Microscopic Relaxation Processes in Branched-Linear Polymer

Article pubs.acs.org/Macromolecules

Microscopic Relaxation Processes in Branched-Linear Polymer Blends by Rheo-SANS N. Ruocco,*,† L. Dahbi,† P. Driva,‡ N. Hadjichristidis,‡ J. Allgaier,† A. Radulescu,# M. Sharp,§ P. Lindner,∥ E. Straube,⊥ W. Pyckhout-Hintzen,† and D. Richter†,# †

Jülich Center for Neutron Science, Forschungszentrum Jülich, 52425 Jülich, Germany Department of Chemistry, University of Athens,15771 Athens Greece # Jülich Centre for Neutron Science, FRM 2, 85747 Garching, Germany § Geesthacht Neutron Facility, GKSS Research Centre, 21502 Geesthacht, Germany ∥ Institut Laue-Langevin, 38042 Grenoble, France ⊥ FB Physik, Martin-Luther Universität Halle, D-06099 Halle, Germany ‡

ABSTRACT: The relaxation time spectrum in blends of architecturally different polymers with strongly disperse time scales has been investigated by their time-dependent small angle neutron scattering signal after a fast uniaxial step strain. Modelhyperbranched dendrimeric polymers of second generation, dilutely dispersed within linear homopolymer matrices, acted like sensitive probes for structurally, though not firmly, established features of the tube model for bidisperse melts. We showed that the equilibration time of the linear matrix determines the size of the fluctuations that the outer and inner arms experience. Within a random phase approximation (RPA) treatment, which accounts for the different degrees of freedom inherent to the broad time scales, the observed loss of anisotropy with time was described in terms of two parameters only, namely the tube diameter and the fraction of relaxed arms of the minority component. The scattering data reveal details of mechanisms, which cannot be extracted from but determine the macroscopic flow properties. At intermediate times, a tube relaxation process was detected. At long times, the dynamic dilution model is confirmed.

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INTRODUCTION In the last few decades, the flow behavior of branched polymers has attracted an increasing attention and the description of their linear rheology by means of the tube model has achieved remarkable progress.1−16 Of particular interest is that the dynamics of branched polymers is, on one hand, much more complex and demanding as compared to that of simple linear chains but, on the other hand, allows to distinguish different processes according to the hierarchy of times in branched polymers. Therefore, the coupling and competition of diffusive reptation and contour length fluctuations, which have dominate until today the understanding of the dynamics of linear chains in detail, does not occur in the same way.17−20 Also, interesting new properties due to this hierarchical approach emerge in the nonlinear rheological response. This bears consequences especially on processing. Blends of short and long linear chains21,22 as well as those consisting of linear and long-chainbranched polymers have well-known additional advantages. Here, we investigated branched polymers immersed in a linear matrix as the major constituent. This allows controlling the time constants of the matrix in relation to the characteristic times of different sections of the branched chains.9,11,19,23,24 Vice versa, the branched polymer may be regarded as a probe for mechanisms of topological constraints and constraint release processes in linear © 2013 American Chemical Society

systems, comparable to probe rheology, which was recently reintroduced by Bailly et al.25 Although probe rheology is a very sensitive and interesting technique for systems with sufficiently different time scales, it is based on the subtraction of large modulus contributions to isolate the response function of the probe without its environment. A rigorous underlying theory like the Random phase Approximation (RPA) for scattering is, unfortunately, is not available.26 Scattering from a dilute incompressible blend is completely dominated by the probe as the minor constituent and is therefore only weakly influenced by the uncertainties related to the majority constituent. Scattering hence yields immediate information about the structure and the dynamics of the probe, while the influence of the matrix is only indirect. Besides serving as a scattering probe and being of fundamental interest, the behavior of branched chains under flow in blends is critically important for processing applications,1,3,11,14,23,24 although it has been only minimally investigated on the molecular level. The hierarchical arm retraction process, i.e., the relaxing the outer generation of branches before the next layer gets mobile, requires a timedependent increase of the constraining tube. This is summarized Received: July 13, 2013 Revised: October 28, 2013 Published: November 18, 2013 9122

dx.doi.org/10.1021/ma4014498 | Macromolecules 2013, 46, 9122−9133

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under the name of “dynamic dilution”, in which the tube diameter “swells” due to relaxed dangling end material, which “dilutes” the topological network. The great benefit of scattering methods precisely relies in the possibility of extracting these fluctuations with high precision from time- and space-averaged measurements, thereby providing valuable additional information.27−32 In general, the improved understanding of nonlinear rheological properties is paramountly, as especially materials for film blowing and similar applications involve similar complex admixtures of different architectures.1,4 In this article, we consider blends of model systems that possess the required ingredients, i.e., interactions and rheological properties, to simulate complex, real branched systems. This work strongly supports the RPA approach by D. J. Read33,34 that deals with the simultaneous influence of incompressibility and topology conservation caused by entanglements in a tube approach and treats quenched and annealed variables simultaneously.34 Our experiments thoroughly investigated the RPA model and provided insight into the problematic of constraint release or dynamic dilution models in dynamically asymmetric systems.4,5,9,10,35 The molecular rheology-based theory accounts for linear chain retraction occurring after step deformation and describes microscopic deformations and the tube diameter as the fluctuation parameter.30,36 The investigated architectural blend systems consist of dendritically branched fully hydrogenous PI, diluted to 10% (≤ϕ*) into deuterium-labeled linear matrix polymers to provide the necessary contrast for small angle neutron scattering (SANS) experiments. Here, the scattering vector q probes length scales across the radius of gyration of the branched polymer down to a few monomer lengths. Because of the present labeling, the outcome is not specifically related to monomer fluctuations in the different hierarchies as in the original theoretical approach. To our knowledge, this is the first SANS study that presents a combinatorial structure−property investigation of asymmetric branched/linear blends. This information is highly needed to quantify the nonlinear viscoelasticity and strategic blending of mixed architectural systems.

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The size of both linear matrix polymers was tuned by synthesis to the dynamical features of the upper hyperbranched systems. As the outer arms are identical, the same short time dynamics is expected. However, both differ in the terminal region. The short matrix of ∼200000 g/mol was therefore selected to reflect a simultaneously relaxing species with the outer branches, whereas in the long matrix (∼800000 g/mol) a comparable terminal behavior was envisaged. In the latter, the outer branches were to experience a strongly elastic behavior, imposed by the matrix. Two blends each consisting of both smallest and largest components were prepared by dissolving in toluene in a mass ratio of 90/10. Thereafter, the mixtures were precipitated in methanol to which 0.1% antioxidant (BHT) had been added. The blends were dried for 1 week under high vacuum at room temperature and then vacuum-molded in different shapes depending of the experiment, namely disks for linear shear rheology and rectangular stripes for extensional experiments. Rheology. Linear oscillatory shear experiments were conducted on a ARES (Rheometrics Sci) rheometer equipped with a 2K FRNT transducer under a nitrogen blanket. Either a 8 or a 25 mm parallel-plate geometries were used and the frequency range 0.01 < ω < 100 rad/s and 0.5−1% strain amplitude were selected to ensure linear response over the full temperature range. Isothermal frequency sweeps at different temperatures comprised between −35 and +145°C were mastered to a reference temperature T0 = 25°C by time−temperature superposition. The WLF shift factors log10 aT = (−C1(T − T0))/(C2 + (T − T0)) with C1 = 4.5 ± 0.1 and C2 = 127 ± 1 K for linear and 4.9 ± 0.1 and 134.0 ± 1 K for the dendrimers at T0 were determined from 2-dimensional shifting using the ORCHESTRATOR software and yield within 10% the same horizontal shifting.30,39 Additionally, the extensional regime was explored using the standard extensional viscosity fixture supplied with the ARES. These exploratory experiments in start-up and stress relaxation were conducted consecutively to simulate comparable conditions of the Rheo-SANS investigation. The temperature was kept constant to −30°C. Here, we only refer to the nonlinearity of the Rheo-SANS experiments.

EXPERIMENTAL SECTION

Materials. All polyisoprene-based polymers (PI) in this study were prepared via the technique of anionic polymerization, which allows a good control of molecular weights and molecular weight distributions. Two deuterium-labeled linear matrix homopolymers were prepared in benzene with molecular weights of Mw = 216000 g/mol with Mw/Mn = 1.03 respectively Mw = 810000 g/mol with Mw/Mn = 1.09, as determined from off-line low angle light scattering (LALS) and size-exclusion chromatography (SEC). Their microstructure was 92% 1,4 cis/trans. Phase separation occurred in combination with a fully deuterated long matrix due to nonfavorable isotope interactions and the mixing of architectures. To suppress this, the longer matrix was a random copolymer of H/D monomers in a ratio of about 2:1, confirmed by 1 H NMR to be (68.2 ± 0.2)% H. The synthesis of the used model hyperbranched polymers by anionic polymerization (cis-1,4, 70%; trans1,4, 23%; and 3,4, 7%; determined from 1H and 13C NMR in CDCl3) was described previously.37,38 The characterization of the final products was obtained from LALS and SEC. The final fractionated branched polymers had Mw = 194000 g/mol with Mw/Mn = 1.03 (hyper 19,28) and Mw = 250000 g/mol with Mw/Mn = 1.07 (hyper 19,47). The labels denote the mass of outer and inner arms. Each outer arm thus had the same molecular weight Mw,a = 18500 g/mol in both branched polymers. The inner parts had 27700 and 46600 g/mol, respectively. From SEC, up to 5−10% of 3-arm star as well as some short linear byproducts remained as principal contaminants, which could be neglected due to the dilute concentration of the dendrimers.

Figure 1. Linear rheology measurements of the blend short, hyper 19,28 and short matrix. See text. Figure 1 and Figure 2 summarize the dynamic moduli G*(ω) for both pure linear, pure dendrimers, and their blends at the same T0. Figure 3 illustrates the step-strain experiment with subsequent relaxation and serves as an approximate mechanical reference to the Rheo-SANS experiment. Small Angle Neutron Scattering (SANS). SANS measurements were performed at the KWS1-FRM2-Garching instrument, at the D11ILL-Grenoble instrument and at the SANS1-GKSS-Geesthacht instrument. The scattering vector range q with q = 4π sin(θ/2)/λN covered 5 × 10−3 Å−1 < q < 1.5 × 10−1 Å−1, where the scattering vector is defined by the scattering angle θ and λN is the neutron wavelength. The twodimensional (2D) scattering intensities in 128 by 128 channels were 9123


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The samples in Table 1 were labeled by means of their matrix component. For comparison reasons, the strain rates at the stretching temperature Ts were converted to the reference temperature of 25 °C using the shift factor aT(Ts) with the WLF parameters determined for the main component. The accuracy and stability of the gas temperature measurement is ∼0.5 °C and the macroscopic sample deformation (i.e., the ratio of stretched-to-isotropic sample length from a grid of marks on the samples) is precise down to about 0.05−0.1 strain units from monitoring it by a video camera. Transmission ratios of stretched and isotropic state as well as counting rates were additionally used to check the effective thicknesses of the samples after straining within the accuracy of the experiments. As the neutron beam cross section is constant, only the thickness of the samples during the relaxation study can be evaluated. Anticipating the nonlinear rheology description, these data indicated an increase of the thickness by a factor of approximately 1.07 ± 0.01 over the full time range which translates itself into a likewise reduction of the width. The corresponding effective lower macroscopic strain is, however, within the accuracy of the experiment and has no influence on the outcome. Immediately after the stretching, each sample was rapidly quenched below its glass transition temperature (less than 3 s) to freeze the chain conformation and avoid further global chain relaxation during the collection of the scattering data.40 To this end, the SANS experiments were always performed at about −90 °C. The relaxation processes were restarted from −90 °C by annealing the sample over a controlled period through consecutive temperature ramps up to −40 °C, (∼1.2 K/min) followed by a period of isothermal soaking at the final temperature. In this way, the full relaxation spectrum of the melt after different annealing times could be investigated.40 During the annealing phase, the temperature was recorded at intervals Δt = 30 s. Using the WLF shift factors, the total equivalent relaxation time at the reference room temperature (RT) was obtained by taking into account the contribution of each step at temperature T:

Figure 2. Linear rheology measurements of the blend long, hyper 19,47, and long matrix. See text.

trelaxation

trelaxation , T0 =

∑ t=0

corrected pixelwise for empty cell scattering and sensitivity using QtiKWS, LAMP, and SANDRA softwares of the host institutes and normalized to absolute scale (cm−1) via calibration with the incoherent scattering from a water standard. Isotropic data were radially averaged. For the anisotropic data, sections along the main strain axes (parallel and perpendicular) were selected and averaged over an opening angle of ±10°. Identical stretching procedures for the in situ deformation of both blend samples were chosen: a rectangular sample of 1 mm or 2 mm thickness, respectively, for long and short matrix blends with approximate initial length-to-width ratio of 6:1 was mounted inside the grips of the stretching machine and stretched with constant strain rate ε̇ at constant T (Table 1). Details of the stretching equipment have

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MODELING Linear Rheology. Figure 1 and Figure 2 display the dynamic moduli of the investigated blends as well as those of the pure components. These Figures show the characteristic features of the 2-level hierarchical structure: one bump in G″ for each level and a broadened relaxation time spectrum, which are characteristic for the activated arm retraction process and are due to the entropic barrier of fluctuating against the branch point. The rheological response of the blends is completely determined by the linear matrix components and only minor details (as expected) from the hyperbranched architectures can be spotted. For evaluation reasons we tested the branch-on branch (BoB) procedures on the dendrimers and blends and used the Likhtman model for the linear matrix.3,17 This computational model calculates the tube model response of pure polymer melts of arbitrary architecture by including the hierarchical relaxation and

Table 1. Summary of the Stretching Conditions λ [−]

tstretch [s]

T [°C]

ε̇(T) [s−1]

ε̇(25 °C) [s−1]

ε̇τR,lin

short matrix long matrix long matrix long matrix

1.7 2 2 2

2 20 3600 6000

−25 −30 −10 −10

∼0.27 ∼0.035 ∼0.000 19 ∼0.000 12

∼215 ∼88 ∼0.01 ∼0.006

∼7 ∼43 ∼0.005 ∼0.003

(1)

This assumption of time−temperature superposition in nonlinear rheology was already adopted in previous works.27,29,40 Anticipating the linear rheology modeling, the short stretching time (2 s at −25 °C and 20 s at −30 °C) in both matrices allowed to investigate the relaxation time scale of the outer generation. In the case ε̇τR,lin ≥ 1, the contour or tube length of the linear chain was stretched and thus the samples were nonlinearly deformed. On the other hand, if ε̇τR,lin ≤ 1, the tube length rapidly re-equilibrated. Additionally, the blend with the long matrix was also investigated in the long time limit. This allowed the investigation of the deeper-lying inner arm relaxation. To this end, the strain rate and temperature were changed to avoid long annealing times in the in situ neutron beam experiment. Herein, ε̇τR,lin ≈ 0.005 and ε̇τd,lin ≈ 2; i.e., the chain is oriented in the strain field. It was assumed that the same state was reached as after relaxation from a faster step strain.

Figure 3. Comparison of the long linear matrix, hyper 19,47 and blend long. *εH represents the theoretical Hencky strain (line) and the apparent stretch (line + symbol) as discussed in the text and basing on the SANS evaluations in close-to-identical conditions. The dashed line is the linear profile of the long matrix. Nonlinearity and strain hardening is seen at εH ∼ 0.7 at t ∼ 5 s. A faster relaxation in the pure dendrimer than in the blend is observed.

label

Δt 10−C1(T(t ) − T0)/ C2 + (T(t ) − T0)

been published elsewhere.13,27,29 Strain hardening effects are expected to occur in the case ε̇ τR,lin ≫ 1. 9124


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at fixed length, contrary to the better filament stretching rheology (FSR) method, the identical SANS conditions (see before) confirm a rather constant Hencky strain to within 10%. From the sample dimensions, the apparent Hencky strain can be determined (right axis of Figure 3). As stated before, this has no impact on the forthcoming SANS description. Further, as the evaluations presented here wanted to provide an exclusively qualitative description, no more in-depth analysis is undertaken. Despite being only EVF results with the discussed disadvantages on the nonconstant Hencky strain or varying width of the sample, the observed relaxation behavior is not unexpected: at the longest times t/τR ∼ 1, the stress has reduced to approximately the level in the limit of linear behavior (i.e., without strain-hardening), which underlines the nonlinear mechanism of retraction. Indeed, a clear nonlinear effect with strain hardening in the stress upturn at ∼5 s could be observed for the pure matrix. For the blend in Figure 3, the mechanical response was again dominated by the matrix. The pure dendrimer, on the other hand, showed comparable strain hardening, but much faster relaxation behavior presumably due to dynamic dilution effects. Therefore, a Rheo-SANS experiment is of utmost importance to uniquely resolve the difficulties in extracting the dynamics of the minority component from the macroscopic rheology. Scattering Model and Parameter Estimation. The scattering of a 2-component incompressible system in thermal equilibrium is usually well modeled by the random phase approximation (RPA).26 However, the simple standard RPA45 is inappropriate and leads to erroneous results when applied to a deformed system with quenched disorder, for example due to cross-links or permanent or transient entanglements. Therefore, an extended RPA version for entangled polymers was recently proposed by D. J. Read,34 which takes into account elastic fluctuations and inhomogeneities related to the quenched disorder caused by entanglements. This quenched RPA approach was successfully applied for the case of pure, stretched, partially labeled H-polymers in a similar study by some of the same authors.27 The main success in the latter study has been the prediction of the enhanced anisotropy and scattering intensity in the direction parallel to the deformation axis with progressing relaxation times. The same physics is very probably responsible for the appearance of the well-known, anomalous butterfly patterns. We refer to the literature for a detailed review on the former experiment and the formal presentation.27,34 Here, just the key elements of the RPA-approach are summarized, which are necessary for the discussion and the comparison of results from scattering and rheological experiments in blends. According to this approach, two levels can be identified: • First, a set of correlation functions of quenched variables, like the entanglements of the system, as well as correlation functions of the annealed variables, like the segment densities at a given entanglement structure represent the system structure factor for an incompressible polymer system. • In the second step, these correlation functions are approximated by tube model results in line with the Warner−Edwards approach of harmonic restoring potentials.46 The deformation enters in via the deforma→ → → ⎯⎯ → ⎯⎯ tion tensor (λ ) of the tube axis ri ⃗ ̂→λ ri ⃗ ̂. Finally, the mean square fluctuations relative to the tube axis in the main

can account for mutual constraint-release events in the case of blends using dynamic dilution concepts. For the description of the linear chains, the Likhtman tube model17 was thus adopted and the resulting characteristic time scales were used for the description of the linear matrices.30 For the predicted curves, the molecular parameters were fixed to the chemical characterization values. The entanglement time τe for PI at 25 °C being ∼ (1.5 ± 0.1) × 10−5 s and corresponding to an entanglement mass Me ∼ 4500 g/mol,30,41,42 needed to be but minimally optimized in the branched case (Table 2 and 3). The dendrimers thus consist of Table 2. Main Parameters of the Linear Chains at 25 °C, Using the Likhtman Approach label

τe [s]

τR [s]

τd [s]

short matrix long matrix

1.5 × 10−5 1.5 × 10−5

0.034 0.49

2.3 205

Table 3. Characteristic Times from BoB for the Hyperbranched Polymers and the Polymer Blends at 25 °C label

τe [s]

τR,outer [s]

τarm [s]

τmax [s]

hyper 19,28 hyper 19,47 blend short blend long

1.4 × 10−5 1.4 × 10−5 1.5 × 10−5 1.5 × 10−5

2 × 10−4 2 × 10−4 − −

∼0.1 ∼0.1 − −

∼20 ∼100 1.4 163

Zo = 4 outer and Zi = 6 (hyper 19,28) and Zi = 10 (hyper 19,47) inner entanglement segments, with Z = Mw/Me, respectively. The linears correspond to Zs = 48 and Zl = 181. The constraint-release parameters Cν were fixed at 0.1. Herewith, the quoted reptation time, corrected for contour length fluctuations, is τdls ∼ 2 s for the short matrix chain and is hence longer than the longest outer arm relaxation time, namely τarm ∼ 0.1 s. From these estimates, it is clear that concurrent equilibration processes will be present in the short matrix blend. The long matrix (with final τdll ∼ 200 s), on the other hand, will act like a quasi-static or elastic environment for the dendrimer outer arms. For the pure branched polymers, the terminal time τmax is estimated from the position of the loss peak in the G″ profile. Nonlinear Rheology. In a nonlinear start-up experiment, the blend comprising of the “hyper 19,47” and the long matrix were compared with the pure linear and with the pure branched component in the same stretching condition, shown in Figure 3. These measurements, performed with the EVF of the ARES instrument, missing a feedback control loop, which is standard for the more refined filament-stretching rheometer (FSR), especially for relaxation, provide, however, a suitable comparison to the presented SANS data in this work. In this context, two phenomena are discussed in the literature, i.e., a possible secondary flow and a relaxation acceleration, which are associated with the start-up and stress relaxation experiment. The former was described by Nielsen et al.,43 who reported an analytical correction for the uniaxial elongation viscosity. The latter was discussed by Yaoita et al.,44 who interpreted stress relaxation acceleration by a reduction of the monomeric friction coefficient. Both these phenomena may play a role for certain values of the deformation. In the present study, however, the amplitude of the applied deformation causes them to be of lesser relevance. Whereas the EVF data in relaxation were obtained 9125


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axes (μ = x, y, z) of the deformation tensor emerge as the parameters of the topological constraints: dμ 2 = ⟨( ri ,⃗ μ − ri ,⃗ ̂ μ)2 ⟩

the inner part of the hyperbranched polymer as a diblock copolymer star with stretched and oriented material characterized by (1−fe)-anisotropic and fe-isotropic material. The effective reduced deformation then provides a reasonably good description of the loss of anisotropy during the relaxation process of the inner region as it decouples the correlation from dangling ends and fluctuations in the tube. The involved math of the full RPA is complex and even for a simple blend of linear chains 15 different correlation functions appear.33 For the dendritically branched polymer investigated here, the seven different contributions from the different subchains will further increase the computational cost. Two limiting cases can be discussed: for unrestricted chains (i.e. having an infinite fluctuation range d → ∞) as well as in the undeformed state the standard equilibrium RPA result, SRPA,eq, is obtained in terms of the bare structure factors: → → → ⎯⎯ → ⎯⎯ → SRPA(q ⃗ , λ )|→ = SRPA(q ⃗ , λ )|d →∞ = SRPA , eq(q ⃗) ⎯⎯

(2)

Here, r ⃗ and r ⃗ ̂ stand for actual and mean configuration of the chains. The Warner/Edwards approach, developed originally for networks,46 allows anisotropic fluctuation ranges to be described, e.g., with the standard ansatz:

dμ = d0λμν

(3)

with ν = {0,1} for isotropic or affinely scaled fluctuations, respectively. • The theoretical framework34 contains, however, only one fluctuation scale for all polymer segments in the system. This assumption is based on the mean field picture that the fluctuations are entirely determined by the environmentin this case, the matrix. • For the description of relaxing melts after a fast step-strain deformation, three additional parameters are required. The first process of tube length retraction is accounted for by the retraction parameter γ(t) = α(λ, 0)/α(λ, t) that describes the shortening of the occupied tube length during the annealing steps. α(λ, 0) is the instantaneous Doi−Edwards tube elongation.36 In this study, α(λ ∼ 2,t = 0) ≃ 1.2, while γ is rather unimportant and its time evolution toward equilibrium is approximated by the longest mode:

λ =1

SAA(q ⃗)SBB(q ⃗) = SAA(q ⃗) + SBB(q ⃗)

A and B are the differently labeled components i.e. linear and dendrimer, thus SAB(q⃗) = 0. In the limit of d → 0, on the other hand, the structure factor for an affinely deformed system results as: → → → ⎯⎯ → ⎯⎯ → → → ⎯⎯ → ⎯⎯ SAA(q ⃗λ )SBB(q ⃗λ ) SRPA(q ⃗ , λ )|d → 0 = SRPA , eq(q ⃗λ ) = → → → ⎯⎯ → ⎯⎯ SAA(q ⃗λ ) + SBB(q ⃗λ )

α(t ) = 1 + (α(λ , 0) − 1) exp( −t /τR ) (4) τR being the Rouse time of the chain under consideration.6,7,30,36,47,48 In the present work, the retraction parameters for both generations of the dendrimer are neglected due to the experimental conditions of stretching. • The progress of relaxation after a step strain is captured by 2 parameters: (i) the time-dependent fluctuation parameters dμ(t) (or d0 if isotropic) introduced by eq 2, which characterize the transversal fluctuations of segments, and (ii) the time-dependent fractions of isotropic chain end material in both dendrimer and linear matrix polymer, i.e. feo(t), fei(t) for outer (o) and inner (i) section and fel(t) respectively, which represent the deoccupation of the deformed tube by retraction or reptation processes. • The contribution of the inner part relaxation to the total structure factor was approximated on the level of the radius of gyration Rg of the inner block, in order to avoid a new large set of correlation functions for the inner part and the outer arms and a further merely speculative interpretation of the experimental data. The following approach is not part of the stringent RPA treatment of D. J. Read.33,34 The relaxing inner block can be thought of as a diblock copolymer, in which both blocks differ by their state of strain i.e. a block with affine deformed and a block having isotropic segment lengths b, in the ratio (1 − fe) to fe respectively. By adopting the general definition of the radius of gyration for a diblock and by distinguishing into intra- and interblock distance contributions as indicated by the textbooks,49,50 one derives for such a deformed copolymer the following expression: R g , tot , λ 2 =

(7)

SRPA,eq can be calculated straightforward for both cases. For all → → ⎯⎯ incompressible systems between both limits, SRPA(q⃗,λ ) is dominated by the minority component and consequently the majority linear component that determines the dynamics of the branched polymer as matrix is near-to-invisible in the scattering. For the linear component the evaluation of chain parameters as determined in previous investigations30 should thus be transferable without affecting much the results. In a permanent network of linear chains the RPA structure factor reduces to the Warner− Edwards form with:33 → → ⎯⎯ SWE(q ⃗ , λ ) = 2

∫0

1

di′

∫0

i′

⎛ i′ − j′ ⎞ dj′ ∏ exp[− Q μ2λμ2⎜ ⎟ ⎝ γ ⎠ μ

− ξQ μ2λμ2(1 − e−i′− j′ / γξ)]exp[− Q μ2ξ(1 − e−i′− j′ / ξ)]

(8)

where the dimensionless contour length variable i′ = i/N (with N the total number of monomers) describes the position along the chain. In eq 8 for the network, γ = 1. The integral explores all i and j monomers and is a function of their distance along the chain. Furthermore, Qμ and ξ are defined as

Q μ = qμR g , μ ξ=

Nb2 3 [f + (1 − fe )λ 2 + 3(fe − fe 2 )(fe + λ 2 − λ 2fe )] 6 e

= R g , iso2λeff 2

(6)

(9)

dμ 2 2 6 Rg 2

(10)

In the case of relaxation, isotropic dangling ends of relative length fe are accounted for by limiting the double integration to the tube-constrained section, [fe − (1 − fe)] for linear chains51 or [0 − (1 − fe)] in the case of a fixed end of an arm. The well-known

(5)

Note that λeff ≤ λ for fe ≠ 0 and becomes 1 at fe = 1, like a fully relaxed diblock. This empirical approach allows to treat 9126


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Figure 4. Ratio between perpendicular and parallel data for different values of the parameters feo (a), fei (b), fel (c), and d0 (d). For each calculation the starting configuration was the affine case with parameters ϕ = 0.1, λ = 2, feo = 0, fel = 0, fei = 0, d0 = 0 Å, and ν = 0.

isotropic Debye function is retrieved for fe = 1 or d0 → ∞. The anisotropy for a given λ is thus entirely determined by fe and d0. Rg is the isotropic value of the chain dimension. However, in this work the full quenched RPA approach by Read was used, although it could be shown afterward that the RPA-corrections due to the quenching of variables were virtually q-independent for q > 0.01 Å. The scattering model involves several parameters. In the following, we discuss its sensitivity to the most relevant ones. This is shown in Figure 4 and Figure 5. We used 2D spectra as well as the ratio S(q)⊥/S(q)∥, which is particularly sensitive to the needed parameters. The starting parameter set for the graphical presentation of the results always was the strictly affine case, i.e., feo, fei, fel, and d0 = 0.

by the inverse length scale at which affinity and tubeconfinement meet; i.e., qpeak ∼ 1/d0. The 2D patterns show a transition from almost purely elliptical shape to lozengelike patterns at high fractions of feo. • fei: the loss of the affine reference plateau if one compares to the sensitivity of the ratio for feo is noted. This is because fei contributions require feo = 1 due to the hierarchical principle. The behavior is similar. No major shift in the position of the peak is predicted, which confirms the former statement that latter marks the length scale at which affinity is lost due to fluctuations in the tube. The effect of fei on the 2D scattering patterns is pronounced and almost isotropic circular intensities can be regained. • fel: the fraction of dangling end material of the linear chain was said to contribute in a minor way to the total RPA structure factor, with the sensitivity being quite constant. In the model, fel can be defined from the relaxation modulus as (1−μ(t))/2, with μ(t) being the occupied tube population function, so that the linear chain is fully relaxed at fel = 0.5. No further explanation for the predicted behavior of the scattering of this hypothetical blend is sought, however, as it does not really conform to the reality of relaxing a matrix chain, whereas the dendrimer does not. • d0: the influence of the tube parameter d0 is strong and can be compared in principle to the reaction of feo. The peakto-plateau ratio is now much higher and the position at which the ratio of S(q) ⊥ /S(q) ∥ is peaked, shifts congruently with ∼1/d0. The larger the tube diameter is, the more chain material undergoes isotropic fluctuations and consequently these approach more the scattering in

• feo: the height of the affine plateau at intermediate-to-high q which is clearly visible in the ratio, can be estimated for simple linear chains to be λ3 for an incompressible system, where λ∥ × λ⊥ × λ⊥ = 1. The process of relaxing the dendrimers from their tips inward leads to a pronounced change of this plateau and induces a peaked structure. The peak-to-plateau ratio is therefore a promising indicator for a fine-adjustment of the fraction of relaxed material. The independence at high q is mainly related to feo, i.e., the isotropic material, its height being in turn correlated to the relative amount of small length scale polymer material present in both extreme deformation states. The peak structure arises due to remaining deformed material at larger length scales in deeper-lying hierarchies. The position of the maximum is roughly given 9127


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Figure 5. 2D representation for different values of the parameters feo (a), fei (b), fel (c), and d0 (d). The calculations were performed at a detector distance of 4m (2 × 10−2 ≤ q ≤ 2 × 10−1 Å−1). The strain direction is vertical. For each calculation the starting configuration was the affine case with parameters ϕ = 0.1, λ = 2, feo = 0, fel = 0, fei = 0, d0 = 0 Å, and ν = 0.

the short, fully deuterated matrix sample. There, this void scattering is more appreciable due to the higher contrast and is less important in protonated matrices. The latter (∼ (CH2)n) have roughly zero coherent scattering length (bC ∼ 6.6 × 10−10 cm, bH ∼ 3.7 × 10−10 cm) like the voids. The contrast situation in the long matrix, which consisted of almost 2/3 of hydrogeneous monomers is better by a factor of approximately 10. Isotopic HD-interaction parameters were neglected for PI, due to either the small segment number in the short matrix blend or the copolymeric structure in the long matrix mixture with a predominantly hydrogen-containing long matrix. • As reference Rg of the dendrimer we used the same definition as for the cited H-polymer investigations,27 i.e., that of a linear polymer of the same Mw. The choice is purely technical as only the ratio Rg,λ/Rg,iso is of importance. • The full scattering function calculation in 2-dimensional-q space is quite time-consuming and involves the numerical evaluation of double integrals. Wherever possible, these were turned into single integrals and were maintained only for the correlations between inner and outer segments. However, a nonlinear fitting of the data was excluded for practical reasons and an extended grid search in two classes of unknown parameters, specifically comprising tube parameter (d0) and fraction of dangling ends (feo(t), fei(t), and fel(t)), led to the optimal parameter set in Tables 4 and 5. fei and feo depend on each other: fei is 0 for all

the isotropic limit. For even larger tube parameters the peak decreases further, which illustrates our conjecture for eq 6. Therefore, the comparison between the fraction of dendrimer dangling ends and the tube parameter is crucial. Dangling ends play a similar role during the relaxation process, although, as we computationally observed, on different length scales. The largest difference in the sensitivity of both was found in the high q-limit, where feo leads to a flattening in the ratio. Instead, the curves did not flatten out in the case of d0. In conclusion, we were able to clearly distinguish the main parameters of the model. Therefore, by combination, it was possible to unambiguously characterize the experimental data in good accuracy. This theory can now be applied in all its facets to the blend of a 2nd generation dendrimer2 with a linear homopolymer.4,19,23 The other parameters necessary for the calculation of the correlation function and structure factor are consequently assumed to be the segment numbers of subchains of the branched and of the linear polymer.

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RESULTS

Before the separate experiments are discussed, some general aspects of the evaluation are to be elucidated: • All SANS data showed some parasitic forward scattering for q < 1/Rg, most probably due to residual voids after molding. In the isotropic state, this additional scattering followed a q−4 -power law, which was more pronounced in 9128


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Table 4. Parameters of the SANS Comparison of the Short Matrix Blenda ti

t [s]

t/τarm

t/τR,ls

feo [−]

fel [−]

d0(t) [Å]

t0s t1s t2s t3s

0.005 0.10 0.20 0.30

∼0.06 ∼1 ∼2 ∼3

∼0.15 ∼3 ∼6 ∼9

0.10 0.50 0.65 0.75

0.01 0.10 0.20 0.20

26 28 32 32

ν = 0, τarm ∼ 0.1s, τR,ls = 0.034 s, and λ = 1.7 are taken. Uncertainties on the fraction of dangling ends and the tube parameter are 0.05 and 2 Å respectively. a

feo < 1, and fei ≠ 0 implies feo = 1. fel values were initially set to rheologically expected values (using the Likhtman model) and were optimized only at the final stage. The adopted strategy is a three step process, which comprises (i) a parameter value-identification using the ratio approach of S(q)⊥/S(q)∥ as described above, (ii) the 2D comparison including the off-axis scattering information on the anisotropic structure factor, and (iii) checking of the data agreement along the principal axes of the deformation tensor. We note that the ratio of the data along both principal axes ∥ and ⊥ had the additional advantage of avoiding possible calibration errors in the strained samples. Relaxation Process: Short Matrix. The short matrix blend can be compared to a pure branched system like the formerly studied pure H-polymer architectures27,29 in a similar combined SANS-rheology approach.40 However, now the relaxation spectrum of the outer section coincides the dynamic time range of the linear polymer (see Figure 1). Here, dynamic dilution effects, leading to time-dependent tube parameters, d0(t) = d00/ (1 − fe(t))1/2, could be expected. d00 is the initial undiluted tube parameter of the entanglement network. Less dilution will appear due to the dilute character of the blend. The isotropic data taken at T = −40 °C yielded a reference Rg = 144 Å, which is in good accordance to expectations.30 The sample was stretched in 2s with a strain rate of ε̇ ∼ 0.27 s−1. at −25°C, i.e., ε̇ ∼ 215 s−1 at T = 25 °C. Since ε̇τR,ls ∼ 7, the tube length of the linear chain is only slightly stretched and hence rapidly relaxes. The normalized data presented in Figure 6 correspond to the principal axes of strain. The parameters of the experiment are summarized in Table 4 and discussed as follows:

Figure 6. 1D representation and S(q)⊥/S(q)∥ of the full relaxation process of the blend composed of short matrix and hyper 19,28.

• The tube parameter d0 increases from 26 Å up to 32 Å, for t > t0s. The ratio-approach of scattering data shows that the accuracy of d0 is about 2 Å, therewith the tendency toward saturation of the increase of d0 in the investigated time scale seems to be well established. For comparison, the undiluted tube parameter value in a pure PI−H polymer was found to be 33 Å40 • his observation also parallels the findings on purely linear polymers of the same length30 that the times shorter than τR,ls shows significantly smaller segmental fluctuations. Latter increase after the linear chain has equilibrated its length. Relaxation Process: Long Matrix. The long matrix blend was prepared and treated exactly in the same way as the former short matrix blend. The structural relaxation of the dendrimer was investigated in the three regimes, i.e., focusing (i) on outer arms, (ii) branch points, and (iii) inner arms region by using

• All scattering data from t0s to t4s can be consistently described by the approach of Read34 by optimizing both parameters feo and d0 directly with isotropic fluctuation ranges dμ = d0 (ν = 0) throughout.

Table 5. Parameters of the SANS Comparison of the Long Matrix Blenda

a

ti

t [s]

t/τarm

t/τR,ll

feo [−]

fei [−]

fel [−]

d0(t) [Å]

λ [−]

t0l t1l t2l t3l t4l t5l t6l t7l

0.01 0.02 0.1 0.5 5 30 100 200

∼0.1 ∼0.2 ∼1 ∼5 ∼50 ∼300 ∼1000 ∼2000

∼0.02 ∼0.04 ∼0.2 ∼1 ∼10 ∼60 ∼200 ∼400

0.20 0.30 0.60 0.80 1.0 1.0 1.0 1.0

− − − − − 0.05 0.2 0.45

0.01 0.02 0.05 0.10 0.10 0.10 0.25 0.25

22 22 22 24 26 26 34 36

2 2 2 2 2 1.98b 1.92b 1.65b

ν = 0, τarm ∼ 0.1s,τR,ll = 0,5s and λ = 2 are taken.

b

Represents the effective deformation λef f on the inner region. 9129


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different stretching conditions. The best-fit parameters of the spectra are reported in Table 5 and the corresponding scattering

relaxed arm segments inward. The latter behave as big friction blobs. Such a motion becomes increasingly possible, if the tube parameter simultaneously allows the branch points to hop and incorporate part of the relaxed chain section. At t ≈ τR,ll, the diameter 2d0 ∼ 60 Å is of the order of Rg of the simple arm and hence it can accommodate part of the blob material. We observed a clear and fast, increase of the tube parameter for relaxation times belonging to the inner part, which we assigned to the motion of the branch points and to tube dilution (see later). In this inner region, the remaining entanglements consist of only those included between the innerarms and the linear matrix. The 2D-detector patterns of some characteristic relaxation times (t0l, t4l, and t7l) of the long polymer blend and fits are reported in Figure 8. The results show an excellent agreement.

■

DISCUSSION AND CONCLUSIONS In this work, model dendrimers were used to probe locally the tube dynamics in a linear matrix and the response to external deformation. As we showed in the modeling chapter by linear rheology, the visco-elasticity of such systems is entirely governed by the matrix alone owing to the dilute character of the blend.25 The SANS technique was used to monitor in detail the dynamics of the minority component, which is coupled to that of the linear matrix, using a description with advanced RPA theory applicable to nonequilibrium systems.34 The obtained results were interpreted in the tube model for branched polymers.4,5 Figure 9 summarizes the main results of both experiments, which differed in the size of the linear matrix. Rouse (∼ Z2) and effective reptation times (∼ Z3.4) of the linear matrices, then vary by factors of (Zl/Zs)2≃ 14 and (Zl/Zs)3.4 ≃ 91 respectively. The comparison between the experiments can be guided by the Rouse relaxation time of the linear matrix: while the long matrix blend allowed both time domains, i.e., below and beyond τR, to be studied, the short matrix blend was limited to t ≥ τR. In a normalized time representation (i.e. (t/τR) diagram) important differences and similarities are revealed. Figure 9 resembles the typical t(x) diagrams found in the rheological literature.4,5,23 Here, x is the relaxed part of arms equivalent to our parameter fe. In these previous reports, the time scales were derived as a function of molecular parameters and assumptions made about the processes. To go beyond what is usually reported in the rheological literature, i.e. thus to include the tube parameter d0(t) in the discussion, Figure 9 now presents fe(t), which is to be compared to t(x). The interpretation of the data in the 2-parameter space is rather complex and interdependencies are observed. • The upper two curves respectively show the liberated chain end fraction for inner and outer arm relaxation of the performed experiments. The reversed order of presentation allows to show the hierarchical relaxation of both levels vs the normalized time axis. While a detailed discussion of the arm retraction in branched polymers can be looked-up in the specialized literature, here we merely draw a validation.4 Our experiments have covered the full arm relaxation spectrum, which, according to theory, consists of early time fluctuations (tearly(feo)) and activated late-time retraction of arms tlate(feo). For t ≪ feo ∼ t1/4 is predicted. The late times tlate, on the other hand, increase exponentially as exp(U(feo)), where U ∼ Zfeo2 is an entropic potential barrier for deep fluctuations. This includes the

Figure 7. 1D representation and S(q)⊥/S(q)∥ of the full relaxation process of the blend composed of long matrix and hyper 19,47.

curves are shown in Figure 7. The above-mentioned regimes are as follows: • t0l − t2l, regime i: in this time domain, the outer arms of the dilute hyperbranched polymer are mainly entangled with the linear matrix (since ϕ ≤ ϕ*), although some interdendrimer interactions cannot be excluded. The time scales concern the sub-Rouse time of the matrix polymer. Since ε̇·τR,ll ∼ 43, the matrix is nonlinearly deformed and needs to perform chain retraction still. This process shortens the elongated tube by which, in a first simple approximation, almost no chain ends are produced since the tube ends follow the chain path. The retraction can be considered “terminated” at ∼τR,ll, which coincides approximately with τarm. • t3l − t4l, regime ii: this time domain comprises the Rouse equilibration time of the linear matrix up to ∼10τR,ll. Whereas the tube length is equilibrated in this range, the transversal tube component shows a clear and significant increase from 22 to 26 Å. This rise corresponds with a tube parameter relaxation time, which is of the order of several times τR,ll. However, the limited number of data points is insufficient to discriminate between models, e.g., the tube pressure concept.52 After this step a steep rise in the fluctuation range is observed. • t5l − t7l, regime iii: If the outer arms relax completely, the inner arms may pull effectively the branch points and the 9130


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Figure 8. SANS data at 4m detector distance (2 × 10−2≤ q ≤ 2 × 10−1 Å−1) for the blend with the long matrix. The representation covers the three main region of the relaxation process: (a) outer arms, (b) branch points, and (c) inner region.

softening due to the dilution of the entanglement network. For linear rheology, the full arm relaxation spectrum was approximated by the Milner−McLeish crossover function:28 t(fe ) =

The relaxation of the inner arm is governed by the same arm retraction modes as for the outer generation. The tube, however, is now wider and the potential is increasingly softened. Since the remaining tube is mainly due to the entanglements with the linear matrix, the latter imposes its dynamics on the arm relaxation. The fraction of liberated chain ends in the linear matrix generated by contour length fluctuation and reptation processes provides local constraint-release (similar to dynamic dilution). In good approximation, the fraction of inner arm relaxed segments hence follows the time dependence of this reptation process. When only the longest reptation mode p = 1 is accounted for, the data points roughly agree with the expected increase with (1 − exp(−t/τd,lin)). Higher modes can be safely neglected due to the 1/p2 weighing of the allowed, odd values for p. Apparently, a better agreement could be obtained for τd ∼ 1.5τd,lin as the tube was elongated and needs longer to “disintegrate”. This was discussed already as a possible nonlinear effect before.22

tearly(fe ) exp( −U (fe )) + tearly(fe )/tlate(fe )

(11)

which couples both time domains. For small Z-arms (as is the case here), the resulting crossover function in fe(t) resembles a straight line in a log−log representation corresponding to an overall ∼ t1/4 behavior. The agreement with the data is fair but suffices at this stage. At the still rather small nonlinear deformation of the samples, the comparison with this linear theory estimate can be justified. At later times t/τR ∼ 10, the outer arms have fully relaxed ( feo ≤ 1) and contributions of the inner arms may become visible. This mixing-in seems to extend over at least one decade in time (10−100 t/τR units) and points at a concerted action, involving the tube parameter. 9131


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within an anisotropic box. The present anisotropy is dynamic, i.e. consisting of a longitudinally relaxed motion vs. a transversally blocked motion, as the tube-forming linear matrix behaves fully elastic in the plateau modulus region. By balancing the three directions of the tube parameters by 1/3 and 2/3 for longitudinal and perpendicular one, Wagner et al.54 recently estimated that the tube relaxation time is ∼3τR instead of τR. Our experimental value ∼5τR is a good compromise over the few data points in this time regime. Concurently, the inner arm (upper curve) does not relax at all since the fat beads of the outer arms are too bulky to fit in the still narrow tube. So, hop processes are suppressed as long as tube dilution has not progressed. At very long times the tube diameter seems to follow a power law. There, reptation strongly contributes and, as for equilibrium systems, isotropic chain end material may be used to dilute the entanglement network. The actual tube parameter d(t) is then described by d00/μ(t)α/2, where d00 is the undiluted tube parameter and the θ−solvent analogueon, α, was chosen equal to 1.5,55 In terms of fel with fel ∼ Ct1/2, wherein C is a lumped constant, we have found a variation in time as d(t) ∼ 1/(1 − Ct1/2)1/2 (red dashed line). This fits in agreeable way to the rise of the tube parameter for t/τR ≥ 50 and yields a-close-to t1/4 power law, as shown by the dashed lines in that time domain. With these findings, we concluded that the present molecular picture of hierarchical relaxation of branched architectures in a linear matrix as the main component is complex. Small angle neutron scattering experiment on such architecturally complex blends, which accessed the basic parameters for predictions in both linear and nonlinear rheology, indicated clear evidence for some modifications to the hierarchical model: • The entanglement network is not just affected by simple dilution (except for long times). • Structural details, related to the simplistic tube pressure model seem to exist. • The friction blobs after outer-arm relaxation only move with the branching point if space allows for it. Thus, SANS experiments combined with elongational or large amplitude shear rheology on partially labeled model architectures in blends are an ideal tool to refine the presently achieved microscopic understanding of the dynamics of entangled melts.

Figure 9. Summary of the results in function of the reduced relaxation time for the polymer blend with short and long matrix. The dot line is the τR of the linear matrices.

• The fraction of “optimized relaxed end material” in both linear matrices is of minor importance for discussion as it constitutes a parameter, which just weakly influences the experimental results. Nevertheless, two separate t1/2 dependencies can be identified: a near t1/2 law for t < 0.1t/τR, i.e., the Rouse dynamical range, and another one for t > 100 t/τR, which could refer to the reptation dynamics and leads to the well-known ω−1/2 slope in the loss peak. The time at which this occurs fits to the transition of the −1/4 to −1/2 slope shown in Figure 2. This coincidence is expected somewhat since in the data evaluation the Likhtman model was initially adopted for the description of the matrix chains, which was optimized independently only in a further refinement step. Furthermore, using the early time-estimate fel = 1.5/Z(t/τe)1/4, the depopulation of the tube by contour-length-fluctuations was calculated to be ∼0.1 at t = τR,53 which is in very good agreement with the experimental outcome. The noted increase of fel for t ≫ τR is thus expected to follow rather closely the predictions for reptation. As Figure 9 shows, this comparison is rather good and provides the right timedependence and substantiates that the experimental τd-value should be somewhat longer than the linear estimate. This result might in turn be related to the stretched and oriented tube or even to the fact that the tube is affected by the branched environment. • The bottom curve in Figure 9 summarizes the time evolution of the tube parameter in both matrices. No considerable differences are found for t/τR ≤ 1. The normalized representation yields an identical value for the tube parameter at t = τR, i.e., 26 Å, which confirms the consistency of the evaluations. Unlike in the short matrix, a 2-step process was detected in the long matrix case. Although hardly visible, an exponential increase from 26 to 29 Å is present, which extends over ∼2 decades with characteristic time ∼5τR, as indicated by the blue dashed line. A similar dependence is expected in the so-called tube pressure model, where the chain is thought to be confined

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AUTHOR INFORMATION

Corresponding Author

*(N.R.) E-mail: [email protected]. Telephone: +49 2461 614681. Fax: +49 2461 612610. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank the financial support of the EU ITN DYNACOP 214627 Marie Curie Network. They also acknowledge Prof. H. Watanabe and Dr. D. J. Read for helpful and productive discussions, throughout the evaluation procedure. A special thanks to the hospitality of C. Bailly in UCL where the extensional rheological experiments were made under the supervision of Drs. D. Auhl and M. Shivokhin. 9132


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Microscopic Relaxation Processes in Branched-Linear Polymer

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