McLeish tube model (τe = 5 μs).

this approach gives an Me value around 3.8 ± 0.4 kDa, which is somewhat low compared to the commonly used value 4.5−5.0 kDa. This low value is probably because we have not allowed any order unity prefactor in the relationship between Ge and Me, although this has been found important by Auhl et al.54 and results of DSM give similar prefactors. However, since agreement was quite good here, we did not explore this difference any further. Nonetheless, we arrive at the same conclusion as Glomann et al., that the Likhtman−McLeish theory can describe the linear monodisperse G* and ε* simultaneously, where ε* is determined by the single-chain relaxation function. Bidisperse Linear Blends. A detailed explanation of DSM for bidisperse linear blends can be found in.24 Here we mention that the number of parameters remains the sameMK, β, and τK. MK is determined independently from static quantities,55 and β and τK were already determined from monodisperse G* data. The mathematical formalism is unchanged. These

Figure 9. G′(ω), G″(ω), ε′(ω), and ε″(ω) for bidisperse blend of linear polyisoprene 94 kDa and 308 kDa (Tr = 40 °C). Numbers indicate the volume fraction of the long chains. Symbols are the data and lines are prediction by DSM (β = 5, NK,s = 160, NK,l = 525, τK = 0.45 μs). 5734


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cases, and prediction for the PI94−PI308 blend is worse than that for the PI21−PI308 blend. We also compare the longest relaxation time of the long chains in the blend τGl and τεl . A similar comparison was made by Watanabe et al.44 In Figure 10 we plot the dielectric τεl and

Figure 11. Ratio of dielectric and viscoelastic longest relaxation times of the long chains in the bidisperse blend of linear polyisoprenes 94 kDa and 308 kDa.

probes a longer length scale than does the mechanical relaxation. Watanabe’s estimation of the longest relaxation time for these data show no such behavior and stay between 1 and 2, but we were unable to reproduce these results. The DSM predicts τεl /τGl ∼ 3 which is consistent with pure 3D Rouse motion of the long chain (see Appendix for details). For very dilute long chains in a sea of short chains, one expects the dominant relaxation motion to be from constraint dynamics caused by only short chains. Since the molecular weights are widely separated, the range of time scales from the short chains is very narrow. The resulting motion of the long chains looks very similar to that of the Verdier−Stockmayer and Orwoll− Stockmayer models,56,57 which were shown to give Rouse-like relaxation. We also looked at the longest relaxation time for the monodisperse melts. In Figure 12, τε and τG are plotted for

Figure 10. Viscoelastic and dielectric longest relaxation times of the long chains in the bidisperse blend of linear polyisoprenes 94 kDa and 308 kDa.

the viscoelastic τGl longest relaxation times of the long chains for three different cases: (1) our estimation from DSM results for different volume fractions (red empty circles and squares), (2) our estimation from experimental data (blue filled circles and squares), and (3) Watanabe’s estimation from experimental data published in44 (orange filled circles and squares). We mention here that the longest relaxation time for DSM was estimated from the (numerical) integration of predicted Gl(t) and Rl(t) for the long chains. For the experimental data the longest relaxation time was estimated from the terminal zone at low frequencies by fitting power laws of the form aω2 and bω to the storage and loss moduli, respectively for both the blends and the monodisperse short chains, both dielectric and mechanical relaxation. The long-chain relaxation times were then estimated by subtracting off the short chain monodisperse values weighted by their volume fraction, and taking the ratio of the prefactors ⟨τ ⟩ =

short abidisperse − ϕSamonodisperse short b bidisperse − ϕSbmonodisperse

(13)

This procedure is equivalent to what was done by Watanabe (see definitions of ⟨τ2⟩G and ⟨τ2⟩ε used by Watanabe et al. in43,44). From Figure 10, we see that there is good consistency between theory and experiment for G* at all volume fractions. Watanabe’s reported estimates are also in good agreement. Agreement of ε* data and theory is also good for ϕ > 0.2, but we start to see a deviation of Watanabe’s reported value and our own estimate from his data at ϕ = 0.2. Also, for ϕ < 0.2, estimated from DSM predictions values disagree with Watanabe’s reported values and our estimates of his data. In Figure 11 the ratio τεl /τGl is plotted. One can see that our estimation of the longest relaxation times from the experimental data gives the ratio τεl /τGl ≲ 1 for diluted long chains. It is a surprising result: since constraint dynamics has very little influence on dielectric relaxation of linear chains (as will be shown further), one would expect the end-to-end autocorrelation function R(t) to relax slower than the viscoelastic relaxation function G(t) and, therefore, τεl always to be greater than τGl . Note further that dielectric relaxation

Figure 12. Viscoelastic and dielectric longest relaxation times of the monodisperse melts of linear polyisoprenes.

different molecular weights (unfilled circles and squares). The symbols have the same meaning as in Figure 10. As expected, we see a power-law dependence (dashed and solid lines), the values on the legend to the plot indicate the slope value for each case. The DSM (red lines, which are power law fits to red symbols) predicts the same slope for viscoelastic and dielectric relaxation τG,DSM ∼ τε,DSM ∼ MW3.6. Our estimation from experimental data are a little different, with 3.61 and 3.68. Watanabe’s estimation for the τG slope is 3.7 (we have not found published results for τε). However, the scatter in theory and data points suggests that our data estimation, Watanabe’s data estimation and DSM dielectric and mechanical predictions, 5735


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Figure 13. G*(ω) and ε*(ω) for bidisperse blend of linear polyisoprene 94 kDa (95%) and 308 kDa (5%). Symbols are the data, red dashed lines are prediction by LM-like tube model (ZLM,s = 24, ZLM,l = 71, τe = 5 μs), blue lines are predictions by DSM (β = 5, NK,s = 160, NK,l = 525, τK = 0.45 μs).

where ϕl is the volume fraction of the long chains in the blend, Zl is the number of entanglements for the long chain, and τmono d is the value for τd, determined by the Likhtman−McLeish model. As in constraint dynamics of the long chains we used a distribution of mobilities from the short chains to calculate the 3-D Rouse autocorrelation function. Details for these calculations can be found in the Appendix. To summarize, our simplifications are an attempt to minimize the effect of long chains on the blendto see if we can obtain a prediction without the long-chain shoulder. Note that relaxation of our simplifying assumptions would have the effect of increasing the prominence of the shoulder. The results for rheology and dielectric relaxation for the blend of 5% long chains in the sea of short chains are shown in Figure 13. One can see that this simplified bidisperse model predicts the same low-frequency shoulder. In these figures the DSM prediction is also present and there is almost no difference between it and our LM-like tube model. Both models deviate from the ε* data, while describing the G* data. The constructed tube-model was simplified, but we repeat that removal of our simplifications would drive the predictions in the wrong directionthe shoulder of long chains would grow even larger. Monodisperse Star-Branched Melts. A star-branched chain is a more complicated structure, but the DSM works well for this type of polymer as well. We consider here a symmetric star-branched polymer with a single branch-point, although calculations may be generalized to other branched structures in a straightforward way. We do not permit the branch-point to pass through an entanglement, though this can easily be relaxed. We assume that the branching point is not fixed in space and is allowed to fluctuate. Therefore, the only difference from the case of linear chains is that for the free energy of a star polymer we need to calculate the free energy Fb of the “branch complex”branch-point plus connected strandsand add it to free energies FS of the arms (which are identical to linear chains)

give essentially the same slope. This is expected, since these scalings are typically assumed to be insensitive to CD. To explore our suspicion that discrepancy between theory and experiment for bidisperse blends is not restricted to the DSM, we constructed a simple tube model for the case of long chains diluted in a sea of short chains. We took the Likhtman− McLeish tube model for monodisperse linear chains,3 and made a few simplifications to construct our blend. First, we assumed factorization of sliding and constraint dynamics for long and short chains. Second, because the concentration of long chains is small, we treat short chains as monodisperse and ignore the influence from the presence of long chains for them. Therefore, we calculated the contribution of short chains Gs(t) from the LM model. Then, because long chains are very dilute, we ignore long−long entanglements, and assumed that long chains are entangled only with short ones. In our modification of the Likhtman−McLeish tube model, estimation of constraint dynamics is done by distributing mobilities of short chains on the long chain. In this way we were able to calculate Gl(t). The resulting G(t) was calculated by a weighted sum of short and long chains. The end-to-end autocorrelation function for the short chains was again assumed to be the same as in the monodisperse case. For the long chains R(t) consists of two parts: a single-chain relaxation function (μ(t) in terms of LM model) and autocorrelation of a 3-D Rouse chain. We further assumed factorization for sliding and constraint dynamics in R(t). We discuss this assumption below. We mention here that the Struglinski−Graessley parameter Gr,58 which is used in tube models to determine if constraint release has an influence on sliding dynamics processes, has a value of 0.0093 for the blend used in this paper. The critical value for the polyisoprene is not strictly knownStruglinski and Graessley use 0.158whereas Park and Larson reported the value of 0.064.7 Nonetheless, we seem to be safe not to take into account the Struglinski−Graessley effect. For the situation when the Struglinski−Graessley parameter is larger than its critical value, the longest relaxation time of the long chains τd,l should be changed. Note that no such consideration is necessary in the DSM, since such effects arise naturally from the model. According to Park and Larson,7 Doi et al. predictions should be used59 mono 2 ⎧ Zl , if ϕlGr < 1 ⎪ τd, s τd, l = ⎨ ⎪ mono α ⎩ τd, l ϕl , if ϕlGr > 1,

Na Z j − 1

F(Ω) =

∑ ∑ FS(Q ij, Nij) + Fb(Ω) j=1 i=2

(15)

where Fs(Qji,Nji) is the free energy of the ith strand on the jth arm, Qji and Nji are the vector and the number of Kuhn steps between two neighboring entanglements on the jth arm, Na is the number of arms, Zj is the number of strands on the jth arm and Fb(Ω) is the free energy of the branch complex.

(14) 5736


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These calculations are given in46 and here we repeat the main steps. To calculate Fb(Ω) with fluctuating branch-point position, we start with calculation of the free energy with the branch-point fixed at some location and treat it similar to an entanglement. From the free energy, using the Maxwell− Boltzmann relationship, we find a probability density for conformations with a known branch-point position. After integrating the distribution over all possible positions of the branch-point we obtain a probability density for conformations with unknown branch-point position. From the distribution it is straightforward to obtain the free energy of these conformations, which can be written in terms of model variables as Fb = kBT

i 2 ⎡ 3(Q̂ − Q̅ )2 3 ⎛ 2πN1aK ⎞⎤⎥ 3 i ⎟ + ∑⎢ + ln⎜ i 2 ⎥ ⎢ 2 ⎝ 3 ⎠⎦ 2 2N1aK i=1 ⎣ n

⎡ 3 ∑n (N i)−1 ⎤ 1 i=1 ⎥ ln⎢ ⎢⎣ ⎥⎦ 2π

(16)

where Q̂ i is the vector between the first and ith entanglement of the branch complex and Q̅ := (ΣNi=1a (Ni1)−1Q̂ i)/Σni=1(Ni1)−1. See Figure 14 for details. Note that the summation is over entangled arms only. See the Appendix for details of the derivation.

Figure 15. G*(ω) and ε*(ω) for 6-armed star-branched polyisoprenes with arm molecular weight 40.7 kDa and 20.2 kDa (Tr = 40 °C). Symbols are the data and lines are prediction by DSM (β = 5, τK = 0.45 μs).

RMM(t ) =

∫0

1

⎛ t ⎞ exp⎜ − ⎟ ds ⎝ τ (s ) ⎠

(17)

where τ(s) is a first-passage time and s is the fractional distance from the tip of the arm.4 The results of simulations of the Milner−McLeish model are presented in Figure 16. In agreement with what was reported

Figure 14. Schematics of the star-branched chain primitive-path with variables used in the DSM. The dashed primitive-path indicate the branch complex and the circle is the branch-point.

The number of parameters for the DSM remains the same, and because we have a single mathematical object for the theory and data for the same polymeronly the chain architecture is differentwe can use the values of parameters determined from linear chains. The end-to-end vector autocorrelation function is just the autocorrelation function for the arm vector: from branching point to the end. Figure 15 show the results for rheology and dielectric relaxation compared to experimental data for 6-armed polyisoprene with arm molecular weight 40.7 kDa and 20.2 kDa. The shape of G* and ε* DSM theory curves match the data very well. Again, we see that for lightly entangled arms the time scale τK would need to be smaller in order to match the data. This is the same effect that we saw for linear polymers of similar molecular weight (PI 21.4). There is no unified mathematical tube model that can describe linear and star-branched polymers simultaneously. However, there also exists a good model for linear viscoelasticity of star-branched polymers by Milner and McLeish.4,5 The autocorrelation function of the end-tobranch-point vector RMM(t) for the Milner−McLeish model, was derived by Watanabe et al.45

Figure 16. G*(ω) and ε*(ω) for six-armed star-branched polyisoprenes with arm molecular weight 40.7 kDa and 20.2 kDa. Symbols are the data and lines are prediction by the Milner−McLeish model (τe = 1.5 μs).

by Watanabe,45,60 the Milner−McLeish model is able to predict rheological data, but fails to describe dielectric relaxationthe time-scale is predicted somewhat larger than observed. Note also that τe must be set much smaller than the value used to fit linear-chain data. 5737


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Examination of Sliding Dynamics Contribution to Dielectric Relaxation. One advantage of the DSM is that we can examine relaxation by sliding and constraint dynamics separately. This allows us to check common assumptions used in tube models to derive eq 6. There are three typical tubemodel assumptions, which we can write as G(t ) ∼ GCD(t )GSD(t )

(18)

GCD(t ) ∼ GSD(t )

(19)

GSD(t ) ∼ μ(t ) ∼ RSD(t ) ≈ R(t )

(20)

bars of the numerical calculation of the full relaxation modulus. The same factorization holds for the end-to-end autocorrelation function. Similar to the monodisperse case, R(t) is determined by sliding dynamics for both short and long chains. This justifies the use of the factorization assumption in our LM-like model. Comparison of the 6-armed polyisoprene stars with Ma = 40.7 kDa is presented in Figure 19. Opposite to what we see for linear chains, constraint dynamics has a significant influence on the autocorrelation function of the end-to-end vector R(t). It is possible that this change is either from the effect that CD has on SD, similar to DTD, or that factorization fails. Whatever the cause, the effect is dramatic, decreasing the longest relaxation time by nearly 2 orders of magnitude.

Equation 18 is a factorization assumption commonly used in tube theories and was already found with DSM to be reasonable.23 Equation 19 says that contributions from sliding and constraint dynamics are equivalent. This assumption was also checked in ref 23, where it was shown that eq 19 is not particularly accurate. The third assumption eq 20 says that the relaxation function of the end-to-end vector R(t) is proportional to tube survival, and that it is not sensitive to constraint dynamics, or, in terms of DSM, R(t) ∼ GSD(t). In this work we check the validity of the third assumption. We make two predictions for linear chains with the same set of parameters. The difference is that in one prediction constraint dynamics is turned off and only sliding dynamics is present, in the other prediction both processes are present. Normalized relaxation functions with sliding dynamics only (GSD(t) and RSD(t)) and full autocorrelation function of the end-to-end vector R(t) (with constraint and sliding dynamics) for linear polyisoprene with molecular weight 180 kDa are plotted in Figure 17. In the inset, the ratio of relaxation

■

CONCLUSION In a recent paper by Glomann et al., it was shown that the Likhtman−McLeish tube model is able to predict rheological and dielectric relaxation experiments for linear monodisperse polymers simultaneously, and the single chain relaxation function μ(t) was assumed to describe dielectric relaxation. However, the entanglement molecular weight Me, plateau modulus Ge, (constant) number of entanglements ZLM and constraint release parameter cν were treated as independent and adjustable parameters. In the present work we use a slightly more strict comparison: we allow ZLM to be an adjustable parameter, while Me and Ge are determined from ZLM. The cν value and time scale τe were determined globallyone value for each parameter that describes all rheological and dielectric data. We show that our stricter application of the Likhtman− McLeish model still gives good results for predicting simultaneously linear rheology and dielectric relaxation of monodisperse linear chains. We also prove with the DSM that the single-chain relaxation function μ(t) can be used to estimate the end-to-end autocorrelation function for predicting dielectric relaxation experiments. We show that the discrete slip-link model is a unified model that can predict rheology and dielectric relaxation for linear (monodisperse and bidisperse), and star-branched (monodisperse) polymers simultaneously. DSM has only two adjustable, molecular-weight-independent parameters that can be determined from the rheological data of a single-molecularweight linear melt and used to predict bidisperse blends or starbranched systems. We were able to predict rheology and dielectric relaxation of linear and star-branched polyisoprenes for a wide range of molecular weights. For bidisperse linear blends, the DSM predicts rheology very well for the entire range of compositions. DSM is able to describe dielectric relaxation experimental data for the volume fraction of long chains from 100% to 10%, but fails to describe 5% and 3%: the model predicts a long-chain shoulder at low frequencies, which is not observed in experimental data. We constructed a simplified bidisperse tube model based on the Likhtman−McLeish model for monodisperse chains in an attempt to describe these data. Predictions of the constructed model are similar to those from DSM and, therefore, are unable to describe the low long-chain-concentration dielectric data as well. Our analysis suggests that the ε* data show relaxation of long chains that is faster than observed for G*. Furthermore, the predicted ratio τε/τG ≈ 3 is consistent with 3D Rouse relaxation. In contrast, the data show τε/τG ≈ 1 for the dilute long chains, which is difficult to rationalize. We believe that

Figure 17. Normalized relaxation functions with sliding dynamics only (GSD(t) and RSD(t)) and full (with constraint and sliding dynamics) autocorrelation function of the end-to-end vector R(t). The DSM parameters are chosen to match linear monodisperse polyisoprene with Mw = 180 kDa, Figure 1.The longest relaxation times are τG = 0.61 s and τε = 1.15 s.

functions GSD(t) and RSD(t) to the full R(t) are plotted. From the plot, we see that constraint dynamics does not affect the autocorrelation function of the end-to-end vector significantly. Also, we can say that for the time range of experiments (1 μs to 1 s) the ratio GSD(t)/R(t) changes less than 20%, which suggests that the assumption R(t) ∼ GSD(t) is reasonable for linear monodisperse chains. We also made a similar comparison for the blend of linear chains PI94 and PI308 with 5% concentration of long chains. Results for long and short chains are presented in Figure 18. The product of the relaxation modulus predicted with CD only and the modulus predicted with SD only lies within the error 5738


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Figure 18. Normalized relaxation functions for the blend of linear chains with 5% of PI308 and 95% of PI94. Empty circles are CD only, empty squares are SD only, black filled squares contain both processes and gray squares are a product of CD only and SD only.

these data remain an unsolved puzzle for all entangled polymer theories. We showed that constraint dynamics has significant influence on the end-to-end vector relaxation function of the monodisperse star-branched melts, showing a behavior somewhat similar to the dynamic tube dilution in tube models.

■

APPENDIX

Rouse Constraint-Release Limit for Bidisperse Linear Blend with Low Concentrations of Long Chains in a Sea of Very Short Chains

If we look at the extreme casea blend of long chains diluted in a sea of very short chains, then we might approximate the relaxation of the long chains with a Rouse constraint release process Gl(t ) = Gl(0)

R l(t ) = R l(0)

1 Zl

Zl

p=1

∑ p = o dd

Figure 19. Normalized relaxation functions for 6-armed stars. The DSM parameters are chosen to match 6-armed monodisperse polyisoprene with Ma = 40.7 kDa, Figure 15.

⎛ 2p2 t ⎞ ⎟ ⎝ τCR ⎠

(21)

⎛ p2 t ⎞ 8 ⎜− ⎟ exp p2 π 2 ⎝ τCR ⎠

(22)

∑ exp⎜−

where τCR is the dielectric Rouse−CR time. If we take these analytic expressions for the Rouse model and integrate them to

Figure 20. Normalized relaxation functions of the long chains for the blend of linear chains with 3% of PI308 and 97% of PI21. Symbols are predictions by DSM, lines are predictions by Rouse-CR limit. τCR = 2 × 106τK. 5739


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Likhtman−McLeish-Like Tube Model for Bidisperse Linear Blend with Very Low Concentrations of Long Chains

calculate the longest relaxation times, the integral of Rl(t) is independent of the number of entanglements and Gl(t) integral is a finite sum that depends on Zl, but is independent of τCR.

In this section, we show the details of our slightly modified tube model for a bidisperse linear blend with low concentration of long chains. The relaxation functions of the blend are given by the weighted sums of the long- and short-chain components:

∞

G

τ =

∫0 tGl(t ) dt ∞

∫0 Gl(t ) dt

(23)

G blend(t ) = ϕlGl(t ) + (1 − ϕl)Gs(t ) R blend(t ) = ϕlR l(t ) + (1 − ϕl)R s(t )

∞

ε

τ =

∫0 tRl(t ) dt ∫0 Rl(t ) dt

(25)

Here indices “l” and “s” are used for long and short chains respectively and ϕ is the volume fraction. For the case of small ϕl we assume that short chains behave as monodisperse, therefore, Gs(t) ≈ Gmono (t), and Rs(t) ≈ s Rmono (t). Gmono (t) can be calculated by the Likhtman−McLeish s s model as

∞

(24)

The ratio of these two times for Zl = 71, which corresponds to PI308, is 2.97. This value should be compared to the ratio plotted in Figure 11 at low volume fraction. Note that the value of τCR is determined by short-chain dynamics. Here we are checking only for self consistency and leave it as adjustable. To check that DSM correctly reproduces the Rouse−CR limit, we make a prediction for the 3% of long (PI308) chains in a sea of very short chains (97% of PI21). The results are presented in Figure 20. One can see that DSM is in good agreement with that limit, taking into account that PI21 is a lightly entangled polymer and also that according to Watanabe’s estimations for Rouse−CR limit we need a blend with 1% or less of the long chains.43 In Figure 21, we plot the comparison of DSM predictions and experimental data for the blend 97% PI21 and 3% PI308. Recall that for monodisperse PI21 there was a discrepancy between data and DSM prediction due to the limitation of the theory for lightly entangled polymers. Thus, the same discrepancy between rheological data and the DSM prediction is expected. From Figure 21, one can see that the shape of G* is in a good agreement between DSM prediction and experimental data, but the time scale is off, as for monodisperse PI21. For the dielectric relaxation, again, we see a discrepancy at low frequencies, similar to the results for PI94−PI308 blend (see Figure 9)

⎛ 1 4 Gsmono(t ) = Ge, s⎜⎜ μs (t )R sLM(t ) + 5Z LM, s ⎝5 Z LM, s − 1

∑ p=1

⎡ p 2 t ⎤⎞ ⎥⎟ exp⎢ ⎢⎣ τR, s ⎥⎦⎟⎠

(26)

where Ge is the entanglement modulus, μ(t) is the single-chain relaxation function, RLM(t) is the relaxation function of the Rouse tube, τR is the Rouse time. For calculation of RLM(t) we use the method described in:23 Z LM − 1

1

RLM(t ) =

Z LM − 1

∑

⟨exp( −tm̂ i)⟩

i=1

(27)

where m̂ i are the eigenvalues of a modified Rouse matrix A̅ = ΣZ−1 k=1 mk(δk,i−1−δk,i)(δk,j−1−δk,j), and mk are mobilities chosen at random from a distribution P(m) given by the inverse Laplace transform of μ(t)3 μ(t ) =

∫0

∞

P(m)exp( −mt ) dm

(28)

The ensemble size for the average in eq 27 was on the order of 1000. The end-to-end autocorrelation function Rs(t) is the same as the fraction of tube survival, and, therefore Rmono (t) = μs(t); i.e., s CD is neglected for short chains. We assume that the long chains are entangled only with the short chains. Therefore, for calculation of RLM l (t) we use the distribution of the mobilities for the short chains R lLM(t )

≈

Z LM, l − 1

1 Z LM, l − 1

∑

⟨exp( −tm̂s , i)⟩

i=1

(29)

and μl(t) is calculated as for the monodisperse case μl (t ) =

p*

8Gf , l

∑

π2

p = 1,odd ∞

+

∫ε*

⎛ tp2 ⎞ 1 ⎜⎜ − ⎟⎟ exp p2 ⎝ τdf , l ⎠

0.306 exp( −εt ) dε Z LM, lε 5/4

(30)

and definitions of Gf, p*, τdf, ε* are given in ref 3. The end-to-end autocorrelation function for long chains Rl(t) is a product of the single-chain tube survival from SD μl(t), and the 3-D Rouse chain autocorrelation function RRouse(t). To calculate RRouse(t), we use a similar procedure for calculation of

Figure 21. G*(ω) and ε*(ω) for bidisperse blend of linear polyisoprene 21 kDa (97%) and 308 kDa (3%). Symbols are the data; red lines are predictions by DSM (β = 5, τK = 0.45 μs). 5740


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Article Z LM, l − 1 ⎛ ⎛ ⎞ 4 1 ⎜ ⎜ Gl(t ) = Ge, l ⎜ μl (t )⎜ ∑ ⟨exp(−tm̂s , i)⟩⎟⎟ ⎝ Z LM, l − 1 i = 1 ⎠ ⎝5 Z LM, l − 1 ⎞ ⎡ p2 t ⎤ 1 + ∑ exp⎢ ⎥⎟⎟ ⎢⎣ τR, l ⎥⎦⎠ 5Z LM, l p = 1

the Rouse relaxation function in ref 23. The evolution equation for the conditional probability density of a Rouse chain of ZLM−1 entanglements with mobilities {mi} is given by ∂p({Q i}; t |{Q i0}; t0})

Z LM, s − 1 ⎛ ⎛ ⎞ 4 1 Gs(t ) = Ge, s⎜⎜ μs (t )⎜⎜ ∑ ⟨exp(−tm̂s ,i)⟩⎟⎟ ⎝ Z LM, s − 1 i = 1 ⎠ ⎝5 Z LM, l − 1 2 ⎤⎞ ⎡ p t 1 + ∑ exp⎢ ⎥⎟⎟ ⎢⎣ τR, s ⎥⎦⎠ 5Z LM, s p = 1

∂t Z LM − 1

=

∑ i,j=1

⎡ ⎤ ∂p ⎥ ∂ ⎢ H A̅ ij Q ip + ∂Q i ⎢⎣ kBT ∂Q j ⎥⎦

(31)

We can diagonalize the matrix A̅ and find the normal modes

Z LM, l

{Q̂ i = ΣZj =LM2−1ΩijQj} and eigenvalues m̂ i with rotation matrix Ωij.

R l(t ) = μl (t )⟨ ∑ Ω†ijΩ†il exp( −tm̂s , i)⟩

Therefore, the autocorrelation function of the end-to-end

ijl

R s(t ) = μs (t ),

vector is

(39)

RRouse(t ) = ⟨∑ Q i(t ) ∑ Q k(0)⟩ i

k

which can be used with eq 25 to predict both the dynamic modulus and dielectric relaxation of the dilute bidisperse blends.

(32)

=⟨∑ Ω†ijQ̂ i(t ) ∑ Ω†klQ̂ k(0)⟩ ij

Free Energy of the Branch Complex for DSM

As mentioned above, in subsection Monodisperse StarBranched Melts, to calculate Fb(Ω) with an arbitrary position of the branch-point, we start with the calculation of the free energy with the branch-point fixed at location R. Then we find the probability density for conformations with known branchpoint position and integrate this distribution over all possible positions of the branch-point to obtain a probability density for conformations with unknown branch-point position and, therefore, the free energy for a fluctuating branch-point. The branch-point is surrounded by ≤ Na entanglements at locations {Ri}, with {Ni1} Kuhn steps from the branch-point to the ith entanglement. We mention here that the number of entangled branches is ≤ Na, because some of the branches may be unentangled and, therefore, have zero free energy. The free energy of the branch complex for Gaussian chains of known branch-point location is

(33)

kl

=⟨∑ Ω†ijΩ†ilQ̂ i(t )Q̂ i(0)⟩ (34)

ijl

For Q̂ i(t) we can write the stochastic differential equation dQ̂ i = −

Hm̂ i Q̂ dt + kBT i

2m̂ i dW it

(35)

whose solution is ⎛ Hm̂ it ⎞ Q̂ i(t ) = Q̂ i(0) exp⎜ − ⎟ + 2m̂ i ⎝ kBT ⎠ t ⎡ Hm̂ i(t − t ′) ⎤ exp⎢ − ⎥ dW it ′ 0 kBT ⎦ ⎣

∫

Fb({R i}, R) = kBT

(36)

giving the autocorrelation function ⟨Q̂ i(t )Q̂ j(0)⟩ = δij

⎛ Hm̂ it ⎞ kBT exp⎜ − ⎟δ H ⎝ kBT ⎠

Na

⎡ 3(R − R )2 3 ⎛ 2 i ⎞⎤ i ln⎜ πN1⎟⎥ + ⎠⎦ 2 ⎝3 2N1i ⎣

∑⎢ i=1

where it is understood that unentangled arms do not contribute. Here we have made all lengths dimensionless by the Kuhn step length. We can re-write the summation of the first term on the right as

(37)

Na

Na ⎡ (R − R )2 ⎤ i ⎥ ( (N1i)−1){[R − R̅ ]2 − R̅ 2} ∑ = N1i ⎦ ⎣ i=1

Therefore, the dimensionless autocorrelation of the Rouse

∑⎢

chain becomes

i=1

Na

+

RRouse(t ) = ⟨∑ Ω†ijΩ†il exp( −tm̂ i)⟩ ijl

(40)

∑ i=1

(38)

R i2 N1i

(41)

where R̅ := (ΣNi=1a (Ni1)−1Ri)/(ΣNi=1a (Ni1)−1). Now it is straightforward to integrate the distribution over branch-point position

The final results are 5741


Macromolecules

∫

■

⎡ F ({R i}, R) ⎤ exp⎢ − b ⎥ dR kBT ⎦ ⎣

( )

(42)

and obtain the free energy of the branch complex independent of the branch-point position Na

i ⎡ 3(R 2 − R̅ 2) 3 3 ⎛ 2πN1 ⎞⎤ i + ln⎜ ⎟⎥ + i ⎢⎣ 2 ⎝ 3 ⎠⎥⎦ 2 2N1

∑⎢ i=1

⎡ 3 ∑Na (N i)−1 ⎤ 1 i=1 ⎥ ln⎢ ⎢⎣ ⎥⎦ 2π

(43)

For the case Na = 2 eq 43 recovers the free energy of a single strand. We can re-write this expression as a free energy for when the branch-point is pinned to R̅ , plus the extra entropy from branch-point fluctuations Fb({R i}) = kBT

Na

i ⎡ 3(R − R̅ )2 3 ⎛ 2πN1 ⎞⎤ 3 i ln + ⎜ ⎟⎥ + ⎢⎣ 2 ⎝ 3 ⎠⎥⎦ 2 2N1i

∑⎢ i=1

⎡ 3 ∑Na (N i)−1 ⎤ 1 i=1 ⎥ ln⎢ ⎢⎣ ⎥⎦ 2π

(44)

Because eq 44 is independent of where the origin is placed, we place it at one of the entanglements. Thus, the dimensional expression for the free energy of the branch complex in model variables is given by Fb = kBT

Na

i 2 ⎡ 3(Q̂ − Q̅ )2 3 ⎛ 2πN1aK ⎞⎤⎥ 3 i ⎟ + ⎜ ln + ⎢⎣ 2N1iaK 2 2 ⎝ 3 2 ⎠⎥⎦

∑⎢ i=1

⎡ 3 ∑Na (N i)−1 ⎤ 1 i=1 ⎥ ln⎢ ⎢⎣ ⎥⎦ 2π

(45)

where Q̂ = Ri − R1 and Q̅ = (ΣNi=1a (Ni1)−1Q̂ i)/(ΣNi=1a (Ni1)−1. See Figure 14 for references. Note that the summation is over entangled arms only. Other details about implementation of the DSM to branched polymers can be found in ref 61, where the model was applied to cross-linked system.

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REFERENCES

(1) Doi, M.; Edwards, S. F. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1789−1801. (2) de Gennes, P. G. J. Phys. Lett. (Paris) 1975, 36, 55. (3) Likhtman, A. E.; McLeish, T. C. B. Macromolecules 2002, 35, 6332−6343. (4) Milner, S. T.; McLeish, T. C. B. Macromolecules 1997, 30, 2159− 2166. (5) Milner, S. T.; McLeish, T. C. B.; Young, R. N.; Hakiki, A.; Johnson, J. M. Macromolecules 1998, 31, 9345−9353. (6) Bishko, G.; McLeish, T. C. B.; Harlen, O. G.; Larson, R. G. Phys. Rev. Lett. 1997, 79, 2352−2355. (7) Park, S. J.; Larson, R. G. Macromolecules 2004, 37, 597−604. (8) Graham, R. S.; Likhtman, A. E.; McLeish, T. C. B.; Milner, S. T. J. Rheol. 2003, 47, 1171−1200. (9) Zimm, B. H. J. Chem. Phys. 1991, 94, 2187−2206. (10) Hua, C. C.; Schieber, J. D. J. Chem. Phys. 1998, 109, 10018− 10027. (11) Likhtman, A. Macromolecules 2005, 38, 6128−6139. (12) Masubuchi, Y.; Ichi Takimoto, J.; Koyama, K.; Ianniruberto, G.; Marrucci, G.; Greco, F. J. Chem. Phys. 2001, 115, 4387−4394. (13) Doi, M.; Takimoto, J.-i. Philos. Trans. R. Soc. London, A 2003, 361, 641−652. (14) Schieber, J. D.; Neergaard, J.; Gupta, S. J. Rheol. 2003, 47, 213− 233. Although all dynamical calculations were verified to agree with analytic results at equilibrium, we later discovered that stress calculations contained a bug. Hence, stress curves in this manuscript should be ignored. (15) Graham, R. S.; Likhtman, A.; McLeish, T. C. B.; Milner, S. T. J. Rheol. 2003, 47, 1171−1200. (16) Likhtman, A. E.; Graham, R. S. J. Non-Newtonian Fluid Mech. 2003, 114, 1−12. (17) Likhtman, A.; McLeish, T. C. B. Macromolecules 2002, 35, 6332−6343. (18) Schieber, J. D.; Neergaard, J.; Gupta, S. J. Rheol. 2003, 47, 213. (19) Schieber, J. D. J. Non-Equilib. Thermodyn. 2003, 28, 179−188. (20) Khaliullin, R. N.; Schieber, J. D. Phys. Rev. Lett. 2008, 100, 188302. (21) Foteinopoulou, K.; Karayiannis, N. C.; Laso, M.; Kröger, M.; Mansfield, M. L. Phys. Rev. Lett. 2008, 101, 265702. (22) Nair, D. M.; Schieber, J. D. Macromolecules 2006, 39, 3386− 3397. (23) Khaliullin, R. N.; Schieber, J. D. Macromolecules 2009, 42, 7504−7517. (24) Khaliullin, R. N.; Schieber, J. D. Macromolecules 2010, 43, 6202−6212. (25) Liu, C.-Y.; Halasa, A. F.; Keunings, R.; Bailly, C. Macromolecules 2006, 39, 7415−7424. (26) Debye, P. Polar Molecules; Chemical Catalog Co: New York, 1929. (27) Glarum, S. H. J. Chem. Phys. 1960, 33, 1371−1375. (28) Stockmayer, W. H. Pure Appl. Chem. 1967, 15, 539−554. (29) Hedvig, P. Dielectric Spectroscopy of Polymers; Wiley & Sons: New York, 1977. (30) Adachi, K.; Kotaka, T. Prog. Polym. Sci. 1993, 18, 585−622. (31) Watanabe, H. Korea-Austr. Rheol. J. 2001, 13, 205−217. (32) Kremer, F.; Schonhals, A. Broadband Dielectric Spectroscopy; Springer-Verlag: Berlin, 2003. (33) Kubo, R. J. Phys. Soc. Jpn. 1957, 12, 570−586. (34) Adachi, K.; Kotaka, T. Macromolecules 1988, 21, 157. (35) Adachi, K.; Kotaka, T. Macrromolecules 1984, 17, 120. (36) Adachi, K.; Kotaka, T. Macromolecules 1985, 18, 466−472. (37) Tsenoglou, C. Polym. Prepr.(Am. Chem. Soc., Div. Polym. Chem) 1987, 28, 185−186. (38) des Cloizeaux, J. Europhys. Lett. 1988, 5, 437−442. (39) Matsumiya, Y.; Watanabe, H.; Osaki, K. Macromolecules 2000, 33, 499−506. (40) Marucci, G. J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 159. (41) Ball, R. C.; McLeish, T. C. B. Macromolecules 1989, 22, 1911.

⎡ ⎤3/2 3 (Na − 1) ⎡ Na 3(R 2 − R̅ 2) ⎤ ⎢ ⎥ 2π i ⎢−∑ ⎥ =⎢ exp Na Na i i −1 ⎥ ⎢ ⎥⎦ 2N1i ∑ ∏ ( N ) ( N ) ⎣ i=1 ⎢ ⎥ 1 i=1 1 i=1 ⎣ ⎦

Fb({R i}) = kBT

Article

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support of this work by Army Research Office Grants W911NF-08-2-0058 and W911NF-09-2-0071 are gratefully acknowledged. We are also grateful to Prof. Hiroshi Watanabe for kindly supplying much of the raw data and for fruitful discussions. 5742


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Article

(42) Milner, S. T.; McLeish, T. C. B. Macromolecules 1998, 31, 7479−7482. (43) Watanabe, H.; Ishida, S.; Matsumiya, Y.; Inoue, T. Macromolecules 2004, 37, 1937−1951. (44) Watanabe, H.; Ishida, S.; Matsumiya, Y.; Inoue, T. Macromolecules 2004, 37, 6619−6631. (45) Watanabe, H.; Matsumiya, Y.; Osaki, K. J. Polym. Sci.: Part B: Polym. Phys. 2000, 38, 1024−1036. (46) Khaliullin, R. N. Ph.D. thesis, Illinois Institute of Technology: 2010. (47) Doi, M.; Takimoto, J. I. Philos. Trans. R. Soc. London , Ser. A 2003, 361, 641−652. (48) Watanabe, H. Macromol. Rapid Commun. 2001, 22, 127−175. (49) Adachi, K.; Wada, T.; Kawamoto, T.; Kotaka, T. Macromolecules 1995, 28, 3588−3596. (50) Glomann, T.; Schneider, G. J.; Bras, A. R.; Pyckhout-Hintzen, W.; Wischnewski, A.; Zorn, R.; Allgaier, J.; Richter, D. Macromolecules 2011, 44, 7430−7437. (51) Watanabe, H.; Matsumiya, Y.; Inoue, T. Macromolecules 2002, 35, 2339−2357. (52) Baumgartel, M.; Schausberger, A.; Winter, H. H. Rheol. Acta 1990, 29, 400−408. (53) Friedrich, C.; Waizenegger, W.; Winter, H. H. Rheol. Acta 2008, 47, 909−916. (54) Auhl, D.; Ramirez, J.; Likhtman, A. E.; Chambon, P.; Fernyhough, C. J. Rheol. 2008, 52, 801−835. (55) Rubinstein, M.; Colby, R. Polymer Physics; Oxford University Press: New York, 2003. (56) Verdier, P. H.; Stockmayer, W. H. J. Chem. Phys. 1962, 36, 227− 235. (57) Orwoll, R. A.; Stockmayer, W. H. In Stochastic Models for Chain Dynamics; John Wiley & Sons, Inc.: 2007; Vol. 15, Chapter 16, pp 305−324. (58) Struglinski, M. J.; Graessley, W. W. Macromolecules 1985, 18, 2630−2643. (59) Doi, M.; Graessley, W. W.; Helfand, E.; Pearson, D. S. Macromolecules 1987, 20, 1900−1906. (60) Watanabe, H. Prog. Theor. Phys. Suppl. 2008, 175, 17−26. (61) Jensen, M. K.; Khaliullin, R.; Schieber, J. D. Rheol. Acta 2011.

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Dielectric Relaxation as an Independent Examination of Relaxation

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